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Engineering the scattering, refraction and polarization of microwaves using Huygens surfaces

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Engineering The Scattering, Refraction and Polarization Of
Microwaves Using Huygens Surfaces
by
Michael Selvanayagam
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Graduate Department of Electrical Engineering
University of Toronto
c Copyright 2016 by Michael Selvanayagam
ProQuest Number: 10189929
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Abstract
Engineering The Scattering, Refraction and Polarization Of Microwaves Using Huygens Surfaces
Michael Selvanayagam
Doctor of Philosophy
Graduate Department of Electrical Engineering
University of Toronto
2016
An electric and a magnetic dipole that are superimposed and orthogonal to each other have interesting
properties where the dipoles can be designed to radiate into one half of space only. Such a configuration
is referred to as a Huygens source. Both active and passive configurations of Huygens sources exist.
Active configurations consist of antennas while passive configurations consist of scatterers engineered to
have induced electric and magnetic dipole moments.
Here, the properties of arrays of passive and active Huygens sources are investigated. These arrays
are referred to as Huygens surfaces. These surfaces are interesting because they can be designed to
implement, thin, low-profile devices which is desirable for current RF/microwave hardware. The design
of active and passive Huygens surfaces is investigated for three specific applications in the microwave
frequency range; scattering, refraction and polarization.
For the problem of scattering, active Huygens surfaces are investigated to implement a cloak which
suppresses scattering from an object in all directions. Both theoretical and experimental aspects are
examined and limitations and possible workarounds of active cloaking are briefly discussed.
On the problem of refraction, the synthesis and analysis of a passive Huygens surface to refract a
plane wave is discussed for the application of thin microwave lenses. Equivalent circuit models and the
implementation of these Huygens surfaces are also discussed.
Finally for polarization control, the design of passive Huygens surfaces is examined from theoretical,
numerical and measurement perspectives to demonstrate polarization manipulation. Passive Huygens
surfaces are constructed to perform polarization conversion and chiral polarization effects (circular birefringence). This idea of implementing chiral surfaces is further examined by looking at the design and
measurement of surfaces consisting of an electric response only.
ii
Dedication
To my daughter:
Elsie
iii
Acknowledgements
Despite the fact that this thesis is credited to only one author, the help of many people was still required
to bring this project to completion. The first person to acknowledge is my supervisor Prof. George Eleftheriades who was nothing but encouraging and supportive throughout the whole process. His patience
was also appreciated as there were many times where my only update was that I needed more time. I
am sincerely grateful for your guidance and mentorship.
Over the course of a graduate degree, the journey is completed alongside many other students who
are on parallel paths, working towards the same goal. To all the current and former students in the
EM group whom I had the privilege of getting to know: Thanks. I have been honoured to share many
intellectually challenging and hilarious conversations (not necessarily mutually exclusive) with many of
you. There were many times where, stuck on a problem, bouncing an idea off of someone in the office
provided the necessary insight to move further in my research. I would especially like to thank Joseph
Wong and Pouya Yasrebi whom I had the privilege of working with on some of the work covered in this
thesis. Your help and collaboration were greatly appreciated.
I would like to thank my parents for instilling in me a love of learning and a curiosity about the world
that was essential to completing this degree. I would also like to thank my sister for being there when I
needed an outlet to think about anything else besides school.
Thanks to all my friends, many who have been around since my elementary school days. Despite
the fact that we never got to hang out as much as we would like, our time together was always treasured
by myself and a source of great comfort in the many stressful days of graduate student life.
I have to acknowledge the consistent and calming presence of my wife, Hillary. Working on a degree that took a large amount of time and energy, you were always patient and understanding with me,
putting up with my late hours and long nights. I cannot thank you enough for the grace that you have
extended to me in my many absences. You have been a great partner throughout this journey and I
treasure all the special memories that we have accrued together whether it has been our travels, or just
a walk in our neighbourhood. As we start the next stage of our life, I look forward to that journey and
I am glad that I get to share it with you.
The biggest accomplishment over the course of this degree for me, is not finishing this thesis but becoming a father. I really don’t have any profound insight into how fatherhood changes you other than
to say that my life is infinitely better with Elsie, who always lights up when she sees me and vice versa.
Elsie, this is probably as far as you will ever read in this thesis so all I have to say is that I hope you
will always enjoy learning new things as you do now, no matter what form that takes or what interests
capture your heart and mind.
-Michael Selvanayagam
iv
Contents
1 Introduction
I
1
1.1
The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Huygens Sources And Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Huygens Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.5
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Controlling Scattering With An Active Huygens Surface
2 Background On Active Cloaking
2.1
14
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1
Transformation Optics Based Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2
Plasmonic Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3
Other Cloaking Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4
Radar Supression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.5
Active Cloaking
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Active Cloaking - Modelling and Simulations
3.1
13
23
Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1
Scattering Off Of A Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.2
The Induction Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2
An Active Cloak Using The Induction Theorem . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3
Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.1
Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4.2
CAD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5
Extending The Active Cloak
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
v
4 Active Cloaking - Implementation and Measurement
4.1
4.2
36
Design and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1
Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2
Designing the Cloak . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Measured Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1
Far-Field Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3
Disguising a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4
Estimating the Incident Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.1
II
Measured Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Refraction At A Huygens Surface
51
5 Background On Diffracting/Refracting Surfaces
52
5.1
Diffraction Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2
Frequency-Selective-Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3
Electromagnetic Bandgap Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4
Metasurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.5
Reflectarrays and Transmitarrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.6
Summary and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Refraction From A Huygens Surface
6.1
6.2
66
Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.1.1
Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.2
Analysis - Floquet Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.1
Method of moments verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.2
Physical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.3
Circuit Models For Huygens Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.4
Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.4.1
Properties of the Lattice Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.4.2
Modelling a Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.5
Modelling a Huygens Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.6
A PCB Design At Microwave Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
III
Polarization Control With A Huygens Surface
7 Background On Polarization
7.1
93
94
Material Parameters Which Alter Polarization
. . . . . . . . . . . . . . . . . . . . . . . . 94
7.1.1
Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.1.2
Chirality
7.1.3
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
vi
7.2
7.3
Engineered Structures Which Alter Polarization . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2.1
Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.2.2
Frequency Selective Surfaces And Reflectarrays/Transmitarrays . . . . . . . . . . . 102
7.2.3
Circular Polarization Selective Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2.4
Metasurfaces And Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8 Tensor Huygens Surfaces
107
8.1
The Boundary Condition At A Tensor Huygens Surface . . . . . . . . . . . . . . . . . . . 107
8.2
Circuit Modelling of Tensor Huygens Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 112
8.1.1
8.3
Determining The Impedance Tensors Using Diagonalization . . . . . . . . . . . . . 109
8.2.1
Tensor Circuit Element And Equivalent Circuit . . . . . . . . . . . . . . . . . . . . 113
8.2.2
S-parameter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3.1
Computational Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.3.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Converting a TE-polarized wave to a circularly polarized wave . . . . . . . . . . . 119
Converting a TE-polarized wave to a TM-polarized wave . . . . . . . . . . . . . . 120
Converting one elliptical polarization to another elliptical polarization . . . . . . . 121
8.4
Implementation Of A Tensor Huygens Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 122
8.4.1
8.5
Designing a TE-polarization to TM-polarization converter . . . . . . . . . . . . . . 123
Cascaded Tensor Huygens Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.5.1
Polarization Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
TLM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.5.2
Circular Polarization Selectivity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
TLM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.5.3
8.6
8.7
Implementing A Polarization Rotator . . . . . . . . . . . . . . . . . . . . . . . . . 130
Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.6.1
Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.6.2
TE-to-TM polarization converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.6.3
Polarization Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
9 Tensor Impedance Transmitarrays
9.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.1.1
9.2
140
Multi-Conductor Transmission-Line Model . . . . . . . . . . . . . . . . . . . . . . 142
Tensor Impedance Transmitarrays For Chiral Polarization Control . . . . . . . . . . . . . 145
9.2.1
Design Procedure Using The MTL model . . . . . . . . . . . . . . . . . . . . . . . 146
9.2.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A Polarization Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
A Circular Polarization Selective Surface . . . . . . . . . . . . . . . . . . . . . . . . 150
9.3
Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.4
Comparing Tensor Huygens Surfaces and Tensor Impedance Transmitarrays . . . . . . . . 152
9.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
vii
IV
Conclusion
154
10 Conclusions and Future Work
155
10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.2.1 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
10.2.2 Refraction and Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
V
Appendices
161
A A Bridged-T Phase Shifter
162
A.1 Basic Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
A.2 Varactor Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.3 Final Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
A.4 Measured Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B Floquet Analysis Of A Periodic Magnetic Current Sheet
168
C A Frequency Domain TLM/MOM Solver
172
D S-parameter Conversion
175
D.1 Converting Linearly-Polarized S-parameters To Circularly Polarized S-parameters . . . . . 175
D.2 Rotating S-parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
D.3 Transfer Matrices and Cascading S-parameters . . . . . . . . . . . . . . . . . . . . . . . . 177
D.3.1 The S-parameters Of Cascaded Tensor Huygens Surfaces
. . . . . . . . . . . . . . 178
E Quasi-Optical Measurements
179
E.1 Horn Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
E.2 Dielectric Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
E.3 Overall Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
F Determining The Number Of Layers In A Tensor Impedance Transmitarray
182
Bibliography
184
viii
CHAPTER
1
Introduction
Wave propagation has long fascinated scientists and engineers who have aimed to study these phenomena
in the many contexts that they arise; from mechanical, to acoustic, to electromagnetic and to the
quantum level. One of the earliest theories of wave propagation put forward was the Huygens principle
of wave-propagation.
The Huygens principle was introduced in 1699 by Dutch scientist Christiaan Huygens. This idea
theorized a way to explain how one wavefront generates the next wavefront, i.e. how a wave propagates.
To explain this, one front of the wave is replaced by an array of sources. These sources have the unique
property that they radiate spherical waves in the forward direction, while not radiating at all in the
backwards direction. Each source is weighted by the point on the existing wavefront that it replaces.
From this array of sources the next wavefront can be constructed based on the individual radiation of
each source. This is illustrated in Fig. 1.1.
This process can then be repeated for this new wavefront, by again replacing the new wavefront with
these special sources to construct the next wavefront. This idea of wave propagation is intuitive as it
ties into well known concepts such as constructive and destructive interference, giving insight into how
each wavefronts propagates.
In this thesis, the focus is on engineering hardware to control the propagation of electromagnetic
radiation, specifically microwaves. While Huygens’ principle of wave propagation preceded the knowledge
that electromagnetic fields propagate as a wave, it can also be generalized to electromagnetic fields as
well. The question is, what special insight does this Huygens principle give in the synthesis of RF and
microwave devices? To begin answering this question, a more rigorous form of the Huygens principle for
electromagnetic waves is discussed, which is referred to as the equivalence principle.
1.1
The Equivalence Principle
The Huygens principle mentioned above, was a fairly qualitative discussion of how a wave propagates.
This idea was put on more solid mathematical footing in the context of electromagnetics with the
1
2
Chapter 1. Introduction
Figure 1.1: Huygens principle illustrated. One wavefront creates the next wavefront via an array of
fictitious sources, weighted by the current wavefront. The sources radiate only in the forward direction.
development of the equivalence principle [1, 2]. The equivalence principle states how an arbitrary electromagnetic wavefront can be decomposed into an array of equivalent currents, which can then be used
to reconstruct the wave at any other point in space.
To begin this discussion, a source of electromagnetic radiation is placed in free-space. This source
can be any source of electromagnetic radiation, but is denoted in Fig. 1.2a as an antenna. This source,
as per Maxwell’s equations, radiates an electric and magnetic field, E and H everywhere in space.
The fields in space are now divided into two arbitrary volumes, V1 and V2 , via an arbitrary closed
surface S separating the two regions of space as shown in Fig. 1.2a. The fields inside V1 are denoted
E1 and H1 while the fields in V2 are denoted E2 and H2 . The goal now is to represent the fields in V2
using equivalent sources on the surface S instead of the original radiating source inside the surface.
To accomplish this, the fields in volume V1 are set to zero1 so that the surface is responsible only for
the radiation in volume V2 . From here, sources can be defined on the surface S as shown in Fig. 1.2c. For
electromagnetic fields the sources are surface current densities. It is assumed that Maxwell’s equations
are defined symmetrically with both electric and magnetic currents as sources of the fields. These surface
current densities are defined as the difference between the fields in volume one and two based on the
boundary conditions derived from Maxwell’s equations [3]. This is given as,
Js = n̂ × (H2 − H1 )
(1.1)
Ms = −n̂ × (E2 − E1 ) ,
(1.2)
where n̂ is a unit vector that is normal to surface and always pointing into V2 . The electric and magnetic
surface current densities are the electromagnetic analogue of the sources defined in the Huygens principle
which radiate to create the next wavefront. Note that the electric and magnetic current densities are
superimposed together on the same surface. Here however, these currents define the wave everywhere
1 This
is not strictly necessary as the fields can be set to any arbitrary value that is different than E1 and E2 , though
zero is simply a convenient choice.
3
Chapter 1. Introduction
E,H
Source
E1,H1
E2,H1
n̂
V1
V2
(a)
(b)
Ms = −n̂ × (E2 − E1),
Js = n̂ × (H2 − H1)
E2,H1
0
µo RR
e−jk(r−r ) 0
A = 4π
S Ms |r−r0| dS ,
0
εo RR
e−jk(r−r ) 0
F = 4π
S Js |r−r0| dS
n̂
(c)
Figure 1.2: (a) A source radiating a field. (b) A surface dividing space into an interior volume and
an exterior volume. The fields inside the surface are removed while the fields outside the surface are
maintained (c) The equivalent electric and magnetic currents on the surface which radiate the same
field outside the surface as the original source. The fields from the currents can be calculated via the
radiation integrals.
in V2 via the radiation integrals,
A=
µo
4π
F=
ZZ
εo
4π
0
Ms
S
ZZ
S
e−jk(r−r ) 0
dS ,
|r − r0 |
(1.3)
0
Js
e−jk(r−r ) 0
dS
|r − r0 |
(1.4)
Note that F and A are the vector potentials from which the fields can be found via the usual definitions
[3].
The surface current densities defined here are fictitious quantities which exist as a mathematical
convenience to define the fields in V2 . Despite their fictitious nature, these currents can be useful for a
variety of numerical problems where the surface current densities can be used to propagate fields forward.
The most common example is to find the far-field radiation from an antenna or to model a source in
computational problems [4, 5].
For example, in finding the radiation from an antenna, the fields of the antenna are computed in some
volume which is in the near-field of the antenna itself. The fields in the far-field can be found by finding
the equivalent current densities on the surface of the computational domain and then propagating them
4
Chapter 1. Introduction
pe
Js
pm
Ms
A
Figure 1.3: A small section of the surface currents which define the equivalent field, where it has been
assumed that the electric and magnetic currents are orthogonal to each other. These currents can be
discretized into electric and magnetic dipoles.
to the far-field using (1.3)-(1.4) [6, 7] .
However, the focus here in this thesis is not on numerical modelling using equivalent currents, but
on implementing hardware to control the scattering and propagation of microwaves. To see how this
equivalence principle can be applied to this task, the current densities on the surface are examined more
closely as discrete sources.
1.2
Huygens Sources And Scatterers
As stated above, the surface currents densities defined for the equivalence principle are conceptually the
same as the sources used in the Huygens principle, in that they radiate the field forward into V2 while not
creating any field interior to the surface in V1 . To probe this more closely, a special case is examined,
where the electric and magnetic surface current densities are assumed orthogonal to each other. In
general, the electric and magnetic surface current densities defined in (1.1)-(1.2) are not orthogonal to
each other because they are dependant on the choice of the surface. However by carefully choosing
the orientation of the surface S, the surface current densities can be made orthogonal2 . With this
assumption, the electric and magnetic surface current densities have some interesting properties upon
closer investigation.
The orthogonal currents on the surface S can be examined more closely by zooming into a small,
sub-wavelength surface of area A where it is assumed that this small area can be treated as a planar
surface. Here the electric and magnetic surface current density vectors can be discretized into electric
2 The
electric and magnetic fields are orthogonal to each other with E · H = 0. However (n × H) · (−n × E) = 0 only if
n · H = 0 or n · E = 0 implying that the surface S is tangential to either the electric field or the magnetic field or both.
5
Chapter 1. Introduction
and magnetic dipole moments, p, m, respectively as shown in Fig. 1.3. This is given by,
jωp
,
A
jωm
Ms =
,
A
Js =
(1.5)
(1.6)
It can be noted that the corresponding dipole moments also have the same properties as their continuous
counterparts in that they are both superposed and orthogonal.
This discrete analogue of an electric and magnetic surface current density has some interesting
properties as an isolated source. The properties of these dipole moments can be calculated using simple
antenna theory using expressions for electric and magnetic dipoles [4]. An illustration of an isolated
electric and magnetic dipole moment that are superposed and orthogonal to each other is shown in
Fig. 1.4a For an electric dipole, pointed along the z-axis, the far-fields are given by,
Eed =
ηjIe l
sin θθ̂,
2λ
(1.7)
where θ is defined in Fig. 1.4a and l is the length of the dipole. For a magnetic dipole pointing along
the z-axis, the far-fields are given by
Emd =
ηIm lλko2
sin θϕ̂
8π 2
(1.8)
These far-fields can be rotated to correspond to a magnetic dipole pointing along the y-axis as shown in
Fig. 1.4a by multiplying the coordinates in (1.8) by a 90◦ rotation matrix,
 0 
x
1
 0 
y  = 0
z0
0
0
0
−1
 
0 x
 
1 y 
z
0
(1.9)
Here the far-fields of a superposed, orthogonal electric and magnetic dipole moment are calculated as
given by,
U (θ, ϕ) =
1
|Eed + Emd |2 ,
2
(1.10)
where U (θ, ϕ) is the radiation intensity of the antenna. To understand how the electric dipole and
magnetic dipole radiate in tandem, the amplitude and phase of the magnetic dipole is varied with
respect to the electric dipole,
Im
= Iejγ
(1.11)
Ie
For this two-dimensional space of the relative amplitude and phase between the magnetic and electric
dipole, some interesting properties can be calculated. First in Fig. 1.4b, the front-to-back ratio can be
calculated across this two-dimensional space where the front-to-back ratio defined here as the power
radiated along the positive x-axis (θ = 90◦ ,φ = 0◦ ) versus the power radiated along the negative x-axis
(θ = 90, φ = 180◦ ) as given by,
FB =
U (θ, ϕ)|θ=90◦ ,ϕ=0◦
,
U (θ, ϕ)|θ=90◦ .ϕ=180◦
(1.12)
It can be seen as the relative amplitude and phase are varied, the front-to-back ratio varies as well from
all the radiation going into the +x direction to all the radiation going into the −x direction. Some
6
Chapter 1. Introduction
of the radiation patterns for different points on Fig. 1.4b are shown in Fig. 1.4c-1.4f. For example, in
Fig. 1.4c it is seen that the orthogonal and superposed electric and magnetic dipole moments radiate
isotropically into one half of space, while not radiating at all into the other half. This occurs when the
relative amplitude of the electric and magnetic dipoles are the same while the magnetic dipole is 90◦ out
of phase with the electric dipole. The same is true in Fig. 1.4d except the half-spaces are flipped while
in Fig. 1.4e-1.4f the sources radiate into both half-spaces with varying magnitudes.
What this example shows is that an orthogonal and superposed electric and magnetic dipole can, with
the right weighting, be arranged to radiate like an electromagnetic Huygens source, a single source that
radiates in one direction only. In the literature, this configuration of dipole moments is often referred to
as a ‘Huygens Source’.
With this definition of a Huygens source it can be seen that this configuration can be realized using
physical dipole antennas. Because an electrically short wire, fed by a source acts like an electric dipole
while a sub-wavelength loop of current also acts like a magnetic dipole, the two superposed in the right
orientation can be made to act like a real Huygens source as shown in Fig. 1.4g. This has been studied
extensively in the literature, for example in [8–11]. In fact, historically, this idea of Huygens source goes
back to [12–14], as a feed for reflectors. In this context of an isolated antenna, these Huygens sources
have been interesting antennas to study especially with regards to the properties of electrically small
antennas [8].
By looking at the radiation from a pair of electric and magnetic dipoles, this can be considered as an
active case of a Huygens source because the source itself is radiating. However a passive case can also
be defined where the scattering pattern from an electrically small cylindrical scatterer illuminated by a
plane wave can act like a Huygens scatterer as shown in Fig. 1.5a. Here the scatterer has both electric
and magnetic dipole moments because it has both a permittivity (εr ) and a permeability (µr ) that differ
from the surrounding medium (free space). Because of the incident plane wave, an electric and magnetic
dipole is induced in the cylinder. These dipole moments are orthogonal to each other because they are
induced by the electric and magnetic fields respectively (which are orthogonal themselves).
The scattering pattern from the cylinder, in the far-field, when illuminated by a plane wave travelling
in the x̂ direction as given in Fig. 1.5a, is defined by,
Es =
∞
X
an Hn(2) (kρ)ejϕ ẑ
(1.13)
−∞
where an is the scattering coefficients calculated by enforcing the continuity of the fields across the
(2)
surface of the cylinder, and Hn (kρ) is the Hankel function of the second kind [3] (See Chapter 3 for
more details). Again, the far-field pattern of the scattered fields can easily found from (1.13) and the
front-to-back ratio of the scattering-pattern can be calculated, similar to the definition in (1.12)
Here the ratio of εr and µr is varied as this controls the weight of the induced dipole moments. The
front-to-back ratio of the scattering pattern is plotted for different values of this ratio in Fig. 1.5b. The
scattering pattern at a few points on this plot are also shown in Fig. 1.5c-1.5f. Again it can be seen that
the scattering from the isolated cylinder also resembles the ideal pattern from the Huygens principle
except that this is a scattering pattern. Thus, this cylinder can be referred to as a Huygens scatterer
when the permittivity and permeability are chosen properly.
When it comes to implementation, using a magneto-dielectric particle is one option and an active area
of research. Some examples of dielectric surfaces can be found in [15–18] where the Mie resonances of
Chapter 1. Introduction
7
dielectric scatterers are engineered to design a surface. However it can be quite challenging to realize due
to limitations on realizable material properties, especially those with a negative value or large dielectric
constant. Because of this, realizing a passive Huygens scatterers, as will be discussed later on in Part 2
and Part 3, magneto-dielectric cylinders are not used. Instead applying ideas developed in the realization
of metamaterials [19, 20], electrically small and passive dipoles are used to realize an electric response
and electrically small and passive loops are used to realize a magnetic response. This results in a passive
analogue to the active Huygens source shown in Fig. 1.4g. The source is now replaced with a reactive
loading to adjust the scattering of the loop and dipole as shown in Fig. 1.5g.
Finally it should be mentioned again that the assumption taken here is that the electric and magnetic
dipoles are orthogonal to each other. Obviously, this is not required by the equivalence principle itself,
but by enforcing this constraint, it allows for the realization of individual sources and scatterers with
the interesting properties shown above.
1.3
Huygens Surfaces
As stated, the goal is to be able to use the equivalence principle/Huygens principle to design RF and
microwave hardware to control the propagation of electromagnetic fields. While it is interesting to note
that a single electric and magnetic dipole can act like a Huygens source or scatterer, the question of how
hardware, capable of controlling microwave radiation, can be constructed using these ideas is still not
clear.
Working toward this idea, the continuous surface current densities from the equivalence principle in
Section 1.1 are returned to again with the assumption that the electric and magnetic surface current
densities are orthogonal to each other. As discussed, the electric and magnetic surface current densities
represent fictitious sources which radiate the desired field in the equivalence principle. However, now
that physical Huygens sources and scatterers have been defined, a path now exists for implementing
these currents. This is done by forming an array of Huygens sources/scatterers which is referred to as
an active or passive Huygens surface. This is shown in Fig. 1.6.
In the active case, the surface consists of arrays of electric and magnetic dipole antennas, each
weighted with a source. By carefully choosing the weights of each source the electric and magnetic dipole
moments form a discrete approximation to the continuous surface current densities of the equivalence
principle. This physical approximation can be designed to radiate a desired field. For the active case, the
orthogonality constraint is not necessary per-se, but for the cases examined in the thesis it is enforced.
For the passive case, when the array of particles are illuminated by a source, their induced electric and
magnetic dipole moments add up to an induced electric and magnetic surface current which re-radiates a
desired scattered field. Here, the weights of each scatterer are tuned so that each scatterer only scatters
into one half of space. The resulting array of scatterers only scatters in one direction, allowing for the
realization of a reflection-less surface as will be discussed in Part 2 and Part 3
The question then, is how to design these Huygens surfaces. In general, this will be done by finding
the equivalent currents required to synthesize the desired radiated or scattered fields. From there, the
passive and active Huygens surface can be synthesized. Further questions regarding the active and
passive Huygens surfaces are also addressed. For example, how are the weights of the active antenna
array chosen and how many sources are needed to approximate the desired continuous surface current
densities. For the passive case, questions surrounding how the passive array of scatterers are modelled
Chapter 1. Introduction
8
and designed, and how the effective dipole moments are implemented using printed technology will have
to be answered.
These questions and more form much of the work discussed in this thesis, and these problems will
be addressed by examining Huygens surfaces in the context of specific applications.
1.4
Applications
Microwave engineers work on a variety of problems designing hardware for communication and imaging
systems. When these problems are reduced to their fundamental physics, the basic concepts involve
trying to control various properties of the electromagnetic field. These basic properties of the electromagnetic field are discussed in many electromagnetics textbooks and include phenomena such as
wave-guiding, radiation, scattering, refraction and polarization. These properties form the basis for
many fundamental technologies used in antenna and microwave engineering. One basic example is radiation, as discussed above, which is used to construct antennas used in wireless communications and
covers everything from basic dipoles to very complex antenna arrays. Another example is wave-guiding,
being able to confine and route electromagnetic fields, which is also important in designing networks for
phasing and filtering of microwaves.
In this thesis, the focus is on three areas, scattering, refraction and polarization. In each of these
areas the concept of a Huygens surface is applied.
On the topic of scattering, the problem of cloaking is examined, which seeks to suppress the scattering
from an object in all areas of space. Here the scattering of a metallic cylinder at microwave frequencies
is studied. To achieve this goal of cloaking a cylinder, the design of an active Huygens surface will be
investigated to form a new kind of cloak. This is discussed in Part 1.
For refraction, the problem of bending an incident plane wave with a single passive Huygens surface is
investigated as a means to constructing flat lenses made up of a single surface of scatterers. A procedure
for synthesizing and analyzing a passive Huygens surface is investigated along with their implementation
and will be discussed in Part 2.
Finally the problem of controlling the polarization of microwaves is discussed where the passive
Huygens surface designed to refract a plane wave are extended to also control the polarization. Both
polarization conversion effects and chiral effects are investigated. A further investigation of chiral effects
using impedance surfaces will also be discussed. This is done in Part 3.
1.5
Notation
For the rest of this thesis, some common definitions will be established so that the geometry and fields
can be discussed without confusion. Unless otherwise stated, wave-propagation takes place in twodimensions as shown in Fig. 1.7 where propagation is confined to the xy-plane. All quantities are
∂
assumed to be invariant in the z-direction ( ∂z
= 0). This two-dimensional approximation simplifies the
problems discussed herein, but none of the results that will be presented are confined to two-dimensions
and can be generalized to a full three-dimensional problem as required.
The fields in two-dimensions can be either TE or TM. The definitions of the TE and TM modes
are also shown in Fig. 1.7, where the TE mode consists of a plane wave where the electric field Ez is
Chapter 1. Introduction
9
transverse to the direction of propagation. Likewise the TM mode has its magnetic field Hz transverse
to the direction of propagation.
10
Chapter 1. Introduction
c
z
θ
f
e
d
pe
pm
x
y
ϕ
(a)
(b)
(c)
(d)
(e)
(f)
pe
pm
(g)
Figure 1.4: (a) A Huygens source consisting of an orthogonal electric and magnetic dipole (b) A plot
of the front-to-back ratio as a function of the relative amplitude and phase of the electric and magnetic
dipoles. The four points marked in the plot are the radiation patterns in the following figures. (c)
◦
◦
The radiation pattern for Im = 1ej90 Ie . (d) The radiation pattern for Im = 1e−j90 Ie . (e) The
−j10◦
j10◦
radiation pattern for Im = 1.75e
Ie . (f) The radiation pattern for Im = 0.4e
Ie . (g) The physical
implementation of a Huygens Source using small wire antennas. Here two independent sources feed each
antenna to control their amplitude and phase.
11
Chapter 1. Introduction
z
E
k
pe
H
d
c
f
pm
µr , εr
x
y
ϕ
e
a
(a)
(b)
(c)
(d)
jXe
(e)
(f)
jXm
(g)
Figure 1.5: (a) A Huygens scatterer consisting of a dielectric cylinder (in 2D). The incident plane
wave induces the electric and magnetic dipole moments as shown. (b) A plot of the front-to-back ratio
of the scattering pattern as a function of εr and µr . The locus of points with a large and positive
front-to-back ratio act like a Huygens source. The four points marked in the plot are the radiation
patterns in the following figures. (c) The scattering pattern for εr = 2.08, µr = 4.89 which acts like a
Huygens scatterer. (d) The scattering pattern for εr = −0.53, µr = 5.74. (e) The scattering pattern
for εr = −6.14, µr = −6.99. (f) The scattering pattern for εr = 9.05, µr = 4.74. (g) The physical
implementation of a Huygens scatterer using passive loops and wires.
12
Chapter 1. Introduction
z
Einc
Hinc
k
x
y
Figure 1.6: A Huygens surface of loops and dipoles. The surface can be either passive or active
z
E
TE
k
x
H
H
TM
k
E
y
Figure 1.7: The definitions for TE and TM fields in a two-dimensional geometry
Part I
Controlling Scattering With An
Active Huygens Surface
13
CHAPTER
2
Background On Active Cloaking
The first application of a Huygens surface in this thesis is on the problem of cloaking. However, before
describing the design of the cloak the existing research on cloaking is reviewed to give some context.
2.1
Literature Review
Cloaking an object from an electromagnetic field represents a major advancement in the ability to
control and manipulate electromagnetic radiation. Since the first work reported in [21], the concept of
cloaking an object from an incident electromagnetic wave has spurred many different perspectives on
how scattering off of an object can be controlled. While cloaking caught a lot of attention due to its
science-fiction implications, the idea of controlling the electromagnetic scattering of an object also has
a long history in the field of radar. Here an overview of some of the key results are presented in both of
these areas.
2.1.1
Transformation Optics Based Cloaking
As stated above, cloaking was first demonstrated in [21], and combined two important and recently developed ideas in electromagnetic research. These were metamaterials and transformation optics. Metamaterals are arrays of sub-wavelength scatterers, engineered to generate an effective material response
[22, 23]. In a metamaterial, the properties of these scatterers can be engineered, including the lattice,
spacing, geometry and material composition, resulting in electromagnetic responses that are either difficult or impossible to find in nature. These phenomena include, negative material parameters [19, 20, 24],
material parameters near zero [25, 26] and bianisotropy [27–29].
Transformation optics is a formulation of Maxwell’s equations that makes a connection between
geometry and material parameters [30, 31]. This is done by transforming Maxwell’s equations into an
arbitrary coordinate system and making the observation that the material parameter tensors in the
equations, the permitivitty and permeability tensors, have the same effect as the geometrical properties
given by the metric tensor. Hence, geometry and material properties are interchangeable and the effect
14
15
Chapter 2. Background On Active Cloaking
of a coordinate transform on electromagnetic waves could also be mimicked by an equivalent set of
material parameters. This is given by the now famous equation [30, 31],
√
0
det G0 ΛG −1 ΛT
¯
¯
ε̄ = µ̄ = √
,
det Γ det Λ
(2.1)
where Λ is the coordinate transform tensor, and G, Γ are the metric tensors of the original space and the
material space. This equation can be used as a way to engineer the bending of electromagnetic fields via
a coordinate transform which can then be implemented by the corresponding set of material parameters.
The one thing not guaranteed by (2.1) is that the required material parameters fall within the spectrum
of materials found in nature. This is where the combination of transformation optics and metamaterials
seems especially enticing as transformation optics can give a description of how one wants to control
electromagnetic fields while metamaterials give a way to realize this.
The cloak was the first demonstrated transformation optics device that was ever proposed [30]. The
idea behind the cloak was to use a coordinate transform to bend an electromagnetic wave around a
cylindrical (or spherical) volume as shown in Fig. 2.1. Because the wave never enters the interior of the
cylinder and instead travels seamlessly around it, this volume is cloaked in this coordinate transformation
for an incident plane wave. The required material parameters using (2.1) are given to be [21, 32],
 ρ−a
ρ

¯Γ =  0
¯Γ = µ̄

0
0
ρ
(ρ−a)
0

0
b
b−a
0
2
ρ−a
ρ

,

(2.2)
where a and b are the inner and outer radii of the cloak. This material is diagonally anisotropic in the
cylindrical coordinate basis used above. It can also be seen that the material parameters along the z-axis
of the cloak range between 1 ≤ {µz , εz } ≥ ∞ and the azimuthal and radial components are bounded
between 0 ≤ {µr,φ , εr,φ } ≥ 1. The material parameters of the metamaterial take on this range of values
between the inner radius a and the outer radius b where the rate of change between the inner and outer
radius is dependent on the thickness of the metamatieral shell. Clearly a thicker cloak allows for a more
gradual change in the material parameters while a thinner cloak requires a very rapid variation.
This material need to construct the cloak can only be implemented by a metamaterial and in [21]
a metal cylinder was covered with a specially engineered metamaterial that realized the material parameters of the cloak. The design and measurement of the cloak discussed [21] is reviewed here, as the
details given there help form the basis for the choices made in the design and measurement of the cloak
discussed in the following chapters.
The metamaterial cloak was designed for a single polarization, in this case a TE polarization, thus
requiring only a εz , µφ and µr response in a two-dimensional environment. This two-dimensionality
was enforced by placing the cloak and cylinder in a parallel-plate waveguide terminated by absorbing
materials. The outer radius of the cloak is made two times thicker than the inner radius to allow for a
gradual change in the material parameters which is easier to implement. To simplify the construction
of the cloak the material parameters of the cloak are approximated. As further detailed in [33], using
approximate material parameters for the cloak is a tradeoff between achieving ideal cloaking performance
and designing a realizable metamaterial. For example, large values εz are very difficult to implement,
and having a controlled material response in both εz , µφ and µr leads to a very complex arrangement
16
Chapter 2. Background On Active Cloaking
(a)
(c)
(b)
(d)
Figure 2.1: (a) An empty cylindrical space. (b) The transformed space. (d) The equivalent material
space shaded in blue in a cylindrical space. (d) The transformed fields.
of subwavelength scatterers due to the anisotropy. Thus, the approximation using in [21] is to relax the
material parameters such that the wavefronts of the incident field are still guided through the cloak but
the cloak itself is no longer impedance matched to its surroundings. This will introduce some scattering
from the cloak but will still allow for the shadow created by the cylinder to be removed. This requires
a constant value of µφ , chosen to be µφ = 1 and a constant value of εz . Only µr is a function of space.
With this simplification, concentric rings of split-ring resonators fabricated on printed circuit boards
are used to implement both µr and εz . In total 12 layers of split-ring resonators are needed to hide a
cylinder with a radius of 0.77λ. To characterize the scattering off of the cloak, the cloak was illuminated
by a point source placed in the waveguide. To characterize the fields surrounding the cloak a probe was
attached to the top-plate of the waveguide and translated using a xyz-translator to map the electric field
in the waveguide. These results are shown in Fig. 2.2 along with the fabricated cloak. It can be seen
17
Chapter 2. Background On Active Cloaking
(a)
(b)
Figure 2.2: (a) The fabricated metamaterial cloak. Taken from [21]. (b) The measured electric field of
the bare cylinder and the cloak. Taken from [21]
that when compared to the ideal cloak the reduction in the scattering off of the cloak is not as dramatic
with this approximated design. But even still, the fields from the point source are restored behind the
cylinder removing the shadow created by the cylinder.
This approach to cloaking is very general as in theory it can cloak an object of any shape or size simply
by choosing the right coordinate transform. The cloak itself is independent of the material properties
of the object and the properties of the incident field. However, as seen from this brief discussion of an
experimentally realized cloak using a metamaterial shell, there are many challenges in realizing this cloak
that cannot be avoided and it quickly becomes clear that despite advances in metamaterial fabrication
an ideal cloak cannot be realized and only crude approximations are possible.
Moreover, there are fundamental theoretical limitations with this cloak that limit its performance no
matter how accurately it is implemented using a metamaterial. These limitations come from studying
the cloak in the time domain when pulses are used to illuminate the cloak as opposed to monochromatic
waves. Under these scenarios it becomes clear that after accounting for realistic dispersion in the material
parameters of the cloak, it is not possible to use a cloak to hide an object from pulsed electromagnetic
fields [34, 35]. In fact a perfect cloak is only possible at a single frequency as it is not possible to cloak
an object over a finite bandwidth without violating causality [36]. This can simply be explained by the
fact that the phase velocity on the inner radius of the cloak is infinite, a requirement that can only be
sustained at single frequencies due to dispersion and causality limitations of passive materials. Still,
the total scattering cross section can be reduced over a finite bandwidth, despite the fact that perfect
cloaking of a pulse is not possible. However, for most realistic scenarios this bandwidth is still very
small due to dispersion and causality. It can also be noted that even reducing the total scattering cross
section over a finite bandwidth means that the scattering off of the cloak and object increases outside
the desired bandwidth [37].
Within the context of transformation optics, various proposals have been put forth to see how cloaking
can be achieved for scenarios that can relax these limitations. This is done by using specific knowledge
about the object being hidden and/or the incident field it is being illuminated with. By studying more
specific problems as opposed to a general cloaking problem, more practical cloaks can be realized. One
example of this is the ground plane cloak which can hide an object placed on a PEC conductor over a
broad bandwidth [38, 39]. Here, the cloak is designed to take the incident field and redirect it into the
specularly reflected direction of a bare ground plane. The coordinate transform for this kind of problem
Chapter 2. Background On Active Cloaking
18
can be found using a quasi-conformal coordinate transform [40]. This quasi-conformal approach gives
rise to material parameters that are isotropic with minimal dispersion since ε > 0 and µ > 0 over a
broad-bandwidth. These kinds of material parameters can easily be implemented by a metamaterial
for the right polarization (i.e. a positive permitivitty) as shown in [38]. This was also demonstrated at
optical frequencies [41]. Another example of this kind of relaxed approach to cloaking is unidirectional
cloaking which takes advantage of knowing the direction of the incident field a priori to relax the
material parameters in (2.2) to isotropic dielectric parameters with minimal dispersion [42, 43]. Doing
this however, increases the scattering of the cloaked object if it is illuminated from a direction other
then the direction it was designed for, though the cloak can be designed for multiple direction by taking
advantage of symmetry considerations [43].
It should be noted that despite the inherent limitations with transformation optics based cloaking, the
idea itself has been a fruitful research topic furthering the understanding of both cloaking and scattering
problems from both theoretical and engineering perspectives. One example of this is the time-cloak
which generalizes the transformation optics cloak, allowing for an object to be hidden in both time and
space, thus cloaking events [44]. Implementing this cloak would require biansiotropic materials but an
experimental demonstration of this idea was done using non-linear optics instead [45]. Cloaking has also
been demonstrated for other kinds of waves and fields including acoustic cloaks [46] and thermal cloaks
[47]. Meanwhile the transformation optics idea has led to novel designs being proposed for a variety of
electromagnetic devices from lenses to antennas [48–50].
2.1.2
Plasmonic Cloaking
Parallel to the development of the transformation optics based cloak, another cloaking mechanism was
proposed around the same time, referred to as plasmonic cloaking [51]. Understanding plasmonic cloaking
requires an understanding of how electromagnetic waves are scattered off of an object in two or three
dimensions. This is done for canonical objects such as cylinders in two-dimensions and spheres in
three-dimensions. Assuming an incident plane wave, the scattered field of the object can be broken
down into a sum of cylindrical or spherical wavefunctions of increasing order as briefly mentioned in
Chapter 1. Plasmonic cloaking proposes covering an object with an homogeneous material shell. By
properly choosing the material properties of the shell one can choose to cancel out exactly one order of
the scattered field. These material properties are found by solving the boundary value problem given
by the object+shell geometry. For an arbitrarily large object this approach would have minimal effect
as the scattered field is a sum of many orders. However for small objects, where the radius of the object
is less than λ/10, the scattering is dominated by the lowest order term which is a monopole term in
two-dimensions and a dipolar term in three-dimensions. Thus an electrically small object covered by a
shell with the right material parameters can be rendered invisible as depicted in Fig. 2.3.
As it turns out, for a TE-polarized incident plane wave, the material parameters for a shell covering
a dielectric or PEC cylinder/sphere are determined to have a negative permittivity, like a plasmonic
material, hence the name plasmonic cloaking. Intuitively this makes sense as a plasmonic material has
a polarizability that is approximately 180◦ out of phase from the polarizability of a dielectric material.
Hence when superimposed together and properly designed the response of both the object and the shell
can cancel out, effectively hiding the object.
Implementing a plasmonic material shell can be done by either using a metamaterial at microwave
and terahertz frequencies or a plasmonic material at optical frequencies. At microwave frequencies plas-
19
Chapter 2. Background On Active Cloaking
1
Complex Mag (a.u.)
0.8
0.6
0.4
0.2
0
−6
−4
−2
0
2
4
6
n (order)
(a)
(b)
0
−5
RCS (dB)
−10
−15
−20
−25
−30
−35
−40
Plasmonic Cloak
Dielectric Cylinder
−150
−100
−50
(c)
0
φ
50
100
150
(d)
Figure 2.3: (a) Scattering off of an electrically small dielectric cylinder from a TE-polarized plane wave.
The dielectric cylinder has a radius of ρ = λ/10 and a dielectric constant εr = 5εo . (b) The different
scattering orders for the dielectric cylinder. For a small cylinder the n = 0 term dominates. (c) The
cloaked cylinder with a plasmonic cover with thickness ac = 1.5a and a dielectric constant εc = −5.46εo .
The scattering has been dramatically reduced (d) The calculated radar-cross-section of the cloaked
dielectric cylinder. With the n = 0 term cancelled the only remaining terms are the n = ±1 terms which
are dipolar but much smaller in magnitude.
monic metamaterials can be implemented by using cut-off parallel-plate waveguides [52]. Experimental
demonstration of plasmonic cloaks at the GHz frequency range have been shown in [53, 54]. At infrared
frequencies and above, naturally occurring plasmonic materials such as silver and gold become viable
options and may be useful for cloaking nano-particles [55].
For the orthogonal polarization or for an object with a significant permeability, the shell would be
required to have a negative permeability also. While this is possible to implement using a metamaterial
it can be more challenging to engineer and is much more constrained by dispersion. Hence plasmonic
cloaking has tended to focus on using negative permitivitty shells only.
When looking at the capabilities of a plasmonic cloak it can be thought of as a specially designed
cloak for electrically small dielectric objects only. While it has been shown that the plasmonic cloak can
be extended to larger arrays of objects, a cluster of spheres for example, size and material composition
are the main limitations. Like the theoretical limitations discussed above for transformation optics based
cloaks, plasmonic cloaks too are limited by the dispersion of the plasmonic material, meaning that they
can reduce the total scattering cross section across a finite bandwidth only, but outside that bandwidth
Chapter 2. Background On Active Cloaking
20
the scattering increases [37]. Further work on the plasmonic cloak has looked at reducing the material
shell to a passive surface impedance [56], as well as using non-Foster surface impedances to broaden the
bandwidth and limit the dispersion [57].
2.1.3
Other Cloaking Designs
With the advent of transformation optics based cloaking and plasmonic cloaking, researchers have also
investigated cloaking designs using more traditional electromagnetic and photonic technologies. One
example of this in the microwave frequency range are transmission-line/waveguide type cloaks [58, 59].
These kinds of cloaks surround the object with transmission-lines or waveguides which couple the incident
free-space wave into a transmission-line type structure and radiate it out again on the other side. By
doing this, the incident field is prevented from scattering off of the object and appears to pass through the
object unperturbed. While the exact properties of the guiding networks have only been found through
empirical methods, no general mechanism has been elucidated for these guided wave networks, though
they have been shown to be broadband in reducing the total scattering off an object.
2.1.4
Radar Supression
Before turning the discussion of cloaking to active cloaking methods, it is worth talking briefly about
radar avoidance/suppression techniques as the frequency range of interest in this thesis is in the microwave band. To begin, note that radar suppression is a slightly different problem than cloaking.
Radar is usually discussed in the context of monostatic and bistatic radars where there is only one transmitter and one reciever. In the monstatic case the receiver and transmitter are co-located and in the
bistatic case they are separate. Hence, radar suppression methods only need to prevent the signal scattered from the target from travelling to the receiver only, unlike in cloaking where the scattered signal is
suppressed in all directions of space. In the monostatic case that means preventing the scattered signal
from reflecting back in the direction of the collocated transmitter/receiver and in the bistatic case from
specularly reflecting to the receiver that is separate from the transmitter. To accomplish this, two main
approaches are taken; the first are absorbing surfaces and materials and the second are non-specularly
reflecting surfaces.
Using absorbing boundaries relies on destructive interference and loss. Examples of designs which
eliminate reflected fields are classic structures such as Salisbury and Jaumann screens [60, 61]. The
basic idea behind these kinds of structures include using resistive sheets placed a quarter wavelength
above a PEC. The resistive sheet will absorb some of the incident field and any of the reflected field
off of the PEC will destructively interfere with the reflected field off of the absorber. Such a design
can dramatically reduce scattering when compared to a bare PEC plate. Lossy materials, either lossy
dielectrics or lossy magnetic materials can also be used to absorb electromagnetic materials [62, 63].
This can include tapering the conductivity of the surface to suppress reflections.
Non-specularly reflecting surfaces either use the physical design of the surface to reflect the incident
field into an undesired direction or use an FSS-type patterning of the surface to redirect the radiation
into undesired directions. [64, 65]
Conceptually, while these approaches are fairly straightforward, they can be effectively engineered
to reduce the scattering cross-section. They can also be easily optimized and incorporated into existing
structures making them practical choices. This has resulted in these designs being readily adopted and
Chapter 2. Background On Active Cloaking
21
used in everyday scenarios. The challenges with these approaches are that absorbers can give rise to
thermal signatures from the heat generated by the absorbed field while non-specularly reflecting surfaces
can be less effective in multi-static radar environments where more receivers are present to detect the
reflected signal. It should also be noted, for completeness, that signal processing techniques such as
radar jamming and digital radio frequency memory are also used to confuse radar receivers and to create
false targets [66].
2.1.5
Active Cloaking
With this understanding of passive cloaking and radar avoidance, active cloaking now becomes the focus
for the rest of this chapter as well as the following two. Active cloaking involves constructing a cloak
using sources which radiate to cancel the scattered field, hence the designation as active. One of the
first papers to discuss this idea of active cloaking is [36] by Prof. David A.B. Miller. In [36] the topic of
active cloaking is introduced, focusing on the constraints of causality and how one would achieve perfect
cloaking for an incident pulse.
From the discussion in Section 2.1.1, it was noted that the transformation optics cloak was fundamentally limited in bandwidth due to causality constraints. In [36], this is defined as true for any
passive cloak because for a pulse to travel around an object, the path in the cloak is always longer than
the distance straight through the object. Thus the pulse would always be delayed due to the material
comprising the cloak. In [36], the question now turns to active cloaks, which have no materials but
instead use sources, and the limitations facing these kinds of cloaks?
In [36], an active cloak is tackled for an arbitrary scalar wave incident on an object. For this scenario
there is an array of sources and sensors surrounding and object. In an electromagnetic context these
sources are antennas with specific radiations patterns. (For other waves such as acoustic waves, these
sources would be transducers). Here, the sources are assumed to be dipolar. The problem of cloaking
in this scenario becomes figuring out what the weights on each source should be as to cancel the fields
scattered by an incident pulse. These weights are determined by the measurements taken by each
sensor. As the sensors sample the incident field all around the object, these measurements are then used
to update the weights on the sources all around the object. Mathematically this requires integrating the
measured fields by the sensors over all of space and all of previous measurements in time. To do this in
‘real-time’ however requires the information from each sensor surrounding the object to be distributed
to all the other sources. For an electromagnetic wave, this information cannot be distributed to all the
sources in real-time without violating causality as the information from each sensor takes a finite amount
of time to be transmitted to all of the sources. Thus, an active electromagnetic cloak for pulses is not
possible either. Miller notes in [36], that other waves, such as acoustic waves, do not face the same
limitations and that an acoustic cloak for incident pulses would be possible. Such a cloak is referred to
as perfect cloaking.
While there is this fundamental limitation restricting both active and passive cloaking, there are still
some interesting things that can be accomplished with active cloaking that is not ‘perfect’. If there is
prior knowledge about the incident field, active cloaking for a pulse is still possible. This kind of cloaking
is referred to as a priori active cloaking. The knowledge of the incident field required is its bandwidth,
direction, amplitude and phase. With this knowledge the weights on the sources can be set ahead of
time. It should be noted that in acoustics these ideas were known in some capacity already [36, 67]. For
a priori cloaking of electromagnetic fields this can be accomplished in one of two ways; using sources
Chapter 2. Background On Active Cloaking
22
Figure 2.4: An example of an active exterior cloak. The multipole sources are contained in the three
dark circles and combined with the radiation of the incident plane wave create a quiet region around the
scatterer at the center of the figure. Taken from [68].
surrounding an object as mentioned above and referred to in the literature as interior cloaking. Or
sources can be arranged in the far-field of the object being hidden, referred to as exterior cloaking.
Exterior methods require an array of multipole sources situated around an object to be cloaked.
These sources create a region of space where, when added to the incident field, the total field is zero
[68–70]. The number of multipole sources required is at least three though there is no proof why that it
is so. The weights and order of the multipole terms of these sources are determined by the properties
of the incident field and the size of the space where the field is desired to be zero. Because the sources
are far away from the location of the object, any object can be placed in this zero field region without
disturbing the interference of the multipole sources and the incident field. This allows for no scattering
off of the object and thus no scattered field. An example solution of an exterior cloak is pictured in
Fig. 2.4. Because of the challenges of implementing multipole electromagnetic sources, the multipole
sources themselves can be implemented as arrays of dipolar sources surrounding the source via the
equivalence principle.
The concept of interior active cloaking is discussed in the following chapter as it forms the basis
of the active cloak discussed in this thesis. It is also important to note that no active cloak has been
demonstrated experimentally up to this point and the experimental details will be discussed in Chapters 3
and 4. While no active or passive cloak can faithfully hide an object from an electromagnetic pulse
due to causality, a priori active cloaking can still be useful in hiding an object from a predetermined
electromagnetic field. In the following chapters, the limitations of this kind of a priori cloak will be
discussed as well.
CHAPTER
3
Active Cloaking - Modelling and Simulations
With an understanding of passive and active cloaking established, the problem of constructing an active
cloak is now considered. As stated in the previous chapter, the kind of active cloak that is under
discussion is an active interior cloak, one where the object being hidden is immediately surrounded by
sources. With this stated, the ‘interior’ term will be dropped for brevity’s sake. The active cloak also
requires prior knowledge of the incident field, mainly its frequency, direction, amplitude and phase.
To show how such an active cloak is constructed, some basic concepts regarding scattering are first
established. Then the design of an active cloak is discussed along with simulations results.
3.1
3.1.1
Problem Definition
Scattering Off Of A Cylinder
Scattering of electromagnetic fields off of an object can be studied for a variety of shapes, though closedform solutions only exist for a handful of canonical shapes. Here the two-dimensional case of an infinite
cylinder is studied.
The geometry is defined in Fig. 3.1 with a cylinder centred at the origin of the coordinate system with
a radius of a and a material composition of εr (Perfectly conducting cylinders are also treated as well
as these are a special case when εr → ∞). The incident field is a monochromatic plane wave travelling
along the +x direction with an amplitude of Eo and constant phase offset of ϕo , Ei = Eo eθo e−j(ko x−ωt) ẑ.
Note that the angle of incidence can be made arbitrary without loss of generality.
With the incident field, the scattered field can be calculated and then summed with the incident
field to give the total field in the space both inside and outside the cylinder. Given the two-dimensional
nature of the problem, the incident plane wave is expanded into an infinite sum of cylindrical harmonics
[3]
Ei = Eo eθo e−jko x x̂ = Eo eθo
∞
X
n=−∞
23
j −n Jn (ko ρ) ejnϕ ẑ.
(3.1)
24
Chapter 3. Active Cloaking - Modelling and Simulations
Figure 3.1: The scattering problem for an infinite dielectric cylinder.
0
−5
RCS (dB)
−10
−15
−20
−25
−30
−35
−40
(a)
(b)
−150
−100
−50
0
φ
50
100
150
(c)
Figure 3.2: (a) The theoretical scattered field for a dielectric cylinder illuminated by a plane wave. (b)
The theoretical total field. (c) The calculated bi-static radar cross section (RCS).
The scattered field outside of the cylinder is also expressed as a sum of cylindrical harmonics as stated
in Chapter 1, given by,
Es =
∞
X
an Hn(2) (ko ρ) ejnϕ ,
(3.2)
n=−∞
along with the field inside of the cylinder,
Eint =
∞
X
bn Jn (k1 ρ) ejnϕ ,
(3.3)
n=−∞
√
1
where k1 = ko εr . The magnetic fields are found from H = − jωµ
∇ × E. To find the weights an and
o
bn of the scattered and interior fields, the boundary condition imposed by the dielectric cylinder can be
used where the continuity of both the electric and magnetic fields is imposed at the perimeter of the
25
Chapter 3. Active Cloaking - Modelling and Simulations
y
Es,Hs
Eint,
Hint
x
εr
Js,Ms
Figure 3.3: The scattering problem for an infinite dielectric cylinder where the scattered and interior
fields are generated by equivalent currents radiating at the boundary of the cylinder.
cylinder. Doing this gives,
0
an = Eo eθo j −n
0
k1 Jn (ka)Jn (k1 a) − kJn (ka)Jn (k1 a)
(2)0
(2)
k1 Hn (ka)Jn0 (k1 a) − kHn (ka)Jn (k1 a)
(2)0
θo −n
bn == Eo e j
0
(2)
(3.4)
,
(3.5)
(2)
k1 Jn (ka)Hn (ka) − kJn (ka)Hn (ka)
(2)0
,
k1 Hn (ka)Jn0 (k1 a) − kHn (ka)Jn (k1 a)
where the primed notation indicates the derivative with respect to the argument. The total field is then
given to be Et = Ei + Es . The scattered and total fields are plotted in Fig. 3.2 for a dielectric cylinder of
radius a = 0.6λ and εr = 10. With these fields, the radar cross section of the cylinder can be calculated
as well. The radar cross section is given by [71],
σRCS = lim 2πr
r→∞
|Es |2
|Ei |2
(3.6)
and the RCS of a dielectric cylinder is also plotted in Fig. 3.2c.
3.1.2
The Induction Theorem
Closely related to the equivalence principle discussed in Chapter 1 is a theorem called the induction
theorem which gives another method of solving the scattering problem discussed above in Section 3.1.1
[3]. Again, as shown in Fig. 3.1 there are some sources incident on a scatterer and the fields can be
decomposed into incident, scattered and interior fields. Another problem can also be defined where the
sources are removed but the scattered field is maintained outside the scatterer and the interior field is
maintained inside. Because of the discontinuity that exists with the removal of the incident field, electric
and magnetic current densities must be defined on the surface of the scatterer. To enforce the fields
given in Fig. 3.3 the electric and magnetic surface current densities are given to be,
Js = n̂ × (Hs − Hint ) = −n̂ × Hi ,
(3.7)
Ms = −n̂ × (Es − Eint ) = n̂ × Ei .
(3.8)
Comparing the two scenarios in Fig. 3.1 and Fig. 3.3 the electric and magnetic surface current densities
on the boundary of the scatterer can be interpreted as generating the scattered fields outside the scatterer
and the interior fields inside. These currents then completely define the scattered and interior field.
26
Chapter 3. Active Cloaking - Modelling and Simulations
(a)
(b)
(c)
Figure 3.4: The superposition of sources on the boundary of an object to form an active cloak which
suppresses the scattering off of a cylinder.
Since these current densities are a function of the incident field only as shown in (3.7)-(3.8), the
scattering problem can now be recast as a radiation problem, with the incident field creating (fictitious)
currents on the boundary of the scatterer which subsequently radiate the scattered field outside and
the total field inside. To solve the scattering problem then can be done by finding the radiation of the
currents in the presence of the scatterer,
Es = (−jωµo +
1
∇∇·)
jωεo
ZZ
S
Js GE,scat (ρ, ρ0 )dS − ∇ ×
ZZ
Ms GM,scat (ρ, ρ0 )dS,
(3.9)
S
where GE,scat and GM,scat are the Green’s functions for electric and magnetic currents radiating in the
presence of a scatterer [72].
In terms of solving the scattering problem, the boundary-value-problem approach as done in Fig. 3.1
or the induction theorem illustrated in Fig. 3.3 are both equally challenging but the insight of the
induction theorem will come in useful in solving the active cloaking problem.
3.2
An Active Cloak Using The Induction Theorem
Given the understanding of scattering from both a boundary value perspective and the inductance
theorem perspective that was established in the previous section, an active cloak can be constructed
using these two basic ideas. This is illustrated in Fig. 3.4. Here these two scenarios are superimposed.
In the first scenario of Fig. 3.4a, is a picture of the traditional scattering problem shown earlier in
Section 3.1.1. In Fig. 3.4b the induction theorem scenario also presented in the previous section, is shown
where the equivalent magnetic and electric currents Ms and Js given by (3.7)-(3.8) are established on
the boundary of the cylinder.
With these two scenarios, the goal of an active cloak is to cancel the scattered fields outside of
the cylindrical object. This can be done by subtracting the two scenarios described in Fig. 3.4a with
Fig. 3.4b. In this situation a known incident field, Ei , Hi , impinging upon the cylindrical object generates
the scattered and interior fields. However, there are also a set of magnetic and electric surface current
densities surrounding the object, Mcloak and Jcloak respectively, with the magnetic and electric surface
current densities given by the negatives of (3.7) and (3.8) with Mcloak = −Ms and Jcloak = −Js . These
surface current densities are imposed on the boundary of the object. Here Mcloak and Jcloak are 180◦
out of phase with the incident field. Because of this, the total field outside the object is simply the
Chapter 3. Active Cloaking - Modelling and Simulations
27
incident field by superposition. This is because the scattered fields generated by the incident field are
cancelled by the fields generated by the magnetic and electric surface current densities, Mcloak and
Jcloak . Likewise, the interior field vanishes. These electric and magnetic surface currents are thus used
to restore the incident field . With these sources in place around the object, the scenario in Fig. 3.4c is
equivalent to the object being cloaked since all that is left are the incident fields Ei , Hi .
3.3
Implementation
While the surface current densities described by Mcloak and Jcloak are continuous functions on the
boundary of the object being hidden, a simple way of approximating such a distribution is by using
an array of electric and magnetic dipoles. These discrete arrays of electric and magnetic dipoles are a
Huygens source as discussed in Chapter 1. These electric and magnetic dipoles can be arranged around
the boundary of the object in such a way as to implement a discrete version of a magnetic or an electric
surface current. When implementing this array of dipoles two questions are raised, 1) How many electric
or magnetic dipoles are needed? and 2) What are the weights required on each dipole?
Beginning with the first question, for an incident plane wave travelling in the x̂ direction with a ẑ
directed electric field, the magnetic and electric currents on the boundary of a cylinder are,
Mcloak = −e−jka cos ϕ ϕ̂
Jcloak =
cos ϕ −jka cos ϕ e
ẑ
η
(3.10)
(3.11)
and are plotted in Fig. 3.5 for a cylinder with radius a = 0.7 λ. On a circular boundary, these current
distributions are periodic and can be decomposed into a set of discrete sinusoidal functions, ejmθ where
m is the order of the sinusoid. The highest non-zero sinusoid needed to reconstruct this surface current
distribution is given by M . This allows for the minimum number of discrete sources needed to reconstruct
the surface current densities, N to be defined as N = 2M as per the Nyquist (sampling) theorem [73].
These sources are placed at equal angular spacings around the circular boundary. This is done by taking
(3.10) and (3.11) and applying a discrete Fourier series (FFT) to determine M . For example, a cylinder
with radius a = 0.7λ requires Nm = 20 samples to reproduce the magnetic surface current distribution
given in (3.10) and Ne = 24 samples to reproduce the electric surface current distribution given in (3.11).
Thus 20 magnetic dipoles and 24 electric dipoles are needed, each surrounding the object. Note that
this method can be carried out for other geometric boundaries and other incident field patterns and can
also be generalized to three-dimensions.
With the number of electric and magnetic dipoles known, the weights of each dipole can be determined by converting the magnetic and electric surface current densities into electric and magnetic dipole
moments. These dipole moments are directed along the same direction as their corresponding surface
current densities. For the magnetic dipole moment its magnitude, |pm |, can be related to the magnetic
surface current density by, |pm | = |Ms |hl where h is the height of the cylindrical volume that Ms resides
on and l is the arc length between adjacent dipoles. Likewise the electric dipole moments, pe is given
by |pe | = |Js |lh.
At microwave frequencies, these electric and magnetic dipoles can be implemented using electrically
small antennas as discussed in Chapter 1. For electric dipoles, electrically small straight wire antennas
radiate as electric dipoles and for magnetic dipoles, electrically small wire loops radiate as magnetic
28
Chapter 3. Active Cloaking - Modelling and Simulations
(a)
(b)
Figure 3.5: (a) The equivalent magnetic surface current on the boundary of the cylinder for a TEpolarized plane wave. (b) The equivalent electric surface current on the boundary of the cylinder for a
TE-polarized plane wave.
dipoles. Placing these straight and loop wire antennas around a cylindrical object and feeding them
with the appropriate current is sufficient to physically realize the active cloak. For the electric dipole,
the dipole moment can be translated into a feed current by dividing |pe | by the height of the cylinder
h. For a magnetic dipole, the dipole moment |pm | can be translated into a feed current on the loop as
given by Im =
|pm |
jωµS ,where
S is the area of the loop [3]. It is important to note that since the surface
current densities only depend on the incident field, the physical currents on each antenna are also a
function of the incident field. For an incident plane wave, which is an appropriate approximation for
far-field sources, this requires only control of the phase of the currents on each electric and magnetic
dipole, resulting in a less complex feed that requires only phase control. However, in Section 3.5 it will
be shown what can be achieved with amplitude and phase control also.
3.4
Simulation
The active cloak is now verified using both closed-form expressions and numerical simulations using
commerical CAD tools.
3.4.1
Theoretical Model
A simple closed-form model of the active cloak can be constructed using two-dimensional solutions to
Maxwell’s equations. For an incident plane wave, the scattered and interior fields are already known
from (3.1)-(3.5). To model the active cloak, the radiation from an array of electric and magnetic dipoles
surrounding the cylinder is calculated. The exact model is shown in Fig. 3.6 where the cylinder is in the
usual location with a radius ρ = a and a dielectric constant εr . The electric and magnetic dipole array
is located just outside the scatter at a radius ρ = ac , this offset is necessary as the dipoles can never be
physically placed directly on the boundary due to their finite size. Ideally the dipoles should be placed
as close as possible to the scatterer for the derivation described above to hold true. The dipoles are
uniformly distributed around the cylinder with Ne electric dipoles and Nm magnetic dipoles as described
in the previous section. For the TE-polarized plane wave the magnetic dipoles are defined in the plane
and the electric dipoles point out of the plane. For the i’th electric dipole located at {ρ0 = ac , ϕ0 = ϕie }
29
Chapter 3. Active Cloaking - Modelling and Simulations
y
ρ=a
Md
Jd
x
εr
ρ = ac
Figure 3.6: A theoretical model for the active cloak consisting of Ne electric and and Nm magnetic
dipoles surrounding a dielectric cylinder.
the current density on a ẑ-directed dipole is given by
Jd,ie = |Jcloak (ac , ϕie )|
δ(ρ0 − ac )δ(ϕ0 − ϕie )
ẑ
ρ
(3.12)
The radiation from the single dipole in the presence of the scatterer is given to be,
Ed,ie

−jωµ RR J G (ρ, ρ0 )dS,
ρ > ac
o
S d,ie E,s
=
RR
−jωµ
J G
(ρ, ρ0 )dS, ρ < ac
o
S d,ie E,int
(3.13)
where the Green’s function for a surface current in the presence of a cylindrical scatterer is,
GE,s (ρ, ρ0 )
0
GE,int (ρ, ρ )
=
=
n=∞
X
0
1 (2)
Ho (k|ρ − ρ0 |) +
an,e Hn(2) (kρ0 )Hn(2) (kρ)ejn(ϕ−ϕ ) , ρ > a,
4j
n=−∞
n=∞
X
0
bn,e Hn(2) (k1 ρ0 )Jn (k1 ρ)ejn(ϕ−ϕ ) , ρ < a,
(3.14)
(3.15)
n=−∞
and k1 =
√k .
εr
The coefficients an,e and bn,e are given by enforcing the continuity of the electric and
magnetic field at ρ = ac , where the magnetic field is found using Hd =
0
0
1
jωµo ∇
× Ed . For the i’th
magnetic dipole, located at {ρ = ac , ϕ = ϕim } the magnetic current on an in-plane magnetic dipole is
given by
Md,im = |Mcloak (ac , ϕim )|
δ(ρ0 − ac )δ(ϕ0 − ϕim )
ϕ̂
ρ
(3.16)
The electric field from the magnetic dipole is then determined to be
Ed,im
 RR
−
Md,im GM,s (ρ, ρ0 )dS,
ρ > ac
S
=
RR
0
−
Md,im GM,int (ρ, ρ )dS, ρ < ac
S
(3.17)
30
Chapter 3. Active Cloaking - Modelling and Simulations
0
−5
RCS (dB)
−10
−15
−20
−25
−30
−35
−40
(a)
Active Cloak
Bare Cylinder
−150
(b)
−100
−50
0
θ
50
100
150
(c)
Figure 3.7: (a) The fields radiated by the dipole array only in the presence of the cylinder. (b) The
total field of the incident plane wave, scattered fields and fields from the dipole array. (c) The calculated
bi-static radar cross section (RCS).
and the Green’s function for the magnetic dipole radiating in the presence of the scatterer is,
GM,s (ρ, ρ0 )
n=∞
X
0
=
ρ cos(ϕ − ϕ0 ) − ρ0
k (2)0
Ho (k|ρ − ρ0 |)
+
4j
|ρ − ρ0 |
0
an,m Hn(2) (kρ0 )Hn(2) (kρ)ejn(ϕ−ϕ ) , ρ > a,
(3.18)
n=−∞
GM,int (ρ, ρ0 )
=
n=∞
X
0
0
bn,m Hn(2) (k1 ρ0 )Jn (k1 ρ)ejn(ϕ−ϕ ) , ρ <(3.19)
a,
n=−∞
and the prime superscript on the Hankel functions indicates the derivative with respect to the argument.
Once again the coefficients, an,m and bn,m are found by enforcing the continuity of the fields at ρ = ac .
With the radiation from both a single electric and magnetic dipole in the presence of the cylindrical
scatterer, the active cloak can be found by simply superposing the array of electric and magnetic dipoles
with the incident plane wave. The radiation from the array of electric and magnetic dipoles is given by

RR
−jωµo
S
Ed,e =
−jωµ RR
o
S
 RR
1
−
S Nm
Ed,m =
− RR 1
S Nm
P
Jd,ie GE,s (ρ, ρ0 )dS,
ρ > ac
P
ie =Ne
1
0
ie =1 Jd,ie GE,int (ρ, ρ )dS, ρ < ac
Ne
P
im =Nm
M
GM,s (ρ, ρ0 )dS,
ρ > ac
d,i
m
Pim =1
im =Nm
Md,im GM,int (ρ, ρ0 )dS. ρ < ac
im =1
1
Ne
ie =Ne
ie =1
(3.20)
(3.21)
The total field then is simply E = Ei + Es + Ed,e + Ed,m .
An example solution is solved at fo = 2.5 GHZ for a dielectric cylinder with radius a = 0.7λ and
dielectric constant εr = 10. A TE-polarized plane wave is incident on the cylinder travelling in the
positive x-direction with an amplitude Ei = 1V /m and a phase of θ = 0◦ . Following the discussion in
Section 3.3, an array of Ne = 20 electric and Nm = 24 magnetic dipoles are equally spaced around the
cylinder at a radius of ac = 0.75λ. Using (3.20)-(3.21), the fields from the electric and magnetic dipole
array alone are plotted in Fig. 3.7a. It can be easily seen that the discrete dipole array generates the
scattered field and interior field as if the scatterer was illuminated by a plane wave. Relative to the
incident field, the field created by the dipole array is out of phase with the known incident plane wave.
Chapter 3. Active Cloaking - Modelling and Simulations
31
When the fields of the array are superposed with the incident plane wave as also shown in Fig. 3.7b, the
scattered and interior fields are cancelled and resultant fields are simply the incident field in the presence
of a scatterer resulting in the cylinder being cloaked for a known incident field. The bistatic RCS as
defined above is also plotted as well in Fig. 3.7c where it is easily seen that the active cloak can reduce
the radar cross section of the cylindrical object. The choice to use RCS to characterize the cloak is a
logical one as it allows for the drop in the scattered field to be quantified. However it is worth noting
that this measure would also work for an absorber. Thus it is the field plots and the RCS which give
the full picture of a cloak as the RCS describes the drop in the scattered field and the field plots show
the restoration of the incident field. An absorber for contrast would also show a drop in the RCS but
would not show the incident field restored, as it too would be attenuated by the absorber.
The drop in scattering in the forward direction is approximately 15 dB in the forward direction. While
this is significant it does raise the question as to what is the limit in what an active cloak can achieve.
For the specific case under discussion here, where the cloaking arises due to destructive interference, the
exact phasing of the sources can effect the the performance of the cloak. In the model shown above, this
manifests itself through the finite displacement of the electric and magnetic dipoles from the surface of
the scatterer which introduces some phase-error in the sources as they are assumed to be on the surface
of the scatterer . In the theory presented here this effect has not been corrected for, but further research
may try to introduce correction terms in the amplitude and phase of the sources that comprise the cloak
to further reduce the RCS. In general, the global bounds on how well a cloak can perform is an active
area of research, and researchers are investigating this problem to understand what are the limitations of
cloaks with respect to scattering, bandwidth, losses and other important variables. While this is outside
the scope of discussion here, the interested reader is referred to recent publications in [35, 35].
3.4.2
CAD Model
The active cloak can also be simulated and verified using commercial CAD tools. Here two different
examples are shown. The first example is the same as the theoretical calculation done above using the
same incident field and cylindrical object and is simulated using COMSOL. The reason for choosing
COMSOL is that it allows for the use of ideal electric and magnetic dipoles as sources. The second
example is modelled using HFSS and is a more realistic implementation of an active cloak using actual
antennas. For this example, the dielectric cylinder is replaced by a PEC cylinder.
Beginning with the first example, the dielectric cylinder is simulated in COMSOL to characterize
this cylindrical object. Here the total electric field (incident electric field plus scattered electric field)
and the bistatic radar cross-section (RCS) of the plane wave hitting the bare circular object are plotted
in Fig. 3.8a and Fig. 3.8c respectively. Now, using COMSOL again, the fields for the dielectric cylinder
covered by an active cloak are solved for. Here the dielectric cylinder is surrounded by point electric and
magnetic dipoles placed at ρ = 0.75λ. Again the weight of each electric and magnetic dipole is found
by the value of the surface current density at the location of each source. The total electric field and
the bistatic RCS are plotted in Fig. 3.8b and Fig. 3.8c respectively. Once again a noticeable change in
the fields can be seen,where the total electric field resembles the incident plane wave. Note also that the
interior fields inside the dielectric have been cancelled as well. In terms of the RCS, there is a 14 dB
drop in the forward scattering due to the presence of the electric and magnetic dipoles making up the
active cloak. Again this limitation is due to the finite distance between cloak itself and the scatterer,
which has not been accounted for in the weights of the cloak.
32
Chapter 3. Active Cloaking - Modelling and Simulations
0
RCS (dB)
−10
−20
−30
−40
−50
(a)
Dielectric Cylinder
Active Cloak
−150
−100
−50
(b)
0
φ
50
100
150
(c)
Figure 3.8: (a) The total electric field of the dielectric cylinder only. (b) The total electric field of the
dielectric cylinder surrounded by the active cloak. (c) The bistatic RCS of the dielectric cylinder and
the dielectric cylinder surrounded by the active cloak.
(a)
(b)
(c)
Figure 3.9: (a) The total electric field of the metal cylinder only. (b) The scattered electric field of the
metal cylinder surrounded by the active cloak. (c) The bistatic RCS of the metal cylinder and the metal
cylinder surrounded by the active cloak.
With the feasibility of the active cloak demonstrated from both closed-form expressions and CAD
models a quasi-three-dimensional example that can be realized in an experimental setting is studied.
Here, using the commercial solver Ansoft HFSS, the scattering and cloaking of a metallic cylinder
placed inside a parallel-plate waveguide is simulated. The purpose of the parallel-plate waveguide is to
emulate an infinite domain along the z-axis as will be done in Chapter 4. In this example, the cylindrical
object has a radius of ρ = 0.7λ but is now a perfect electrical conductor (PEC). The reason for this is
that a PEC cylinder only requires magnetic dipoles to implement the active cloak as the electric dipoles
are all shorted out by the presence of the PEC object.
For comparison, the total electic field along a cross-section of the waveguide as well as the bistatic
RCS are plotted in Fig. 3.9a and Fig. 3.9c respectively. Again, the specific signature of the forward and
backward scattering of a PEC cylinder as well as a quantitative measure of it in the RCS can be seen in
either plot respectively.
To implement the active cloak, small circular loops are placed around the cylindrical object at a
distance of λ/20 away from the PEC object. The radius of each loop is also λ/20 and each loop is
fed with a current source which is weighted to implement the appropriate magnetic dipole moment as
given in Section 3.3. It is noted that the feed is ideal and the simulation does not take into account the
bandwidth and efficiency of the loop antennas. Another concern is the fact that every antenna acts like
33
Chapter 3. Active Cloaking - Modelling and Simulations
y
Es=Ed
Hs=Hd
Ei
Hi
k
Eint
Hint
x
εr
Jcloak
Mcloak
Figure 3.10: A schematic of the active cloak which is constructed by enforcing electric and magnetic
dipoles on the boundary of the dielectric cylinder. In this section two cloaking schemes are demonstrated
by altering the weights of the electric and magnetic dipoles on the boundary. In this example the fields
are cancelled outside of the scatterer as well as making the object look like a metallic cylidner while also
preserving the interior fields {Eint , Hint }.
a scatterer also. However this is not a concern as the scattering from the electrically small antenna is
much smaller than the scattering from a PEC object that is much larger. However, if the object was
electrically small and on the same order with regards to size as the antennas than this assumption would
no longer be true and the cloaking method described here would not work. Thus an implicit assumption
in all this is that the scatterer is much larger than the antennas such that the scattering of the antennas
can be neglected.
Simulating this environment in Ansys’ High-Frequency Structural Simulator1 (HFSS), the total electric field as well as the RCS are shown in Fig. 3.9b and Fig. 3.9c respectively. Once again the total field
of the active cloak resembles the incident plane wave. Likewise, the RCS shows a dramatic 20dB drop
in the forward scattering. This demonstrates that realizable physical sources such as small antennas can
be used in appropriate configurations to cancel scattered fields. Also, this array of antennas can be fed
using well-known feed designs to control the magnitude and phase of the current on each antenna as will
be demonstrated in the next chapter.
3.5
Extending The Active Cloak
Because the active cloak imposes sources on the boundary of the object being hidden and because the
weights on those sources can be set as desired, it should be possible then to set those weights to not
only cancel the scattered field but to also alter the field outside and inside the object. This makes sense
intuitively because the sources that are placed around the object are implementing electric and magnetic
currents which can implement an arbitrary discontinuity in the field not just one that is appropriate
for cloaking. This is achieved once amplitude and phase control of the dipoles forming the cloak is
introduced. In this section an example is shown which alters both the interior and exterior fields of the
object being cloaked.
1 http://www.ansys.com/Products/Electronics/ANSYS+HFSS
Chapter 3. Active Cloaking - Modelling and Simulations
34
Figure 3.11: An active cloak which cancels the scattered field of the dielectric cylinder while making the
dielectric cylinder look like a metallic cylinder. On the left is a plot of the total out-of-plane electric
field, Ez for a bare metallic cylinder. The boundary is marked with a black circle and is the same size as
our dielectric cylinder. The right plot shows the dielectric cylinder with a cloak made up of electric and
magnetic dipoles. We can see that the field pattern now resembles the total field of our metallic cylinder,
which disguises the dielectric cylinder. Note also that the fields in the dielectric are undisturbed as well.
Finally the 2D radar cross section is shown on the bottom plot demonstrating quantitatively how the
dielectric cylinder resembles a metallic cylinder when cloaked with the active dipole array.
In this example, the electric and magnetic currents are set to cancel the scattered field while also
superimposing a set of fields outside the scatterer which are the scattered fields of a different object
[70]. This makes the dielectric scatterer look like a different object. Meanwhile the interior fields are
unchanged as opposed to the cloak in the previous section where the interior fields are cancelled out.
With the interior fields unchanged this creates a cloak that allows for the incident field to interact with
the interior of the object similar to the anti-cloak proposed in [74, 75].
The object being cloaked is the same dielectric cylinder situated at the origin as in Section 3.4. The
incident field is again a plane wave travelling in the positive x direction. Here the cloak is designed to
make the dielectric cylinder look like a perfectly conducting cylinder. The electric and magnetic currents
surrounding the cloak are given to be,
Chapter 3. Active Cloaking - Modelling and Simulations
35
n=∞
h
X
Ms = −n × (−Es + Esm ) = − k 2
an Hn(2) (kρ)e−jn(ϕ) +
k2
n=−∞
n=∞
X
i
Asn Hn(2) (kρ)e−jn(ϕ) ϕ̂,
n=−∞
h n=∞
X
Js = n × (−Hs + Hsm ) = jωk
n=−∞
n=∞
X
n=−∞
(3.22)
0
an Hn(2) (kρ)e−jn(ϕ) −
i
0
Asn Hn(2) (kρ)e−jn(ϕ) ẑ.
(3.23)
The first terms in these expressions are Es and Hs , respectively which are used to cancel the scattered
field of the dielectric cylinder while the interior fields are unchanged. The second terms, Esm and Hsm are
the scattered fields of a perfectly conducting cylinder of the same radius as the dielectric scatterer and Asn
are the scattering coefficients of the perfectly conducting cylinder, given by Asn = −Jn (ka)/(k 2 Hn2 (ka))
[3]. It can easily be seen that control of the amplitude and phase is required to implement this electric and
magnetic current distribution. This is different than the currents found in (3.11)-(3.10) which depend
only on the the phase of the incident plane wave, Ei and Hi . Here, Ne = Nm = 24 electric and magnetic
dipoles are required to sufficiently sample the currents for the same geometry.
This configuration is also simulated using COMSOL multiphysics and the results are shown in
Fig. 3.11. Here the total field of a bare perfectly conducting cylinder is shown along with the total
field of the active cloak. It can be seen from the electric field pattern, that the dielectric cylinder has
been made to look like a conducting cylinder. Also the field inside the dielectric cylinder is undisturbed.
Finally the 2D bistatic radar cross sections is shown and with a good agreement between the scattering
of a metal cylinder and the cloak is demonstrated as well as the noticeable difference between the cloak
and a bare dielectric cylinder. This demonstrates the ability of this array of electric and magnetic dipoles
to disguise a dielectric cylinder as a metallic cylinder by imposing an arbitrary discontinuity in the field
at the boundary.
3.6
Summary
As mentioned in the previous chapter, for this method of active cloaking to function correctly, the
magnitude and phase of the incident field at the object must be known. This is not an insurmountable
challenge as there are many practical scenarios where the fields incident on a scatterer are known, such as
in communication systems where an object is obstructing an antenna. For scenarios where the incident
field is not known, further work would be required to somehow figure out the incident field in a causal
fashion. This is elaborated on in the following chapter as well. With regards to polarization, the designs
discussed here focused on TE-polarized waves. However in principle, polarization independent cloaks
can be constructed by using dual-polarized antennas to surround the object being hidden.
With these basic examples of active cloaking it becomes clear that being able to impose a discontinuity
in the electromagnetic field is a powerful concept which allows for the realization of some novel applications, including both cloaking and disguising scenarios. In the next chapter, constructing a prototype of
an active cloak is discussed
CHAPTER
4
Active Cloaking - Implementation and Measurement
In the previous chapter, active electromagnetic cloaking was proposed for a known incident field. Specifically, this was achieved by imposing appropriate orthogonal electric and magnetic currents (or Huygens
sources) around a scatterer. This idea was also extended to include disguising a cylinder to appear as
another object.
From the previously discussed theory, the experimental realization of an active cloak will be demonstrated for a perfectly electric conducting (PEC) cylinder. For this scenario, the cloak will be designed
for the microwave frequency range at a frequency around 1.5 GHz and the size of the PEC cylinder is
given by a radius of a = 0.56λ. Note that this is an electrically large object with respect to the incident
wavelength. The PEC scatterer is illuminated by a TE-polarized monopolar point source 3λ away from
the object. The reason for using a point source will become apparent below. Because the object is
a perfectly conducting cylinder only magnetic currents are needed as per the equivalence principle [3].
Using (3.8) then, the required magnetic currents which cloak the cylinder are given by,
Ms,cloak = Eo ejθ
n=∞
X
0
An Hn(2) (ko ρ0 )Hn(2) (ko a)ejn(ϕ−ϕ ) ϕ̂,
(4.1)
n=−∞
where the monopolar point source is described by a Hankel function and ρ0 and ϕ0 describe the location
of the source and Eo ejθ is the complex weight of that source. The magnetic current distribution is
plotted in Fig. 4.1. With the magnetic currents on the boundary of the PEC scatterer defined, this
current distribution can be constructed using a magnetic dipole array which will be discussed in the
following section.
4.1
Design and Measurement
To demonstrate this form of active cloaking for a PEC cylinder, the cloak will be constructed from an
array of elementary loop antennas. The measurement setup of the cloak also needs to be considered as
well. Similar to other experimental demonstrations of cloaking and lensing [48, 53, 76], the cloak will be
36
37
Magnitude (dB)
Chapter 4. Active Cloaking - Implementation and Measurement
0
−2
−4
−6
−8
−10
0
50
100
150
0
50
100
150
φ
200
250
300
350
200
250
300
350
Phase (deg)
150
100
50
0
−50
−100
−150
−200
φ
Figure 4.1: The magnitude and phase of the magnetic currents required to cloak a PEC cylinder. The
magnetic currents are determined by the scattered field at the boundary of the PEC cylinder.
xy translation
VNA
coaxial
probe
2
1
3λ
3λ
Attn.
2-way
Splitter
loop
antenna
ϕ
monopole
x12
Attn.
16-way
splitter
Attn.
z
ϕ
x12
ϕ
waveguide
y
x
Figure 4.2: A schematic of the measurement setup used to measure scattering and cloaking off of an
aluminium cylinder. Note that the waveguide plates sit flush against the aluminium cylinder. Also note
that absorbers are used to close off the waveguide (not shown).
situated in a parallel-plate waveguide which allows for the emulation of an environment with an infinite
extent along the z-axis. To discuss these details, a brief overview of the measurement setup is given
followed by some details regarding the design of the cloak.
4.1.1
Measurement Setup
The measurement setup is designed for the microwave frequency range with an operating frequency
around 1.5GHz as stated and is illustrated in Fig. 4.2 along with the corresponding coordinate system.
The waveguide is fabricated from two rectangular steel sheets which are held parallel to each other via
Chapter 4. Active Cloaking - Implementation and Measurement
38
plastic spacers around the perimeter of the waveguide. The sheets are 1626 mm×1092 mm. The height
of the waveguide is 38 mm which allows for the waveguide to support only a TEM mode as all other
modes are cut-off. On the periphery of the waveguide foam absorbers (Emerson-Cummings, AN-77) are
placed to terminate the guide and reduce reflections from its ends.
As shown in Fig. 4.2 a small monopole antenna is placed at one end of the waveguide to excite
the waveguide itself. This is a point-source like excitation and gives rise to cylindrical wavefronts in the
waveguide. For a parallel-plate waveguide, a point source is a simpler excitation to realize as opposed to a
more directive excitation like a plane wave or gaussian beam. The amplitude and phase of the current on
the monopole are controlled through a voltage-controlled phase shifter in series with a voltage-controlled
attenuator, (Mini-circuits ZX73-2500-S+). The purpose of the attenuator and phase shifter is to control
the magnitude and phase of the source with respect to the magnitude and phase of the antenna array
that forms the cloak. This is because the signal delivery paths for the cloak and for the source exciting
the waveguide experience different delays, losses and reflections.
The scatterer in the setup is an aluminium cylinder also placed in the waveguide at 3λ away from
the monopole antenna (600 mm, center to center). This approximates a PEC scatterer. The aluminium
cylinder has a height of 38 mm and a radius of 112 mm.
To measure the fields inside the waveguide, the top plate of the waveguide is perforated with small
subwavelength circular holes with a radius of 5 mm. These holes are repeated in a square grid with a
period of 15 mm. Because the holes are subwavelength, they have a minimal perturbation on the fields
inside the waveguide as confirmed through fullwave simulation. To probe the fields, a small semi-rigid
coaxial cable is used with its center conductor exposed. This picks up the electric field in the waveguide
(Ez in the given coordinate system). This cable is placed on a translation stage which moves the probe
into and out of each hole at a predetermined height. The range of motion of the translator allows for
an area of 600 mm by 600 mm to be scanned, or in terms of wavelengths 3λ × 3λ as shown in Fig. 4.2.
The measurements are recorded using a Vector Network Analyzer (VNA, Agilent E8364B) with port 1
of the VNA connected to the monopole antenna and port 2 connected to the probe. The measured S21
(transmission from port 1 to port 2) at each position yields a relative map of the complex electric field
(Ez ) as a function of space. These field plots are rescaled to fit in a -1 to 1 scale. Note that for the field
plots, it is not the absolute value that is important but the relative values in space which map out the
wave behaviour in the waveguide.
To demonstrate this measurement setup, the field plot of the monopole source scattering off of the
aluminium cylinder is shown in Fig. 4.3a with the real part of the field plotted. Here the distinctive
shadow region created by the object is seen in the field plot (as well as the scattering in the other
directions). This has good agreement with the ideal PEC case shown in Fig. 4.3b (found using COMSOL
multiphysics).
4.1.2
Designing the Cloak
The continuous magnetic currents required to form the active cloak can be approximated by a discrete
array of sources as stated earlier. As shown in Fig. 4.1 the magnetic currents required for the various
cloaking schemes in general vary in both amplitude and phase. Thus to effectively implement the cloak,
a discrete source and a way of controlling the weight of that source is required for both its magnitude
and phase. The source is a magnetic dipole in the form of an electrically small loop antenna (a loop with
radius, r < λ/10). As stated earlier, the current on the loop is related to the effective magnetic dipole
39
Chapter 4. Active Cloaking - Implementation and Measurement
(a)
(b)
Figure 4.3: (a) The real-part of the total measured electric field for a point source scattering off of an
aluminium cylinder. The inner circle shows the boundary of the aluminium cylinder. (b) The real-part
of the total electric field for a point source scattering off of a PEC cylinder as found using COMSOL
multiphysics.
−10
−15
Gain (dB)
−20
−25
−30
−35
−40
−45
0
(a)
H−plane
E−plane
100
200
Angle (Deg)
300
400
(b)
Figure 4.4: (a) The fabricated loop antenna. (b) The measured radiation pattern of the loop antenna.
Two cuts of the 3D pattern are shown indicating the dipolar nature of the antenna.
moment by I = |pm |/(µS), where S is the area of the loop, and the dipole moment perpendicular to
the face of the loop. To construct these antennas, printed loops are fabricated on FR-4 circuit boards
with a radius of 5 mm. These antennas are designed for a frequency around 1.5 GHz and are pictured in
Fig. 4.4a. Also seen in Fig. 4.4a is the feed for the antenna itself which is one of the biggest challenges
when using a loop antenna. Because a loop antenna requires a differential feed, a chip balun (Johanson
1600BL15B050E) is used to convert the single-ended coaxial feed to a differential feed. The other biggest
challenge is the mismatch between the antennas input impedance and the 50Ω feedline, however in the
context of this design, a matching network was not utilized. To confirm the dipole-like properties of the
antenna, the measured radiation pattern is shown in Fig. 4.4b, where the two-lobe dipolar pattern is
clearly visible. The efficiency of the antenna is not a concern here as the antenna is being connected to
a VNA, but a more practical system would require more efficient antennas which is an area of future
investigation.
To determine the number of antennas that are needed, the magnetic current distribution in Fig. 4.1
is sampled. Taking the Fourier transform numerically (FFT) of Fig. 4.1 it is found that a minimum of
Chapter 4. Active Cloaking - Implementation and Measurement
(a)
40
(b)
Figure 4.5: (a) The exterior of the waveguide setup. Pictured are the phase shifter, translator and
coaxial probe. The VNA is just outside of the picture and to the left. (b) The interior of the waveguide
setup. The loop antennas conformally surround the scatterer.
12 dipoles is required to effectively sample the current shown in Fig. 4.1.
With the number of dipoles given, the magnitude and phase for the current on each element of the
12-dipole array needs to be set. To do this, custom designed phase-shifters modelled after [77] are used
to control the phase on the antenna and off-the-self SMA attenuators to alter the magnitude. The phase
shifters are printed on a Rogers 3003 circuit board using varactor diodes (Aeroflex-MGV125-26-E28X),
with each phase shifter controlled by an external voltage source. By varying the voltage between 010V the phase shifter can cover over a 360◦ phase range thus allowing for the proper phasing of the
current on each loop. See Appendix A for further details. For the cloak, because the amplitude of
the magnetic current varies very approximately by ±1 dB across the cylindrical boundary, as shown in
Fig. 4.1, the magnetic current can be approximated with a constant amplitude, without the need for
attenuators. This is essentially a far-field approximation where the driving point source is assumed to
radiate a plane wave at the boundary of the cylinder, as a plane wave would have a uniform amplitude
over the cylinder. However, it will be demonstrated in the following sections that amplitude control is
useful for other magnetic current distributions. In those scenarios, attenuators are used which have a
fixed attenuation in the -1 dB to -20 dB range (Crystek Corporation). For the required magnitude the
appropriate attenuator is used.
The complete setup is pictured in Fig. 4.5 with the loop antennas placed next to the cylindrical
scatterer in foam stands. The phase shifters and attenuators are placed outside of the waveguide as
shown in Fig. 4.5a, due to the discrete nature of the setup. It is important to emphasize that a more
compact design could be envisioned in the microwave regime using an integrated circuit approach [78].
To feed the dipole array, port 1 of the VNA is split through a 2-way power splitter (Mini-Circuits
ZAPD-2-21-3W-S+), with the first feed going to the monopole antenna and the second feed to the dipole
array. This second feed is then passed through another 16-way power splitter (Mini-Circuits ZC16PD-
Chapter 4. Active Cloaking - Implementation and Measurement
(a)
41
(b)
(c)
Figure 4.6: (a) The measured real part of the electric field created by the magnetic dipole array alone
in the presence of the aluminium cylinder. The larger circle is the circle on which the near field data
is used to calculate the far-field patterns. (b) The measured scattered field of the aluminium cylinder
found by subtracting the total field shown in Fig. 4.3a from the measured incident field (not shown).
The larger circle is the circle on which the near field data is used to calculate the far-field patterns. (c)
The simulated real part of the scattered electric field of the cloak computed using COMSOL.
24+) to feed each of the 12 phase-shifters-attenuators-loop antenna blocks (with the remaining ports
terminated). To demonstrate the cloaking effect, the voltages on each phase-shifter are set to generate
the desired phase.
4.2
Measured Results
With the complete description of the experimental setup, the performance of this cloak can now be
examined. Before looking at the full cloaking effect, the capabilities of the magnetic dipole array in the
presence of the aluminium cylinder are discussed first.
If the magnetic dipole array is properly phased in the presence of the aluminium cylinder, a field
distribution which resembles the scattered field of the cylinder illuminated by a point source is expected.
In Fig. 4.6a the real part of the measured field generated by the magnetic dipole array in the presence
of the cylinder is shown. Also shown in Fig. 4.6 are the scattered fields for the ideal PEC cylinder found
from fullwave simulation and the scattered field found from the measured total field of the aluminium
cylinder by itself. The latter is determined by subtracting the measured total field from the measured
incident field (the field in the empty waveguide). As shown in Fig. 4.6 good agreement is observed in
all three scenarios. There are some discrepancies and variations in the amplitude of the measured fields.
Chapter 4. Active Cloaking - Implementation and Measurement
(a)
42
(b)
(c)
Figure 4.7: (a) The measured real part of the total electric field created by the magnetic dipole array
and the incident field from the monopole antenna in the presence of the aluminium cylinder. Note the
shadow region has disappeared and cylindrical wavefronts have been restored. (b) The measured total
electric field for the scattering off of the bare aluminium cylinder, replotted again from (a). Note the
shadow region. (c) The simulated cloak using ideal magnetic dipoles, found using COMSOL.
This is due to inherent limitations in the cloak itself due to amplitude and phase variations in the loop
antennas and phase shifters. This arises from the fabrication tolerances and variations in these devices
and in the measurement setup itself.
With this confirmation of the capability of the magnetic dipole array, the phase and amplitude of the
monopole source can be aligned with those of the magnetic dipole array to demonstrate the behaviour of
the cloak. The measured fields in this scenario are plotted in Fig. 4.7a where the wavefronts surrounding
the aluminium cylinder are restored to their original state without the scatterer present, which in this
case corresponds to cylindrical-like wavefronts. This can be especially observed when comparing the
shadow region of the cloak to the shadow region of the bare PEC cylinder which is plotted again in
Fig. 4.7b for comparison. The simulated behaviour of the cloak using COMSOL multiphysics is also
shown in Fig. 4.7c and has good agreement with the measured results as well. Note again that there
are discrepancies between the simulated and measured results as a result of the amplitude and phase
variations in the loop antennas and phase shifters mentioned above.
4.2.1
Far-Field Plots
To further characterize the cloak, one can take the measured fields and find the bi-static radar cross
section (RCS) of the bare aluminium cylinder compared to the cloak itself. However, this assumes
knowledge of the far-field whereas for this measurement setup the near-field of the scatterer is being
Chapter 4. Active Cloaking - Implementation and Measurement
43
measured. Instead, given that near-field measurements are available, the scattered field determined
from the measured data are projected onto cylindrical harmonics thus enabling the computation of the
far-field [79].
This is carried out by finding the measured scattered field on a circle concentric with the scatterer
itself with radius ρ = b as shown in Fig. 4.6b. Again the scattered field is determined by subtracting the
measured incident field in the empty waveguide from the measured field in the presence of the scatterer.
Following [79], this scattered field is expanded in terms of cylindrical harmonics, assuming a point-source
illumination, which gives,
Es,z =
n=∞
X
an Hn(2) (ko ρo )Hn(2) (ko b)ejn(ϕ−π) ,
(4.2)
n=−∞
where an are the coefficients for the expansion, and ρo the distance between the scatterer and the source.
The scattering coefficients are found by treating the above as a Fourier series which gives,
an =
1
(2)
(2)
2πHn (ko b)Hn (ko ρo )
Z
2π
Es,z exp (−jnϕ)dϕ
(4.3)
o
Finally, the RCS in the far-field is given by
σ(ϕ) = |
n=∞
X
n=−∞
exp (jnϕ)an |.
(4.4)
Before presenting the measured results, a couple of limitations with this calculation using the measured data are noted. Given the measurement setup, there are limitations due to the quality of the
absorbers (approximately -20dB of attenuation), the sensitivity of the probe [79], and finally the directivity of the scattered field itself, which for the given scatterer, are not very directive patterns. With
these limitations there are constraints on the accuracy at which one can extract the RCS data. Thus for
the cloak, the RCS data is used to simply examine the relative change between the uncloaked cylinder
and the cloaked cylinder to get an estimate of how well the cloak performs.
With this in mind, the RCS patterns for the bare aluminium cylinder and the cloak are presented in
Fig. 4.8. For comparison, the RCS of the magnetic dipole array only in the presence of the cylinder is also
plotted along with the theoretical calculation for a point source scattering off of a bare PEC cylinder.
Note that the RCS values for each trace have each been normalized with respect to their peak value
so that each trace is referenced to 0dB. Comparing at first the measured RCS for the bare aluminium
cylinder to the theoretical value it is found that the general shape is tracked by the measured data,
with an increase in the scattering in the forward direction due to the shadow created by the scatterer.
However, there are variations in the measured data along the back scattering directions. There is also
an overall agreement between these two plots with the RCS of the magnetic dipole array only.
Nonetheless, as stated earlier, this data is used to compare to the cloaked RCS which is also shown
in Fig. 4.8 and a dramatic drop in the RCS is seen. Relatively, in both the forward and backward
directions, a drop of 11dB and 8dB is observed respectively. Overall, the cloak suppresses the scattering
of the object across most angles, though it is noticed that at −55◦ the scattering increases slightly. As
stated above, such an effect is due to the overall variation in the fabricated antennas and phase shifters.
Overall, this indicates that the active cloak is effective at reducing the scattering of an object for a
Chapter 4. Active Cloaking - Implementation and Measurement
44
Magnitude (dB)
0
−5
−10
−15
−20
Theoretical RCS, PEC Cylinder
Measured PEC Cylinder
Magnetic Dipole Array
−25
−30
−200
−150
−100
−50
0
φ
50
100
150
200
Magnitude (dB)
0
−5
−10
−15
−20
Cloak
PEC Cylinder
−25
−30
−200
−150
−100
−50
0
φ
50
100
150
200
Figure 4.8: The calculated RCS patterns for various scenarios. The upper plot shows the measured RCS
for the bare aluminium cylinder, the magnetic dipole array and the theoretical RCS for a PEC cylinder.
The lower plot shows the measured RCS for the bare aluminium cylinder and the active cloak.
known incident wave.
4.3
Disguising a Cylinder
Because the weights of the magnetic dipole array are purposefully set, other configurations of magnetic
currents are conceivable which can create interesting field distributions. Here the weights of these dipoles
will be configured to impose a discontinuity in the field that causes the cylinder to appear as another
object as described in Chapter 3. To accomplish this, the fields scattered by the object must continue
to be cancelled, but on top of that the required field to cause the scatterer to look like a different object
must be superimposed. From this, the equivalent magnetic currents are given by
Ms = n̂ × (Ei − Ed ),
(4.5)
where Ed are the fields that cause the cylinder to look like another object. Here two ‘disguises’ are
demonstrated. The first disguise makes a PEC cylinder look like a smaller PEC cylinder with radius
ad = 0.2λ. The second disguise makes a PEC cylinder look like a dielectric cylinder with εr = 10 and
radius ad = 0.4λ. Given a point-source incident field, this gives a magnetic current distribution of
Ms,d = Eo ejθ
−
"
n=∞
X
n=−∞
n=∞
X
n=−∞
0
An Hn(2) (ko ρ0 )Hn(2) (ko a)ejn(ϕ−ϕ )
0
Bn Hn(2) (ko ρ0 )Hn(2) (ko ad )ejn(ϕ−ϕ )
#
ϕ̂
,
(4.6)
45
Chapter 4. Active Cloaking - Implementation and Measurement
Figure 4.9: The magnitude and phase of the magnetic currents required to disguise a PEC cylinder.
The blue curves show the required magnetic currents for disguising the PEC cylinder as a smaller PEC
cylinder. The red curves show the required magnetic currents for disguising a PEC cylinder as a smaller
dielectric cylinder.
where Bn are the coefficients for the scattered field for the desired disguise. To make a PEC cylinder
look like a smaller PEC cylinder the coefficients Bn are given by,
Bn =
−Jn (ko ad )
(2)
Hn (ko ad )
.
(4.7)
Similarly to make a PEC cylinder look like a dielectric cylinder the coefficients for the scattered field of
a dielectric cylinder are given by
0
Bn = −
0
k1 εr ko2 Jn (k1 ad )Jn (ko ad ) + k12 ko Jn (k1 ad )Jn (ko ad )
(2)
0 (2)
ko2 k1 εr Hn (ko ad )Jn0 (k1 ad ) − k12 ko Jn (k1 ad )Hn (ko ad )
,
(4.8)
where k1 is the wavenumber in the dielectric and the primes indicate a derivative with respect to the
argument. The magnetic currents for both of these scenarios are plotted in Fig. 4.9 where it is seen that
both amplitude and phase of the magnetic currents vary across the boundary of the PEC scatterer.
4.3.1
Measured Results
Using the same setup described previously, the complex weights of the 12 magnetic dipole antennas are
set using both the voltage-controlled phase shifters and the fixed attenuators.
Measuring the fields inside the waveguide, the changes in the scattering around the aluminium
cylinder can be monitored when the magnetic currents given in Fig. 4.9 are imposed on the boundary.
Examining first the weights which make the cylinder appear to be a smaller cylinder with a radius of
ad = 0.2λ, the measured fields are plotted in Fig. 4.10a. Also plotted are the fullwave simulation for the
same setup and for convenience, the measured fields of the bare PEC cylinder are shown again. Here
it can be noted that both the measured and simulated results show good agreement. Examining the
measured results between the disguised cylinder and the bare cylinder it is seen that the main difference
between the two is the narrowing of the shadow region around the cylinder. This narrowing is indicative
Chapter 4. Active Cloaking - Implementation and Measurement
(a)
46
(b)
(c)
Figure 4.10: (a) The measured real part of the total electric field with the weights of the magnetic dipole
array setup to disguise the cylinder as a smaller metallic cylinder. Note how the shadow region narrows.
(b) The measured real part of the total electric field of the bare aluminium cylinder (repeated from
Fig. 4.3a for ease of comparison). (c) The simulated electric field found using COMSOL for disguising
a PEC cylinder as a smaller PEC cylinder.
of the smaller appearance of the 0.56λ cylinder through the active modification of the scattered field
created by the magnetic dipole array. A more quantitative analysis through the RCS is not shown here
as the there were experimental limitations due to the absorbers that were used. This limited the RCS
to only being useful to quantify cloaked versus bare cylinders, not disguised cylinders which required a
more sensitive RCS measurement.
For the second example, the measured fields are plotted in Fig. 4.11 along with the fullwave simulated
results of the same setup as well as a fullwave simulation of ereeeeerthe tyes rbice done scattering off of
a dielectric cylinder with radius, ad = 0.4λ and ε = 10. Here again a good agreement between all three
plots is seen. By inspecting the shadow region of the cylinder, there is a clear modification of the fields
in this region with the fields now resembling those of the dielectric cylinder shown in Fig. 4.11.
From these two examples it is qualitatively shown that by having control of the amplitude and phase
of the magnetic dipoles that form the cloak, the scattering of the aluminium cylinder can be tailored in
a number of ways to disguise an object instead of simply cloaking it.
4.4
Estimating the Incident Field
There are practical scenarios where the incident field is known a priori such as when an object is
blocking the line-of-sight of an antenna array [80]. However, in many situations it is desirable to do
Chapter 4. Active Cloaking - Implementation and Measurement
(a)
47
(b)
(c)
Figure 4.11: (a) The measured real part of the total electric field with the weights of the magnetic dipole
array setup to disguise the cylinder as a smaller dielectric cylinder. Note how the shadow region has
changed. (b) The simulated total electric field found using COMSOL for disguising a PEC cylinder as
a smaller dielectric cylinder. Note the good agreement with Fig. 4.11a. (c) The simulated total electric
field for a dielectric cylinder of radius r = 0.4λ and dielectric constant ε = 10 found using COMSOL.
cloaking without a priori knowledge of the incident field to determine the weights of the magnetic
dipole array. Despite not being able to respond to the field in a causal manner as established, it is still
interesting to see if it is possible to estimate what the incident field is. Such an estimate could still be
used to causally set the weights for the cloak after letting the fields scatter off of the cylinder. This
kind of scenario would be more akin to camouflaging then cloaking. Here a possible way to estimate the
incident field for a cylindrical PEC object is described.
Considering again the cloak described above, one can observe that the magnetic dipole array operates
in a transmitting mode of operation to cloak the PEC cylinder from a prescribed incident field. However
the same array, which surrounds the object, can potentially be used as a receiver to detect the incident
field scattering off of the PEC object. To examine this in more detail the unknowns surrounding the
incident field are first detailed.
It is first assumed that the cylinder lies in the far-field for any source and that the incident field is a
plane wave like excitation. This is a fair assumption as for any reasonable scenario an unknown source
would have a plane wave like appearance within the vicinity of the object. Due to the narrow-band nature
of the setup and the fact that this is a hardware based solution, it is also assumed that the frequency is
known. With this in mind one can observe that the plane wave, described by Eo exp{j(ωt − k · r + θ)},
has three unknowns. The amplitude, Eo , the phase θ and the direction k · r. Thus, the signal received
by the magnetic dipole array must allow for these three qualities to be determined.
Magnitude (dB)
Chapter 4. Active Cloaking - Implementation and Measurement
48
0
−10
−20
−30
Total Hφ Field
−40
Incident Hφ Field
−50
0
50
100
150
φ
200
250
300
350
Phase (deg)
200
100
0
−100
No Phase Offset
Phase Offset
−200
0
50
100
150
φ
200
250
300
350
Figure 4.12: The tangential magnetic field, Hϕ , (magnitude and phase) at the boundary of a PEC
cylinder when illuminated with a plane wave (blue-curve). The black curve in the upper plot is the
magnitude of the magnetic field of the incident plane wave when compared to the magnitude of the
the total magnetic field at the boundary. The red curve in the lower plot is the change in phase for a
constant 200◦ phase offset.
To accomplish this, knowledge about the object that is being cloaked can also be used advantageously.
Here in this example, the specific characteristics for scattering off of a PEC cylinder can be used. Note
again that the receiver is a magnetic dipole array around the boundary of this cylinder. The orientation
of this array, as seen in Fig. 4.2, is such that each loop antenna is excited primarily by the Hϕ field
component. (Note that while the boundary condition of the PEC cylinder requires the Ez component
to be zero, the magnetic field is non-zero at the boundary). Thus one can at least predict the general
structure of the total Hϕ field (incident plus scattered) that would be received by the antenna at the
boundary of the cylinder from general knowledge of how a plane wave scatters off of a PEC cylinder.
This is given for an arbitrary incident plane wave as,
ko cos ϕ
Eo ejθ ej(ωt−k·r) +
jωµ
∞
X
0
j −n An Hn(2) (ko a)e(jnϕ)
Hϕ = −
Eo ejθ
ko
jωµ n=−∞
(4.9)
where the primes indicate derivatives with respect to the argument. As it will be shown, one does not
need to measure other field components as information about the incident plane wave is contained in
the Hϕ field.
To see how this can be done, the field given in (4.9) is first plotted for arbitrary incident field
parameters (magnitude, phase and direction) near the boundary of the PEC cylinder (specifically λ/15
away from the boundary which is the location of the loop antennas). The direction of the plane wave
is examined first. For a simple and symmetric object like a PEC cylinder, the magnitude of the Hϕ
field tells us the direction of the incident plane wave due to the strong shadow region created by the
object. Thus the point at which the minimum in the magnitude of Hϕ field occurs, the corresponding
Magnitude (dB)
Chapter 4. Active Cloaking - Implementation and Measurement
49
0
−10
−20
−30
−40
−50
Measured
Theoretical
0
50
100
150
φ
200
250
300
350
Phase (deg)
200
100
0
−100
Measured
Theoretical
−200
0
50
100
150
φ
200
250
300
350
Figure 4.13: The theoretical total tangential magnetic field, Hϕ at the boundary of a PEC cylinder (blue
curve) compared to the measured values found using the loop-antennas in the waveguide measurement
(red curve).
outward surface normal at that point tells us the direction of the incident plane wave. In the plot shown
in Fig. 4.12, since the minimum occurs at 0◦ , and the outward surface normal points in the x̂ direction it
can be deduced that the incident plane wave travels in that direction. Obviously, for a more complicated
object more complicated signal processing would be needed to deduce the direction of arrival [81], but
it can be seen that for at least a symmetrical object simple assumptions can be made.
To calculate the phase of the incident field one can simply fit a constant phase-offset (with respect
to a 0◦ reference at the center of the object). This is seen in Eq. (4.9) where the phase of the incident
field, θ, is common to all terms. This phase-offset then determines the overall phase of the incident field
and the corresponding phase of the required magnetic dipole array. To see how the phase changes with
a phase offset, the phase is plotted in Fig. 4.12 with a phase offset of θ = 200◦ .
Finally, the magnitude of the incident plane wave can be compared to the magnitude of the total
field as shown in Fig. 4.12. Here it is shown that given an incident plane wave, the offset between
the measured total field and amplitude of the plane wave at broadside is approximately -5dB. Thus by
measuring the magnitude of the total Hϕ field at the boundary, information about the magnitude of the
incident field can be determined by assuming a plane wave incidence. From these simple observations
the key properties of the incident plane wave have been deduced.
The theoretical observed field given by (4.9) can be compared with the measured field using the
waveguide setup of the previous sections. Here the (normalized) transmission (S21 ) from the monopole
antenna to each loop antenna is measured and also plotted in Fig. 4.13. Note that the general trend
in both the magnitude and phase of the received signal at each element of the magnetic dipole array
corresponds well with the predicted theoretical result. Note also that in the measured scenario there
is a point source, not a plane wave due to limitations with the measurement setup. This shows that
from these measured received signals the properties of the incident field are calculable. A realistic realtime setup of this scenario would use these received signals found using a standard receiver circuitry to
calculate the incident field and dynamically set the weights to setup the cloaking effect.
Chapter 4. Active Cloaking - Implementation and Measurement
50
In this example the specific knowledge about the size, shape and material composition of the scatterer
has been used to predict the incident field and thus the required parameters for the cloak. As discussed in
the introduction, other cloaking schemes have attempted more feasible cloaking mechanisms by making
assumptions about the object being cloaked and/or the incident field itself [42, 43, 51] and it has been
shown here how these assumptions can also be useful in the active regime.
As mentioned, causal electromagnetic cloaking is not possible. Thus determining the parameters
of the incident field cannot be accomplished by the magnetic dipole array without letting any of the
incident field scatter off of the object. However, once the properties of the incident field are found, the
appropriate weights on the magnetic dipole array can be set and the magnetic dipole array can conceal
the object from the incident field (assuming steady-state). Here also note that the receive and transmit
functions are assumed to be operating at separate times, where the receiver senses the environment and
the transmitter responds to it. Hence, this is the reason this functionality is perhaps better termed as
camouflaging as opposed to cloaking.
What has been shown here is a simple example of how the incident field could be determined by
an antenna array that can then be used to hide the object itself. For more arbitrary geometries, more
thorough knowledge of how the object scatters the electromagnetic field would be required, perhaps
through numerical analysis; More advanced signal processing of the received signal would also be required
to determine the magnitude, phase and direction of the incident field. Another implicit assumption is
the narrow-band steady-state properties of the incident (and scattered) field. Here again, more advanced
signal processing would be required to handle transient, pulsed or wideband signals.
4.5
Summary
In this chapter an experimental demonstration of active interior cloaking for a predetermined electromagnetic field has been presented. Using a formula based on the equivalence principle a cloak for an
aluminium cylinder made from a magnetic dipole array has been constructed. Measuring the cloak in a
parallel-plate waveguide demonstrated both the cloaking effect as well as other possible configurations
which allow the cylinder to be disguised as another object. Finally, using the magnetic dipole array as
a receiver, it was shown how the incident field could be determined which could lead to a camouflaging
type of behaviour.
By using these active sources the appeal of this approach is that it requires only a thin conformal layer
to construct the cloak, using well known technology from phased-array theory. No artificial materials or
surfaces with extreme parameters are required.
Part II
Refraction At A Huygens Surface
51
CHAPTER
5
Background On Diffracting/Refracting Surfaces
The idea of using a surface to scatter or diffract electromagnetic radiation in a controlled manner has a
long history in both the microwave and photonic research. There are many different kinds of surfaces
that have been researched from a variety of perspectives. In this chapter, some of these topics are
reviewed, and some will be discussed in more detail than others based on their relevance to passive
Huygens surfaces which are introduced in Chpater 6.
5.1
Diffraction Gratings
The first topic that is considered are diffraction gratings, which have a long history in both electromagnetics and photonics and are some of the simplest surfaces which coherently scatter radiation. A grating
is shown in Fig. 5.1 and traditionally consists of periodic slits in a PEC screen. The slits themselves can
range from subwavelength in size to on-the-order of a wavelength and the periodicity itself has the same
range. Depending on the exact size, the grating can be described either from a ray-optic perspective or
from an electromagnetic perspective using Maxwell’s equations.
y
N = −2
N = −1
N =1
N =2
N =0
z
a
x
W
Einc
N = −2
N =0
N = −1
N =1
N =2
Figure 5.1: A plane wave incident on a metallic grating. The incident field is scattered into plane waves
propagating in discrete directions.
The grating is an interesting research topic because it is a conceptually simple problem that leads to
52
Chapter 5. Background On Diffracting/Refracting Surfaces
53
rich results in understanding the diffraction of electromagnetic waves through periodic structures. For
the simple diffraction grating shown in Fig. 5.1 with periodicity d a solution for the fields can be found
using simple periodic analysis. For this scenario it is assumed that the grating is an array of slits in
a one-dimensional PEC screen in a two-dimensional space and is illuminated by an obliquely incident
TE-polarized plane wave along the angle ϕi . Because the grating is periodic, the fields scattered by the
grating are described using the Bloch-Floquet theorem [82]. This means that for each unit cell of the
grating, the fields are the same except for a phase shift from cell to cell that is related by wavenumber
k in the surrounding medium. This field distribution is thus also periodic with the same period as the
grating itself and this allows for the field to be expressed using a Fourier Series. This results in the
following expression for the transverse electric field scattered on either side of the grating
Ez,s

P∞
An e−jkx x−jky y , y < 0,
= Pn=−∞
−jkx x+jky y
 ∞
,y > 0
n=−∞ An e
2nπ
kx = k sin ϕi +
,
d
p
ky = k 2 − kx2 ,
(5.1)
(5.2)
(5.3)
where An is the weight of each Fourier series components. This Fourier series expansion of the field
scattered by the grating can be interpreted as a sum of plane waves travelling in a discrete set of
directions determined by the periodicity of the grating. These plane waves, in the context of a periodic
structure are referred to as the Floquet harmonics and also arise in other periodic structures such as
antenna arrays. This summation can also be expanded to two-dimensional periodic structures as well
using similar reasoning. The index of each of these plane waves is referred to as the order and for higher
orders, the plane waves become evanescent and travel along the surface of the grating.
To precisely determine the scattered field of the grating, the coefficients An in (5.1), have to be found.
This can be done by enforcing the boundary condition of the PEC screen on the fields as well as using
approximate expressions for the fields in the slits [82]. Other methods include numerical procedures
such as the periodic method of moments. The key understanding of these Floquet harmonics however
is that despite the simple illumination of a grating by a plane wave, the periodic nature of the structure
always scatters a field into discrete number of transmitted and reflected plane waves. While the number
of propagating plane waves scattered by the grating can be controlled by the periodicity, the amplitude
of these plane waves is not easily controllable. While such gratings often find application in sensing
and communication systems [83], constructing more involved gratings that give more control over the
scattered fields is often desired.
This has led to the development of gratings with a more complicated structure that can alter the
weights of the Floquet harmonics that are excited. One example of this are blazed gratings which can
be carefully designed to excite only the n = ±1 harmonic . This is achieved by using a sawtooth
profile along the grating as shown in Fig. 5.2. From a ray-optic understanding, the sawtooth profile is
understood to provide a linear phase shift across the plane of the grating [84]. If that phase shift can be
engineered to cover a 2π phase range then only the n = ±1 harmonic is scattered. The 2π phase shift
can be understood from Fourier optics where the transmission phase in the spatial domain corresponds
to a shift of the diffraction order (See [85] for further details). Of course this harmonic is still scattered
in both the transmitted and reflected directions as well. A physical sawtooth profile can be used [86]
54
Chapter 5. Background On Diffracting/Refracting Surfaces
y
n=1
εr
z
Einc
Figure 5.2: A plane wave incident on a dielectric blazed grating. The sawtooth profile of the grating can
scatter the field into the N = 1 harmonic only. Reflections off of the grating still have to be managed
as well.
but is not necessarily needed to form a blazed grating as many other physical structures which mimic
the sawtooth effect can be used to build this kind of grating including graded index structures [87, 88].
However, due to material limitations it is often difficult to realize the desired 2π phase range across the
period of the grating.
5.2
Frequency-Selective-Surfaces
At the microwave-frequency range, the use of surfaces to filter and redirect electromagnetic radiation
led to research on structures known as frequency selective surfaces (FSS). The basic idea of an FSS is
to use a periodic array of scatterers to tailor the frequency response of the surface to an incident plane
wave [60]. Like a guided-wave filter, the FSS will have regions of pass-band and stop-band behaviour for
an incident wave over a given bandwidth. The FSS, being a periodic structure in free-space, must be
designed to prevent unwanted Floquet harmonics as well as working for a variety of angles of incidence.
To achieve both the frequency response as well as these added constraints, FSS’s are often designed as
a cascade of surfaces stacked together to achieve the desired response as shown in Fig. 5.3a.
The key to the FSS are the unit cells that are used to design each surface. By carefully designing the
parameters of the unit cell, including its periodicity, size and geometry the desired design parameters
can be achieved. A variety of FSS-type unit cells are shown in Fig. 5.3b. Each unit cell can also be
modelled using equivalent LC-circuits to capture their frequency response to form compact models for
each surface comprising an FSS.
Research into FSS’s has focused both on the numerical methods used to model them as well as
design and fabrication of actual devices. Numerical methods have focused on algorithms such as the
periodic method of moments [90]. A variety of designs have been proposed including broadband FSS’s
[91], reconfigurable and tuneable FSS’s, which are loaded with varactors to alter their behaviour over
frequency [92, 93], flat lenses which can alter phase fronts [89, 94], and miniaturized and improved
elements [95]. It should also be noted that the Salisbury screens and Jaumann absorbers mentioned in
Chapter 2 arose from research into FSS’s, and FSS’s can also be designed to control polarization which
will be discussed in Chapter 7.
5.3
Electromagnetic Bandgap Surfaces
While gratings and FSS’s are research areas with a lot of history, relatively newer research areas on
periodic structures have emerged such as photonic crystals and metamaterials. Both of these have led to
x
Chapter 5. Background On Diffracting/Refracting Surfaces
55
(a)
(b)
Figure 5.3: (a) An example of a stack of surfaces comprising a Frequency Selective Surface. Taken from
c
[89], 2011
IEEE. (b) Some example geometries for FSS unit cells.
interesting research on periodic surfaces. Photonic crystals were focused on the design of two-dimensional
and three-dimensional structures which supported an electromagnetic band-gap. This idea of creating a
bandgap for propagating waves was extended to surface waves with the invention of an electromagnetic
bandgap (EBG) surface. Here a properly engineered periodic surface can create a bandgap for surface
waves propagating in all directions on that surface [96]. The basic design of an EBG surface consists of
metallic patches connected to a groundplane through a via as shown in Fig. 5.4. The size of the patch
is usually less than a wavelength. The inductance of the metallic patch and via, combined with the
capacitance between patches can be resonated together to create a bandgap over a range of frequencies.
These ideas were confirmed both in numerical studies and measurement.
There are two interesting properties of the EBG surface, the first is that it suppresses surface waves,
while the second is that the reflection phase can be tuned to 0◦ [96]. Both of these ideas lead to novel
designs of low-profile antennas. Using the surface-wave suppressing property of the EBG, antennas which
generate unwanted surface waves can be placed above an EBG to suppress their propagation. This was
done for example, using patch antennas, where the groundplane is replaced by an EBG. The reflection
Chapter 5. Background On Diffracting/Refracting Surfaces
56
Patch
Via
Figure 5.4: An electromagnetic bandgap surface. A homogeneous array of patches and via’s sit above a
groundplane creating a stopband for surface wave propagation.
phase properties of an EBG allow for the surface to act like a perfect magnetic conductor (PMC). This
allows for antenna engineers to use an EBG as a groundplane for a horizontal electric dipole which
enhances the gain of the antenna as shown using image theory [97]. Using EBGs in the reflection phase
mode also allows for the realization of other devices such as TEM waveguides [98].
5.4
Metasurfaces
While EBG’s had unique electromagnetic properties for surface waves, further research on periodic
structures would lead to the development of surfaces which also control propagating waves. These kinds
of surfaces are generally referred to as metasurfaces. By analogy, just as EBGs applied the idea of a
photonic crystal idea to a surface, metasurfaces apply the idea of a metamaterial to a surface as well.
Metamaterials, as discussed in Chapter 2 deal with extremely subwavelength unit cells (< λ/10) arranged
in a volumetric configuration that are homogenized into effective material parameters. Metasurfaces
take these subwavelength scatterers and investigate their properties when configured on a surface. Here,
instead of characterizing the surface with effective parameters such as the permitivitty and permeability,
properties specific to a surface are used such as effective polarizabilites [99, 100] or surface impedances
[101]. Thus a metasurface can be defined as a subwavelength array of scatterers, situated on a surface,
that can be treated by a homogenized quantity describing their collective behaviour.
Research into metasurfaces followed soon after the idea of a metamaterial was proposed with some of
the first work done by Holloway and Keuster in [100]. The problem that was originally tackled was how
does one assign a homogenized quantity to a surface of subwavelength scatterers. In a series of works
[100, 102–104], this led the authors to look at a variety of canonical structures (surfaces comprising of
spheres or patches) and derived closed form expressions on how these surfaces can be homogenized. This
involved using boundary conditions known as Generalized Sheet Transition Conditions to model these
surfaces. This boundary condition can be derived by finding the local fields at each scatterer and thus
the effective polarizability of the surface. With this homogenized model, the reflection and transmission
of plane waves off of the surface can be calculated along with the inverse problem where the effective
polarizabilites are found from reflection and transmission measurements. These models can also be used
to study other waves supported by the metasurface including surface waves and leaky waves [105].
Parallel to this work, other research groups were also looking at the same problem of deriving bound-
Chapter 5. Background On Diffracting/Refracting Surfaces
57
ary conditions and effective polarizabilities of surfaces comprised of periodic structures. This included
modelling higher-order boundary conditions as well as other homogenization methods to model surfaces
of wires and nano-particles [99, 106]. Also investigated, were surfaces of bianisotropic particles and how
electromagnetic waves are reflected and transmitted off these kinds of surfaces [107, 108]. These surfaces
will be discussed further in Chapter 7 in the context of polarization manipulation.
It is worth stopping here to discuss in further detail the homogenization process of a metasurface,
for without a workable description of how a metasurface collectively scatters electromagnetic radiation,
designing a metasurface is impossible. While the papers cited above all have specific methods to analyze
a metasurface there are some generalities between all these methods which give some insight into how
one approaches this problem. Here, we try to summarize the commanalities of the work in [99, 102, 106]
by looking at a general procedure for Homogenization.
In Fig. 5.5 an array of identical scatterers are shown with a plane wave incident on the surface, given
by Einc . The physics of the problem is as follows, the incident plane wave polarizes each scatterer with
an electric/magnetic polarizabilitiy depending on the geometry and composition of scatterer. (Here only
an electric polarizability is assumed). Because each scatterer is polarized, each individual scatterer also
sets up a field everywhere in space from the induced dipole moment which subsequently interacts with
every other scatterer in the array. Therefore the local field at every scatterer is given by
Eloc = Einc + Eint ,
(5.4)
where Eint is the interaction field between a specific scatterer and all the other scatterers in the array.
With the local field defined for each scatterer the dipole moment of each scatter is well defined by the
polarizability of the scatterer which is a function of its geometry and material composition and is given
to be,
p = αe · Eloc ,
(5.5)
where αe is the electric polarizability of the scatterer1 . Thus, to solve the problem of how the metasurface
can be homogenized two things are needed, the value of the individual polarizability αe and the effect of
the array from Eint on every individual scatterer. The individual polarizability of each scatterer can be
found, either from analytical methods, if the scatterer has a simple geometry, or from numerical methods
by modelling an isolated scatterer illuminated by a plane wave [109].
The challenging part of this problem however is finding an expression for Eint . This is where different approaches are taken. In [106], a dyadic Green’s function is used to sum all of the contributions
from each scatterer2 . In [100] this is done by finding the effect due to the neighbouring scatterers.
A similar approach is given in [108] which uses approximate expressions for the interaction between a
single scatterer and the array based on an assumption that the array is dense and subwavelength and
that each scatterer acts like a small dipole. In general these approaches are all nearest neighbour(s)
approximation that arrive at close form expression for the interaction field. The method that is most
useful or appropriate is very much dependant on the geometry and structure being analysed as well as
1 It has been assumed that the scatterer only has an electric dipole moment here, but this can be easily generalized to
include a magnetic dipole moment and polarizability can also include cross-coupling between the electric and magnetic
dipole moments (bianisotropy) [99].
2 It should be noted that this sum using the Green’s function is slowly converging and to actually calculate the contribution from each scatterer it is more efficient to treat a two-dimensional array of scatterers as an array of one-dimensional
arrays. This however leads to more complicated expressions that have closed form expressions for surfaces comprised of
simple geometries such as spheres.
Chapter 5. Background On Diffracting/Refracting Surfaces
58
the level of generality that is desired. The interaction field Eint is then reduced to a single scalar value
βe , referred to as a the interaction constant which is proportional to the dipole moment p.
Using these ideas, the local field Eloc can then be expressed as a function of the incident field, the
interaction constant βe , and the dipole moment p,
Eloc = Einc + β · p.
(5.6)
The expression allows for the local field to be combined with (5.5). From here the effective polarizability,
which is a homogenized quantity, can be found and is given to be,
p = αef f Einc =
αe Einc
.
1 − αe βe
(5.7)
This effective polarizability can be used to study the surface or it can be further abstracted into other
homogenized quantities such as an induced electric surface current given to be
Js =
jωαef f
E,
S
(5.8)
where S is the size of the unit cell. This can be further used to calculate properties such as the surface
impedance and the reflection/transmission through the surface.
The procedures discussed here are one way to characterize and study a metasurface, but it is clear
from this discussion that despite the elegance of the theory, these procedures are limited in dealing with
geometrically simple structures with closed form expressions for the interaction field Eint . Obviously,
surfaces with more geometrically or materially complicated scatterers would have to resort to brute-force
numerical techniques. When using numerical techniques the homogenized quantities such as the effective
polarizability or surface impedance, can be found without resorting to characterizing both an isolated
scatterer and an array of them. However, some of the rigorous insight provided by the homogenization
procedure above is lost. In this thesis, full-wave methods using commercial solvers such as HFSS will
be used to characterize metasurfaces. However as will be discussed in Chapters 6 and 8, circuit models
will be used to help interpret these results and give some intuition as well.
While the homogenization of a metasurface was one of the first problems that was tackled, research
into the applications of metasurfaces to various electromagnetic problems was also being studied. A
wide range of applications have been proposed including using these surfaces as wide-angle impedance
matching sheets, waveguides and sensors, and while a detailed discussion is beyond the scope of this
thesis more information can be found in [104].
Some of the applications of metasurfaces that are more relevant to this thesis include those metasurfaces which are able to refract or bend electromagnetic radiation. One example of this are metasurfaces
which are able to focus electromagnetic radiation into sub-wavelength spots as proposed in [101, 110].
The method for implementing this kind of focusing can be done using different techniques. In [101], the
subwavelength focus is achieved by patterning an impedance surface with a specific impedance profile
using a technique referred to as radiation-less interference [111]. Variations of this concept have been
applied to other kinds of surfaces including corrugated surfaces excited by a monopole antenna [112].
This idea can then be further extended to other kinds of beam generation techniques such as Bessel
beam generation using impedance surfaces [113]. Instead of using impedance surfaces to create subwavelength spots, arrays of spatially shifted antennas can also be used by tuning the impedance of each
59
Chapter 5. Background On Diffracting/Refracting Surfaces
antenna [110]. This focusing mechanism can be implemented in planar geometries such as slots etched
in a groundplane or their complement [114, 115] and can also be used to detect buried objects [116].
The kinds of metasurfaces described above, focus incident radiation in the near field and the required
spatial variation in the surface can be appropriately designed for (impedance, corrugation depth, slot
size). However these surfaces are confined to near fields (due to their use of evanescent waves) but it is
clear from the above that spatially varying metasurfaces have interesting properties and should be able
to tailor the distribution of waves in the far-field as well. The question then is what spatial variation in
a metasurface is required to alter the propagating characteristics of an incident field.
One of the first solutions to this problem was proposed in the optical frequency range using an
inhomogeneous array of sub-wavelength nano-antennas arrange on a surface [117]. To properly design
the array of antennas the authors in [117] propose a generalized refraction/reflection law given to be
nt sin θt − ni sin θi =
1 ∂Φ
ko ∂x
(5.9)
where it can be seen that the usual Snell’s law for a dielectric interface has been augmented with a
spatially varying phase profile. Thus the total refraction and reflection from the surface is a result of
both the change in index of refraction as well as any gradient phase shift that exists on the surface. The
design of the nano-antennas that make up the metasurface are done in such a way as to implement the
desired phase gradient to refract and reflect an incident plane wave in an arbitrary direction. This is
depicted in Fig. 5.6 where each nano-antenna is designed to introduce a different phase shift to refract
a plane wave and the spatial variation of the nano-antennas across the metasurface can be seen. When
implemented and measured, the metasurface does indeed refract an incident plane wave into a new
direction as also seen in Fig. 5.6. However, it is clear that other plane waves in other directions are
also present. Also, given the orientation of the nano-antennas, vertically and horizontally polarized
components are present despite being excited by a vertically polarized plane wave only.
While the generalized refraction law in (5.9) does indeed describe a surface capable of refracting a
plane wave, it is clear that this metasurface of nano-antennas does not follow this description exactly.
In fact, given the orientation and the dipolar scattering pattern of each individual nano-antenna, an
incident plane wave will always be transmitted and reflected by a surface of these nano-antennas with
both vertically and horizontally polarized components. Fundamentally then, the metasurface acts more
like a sub-wavelength diffraction grating (albeit a grating supporting two polarizations) as opposed to a
homogenized surface that bends light into a single direction. Thus, this surface would never be able to
transmit a field into the ±1 spatial harmonic only.
It should be noted that the generalized refraction law proposed in [117] is still a valid description
for a surface that refracts a plane wave. However, additional constraints on the unit cells making up
the metasurface are needed. In antenna theory it is known from the context of transmitarrays that for
a surface to bend incident electromagnetic radiation, the transmission amplitude across the array must
be unity. To realize a surface that refracts a plane wave, a unit cell with unity transmission and varying
phase is needed and will be discussed more in Chapter 6.
Another example of a metasurface that bends light is found by applying the metasurface idea to a
blazed grating, mentioned above in Section 5.1. One way of implementing a blazed grating is using a
gradient refractive index profile. However, given the advent of metamaterials and the ability to engineer
desired material parameters using subwavelength unit cells, it should be possible to manufacture more
Chapter 5. Background On Diffracting/Refracting Surfaces
60
compact blazed grating which are able to bend light efficiently. This was the approach taken by the
authors in [118, 119], where a metamaterial grating was designed to act like a graded index material.
Only a small handful of layers of sub-wavelength unit cells, shown in Fig. 5.7, were needed to realize the
effective refractive index which could be varied from 2 < n < 5.5 to modulate the effective path length.
This is thinner than traditional gradient index devices which are thicker due to the limitations on the
material parameters required to realize the desired optical path length. Combined with a numerical
algorithm to encode a hologram on the metamaterial grating, the metamaterial can be used to generate
an arbitrary image in the far-field as also shown in Fig. 5.7 by using an inhomogeneous array of unit
cells. It should be noted that due to fabrication challenges this approach had trouble achieving the full
2π phase shift, resulting in light scattered into other diffraction orders. The metamaterial also reflects
some light as well as it is not impedance matched.
It is interesting to note that both the metamaterial grating in [119] and the metasurface in [117] can
be thought of as describing the same effect as discussed in [85]. However, the authors in [85], suggest the
grating terminology is more useful as it can encapsulate more complex phase profiles while the refraction
law in (5.9) is a more specific case.
Other interesting metasurfaces have been proposed to alter the propagation of electromagnetic waves
such as extending the generalized refraction metasurface to visible light as well as metasurfaces fabricated
using Babinet’s princple [120]. Other kinds of infrared metasurfaces have been proposed using plasmonic
structures to generate a −π to π phase shift [121]. Incorporating loss on purpose into the metasurface
can also be beneficial as shown in [122] where carefully tuning the real and imaginary parts of thin
films of dielectrics can be used to alter the optical path length. Various kinds of synthesis procedures
have been proposed for metasurfaces to carefully design the required inhomogeneity of a metasurface
including complex wave transformations for taking a plane wave and turning it into a complex beam
such as a hypergeometric bessel beam [123, 124]. A summary and review of various metasurfaces can
also be found in [125].
It is clear from the examples above that inhomogeneous metasurfaces can be designed to powerfully
control electromagnetic radiation. This has been also extended to grounded metasurfaces which support
propagating surfaces waves. Grounded impedance surfaces have been known to support surface and/or
leaky waves for a long time [126], however these metasurfaces are designed with an inhomogeneous
impedance that leads to new results. These kinds of metasurfaces are implemented by varying the surface
impedance to either guide surface waves or to become leaky and radiate them in a coherent manner. One
of the first examples of this was an impedance surface designed using a holographic patterning method
[127]. Here by interfering a point source with a desired radiation pattern, a impedance modulation
is found for the surface. This impedance modulation was superimposed over an existing sinusoidal
modulation as to guarantee that the surface supported leaky waves. This allowed for the generation
of directive radiation from two-dimensional apertures when excited by a point source. However, the
polarization of this radiation from the two-dimensional surface impedance is hard to control and is not
in a coherent state. To rectify this, the impedance modulation is made into a tensor which introduces
extra degrees of freedom and allowed for the generation of circular polarization.
Similar ideas for impedance surface patterning were also proposed in [128], where non-holographic
methods of patterning the surface were discussed as well as novel simulation techniques which can be used
to simulate entire surfaces using method of moments techniques [129, 130]. Other analytical approaches
to these kinds of impedance surfaces were also proposed to derive closed form dispersion equations
Chapter 5. Background On Diffracting/Refracting Surfaces
61
[131, 132]. This led to finding techniques to synthesize the impedance surfaces to guide surface waves
and build devices such as beam splitters [133]. For these kinds of devices, exciting these surface waves
is a crucial aspect as well and various surface wave launchers have been proposed [134].
5.5
Reflectarrays and Transmitarrays
The last example of electromagnetic surfaces are specific kinds of antennas referred to as transmitarrays
and reflectarrays. Here, arrays of antennas are passively arranged on a surface. The surface is then
illuminated by a source (horn, dipole, waveguide, etc...) and the surface of antennas is used to collimate
the incident field into a directive beam. The surface can operate as either a reflector or a transmitter,
which in the latter case is sometimes referred to as a lens antenna.
The design of this surface can take one of two main configurations each consisting of arrays of unit
cells. The first configuration consists of unit cells made up of antennas interspersed by phase-shifters as
shown in Fig. 5.8a for a transmitarray antenna [77, 135]. The phase shift is varied across the surface
to allow the optical path length of each ray emanating from the source and collimate the beam. The
second configuration tends to overlap with some of the previous work highlighted above, where FSS’s
and/or impedance surfaces are used to form the transmitarray and the line distinguishing the two can
be blurry. Here, these impedance surfaces or FSS’s are used to form a unit cell with multiple surfaces
stacked together in a number of layers as shown in Fig. 5.8b to achieve both impedance matching to the
incident field as well as the desired phase shift at each point on the surface [136, 137]. Because the phase
shift varies across the surface, the FSS or impedance surface varies as well. This concept has also been
proposed at optical designs using surfaces made up of varying ratios of dielectric and semiconductor
material to vary the impedance of each surface [138].
Compared to parabolic reflectors and traditional dielectric lenses it is clear that the appeal of these
structures is that they can be made flat and low-profile while still providing the desired gain. Another
major thrust in the design of these arrays is their ability to be made reconfigurable. The incorporation
of tunable elements such as varactors, PIN diodes, or MEMS into the phase shifter between antennas or
in directly loading the impedance surface/FSS allows for the reflectarray or transmitarray to steer its
beam. A thorough review of these kinds of devices is given in [139].
Leaving aside reflectors as their configuration is outside the scope of this thesis and focusing on
transmitarrays, it is interesting to compare the performance of the two different kinds of architectures
for transmitarrays. This was done thoroughly in [77, 140] and the main results are reviewed here. With
the back-to-back antenna configuration it was shown that the transmitarray is capable of generating
a directive beam with low sidelobes as the size of the aperture gets larger, as with any traditional
antenna array. While the unit cells can become complex trying to realize a 2π phase shift, synthesis of
back-to-back antennas configurations are very feasible with modern fabrication processes.
However, there are significant challenges with the FSS/impedance surface approach. In [77, 140]
the problem is analyzed using Floquet theory as a stack of impedance surfaces can be treated as a
periodic structure, where the problem is again reduced to bending an incident plane wave into the ±1
Floquet mode at the exclusion of the other modes. As mentioned, the stack of impedance surfaces varies
across the aperture so the problem is broken down into two steps. First, find the reflection/transmission
through a uniform stack of impedance surfaces, which models each unit cell, and then combine each unit
cell to model a period of the array.
Chapter 5. Background On Diffracting/Refracting Surfaces
62
From the first step, the impedance of the stack of surfaces is varied until the transmission is maximized
and a 2π phase shift across the different solutions is achieved. As the number of layers is increased the
phase range that can be covered increases, but the transmission cannot be maintained at unity for each
unit cell. Thus a tradeoff must be realized between the number of layers and the the maximum and
minimum of the impedance that must be implemented. This is a consequence of the filter-like topology
of the structure which results in ripple in the passband.
The second step has these unit cells combined to form a linear-phase shift across the surface and
the Floquet-modes that are scattered off of the surface are analyzed. Here it is found that no matter
what, multiple Floquet modes are excited off of the surface, and while the +1 Floquet mode is dominant,
other modes are transmitted through the structure even for modest angles of beam steering. From the
results presented in [77, 140], this arises from the fact that the transmission through one period of the
transmitarray has a 2π phase shift but a non-negligible amplitude variation. From a Fourier optics
understanding, this will lead to multiple diffraction orders. This means that for finite structures nonnegligible sidelobes will be present even in very large arrays when bending an incident wave a modest
amount, i.e. 20◦ for example.
Compared to a grating, which is also periodic and scatters an incident plane wave into different
Floquet modes, a transmitarray has some similarities. However, it can be seen that the reflections from
the transmitarray are smaller than the grating due to the presence of multiple layers which are used to
cancel out the reflections. Instead the bigger challenge for a transmitarray is to reduce the transmitted
Floquet modes to a single desired mode. Clearly, improving on this design would involve synthesizing
an element that allows for control of both the amplitude and phase of each unit cell.
5.6
Summary and Comments
From this overview of a variety of surfaces used to control the scattering and radiation of electromagnetic
fields, it is clear that all of these surfaces have interesting properties and are capable of refracting
electromagnetic fields to some degree or another. However, an open questions remains: how can a single
surface refract microwaves into a single direction with minimal reflections?
Clearly from the discussion above, only some of these technologies present a path forward to this
problem. The solutions using FSS’s and transmitarrays could potentially be further refined to more
successfully diffract microwaves into one direction while being impedance matched, but their multi-layer
structure is less than ideal. Gratings are another mature technology, however they do not offer enough
degrees of freedom to realize a single layer design that can be matched and diffract into a single order only.
Thus the most appropriate choice is a metasurface consisting of an engineered array of subwavelength
scatterers. Clearly the first attempts at using a metasurface to solve this problem in [117] only partially
achieve this goal as they generate reflections and multiple Floquet modes. However this can be improved
upon by using the Huygens scatterer idea from Chapter 1 as will be shown next.
63
Chapter 5. Background On Diffracting/Refracting Surfaces
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
p
Eint
p
p
Einc
p
p
p
p
k
p
p
p
p
p
p
p
p
p
p
p
p
Hinc
p
p
p
z
p
y
y
x
x
(a)
(b)
Einc
k
Hinc
z
αef f , Js, Zs
z
y
x
(c)
Figure 5.5: (a) A metasurface consisting of an array of identical scatterers illuminated by an incident
plane wave. The incident plane wave polarizes each scatterer with a dipole moment p. (b) At any given
point on the metasurface, the polarized scatterers also contribute a field, referred to as the interaction
field Eint . This interaction field plus the incident field create the local field at each scatterer which
results in the net dipole moment. (c) The net dipole moment can then be homogenized into a an
averaged quantity such as the effective polarizability αef f , surface current Js or surface impedance Zs .
Chapter 5. Background On Diffracting/Refracting Surfaces
64
Figure 5.6: An array of fabricated nano-antennas used to implement a phase gradient on a surface. Each
antenna provides a different phase shift than its neighbours creating the required gradient. The bottom
two plots show the measured transmission through the surface. The n = 0 and n = 1 diffraction orders
are both excited though they are of a different polarization. Figure taken from [117].
Figure 5.7: A metamaterial blazed grating fabricated using 3-6 surfaces stacked back-to-back. A simulation of the multi-layer structure is also shown. Here, the grating can be designed,using an algorithm,
to generate an arbitrary image in the far-field in the n = 1 diffracted beam. A small component remains
in the n = 0 beam as well. Taken from [119].
Chapter 5. Background On Diffracting/Refracting Surfaces
(a)
65
(b)
Figure 5.8: (a) A unit cell of a transmitarray comprised of back-to-back antennas with phase-shifting
c
circuitry in-between. Taken from [77], 2012
IEEE. (b) A fabricated transmitarray consisting of layers
of FSS-type unit cells, with the top layer shown only. The FSS unit cells vary across the aperture to
c
implement the desired phase gradient. Taken from [94], 2013
IEEE.
CHAPTER
6
Refraction From A Huygens Surface
In the previous chapter, various types of electromagnetic surfaces were discussed and the effects of
electromagnetic fields diffracting off of these surfaces were examined. In this chapter the problem of
refracting microwave radiation with a single surface of scatterers is examined. With a few exceptions,
most of the surfaces discussed in the previous chapter utilized an electric response only. However, given
that both electric and magnetic responses can be engineered in metamaterials, it will be shown how a
surface with both an electric and magnetic response can be used to coherently scatter microwaves into
a desired direction.
The electric and magnetic responses are incorporated into a surface in the form of a Huygens scatterer
introduced in Chapter 1. The array of Huygens scatterers comprising the surface will be analyzed and
modelled to see how incident electromagnetic radiation can be refracted/diffracted1 into a single order
with minimal reflections.
6.1
Concept
A surface of Huygens scatterers is shown in Fig. 6.1, situated on the yz-plane. The scatterers that
comprise the Huygens surface are unit cells of co-located and passive loops and dipoles each loaded
with a reactive element as shown in Chapter 1. In this two-dimensional scenario, the incident plane
wave is TE-polarized given the orientation of the loops and dipoles, though all the results below can
be derived for a TM-polarized wave with Huygens scatterers in the orthogonal orientation without any
major modifications. The dipoles have an electric dipole moment pz and the loops have a magnetic
dipole moment my . The intuition in using such a configuration is that the scattering off of the surface
1 In this chapter, either refraction or diffraction could be used when describing the functionality of the Huygens surface.
Physically, the electromagnetic field is diffracting off of the individual scatterers of the Huygens surface, but because of
its unique properties the resulting behaviour appears like refraction when looking at the homogenized interface. Note that
this is true for refraction at an interface between two material half-spaces. The incident electromagnetic field is actually
diffracting off of the atoms in the material and constructively interfering to form, what is referred to as, the refracted beam.
The overall phenomena is referred to as refraction as it is described simply by the boundary conditions at the interface.
Likewise with the Huygens surface.
66
67
Chapter 6. Refraction From A Huygens Surface
Reactive
loading
m
y
Ei
Hi
p
z
z
φ k
i
y
x
1
z
y
ZeZm 2
^
n
x
Figure 6.1: A Huygens surface consisting of co-located loops and dipoles. The surface is homogenized
as an electric surface impedance, Ze and a magnetic surface impedance, Zm .
will be in the forward (transmitted) direction only just as a single Huygens source or scatterer radiates
in one direction only as discussed in Chapter 1. To deal with the surface as whole, these dipole moments
have to be homogenized into an averaged quantity. This can be done by treating the surface using an
impedance boundary condition. For a homogenized array of electric dipole moments only the boundary
condition is,
n̂ × [E2 − E1 ] = 0,
E2 |t = Ze n̂ × (H2 − H1 ) = Ze Js ,
(6.1)
(6.2)
where the subscripts 1, 2 are defined in Fig. 6.1 as the two different sides of the surface. For an array of
magnetic dipole moments only, the boundary condition is
n̂ × [H2 − H1 ] = 0,
−1
−1
H2 |t = −Zm
n̂ × (E2 − E1 ) = −Zm
Ms ,
(6.3)
(6.4)
In both of these scenarios, one field component is continuous, while the other is not. This difference in the
electric or magnetic field for either of these boundary conditions can be interpreted as an induced electric
or magnetic current respectively as shown. Individually then, the induced currents for either boundary
condition would radiate equally into the space on side 1 or 2. For a Huygens surface however, the electric
and magnetic responses are superimposed. When this happens, the overall boundary condition can be
expressed as [99, 100],
E2 |t + E1 |t
= Ze n̂ × (H2 − H1 ) = Ze Js ,
2
H2 |t + H1 |t
−1
−1
= −Zm
n̂ × (E2 − E1 ) = −Zm
Ms ,
2
(6.5)
(6.6)
Chapter 6. Refraction From A Huygens Surface
68
where the fields on the right-hand-side are now averaged across the boundary. From this superposition,
the electric and magnetic responses create a discontinuity in both the electric and magnetic fields respectively, which corresponds to both induced electric and magnetic currents existing on the surface. These
induced electric and magnetic currents allow for the realization of the Huygens scatterer behaviour,
introduced in Chapter 1 (scattering fields into one-half of space only). The question is what values of
Ze and Zm are required for this to be realized.
Note that for the derivation that follows, the boundary conditions in (6.1)-(6.4) will be used initially
as the electric and magnetic impedance surfaces will be treated separately, being superimposed after.
Ultimately for these boundary conditions to hold, the loops and dipoles must be subwavelength with
the unit cell size d < λ/10. It will also be shown in Section 6.3 how these surface impedances can be
extracted from numerical simulations of individual unit cells using an equivalent circuit model.
With the Huygens surface characterized by electric and magnetic impedances there are two problems
that need to be solved. The first is a synthesis problem; What values of Ze and Zm are needed such that
an incident plane wave is refracted into a different direction with minimal reflections off of the surface
(i.e. how to make the surface act like a Huygens surface?). Constraints on the electric and magnetic
surface impedance must also be taken into account which force the surface to be passive and lossless,
(<(Ze ) = 0 and <(Zm ) = 0). These constraints will also affect the amplitude of the scattered fields so
that power conservation is maintained.
The second problem is an analysis problem; given values for Ze and Zm what are the fields scattered
by the surface? While this may seem secondary to the goal of synthesizing an appropriate impedance
surface, it is in fact necessary as it helps to verify the solution as well as to explain how the Huygens
surface works.
In Section 6.1.1 the synthesis problem is dealt with by studying how a discrete number of plane
waves scattering off of the Huygens surface determines the values of Ze and Zm and how this results in a
surface capable of refracting an incident plane wave. In Section 6.1.2, the scattering off of the Huygens
surface with a known impedance is solved for using Floquet theory.
6.1.1
Synthesis
When electric and magnetic impedance surfaces are illuminated by a plane wave, a set of fields are
scattered off of the surface to satisfy the boundary conditions given in (6.1)-(6.4). For a Huygens surface
to refract a plane wave this scattering must be very precise. From a total field/scattered field perspective,
an incident plane wave scattered by a Huygens surface on the transmitted side must cancel the incident
plane wave, as well as scattering the refracted plane wave. On the reflected side, the scattering from both
the electric and magnetic impedance surface should destructively interfere due to the Huygens source
characteristic of the surface.
Given the desired behaviour then, an incident plane wave scattering off either the electric or magnetic
impedance surface must generate four-scattered plane waves, two on either side of the interface. This
is because the fields radiated by the induced electric and magnetic currents given in (6.1)-(6.4) for the
two-dimensional case are symmetric with respect to the y-axis and thus are equal in the transmitted
and reflected directions (implicitly enforcing (6.1) and (6.3), the continuity of the electric and magnetic
fields respectively). Thus whatever desired plane waves are wanted in the transmitted direction, a
reflected plane wave must also exist to satisfy the boundary condition. This is pictured in Fig. 6.2 for
the electric impedance surface and in Fig. 6.3 for the magnetic impedance surface. The scattered fields
69
Chapter 6. Refraction From A Huygens Surface
Hise
Ei
Hi
φ
x
φt
Eise
φ k
i
k
i
z
Hte
Zs
Ete
Hte
x
2
^
n
x
φ k
i
1
y
Hise
Eise
Ete
k
φt k
Figure 6.2: A plane wave incident on an impedance surface. Shown are the incident and the scattered
fields. On side 2 of the boundary the total fields are the scattered fields summed with the incident plane
wave (not-shown).
Eism
Ei
Hi
φ
φ
k
i
Htm
Hism
ik
x
z
Eism
Hism
φt
Etm
Ys
Etm
x
1
y
Htm
2
^
n
x
φ k
i
k
φt k
Figure 6.3: A plane wave incident on an admittance surface. Shown are the incident and the scattered
fields. On side 2 of the boundary the total fields are the scattered fields summed with the incident plane
wave (not-shown). To create a surface which primarily reflects the incident plane wave as opposed to
refracting it, the sign of Etm should be flipped.
in Fig. 6.2 form a set of continuous electric fields and discontinuous magnetic fields which will define the
electric impedance surface. The scattered fields in Fig. 6.3 form a set of continuous magnetic fields and
discontinuous electric fields which correspondingly define the admittance surface.
These fields can now be used to analyze each impedance boundary condition respectively. Starting
with the electric impedance boundary condition the fields given in Fig. 6.2 are inserted into (6.2) to give,
Ei e−jky sin ϕi − Eise e−jky sin ϕi + Ete e−jky sin ϕt =
1
1
1
Ze − cos ϕi Ei e−jky sin ϕi + cos ϕi Eise e−jky sin ϕi − cos ϕt Ete e−jky sin ϕt
η
η
η
i
1
1
1
−jky sin ϕi
−jky sin ϕi
+ cos ϕi Ei e
+ cos ϕi Eise e
− cos ϕt Ete e−jky sin ϕt .
η
η
η
h
(6.7)
Note that the boundary condition is defined at x = 0. From here, Eise and Ete , can be solved for which
are the amplitudes of the scattered fields, by forcing <(Ze ) = 0. This will also give an expression for
Chapter 6. Refraction From A Huygens Surface
70
=(Ze ) = Xe . Doing this gives,
Ei cos ϕt
,
cos ϕi + cos ϕt
Ei cos ϕi
=
,
cos ϕi + cos ϕt
Eise =
(6.8)
Ete
(6.9)
and the expression for the electric surface reactance is,
Xx = −
η
cot(Φ/2),
2 cos ϕt
(6.10)
where Φ = ky(sin ϕt − sin ϕi ), the phase shift between the refracted and incident plane waves. Note
that the expression for Xx has cotangent variation with respect to y (or Φ). Such an impedance surface
imposes a linear phase shift along the boundary, similar to the nano-antenna arrays and transmitarrays
in [77, 117]. However it only scatters fields into the plane waves shown in Fig. 6.2 with the amplitudes
given in (6.8)) and Eq. (6.9). This kind of scattering from an impedance surface on its own is not entirely
useful, however when combined with the magnetic surface impedance below it will lead to the desired
result. Also note that this is the best possible case for a single impedance surface as the incident plane
wave can be diffracted into a minimum of four other plane waves, and thus only succeeds in partially
refracting the beam.
The same analysis can be done for the magnetic impedance surface at x = 0. Taking the fields in
Fig. 6.3 and inserting them into (6.4) and forcing <(1/Zm ) = 0 results in
Eism = Etm =
Ei cos ϕi
,
cos ϕi + cos ϕt
(6.11)
Again the surface susceptance is found to be,
Bs = −
cos ϕt
cot(Φ/2)
2η
(6.12)
which is also a cotangent function of y. Note also that both (6.10) and (6.12) depend on Φ, the phase
shift between the incident and refracted wave.
As stated before, the complete effect comes when both the electric and magnetic impedance surfaces
are superimposed creating a discontinuity in both the electric and magnetic field which results in the
plane wave being refracted. Superimposing the scattered fields off of both impedance surfaces results
in the following set of fields shown in Fig. 6.4. The incident plane wave scattered by both impedance
surfaces yields a refracted field with an amplitude given by,
Et = Ete + Etm =
2Ei cos ϕi
,
cos ϕi + cos ϕt
(6.13)
and a reflected field with an amplitude of,
Eis = Eism − Eise =
Ei (cos ϕi − cos ϕt )
,
cos ϕi + cos ϕt
(6.14)
where the reflected field is in the specular direction. Note however, that this reflected field is much
71
Chapter 6. Refraction From A Huygens Surface
φ
Eis
Ei
Hi
i
k
His
φ k
i
z
S
Zs Ys
Et
x
Ht
1
y
2
^
n
φt k
Figure 6.4: A plane wave incident on a combined impedance and admittance surface. Shown are the
incident and the total fields on either side of the boundary. Because the impedance and admittance
surfaces create induced electric and magnetic currents they can successfully refract the incident plane
wave without any other fields on side two of the boundary. Also note that the majority of the power is
scattered into Et as the amplitude of Eis is much smaller as shown in Fig. 6.5
smaller than the refracted field as will be demonstrated below.
To further show that these are the only fields generated by the combined surface, the power entering
and leaving the scattered surfaces can be found by integrating the fields over an area enclosed by the
surface shown in Fig. 6.4. The power entering and leaving the surface is given by,
P =
1
2
ZZ
S
< [E × H∗ ] · n̂dS.
(6.15)
For plane waves this reduces to the power in the normal component of the field that enters or leaves a
surface with area, A, which gives,
P =
A
2
(|Ei |2 cos ϕi − Et2 cos ϕt − Eis
cos ϕi ) = 0
2η
(6.16)
which demonstrates that all the power from the incident plane wave is scattered into the refracted and
reflected beam.
An equivalent setup for this surface that reflects an incident plane wave along an arbitrary angle
while minimizing the transmitted field can easily be extended from the derivation given here by simply
negating the direction of Etm in Fig. 6.3 and re-deriving the subsequent equations.
These results can be summarized in a set of design curves which describe how a plane wave refracts
through a combined impedance surface or Huygens surface. This is plotted in Fig. 6.5 and describes
the solutions of Et , Eis , Xs and Bs for all possible combinations of ϕi and ϕt . The plots in Fig. 6.5
give the amplitude of Et and Eis respectively for different values of ϕt and ϕi . Note that for most
reasonable angles of incidence the amplitude of the refracted beam, Et is close to 1 while the reflected
beam is closer to 0. Specifically, for both incident and refracted angles between ±50◦ the amplitude of
the reflected field is kept between ±0.2 while the amplitude of the transmitted field is kept above 0.8.
This shows that the combined passive surface is ideal for refraction with minimal reflection. Of course
as either the incident or refracted beams approach a grazing angle the reflections worsen. In Fig. 6.5 the
contours of constant impedance and susceptance are also given for different combinations of ϕt and ϕi
at a single point on the surface showing the full-range of values covered by the impedance surfaces for
72
Chapter 6. Refraction From A Huygens Surface
1.6
0.6
50
1.4
1.2
φt
0
0.4
0.2
0
1
0
0.8
−50
−0.2
−50
0.6
−50
0
φi
−0.4
50
−50
0
φi
50
1000
0
0
−500
−50
−50
0
φi
50
50
−1000
φt
500
Impedance Ω
φt
50
0.5
0
0
−50
−50
0
φi
50
Admittance Ω−1
φt
50
−0.5
Figure 6.5: Design curves summarizing refraction through a combined impedance and admittance surface
for an incident plane wave of amplitude Ei = 1. All plots are drawn with incident angle, ϕi , on the
horizontal axis and the refracted angle, ϕt , on the vertical axis. The top-left plot gives the amplitude
of the refracted beam, Et while the top-right plot gives the amplitude of the reflected beam Eis . The
bottom two curves are contour plots of the impedance and admittance of the screen at a given point,
(y = λ) for different incident and reflected angles.
different incident and refracted angles.
Had this analysis not been careful to constrain the amplitudes of the refracted fields the solution
for the electric and magnetic impedance surfaces would have had a non-zero and possible negative real
part. It also would have been possible to use the real parts of the electric and magnetic impedance
surfaces to suppress the reflected field completely or even more interestingly to shape the amplitude of
the refracted field. This is because the real part of the impedance adds another degree of freedom to
control the scattered field.
Finally, note that this construction of a Huygens surface gives the full description of how to refract a
plane wave across a single surface. As a point of comparison, examine how the refraction and reflection
of a plane wave across a material interface is derived. There are three attributes about refraction at a
material interface. First it is derived from the boundary conditions at a material interface. Secondly,
it tells us how the field will refract and reflect (Snell’s Law). Third, it tells us the amplitudes of the
field (the Fresnel reflection coefficients). Here the same attributes have been realized using a Huygens
surface. Starting from the boundary conditions given by impedance and admittance surfaces, the field
has been found for surfaces of different impedances (6.10) and (6.12) as well as the amplitudes of those
fields in (6.13) and (6.14). This gives us a complete picture for how to tailor a surface to refract (or
reflect) an incident plane wave.
In comparison, the generalized refraction law in (5.9) simply tells us the phase gradient between the
two refracted plane waves. It does not tell us how to synthesize a surface. (As shown, trying to synthesize
a surface which simply mimics the phase gradient is incomplete). Thus this equation is descriptive, as
it describes how a plane wave refracts across a surface, but it is not prescriptive in that it does not
73
Chapter 6. Refraction From A Huygens Surface
describe how to design the surface. Equation (5.9) is useful as a preliminary step as one can envision an
arbitrary gradient along a surface which encodes some functionality (focusing, beam steering, vortexes)
with respect to some incident field. However the equations given in (6.10) and (6.12) are useful to
synthesize the required surface using both electric and magnetic impedances.
6.1.2
Analysis - Floquet Modes
From the synthesis of the Huygens surface described above, the impedances of both the electric and
magnetic surfaces are periodic functions. This is expected of course as both surfaces of electric and
magnetic scatterers are diffracting the electromagnetic radiation from a plane wave in one discrete
direction into a plane wave into another discrete direction. To synthesize such a surface the constraint
of having only one other plane wave propagating in a different direction, besides the incident direction,
was imposed when solving the boundary conditions in (6.2) and (6.4). From Floquet theory, as discussed
in Chapter 5, this can be understood as scattering the incident field into the n = 1 Floquet modes only.
However, as also discussed in Chapter 5 a general periodic surface, tends to scatter fields into multiple
Floquet orders and a lot of effort is required to suppress these spurious modes. This raises the question
then, why does the cot(Φx) impedance variation in the electric and magnetic surface suppress scattering
from multiple Floquet orders leaving only the n = 1 mode. To do this a simple analysis of the problem
is presented here, following [140], where the induced currents and scattered fields are expanded into a
sum of 2P − 1 Floquet modes to see how the impedance defined in (6.10), (6.12) cancels all but the
n = 0 and n = 1 modes.
N =1
N =0
J(x)
N =1
N =0
kin
Z(x)
N = −1
z
y
N = −1
x
W
Figure 6.6: The Floquet mode model of an electric impedance surface. Since the impedance function
Ze is periodic the problem can be analyzed in a periodic environment. Here, given an incident plane
wave and the impedance, Ze of the surface, the induced currents can be decomposed into Floquet modes
as shown. The scattered fields are also decomposed into 2P − 1 Floquet modes as shown and their
coefficients are solved for.
The geometry of the problem is shown in Fig. 6.6 where the impedance surface is now placed in a
74
Chapter 6. Refraction From A Huygens Surface
2D periodic waveguide of width W . This periodic waveguide only supports propagation of plane waves
P∞
with a discrete spectrum, given by p=−∞ e−jkx,p x e−jky,p y where the kx,p and ky,p are defined to be,
2πp
kx,p = k sin ϕi +
,
W
q
2
ky,p = k 2 − kx,p
(6.17)
(6.18)
To begin, only the electric impedance surface with an impedance Z(x) is dealt with and it is illuminated
by an incident plane wave Einc = Eo e−jk sin ϕi x e−jk cos ϕi y ẑ. With the impedance of the surface given
to be a known quantity here, the unknowns that are being solved for are the induced surface currents
and the scattered fields. For the electric impedance surface, the induced surface current density on the
impedance surface can also be expanded into a sum of Floquet modes,
P
−1
X
Jx,y = δ(y)
Ap e−jkx,p x ẑ
(6.19)
p=−(P −1)
where 2P − 1 modes are used to expand the unknown current distribution with the implicit assumption
that a sufficient number of propagating and evanescent waves are included in the summation to accurately
represent the unknown current distribution. These induced surface currents radiate the scattered field
created by the impedance surface . To find the radiation of these surface currents, the tensorial Green’s
function for the periodic waveguide is defined to be [140, 141],
Ḡe (x, y|x0 , y 0 ) =
1
2jW
0
∞
X
e∓jky,m (y−y ) −jkx,m (x−x0 )
e
Ī
ky,m
m=−∞
(6.20)
where Ī is the identity matrix. The scattered fields are then found to be,
Es = −jωµo
Es = −
jωµo
ẑ
2jW
Z
∞
−∞
Z
0
W
Z
Ḡe (x, y|x0 , y 0 )J(x0 , y 0 )dx0 dy 0 ,
(6.21)
S
0
∞
X
e∓jky,m (y−y ) −jkx,m (x−x0 )
e
δ(y)
ky,m
m=−∞
P
−1
X
Ap e−jkx,p x dx0 dy 0 .
(6.22)
p=−(P −1)
Using the orthogonality of the complex exponentials, the scattered field reduces to,
Es = −
ωµo
2
P
−1
X
p=−(P −1)
e∓jky,p y
Ap e−jkx,p x ẑ,
ky,p
(6.23)
where the ∓ indicates forward propagating waves for y > 0 and backward propagating waves for y < 0.
The total field then is simply Etot = Ei + Es . With the total field and the surface currents each defined
as a sum of Floquet modes, they can be inserted into the boundary condition imposed by the impedance
surface given in (6.2) which results in,
Etot (x, y = 0) = Z(x)J(x, y = 0),
Eo e−jkx,0 x −
ωµo
2
P
−1
X
p=−(P −1)
1
Ap e−jkx,p x = Z(x)
ky,p
P
−1
X
p=−(P −1)
Ap e−jkx,p x
(6.24)
(6.25)
75
Chapter 6. Refraction From A Huygens Surface
The periodic expression for the electric impedance surface in (6.10) can now be inserted into the equation
above. First, however, (6.10) is modified to make it more amenable to the Floquet expansion above.
The argument for the cotangent term in the impedance, Φ/2 =
k sin ϕo x−k sin ϕi x
2
can be rearranged in
terms of the Floquet modes defined above where,
k sin ϕo = k sin ϕi +
and
2π
2π
= kx,0 +
,
W
W
(6.26)
πx
Φ
=
2
W
(6.27)
This formally states that the shift in the angle between the incident and transmitted fields is a shift
between the the p = 0 Floquet mode and the p = 1 mode. Whereas in the infinite screen synthesis
in the previous section the difference between the incident and transmitted fields was controlled by the
difference between the angles ϕi and ϕo , in the Floquet analysis it is controlled simply by the periodicity
of the screen, W . Likewise the cos ϕo term in (6.10) can also be expressed as,
cos ϕo =
ky,1
.
k
(6.28)
The impedance of the surface is now given to be,
Z(x) =
−jηk
πx
cot( ),
2ky,1
W
πx
Z(x) =
(6.29)
πx
ωµo ej W + e−j W
πx
πx
2ky,1 ej W − e−j W
(6.30)
This expression can now be inserted into (6.25) which, after rearranging, results in
Eo e−j (kx,0 x− W ) − Eo e−j (kx,0 x+ W ) =(6.31)
ωµo
ωµo
−j (kx,p x+ πx
−j (kx,p x+ πx
−j (kx,p x− πx
−j (kx,p x− πx
)
)
)
)
W
W
W
W
−e
+e
+
.
Ap e
Ap e
2ky,p
2ky,1
πx
P
−1
X
p=−(P −1)
πx
πx
It can be seen now that the impedance term has shifted every Floquet plane wave by a factor of e±j W .
The resultant wave vectors after such a shift can now be expressed as,
0
kx,2p∓1 = ko sin ϕi +
π(2p ∓ 1)
.
W
(6.32)
Substituting this into (6.31) gives
P
−1
X
p=−(P −1)
Eo e−jkx,−1 x − Eo e−jkx,1 x =
0
0
0
0
ωµo
ωµo
Ap e−jkx,2p−1 x − e−jkx,2p+1 x +
Ap e−jkx,2p−1 x + e−jkx,2p+1 x .
2ky,p
2ky,1
(6.33)
To solve for the 2P − 1 unknown coefficients, Ap , of this Floquet expansion, (6.33) needs to be expanded
into 2P − 1 equations. This can be done by projecting the left-hand side and the right-hand side
76
Chapter 6. Refraction From A Huygens Surface
0
of (6.33) onto 2P − 1 complex exponentials2 . The complex exponentials used are e−jkx,2n−1 x where
n ∈ {−(N − 1), N − 1} and N = P . The projection of each side of (6.33) is done by using the inner
product defined as,
0
−jkx,2n−1 x
hf (x), e
1
i=
W
Z
W
0
f (x)ejkx,2n−1 x dx.
(6.34)
0
Again because of the orthogonality of the complex exponentials the resulting system of equation takes
the form
0=
ωµo
0=
2
1
ky,1
−
1
ky,−(P −1)
ωµo
2
ωµo
A−(P −1) +
2
1
ky,1
1
ky,1
+
+
1
1
ωµo
−
A−1 +
2
ky,1
ky,−1
1
ωµo
1
−Eo =
−
A0 +
2
ky,1
ky,0
ωµo
1
1
0=
−
A1 +
2
ky,1
ky,1
1
ωµo
1
0=
−
A2 +
2
ky,1
ky,2
1
ky,−(P −1)
1
ky,−(P −2)
A−(P −1)
(6.35)
A−(P −2)
..
.
ωµo
1
1
+
A0
2
ky,1
ky,0
1
ωµo
1
+
A1
2
ky,1
ky,1
1
ωµo
1
+
A2
2
ky,1
ky,2
1
ωµo
1
+
A3
2
ky,1
ky,3
..
.
1
1
1
ωµo
1
ωµo
−
AP −2 +
+
AP −1
0=
2
ky,1
ky,P −2
2
ky,1
ky,P −1
Eo =
By inspection it can be determined that the coefficients Ap = 0 for p 6= {0, 1}. Thus the only non-zero
coefficients are A0 and A1 . It can easily be determined that when A0 and A1 are substituted into
the expression for the scattered fields in (6.23), the amplitude of the scattered field is the same as in
(6.8)-(6.9) for the infinite impedance screen.
From this analysis, it is clear that the specific impedance variation in (6.10) and (6.12) allows for
only the p = 0 and p = 1 Floquet modes to be scattered by the surface. The specific construction of
the impedance function is what allows for this, and can be attributed to two characteristics of Z(x),
specifically the cot(Φ/2) variation of the impedance as well as its
1
ky,1
weighting given in (B.10). The
cot(Φ/2) variation of the impedance allows for the spectrum of the Floquet modes to be shifted into two
discrete modes. Any periodic structure can shift an incident plane wave into multiple Floquet modes,
and the cot(Φ/2) variation distributes each Floquet mode into two distinct modes only. However, to
reduce this theoretically infinite summation of Floquet modes to the remaining p = 0 and p = 1 modes
only, the
1
ky ,1
weighting of the impedance is key. The
1
ky,1
weighting of the impedance allows for the A1
mode to dominate the scattering
along with
the A = 0 mode. This can be observed from the fact that
1
1
each coefficient is weighted by ky,1
± ky,p
and when p = 1 it forces the other modes to be cancel. For
other possible functions of the impedance Z(x), it is impossible to end up with the behaviour observed
here with Z(x) given by (6.10). For example, if Z(x) had a sawtooth (linear) variation as discussed
2 The steps taken to solve (6.33) are essentially a periodic method of moments procedure using entire-domain basis
functions combined with Galerkin’s method [71]. However, as it will be shown, a closed form solution can be found in this
special case without requiring a numerical solution.
77
Chapter 6. Refraction From A Huygens Surface
in [140], when forming a system of equations using the inner product in (6.34), multiple coefficients
of each Floquet mode would be needed to expand this impedance in the Floquet mode basis requiring
a non-trivial and non-sparse matrix in the system of equations. When (numerically) solved, it would
require multiple non-zero Floquet modes. Thus the intuition in the impedance function of (6.10) is its
shifting of the Floquet mode basis and its appropriate weighting.
Again as a reminder, from a total field/scattered field perspective the impedance surface must scatter
into both the p = 0 and p = 1 modes. The p = 0 mode destructively interferes with the incident field on
the transmitted side of the surface while the p = 1 mode is the desired refracted field. By a simple appeal
to duality, a similar argument can be put forth for the magnetic currents, where the magnetic currents
are also expanded into a Floquet series and the dyadic Green’s function for the magnetic currents is the
same as (6.20). The same procedure to form a system equations can also be used to show that the p = 0
and p = 1 modes are the only scattered fields for the magnetic impedance surface given in (6.12) and is
detailed in Appendix B. When both of these surfaces are combined it is the superposition of the p = 0
and p = 1 modes from both of these surfaces that implement the Huygens scattering effect. This allows
for the fields scattered by both the electric and magnetic impedance surfaces to destructively interfere
in the reflected direction while constructively interfering in the transmitted direction.
This Floquet model of the Huygens surface can also be used to study how plane waves with an
incident angle that differs from the incident angle the surface was designed for, scatter off of the Huygens
surface. Such an analysis gives insight into how the Huygens surface functions when used beyond its
intended design. With analogy to a dielectric interface, as the incident angle varies the angle of the
refracted plane wave bends in accordance with snells law. The generalized refraction law from [117] also
predicts a similar behaviour where for a given phase-gradient, an arbitrary incident field will refract into
a transmitted field given by (5.9). However, since it has been established that this generalized refraction
law is an incomplete description of the problem at hand, the Floquet theory described here will give
better physical insight as to how the Huygens surface functions in this kind of scenario.
The problem now begins with a given electric and magnetic impedance Ze and Zm designed for some
known incident field and a desired transmitted field following Section 6.1.1. The incident angle ϕi and
transmitted angle ϕo determine the periodicity of the surface W ,
W =
2π
,
ko sin ϕo − ko sin ϕi
(6.36)
where it is assumed that the transmitted angle is the p = 1 mode. With a given periodicity W , an
arbitrary incident angle for an incident plane wave can be chosen, denoted by ϕa . This incident angle
is then inserted into (6.17)-(6.18), which determines the wavevectors for each Floquet mode and the
system of equations derived from (6.33) can be properly setup. Since ϕa 6= ϕi , the angle of the p = 1
mode is no longer equal to ϕo . Thus in the system of equations given in (6.35),
with
1
cos ϕo .
1
ky,1
is now replaced
This system of equations must be solved numerically as it can be seen that Ap is non-zero
for other modes besides p = {0, 1}.
It is clear from having to solve the system of equation numerically that for deviations from the
designed incident angle, ϕi , the Huygens surface loses its refraction like behaviour and starts to scatter
fields into other modes. This is different than refraction at a dielectric half-space or even the behaviour
predicted by the generalized refraction law. The remaining question then is how severe is this behaviour.
To illustrate this, two examples are studied here. In the first example the Huygens surface is designed
78
20
1.4
15
10
1.2
0.6
10
0.5
5
0.4
5
1
0
0.8
−5
0.6
−10
0.4
−15
0.2
−15
0
−20
−40
−20
0
∆φi (degrees)
20
0.3
−5
0.2
−10
0.1
−20
0
∆φi (degrees)
(a)
0
(b)
20
20
2
1.5
5
0
1
−5
−10
0.5
−15
p (Floquet Mode)
10
1
15
Amplitude (a.u.)
15
p (Floquet Mode)
20
0.8
10
5
0.6
0
0.4
−5
−10
0.2
Amplitude (a.u.)
−20
−40
0
Amplitude (a.u.)
1.6
15
p (Floquet Mode)
20
Amplitude (a.u.)
p (Floquet Mode)
Chapter 6. Refraction From A Huygens Surface
−15
−20
−40
−20
0
∆φi (degrees)
20
−20
−40
−20
0
∆φi (degrees)
(c)
20
0
(d)
Figure 6.7: (a) The transmitted Floquet modes for a Huygens surface designed for ϕi = 10◦ and ϕo = 40◦
when the angle of the incident field varies from ∆ϕi = {−40◦ , 20◦ }. (b) The corresponding reflected
Floquet modes. (c) The transmitted Floquet modes for a Huygens surface designed for ϕi = 10◦ and
ϕo = −40◦ when the angle of the incident field varies from ∆ϕi = {−40◦ , 20◦ }. (d) The corresponding
reflected Floquet modes.
for ϕi = 10◦ and ϕo = 40◦ , while in the second example the incident angle is again ϕi = 10◦ , while ϕo =
−40◦ . For both of these scenarios a variety of incident angles are considered where ϕa = {−30◦ , 30◦ }.
The reflected and transmitted Floquet modes are then found,
Et = Eo e−jko sin ϕa x e−jko cos ϕa y ẑ −
ωµo
2
p=−(P −1)
1
2
Er = −
ωµo
2
P
−1
X
p=−(P −1)
P
−1
X
Ap,e −jkx,p x +jky,p y
1
e
e
ẑ −
ky,p
2
Ap,e −jkx,p x −jky,p y
e
e
ẑ+
ky,p
P
−1
X
(6.37)
Ap,m e−jkx,p x e−jky,p y ẑ,
p=−(P −1)
P
−1
X
Ap,m e−jkx,p x e+jky,p y ẑ,
(6.38)
p=−(P −1)
where Ap,e are the Floquet modes coefficients for the electric impedance surface and Ap,m are the
coefficients for the magnetic impedance surface. Again refer to Appendix B for the derivation of the
magnetic impedance surface’s Floquet modes. In this analysis P = 20 modes are considered and the
amplitude of the transmitted and reflected modes in (6.37)-(6.38) are plotted in Fig. 6.7 for both examples
considered.
79
Chapter 6. Refraction From A Huygens Surface
For the transmitted fields in either case it can be seen that as the incident angle shifts by ∆ϕi from
the designed incident angle the amplitude of the p 6= 1 Floquet modes increase. However near ∆ϕi = 0
the transmitted field is still mainly contained in p = 1 modes. The p 6= 1 Floquet modes increase more
rapidly for ∆ϕi > 0 in the first case while the opposite is true in the second case. This is because for
∆ϕi > 0 in the first case and ∆ϕi < 0 in the second case, the transmitted fields are pushed towards
grazing angles, requiring more Floquet modes to satsify the boundary condition as the tangential fields
begin to vary rapidly at the surface.
However, for the reflected fields, the p 6= 0 Floquet modes in either example increase more rapidly
as ∆ϕi moves away from zero in either direction. This indicates that while the Huygens surface can
maintain a relatively pure transmitted field for some range of angles around ϕi the reflected fields are
compromised very quickly as |∆ϕi | increases with higher-order reflected Floquet modes being excited
for non-designated angles of incidence.
For completeness, it should be noted that a related analysis of this problem is presented in [142],
where a different perspective on Floquet modes propagating through a Huygens surface is presented and
the reader is encouraged to review it.
6.2
Examples
These results can be verified using two simple full-wave simulations. The first example is to use a onedimensional method of moments code to model the Huygens surface, including both the synthesis and
analysis of the impedance surface. The second example is to use a commercial full-wave solver (HFSS)
to demonstrate a very basic physical implementation of this concept above.
6.2.1
Method of moments verification
To analyze the refraction through a Huygens surface a one-dimensional method of moments procedure
to numerically synthesize the required admittance and impedance can be implemented. This can also be
used to find the radiation from the induced currents on the surface, confirming the results above. Similar
approaches are discussed in [71, 101] and the work that follows is modelled off of them. Because of the
numerical method chosen, the surface under consideration is a finite surface with a width of L = 10λ.
For the electric impedance surface the induced electric currents on the surface can be found from
examining the relationship between the incident, scattered and total fields at the boundary given by
(6.3),
Etot |x=0
= Ei |x=0 + Es |x=0 .
(6.39)
The scattered field is created by the radiation of the induced electric current on the surface which is
given by [71],
Etot |x=0
= Ei e−jky sin ϕi ẑ −
ωµ
4
Z
L/2
−L/2
Js (y 0 )Ho2 (k|y − y 0 |)dy 0 .
(6.40)
Likewise, for the magnetic impedance surface, the magnetic current induced on the surface can be
80
Chapter 6. Refraction From A Huygens Surface
found by examining the incident, total and scattered field at the boundary given by (6.3),
n̂ × Htot |x=0
= n̂ × Hi |x=0 + n̂ × Hs |x=0 .
(6.41)
Here the tangential magnetic field is created by the radiation of the induced magnetic currents on the
surface which are given to be,
n̂ × Htot |x=0 =
1
Ei
cos ϕi e−jky sin ϕi x̂ −
η
4ωµ
Z L/2
∂2
Ms (y 0 )Ho2 (k|y − y 0 |)dy 0 .
k2 + 2
∂y
−L/2
(6.42)
Both (6.40) and (6.42) can be solved for Js and Ms respectively using the method of moments since
the desired total fields and incident field at the boundary are both known. To solve, the impedance
surface is discretized into N segments and is solved for using pulse-basis functions and point matching
[71]. Once Js and Ms are found, these can be respectively inserted into (6.2) and (6.4).
0.02
Susceptance Ω−1
Reactance Ω
1000
500
0
−500
−1000
−5
0
5
0.01
0
−0.01
−0.02
−5
0
y/λ
2
Electric Field (V/m)
Electric Field (V/m)
2
1
0
−1
−2
−1
5
y/λ
0
1
2
y/λ
3
4
1
0
−1
−2
−2
−1
0
1
2
y/λ
Figure 6.8: Method of moments verification of a combined impedance and admittance surface to refract
an incident plane wave. Solid, coloured curves are calculated from the method of moments procedure.
Dashed black curves are theoretical results from the previous section. The top-left and top-right figures
are the calculated reactance and susceptance of the two surfaces. Good agreement can be seen with the
theoretical results as well as their non-linear dependence on the spatial coordinate of the surface. The
bottom-left and bottom-right plots show the calculated and theoretical electric field on side-two and
side-one of the surface respectively (real and imaginary parts in red and blue respectively ). Note the
much larger amplitude of the the refracted beam compared to the reflected beam.
To demonstrate this method the following example is used. First the incident field is in the normal
direction, ϕi = 0◦ and the desired refracted beam is along the ϕt = 30◦ direction. The electric and
magnetic impedance surfaces are solved at f = 1.5 GHz. The surface is divided up into N = 1000
segments that are λ/100 long. The calculated reactance and susceptance are plotted in Fig. 6.8 along
with the theoretical values given by (6.10) and (6.12) where good agreement can be seen between the two
Chapter 6. Refraction From A Huygens Surface
81
sets of curves. The total field on either side of the screen can also be found by finding the fields radiated
from both sets of currents. On the transmitted side of the boundary, these fields must be summed with
the incident field. These radiated fields are given by,
Es,elec
=
Es,mag
=
Es
=
Z
p
ωµ L/2
Js (y 0 )Ho2 (k| (y − y 0 )2 + z 2 |)dy 0 ,
4 −L/2
Z
h
i
p
j L/2
∇ × Ms (y 0 )Ho2 (k| (y − y 0 )2 + z 2 |) dy 0 ,
−
4 −L/2
−
Es,elec + Es,mag
(6.43)
(6.44)
(6.45)
Plotting these radiated fields in Fig. 6.8, reveals a field that resembles the theoretical prediction fairly
well on either side of the screen with the discrepancies coming from the finite nature of the screen. On
the transmitted side of the screen a field which resembles a plane wave propagating along ϕt = 30◦ is
found. On the reflected side of the screen there is a small reflected field as predicted by Eq. (6.14) which
is a plane wave along the ϕi = 0◦ direction. Note however that the amplitude of these fields is much
smaller than the refracted fields on the other side of the screen. This confirms numerically that a surface
of electric and magnetic impedances supports almost total refraction of an incident plane wave.
6.2.2
Physical implementation
To physically implement the Huygens surface the most logical approach, as discussed in Chapter 1, is to
use a discrete array of electrically small dipoles and loops. Here, the impedance of each dipole and loop
is controlled by varying the reactive loading on each loop/dipole. Again these loops and dipoles are all
electrically small with a unit cell size less than λ/10 to allow for the homogenization of this array as an
impedance surface.
To model such a surface, Ansys’s HFSS is used. To compare the method of moments example, the
same situation is used as before at 1.5 GHz with an incident field at ϕi = 0◦ and a refracted field
at ϕt = 30◦ . Here to mimic an environment with an infinite transverse extent along the z-axis, the
simulation is placed in a parallel-plate waveguide to simplify the computation since only TE fields are
considered. Taking the impedances given in Fig. 6.8, each loop/dipole is designed to implement the
desired impedance at its location along the surface, with the loops/dipoles spaced every λ/10. This is
shown in Fig. 6.9. To illuminate the surface, a Gaussian beam is used with a 3λ focal spot designed to
occur at the surface. With this design there are a couple of caveats. First the unit cells are not optimal
and better designs are possible. This setup is used simply to demonstrate the idea. Secondly any edge
effects from using a finite surface along with a finite incident field have not been included in the analysis
and represent a non-ideality that is beyond the scope of the analysis. Nonetheless the main point can
be demonstrated here with this array of passive loops and dipoles.
The simulated results are shown in Fig. 6.9 where the Ey field plot at 1.5GHz is shown. Here the
Gaussian beam is refracted by the surface from an incident direction of ϕi = 0◦ to a refracted direction
of ϕt = 30◦ . Reflections are also minimal though any discrepancies can be attributed to the finite nature
of the screen as well as the discretization of the impedance surface. The far-field pattern of the scattered
field is also plotted in Fig. 6.9 where the main refracted beam at ϕ = 30◦ is shown. The peak at ϕ = 0◦
is the field radiated by the surface to cancel out the incident field on the transmitted side of the surface
(since the scattered field is plotted). There is also a reflected peak at ϕ = 180◦ due to the small reflected
82
Chapter 6. Refraction From A Huygens Surface
E
H
180
210
150
240
k
120
−5
270
90
5
15
300
25
330
30
0
z
x
60
y
Figure 6.9: Simulation results from HFSS for a physically realized impedance and admittance screen.
On the left a schematic of the surface. The impedance screen is made of reactively loaded dipoles with
the reactive loading shown in red. The admittance screen is made of reactively loaded loops with the
loading shown in blue. In the middle is a plot of the total vertical electric field for a simulation of the
impedance screen for an incident Gaussian beam. The plot on the right is a far-field plot of the scattered
electric field.
field that is generated by the surface and a reflected peak at ϕ = 150◦ that is not completely cancelled
out due to the imperfections in the surface. This can be improved upon further optimization of the
screen.
6.3
Circuit Models For Huygens Surfaces
One aspect of modelling a surface of scatterers that is important is the use of equivalent circuits. This
allows for further understanding of the unit cells that comprise these surfaces, as well as a way to model
the entire surface using a network. These equivalent circuits also give a simple model of a surface that
can be encapsulated by its transmission/reflection properties (S-parameters). The problem then can be
reduced from a field-theory problem to a circuit theory problem. Many equivalent circuit models have
been derived for existing surfaces such as FSS’s [95] and transmitarrays [77, 138]. With the development
of a Huygens surface which has a unique combination of a magnetic response superimposed with an
electric response, it is clear that a different circuit model is needed to model the unit cells which comprise
the surface.
6.4
Derivation
To find the equivalent circuit model for each unit cell a simple derivation follows building off of existing
circuit models. This is done by mapping the impedance boundary conditions for the electric and magnetic
responses given in (6.1)-(6.6) to their corresponding circuit. From each of these boundary conditions
it is clear that the electric boundary condition imposes a discontinuity in the magnetic field, while the
magnetic boundary condition imposes a discontinuity in the electric field. The Huygens surface, of
course, has a discontinuity in both the electric and magnetic field. This boundary condition is enforced
by each unit cell that comprises the surface as the impedance varies across the surface.
83
Chapter 6. Refraction From A Huygens Surface
+
Ze
v1
-
i1
i2
Ze
+
i1
+
v
2
i2
Zm
v1
-
v
(a)
(b)
+
Z1
i1
Z
2
-
-
Zm
+
v1
Zm
-
Ze
i2
Z
2
2
Z1
(c)
The circuit model begins with a two-port network where the electric and magnetic fields of an
incident/reflected and transmitted plane wave of the same direction are mapped to the voltages and
currents of the two-port network respectively as shown in Fig. 6.10. This circuit model assumes the
same angle or propagation for the incident, reflected and transmitted fields at each unit cell, since the
circuit model is looking at the local response of the surface . Note that keeping with the convention that
the fields incident from one side of the surface are transmitted to the other side, the current on port two
of the circuit is also leaving the port as opposed to the usual convention of having it enter the port. The
η
cos ϕ ,
where ϕ is the angle of the incident, transmitted and
reflected plane waves at each unit cell. For the examples below, it is assumed that these plane waves are
normally incident.
For the electric impedance surface it is well known that a passive sheet of these structures can be
treated as a shunt impedance as shown in Fig. 6.10a [143]. Here the surface impedance of the dipole,
Ze , corresponds to a shunt impedance in the circuit. The Z-matrix of this circuit is given to be,
"
V1
V2
#
=
"
Ze
Ze
Ze
Ze
#"
I1
−I2
#
.
(6.46)
Note the negative sign in the current due to the sign convention adopted here. This circuit model
corresponds to the boundary condition given for the electric impedance surface in (6.2). This can be
found by adding or subtracting one row to the other in the Z-matrix definition given in (6.46) which
gives,
V1 = V2 ,
(6.47)
V1 = Ze (I1 − I2 )
(6.48)
where the voltages are continuous across the shunt impedance while the current is discontinuous as
shown in (6.1)-(6.2).
For a magnetic impedance the same statement can be made simply by duality (as well as from
previous circuit models of metamaterials [144]). Here the unit cell of a magnetic impedance surface,
can be modelled as a series impedance where the series impedance itself corresponds to the surface
impedance Zm as shown in Fig. 6.10b. This circuit is described by a Y -matrix given by
"
I1
−I2
#
=
"
Ym
−Ym
−Ym
Ym
#" #
V1
V2
,
v
2
-
Figure 6.10: (a) The equivalent circuit model for a dipole in free space. (b) The equivalent circuit model
for a loop in free space. (c). The equivalent circuit for the unit cell of Huygens surface. Z1 and Z2 are
defined in the text.
impedance of each port is also set to Zo =
+
(6.49)
84
Chapter 6. Refraction From A Huygens Surface
−1
where Ym = Zm
. Here again, the circuit model maps the boundary condition given in (6.3)-(6.4) which
is found by adding and subtracting both rows together in (6.49),
I1 = I2 ,
(6.50)
I1 = Ym (V1 − V2 )
(6.51)
Again, the current is continuous across the series impedance while the voltage is not.
For the Huygens surface which is a combination of electric and magnetic surface impedances, the
equivalent circuit model for each unit cell should capture the boundary condition given in (6.5)-(6.6)
where the averaged fields at the boundary are related to the discontinuity in the fields at the boundary.
Given that electric and magnetic surface impedances map to series and shunt circuit models respectively
it would seem obvious that the combined circuit model should include both series and shunt branches.
The most obvious configuration would be some form of an L/T/Π-model. However a quick examination
of these circuits reveals that neither their elements nor their voltage and current definitions map to
those of a Huygens surface unit cell. This is due to the fact that circuit models such as an L/T/Π-model
separate the series and shunt impedances with non-physical voltage nodes or current branches existing
in the circuit model. Because a Huygens source is a superposition of electric and magnetic currents on a
surface, they exist simultaneously together on the surface and this needs to be captured by the circuit.
To model this scenario the appropriate model that is proposed here is a lattice network shown in
Fig. 6.10c [145–147]. This circuit consists of a crossed network of series and shunt impedances. Here
the lattice network allows for the series and shunt impedances to exist simultaneously in the circuit as
the analogous electric and magnetic surface currents do in a physical Huygens surface. The two-port
parameters of the lattice network can be conveniently represented by an impedance matrix,
" #
V1
V2
"
1 Z1 + Z2
=
2 Z2 − Z1
Z2 − Z1
Z1 + Z2
#"
I1
−I2
#
,
(6.52)
where Z1 and Z2 are shown in Fig. 6.10c. The circuit elements Z1 and Z2 can be mapped to Ze and Zm
by rearranging the impedance matrix in (6.52) to resemble the boundary conditions given in (6.5)-(6.6)
by adding/subtracting one row from another. This is given by
V1 + V2 = Z2 (I1 − I2 ),
(6.53)
I1 + I2 = (1/Z1 )(V1 − V2 ),
(6.54)
where it can be inferred that the impedances of the lattice network is equal to the surface impedances
if Z1 = Zm /2 and Z2 = 2Ze . From this basic relation is can be seen that a lattice network is the circuit
network that is able to represent the boundary condition that a Huygens surface represents.
6.4.1
Properties of the Lattice Network
The lattice network is an interesting network with some special properties. By carefully adjusting the
impedances of the series and shunt branches the circuit can be made either an all-pass or an all-stop
network [148]. This can be illustrated by plotting the magnitude and phase of S21 and S11 as a function
of both Z1 and Z2 where Z1 = jX1 and Z2 = jX2 , where it is assumed that Zo = η (normal incidence).
This is seen in Fig. 6.12 where different imaginary values of Z1 and Z2 allow for S11 and S21 to cover
85
Chapter 6. Refraction From A Huygens Surface
(a)
(b)
200
150
100
deg
50
0
−50
−100
−150
−200
1
0
−1
4
x 10
0.5
1
−1
−0.5
4
Im(Z1) Ω
(c)
0
Im(Z2) Ω
x 10
(d)
Figure 6.11: (a) The |S11 | for different values of Z1 and Z2 . The black line is a contour when |S21 | = 1.(b)
The |S21 | for different values of Z1 and Z2 . The black line is a contour when |S21 | = 1. (c) The ∠S21
for different values of Z1 and Z2 . The black line is a contour when |S21 | = 1. (d). The phase of ∠S21
when |S21 | = 1 only. Note that a 2π phase range is covered.
a range of values, between the circuit either totally reflecting or totally transmitting an incident wave.
For a fixed value of |S21 |, the phase of the lattice network, ∠S21 can cover a 2π phase range. This is
illustrated when |S21 | = 1 where the contours of constant amplitude are traced out in Fig. 6.11a-6.11c
and plotted on a separate scale for ∠S21 in Fig. 6.11d. It can be seen in Fig. 6.11d that for the values
of Z1 and Z2 for which |S21 | = 1, a 2π phase range is indeed covered. This is also a unique property of
the lattice network with its ability to cover a 2π transmission phase for a fixed value of |S21 |.
For the Huygens surface, this lattice cell model allows for the impedances derived in Section 6.1 to
be interpreted as a transmission screen with a linear phase shift. As described in the previous chapter
one common understanding of a surface capable of refracting a plane wave, either from a transmitarray,
blazed grating, or FSS perspective, is that of a surface with a transmission of unity and a periodic and
linear phase gradient of 2π across the surface. The total transmission and phase gradient property is a
local condition at each point on the surface, where regardless of the angle of the incident or transmitted
plane wave, the surface simply phase delays or advances the normally propagating component of the
plane wave. This can be demonstrated for the impedances given in (6.10) and (6.12) by calculating
the S-parameters for the Huygens surface using the lattice cell. However a discrepancy arises when this
transmission-screen perspective is compared to the actual transmitted and reflected amplitudes of the
plane wave found in Section 6.1.1 where the reflections off of the Huygens surface are non-zero. To
harmonize these two perspectives the S-parameters for the lattice cell are examined for different port
86
Chapter 6. Refraction From A Huygens Surface
0
150
−10
100
−15
50
Angle (deg)
Magnitude (dB)
−5
−20
−25
−30
−40
0
−100
|S21|
−35
−150
|S11|
20
40
60
Unit Cell Number
80
0
−50
100
0
20
(a)
40
60
Unit Cell Number
80
100
80
100
80
100
(b)
0
150
−100
100
−150
50
Angle (deg)
Magnitude (dB)
−50
−200
−250
−300
−400
0
−100
|S21|
−350
−150
|S11|
20
40
60
Unit Cell Number
80
0
−50
100
0
20
(c)
40
60
Unit Cell Number
(d)
0
150
−10
100
−15
50
Angle (deg)
Magnitude (dB)
−5
−20
−25
−30
|S21|
−35
−40
0
|S11|
20
40
60
Unit Cell Number
80
(e)
100
0
−50
−100
−150
0
20
40
60
Unit Cell Number
(f)
Figure 6.12: The magnitude and phase of the S-parameters of a Huygens surface at each point on the
surface found using the lattice cell. The Huygens surface is designed to refract a normally incident plane
wave to 30◦ . The S-parameters are plotted for three different port impedances. (a). The magnitude of
the S-parameters when Zo = η. (b) The ∠S21 when Zo = η. (c). The magnitude of the S-parameters
when Zo = η/ cos ϕt . (d) The ∠S21 when Zo = η/ cos ϕt . (e). The magnitude of the S-parameters when
Zo = η at port 1 and Zo = η/ cos ϕt at port 2. (f) The ∠S21 when Zo = η at port 1 and Zo = η/ cos ϕt
at port 2.
impedance terminations for the specific case when ϕi = 0◦ and ϕt = 30◦ . The first case is when Zo = η,
the second case is when Zo = η/ cos ϕt and the third case is when the impedance at port 1 is Zo = η
and the impedance at port 2 is Zo = η/ cos ϕt .
In the first example, the S-parameters are plotted in Fig. 6.12a-6.12b with Zo = η. This is an obvious
choice where each point on the surface is treated as a unit cell where the S-parameters characterize the
Chapter 6. Refraction From A Huygens Surface
87
local transmission and reflection of normally propagating plane waves. As will be shown below, this is
the most common setup especially with respect to designing a unit cell in full-wave simulation. It can be
seen that the transmission is almost unity and that the reflection varies around -20dB and below, while
the transmission phase is linear. However, while the transmission-phase is as predicted the transmission
and reflection amplitudes deviate slightly from either the transmission-screen model or the Huygens
surface model in Section 6.1.1. Clearly then a different port impedance is needed.
In the second case, the characteristic impedance is set to Zo = η/ cos ϕt . Here the model now
represents each point on the surface as a unit cell. The S-parameters characterize the locally reflected
and transmitted plane waves all travelling in the ϕt direction. A similar result is found in Fig. 6.12c-6.12d.
As in the previous example the transmission phase is linear, however the reflections have now dropped
to negligible levels with a truly unity transmission amplitude across the surface. This choice of portimpedance termination corresponds to the transmission screen model. Thus when the port impedance
of the S-parameters is set to this value, the S-parameters of the surface are truly reflectionless with a
linear phase variation. This makes sense since the lattice cell is reflectionless when Z2 Z1 = Zo2 , and it
can easily be seen from (6.10) and (6.12) that when multiplied together the result is Zo = η/ cos ϕt .
In the third case, the reflections off the Huygens surface predicted in Section 6.1.1 can be captured
by the lattice circuit model. Here the port impedance at port one is that of the input plane wave,
Zo = η in this case, and the port impedance of the second port is that of the transmitted plane wave
Zo = η/ cos ϕt . This model finds the local S-parameters at each point on the surface with incident and
reflected plane waves travelling in the ϕi direction and the transmitted plane waves in the ϕt direction.
For the same values of Z1 and Z2 as before, the S-parameters are plotted in Fig. 6.12e-6.12f. Again,
the transmission-phase is linear across the surface. The difference now is that the transmission and
reflection amplitudes are at a constant amplitude that is consistent with the transmission and reflection
coefficients derived in (6.13)-(6.14). This makes sense intuitively given the port impedances. With these
port impedances the lattice network does indeed model the reflection and transmission as derived in
Section 6.1.13 .
What this shows is that the lattice network does capture the transmission/phase perspective and
the perspective in Section 6.1.1, but that it is dependant on the port impedances used to terminate
the network. It should be noted that the port impedances do not affect the values of Z1 and Z2 in
the network itself but the derived S-parameters. Thus the most common approach is to set the port
impedance to Zo = η at both ports as this still allows for the impedance of the surface to be verified
and is more amenable to utilizing the lattice network in conjunction with full-wave analysis as will be
shown below.
Finally it is important to note that this circuit model also helps to differentiate the topology of the
Huygens surface from other metasurfaces or transmitarrays. For example a three-layer transmitarray,
each characterized by an electric surface impedance as in [138], gives the necessary degrees of freedom
required to minimize reflections, maximize transmission and alter the phase of the surface. This structure
has an equivalent circuit of three shunt impedances, each spaced by some distance d along a host
transmission-line. Here it is recognized how a combination of electric and magnetic dipoles can be used
to accomplish the same functionality in a single layer.
3 This
observation arose in discussion with Min Seok Kim and is credited to him.
88
Chapter 6. Refraction From A Huygens Surface
5000
Zm
Impedance Ω
Ze
Einc
k
0
Periodic
Boundaries
Hinc
−5000
0
50
100
150
200
x (Cell Number)
(a)
(b)
0
500
Circuit Model (S21)
Circuit Model (S11)
Full−Wave (S21)
Full−Wave (S11)
60
−15
Impedance (Ω)
−10
0
40
Angle (deg)
Magnitude (dB)
−5
Circuit Model (S21)
Full−Wave (S21)
80
20
0
−20
−500
−1000
Zm (Ideal)
−40
−20
Ze (Ideal)
−1500
−60
Zm (Full−wave)
Ze (Full−wave)
−80
−25
1.2
1.25
1.3
1.35
1.4
Freq (GHz)
(c)
1.2
1.25
1.3
Freq (GHz)
(d)
1.35
1.4
−2000
1.2
1.25
1.3
1.35
1.4
Freq (GHz)
(e)
Figure 6.13: (a) The required screen impedance for the electric and magnetic dipoles to refract a plane
wave from 20◦ to −30◦ . The black dots denote the impedance of the unit cell at a point on the screen
for which the corresponding S-parameters are shown. (b) A reactively loaded loop and dipole unit
cell in HFSS. The unit cell is surrounded with periodic boundaries to simulate the transmission and
reflection of a normally incident plane wave on an infinite array of unit cells. (c) The magnitude of
the S-parameters found from full-wave simulation and the lattice circuit model. (d). The phase of the
transmission through the unit cell. Note that this phase corresponds to the phase shift between incident
and refracted plane waves at that point on the screen. (e) The impedances of the unit cell found from
the lattice unit cell and the full-wave model.
6.4.2
Modelling a Unit Cell
As an example, an individual unit cell can be examined which comprises a Huygens surface to see how
this circuit model fits the unit cell itself. The Huygens surface is designed to refract a plane wave at
1.3 GHz from an angle of 20◦ to −30◦ . The required impedances for the electric and magnetic dipoles
are shown in Fig. 6.13a. Picking a point on the screen as shown in Fig. 6.13a, a unit cell consisting
of an electric and a magnetic dipole, shown in Fig. 6.13b is designed to implement an impedance of
Ze = −j719Ω and Zm = j263Ω. This design is carried out using Ansys’s HFSS. The individual cell can
be characterized by S-parameters for a normally incident and transmitted plane wave where the unit
cell is contained by periodic boundaries in the full-wave model as also shown in Fig. 6.13b. These Sparameters as well as the extracted impedances using (6.52) can be compared to the ideal S-parameters
and impedances of the equivalent lattice network. This is shown in Fig. 6.13 where the S-parameters of
the circuit model are overlaid with the full-wave S-parameters of the unit cell. Here, a good agreement
can be seen between both sets of data. The extracted impedances from the full-wave data using the
Chapter 6. Refraction From A Huygens Surface
89
lattice network topology are also shown in Fig. 6.13e and agree well with the desired impedances for
this unit cell. Note that in this ideal simulation loss has been neglected. However to include and model
losses, resistors can be added to the series and shunt branches if necessary.
With regards to the frequency dependence of the model, the Huygens surface is designed for a single
frequency. With this in mind, note that for the lattice circuit model the impedances have then been
treated as simple inductors or capacitors depending on the sign of the impedance. This choice is justified
by Fig. 6.13e where the dispersion of the extracted impedances of the loop and dipole matches well with
the inductor/capacitor assumption. However, note that the physical structure of the unit cell, consisting
of dipoles and loops, inherently has a more involved dispersion. A more complicated (or broadband)
circuit model would take that into account by modelling the series and shunt branches of the lattice
network with an appropriate configuration of LC-resonators.
6.5
Modelling a Huygens Surface
As a final demonstration of this circuit model, the lattice network can be used to model an entire Huygens
surface by using an array of lattice network unit cells. To demonstrate this, a two-dimensional circuit simulator developed in [149] which is based on a hybrid frequency-domain transmission-line method/method
of moments solver is used. Here free space is discretized as a two-dimensional transmission-line network
(a shunt-node network for the TE-polarization) while the boundaries of the two-dimensional domain are
handled using a method of moments formulation. For more details on the solver refer to Appendix C.
In this transmission-line model of free space the Huygens surface as a 1D-array of lattice unit cells at
an interface as shown in Fig. 6.14a.
Using the same design discussed above in Section 6.4.2 where an incident plane wave at 20◦ refracts
to an angle of −30◦ , the two-dimensional circuit model is simulated at 1.3 GHz with the impedances of
the lattice network as shown in Fig. 6.13a. The incident field is represented as a Gaussian beam.
The simulated transmission-line grid is shown in Fig. 6.14b and Fig. 6.14c where both the real part
of the voltage on the transmission-line grid (or electric field) as well as the far-field are plotted. In
both plots the incident Gaussian beam refracting from 20◦ to -30◦ is demonstrated. It is also observed
that modelling the Huygens surface as an array of lattice circuit captures the desired behaviour of the
Huygens surface itself. Such circuit models can be useful for rapid modelling and study of Huygens
surface designs without resorting to full-wave simulations.
6.6
A PCB Design At Microwave Frequencies
The design in Section 6.2.2 was used to demonstrate the concept of a Huygens surface using an array
of discrete wires and loops loaded reactively to implement electric and magnetic surface impedances.
However, implementing this design using a layout that is amenable to PCB/lithographic approaches in
the microwave/millimetre-wave frequency range is key to actually realizing this concept. This problem
was addressed by my colleague Joseph Wong as part of his Master’s thesis [150, 151] and the main results
are briefly summarized here.
An example of the kinds of unit cells that implement a superimposed electric and magnetic surface
impedance in a lithographic friendly process are shown in Fig. 6.15a. Here, a PCB with three metal
layers is used, with the top and bottom layers being used to define an electrically small loop connected
90
Chapter 6. Refraction From A Huygens Surface
y
x
z
Lattice cell model
of a Huygens surface
(a)
0o
0 dB
−30o
30o
−8
−16
−60o
60o
−24
−32
−90o
90o
−32
−24
−120o
120o
−16
−8
−150o
150o
0 dB
180o
(b)
(c)
Figure 6.14: (a) Circuit model to simulate a Huygens surface in two-dimensions. The lattice unit cells
form the surface itself (b). The real part of the voltage in the TL grid. (c) The far-fields of the simulated
voltages (absolute value of the voltage).
by plated vias, and the middle layer being used to pattern a dipole. Printed inductors and capacitors
are used to define the reactive loading as well. These unit cells are designed for X-band frequencies
with the total thickness being λ/10 and the unit cell spacing also λ/10. This subwavelength design
allows for a compact surface to be fabricated as well as for the homogenization assumption to hold. The
lattice cell model was used in conjunction with full-wave simulation to extract and verify the electric and
magnetic impedance of each unit cell as also shown in Fig. 6.15b. These unit cells are then patterned on
an entire board to form a surface capable of refracting an incident Gaussian Beam in two-dimensions.
A fabricated sample and measured results are shown in Fig. 6.15c-6.15d demonstrating a experimental
implementation of the concepts introduced above.
91
Chapter 6. Refraction From A Huygens Surface
(a)
(c)
(b)
(d)
Figure 6.15: Example of a printed Huygens surface designed at X-band to refract a plane wave. All
c
images taken from [151], 2015
IEEE. (a) The layout of a unit cell comprising the Huygens surface.
A three-layer PCB where the top and bottom layer form a magnetic dipole (loop) using vias. The
middle layer acts like an electric dipole. (b) The theoretical and extracted impedances for one-period.
The extracted impedances are found using the lattice cell circuit model. (c) A picture of the fabricated
board. (d) The measured refracted beam from the surface in the far-field. A normally incident Gaussian
beam is used to illuminate the fabricated Huygens surface. The transmitted beam is refracted at −22◦
in the elevation plane and at 30◦ in the azimuthal plane.
6.7
Summary
In this chapter, the concept of a passive Huygens surface was introduced, characterized by an electric and
magnetic surface impedance. This surface was shown to be able to refract an incident plane wave with
very small reflections and the amplitude of the reflected and refracted fields as well as the impedance of
the surface was analytically derived. Unlike a grating, this surface is capable of refracting a plane wave
into one Floquet mode only, thanks to the presence of both magnetic and electric scatterers. Equivalent
circuit models were developed based on lattice networks.
This passive Huygens surface shows how both an electric and a magnetic response can be used to
tailor a single surface to completely control plane wave propagation. The Huygens property inherently
allows for the fields to be scattered into the forward direction only and by tailoring the impedance of each
Chapter 6. Refraction From A Huygens Surface
92
scatterer into a single plane wave as well. This is in contrast to both the FSS and transmitarray designs
which require multiple layers or gratings with multiple diffraction orders or large reflections. With the
ability to bend a plane wave, the ideas presented here can be used to construct more complicated devices
such as a thin, planar printed lenses. However, this is left for future work and is outside the scope of
this thesis.
As this idea was being developed as part of this thesis, similar work was published around the same
time, also proposing a surface of electric and magnetic dipoles to refract a plane wave [152]. The reader
is encouraged to refer to this work as well since it presents a different perspective to the ideas discussed
here. Their approach is to take the homogenized surface polarizabilities and use the corresponding
reflection and transmission coefficients to find the desired surface impedances for the Huygens surface.
With a scalar impedance quantified, the next challenge is to examine how these superimposed electric
and magnetic boundary conditions behave when the impedance is a tensor quantity and the effect of
this on the polarization of the incident field.
Part III
Polarization Control With A
Huygens Surface
93
CHAPTER
7
Background On Polarization
In this final review chapter, various concepts related to polarization are addressed to give the reader an
overview of how the polarization state of the electromagnetic field is altered using microwave and optical
devices. Polarization is one of the defining characteristics of the electromagnetic field. The ability to
control the polarization of the electromagnetic field is key to many applications such as sensing and
communication links. For example, in radio astronomy where polarization sensitive detectors are used
to measure radio signals from the universe, the imaging equipment must be carefully designed to sense
different polarizations.
Another important example is in satellite communication links where controlling the polarization of
radio waves and microwaves as they travel through the ionosphere and into outer space is crucial to
maintaining a viable communication link. This requires designing antennas very carefully to transmit
and recieve the desired polarization.
Of course, moving higher up to millimetre, terahertz and optical frequencies, polarization continues
to play an important role in communication and imaging systems at these bands as well.
From these examples, it is clear that polarization control is a fundamental aspect of designing microwave (and photonic) systems. To start a discussion on polarization control, the use of materials to
alter polarization is investigated. This is because understanding how materials affect the polarization
of electromagnetic waves helps frame the problem of engineering polarization controlling systems. From
there, a brief review of a variety of polarization controlling devices across the electromagnetic spectrum
is reviewed.
7.1
Material Parameters Which Alter Polarization
One of the most fundamental ways in which the polarization state of the electromagnetic field is controlled
is through the use of materials. The materials in question have unique constitutive parameters which
affect the propagating of polarized waves differently. There are two specific kinds of constitutive relations
which can be understood to affect the polarization of electromagnetic fields. These are anisotropy and
94
95
Chapter 7. Background On Polarization
chirality.
7.1.1
Anisotropy
Anisotropic materials are materials whose permitivitty and/or permeability are described by tensors as
opposed to scalars. The constitutive relations for such materials are generally given by,
  
εxx 0
Dx
  
Dy  =  0 εyy
0
0
Dz
  
Bx
µxx
0
  
µyy
 By  =  0
Bz
0
0
 
Ex
 
0  Ey  ,
Ez
εzz
 
0
Hx
 
0  Hy  .
0
µzz
(7.1)
(7.2)
Hz
The assumption has been made that these tensors are aligned with the underlying coordinate system
and are thus diagonal. To see how an anisotropic permitivitty can be used to alter the polarization of
a propagating wave, a slab of such a material is analyzed. Here a couple of assumptions are made to
simplify the analysis. First it is assumed that the permeability tensor is chosen, such that the material
is impedance matched. Thus from here on only the permitivitty tensor will be discussed. The second
assumption is that this diagonal permittivity tensor is uniaxial meaning only one of the diagonal entries
is different from the others [153], resulting in the following material parameter defintion

εT M

¯
ε̄ =  0
0
0
εT M
0
0


0 
(7.3)
εT E
where the T E and T M suffixes have been used the describe their effect on the corresponding TE and
TM modes with regards to 2-D wave propagation defined in Chapter 1. This permitivitty tensor is used
to describe a slab of material with length d, a simple scenario for which the effect of a normally incident
plane wave on the slab is examined as shown in Fig. 7.1. First, the polarization of this plane wave must
be defined. For 2D wave propagation, the polarization of the incident field can be expressed as a sum of
TE and TM components,
Einc = EoT M e−jko x ŷ + EoT E e−jko x ẑ.
(7.4)
These TE and TM components can be understood as two orthogonal linear polarizations, which subsequently form a basis for all possible polarization states [153]. The amplitudes EoT M and EoT E are
complex weights which define the polarization state of the incident field.
With this normally-incident plane wave defining the incident polarization state, the propagation of
the plane wave through the uniaxial slab is now discussed, with a focus on how the polarization state
changes. Because of the orientation of the anisotropic slab, the problem can be decomposed into finding
transmission of both normally incident TE and TM waves through the slab. To do this requires defining
reflected, transmitted and interior electric and magnetic fields for both TE and TM fields. For the TE
96
Chapter 7. Background On Polarization
H
E
E
TM
H
d
k
TE
k

µT E
 0
0


εT M
0
0
0
µT E
0
0
0
µT M



0
0 
0
εT M
0
εT E
z
y
x
Figure 7.1: An anisotropic slab demonstrating linear to circular polarization conversion.
case, the reflected and transmitted fields are defined to be,
Er T E = ErT E ejko x ẑ,
(7.5)
EtT E e−jko x ẑ,
(7.6)
Et
TE
=
while the fields inside the slab are described by a forward and backward propagating TE-polarized plane
waves given to be,
Ea T E = EaT E e−jkT E x ẑ,
(7.7)
Eb T E = EbT E ejkT E x ẑ,
(7.8)
√
where kT E = ko εT E µT E . Likewise, for the TM case, the fields are,
Er T M = ErT M ejko x ẑ,
(7.9)
EtT M e−jko x ẑ,
(7.10)
= EaT M e−jkT M x ẑ,
(7.11)
Et
Ea
TM
TM
Eb
TM
=
=
EbT M ejkT M x ẑ,
(7.12)
√
where kT M = ko εT M µT M . To solve for the coefficients of the TE and TM planes waves the continuity
of the electric and magnetic fields is enforced resulting in a system of equations. For the TE case, the
resulting equations are given to be,

−1
 1
− η

 0

0
e
1
1
1
η1
−jkT E d
1
η1
jkT E d
e−jkT E d
η1
e
ejkT E d
η1

 

EoT E
ErT E
  T E   EoT E 
 Ea  

0

= η 




−e−jko d  EbT E   0 

−jko d
−e η
EtT E
0
0
(7.13)
Chapter 7. Background On Polarization
97
Figure 7.2: An anisotropic slab demonstrating linear to circualr polarization conversion.
A similar system of equations can be constructed for the TM case. The key insight from this procedure
is that the transmitted coefficients for the TE and TM fields, EtT E and EtT M have a different transmitted
phase, where the transmitted amplitude is unity for both TE and TM components if the anisotropic slab
is matched as is assumed. This transmitted phase is different because the TE and TM waves inside the
anisotropic medium each propagate with a different phase velocity as determined by kT E and kT M . This
phase difference between the TE and TM fields inside the slab accumulates over the length of the slab
and results in the phase difference between the transmitted TE and TM fields. This phase difference
then results in a different polarization state of the transmitted field with respect to the incident field as
seen from the polarization state defined in (7.4).
To demonstrate how this material anisotropy can convert polarization, a simple example is constructed to convert a 45◦ slanted linear polarization to circular polarization. Such an anisotropic material is referred to as a quarter-wave plate in photonics[153]. In this specific case, the incident field has a
frequency of 10 GHz and the anisotropic slab has a length of 3λ. The permitivitty and permeability of
the slab along the TE and TM axes, are defined to be εT E = µT E = 2 and εT M = µT M = 3.048. These
specific material parameters are chosen because they introduce a 90◦ phase shift between the TE and
TM components. Note again that the permeability is used to keep the slab impedance matched for both
TE and TM fields, resulting in a transmission amplitude of unity for both TE and TM waves. Since
the incident 45◦ slanted linear polarization is of equal weight with respect the TE and TM components,
the resulting transmitted field is circularly polarized because of the 90◦ phase shift. The propagating
plane waves in this scenario are plotted in Fig. 7.2 where the incident linear polarization is decomposed
into TE and TM fields which propagate with different phase accumulation in the anisotropic slab and
emerge together as a circularly polarized field.
From (7.4) it can be seen that the polarization state can be altered not only by the phase difference
between the TE and TM components but also by their relative amplitude. This can also be controlled
if the orientation of the anisotropic slab is rotated and does not align with the underlying coordinate
system. This is beyond the scope of the discussion here. However this shows that to arbitrarily convert
a polarization state, both the relative amplitude and relative phase of the TE and TM waves need to be
altered and for an anisotropic slab both its orientation and anisotropy are the key variables.
In this example it should be noted that the material parameters, while not extreme (in the sense
98
Chapter 7. Background On Polarization
that they are large, or negative or close to zero), are very difficult to achieve at microwave frequencies
and require a slab with an appreciable thickness. At optical frequencies the materials only have an
anisotropic permittivity with a very slight difference between the two optical axes, which requires a slab
that is many wavelengths long. However, the much smaller wavelength at optical frequencies allows this
to be of lesser concern. Thus while the example is illustrative of how polarization conversion functions in
an anisotropic material it is not a practical example at microwave frequencies. As will be shown below
and in the following chapter, different approaches are required at microwave (and terahertz) frequencies
to implement polarization conversion.
7.1.2
Chirality
While polarization conversion is one interesting property that arises due to the anisotropy of a material, another class of polarization effects arise from chirality. To discuss this, the effect of plane wave
transmission and reflection through a chiral slab is examined here.
A bi-isotropic material has scalar material parameters which relate the electric and magnetic flux
densities to the electric and magnetic fields,
D = εo E + ξH,
(7.14)
B = µo H + ζE
(7.15)
where ξ and ζ are the bi-isotropic material parameters along with the usual permittivity and permeability
parameters. In its most general form ξ and ζ can also be tensors and such a material would then be
bianisotropic [82] but this is also outside the scope of discussion. It can be seen that the material
parameters ξ and ζ introduce a cross coupling between the electric and magnetic fields. In general, ξ
and ζ are complex which include both chiral and non-reciprocal effects [154]. In this discussion, nonreciprocity is beyond the scope of discussion. By forcing ξ and ζ to be imaginary, where ξ = jκ and
ζ = −jκ only chiral effects can be considered and the constitutive relations can be rewritten to be
D = ε0 E + ε0 β∇ × E,
(7.16)
B = µ H + µ β∇ × H
(7.17)
0
0
κ
0
2
where β = − ko (1−κ
2 ) is the chirality parameter and has the units of meters and ε = εo (1 − κ ) and
µ0 = µo (1 − κ2 ).
For a chiral material with the constitutive relations given by (7.16)-(7.17), it can easily be shown
that the eigenmodes of the slab are not linearly polarized plane waves but circularly polarized plane
waves [82]. Thus to understand transmission and reflection of normally incident plane waves through a
chiral material, the polarization basis must be with regards to right-handed and left-handed circularly
polarized fields (as opposed to linearly polarized TE and TM fields in the previous section).
As in the previous section, the discussion here focuses on a slab of chiral material of thickness d with
the goal of finding the transmission of a normally incident plane wave through the slab. There are two
scenarios to consider, when the incident field on the slab is right-handed circular polarization (RHCP)
and when the incident field is left-handed circular polarization (LHCP). The two cases are illustrated
in Fig. 7.3 where it can be seen that a normally incident RHCP wave transmits a RCHP wave, reflects
a LHCP wave and supports a forward RHCP and backward LHCP wave in the slab. For the incident
99
Chapter 7. Background On Polarization
d
d
Er ,LCP
Er ,RCP
Ei ,RCP
Ei ,LCP
Eb ,RCP
Eb ,LCP
Ea ,LCP
Ea ,RCP
z
z
y
y
Et ,RCP
x
Et ,LCP
x
(a)
(b)
Figure 7.3: (a) The fields when a chiral slab is illuminated by a normally incident RHCP wave. (b) The
fields when a chiral slab is illuminated by a normally incident LHCP wave.
LHCP wave, the opposite is true [155].
Using the coordinate system defined in Fig. 7.3, the fields for the incident RHCP case are defined to
be,
R −jko z
ER
(ŷ + jẑ),
i = Ei e
(7.18)
R +jko z
ER
(ŷ + jẑ),
r = Er e
(7.19)
EaR e−jkR z (ŷ
+ jẑ),
(7.20)
R jkL z
ER
(ŷ + jẑ),
b = Ei e
(7.21)
ER
a
ER
t
=
=
EtR e−jko z (ŷ
+ jẑ),
(7.22)
L −jko z
EL
(ŷ − jẑ),
i = Ei e
(7.23)
− jẑ),
(7.24)
while the fields for the incident LHCP case are given by,
EL
r
=
ErL e+jko z (ŷ
L −jkL z
EL
(ŷ − jẑ),
a = Ea e
(7.25)
− jẑ),
(7.26)
EL
b
=
EiL ejkR z (ŷ
L −jko z
EL
jŷ − jẑ),
t = Et e
(7.27)
and kL and kR are the propagation constants of the left-handed and right-handed circularly polarized
waves inside the chiral slab and are defined to be,
ko
,
1 + ko β
ko
kR =
.
1 − kβ
(7.28)
kL =
It should also be noted that if the chiral slab is lossless, the chirality parameter β must be
(7.29)
−1
ko
<β<
1
ko .
To find the coefficients of the reflected, transmitted and interior fields inside the chiral slab a system of
equations can be setup, similar to (7.13), where the equations are found by enforcing the continuity of the
100
Chapter 7. Background On Polarization
(a)
(b)
(c)
Figure 7.4: A chiral slab demonstrating 90◦ polarization rotation. (a) TE to TM rotation (b) TM to
TE rotation (c) 45◦ slanted linear to 135◦ slanted linear.
electric and magnetic fields [155]. Again it can be seen from the fields defined in (7.28) and (7.29) that
the circularly polarized waves inside the chiral slab have different propagation constants. This allows
the polarization of an arbitrarily polarized incident field to be altered. For a chiral slab, a very specific
polarization effect is associated with this, referred to as polarization rotation [156]. Here, any linearly
polarized wave incident on the chiral slab, is rotated by the same degree due to the different propagation
constants of the RHCP and LHCP waves. Because RHCP and LHCP states also form a basis for all
polarization states, an incident linear polarization is decomposed into equal amplitude RHCP and LHCP
modes with different phases when entering the chiral slab. Assuming that the transmission amplitudes
of each CP mode are equal, the different phase then accumulated by the RHPC and LHCP modes
propagating through the chiral slab causes a linear polarization of a different orientation to emerge.
This can be seen as rotating the entire plane of polarization.
To illustrate this polarization rotation effect the fields inside a chiral slab are solved for. The parameters of the slab are d = 3λ, ε = εo , µ = µo and β = 0.3621/ko at 10 GHz which are chosen so that
the transmission of the RHCP and LHCP modes are unity and that the polarization rotation is 90◦ .
Three different cases are examined, a TE polarized normally incident plane wave, a TM polarized plane
wave and a 45◦ slanted plane wave. Each linearly polarized wave is decomposed into RHCP and LHCP
fields and the transmission is solved for. The output polarization is then constructed from the weights
of the transmitted LHCP and RCHP waves. The results are plotted in Fig. 7.4 for each of the three
different cases. The 90◦ polarization rotation can be seen where a TE-polarized plane wave becomes
TM-polarized, a TM-polarized plane wave becomes TE-polarized and a 45◦ -polarized plane wave becomes slanted along a 135◦ axis. Thus any linear polarization incident on this chiral slab will be rotated
by 90◦ . This is one of the defining characteristics of a chiral material and is a distinct phenomenon when
compared to the anisotropic materials in the previous section.
Again the material parameters discussed here are not very feasible at microwave or optical frequencies
using a natural material. Another limitation is the size of the material which requires an appreciable
thickness to achieve any sort of polarization rotation. But the example is illustrative of the interesting
Chapter 7. Background On Polarization
101
properties that chiral materials posses. Another interesting polarization effect associated with chiral
materials is circular dichroism [156]. This effect involves one hand of circular polarization being absorbed
more than the other hand, resulting in a chiral slab acting like a circular polarized filter. An analysis of
this behaviour involves a lossy chiral slab and is outside the scope of this discussion, but this is another
example which requires the circularly polarized eigenmodes of a chiral slab to be realized.
7.1.3
Comments
From Sections 7.1.1 and 7.1.2 it is clear that material properties can have a major effect on the polarization state of a propagating electromagnetic wave. From an engineering point of view, how these
effects can been harnessed to construct devices which perform precise polarization control is the relevant
question. From the discussions above some major drawbacks when using these material slabs have already been highlighted. It is also clear that reflection from the material slab has been minimized using
fictitious materials and may be hard to realize. Moreover the only real degrees of freedom to engineer
a response are the material parameters and the thickness of the material slab. The choice of material
parameters are what can be found in nature and for more exotic effects like chirality there can be limited
choices. For the thickness, the major drawback is if the material response is weak then electrically thick
structures are required.
As said, at microwave, millimetre and terahertz, frequencies, unlike optical frequencies, material
responses are not so easily engineered due to the lack of materials available. Even at optical frequencies,
more integrated solutions are sometimes desired as opposed to discrete material slabs. This of course
has led to research on artificial structures which implement the polarization effects discussed here. As
will be discussed below, there is a variety of ways to implementing these kinds of polarization control,
but the underlying physics discussed here gives insight into the desired properties of these devices.
7.2
Engineered Structures Which Alter Polarization
Because the focus of this thesis is on Huygens surfaces, a review of other kinds of engineered structures
which alter the polarization state of an electromagnetic wave are discussed here. Given the discussion in
Chapter 5, the work reviewed in this section covers many of the same classes of devices such as gratings,
FSSs, transmitarrays and metasurfaces, except in the context of polarization control.
7.2.1
Gratings
Along with diffracting electromagnetic radiation, gratings can also be engineered to alter the polarization of the electromagnetic field. To accomplish polarization conversion however, the gratings are
engineered differently than the stereotypical slitted screen structure or the blazed grating structures discussed in Chapter 5. Instead, different fabrication technologies are used to essentially make the grating
anisotropic to two orthogonal linear polarizations, allowing for polarization conversion to occur when
a phase difference is imposed between the two orthogonal polarizations by the grating. This includes
liquid crystals [157] and patterned dielectrics [158]. Also, depending on the size of the grating unit cell,
the polarized beam may stay in the 0th diffraction order, if the grating is subwavelength, or also diffract
into a higher order mode depending on the desired behaviour. However, these are all techniques that
Chapter 7. Background On Polarization
102
c
Figure 7.5: A schematic of a meander-line polarizer. Taken from [161], 2012
IEEE.
are more applicable in the optical frequency range and are not very relevant to microwaves and thus are
not discussed here any further.
7.2.2
Frequency Selective Surfaces And Reflectarrays/Transmitarrays
While gratings can be designed to alter the polarization of electromagnetic fields, the more common
approach at microwave frequencies is to use frequency selective surfaces. At microwave frequencies,
there are two commonly used frequency selective surfaces which can affect the polarization state. They
are the linear polarizer and the meander-line polarizer.
Linear polarizers are a relatively simple device which consist of long metal strips which are used to
filter one linear polarization while letting the orthogonal linear polarization pass. The linear polarization
that is filtered is the one whose electric field component is aligned with the metal strips. Examples of
linear polarizers can be found commercially in [159] and can be used to improve the polarization purity
of a feed. Linear polarizers can also be applied in reflectarrays as well [160].
At microwave frequencies, anisotropic materials which can convert polarization are not easily found
and would be prohibitively thick compared to the anisotropic crystals used at optical frequencies. To
get around this, FSSs have been used to create artificial polarization converters. For example, a linear
to circular polarization converter works by introducing a 90◦ phase shift respectively between the TE
and TM components as discussed above. An FSS can be designed to do the same provided that it is
made artificially anisotropic using a design called a meander-line polarizer [162, 163]. The approach uses
a meander-line pattern printed on a thin dielectric which has an inductive branch due to the meanderline structure and an orthogonal capacitive branch between each meandered-line as shown in Fig. 7.5.
Intuitively, there is a phase difference from the inductive and capacitive branches for the fields aligned
along the inductive branch and the fields aligned along the capacitive branch. Since each layer partially
reflects and transmits an incident field, three or more layers of these meander-line patterns are required
so that the reflections from the FSS are minimized and the transmission is maximized. The spacing
between layers is typically on the order of λ/4. Much work has gone into the modelling, computation
and fabrication of meander-line polarizers for linear-to-circular polarization conversion as they are useful
for communication systems where circular polarization is required such as in satellites. Examples of a
variety of meander-line polarizers can be found in [161, 164–166]. These designs can also be used to
Chapter 7. Background On Polarization
103
c
Figure 7.6: A fabricated transmitarray consisting of rotated patches Taken from[168], 2011
IEEE.
convert TE-to-TM fields as well [167].
The other common way of controlling the polarization of microwaves is using transmitarrays and
reflectarrays. As previously discussed, transmitarrays and reflectarrays can be designed to beam-shape
an incident field by providing a phase-shift across the surface of the array, however they can be carefully
designed so that they can also convert the polarization of the incident field. One common design
technique for implementing polarization controlling transmitarrays is to use rotated elements [168, 169].
By rotating the elements, a different phase shift is assigned to TE and TM fields allowing for a linear
polarization to be converted into a CP polarization as well as creating a directive beam as discussed in
[168] and shown in Fig. 7.6. Another way of implementing a polarization controlling transmitarray is to
use impedance surfaces [170]. Many of these techniques also apply to reflectors as well and can be used
to engineer reflectors that can control polarization [171–173].
7.2.3
Circular Polarization Selective Surfaces
The FSSs and transmitarrays discussed above were mainly designed to alter the polarization state.
However being able to control the transmission and reflection of circularly polarized waves is also key
for satellite applications, as shown in Fig. 7.7a. To do this a class of devices, referred to circularly
polarization selective surfaces (CPSSs) are used. Circularly polarized selective surfaces are devices which
are designed to reflect one-hand of circular polarization while transmitting the other. There are many
kinds of circular-polarization selective surfaces including asymmetric designs (which change the hand of
the transmitted and/or reflected CP field) and symmetric designs which don’t [177]. Here symmetric
designs are focused on which preserve the handedness of the transmitted and reflected CP fields.
CPSSs can be designed in a variety of ways that can be reduced to two main topologies. The first is
to use meander-line polarizers. Here, two linear-to circular polarization converters made using meanderline polarizers are stacked together, separated by a linear polarizer. By properly rotating each of the
polarizers with respect to the other, the resultant transmission and reflection coefficients are those of a
CPSS. This is shown in Fig. 7.7b. A variety of different implementations of this kind of CPSS exist in
the literature [175, 177, 178]. These kinds of CPSSs can be made quite broadband as in [175], but are
also quite electrically thick due to the number of layers and the spacing between layers as well.
The other kind of CPSS are referred to as ‘Pierrot-type’ cells and consist of two orthogonal monopoles
104
Chapter 7. Background On Polarization
(a)
(b)
(c)
(d)
Figure 7.7: (a) CPSSs can be used to broadcast LHCP and RHCP waves from the same aperture as
shown here, allowing for two different broadcast regions to be covered by the same antenna. Taken from
c
c
[174], 2012
IEEE. (b) A CPSS fabricated using meander-line polarizers. Taken from [175], 2014
c
IEEE. (c) A Pierrot-type cell. Taken from [174], 2012 IEEE. (d) A Pierrot-type cell fabricated on a
c
folded substrate. Taken from [176], 2014
IEEE.
separated by a quarter-wavelength transmission-line shown in Fig. 7.7c [176, 179]. Again by analysing
the transmission and reflection from this array of dipoles, transmission of one-hand of CP is achieved
while the other is reflected. A variation of this design also includes the Tilston cell where dipoles are
used instead, separated by a half-wavelength long transmission-line [180]. While these are vary simple
structures, the challenge with them is the fact that a section of transmission-line is required in the
direction of propagation, resulting in electrically long vias which can be challenging to fabricate. Thus
much of the work on realizing these kinds of CPSSs is finding fabrication methods which can reliably
realize this kinds of designs in printed cells as shown in [174, 176, 181]. For example in [176], Pierrot
cells are fabricated using a novel folded substrate which eliminates the use of vias as shown in Fig. 7.7d.
7.2.4
Metasurfaces And Metamaterials
One of the main thrusts of research into metamaterials involved the realization of bianisotropic materials.
In fact, this research historically preceded much of the current research in metamaterials and metasurfaces, and was in fact a key building block for later research into negative-index metamaterials. Some of
this earlier research investigated how chirality could be implemented using artificial structures. This included research into arrays of spirals, and Ω-like inclusions, as shown in Fig. 7.8a [23, 155, 156, 185, 186].
These Ω particles introduce a chiral response because the magnetic response of the loop couples to the
105
Chapter 7. Background On Polarization
(a)
(b)
(c)
(d)
Figure 7.8: (a) An Ω-partice unit cell. One of the first examples of a chiral metamaterial. Taken
c
from [156], 1988
IEEE. (b) A more sophisticaed example of a chiral metamaterial consisting of a
3D-fabricated spiral. Taken from [182] (c) A bi-layer surface with chiral properties [183] (d) A stack of
rotated dipoles which act like a chiral structure. Each dipole is described by a (tensor) surface impedance
[184].
electric response of the ‘tail’ of the Ω structure. This idea of artificially inducing chirality has been a
driving force in research into metamaterials
Of course one of the challenges in realizing artificial chiral materials is that they are difficult to
analyze, characterize and fabricate. Because of these challenges, realizing a ‘useful’ chiral material,
(one that could control CP wave propagation), has proved difficult. Nonetheless, research into chiral
metamaterials has continued [28, 187–189]. These include fabricating three-dimensional spirals as shown
in Fig. 7.8b from [182] which provide different transmission amplitudes for LHCP and RHCP fields at
optical frequencies due to the handedness of the spiral.
Despite the ability to analyze and realize bulk chiral metamaterials, research into simpler structures
that don’t require 3-D patterning like spirals or Ω shaped inclusions has been investigated. This led to
work on bi-layer surfaces to realize chiral metamaterials and metasurfaces. These are surface where the
pattern on one side of the surface is rotated with respect to the pattern on the other side of the surface.
One of the first works to realize this idea was in [183] where the design shown in Fig. 7.8c realized
a chiral response at microwave frequencies. This led to research into the properties of these kinds of
metasurfaces including how the chirality is affected by oblique incidence [190, 191] and the realization
of polarization rotators [192, 193]. However no complete analysis of these kinds of structures exists to
explain the electromagnetic wave propagation through these structures.
Chapter 7. Background On Polarization
106
One attempt to further refine this idea and to understand the wave propagation through this structure
was to model each layer as an impedance surface that is rotated with respect to each other [143, 194]
and shown in Fig. 7.8d. This led to the design of a circular polarizer which transmits one hand of CP
only [184]. However, while the work in [184], is one of the first to introduce the idea of rotated tensor
impedance surface as well as a circuit model, it is still an open question as to how all the properties of
the impedance tensor can be used to design a chiral structure. This problem is completely addressed in
Chapter 9.
This idea of using impedance surfaces can also be used to design polarization converters. In these
examples, the impedance surfaces are not rotated with respect to each other but are simply anisotropic.
This was demonstrated in a variety of designs ranging from microwave to optical frequencies [17, 195–
198]. These designs generally used a stack of impedance surfaces, usually containing an electric response
only, to eliminate reflections and maximize transmission. As will finally be discussed in the next chapter,
by using an electric and magnetic response these designs can be reudced to a single Huygens surface.
The final type of surface worth noting are bianisotropic surfaces, made up of surfaces of coupled
electric and magnetic dipoles [108, 199, 200]. These kinds of designs also realize chiral responses and
have a complete theoretical framework describing the electric and magnetic dipole moments and their
coupling. However the challenge is being able to design a surface that implements the desired coupling.
7.3
Summary
From a materials perspective, it has been established as to what kinds of polarization control exist.
Anisotropic materials which convert one polarization state to another and chiral materials which control
the propagation of LHCP and RHCP fields. While natural materials are not practical to use in the
microwave spectrum for polarization control they do give insight into the physics of the problem. The
kinds of devices that can be engineered to control the polarization of microwaves build on that intuition.
These include FSSs and transmitarrays as well as new kinds of metamaterials and metasurfaces.
In Chapter 8, the use of Huygens surfaces to control the polarization of microwaves will be discussed
and it will be shown how a single surface of electric and magnetic dipoles can be engineered to alter
the polarization of microwaves. In Chapter 9 the use of an electric response only to generate chiral
polarization effects will also be discussed.
CHAPTER
8
Tensor Huygens Surfaces
Being able to control the polarization of the electromagnetic field was established in the previous chapter
as an important aspect of a variety of communication and imaging systems. Here, the passive Huygens
surface introduced in Chapter 6 are extended to be able to also alter the polarization of the electromagnetic field. This is done by generalizing the impedances used to describe the Huygens surface from
scalars to tensors, hence the title of this chapter.
As discussed in Chapter 7, the polarization controlling effects that are investigated include polarization conversion as well as chiral polarization effects (circular birefringence) and it will be shown how
tensor Huygens surfaces can be designed to implement both of these concepts.
8.1
The Boundary Condition At A Tensor Huygens Surface
As established in Chapter 6, Huygens surfaces are a superimposed array of passive electric and magnetic
dipoles on a plane. Earlier, the boundary condition for a Huygens surface was defined in (6.5)-(6.6)
using scalar electric and magnetic impedances for TE-polarized fields only. In the two-dimensional space
under discussion, both TE and TM fields can be defined in free space as stated in Chapter 1. Thus
it would be expected that the boundary conditions defined in Chapter 6 must be generalized to also
account for both TE and TM fields.
This requires defining electric and magnetic surface impedance boundary conditions for both TE and
TM fields. Looking at the case of an electric surface impedance the boundary condition is given by,
"
Ey
Ez
#
=
"
Zyy,e
Zyz,e
Zyz,e
Zzz,e
#"
−(Hz,2 − Hz,1 )
Hy,2 − Hy,1
#
,
(8.1)
where 1, 2 indicate the two sides of the surface as in Chapter 6. It can be seen that compared to (6.5)(6.6) the fields in the boundary condition have now become vectors while the impedance, Ze , which was
originally a scalar, now becomes a tensor to be able to relate the two field vectors. The fields Ey and Hz
are the tangential TM fields at the surface while the fields Ez and Hy are the tangential TE fields. This
107
108
Chapter 8. Tensor Huygens Surfaces
Ei
Zy,e
γ
Zm
Zz,e
Zy,m
γ
Zz,m
Zyy,e
Z
zy,e
Zyy,m
=
Zzy,m
Ze =
Zyz,e
Zzz,e
Zyz,m
Zzz,m
TM
~
H
~
E
E
~
E
~k
~
H
TE
~k
t
z
y
x
Figure 8.1: A schematic of a tensor Huygens surface consisting of a superposition of passive electric and
magnetic tensor impedances. The tensor Huygens surface can convert one input polarization state to
another. The TE and TM fields are also defined again.
boundary condition can be interpreted as inducing an electric surface current from an incident field that
is some linear combination of TE and TM fields. This induced surface current flows along the yz-plane
in an arbitrary direction (i.e. not aligned with either the y or z axis) and re-radiates a scattered field
which is also a linear combination of TE and TM fields.
Similarly, the magnetic surface impedances boundary condition is described as follows,
"
Hy
Hz
#
=
"
Zyy,m
Zyz,m
Zyz,m
Zzz,m
#−1 "
Ez,2 − Ez,1
−(Ey,2 − Ey,1 )
#
,
(8.2)
where Zm is the magnetic impedance tensor. The same interpretation of this boundary condition applies,
where a magnetic surface current is induced from an incident field that is a combination of TE and TM
components. This induced magnetic current is directed along the yz-plane in an arbitrary direction and
also re-radiates a scattered field that is a sum of TE and TM fields.
Here the electric and magnetic surfaces on their own are capable of creating a discontinuity in the
magnetic and electric fields respectively. When combined on the same surface, as in a Huygens surface,
a complete discontinuity in the field for both TE and TM fields is created and is described by,
1
2
1
2
"
"
Ey,2 + Ey,1
Ez,2 + Ez,1
Hy,2 + Hy,1
Hz,2 + Hz,1
#
#
=
=
"
"
Zyy,e
Zyz,e
Zyz,e
Zzz,e
#"
Zyy,m
Zyz,m
Zyz,m
Zzz,m
−(Hz,2 − Hz,1 )
#
Hy,2 − Hy,1
#−1 "
Ez,2 − Ez,1
,
−(Ey,2 − Ey,1 )
(8.3)
#
.
(8.4)
This is illustrated in Fig. 8.1.
Given that the discussion is focused on polarization it is clear that these boundary conditions are
able to alter the polarization state as the field scattered by the tensor impedances are a sum of TE and
109
Chapter 8. Tensor Huygens Surfaces
2Φ
2Ψ
Figure 8.2: The definition of the Poincaré sphere which maps out all possible polarization states on the
surface of a sphere.
TM fields. By using a Huygens surface, it is clear that the surface can be made reflectionless based on
the Hugyens principle and that only a transmitted field with a different polarization state should exist.
The question is what are the required tensor impedances to achieve these goals.
Before figuring out what the required impedances are, it is important to define the polarization
state of the fields at the tensor Huygens surface. To do this, the TE and TM fields are defined as a
basis for the polarization state as they are two orthogonal linear polarizations. The TE fields can be
defined as a vertical polarization (E-field perpendicular to the plane of propagation) and the TM fields
as a horizontal polarization. From here any linear combination of TE and TM fields can represent an
arbitrary polarization state. This can be expressed as,
Epol = sin Ψe−j2Φ ŷ + cos Ψẑ
(8.5)
The variables Ψ and Φ define the relative amplitude and relative phase respectively of the TE and TM
fields. It can easily be seen that Ψ and Φ correspond to the variables defined on the Poincaré sphere
which maps the polarization state on the surface of a unit sphere [201]. A picture of the Poincaré sphere
is shown in Fig. 8.2 for reference. For the rest of this discussion, the polarization state of the TE and
TM fields at the tensor Huygens surface will be defined using Ψ and Φ.
8.1.1
Determining The Impedance Tensors Using Diagonalization
The problem being solved again is a synthesis problem. Given an incident field in some polarization
state, find the electric and magnetic tensor impedances which convert the transmitted field into a different
polarization state. To do this the impedance tensors in (8.3) and(8.4) are diagonalized.
To simplify this procedure it is assumed that the incident and transmitted fields are propagating
normally with respect to the surface (along the x̂ direction with ϕi = ϕt = 0). This implies that
refraction, as discussed in Chapter 6, is being neglected in this analysis.
Given the illustration of a tensor Huygens surface shown in Fig. 8.1, an incident plane wave of
arbitrary polarization is incident on the surface. The polarization state is indicated by Ψi , Φi , the
relative amplitude and phase of the incident TE and TM fields shown in Fig. 8.1. The polarization
state of the transmitted field is designated by Ψt , Φt . The first thing that has to be determined is the
amplitudes of these plane waves. For normally incident and transmitted plane waves, which is the stated
assumption, it can easily be shown that the incident and transmitted waves have the same amplitude as
power is conserved through the Huygens surface.
110
Chapter 8. Tensor Huygens Surfaces
With the amplitudes known, the equations to solve for the impedance tensors are then derived by
diagonalizing the impedance tensors themselves. The electric impedance tensor, Ze is examined first,
noting that the exact same procedure can be applied to the magnetic impedance tensor, Zm also.
The incident and transmitted fields on either side of the Huygens surface are substituted into (8.3).
The change in the polarization across the surface introduces a discontinuity in the field that is enforced
by the electric and magnetic impedance boundaries. This is given by,
1
2
"
"
Eo sin Ψt e−j(ξt +2Φt ) + Eo sin Ψi e−j(2Φi )
Eo cos Ψt e−jξt + Eo cos Ψi
"
Zyy,e
Zyz,e
Zyz,e
− η1 (−Eo sin Ψt e−j(ξt +2Φt )
1
−jξt
η (−Eo cos Ψt e
− Eo sin Ψi e
#
Zzz,e
−j(2Φi )
)
+ Eo cos Ψi )
=
#
#
.
(8.6)
where ξt is an arbitrary phase shift equally imposed by the impedance boundary on both the transmitted
TE and TM fields. Note that because the surface is reciprocal, the impedance tensor, Ze , must be
symmetric. Because the surface is also lossless the entries of Ze are purely imaginary. This means that
Ze is anti-Hermitan and it can be diagonalized with imaginary eigenvalues. This is expressed as,
"
Zyy,e
Zyz,e
Zyz,e
Zzz,e
#
= R(γ)
"
Zy,e
0
0
Zz,e
#
R−1 (γ),
(8.7)
where R(γ) is the diagonalization matrix. For a 2 × 2 matrix the geometric interpretation of the
matrices that diagonalize Ze is that they are rotation matrices [132, 202]. These rotation matrices then
are characterized by an angle γ and are given by
R(γ) =
"
cos γ
sin γ
− sin γ
cos γ
#
.
(8.8)
Thus the eigenvectors which form the rotation matrix can be thought of as (orthogonal) eigenvectors,
each rotated by an angle γ with respect to the underlying coordinate system. These eigenvectors each
point in the direction of the homogenized electric dipole moments of the surface. The strength of this
homogenized dipole moment is given by the eigenvalues of the impedance tensor as depicted in Fig. 8.1.
To determine the impedance tensor then, the eigenvalues and the rotation angle γ need to be solved for.
To achieve this, (8.7) is substituted into (8.6) and rearranged which gives,
1
2
=
"
Zy,e
0
0
Zz,e
"
#"
cos γ
− sin γ
"
cos γ
cos γ
#"
Eo sin Ψt e−j(ξt +2Φt ) + Eo sin Ψi e−j(2Φi )
#
cos γ
Eo cos Ψt e−jξt + Eo cos Ψi
#"
#
− η1 (−Eo sin Ψt e−j(ξt +2Φt ) − Eo sin Ψi e−j(2Φi ) )
sin γ
− sin γ
− sin γ cos γ
#"
#
sin γ
Ey,av
cos γ
sin γ
Ez,av
=
"
1
η (−Eo
Zy,e
0
0
Zz,e
cos Ψt e−jξt + Eo cos Ψi )
#"
#"
#
cos γ
sin γ
−Hz,diff
− sin γ
cos γ
Hy,diff
,
(8.9)
.
(8.10)
111
Chapter 8. Tensor Huygens Surfaces
These are two independent equations which can be simplified as,
Zy,e =
Zz,e
cos γEy,av + sin γEz,av
,
− cos γHz,diff + sin γHy,diff
− sin γEy,av + cos γEz,av
=
,
sin γHz,diff + cos γHy,diff
(8.11)
(8.12)
where the fields at the surface are the averaged and differenced fields in (8.9) that have been rotated by
R(γ). Each component of these averaged electric and magnetic field vectors is related by the eigenvalues,
Zy,e and Zz,e . Note that these fields are implicitly a function of Φt and Ψt .
Because the fields are complex quantities it must be ensured that Zy,e and Zz,e remain imaginary. To
do this the value of γ must be chosen such that Re(Zy,e ) = 0 and Re(Zz,e ) = 0 in the above equations.
Using some algebra then, γ can be expressed by rearranging (8.11) or (8.12) into the following quadratic
equation,
∗
∗
∗
Re(Ez,av Hy,diff
) tan2 γ + Re(Ey,av Hy,diff
− Ez,av Hz,diff
) tan γ
∗
+Re(−Ey,av Hz,diff
) = 0,
(8.13)
where ∗ is the complex conjugate. By solving for tan γ and choosing the appropriate root, γ can be found.
From here the eigenvalues can be found substituting γ into (8.11) and (8.12). With the eigenvalues and
rotation angle found the full impedance tensor can be determined by substituting Zy,e , Zz,e and γ into
(8.7).
This procedure can be repeated for the magnetic impedance tensor Zm , where the eigenvalues Zy,m
and Zz,m can be found as well, as given by,
cos γEy,diff + sin γEz,diff
,
− cos γHz,av + sin γHy,av
− sin γEy,diff + cos γEz,diff
=
,
sin γHz,av + cos γHy,av
Zz,m =
Zy,m
(8.14)
(8.15)
It should be noted that γ has the same value for both Ze and Zm . This is because the electric impedance
tensor represents two orthogonal dipole moments Zy,e and Zy,m rotated by an angle γ as stated above.
Likewise for the magnetic impedance tensor Zy,m , Zz,m . Because the superimposed impedance surfaces
represent a Huygens surface, designed to be reflectionless, then both the electric and magnetic dipole
moments must be rotated by the same angle so that the dipole moments are orthogonal to each other,
as this is the definition of a Huygens surface. The implementation of these impedance tensors will be
discussed more in Section 8.4.
To summarize, to find the impedance tensor from the fields at the tensor Huygens surface a simple
two-step procedure can be followed:
1. Solve for γ by forcing the real part of either (8.11) or (8.12) to zero.
2. With γ found, solve for Zyy,e , Zzz,e , Zyz,m which is given by (8.7).
The solution space given by (8.11),(8.12),(8.14)(8.15) can be illustrated by plotting the impedance
tensors for converting an incident field in a given polarization state to a desired transmitted state. Here a
TE-polarized incident plane wave (Ψi = 0 and Φi = 0, or a vertical polarizations) is incident on the tensor
Huygens surface. The arbitrary phase shift imposed by the surface is also set to ξt = 30◦ . The electric
112
Chapter 8. Tensor Huygens Surfaces
(a)
(b)
(c)
(d)
(e)
(f)
Figure 8.3: The required tensorial impedance for a surface to map a (vertical) linear polarized wave to
any other polarization state as plotted on the Poincaré sphere. The black dot on the sphere indicates
the input polarization state. Note all values are imaginary. (a) Zyy,e , (a) Zzz,e , (a) Zzy,e , (a) Zyy,m , (a)
Zzz,m , (a) Zzy,m
and magnetic impedance tensors are then found for all the different combinations of 0 < 2Ψt < 360◦ and
−90◦ < 2Φt < 90◦ . These values of Ψt and Φt span all the different possible transmitted polarizations
states on the Poincaré sphere shown in Fig. 8.2.
To illustrate the values of the impedance tensors found for each combination of Ψt and Φt the electric
and magnetic impedance tensors are plotted in Fig. 8.3 on the surface of the Poincaré sphere. Here it can
be seen how this TE-polarized plane wave can be transformed into any polarization state by choosing
the appropriate values for the impedance tensors at the corresponding point on the Poincaré sphere.
Note that all the values in Fig. 8.3 are imaginary with the real part equal to zero. The main takeaway
from this plot is that any polarization state for the transmitted field is achievable for a tensor Huygens
surface with the correct impedances. This exercise can be repeated for an incident field in an arbitrary
polarization state as well (0 < Ψi < 360◦ and 0 < Φi < 90◦ ). This means that the anisotropy of
the anisotropy of the tensor Huygens surface can be completely controlled to engineer any polarization
conversion desired with a single surface.
8.2
Circuit Modelling of Tensor Huygens Surfaces
Analyzing the fields at the electric and magnetic tensor impedance boundary condition gives a complete
picture of the polarization conversion from the incident to the transmitted fields. However, these Huygens
surfaces are engineered as discrete structures made up of unit cells composed of electric and magnetic
dipole moments. To successfully design the surface then, the unit cells must be designed to implement
the desired tensor impedances across the entire screen.
It was established in Chapter 6 that a lattice cell circuit can be used to model a scalar impedance
Huygens surface. This circuit model also allows for S-parameters to be used to define the transmission/reflection properties of a Huygens surface. Having a circuit model to capture this behaviour of
113
Chapter 8. Tensor Huygens Surfaces
the unit cell is very useful in designing the required unit cells. For the tensor Huygens surfaces under
discussion, the lattice cell model from Chapter 6 is generalized to establish an equivalent circuit model
for a tensor impedance surface. This circuit model is then used to see how the S-parameters tie to the
impedance tensors derived above.
8.2.1
Tensor Circuit Element And Equivalent Circuit
The equivalent circuit model for a tensor Huygens surface requires a four-port network (as opposed to a
two-port network for scalar Huygens surfaces) [138, 168]. This is because the incident and transmitted
fields are defined as a function of both TE and TM polarized waves. Thus, a four-port network maps two
ports to the incident and reflected TE and TM waves and two ports to transmitted TE and TM waves.
This is defined in Fig. 8.4 with the port numbering defined also. With this definition, a transmissionline model for free-space can be constructed where both TE and TM fields are supported. This is
shown in Fig. 8.4. Note that the transmission-lines have a common ground that is repeated twice. This
transmission-line network can be considered to be a multi-conductor transmission-line (MTL) network
where the TE and TM modes on each transmission-line are orthogonal to each other and do not couple.
This leads to defining the propagation constant and impedance of the transmission-line network to be
2π
I,
λ
1
Yo = I,
η
β=
(8.16)
(8.17)
where I is a 2×2 identity matrix and η is the impedance of free space. This multi-conductor transmissionline framework will be expanded on in Chapter 9, but further basics on MTL theory can be found in
[203].
On these orthogonal transmission-lines, voltages and currents can be mapped to the electric and
magnetic fields of the TE and TM plane waves. Again, electric fields map to voltages and magnetic fields
to currents. As shown in Fig. 8.4, Vz,1 , Iz,1 corresponds to the electric and magnetic fields respectively
on side 1 of the screen for the TE fields, and Vy,1 , Iy,1 to the electric and magnetic fields on side 1 of the
screen for the TM fields. On side two of the screen, Vy,2 , Iy,2 and Vz,2 , Iz,2 correspond to the the electric
and magnetic fields for the TE and TM fields respectively.
In Chapter 6, the impedances of the boundary conditions which defined the scalar Huygens surface
were mapped to series and shunt impedances in the equivalent circuit. Here, the impedances in the
boundary condition are tensor impedances. So to be able to map this boundary condition to a circuit
element, a tensor circuit element is defined. By analogy to two-terminal circuit elements, these tensor
circuit elements are defined as four terminal elements as shown in Fig. 8.5. Here currents applied along
either the y or z axis of the element generates a corresponding voltage difference across the element in
the y and z directions as defined by the impedance tensor. This is expressed as,
"
Vy
Vz
#
=
"
Zyy
Zyz
Zzy
Zzz
#"
Iy
Iz
#
(8.18)
Inverting the impedance gives an equivalent definition for an admittance tensor element.
These tensor circuit elements can be inserted into a four-port network by loading the MTL network in
various series and shunt configurations. As in Chapter 6, three configurations will be examined; a shunt
114
Chapter 8. Tensor Huygens Surfaces
2
TM
+
TE
Yo,β
TM
1
+
Vz,1
TE
y
Iz,2
=
Z
-
-
z
+ Vz,1
Iz,1
x
-
(a)
(b)
=
Zsh
+
Vy,2
Iy,2
=
Zsh
+
=
Z
Vz,1
Iz,2
Iy,1 +
Vy,1
-
=
Zse
Iy,1 +
Vy,1
-
+ Vz,1
Iz,1
-
4
V
Iy,1 + y,1
3
Vy,2
Iy,2
-
Vy,2
+
Vz,2
Iz,2
-
=
Zse
+ Vz,1
Iz,1
+
Iy,2
-
(c)
(d)
Figure 8.4: Different configurations of a tensor impedance. (a) The MTL model of free space (b) Shunt
loading. (c) Series loading. (d) Lattice Network.
i
z
i
+
+
=
+ Z= [ ] Z
Z
yy
Zzy
V
-
Zzy
Zzz
Vz
Vy
Figure 8.5: A scalar impedance circuit element and a tensor impedance element. A current applied to
one terminal causes a voltage across both terminals.
network, a series network and a lattice network, to see how these three circuits map to the boundary
conditions given in Section 8.1.
A shunt tensor impedance element in a four-port network is shown in Fig. 8.4b. By applying the
voltage-current relations in (8.18) the four port impedance matrix describing the circuit is given by,

Vy,1

 Vy,2

 V
 z,1
Vz,2


Zyy
 
  Zyy
=
  Z
  zy
Zzy
Zyy
Zyz
Zyy
Zyz
Zzy
Zzz
Zzy
Zzz
Zyz

Iy,1


Zyz 
  −Iy,2

Zzz  
 Iz,1
Zzz
−Iz,2






(8.19)
This Z-matrix can be thought of as a generalization of the Z-matrix for a shunt element in a two-port
network. By rearranging this matrix the boundary condition given in (8.1) can be recovered which is
115
Chapter 8. Tensor Huygens Surfaces
expressed as
"
Vy,2
Vz,2
#
=
"
Zyy
Zyz
Zzy
Zzz
#"
Iy,1 − Iy,2
Iz,1 − Iz,2
#
(8.20)
in terms of voltages and currents. Again the voltages and currents map to the TE and TM fields as
described above. Here a discontinuity in the currents is related to the voltage which is continuous across
the four-port network. This shows how a shunt tensor impedance maps to the electric surface impedance
tensor. This example was first studied in [143].
This can be repeated for a series tensor impedance element as shown in Fig. 8.4c by applying (8.18)
again. Here a four-port admittance matrix can be derived


Iy,1

Yzz
 
  −Yzz
=
  Y
  zy
−Yzy

 −Iy,2

 I
 z,1
−Iz,2
−Yzz
Yzz
−Yzy
Yzy
−Yyz
Yyz

Vy,1


Yyz 
  Vy,2

−Yyy  
 Vz,1
Vz,2
Yyy
−Yyz
Yyy
−Yyy



,


(8.21)
where the elements of the admittance matrix are found from the inverse of the impedance matrix in
(8.18). This Y-matrix can be thought of as a generalization of the Y-matrix for a series element in a
two-port network. Again this can be rearranged to get the boundary condition given in (8.2) which is
given by,
"
−Iy,2
−Iz,2
#
=
"
Yzz
Yzy
Yyz
Yyy
#"
#
Vy,2 − Vy,1
Vz,2 − Vz,1
(8.22)
The currents in this scenario are continuous across the network while the voltages are discontinuous and
are related by the impedance tensor again. Note that the rows of the admittance matrix are reversed as
currents along y correspond to z-directed magnetic fields and vice versa. Thus a series tensor impedance
is equivalent to the surface impedance of magnetic dipoles.
Finally, the most important network in this discussion is a lattice network of tensor impedances.
Just as a lattice network of scalar impedances consists of crossing shunt impedances separated by series
impedances, the tensor equivalent has a similar configuration as shown in Fig. 8.4c. The four port
impedance matrix for this circuit is then given by,



Vy,1
Z2 + Z1



Vy,2  1 Z2 − Z1



V  = 2 Z + Z
5
 z,1 
 6
Vz,2
Z6 − Z5
Z2 − Z1
Z6 + Z5
Z6 − Z5
Z4 + Z3
Z2 + Z1
Z6 + Z5
Z6 − Z5

Iy,1





Z6 + Z5 
 −Iy,2  ,


Z4 − Z3   Iz,1 

Z4 + Z3
−Iz,2
Z6 − Z5
Z4 − Z3
(8.23)
where the series and shunt tensor circuit elements are given by
Zse =
"
Z1
Z5
Z5
Z3
#
, Zsh =
"
Z2
Z6
Z6
Z4
#
,
1
1
Zzz,m , Z2 = 2Zyy,e , Z3 = Zyy,m ,
2
2
1
Z4 = 2Zzz,e , Z5 = Zzy,m , Z6 = 2Zzy,e ,
2
(8.24)
Z1 =
(8.25)
116
Chapter 8. Tensor Huygens Surfaces
Rearranging this set of equations gives us
"
Vy,2 + Vy,1
Vz,2 + Vz,1
"
#
−(Iz,2 + Iz,1 )
−(Iy,2 + Iy,z2 )
#
=
=
"
"
Z2
Z6
Z1
Z3
Z6
#"
Iy,1 − Iy,2
#
,
Z4
Iz,1 − Iz,2
#−1 "
#
Z3
Vz,2 − Vz,1
Vy,2 − Vy,1
Z5
(8.26)
(8.27)
which is the boundary condition of a tensor Huygens surface given in (8.3) -(8.4) with the average voltage
at the terminals of the four port network related to the discontinuity in the currents via the shunt tensor
impedances and vice versa for the series tensor impedance. This four-port impedance matrix then forms
the equivalent circuit of a tensor Huygens surface with its structure understood as a lattice network of
tensor impedance elements. Note that values of the elements in the impedance matrix of (8.23) can be
found by substituting the expression of the impedance tensor given in (8.18). Such a circuit model is
useful in full-wave design of the tensor unit cells as it will be shown in Section 8.3.
8.2.2
S-parameter Model
With the 4 × 4 Z-matrix derived above for a tensor Huygens surface, it can be readily converted to
S-parameters (or any set of network parameters). However such a conversion would not be very useful
as the resulting expressions would not be very intuitive. It is worth asking instead if the S-parameters of
the unit cell (and thus the tensor Huygens surface) can be derived independently. For the scalar Huygens
surface, the S-parameters model for each unit cell was understood as having perfect transmission with a
phase that varies linearly from cell to cell (for refraction) as stated in Chapter 6. Here a similar model
is derived for a tensor Huygens surface by examining the properties of a general 4 × 4 S-matrix. Note
that other polarization controlling devices have used 4-port S-parameters but the S-parameter matrix
presented here is specific to polarization conversion [143, 168].
For a tensor Huygens surface, the 4-port S-parameter matrix can be constructed for each unit cell
by constraining each cell to be symmetric, lossless and reflection-less and reciprocal just as the Huygens
surface is also. In that scenario the S-parameter matrix has its entries constrained such that, Sii = 0,
Sij = Sji and SS∗ = I, where ∗ here is the Hermitian conjugate of a matrix and I is the identity matrix.
Using these constraints and following [204], the S-matrix for a general polarization controlling device
such as a tensor Huygens surface takes on the following structure,



b1,T M
0



−jζ
b2,T M 


 = e−jξ cos Ψe
b


0
 3,T E 

b4,T E
sin Ψe−jΦ
cos Ψe−jζ
0
0
sin Ψe−jΦ
sin Ψe
0
−jΦ
0
cos Ψ


a1,T M


 a2,T M 
0




cos Ψ 
  a3,T E 
0
a4,T E
sin Ψe−jΦ
(8.28)
where ξ is again an arbitrary phase shift, and Ψ and Φ are defined in Section 8.1 as the relative amplitude
and phase difference between the the transmitted TE and TM waves (from here on the subscript ‘t’ is
dropped as it is always the relative phase and amplitude of the transmitted field that is being referred
to). To satisfy the constraints on the S-matrix the following relationship between the amplitudes and
117
Chapter 8. Tensor Huygens Surfaces
phases of the transmitted waves must hold
cos2 Ψ + sin2 Ψ = 1,
(8.29)
ζ = 2Φ − (π + 2nπ), n ∈ Z,
(8.30)
so that S remains unitary. It can be seen that from the definition of the port numbers given in Fig. 8.4,
that S21 and S43 are the coupling between like polarizations and S31 and S23 are the coupling between
cross polarizations. The constraints on the amplitude of the transmission coefficients given in (8.29)
specify the relative amplitude of the transmitted TE and TM polarizations. Meanwhile, the constraints
on the phase shift is related to the phase between the transmitted TE and TM polarizations. This
reflection and transmission model of a tensor Huygens surface explains exactly how an input polarization
in a given state can be converted to a different polarization state through the proper selection of the
co- and cross-polarized reflected and transmitted waves. This also generalizes the transmission model
for anisotropic transmitarrays given in [161, 197, 205] which were designed to work only for a 45◦ slant
linear polarization to work for any input polarization.
Finally, these S-parameter results can be used for the inverse problem of finding the entries of the 4×4
impedance matrix of the lattice model given in Section 8.2.1. This can be accomplished by converting
the S-parameter matrix to a Z-matrix using, Z = Zo (I − S)−1 (S + I), where Zo = η is the free space
wave impedance. The circuit model developed here becomes an effective tool in engineering the surface
as it can be used to design and extract relevant impedances for the individual unit cell. The S-parameter
model also gives an effective way of thinking about the tensor Huygens surface in terms of reflected and
transmitted waves.
8.3
Numerical Modelling
To numerically validate the tensor impedances derived above, the TLM method introduced in Chapter 6
is used to solve some examples. First the computational model is discussed to show how the TLM method
can be extended to support both TE and TM modes as well as how the circuit model proposed above is
used to model the tensor Huygens surface. Then three different examples using a tensor Huygens surface
are discussed. They are:
1. Converting a TE polarized wave to a circular polarized wave.
2. Converting a TE polarized wave to a TM wave.
3. Converting one elliptical polarization to another elliptical polarization.
8.3.1
Computational Model
As stated, it was demonstrated that a scalar Huygens surface is modelled using a two-dimensional hybridTLM/MOM method where the Huygens surface is represented using the lattice cell equivalent circuit.
In the TLM method used here, free-space is discretized into a two-dimensional array of transmissionline unit cells. For a network that supports both TE and TM fields, the transmission-line unit cell
consists of a two-dimensional arrangement of the MTL network discussed in Section 8.2 and shown in
118
Chapter 8. Tensor Huygens Surfaces
z
X
per per
per per
int
per
int
int
Y
per
per
per
int
int
int
int
int
int
int
int
int
int
int
MOM
Tensor
Huygens
Surface
3
TLM
int
int
2
1
Shunt
Node
(TE)
MOM
Series
Node (TM)
TLM
4
TLM MOM
TLM
3
MOM
1
2
4
Figure 8.6: The computational domain for a 2D TE/TM TLM solver.
Fig. 8.6. Now a two-dimensional transmission line supporting the TE-polarization exists orthogonally
to a two-dimensional transmission line supporting a TM-polarization. The transmission-line network for
the TE-polarization is a shunt-node network, while for the TM-polarization it is a series-node network.
As mentioned above, the two modes are decoupled from one another. To model a tensor Huygens surface,
it can be modelled as an array of tensor lattice networks, using the impedance matrix defined in (8.23)
as shown in Fig. 8.6. This array of 4-port lattice networks defines the necessary boundary condition
at the interface to model the tensor Huygens surface. Again, a matrix of the impedances/admittances
for each point in the two-dimensional array of unit cells is constructed and is inverted to find the
voltages and currents in the transmission-line grid. To understand this in more detail along with how
the transmission-line grid is terminated using the method-of-moments see Appendix C.
It is worth noting, that this approach has some conceptual similarities to the generalized scattering
matrix (GSM) methods used in reflectarray and transmitarray design [206]. Here the hybrid TLM/MOM
method is dealing with unit cells which support two different modes and is solving for the fields scattered
off these unit cells. Likewise the GSM method also incorporates multiple modes in the S-parameters
which characterize a unit cell. These S-parameters can then be inserted into a method of moments solver.
However, the TLM/MOM method is best suited (in its current form) to supporting two-modes only (the
dominant TE and TM modes). While the GSM method can be extended to an analysis which includes
many (tens to hundreds) of modes. This TLM/MOM method is well suited for the Huygens surface
under discussion here or for other structures where only the dominant modes are under consideration.
119
Chapter 8. Tensor Huygens Surfaces
(a)
(b)
(c)
Figure 8.7: (a) The Ez component of the TE field. A TE polarization is incident on the screen. The
screen is indicated by the black line. (b)The Hz component of the TM field. (c) The polarization as
traced out by the electric field.
8.3.2
Examples
All of the examples discussed below are carried out at 10 GHz. The unit cell size for both free-space and
the Huygens surface is λ/36. The incident field is a Gaussian beam with a waist of 2.5λ and amplitude
Eo = 1 V/m. The polarization of the incident Gaussian beam will be specified as necessary.
Converting a TE-polarized wave to a circularly polarized wave
Here a TE polarized (vertically polarized) Gaussian beam can be converted into a right-handed circularly
polarized wave. This is an important example for satellite communications where such polarization
screens are used to generate CP as mentioned in Chapter 5. However, this tensor Huygens surface is a
single layer structure that is thinner than the meander-line polarizers discussed previously allowing for
the realization of thin polarization screens. In this example, the transmitted field is given by Ψ = 45◦
and Φ = 90◦ . This defines a tensor Huygens surface with an S-parameter matrix given by
S=e
−jπ/6






0
√1
2
0
√1 e−jπ/2
2
√1
2
0
√1 e−jπ/2
2
0
0
√1 e−jπ/2
2
0
√1
2
√1 e−jπ/2
2
0
√1
2
0



.


(8.31)
From this S-parameter matrix the impedance tensors can be found and are given by (8.23),
Ze = j
"
593.06
−838.71
−838.71
593.06
#
Ω, Zm = j
"
239.64
338.91
#
338.91
239.64
Ω
(8.32)
One way of interpreting the S-parameter matrix/impedance tensor is to look at the rotation angle
γ of the impedance tensor. It can easily be calculated that γ = 45◦ , for both the electric and magnetic
impedance tensors. The homogenized electric and magnetic dipole moments are thus rotated by 45◦ .
The eigenvector for the tensor impedance along the 45◦ axis imposes a 90◦ phase shift with respect
120
Chapter 8. Tensor Huygens Surfaces
to the eigenvector along the −45◦ axis. Because the surface is a Huygens surface, the transmission is
of-course unity. This is confirmed from an S-parameter perspective if the S-parameter matrix is rotated
by γ = 45◦ as discussed in Appendix D and given by,
Srot = R(γ)SR−1 (γ).
(8.33)
This results in (8.31) being recast as,

0
 −jπ/4
 e
SRot = e−jπ/6 

0

0
e−jπ/4
0
0
0
0
0
0
0
ejπ/4
0
ejπ/4
0



.


(8.34)
This understanding of the Huygens surface is analogous to the discussion in Chapter 7 about anisotropic
materials. There it was shown how anisotropic materials can convert polarization by imposing a different
phase-shift between two orthogonal axes. Here the tensor Huygens surface has anisotropic axes that are
rotated 45◦ , with respect to the y and z-axes. On these rotated axes, a 90◦ phase difference is imposed
to create a TE-to-circular polarization converter. This is also how meander-line polarizers are designed
for linear-to-circular polarization converters, however both the impedance boundaries and S-parameter
matrix give a more complete framework to engineer polarization controlling surfaces. Also, compared
to either anisotropic materials or FSS’s, the tensor Huygens surface is able to implement polarization
conversion in a single surface.
With the four-port lattice network, the tensor Huygens surface can be simulated using the TLM
method. In Figs. 8.7a and 8.7b the simulated Ez component of the TE-waves and the Hz component
of the TM-waves from the TLM solver are plotted and it can be seen that the incident TE field is
transmitted through the surface (shown as the black line) into both TE and TM components with
different amplitude and phase. It can also be seen that there are no reflections from the surface as
intended. In Fig. 8.7c the polarization traced out by the electric field on both sides of the screen are
plotted and it is clear that the incident vertically polarized field is indeed converted to a RHCP polarized
field. Note that this TLM method is simply used to understand the properties of these unit cells. A
more quantitative analysis can be done using more comprehensive full-wave solvers as will be shown
below by extracting full-wave S-parameters.
Converting a TE-polarized wave to a TM-polarized wave
In this example a TE-polarized Gaussian beam can be converted into a TM-polarized Gaussian beam.
The transmitted field is given by Ψ = 90◦ and Φ = 0◦ . This defines a tensor Huygens surface with an
S-parameter matrix given by
S=e
−j70◦

0

 0

 0

1
0
0
0
1
1
0
0
0
1


0 
,
0 

0
(8.35)
121
Chapter 8. Tensor Huygens Surfaces
(a)
(b)
(c)
Figure 8.8: (a) The Ez component of the TE field. A TE polarization is incident on the screen. The
screen is indicated by the black line. (b)The Hz component of the TM field. (c) The polarization as
traced out by the electric field.
which is again converted into the impedance matrix for a lattice network given by (8.23). From the
impedance matrix the impedance tensors can be found which are given to be,
Ze = j
"
−68.61
−200.60
#
−200.60
−68.61
Ω, Zm = j
"
−274.40
802.4
802.40
#
−274.40
Ω.
(8.36)
These values are again used in the TLM solver to solve for the fields transmitted by the tensor Huygens
surface.
In this design, γ = 45◦ again for both the electric and magnetic impedance tensors. This implies
that the eigenvectors of the impedance tensor align along ±45◦ . When the TE-polarized incident field
is projected along those axes, a phase shift of 180◦ exists between both axis, which can also be seen
by rotating the S-parameter matrix 45◦ . Again, this acts like an anisotropic material where a different
phase shift exists along two orthogonal axes.
In Figs. 8.8a and 8.8b the simulated Ez component of the TE-waves and Hz component of the TMwaves from the TLM solver are plotted and it can be seen that the incident TE field is transmitted
through the surface (shown as the black line) into the TM component only with no reflections. In
Fig. 8.8c the polarization traced out by the electric field on both sides of the screen is plotted; it is clear
that the incident vertically polarized field is indeed converted to a TM field.
Converting one elliptical polarization to another elliptical polarization
Building off of the previous examples, the last example is the most general example in terms of polarization, where a surface is designed to convert an incident elliptical polarization of a certain tilt and
eccentricity to another elliptical state of a different tilt and eccentricity. The surface is specifically designed to take an input beam which corresponds to an ellipse with a tilt of 0◦ and an ellipticity of −0.50
and convert it to an elliptical polarization with a tilt of −45◦ and an ellipticity of 0.461 . This is given
1 Note
that the ellipticity is the ratio of the amplitude of the TE to TM components.
122
Chapter 8. Tensor Huygens Surfaces
Figure 8.9: The polarization traced out by the electric field for elliptical to elliptical conversion.
by an S-parameter matrix of
S=e
−jξ

◦
0

 0.74ej122◦


0

0.67e−j
0.74ej122
0
0
0.67e−j61
◦
◦
0.67e−j61
0
0
0.74e−j65
◦
0.67e−j61
◦
61◦



,
−j65◦ 
0.74e

0
0
(8.37)
which again gives the corresponding surface impedances using (8.23). Simulating this design using the
TLM solver when illuminated with the appropriate elliptically polarized beam, the polarization traced
out by the electric field is plotted on both sides of the surface in Fig. 8.9. Here the input elliptical
polarization changes its tilt and eccentricity as it is transmitted through the Huygens surface. This
demonstrates the control the tensor Huygens surface has over both the relative amplitude and phase of
the TE and TM fields when properly designed.
8.4
Implementation Of A Tensor Huygens Surfaces
The last detail to fill in is how a tensor Huygens surface is constructed. It was discussed in Chapter 6 how
scalar Huygens surfaces are implemented on three-layer PCB’s by pattering printed loops and dipoles
on the metal layers of the board that are loaded with printed inductors and capacitors to control their
surface impedance. To design a tensor Huygens surface, the design for scalar Huygens surfaces previously
shown can be extended. The simplest way to demonstrate this is to use the diagonalization analysis in
Section 8.1.
As stated in (8.7) the electric and magnetic impedance tensors can be diagonalized such that they
are reduced to two eigenvalues, Zy,e/m , Zz,e/m and a rotation angle γ. For the scalar Huygens surface
shown in Fig. 6.15a that was designed for a TE-polarization, it can be seen as only implementing the
scalar impedance Zz,e and Zy,m .
123
Chapter 8. Tensor Huygens Surfaces
z
Zz,e
z
Zy,e
γ
y
γ
γ
Zy,m
y
Zz,m
(a)
(b)
Figure 8.10: The components of a tensor Huygens surface unit cell. The superposition of the two gives
a tensor Huygens surface unit cell. (a) Rotated Dipoles (b) Rotated Loops.
To be able to implement the full impedance tensor, the geometry of the PCB design can be extended to
a crossed-dipole/loop design. This is shown in Fig. 8.10 for both loops and dipoles. Here, the parameters
of the tensor impedances map to the geometry of the crossed-loops and dipoles. The eigenvalues of the
tensor impedances correspond to the impedance of each arm of the crossed loop/dipole as shown in
Fig. 8.10. For the dipoles, the impedance can be controlled by the printed inductor or capacitor loading
each arm of the dipole. While for the loops, a printed capacitor loading each loop provides the necessary
control. The rotation angle γ maps to the physical rotation of the loops and dipoles also. This unit cell
can thus be used to implement a tensor Huygens surface surface when superimposed together.
While other geometries may be used to implement tensor impedances, see for example [127, 129],
this approach has the simplicity of being able to map one-to-one the eigenvalues and rotation angle
associated with the tensor impedance value. This makes the design more straightforward, especially
when working with both an electric and magnetic response.
To demonstrate this, the TE to TM polarization converter in Section 8.3.2 is implemented using the
printed geometry proposed here.
8.4.1
Designing a TE-polarization to TM-polarization converter
The S-parameters for a TE-to-TM polarization converter are given in (8.35) where it was shown using
a TLM model that the tensor Huygens surface is capable of implementing the desired polarization
conversion. The corresponding tensor impedances for the electric and magnetic response are also defined
in (8.36). Using the rotated crossed loop/dipole geometry proposed in Fig. 8.10, these specific tensor
impedances and corresponding S-parameters can be implemented. This is done around X-band in the 911 GHz frequency range. The precise layout of the tensor Huygens surface unit cell is shown in Fig. 8.11a
where the superimposed crossed loop and dipole are shown. The design is on a Rogers 3000 substrate
that is 3.04 mm thick with three metal layers. At 10 GHz this is λ/10 thick, which is thinner when
compared to traditional meander-line polarizers which are generally on the order of λ/2 thick.
The top and bottom layers of the substrate are used to form the crossed loops, which are connected
by vias, while the middle layer is used to form a crossed dipole. The printed reactances loading each
124
Chapter 8. Tensor Huygens Surfaces
(a)
(b)
Transmission
0
−5
−5
−10
−10
Magnitude (dB)
Magnitude (dB)
Reflection
0
−15
−20
−25
|S11|
−30
−15
−20
−25
|S41|
−30
|S23|
|S |
|S |
31
−35
21
−35
|S |
|S |
33
−40
9
9.5
10
10.5
43
−40
9
11
9.5
Freq (GHz)
11
(d)
150
4000
100
3000
50
2000
Impedance (Ω)
Impedance (Ω)
10.5
Freq (GHz)
(c)
0
−50
−100
−150
−200
Im(Z
−250
Im(Zzz,e))
−300
9
10
)
yy,e
1000
0
−1000
−2000
Im(Z
−3000
Im(Zyy,m))
Im(Zzy,e)
9.5
10
Freq (GHz)
(e)
10.5
)
yy,m
Im(Zzy,m)
11
−4000
9
9.5
10
10.5
11
Freq (GHz)
(f)
Figure 8.11: A TE-to-TM converter designed on a three-layer Rogers 3000 substrate. (a) The unit cell
of a tensor Huygens surface showing how the rotated and crossed dipoles and loops are superimposed.
The loops and dipoles are loaded reactively with printed inductors and capacitors where neccessary. (b)
An array of the unit cell. Note that only one unit cell is simulated in periodic boundary conditions.
(c) The S-parameters for the reflected fields. (d) The S-parameters for the transmitted fields. (e)The
components of the extracted electric impedance tensor. (f) The components of the extracted magnetic
impedance tensor.
125
Chapter 8. Tensor Huygens Surfaces
Ze,1
Ze,2
Zy,e
γ
Zz,e
Zy,m
γ
Zz,m
Zz,e
Zy,e
Ze,2
Zm,2
Zz,m
Zy,m
z
y
x
d
λ
Figure 8.12: A cascaded tensor Huygens surface.
arm of the crossed loop and dipole can also be seen in Fig. 8.11a. The full design, of course, consists of
an array of these unit cells across a two-dimensional aperture on the yz-plane as shown in Fig. 8.11b. It
is immediately obvious that is a homogeneous surface as only normally incident and transmitted waves
are being dealt with, unlike the refractive Huygens surface in Chapter 6.
To model this geometry, HFSS is used. The HFSS model consists of the unit cell, placed in a
waveguide constructed with periodic boundary conditions. The periodic boundary conditions are set to
support the propagation of normally incident TE and TM plane waves, excited by ports at either end
of the waveguide. This allows for the simulation of four-port S-parameters of an infinite surface. Given
the square-lattice geometry of the unit cell, the rotation of the unit cell, given by γ, is not applied in
simulation but in post-processing using (8.33). Combined, this allows for the desired S-parameters and
tensor impedances to be determined.
After fine-tuning the printed geometry in HFSS, the post-processed S-parameters and impedance
tensors are plotted in Fig. 8.11c and 8.11d. From the S-parameters in Fig. 8.11c and 8.11d it can be seen
that the associated reflection terms are smallest around 9.88GHz while in the transmission parameters
the cross-pol transmission (TE-to-TM and vice versa) also peaks around the same frequency at -0.38dB
(note that both conduction and dielectric loss are included in the simulation ). This indicates that the
TE to TM polarization conversion is successful using this unit cell design. This is verified by looking at
the tensor impedances extracted from the S-parameters in Fig. 8.11e and 8.11f where a good agreement
is found at 9.88 GHz with the values given in (8.36).
Chapter 8. Tensor Huygens Surfaces
8.5
126
Cascaded Tensor Huygens Surfaces
The tensor Huygens surfaces discussed up to this point have been designed to perform polarization
conversion. As discussed in Chapter 7, chirality is another important polarization effect which can be
achieved by either introducing electric and magnetic coupling or by designing structures which have
an inherent handedness (like a spiral) along the direction of propagation, such that LHCP and RHCP
modes propagate with a different transmission coefficient. While the tensor Huygens surfaces defined
in Section 8.1 have both electric and magnetic responses, these electric and magnetic responses are
superimposed so that they do not couple as given by the two separate impedance boundary conditions.
Because of this, a single tensor Huygens surface is not capable of implementing chiral polarization effects.
However, a cascade of two tensor Huygens surfaces separated by a spacer is proposed here as a way
of implementing chiral polarization effects. This is done by building off of work done in [183, 184, 192].
It was shown there that chiral behaviour could be achieved using electric impedance surfaces that are
rotated with respect to each other, breaking the symmetry between layers and allowing RHCP and LHCP
modes to propagate with different transmission parameters as discussed in Chapter 7. In [183, 184] it
can be seen that while chirality is achieved, being able to precisely control the transmission and reflection
of RHCP and LHCP modes is not obvious or trivial using electirc tensor impedance surfaces, a subject
that is also tackled in the next chapter.
By stacking two Huygens surfaces, back-to-back and rotated with respect to each other, chiral effects
can be implemented. A diagram of back-to-back and rotated Huygens surfaces is illustrated in Fig.8.12.
What is meant by having each Huygens surface rotated with respect to each other? From the definition
of the tensor Huygens surface and their implementation as discussed above, the electric and magnetic
tensor impedances can be characterized by its eigenvalues Zy,e/m , Zz,e/m and rotation angle γ. The
rotation from layer-to-layer then implies that the value of γ changes from the first Huygens surface to
the second Huygens surface as shown in Fig. 8.12. This change in γ from the impedance tensors in the
first layer to the second layer allows for CP waves to propagate differently imparting a handedness to
the structure.
Note that to precisely control the transmission and reflection through cascaded Huygens surfaces,
the transmission and reflection of each layer must be precisely defined. This of course is trivial now with
the S-parameter model defined in Section 8.2, the tensor impedances of each layer can be found. The
trade-off of using cascaded Huygens surfaces is that they require more layers during fabrication when
compared to single layer bianisotropic surfaces [108]. But this tradeoff may be acceptable given the
control that a Huygens surface offers and the simplicity in realizing the structure as will be discussed
below in Section 8.5.3. A more general case of cascaded tensor impedance surfaces is also dealt with in
Chapter 9.
A key assumption in this approach is that the surfaces must be separated enough such that the fields
between the Huygens surfaces can be described by the fundamental propagating modes and no higher
order modes are needed [184]. This is another way of saying that each layer is decoupled. This implies
spacings between layers greater than the lattice spacing of the unit cells comprising the surface. Because
Huygens surfaces generally have a lattice size on the order of λ/10, the spacing between layers can also
be on the same order which means that such cascaded structures can still be made compact. While it is
not discussed here any further, the spacing between layers is a potential degree of freedom that could be
used to explore how higher-order modes propagate between two tensor Huygens surfaces and how they
effect the coupling between surfaces. Such investigations are left to future work.
127
Chapter 8. Tensor Huygens Surfaces
As discussed in Chapter 7, there are two main chiral effects that are of interest. They are polarization rotation and circular polarization selectivity. Polarization rotation involves the plane of polarization
being rotated by some angle for any linear polarization. Circular polarization selectivity involves transmitting one hand of CP while reflecting the other hand. The requirements for a cascaded Huygens
surface which implements these two chiral effects are discussed here.
8.5.1
Polarization Rotation
Polarization rotation is described by an S-parameter matrix given by

0
cos Ψ
0
sin Ψ



cos Ψ
0
− sin Ψ
0 

S = e−jξ 
 0
− sin Ψ
0
cos Ψ


sin Ψ
0
cos Ψ
0
(8.38)
where Ψ is the rotation angle. Compared to (8.28) it can be seen that there is a π phase shift between the
cross-coupled transmission terms which is what allows for the linear polarization rotation. It can easily
be seen that converting these S-parameters into a circularly polarized basis results in an S-parameter
matrix that has perfect transmission for RHCP and LHCP modes but with a different phase shift (See
Appendix D for more details). It can also be seen that if this S-parameter matrix were converted into a
Z-parameter matrix, the resulting Z-matrix cannot be reduced into the four-port lattice network given
in Section 8.2. This is why a single Huygens surface is not capable of implementing such a design.
Instead a cascade of two Huygens surfaces are designed each with the following S-parameter matrices.
The S-parameter matrix of the first layer is given by,

0

− cos Ψ
S1 = e−jξ1 
 0

sin Ψ
− cos Ψ
0
sin Ψ
0
0
sin Ψ


0 

0
cos Ψ

cos Ψ
0
sin Ψ
(8.39)
which rotates a TM polarization by Ψ and a TE polarization by Ψ + π. The second layer is given by an
S-parameter matrix which imposes the π phase shift between the cross coupled terms and is given by
S2 = e
−jξ2

0

−1

0

0
−1
0
0
0

0

0 0
.
0 1

1 0
0
(8.40)
To understand how these S-parameters are cascaded see Appendix D. The tensor impedances for each
layer are given by converting the S-parameter matrices to Z-matrices using (8.23). It can easily be seen
that γ, the rotation angle for the electric and magnetic impedance tensors of each layer is different.
Because there are no reflections from each surface the spacing, d, can be made arbitrary assuming that
it is free-space.
128
Chapter 8. Tensor Huygens Surfaces
(a)
(c)
(b)
(d)
(e)
Figure 8.13: (a) The Ez component of the TE field. A TE polarization is incident on the screen. The two
surfaces are indicated by the black lines. (b)The Hz component of the TM field. (c) The polarization as
traced out by the electric field when illuminated by a TE-polarized beam. The spike in the fields is due to
the boundary condition enforced by the surfaces. (d) The polarization as traced out by the electric field
when illuminated by a TM-polarized beam. (e) The polarization as traced out by the electric field when
illuminated by a 45◦ slanted beam. The spike in the fields is due to the boundary condition enforced by
the surfaces.
TLM Model
Here a TLM model is demonstrated for a polarization rotator with Ψ = 90◦ . The two surfaces are
separated by a spacing of d = λ/9. Note that the spacing is chosen to be larger than the unit cell
size so that it can be assumed that only the fundamental Floquet modes are propagating between the
two surfaces. The impedances of each surface is found by converting the S-parameters in (8.39),(8.40)
into their corresponding Z-parameters. This design is simulated for three different incident Gaussian
129
Chapter 8. Tensor Huygens Surfaces
beams each with a different linear polarization. The first simulation is with a TE-polarized Gaussian
beam (vertical), the second simulation is with a TM-polarized Guassian beam (horizontal) and the third
simulation is with a linear polarization slanted 45◦ with respect to the z-axis.
Looking at the simulated results in Fig. 8.13a, it can be seen from the plotted TE and TM fields
that when the two surfaces, highlighted by the black lines, are illuminated by a TE-polarized Gaussian
beam the transmitted field is rotated by 90◦ . It is also noted that there are minimal reflections from
either surface. Plotting the polarization traced out by the beam in Fig. 8.13c, the rotation of the
linearly polarized beam by an angle of 90◦ is also clearly demonstrated. The same polarization traces
in Fig. 8.13d and Fig. 8.13d for the other two incident Gaussian beams are plotted and again the 90◦
polarization rotation in both cases is obvious. This demonstrates that by cascading two tensor Huygens
surfaces chiral behviour is easily realized.
8.5.2
Circular Polarization Selectivity
Likewise, circular polarization selectivity can also be realized using two Huygens surfaces. Here however,
one of the Huygens surfaces must be designed to be partially reflective. This is a simple extension of
the S-parameter model in (8.28). It follows that if a Huygens surface can be made to have perfect
transmission, it can also be made to have any value between zero and one (An equivalent derivation
can also be done by applying the boundary conditions in (8.3),(8.4)). A circular polarization selecting
surface, or CPSS, that transmits RHCP while reflecting LHCP, has the following S-parameters in a
linear basis [177],

−1


1 1
S= 
2
−j
j
1
−1
−j
j
−j
−j
1
1
j


j

1

1
(8.41)
This can be broken down into two Huygens surfaces each with the following S-parameters. Again to
understand this see Appendix D. For the first surface the partially reflective S-matrix is given by,

−1

1 −1
S1 = 
2
−j
j
−1
−j
j
1
−j
1
−1
j
j


−j 

1

1
(8.42)
Note that this matrix is still symmetric, reciprocal and lossless allowing for (8.23) to be applied. The
second surface meanwhile has perfect transmission but again with a π phase shift between TE and TM
components.

0

−1
S2 = 
0

0
−1
0
0
0

0

0 0

0 1

1 0
0
(8.43)
When the two are placed together, separated by a distance of length d, this gives a CPSS with the Sparameters in (8.41). Again, the spacer size doesn’t really matter here either as only one of the surfaces
are reflective and there are no standing waves between the two layers.
Chapter 8. Tensor Huygens Surfaces
130
Figure 8.14: The polarization traced out by the reflected and transmitted fields when the cascaded
surfaces are illuminated with a TE-polarized beam. A right-handed CP wave is reflected while a lefthanded CP wave is transmitted. The spike in the fields is due to the boundary condition enforced by
the surfaces.
TLM Model
Using the same setup, the TLM solver is used to simulate the cascaded polarization structure described
in above. Again the impedances of both surfaces are given by converting the S-parameters in (8.42) and
(8.43) into their Z-parameters. The spacing between both surfaces is again d = λ/9. This structure
is simulated when it is illuminated by an incident, TE-polarized, Gaussian beam. Because a linear
polarization can be seen as a sum of both left-handed and right-handed circularly polarized beams it is
expected that part of the wave will be transmitted (the LHCP component) while the other part to be
reflected (the RHPC component).
This is simulated in the TLM solver, and the polarization traced out by the reflected and transmitted
fields is plotted in Fig. 8.14. Here it is seen that incident linear polarization is indeed reflected into a
RHPC component while a LHCP component is transmitted as designed. Again this demonstrates how
cascading two tensor Huygens surfaces can indeed extend the polarization control of a single surface to
include chiral effects while still maintaining a compact thickness to the design by using only two surfaces
of subwavelength thickness.
8.5.3
Implementing A Polarization Rotator
Using the rotated, crossed dipole/loop structure introduced in Section 8.4, the polarization rotator can
be designed using two PCB boards stacked back-to-back. Again the design being implemented is a 90◦
polarization rotator. A careful examination of the S-parameters and tensor impedances of the two tensor
Huygens surfaces required for this case, reveals that the first surface is simply the TE-to-TM polarization
converter shown in Section 8.4.1. (This is given by (8.39)). The second surface is implemented with the
same crossed dipole/loop geometry but with γ = 0◦ . (This is given by (8.40)).
Chapter 8. Tensor Huygens Surfaces
131
Figure 8.15: The layout of a cascaded tensor impedance surface implementing a polarization rotator.
A schematic of the polarization rotator is shown in Fig. 8.15, where the two unit cells are separated
a distance of d = 4.8mm in air. Each unit cell has the exact same geometry but the rotation angle γ
changes from layer-to-layer from 45◦ to 0◦ . Again, each board is a 3.04mm thick Rogers 3000 substrate
with three metal layers. The total thickness of this design is 10.8mm which has an electrical length
of λ/2.77. While this is thicker than a single surface, for a polarization screen that exhibits chirality,
along with complete control of the transmission and reflection through the screen, this is an acceptable
tradeoff.
Simulating this design in a commercial electromagnetic solver is not directly possible due to the
geometry seen in Fig. 8.15. Each layer is homogeneous and thus periodic. However there is no global
periodic boundary conditions that satisfy both tensor Huygens surface since the periodicity differs from
either surface. Thus to model the geometry of this cascaded tensor Huygens surface, each surface is
modelled individually in HFSS, to solve for the S-parameters of each surface. Then the S-parameters of
each surface are cascaded with the S-parameters of the air-spacer to find the S-parameters of the overall
geometry in Fig. 8.15. This is further detailed in Appendix D.
Following this procedure, the S-parameters of the cascaded tensor Huygens surface are plotted in
Fig. 8.16. Here it can be seen that the reflected S-parameters in Fig. 8.16a are below -10dB around
9.83 GHz. Meanwhile the transmitted S-parameters indicate a high cross-polarized transmission in the
same frequency range in Fig. 8.16b. However, what differentiates this design between the TE-to-TM
polarization converter in Section 8.4.1 is that for any linear polarization incident on the design, the
transmitted polarization state is rotated by 90◦ . This is demonstrated by the fact that there is a −180◦
phase shift between the cross-polarized transmission terms (S31 and S42 ) as given in (8.38) and shown
in Fig. 8.16c around 9.83 GHz.
To visualize this polarization rotation, the polarization state of the transmitted S-parameters can be
plotted for different input parameters. Assuming an input linear polarization given by,

 

a1,T M
cos Ψi

 

a2,T M   0 

=
.
a
 

 3,T E   sin Ψi 
a4,T E
0
(8.44)
The transmitted polarization state can be found from the complex transmitted S-parameters as given
132
0
0
−5
−5
−10
−10
−15
−20
−25
|S |
11
−30
Magnitude (dB)
Magnitude (dB)
Chapter 8. Tensor Huygens Surfaces
−20
−25
|S |
21
−30
|S |
31
|S |
13
−35
−15
|S |
41
|S |
23
−35
|S |
|S |
33
−40
9
9.5
10
10.5
43
−40
9
11
9.5
Freq (GHz)
10
10.5
11
Freq (GHz)
(a)
(b)
150
θ (deg)
Angle (deg)
100
50
0
−50
−100
angle(S41)
−150
9
angle(S
)
2,3
9.5
10
10.5
11
Freq (GHz)
9
9.5
(c)
10
Freq (GHz)
10.5
11
(d)
Figure 8.16: (a) The S-parameters for the reflected fields. (b) The magnitude of the S-parameters for
the transmitted fields. (c) The phase of the cross-polarized S-parameters for the transmitted fields. (d)
The transmitted polarization state as a function of the incident polarization state and frequency
by,
b2,T M = S21 a1,T M + S23 a3,T E ,
(8.45)
b4,T E = S41 a1,T M + S43 a3,T E
This is plotted in Fig. 8.16d for different input linear polarizations ranging from TE to TM as a function
of frequency. The transmitted polarization state has been normalzed as a function of frequency. It
can be seen that around 9.83 GHz any input linear polarization state is rotated 90◦ indicating that the
surface is indeed acting like a chiral structure.2
2 The S-parameters of this design can easily be converted to circularly polarized S-parameters as given in Appendix D.
Doing this for the S-parameters found in Fig. 8.16 shows that the transmitted LHCP and RHCP fields are almost unity
with a 180◦ phase shift between them, demonstrating the different transmission parameters for CP fields indicative of a
chiral structure. However, the simplest way to illustrate linear polarization rotation is demonstrated in Fig. 8.16d using
the complex transmitted S-parameters.
Chapter 8. Tensor Huygens Surfaces
8.6
133
Measurement
The designs demonstrated in Sections 8.4.1 and 8.5.3 using a rotated crossed-dipole/loop geometry
on three-layer PCB’s are fabricated using standard, commercial PCB processes3 . Since these designs
are homogeneous polarization screens, the challenge then is to be able to measure the full four-port
S-parameters which characterizes each design. This involves measuring both TE-to-TE and TM-to-TM
transmission and reflection but also TE-to-TM and TM-to-TE transmission and reflection. In general,
there are two main ways to characterize the S-parameters for this kind of polarization screen. They are
anechoic chamber measurements and quasi-optical techniques, both of which are briefly reviewed below.
Anechoic chamber measurements involve measuring the screen in a chamber and characterizing either
its near-field or far-field to extract radiation patterns and S-parameters data. One way of measuring
polarization sensitive devices is to use near-field chambers which allows for the transmitted fields to be
measured using a probe which is raster scanned over the aperture of the screen. The screen itself is
illuminated by a horn antenna, which is usually linearly polarized and thus can excite the screen with
either TE or TM polarized waves. Depending on the exact measurement setup, the beam from the horn
can be collimated to better approximate plane wave incidence. The measurement probe is typically
a rectangular waveguide which is mounted on a stage which rotates. This allows the waveguide to
measure both transmitted TE and TM fields (as defined here) depending on its orientation. Thus from
the horn and probe configuration, the near-field setup is capable of measuring the full set of transmitted
S-parameters. This creates a map of the near-fields which can also be converted into a far-field radiation
pattern using the near-to-far-field transform [207]. Examples of near-field measurement systems used to
characterize polarization can be found in [208].
The challenge with near-field measurements are that they are time consuming as the entire aperture
has to be raster-scanned and for polarization sensitive devices multiple measurements are required (once
for incident TE-polarized fields and once for incident TM-polarized fields.). The other major drawback
is that only the transmitted field is measured, leaving the reflected fields unknown. While the total
power transmitted can still be measured and thus the amount reflected inferred indirectly, the exact
reflected S-parameters are unknown. This is a problem when measuring polarization devices such as
circular polarization selective surfaces as will be shown in the next chapter.
Far-field measurement setups are also possible for such polarization screens where the polarization
state is determined by measurements taken directly in the far-field[108, 168]. Many of the same challenges in far-field systems are present in near-field systems. In theory it would be possible to characterize
the reflected fields in a near field chamber using the transmitting horn antenna. However the measured
reflections would not be calibrated and the cross-polarized reflection coefficient would still not be measured. In the measurements discussed here, near-field measurements are not presented. While near-field
measurements of the designs in Sections 8.4.1 and 8.5.3 were carried out in the near-field chamber at the
University of Toronto as shown in Fig. 8.17. The main challenge with the results, besides the ones already
elucidated is the fact that time gating was not available in the VNA used to record the measurements.
Because the setup shown in Fig. 8.17 has a dielectric lens in front of the horn, to collimate the beam
emanating from the horn, multiple-reflections were present in the system. This lead to a non-negligible
ripple in the measured results. Thus, a quasi-optical measurement system was used instead.
To begin, a brief overview of quasi-optical measurements is presented first. Free-space quasi-optical
3 Candor
Industries http://www.candorind.com
134
Chapter 8. Tensor Huygens Surfaces
Dielectric
Lens
Horn
Huygens Probe
Surface
Figure 8.17: The measurement setup in the near-field chamber to characterize tensor Huygens surfaces.
measurement is a measurement techinique used from X-band and above to measure the transmission
and reflection of fields off of material samples, lenses, transmitarrays and more. An overview of this
measurement method can be found in [209]. The basic idea of quasi-optical measurement systems is
that a properly chosen horn antenna couples most of its energy to a fundamental Gaussian beam. Thus,
borrowing from Gaussian optics literature in photonics, the beam from a horn can be collimated onto
a device-under-test (DUT). From there, the reflected and transmitted fields can be measured using a
VNA. The calibration of this setup is usually done using the thru-reflect-line (TRL) method.
In general, these systems have been singly polarized (TE or TM only) and can only measure 2 × 2 S-
parameter matrices. The question then is how can quasi-optical systems be used to characterize the 4×4
S-parameter matrix of a polarization controlling device. In the literature, when this question has arisen,
either for circular-polarization selective surfaces [175, 176] or for artificial chiral materials [155], the singly
polarized quasi-optical systems have been adapted. For example, the full set of transmission parameters
can be characterized using four sets of measurements where the polarization of the transmitting and
receiving horns are setup to measure TE-to-TE, TM-to-TM, TE-to-TM and TM-to-TE transmission.
However, the full set of reflection parameters is still difficult to measure. In general, TE-to-TE and
TM-to-TM can be measured in a quasi-optical system. However the most difficult measurement is the
TM-to-TE measurement and vice versa. One way of attempting to do this is to measure the reflection
coefficient parameters with respect to a known quantity such as a linear-polarizer as in [176]. However
such a measurement is difficult to calibrate as a linear-polarizer is not an ideal calibration reference
for a VNA where methods based on shorts/opens and transmission-lines are preferred (as in the TRL
method).
Thus, it can be seen that trying to measure the full 4 × 4 S-parameter matrix of a free-space polar-
ization controlling device is still a challenging problem. In this thesis, the goal is to be able to measure
the full 4-port S-pamrameter matrix in a single measurement. To do this a four-port, dual-polarized
135
Chapter 8. Tensor Huygens Surfaces
Lens
OMT
DUT
Lens
Circular Horn
V
H
H
V
V
H
OMT
Circular Horn
V
H
z
1 2
y
x
3 4
4-Port VNA
(a)
Lens
OMT
DUT
Lens
OMT
Circular
Horn
Circular
Horn
VNA
(b)
Figure 8.18: (a) A schematic of the four-port quasi-optical setup. (b) A picture of the measurement
setup.
quasi-optical system is proposed as will be discussed below. The measurement setup described here
will be used to characterize the tensor Huygens surfaces under discussion in this chapter as well as the
designs presented in the following chapter.
8.6.1
Measurement Setup
To measure 4-port S-parameters in free-space that characterize the reflection and transmission of both
TE and TM polarized fields, the hardware required includes dual-polarized sources and a four-port
VNA. The dual-polarized sources are required so as to be able to transmit and receive both vertical
and horizontal polarizations. The four-port VNA is needed to enable the measurement of the full Sparameter matrix with a single measurement. Both of these hardware requirements are different when
compared to a traditional free-space quasi-optical setup.
A schematic of the free-space, four-port, quasi-optical setup is shown in Fig. 8.18a. The dualpolarized source consists of X-band Orthomode-Transducers (OMT) connected to X-band conical horns
(both manufactured by Cernex). The OMT’s allow for the conical horn to transmit and receive both
Chapter 8. Tensor Huygens Surfaces
136
TE and TM polarizations. The horns and OMT are connected to an Agilent 4-port VNA where each
port of the VNA transmits and receives a TE or TM polarization. The rest of the measurement setup
consists of a standard quasi-optical design based on [210]. Further details can be found in Appendix E.
A picture of the quasi-optical system is also shown in Fig. 8.18b.
To calibrate this system a standard four-port TRL calibration is used. Here, a metal plate is used
as the reflect standard, a quarter-wavelength line is used as the line standard, while the thru is defined
as the distance between the reference planes of the DUT. To make sure that the quarter-wavelength
line is implemented accurately, the horn, OMT and lens are placed on micrometer translation stages
(Newport Corporation). The overall calibration procedure then is done using the built-in four-port TRL
calibration routine of the VNA. First the reflect standard is measured at all four ports. Next, line and
thru measurements are taken between co-polarized ports (ports 1-3 and ports 2-4). One of the horns is
then rotated 90◦ and a thru and line measurement is taken between either ports 1-4 or ports 2-3. Once
the horn is rotated back, the system is in its calibrated state.
In this calibrated state however, multiple reflections still exist between the ports of the VNA and
calibrated reference planes. This includes reflections between the cables, OMT’s, horns, and dielectric
lenses. Measuring a DUT in this setup, results in significant ripple in the measured S-parameters. To
minimize this, the measured S-parameters are time-gated using the VNA4 allowing for the reflected and
transmitted waves off of the DUT itself to be isolated from the rest of the measurement setup. This is
common in many quasi-optical setups [155].
8.6.2
TE-to-TM polarization converter
The fabricated TE-to-TM polarization converter designed in Section 8.4.1 is shown in Fig. 8.19a. Again,
it can be seen how the unit cells comprising the polarization screen are rotated 45◦ with respect to the
y and z-axis and that the overall thickness of the screen is approximately λ/10 at 10 GHz.
The full four-port S-parameters are characterized in the quasi-optical measurement setup shown
in Fig. 8.18. The measured S-parameters of the polarization converter are plotted in Fig. 8.19b and
8.19c. It can be seen in Fig. 8.19b that the reflected S-parameters are below -10dB above -9.57 GHz.
Meanwhile, the TE-to-TM transmission in Fig. 8.19c is at -0.86 dB when the TE-to-TE and TM-to-TM
transmission is at its minimum at 9.72 GHz. Compared to the simulated results in Fig. 8.11, there is a
good agreement with the exception of a small frequency shift. This shows that tensor Huygens surface
can easily be designed, fabricated and measured.
8.6.3
Polarization Rotator
The polarization rotator is also measured in the four-port quasi-optical setup. Here the polarization
rotator consists of the fabricated TE-to-TM polarization converter shown above along with a second
tensor Huygens surface fabricated with the same printed loop an dipole geometry, except with the unit
cells having γ = 0◦ as discussed above in Section 8.5.3. This is shown in Fig. 8.20a. The two boards are
held together, separated a distance d = 4.8mm by screws and plastic spacers as shown.
The measured S-parameters are also plotted in Fig. 8.20b-8.20d where it is seen that all the reflected
S-parameters are below -10dB from 9.54 GHz to 10.52 GHz and that the TE-to-TM (S41 ) is at -1.69dB
4 The
VNA uses a Kaiser-Bessel window on the time-domain parameters, of which the width and roll-off of the window
can be chosen
137
Chapter 8. Tensor Huygens Surfaces
λ/10
0
0
−5
−5
−10
−10
Magnitude (dB)
Magnitude (dB)
(a)
−15
−20
−25
|S |
11
−30
|S21|
−35
|S12|
−15
−20
−25
|S |
31
−30
|S41|
−35
|S32|
|S |
|S |
22
−40
9
42
9.5
10
10.5
11
−40
9
9.5
Freq (GHz)
(b)
10
10.5
11
Freq (GHz)
(c)
Figure 8.19: (a) The fabricated TE-to-TM polarization converter. (b) The measured S-parameters of
the reflected fields. (c)The measured S-parameters of the transmitted fields.
and the TM-to-TE transmission is at -1.08dB at 9.73 GHz where the co-polarized transmission is at a
minimum. However the phase difference between the TE-to-TM transmission and TM-to-TE transmission is only 130◦ in this frequency range which affects the polarization rotation. To demonstrate this,
the polarization rotation, the transmitted polarization state is plotted for different linearly polarized
inputs using (8.45). (Again this can be done because the quasi-optical system is able to measure the
full complex S-parameters). This plot is shown in Fig. 8.20e where it can be seen that the transmitted
polarization state does indeed rotate for different linear inputs around 9.73 GHz. However, the transmitted state is more elliptical than desired, due to the variation in the phase shift between the TE-to-TM
transmission and TM-to-TE transmission from the ideal 180◦ .
This discrepancy can be attributed to the second tensor Huygens surface in the cascaded structure
with γ = 0 . The (ideal) S-parameters for this surface are given by (8.40). The measured S-parameters
for this tensor Huygens surface alone are also plotted in Fig. 8.20f-8.20g. While the magnitude of
the transmitted S-parameters are close to ideal, the phase shift between the transmitted TE and TM
polarization is again around 130◦ as opposed to the desired 180◦ phase shift. This discrepancy can be
attributed to some difficulties in fabricating the vias used to form the printed loops in the second layer5 .
Nonetheless, this cascaded tensor Huygens surface does demonstrate that by stacking two tensor
Huygens surfaces back-to-back with different values of γ, chiral polarization effects can indeed be realized.
5 This
was confirmed in a discussion with the vendor who fabricated the boards
138
Chapter 8. Tensor Huygens Surfaces
Layer 1
λ/3
Layer 2
0
200
−5
150
−10
−10
100
−15
−20
−25
|S11|
Magnitude (dB)
0
−5
Magnitude (dB)
Magnitude (dB)
(a)
−15
−20
−25
|S31|
50
0
−50
−30
|S21|
−30
|S41|
−100
−35
|S12|
−35
|S32|
−150
|S22|
−40
9
9.5
10
10.5
angle(S41)
|S42|
−40
9
11
9.5
Freq (GHz)
10
10.5
angle(S23)
−200
9
11
9.5
Freq (GHz)
(b)
10
10.5
11
Freq (GHz)
(c)
(d)
0
150
−5
100
−15
−20
−25
|S31|
−30
|S41|
−35
|S32|
|S42|
9
9.5
10
Freq (GHz)
10.5
(e)
11
−40
9
9.5
10
Freq (GHz)
(f)
10.5
11
Magnitude (dB)
Magnitude (dB)
θ (deg)
−10
50
0
−50
−100
angle(S31)
−150
9
angle(S42)
9.5
10
10.5
11
Freq (GHz)
(g)
Figure 8.20: (a) The fabricated polarization rotator. (b) The measured S-parameters of the reflected
fields. (c) The measured magnitude of the S-parameters of the transmitted fields (d) The measured
phase of S-parameters of the transmitted fields. (e) The polarization state of the transmitted fields as a
function of frequency and input polarization state. (f) The measured magnitude of the S-parameters of
the transmitted fields for the second Huygens surface only. (g) The measured phase of S-parameters of
the transmitted fields for the second Huygens surface only.
8.7
Summary
In this chapter, the theory, design and measurement of tensor Huygens surfaces has been investigated.
By using the more generalized tensor impedance boundary condition, both TE and TM fields can be
controlled in a tensor Huygens surface resulting in polarization controlling effects. This was demonstrated
by designing several polarization controlling surfaces. Many of the ideas in Chapter 6 were extended
to this chapter, including generalizing a lattice circuit to an MTL network, and constructing numerical
models using the TLM method.
Chiral effects were also investigated using cascaded tensor Huygens surfaces which introduce a handedness for propagating CP waves and allow for effects such as polarization rotation and circular polar-
Chapter 8. Tensor Huygens Surfaces
139
ization selectivity to be realized.
Finally measured results were also generated using a novel four-port quasi-optical testbench to completely characterize the transmitted and reflected S-parameters of a tensor Huygens surface. This was
demonstrated for both a TE-to-TM polarization converter and a polarization rotator.
CHAPTER
9
Tensor Impedance Transmitarrays
This chapter is the exception to the rest of the thesis, where the focus here is not on Huygens surfaces
but on electric tensor impedance surfaces. The motivation for looking beyond Huygens surfaces in this
chapter arises from the discussion in Chapter 8 on cascaded tensor Huygens surface used for implementing
chiral polarization control.
The idea of cascading two tensor Huygens surfaces that are rotated with respect to each other,
arose from other cascaded impedance surfaces in the literature. For example in [184], electric tensor
impedance surfaces were cascaded together with each layer rotated with respect to the other. In this
case, by rotating each surface, chiral effects such as linear polarization rotation or circular polarization
selectivity can be realized. As discussed in Chapter 7 these chiral effects are due to circularly polarized
waves propagating with different transmission and reflection coefficients. This is because the rotated
impedance surfaces imparts a handedness to the structure in the direction of propagation and as a result
are circularly birefringent.
In the previous chapter, it was demonstrated how the exact reflection and transmission coefficients can
be controlled in a cascaded tensor Huygens surface by precisely engineering the electric and magnetic
tensor impedances of each layer. However for the case of cascaded electric tensor impedances, as in
[184], it has not been demonstrated how to precisely specify the reflection and transmission coefficients
by choosing the tensor impedance of each layer.
In this chapter, the problem of synthesizing cascaded electric tensor impedances for chiral polarization
control is investigated. The reason that this problem is tackled here is because of some of the ideas
presented in the previous chapter can be adapted to this problem. In Chapter 8, a multi-conductor
transmission-line circuit model was introduced to model tensor Huygens surface. Here, this MTL model
is adapted to show how cascaded tensor impedances can be designed to implement the desired reflection
and transmission coefficients to implement chiral polarization effects. For the remainder of this chapter,
the cascaded tensor impedance will be referred to as a tensor impedance transmitarray, as a stack of
impedance surfaces is a common configuration for a transmitarray.
140
141
Chapter 9. Tensor Impedance Transmitarrays
Ze,1
Zy,e
γ
Zz,e
Ze,2
Zz,e
Zy,e
z
y
x
Ze,3
d
Zz,e
γ
Zy,e
λ
d
TE
E
H
k
H
E
k
TM
Figure 9.1: A schematic of an N = 3-layer chiral tensor impedance transmitarray. Each sheet of the
transmitarray consists of a tensor admittance. The 2×2 tensor admittance is described by two eigenvalues
and a rotation angle γ which maps to a crossed dipole geometry. The TE and TM fields are also shown
again for reference.
9.1
Theory
A tensor impedance transmitarray consists of multiple tensor impedance surfaces separated by a dielectric
spacer. This is depicted in Fig. 9.1. Here, each impedance surface sits on a plane parallel to the yz-plane
separated by dielectric spacers with spacing d and dielectric constant εr . It is assumed, without loss of
generality, that the spacers are identical between each surface.
To simplify the problem, plane wave propagation through this stack of impedance sheets is examined,
with the propagation direction confined to the x-axis (i.e normal incidence only) as in the previous
chapter. Again the same polarization definitions apply from Chapter 8 with regards to TE and TM
fields and the polarization state given by Ψ and Φ.
In the previous chapter, the boundary condition for an electric response from an array of subwavelength scatterers was given by (8.1). When these tensor impedance surfaces are cascaded together
and rotated with respect to each other (as defined by γ, the rotation angle of the impedance tensor),
the waves propagating through each layer of the transmitarray cannot be separated into orthogonal
TE and TM fields that do not couple as done with many transmitarrays and FSS’s [161, 170, 197].
Instead the waves propagating in the transmitarray must include the transmitted/reflected TE waves,
transmitted/reflected TM waves and transmitted/reflected TE-to-TM waves and vice versa. If the tensor
impedances are designed properly these will result in circularly polarized modes and is what allows for
chiral polarization control to be realized.
142
Chapter 9. Tensor Impedance Transmitarrays
3
Yo,β
1
3
4
d
2
Ysh
1
4
2
Γ ,Y
1 in,1
Γ ,Y
2 in,2
Γ ,Y
2 in,2
(a)
Γ1,Yin,1
(b)
3
Y1,sh
Yo,β
Y2,sh
1
2
a
Y a, Γ
a
b
d
Y c, Γ
4
d
c
c
Y d, Γ
d
Y ,Γ
b b
(c)
Figure 9.2: The basic elements for an MTL model of a tensor impedance transmitarray. (a) The MTL
model of free space. The MTL supports a vertically-polarized mode and a horizontally polarized mode
as defined previously and as annotated above. The two modes are orthogonal in the MTL. Note that
we repeat the ground wire for clarity. (b) A shunt tensor admittance loading the MTL. This models the
surface admittance of each layer of the transmitarray.(c) A two-element, tensor admittance matching
network separated by an MTL spacer. Ya and Yd are given. We can then solve for Ysh,1 and Ysh,2 .
The key concept here is extending the multi-conductor transmission-line (MTL) framework proposed
in Chapter 8 to semi-analytically design a tensor impedance transmitarrays. This MTL model allows
for a way to solve for each tensor impedance in the transmitarray to achieve a desired set of reflection
and transmission coefficients that can implement chiral polarization effects.
9.1.1
Multi-Conductor Transmission-Line Model
Transmission-line models are a commonly used method to understand and design transmitarrays. For
transmitarrays consisting of scalar surface impedances, a two-wire transmission-line model is used to
model free-space, where each layer of the transmitarray is a shunt impedance along the transmissionline. By using some form of transmission-line analysis such as filter theory [89], ABCD-matrix analysis
[138, 170] or Smith Chart design [77, 161], a suitable value for the scalar impedance of each layer can
be found which minimizes reflections while varying the phase of the transmitted wave.
This kind of modelling of scalar transmitarrays can be extended to tensor transmitarrays by using
the MTL model introduced in Chapter 8 as there are both TE and TM waves propagating through the
transmitarray. For convenience some of the basic MTL definitions in Chapter 8 are repeated here for
convenience. Free-space in the model consists of a multi-conductor transmission-line, shown again here
in Fig. 9.2a for convenience. The MTL is characterized by a propagation constant and characteristic
Chapter 9. Tensor Impedance Transmitarrays
143
impedance given by (8.16)-(8.17).
For a dielectric spacer between each surface admittance, a similar MTL network can also be used
√
except with (8.16) and (8.17) multiplied by εr , the dielectric constant of the material. Note that the
length of the spacer, d would also correspond to the length of the MTL.
The last part of the model is the admittance surface described by (8.1). Again from Chatper 8, the
admittance surface is modelled in the MTL network as a shunt admittance loading the MTL itself as
shown again in Fig. 9.2b. Because, these circuit elements are in shunt with the host MTL, characterizing
them as admittances is more convenient as they simply add as in the two-wire transmission-line case.
Thus, the MTL model of a tensor impedance transmitarray can be constructed by cascading these
basic circuit elements together as will be shown in the following section. Before looking at the exact
MTL model for the transmitarray, some basic quantities to characterize the waves in an MTL model of
a transmitarray can be defined.
The two quantities that are of concern are the reflection coefficient matrix, Γ and the
" input admittance
#
Γyy Γyz
matrix Yin . At any point on the MTL line the reflection coefficient matrix, Γin =
and
Γzy Γzz
"
#
Yin,yy Yin,yz
input admittance matrix Yin =
are quantities that are related to each other by,
Yin,zy Yin,zz
Yin = Yo (I − Γ)(I + Γ)−1
(9.1)
Γin = (Yo − Yin )(Yo + Yin )−1 .
(9.2)
With the input admittance/reflection coefficient matrices defined, the effect of the shunt admittance
tensor and dielectric spacer on these quantities can also be set. As it will be discussed below, the purpose
of the MTL model is to be able to choose the tensor admittances of each layer so that the reflection
and transmission of the whole transmitarray meet the design criteria. To guarantee that the reflection
coefficient has the desired value at the input of the transmitarray, the shunt admittance and dielectric
spacer are used to engineer these quantities. These scenarios are illustrated in Fig. 9.2a-9.2b. For a
shunt tensor admittance element with an admittance of Ysh , the input admittance matrix looking on
either side of Ysh as shown in Fig. 9.2b are related to each other by simple subtraction,
Yin,1 = Yin,2 − Ysh .
(9.3)
This is analogous to a shunt load on a two-wire transmission-line. The reflection coefficient matrix can
then be found by using (9.2). Because Ysh is assumed to be imaginary, only the imaginary part of the
input admittance is altered.
For an MTL spacer, the reflection coefficient matrix on either side of the MTL is related via the
Chapter 9. Tensor Impedance Transmitarrays
144
phase shift of the MTL spacer1 ,
Γin,2 = e−j2βd Γin,1 ,
(9.4)
where β = 2π/λ. Correspondingly the input admittance matrix on either side of the MTL spacer is
given by
Yin,2 =
1
−1
(Yin,1 + jYo tan βd) (Yo + jYin,1 tan βd) .
η
(9.5)
Note that the MTL spacer affects both the real and imaginary parts of the input admittance matrix.
Now that the input admittance and reflection coefficient matrices can be controlled using these basic
MTL elements, more complex MTL networks can be constructed to control the value of the reflection
coefficient matrix. An example of a more involved network is pictured in Fig. 9.2c which shows two
shunt tensor admittance elements Ysh,1 and Ysh,2 separated by an MTL spacer of length d. Such a
network corresponds to two tensor impedance surfaces cascaded back-to-back. For this MTL network,
it is assumed that the input admittance matrices Ya and Yd at points a and d in Fig. 9.2c are known
quantities. The question then is how are the values of Ysh,1 and Ysh,2 determined, assuming that the
properties of the MTL spacer are fixed, so that the input admittance Yd at point d can be transformed
to Ya at point a. It can easily be seen that this problem is the generalization of the two-element shunt
matching network from basic transmission-line theory.
To find the desired tensor admittances,Ysh,1 and Ysh,2 , the known input admittances at points a
and d can be used to find the admittance at points b and c of the MTL network as a function of Ysh,1
and Ysh,2 respectively, using (9.3). From there, given that the MTL segment between b and c is of a
known and fixed length, Yb and Yc can be substituted into (9.5). Applying some linear algebra, results
in an equation for Im {Yc } = Bc , given to be
1
1
− Gd −1 Bc − Bc Gd −1 + tan(βd)Bc Gd −1 Bc +
η
η
1
Gd −1 + tan(βd)Gd −
2
η tan(βd)
1
tan(βd)
1
Gb −1
I+
I = 0,
η
η tan(βd)
η
(9.6)
To solve this equation, it is recognized that it is of the form of
0 = AT Bc + Bc A − Bc CBc + Q
(9.7)
where the coefficients A, C and Q can be seen from (9.6). This equation is known to be the algebraic
Riccati equation which has applications in control theory [211, 212]. Solutions to this equation can
be constructed explicitly from the eigenvectors of a block matrix of the coefficients [211] or through
βd
β d −1
−jβ
speaking the reflection
coefficient is
Γ (0)(ejβ
) where the matrix exponential operation
"
# given by Γ (d) = e
jβ1 d
e
0
βd
βd
is defined to be ejβ
= P−1
P, and P is the matrix that diagonalizes ejβ
. However since the vertically0
ejβ2 d
polarized and horizontally-polarized modes in free space and the dielectric spacer are orthogonal, this reduces to simply
a scalar multiplication of the reflection coefficient matrix by a complex exponential. A similar statement applies to the
tan βd term in (9.5).
1 Strictly
145
Chapter 9. Tensor Impedance Transmitarrays
3
3
1
Y1
Yo,β
Y2
Yo,β
d
Y3
Yo,β
Y4
4
d
d
2
1
Y1
2
a
Y2
b
c
Y3
d
e
Γe,Ye
Γc,Yc Γd,Yd
Γa,Ya Γb,Yb
(a)
Y4
f
Γf,Yf
g
Γg,Yg
h
Γh,Yh
(b)
Figure 9.3: (a) An equivalent circuit model for an asymmetric tensor admittance transmitarray. (b) The
input admittance and reflection coefficient tensor defined at eight different points within the asymmetric
tensor admittance transmitarray circuit model.
numerical tools 2 .
With a solution for Bc , this allows for Ysh,1 to be found using (9.3)
Ysh,1 = Yc − Yd = jBc − jBd .
(9.8)
Then using (9.5) again, Yb is found, which can be used to find Ysh,2 ,
Ysh,2 = Ya − Yb = jBa − jBb .
(9.9)
Equations (9.6)-(9.9) provide the closed form expressions for the shunt tensor admittances in the MTL
network in Fig. 9.2c. As it will be shown below, this solution forms a key step in designing tensor
impedance transmitarrays by allowing for a desired reflection coefficient to be enforced.
It is also noted that the transmission and reflection through a network of tensor admittance layers
separated by MTL spacers needs to be found as well. This is done by finding the S-parameters of the MTL
spacer and the shunt admittance tensors separately and converting them to 4-port transfer matrices.
By multiplying the transfer matrices and converting them back to S-parameters, the transmission and
reflection through the entire network can be determined. This is described in more detail in Appendix D.
9.2
Tensor Impedance Transmitarrays For Chiral Polarization
Control
With the MTL theory developed above to model multiple tensor surface impedances cascaded together,
these concepts can be used to create a transmitarray capable of implementing chiral polarization effects.
As discussed in the beginning of this chapter, the chiral polarization effects that are of interest are circular
polarization selectivity and polarization rotation. To design transmitarrays capable of doing this, the
MTL model is used to implement the desired S-parameters of a tensor impedance transmitarray. This is
demonstrated below by developing a design procedure for a tensor impedance transmitarray. With this
design procedure, some examples will also be discussed.
2 For
example, in MATLAB this equation can be solved using the care command
4
146
Chapter 9. Tensor Impedance Transmitarrays
9.2.1
Design Procedure Using The MTL model
Before discussing the design procedure, some preliminary concepts must be refreshed. First to implement
the transmitarray, the S-parameter matrix must be defined. Based on the discussion, in Chapter 8, the
S-parameter matrices for chiral polarization controlling effects were defined and are repeated here for
convenience using the same conventions as the previous chapter. For a polarization rotator, the desired
S-parameters in the linearly-polarized basis are given by,

0
cos Ψ
0
sin Ψ



cos Ψ
0
− sin Ψ
0 


S=
− sin Ψ
0
cos Ψ

 0
sin Ψ
0
cos Ψ
(9.10)
0
Likewise for the circular polarization selective surface (CPSS) in a linearly-polarized basis, the Sparameters can be described by [177]

−1


1 1
S= 
2
 j
−j
1
j
−1
j
j
1
−j
1
−j


−j 

1

1
(9.11)
For this set of parameters, a RHCP wave is reflected into a RHCP wave while a LHCP wave is transmitted
as a LHCP wave. The first case is a reflectionless transmitarry while the second case has both non-zero
reflection and transmission coefficients.
The question then arises as to how many tensor impedance surfaces, N , are needed to implement
the desired S-parameters given above. Ultimately, this is answered by the fact that the relationship
between the impedance tensors and the S-parameters are given by some algebraic relationship. Thus
the number of unknowns in the S-parameter matrix can be related to the number of variables provided
by each impedance surface.
For the S-parameter matrix, which is assumed to be lossless and reciprocal, it can easily be seen that
there are three reflection coefficient terms (S11 = Γyy , S33 = Γzz , S31 = Γyz ) which are complex for a
total of six variables. From the constraints of the S-parameter matrix being lossless and reciprocal the
transmission matrix has three remaining variables [204], an overall phase shift of the transmitted field,
and the polarization state of the transmitted field which is described by Ψ and Φ defined in Chapter 8.
Thus in general, a transmitarray which alters the polarization has nine variables to satisfy.
A single tensor admittance surface which is imaginary and reciprocal has three unknowns, which are
Yy , Yz , and γ, in the diagonalized basis as discussed in Chapter 8. Thus it can easily be seen that, at
least N = 3 admittance surfaces are needed. As it will be shown below, the S-parameters for a CPSS
can indeed be implemented using N = 3 cascaded tensor admittance surfaces separated by dielectric
spacers. However for a polarization rotator, N = 4 tensor admittance surfaces are required. This is
because the polarization rotator is reflectionless which is a more stringent requirement. A more detailed
explanation for why a three-layer tensor impedance transmitarray cannot be made reflectionless is given
in Appendix F.
Finally, as a reminder, each tensor impedance surface is implemented using crossed dipoles which
map, one-to-one, to the diagonalized values of the tensor impedance as discussed in Chapter 8.
Chapter 9. Tensor Impedance Transmitarrays
147
With the S-parameters and structure of the transmitarray defined, the MTL model can also be
constructed. For an N = 4 transmitarray, the MTL circuit model is shown in Fig. 9.3. The equivalent
circuit for the N = 3 case can easily deduced from Fig. 9.3 as well. To give some further context for the
design procedure that will be used it is helpful to briefly look at how scalar impedance transmitarrays
are designed.
For a scalar impedance transmitarray, one common design method is to use ABCD/Transfer matrices.
For a three layer design, the transfer matrices for each layer are multiplied together and converted to
S-parameters which sets up a system of equations to solve. Usually, because there are only two unknowns
total, the system of equation can be easily solved using closed-form expressions. However, if this approach
is taken with the MTL model in Fig. 9.3, by trying to use transfer matrices multiplied together, challenges
are encountered with regards to the number of unknowns and the number of equations. The transfer
matrices are now 4×4 and multiplying them together and converting them to S-parameters leads to very
complicated expressions as well as a system of 9 or more equations for which no closed form solution
can be found.
It is clear that transfer matrices alone are not helpful. Thus, the approach taken here is to use the
MTL network and the associated operations defined in Section 9.1.1. This leads to a semi-analytical
method that allows for Y1−4 to be solved for. This approach is based on enforcing the desired reflection coefficient at the input, Γin . This is done by tracking the value of Γin at different points in the
transmitarray as shown in Fig. 9.3b for different values of Y1 , Y2 , Y3 and Y4 . For the N = 4 case, the
following design procedure is used:
1. A value is assumed for Y1 and Y4 .
2. Given the desired S-parameters in either (9.10) or (9.11) the reflection coefficient Γ can be calculated looking into the transmitarray at a and the reflection coefficient looking into ports 3 and 4 at
h (which is simply Yo ). From there, using (9.3) and (9.5), the input admittance can be calculated
at b and g and subsequently c and f using the basic MTL operations.
3. With the input admittance known at points c and f the susceptance, Be , at point e can be found
using (9.6).
4. Finally with the admittance at e known, (9.8) can be used to find Y3 . Then, the admittance at e
is used to determine the admittance at d which allows for (9.9) to be used to find Y2 . This gives
the tensor admittance for all four layers.
5. With the admittance tensors Y1−4 found, the full S-parameters of the transmitarray, including
the transmission, can be evaluated as described in Appendix D. If the desired transmission values
are not achieved steps 1-4 are repeated until they are.
The semi-analytical nature of this approach can be seen where Y2 and Y3 are solved for while
iteratively choosing Y1 and Y4 . This approach reduces the solution space of the problem and only
looks for solutions with the desired reflection coefficient, Γ. Conceptually, this approach is similar to
graphical techniques for scalar transmitarrays where solutions are found by treating the transmitarray
as a matching problem [77, 161]. For a transmitarray with N = 3 layers, this procedure can easily be
adapted to solve for the tensor admittances, where only Y1 needs to be chosen while solving for Y2 and
Y3 .
148
Chapter 9. Tensor Impedance Transmitarrays
While it is left for future research, this approach could potentially be extended to use the tensor
impedances to improve other metrics such as the bandwidth of the transmitarray. This, of course, would
require using this model over a given bandwidth and picking values which bound the reflection and
transmission matrices within a desired tolerance to achieve a desired behaviour.
9.2.2
Examples
To tie this all together two tensor impedance transmitarrays are designed which implement the polarization rotator given in (9.10) and the CPSS given in (9.11).
A Polarization Rotator
The design for a polarization rotator is done at 10 GHz and again implements a Ψ = 90◦ polarization
rotator which rotates any linear polarization by 90◦ . The spacing between the tensor admittance surface
is set to be d = λ/7 at this frequency. Having established that N = 4 layers are needed, the design
procedure defined above is used to find the admittances of Y1−4 .
One possible solution is given by,
Y1 =
Y3 =
"
"
j2.72
0
0
j10.50
#
−j5.25 −j7.78
−j7.78 −j4.93
mf,
#
mf,
Y2 =
"
−j2.65
Y4 =
j1.68
#
j1.68 −j6.61
"
#
j5.05 j1.49
j1.49 j2.12
mf,
mf.
It is noted that the rotation angle of each surface is given by γ1 = 0◦ , γ2 = −20.18◦ , γ3 = −45.58◦ and
γ4 = −67.20◦ , showing the progressive twist along each admittance tensor which is fundamentally what
allows for LHCP and RHCP waves to propagate differently [183, 184].
To implement these admittance surfaces, crossed dipoles are designed on four 0.75 mm Rogers 3203
substrates that are patterned with a single layer of copper. Note that no vias are required. Each arm
of the crossed dipole consists of a printed inductor or capacitor to implement the desired reactance
of the eigenvalues of the admittance tensor with a unit cell size of 4mm×4mm. The crossed dipole
is then rotated to achieve the desired admittance tensor. Each surface is simulated individually in
HFSS using periodic boundaries with metal and dielectric losses included as in Chapter 8. The tensor
admittance of each layer is verified using the equivalent Z-parameter model also defined in Chapter 8. The
corresponding printed dipoles of each layer are shown in Fig. 9.4 along with their stacked configuration.
The overall S-parameters through the four-layer structure are calculated by numerically cascading the
S-parameters of each simulated layer with those of the air spacer as done in the previous chapter. This
is done because each layer has a different rotation angle and thus a different set of periodic boundary
conditions (lattice vectors) and there is no global period for the full four-layer structure in the transverse
direction (yz-plane) despite each layer being periodic.
The S-parameters are shown in Fig. 9.4 and it can be observed that the large transmission for the
cross-polarized transmission terms as well as a 180◦ phase shift between S41 and S32 as predicted for a
90◦ polarization rotator, which is what allows for the 90◦ polarization rotation. The individual entries
of the reflection coefficients are also below -10 dB around the design frequency. This realization of a
polarization rotator shows that LHPC and RHCP modes are both transmitted with high transmission
but different phases. Using the same kind polarization map in Chapter 8, the transmitted polarization
149
Chapter 9. Tensor Impedance Transmitarrays
λ/7
λ/7
2.3mm
0.3mm
0.1mm
1.35mm
0.1mm
4mm
0.2mm
0.1mm
0.2mm
2.35mm
0.1mm
0.2mm
2mm
1.5mm
1.8mm
4mm
λ/7
0.3mm
0.1mm
1.4mm
(a)
(b)
(c)
(g)
(e)
(h)
Ψ (deg)
(f)
(d)
9
9.5
10
Freq (GHz)
10.5
11
(i)
Figure 9.4: The layout and S-parameters of a polarization rotator designed using printed and rotated
crossed-dipoles. Each layer sits on a 0.75mm Rogers 3203 substrate. Depending on the eigenvalues
of Y1−4 , each arm of the crossed-dipole is loaded with a printed inductor or capacitor. (a) Layer 1
which implements Y1 . (b) Layer 2 which implements Y2 . (c) Layer 3 which implements Y3 . (d)
Layer 4 which implements Y4 . (e) The cascaded layers showing the relative rotation between layers.
The substrate is not shown. Each layer is separated by λ/7. (f) The reflected S-parameters. (g)
The transmitted S-parameters. (h) The phase shift between the cross-polarized transmission. (i) The
transmitted polarization state as a function of frequency and input polarization state.
state is plotted as a function of both frequency and input polarization, in Fig. 9.4i. Here it can be
seen that as the input polarization varies from a linear TE polarization to a linear TM polarization, the
corresponding transmitted state is rotated by 90◦ around 10 GHz as expected.
150
Chapter 9. Tensor Impedance Transmitarrays
4.8mm
0.2mm
0.4mm
0.2mm
4mm
1mm
0.47mm
4mm
γ=9.5o
γ=-8.5o
γ=0o
(a)
4.8mm
(b)
(c)
(e)
(d)
(f)
Figure 9.5: The layout and S-parameters of a CPSS designed using printed and rotated crossed-dipoles.
Each layer sits on a 0.8mm FR-4 substrate. (a) Layer 1 which implements Y1 . (b) Layer 2 which
implements Y2 . (c) Layer 3 which implements Y3 . (d) The cascaded layers showing the relative
rotation between layers. The substrate is not shown. Each layer is separated by 4.8mm. (e) The
transmitted and reflected LHCP-to-LHCP fields and RHCP-to-RHCP fields. (f) The transmitted and
reflected LHCP-to-RHPC and RCHP-to-LHCP fields.
A Circular Polarization Selective Surface
Unlike the polarization rotator, a circular polarization selective surface (CPSS), is not reflectionless but
is designed to reflect one hand of circular polarization while passing the other [108, 177, 184]. The
S-parameters for a CPSS which rejects RHCP and transmits LHCP are given in (9.11). Here N = 3
layers with an air spacing of 4.8mm between each layer at a design frequency of 10 GHz are chosen.
Using the design procedure given in Section 9.2.1, the admittance tensors for all three layers are
found to be,
Y1 =
"
−j12.88
−j2.63
−j2.63
j1.366
#
mf, Y2 =
"
j1.75
0
0
−j319.16
#
mf,Y3 =
"
−j13.08
j2.93
j2.93
j1.73
#
mf.
Again the value for the rotation angle of each surface is γ1 = 10.13◦ , γ2 = 0◦ and γ3 = −10.79◦ , showing
the progressive twist in the rotation angle through the transmitarray.
To implement the CPSS, three 0.8mm FR-4 substrates are used that are patterned on a single layer
with copper. The choice of FR-4 here is not ideal for 10 GHz in terms of its high-loss, but is used because
of its mechanical stability as discussed below in Section 9.3 when fabricating this design. The rotated
Chapter 9. Tensor Impedance Transmitarrays
151
dipoles are designed again in HFSS using periodic boundaries and the pattern for each rotated dipole is
shown in Fig 9.5. The simulation of each unit cell includes both conductor and dielectric losses. Note
that for each layer, one arm of the crossed-dipole is ‘missing’ because the capacitance between adjacent
dipoles is sufficient that an orthogonal dipole is not needed. Again, the overall S-parameters of the CPSS
are found by numerically cascading the simulated S-parameters for each layer and the corresponding air
spacer.
To plot the simulated S-parameters, they are first converted to a circular polarization basis as that is
the simplest way to see how the RHCP and LHCP modes are reflected and transmitted (See Appendix 4
for further details). In Fig. 9.5e the transmission and reflection of RHCP-to-RHCP and LHCP-to-LHCP
are plotted where it can be seen that the transmission of LHCP waves and the reflection of RHCP waves
is large at the design frequency of 10GHz. Fig. 9.5f shows the transmission and reflection of RHCP-toLHCP and LHCP-to-RHCP which are all less than -15dB. Compared to meander-line polarizers which
are used to implement CPSS’s [175], this design is much thinner, requiring only three PCB layers.
When compared to Pierrot-type cells [179, 180], it is clear that this approach does not require vias, or
transmission-lines along the direction of propagation either.
From these two examples it is seen that the design procedure is successful in synthesizing cascaded
tensor impedance surfaces, allowing for the realization of chiral polarization effects with these kinds of
transmitarrays.
9.3
Measurement
Using the four-port quasi-optical system described in Chapter 8, the four-port S-parameters of a tensor
impedance transmitarray can be characterized as well. Of the two examples described above, the CPSS
was fabricated based on the design given in Section 9.2.2 using standard PCB fabrication on three 0.8mm
FR4 boards 254mm×254mm in size and held together using 4.8mm plastic spacers with screws around
the edge of the board as shown in Fig. 9.6. The total thickness is 12mm or 0.4λ at the design frequency
of 10 GHz. The choice of FR-4 was motivated by the fact that it was mechanically stiff enough to stack
each PCB in this manner while keeping the boards parallel to each other.
After the quasi-optical system is calibrated, the measurement simply involves placing the fabricated
CPSS at the reference plane of the measurement system. Again,to minimize the reflections between the
VNA, horn, lenses, and OMT, the measured S-parameters are time-gated. The measured S-parameters
in the linear polarization basis are converted to a circular polarization basis as well and plotted in
Fig. 9.6c and Fig. 9.6d. Again it can be seen that the LHCP wave is transmitted, while the RHCP wave
is reflected. The cross-polarized transmission and reflection in Fig. 9.6d is also small, and below -15dB.
Compared to Fig. 9.5e-9.5f, a general agreement can be found between the simulated and measured
results. However, note that the transmission of the LHCP wave peaks at −3.7dB, compared to −2.0dB
in simulation which is lower than desired. This can be attributed to the loss of the FR4 at X-band.
This is confirmed because the cross-polarized transmission and reflection between LHCP and RHCP is
small as shown in Fig. 9.6d and the reflection of the LHCP-to-LHCP wave is also below -10dB indicating
that the transmitted LHCP wave is not being lost to the other scattering parameters. In the design of
these kinds of tensor impedance transmitarrays, if a lower-loss substrate was used the transmission of
the LHCP wave would be higher. This is an area for further optimization in designing and fabricating
these kinds of transmitarrays.
152
Chapter 9. Tensor Impedance Transmitarrays
(a)
(b)
0
0
−10
−10
|Tlhrh|
−20
|Rlhcp|
|Tlhcp|
−30
Magnitude (dB)
Magnitude (dB)
|Trhlh|
|Rlhrh|
|R
|
rhlh
−20
−30
|Rrhcp|
−40
9.5
|Trhcp|
10
Freq (GHz)
(c)
10.5
−40
9.5
10
10.5
Freq (GHz)
(d)
Figure 9.6: (a) A picture of the three layers of the fabricated CPSS . (b) A side view of the CPSS. (a)
The measured transmission and reflection of LHCP-to-LHCP, and RHCP-to-RHCP. Overlaid in grey
are the simulated S-parameters. (d) The measured transmission and reflection of LHCP-to-RHCP, and
RHCP-to-LHCP. Overlaid in grey are the simulated S-parameters.
Nonetheless, this measurement verifies the synthesis method discussed above and shows that this
MTL model is successful in designing transmitarrays which can control CP polarization.
9.4
Comparing Tensor Huygens Surfaces and Tensor Impedance
Transmitarrays
At this point, two methods for synthesizing impedance surfaces capable of chiral polarization control
have been presented in this thesis; these are the cascaded tensor Huygens surfaces discussed in Chapter 8
and the tensor impedance transmitarray presented here in the current chapter. It is worth pausing at
this point to compare these two methods in terms of both their synthesis, design and performance to
discuss their respective tradeoffs.
With regards to the synthesis of the required tensor impedances, the tensor Huygens surfaces are
fairly easy to solve as they consist of two surfaces cascaded together with at least one of them being
reflectionless. Once the S-parameters are found, as discussed in Appendix D, the tensor impedances
are easily found using the equivalent circuit models given by the Z-parameter matrices. For the tensor
impedance transmitarray the synthesis is much more involved, to the point that closed-form expressions
do not exist. As discussed above, in Section 9.2.1, the design procedure used to find the tensor impedances
for each surface is iterative and while it could potentially be sped up by combining the presented
Chapter 9. Tensor Impedance Transmitarrays
153
procedure with optimization techniques it still requires a search of all the possible solutions. With
regards to synthesis then, the tensor Huygens surfaces are much more simpler to solve for.
For the physical design of the surfaces, the tensor impedance transmitarrays have an edge. Each
surface in the tensor impedance transmitarray consists of a crossed-dipole type geometry to implement
the required electric response. These crossed-dipole geometries are very easy to model in commercial
CAD tools, such as HFSS, and run quickly even when including substrate and conductor effects. They
are also relatively simple to design as there are only a small number of variables to play with for each
impedance surface. For the tensor Huygens surface the design is more involved as the surfaces consists of
both an electric and magnetic response requiring both crossed-dipoles and crossed-loops. This is a more
involved layout and thus a more involved computational model with more variables to refine to reach
the desired impedance. Considering this, it is clear that designing a tensor impedance transmitarray is
simpler.
Finally with regards to performance, the measured results of the tensor Huygens surfaces and tensor
impedance transmitarrays are both comparable given that the design presented in this thesis are both
prototypes. The limitations in the performance of the tensor Huygens surface is due to the fabrication
challenges of the vias used to implement the loops. For the tensor impedance transmitarray it was
the losses in the substrate used in the design. Both of these challenges can be potentially overcome
with the substrate losses being improved upon using lower-loss substrates as stated above, while the
vias can be eliminated from the design using a different layout for the loops. However with regards to
further refinements of these designs when including metrics such as bandwidth, the tensor impedance
transmitarrays become more desirable. This is due to the fact that a magnetic response is always
narrowband due to the use of subwavelength loops. An electric response can be made broadband by
judicious design and layout of the impedance surface as demonstrated in the FSS literature.
As the designs presented here are refined and improved in future works, both tensor Huygens surfaces
and tensor impedance transmitarrays may find use in applications where chiral behaviour is needed in
microwave applications.
9.5
Summary
In this chapter, the analysis, design and measurement of a transmitarray capable of chiral polarization
control has been demonstrated. The transmitarray is made up of a stack of tensor admittance surfaces
each rotated with respect to each other. The use of these tensor admittance sheets allows for control of
transmitted and reflected circularly-polarized waves. In the examples presented here, this includes effects
such as polarization rotation and circular polarization selectivity. By using the MTL model proposed
here, the admittance tensors can be carefully chosen to achieve a desired transmitted and reflected field
as confirmed by both simulation and measurements.
It is worth noting again, that during the development of the ideas in this chapter, that other research
groups were working on similar concepts. This can be found in [178, 213] and the reader is encouraged
to examine these works again to gather a different perspective on the problems discussed here.
Part IV
Conclusion
154
CHAPTER
10
Conclusions and Future Work
10.1
Summary
From the previous nine chapters, the design and measurement passive and active Huygens surfaces as
applied to the scattering, refraction and polarization of microwaves has been discussed.
Beginning with the active Huygens surface, the design of an active Huygens surface was established to
form a conformal surface which can cancel out the scattered field of an object illuminated by microwave
radiation. It was shown how both electric and magnetic dipoles can synthesize the desired field. Both
the design and modelling problems were addressed along with the demonstration of an experimental
prototype at 1.5 GHz.
For a passive Huygens surface both refraction and polarization control of microwaves was demonstrated. The passive Huygens surface concept was introduced by superposing both homogenized electric
and magnetic surface impedances. These superposed impedance surface, form a single surface in which
high transmission, can be achieved. For the refraction problem, these were scalar quantities but were
generalized to tensor quantities when dealing with the polarization of microwaves. These impedances
were analyzed theoretically using Floquet analysis, circuit theory and numerical methods. Prototypical devices were shown for both refracting a plane wave and converting the polarization of an incident
microwave field. The concept of a chiral surface using cascaded Huygens surfaces was also developed.
Experimental verification using quasi-optical measurement techniques was also developed demonstrating
prototypes of a TE-to-TM polarization converter and a polarization rotator.
The idea of cascaded electric tensor impedance surfaces was also discussed here, where the MTL
model developed for a tensor Huygens surface was generalized so that a matching network approach
could be used for a tensor impedance transmitarray. This idea was also verified by constructing a
prototypical circular polarization selective surface.
155
156
Chapter 10. Conclusions and Future Work
(a)
(b)
Figure 10.1: (a) A connected array, which is an array of broadband slots that radiate well in the
presence of a groundplane. Taken from [214] (b) The schematic of an active cloak which both receives
and transmits a signal, using signal processing algorithims.
10.2
Future Work
Like any interesting research problem, the work developed here leads to further problems to be investigated. While there are many possible directions to go from, the more promising or interesting ones are
addressed here.
10.2.1
Scattering
The idea of a an active cloak is especially attractive for a radar suppression scheme where the conformal
nature of the cloak presents an attractive opportunity to implement a realistic cloak. Of course, there are
many complications to deal with. From a hardware perspective, the problem that needs to be addressed
is the use of a loop antenna. Loop antennas are poor radiators due to their poor efficiency and poor
bandwidth. Their use here was motivated by their one-to-one mapping between the theory and the
experimental realization. However, the only real constraint on the antenna is that it be a good radiator
in the presence of a PEC1 . There are in fact many antennas that are indeed good at that such as slots
and patches. One of the more interesting examples which would also offer good bandwidth performance
are connected arrays shown in Fig. 10.1a [214, 215]. To adopt a connected array for active cloaking
purposes, the main problem that needs to be addressed is the synthesis of the scattered field pattern
using the array. For a magnetic dipole array this is one-to-one as stated, but for a more complex antenna
array like a connected array this would have to be addressed via the magnitude and phase of the array
elements. As the cloak would move towards a truly three-dimensional object, work would also involve
synthesizing antennas that support both TE and TM free-space modes as it would need to work for an
arbitrary plane-wave incidence.
The concepts put forward to synthesize the active cloak as derived from the equivalence principle,
may also be relevant to the synthesis of radiation patterns for various phased arrays. The use of a
Huygens source may also be relevant to synthesizing phased arrays which scan closer to the horizon as
1 We
are assuming, of course, that the scatterer is much larger than any of the individual antennas and that any
scattering from the antennas is much smaller than that of the scatterer.
157
Chapter 10. Conclusions and Future Work
z
y
x
θt=30o
(a)
(b)
Figure 10.2: (a) A lens designed using a Huygens surface to focus a Gaussian beam. The lens is
approximately λ/10 thick at 10 GHz. Design by Joseph Wong. Photo courtesy of Joseph Wong. (b)
A simulation of an ideal Huygens surface which can refract a Gaussian beam along with converting its
polarization.
well.
However, the biggest challenge facing the realization of a practical active cloak is the problem of
causality as mentioned in Chapter 5. While causality is a fundamental limitation as shown in [36], an
adaptive or camouflaging approach may be possible. As mentioned in Chapter 5 this would involve
using the antenna array surrounding the object as a receiver as well as a transmitter so that the active
cloak acts like an ‘electronic’ camouflage. To do this realistically would involve using the configuration
shown in Fig. 10.1b, where the received signals from the antenna array are processed using digital signal
processing to find the direction, amplitude and phase. Figuring out these quantities would involve using
hardware to analyze the direction, amplitude and phase of the received signal as stated in Chapter 5. In
this case, phased-locked loops and direction-of-arrival algorithms may be needed requiring further work
on both hardware and software.
Of course, given the 2.5-dimensional implementation discussed here, further work will also involve
designing and measuring an active cloak in a three-dimensional environment. This will involve the
further refinement of this idea to a more practical and realizable prototype that is capable of reducing
the scattering cross-section of a ‘real’ object.
10.2.2
Refraction and Polarization
Both the refraction and polarization problems are lumped together here because they involve the further
development of impedance surfaces, whether tensor or scalar, to implement flat and thin lenses. These
kinds of surfaces are suited to directive communication links where flat lenses are desired and a few steps
can be taken to improve and tailor the designs presented here for these applications.
The construction of more inhomogeneous scalar impedance surfaces for incident fields that are not
plane-wave-like is required. One example of this was developed in [150] for Gaussian-to-Gaussian focusing and a prototype was constructed at the University of Toronto. This design, which worked at
X-band, is much thinner and compact compared to the equivalent dielectric lens used in Fig. 8.18. This
Chapter 10. Conclusions and Future Work
158
shows the potential for lenses built using printed impedance surfaces in the microwave and millimetre
wave frequency as they are much more compact then their dielectric counterparts. More complicated
synthesis procedures for Huygens surfaces can also be investigated, an example of which has already
been demonstrated at the University of Toronto in [216].
Combining both wavefront and polarization control is another potential avenue for more complex
lenses. This can be done by using an inhomogeneous tensor impedance screen. An example of this is
shown in Fig. 10.2b where a normally incident, TE-polarized Gaussian beam is converted into a circularly
polarized field that is refracted by 30◦ using a tensor Huygens surface. The example shown is only a
simulation using the TLM solver discussed in Chapter 6. However using the rotated loops and dipoles
shown in Chapter 8 such a design could also be implemented as well using printed circuit technology
across the microwave and millimetre wave frequency bands, demonstrating a novel way to create thin
lenses with advanced wavefront and polarization control.
This same idea is also applicable to the cascaded tensor impedance surfaces discussed in Chapter 9.
Here using an inhomogeneous array of cascaded impedance surfaces would allow for the combination of
beam-shaping designs along with chiral polarization control. This would be especially useful in circularlypolarized focused applications such as satellite communications, where beam-forming LHCP and RHCP
waves differently are often required.
One of the most critical improvements to the designs presented here is improving the bandwidth
of the impedance surfaces used. Ultimately this involves engineering the dispersion of the impedances
surface such that the dispersion of the printed elements can successfully be used to implement more
broadband lenses and polarization screens. This would be most successfully accomplished using the
tensor impedance transmitarrays framework discussed in Chatper 9 where the electric response of the
surface can be more easily engineered. An example of this for reflectarrays has been done in [217], and
trying to extend these ideas for transmit scenarios, like those discussed in this thesis, will prove to be
extremely useful.
For Huygens surfaces which have both a magnetic and electric response the bandwidth improvements
may be more challenging due to the limitations in engineering the dispersion of the magnetic response.
Though adding more resonances to the unit cell may prove to be a promising avenue.
Finally extending these these refracting and polarization controlling surfaces to incorporate more
complex ideas such as tuneability and non-reciprocity would further increase the potential application of
these surfaces. Non-reciprocal surfaces would be interesting in their application to various communication
links while tuneability would allow for realizing systems that could be reconfigured in real-time. Nonreciprocity could be accomplished either by incorporating ferrite materials or active components, while
tuneability could be realized by using varactor diodes or more exotic material such as liquid crystals
and vanadium dioxide. In fact both of these areas are currently being investigated by various research
groups including recent works published in [218–220].
10.3
Contributions
As stated above the work in this thesis put forth contributions on the first demonstration of an active
cloak prototype using an active Huygens surface, and the theory, design and measurement for refracting
microwaves and controlling their polarization state with both scalar and tensor Huygens surfaces. Also
developed was the theory and measurement of tensor impedance transmitarrays. The work in this thesis
Chapter 10. Conclusions and Future Work
159
has led to the following contributions in the peer-reviewed literature:
Journal Publications
J1 M. Selvanayagam and G.V. Eleftheriades, “Experimental Verification of the Effective Medium Properties of a Transmission-Line Metamaterial on a Skewed Lattice,” IEEE Antennas and Wireless
Propagation Letters, vol.10, no., pp.1495-1498, 2011
J2 M. Selvanayagam and G.V. Eleftheriades, “Transmission-Line Metamaterials on a Skewed Lattice
for Transformation Electromagnetics,” IEEE Transactions on Microwave Theory and Techniques,
vol.59, no.12, pp.3272-3282, Dec. 2011
J3 G.V. Eleftheriades and M. Selvanayagam, “Transforming Electromagnetics Using Metamaterials,”
IEEE Microwave Magazine, vol.13, no.2, pp.26-38, March-April 2012
J4 M. Selvanayagam and G. V. Eleftheriades, “An Active Electromagnetic Cloak Based On The Equivalence Principle,” IEEE Antennas and Wireless Propagation Letters, vol. 11, pp.1226-1229, 2012.
J5 M. Selvanayagam and G.V. Eleftheriades, “Dual-Polarized Volumetric Transmission-Line Metamaterials,” IEEE Transactions on Antennas and Propagation, vol. 61, pp.2550-2560, 2013.
J6 M. Selvanayagam and G.V. Eleftheriades, “Discontinuous Electromagnetic Fields Using Orthogonal
Electric And Magnetic Currents For Wavefront Manipulation,” Optics Express, vol. 21, pp.1440914429, 2013.
J7 M. Selvanayagam and G.V. Eleftheriades, “Experimental Demonstration of Active Electromagnetic
Cloaking,” Physical Review X, vol. 3, p.041011, 2013.
J8 M. Selvanayagam and G.V. Eleftheriades, “Circuit Modelling of Huygens Surfaces,” IEEE Antennas
and Wireless Propagation Letters, vol. 12, pp. 1642-1645, 2013.
J9 M. Selvanayagam and G.V. Eleftheriades, “Polarization control using tensor Huygens surfaces,”
IEEE Transactions on Antennas and Propagation, Accepted, 2014.
J10 M. Selvanayagam and G.V. Eleftheriades, “Design of tensor impedance transmitarrays for chiral
polarization control,,” IEEE Transactions on Microwave Theory and Techniques, Submitted, 2015.
Conference Presentations
C1 M. Selvanayagam and G.V. Eleftheriades, “A rotated transmission-line metamaterial unit cell for
transformation-optics applications,” 2011 IEEE MTT-S International Microwave Symposium Digest (MTT), pp.1-4, 5-10 June 2011
C2 M. Selvanayagam and G.V. Eleftheriades, “A sheared transmission-line metamaterial unit cell with
a full material tensor,” IEEE International Symposium on Antennas and Propagation (APSURSI),
2011 pp.2872-2875, 3-8 July 2011
Chapter 10. Conclusions and Future Work
160
C3 M. Selvanayagam and G.V. Eleftheriades, “A Dual-Polarized Transmission-Line Metamaterial Unit
Cell,” 2012 IEEE MTT-S International Microwave Symposium Digest (MTT), pp.1-4, 17-22 June
2012
C4 M. Selvanayagam and G.V. Eleftheriades, “Dual Polarized Negative Refraction In A Volumetric
Transmission-Line Metamaterial,” IEEE International Symposium on Antennas and Propagation
(APSURSI), 2012 pp.1-2, 8-14 July 2012
C5 M. Selvanayagam and G.V. Eleftheriades, “New transmission-line unit cells for building new transmissionline unit cells for building transformation electromagnetics devices,” Meta-materials ’2012: The
Sixth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics,,
2012 pp.1-3, 17-22 September 2012
C6 M. Selvanayagam and G.V. Eleftheriades, “A surface cloak using active Huygens sources,” Metamaterials’ 2013: The Seventh International Congress on Advanced Electromagnetic Materials in
Microwaves and Optics, 2013 pp.1-3, 16-19 September 2013
C7 M. Selvanayagam and G.V. Eleftheriades, “Tensor Huygens Surfaces,” IEEE International Symposium on Antennas and Propagation (APSURSI), 2014 pp.1-2, 7-11 July 2014
C8 M. Selvanayagam and G.V. Eleftheriades, “Implementing Tensor Huygens Surfaces For Polarization Control Using Rotated Loops and Dipoles,” Metamaterials’ 2014: The Eighth International
Congress on Advanced Electromagnetic Materials in Microwaves and Optics, 2014 pp.1-2, 22-26
August 2014
C9 M. Selvanayagam and G.V. Eleftheriades, “Chiral Polarization Control Using Cascaded Tensor
Impedance Surfaces,” IEEE MTT-S International Microwave Symposium Digest (MTT), 2015
pp.1-3, 18-23 May 2015
Part V
Appendices
161
APPENDIX
A
A Bridged-T Phase Shifter
In Chapter 4, a tunable phase shifter is used to control the phases of the currents on each loop antenna
to properly phase the entire array to conceal a metallic cylinder. Here, the phase-shifter is a custom
designed circuit and the main criteria when designing this circuit is realizing a 360◦ phase range with
maximal transmission at the desired frequency (around 1.5GHz here). The proposed topology for the
circuit is a bridged-T network and the phase-shifter consists of two of these networks cascaded back-toback as shown in Fig. A.1. This kind of phase shifter was borrowed from [77] where a similar topology
was used around 5-GHz in the unit cell of a transmitarray. The bridged-T network is an ideal choice for
a phase shifter as it is an all-pass network, making it a good fit for a transmission-type phase shifter (a
bridged-T network can be transformed into a lattice network and vice-versa). Because this is a custom
designed circuit, as stated, the design of this phase shifter is briefly reviewed here.
A.1
Basic Design
Vdc
Rrf
Cdc
Zo,β
Cvar
Cvar
Cdc L
s
Rrf
Cdc
Ls
Zo,β
Cvar
Ls
Ls Cdc Zo,β
Cvar
Ls
Lsh
Figure A.1: The circuit schematic for a bridged-T phase shifter. The DC bias lines are included to show
how each varactor is biased. The transmission-line segments are all 50Ω microstrip lines on a 1.524mm
Rogers 3003 substrate.
Based on the schematic of the circuit shown in Fig. A.1, the design involves finding values for the
162
Appendix A. A Bridged-T Phase Shifter
163
loading reactances to optimize both the tuning phase and transmission of the circuit. The transmissionline segments are 50Ω microstrip transmission-lines on a Rogers 3003 substrate with a height of 1.524mm.
The inductors are implemented as printed inductors by meandering the transmission-line while the
capacitors are first modelled as ideal tunable capacitors and then later as a varactor. The circuit is
modelled in a commercial CAD tool, ADS, where the model is constructed as a hybrid full-wave/circuit
model. The full-wave model consists of the transmission-line segments, the printed inductors and the
DC bias and grounding lines which are needed to bias the varactor. The parts modelled in the circuit
simulator are the tuning capacitors and/or varactors, the DC blocking capacitors and the RF choke
resistors. The DC blocking capacitors are fixed at a value of 0.1µF which are almost a short at the
design frequency of 1.5GHz and the RF choke resistors are set at 75kΩ which act like an open at 1.5GHz
as well. Note that the variable capacitors are all tuned in tandem as a single DC-bias is used to vary
both the capacitors in the series and parallel branches. The two models are combined using ADS’s
co-simulation tools which allow for the overall S-parameters of the combined full-wave model and circuit
elements to be found.
The design procedure consists of optimizing the printed inductors while the capacitances are varied
over a range of values. This is done iteratively using parametric sweeps until a suitable printed design is
found for the range of values for the capacitors. A suitable printed design is one with minimal area and
acceptable feature size. The range of values that are taken on by the tunable capacitor are dictated by the
choice of varactor. As stated in Chapter 2 the varactor chosen for the phase shifter is made by Aeroflex
(MGV125-26-E28X). This specific varactor has a tuning range between approximately 0.5pF − 3pF . In
a basic model of the phase shifter, the ideal tunable capacitor simply takes on this range. However, to
fully model this design the whole varactor must be modelled to take into account parasitic reactances
and resistances.
A.2
Varactor Modelling
To fully model this specific varactor diode, the simplest way to capture its properties is to measure it.
This is done by characterizing its two-port S-parameters. To measure the two-port S-parameters of the
varactor, the varator is placed on a microstrip testbed and measured with a two-port vector network
analyzer (VNA). To calibrate out the effects of the cables, transitions and host microstrip lines a thrureflect-line (TRL) calibration procedure is used. This is shown in Fig. A.2a where the calibration lines
as well as the measurement lines are shown on the same Rogers 3003 board. The TRL lines consist
of a microstrip thru of the same length as the microstrip lines leading up to the varactor diode, a
microstrip line which is of an arbitrary length longer than the microstrip thru and a microstrip open
which implements a reflect standard. From these three standards the VNA can be calibrated. Note
that in the calibration standards defined in the VNA, the phase delay of the thru is defined (found from
a full-wave simulation in ADS ) and the fringing capacitance of the thru is modelled by a third-order
polynomial model as function of frequency over the calibration bandwidth (1-2GHz in this case), again
using full-wave models in ADS.
The measured S-parameters of the varactor diode are shown in Fig. A.2b- A.2c. A tuning voltage
is applied to the varactor, via an SMA bias-tee, to vary the capacitance of the varactor diode. A close
observation of these parameters indicate that they correspond to those of a series capacitor however they
do differ from an ideal tuneable capacitor due to parasitics present in the varactor and the packaging of
164
Appendix A. A Bridged-T Phase Shifter
2
5
1.8
4
1.6
3
1.4
2
Capacitance (pF)
Freq (GHz)
(a)
1.2
1
S11
S21
1
5
10
15
DC Bias(V)
(b)
(c)
Figure A.2: (a) A picture of the fabricated TRL board. (b) The complex S-parameters plotted on the
Smith Chart as a function of both frequency and bias voltage. Note that the S11 is mainly located
around the 1 − j smith chart circle as expected for a series capacitor though parasitics alter that precise
location. (c) The extracted series capacitance of the varactor as a function of frequency and bias voltage.
In the top left corner of the plot, the extracted reactance is actually inductive and is set to zero in this
plot.
the varactor itself.
A.3
Final Design
With a full characterization of the varactor, the design of the phase shifter can be completed. Here, the
measured two-port S-parameters of the varactor are inserted into the CAD model of the phase shifter.
The S-parameters of the varactor are inserted into the circuit simulator along with the RF and DC chokes
while the rest of the circuit is modelled in full-wave simulation. The circuit is then optimized again, by
varying the geometry of the printed inductances to account for any parasitics or unwanted loading by
the varactor model. The final layout is shown in Fig. A.3a. The simulated S-parameters are plotted
Appendix A. A Bridged-T Phase Shifter
165
again in Fig. A.3 as a function of both frequency and the tuning voltage. Near the design frequency of
1.5GHz, a high |S21 | is seen again along with a correspondingly small |S11 | while the transmission phase
covers the desired 360◦ phase range. This can also be seen in Fig. A.3e where the S-parameters as a
function of voltage are plotted at the measured frequency of the cloak at 1.48GHz.
A.4
Measured Results
The microstrip phase shifter is fabricated in a commerical PCB fabrication process and the lumped
elements are populated by hand as shown in Fig. A.4a. The S-parameters of the phase-shifters are
measured on a calibrated VNA and the measured S-parameters are plotted in Fig. A.4 as a function
of both frequency and tuning voltage. It can be seen that measured S-parameters have the desired
transmission and reflection properties, including the desired phase coverage as a function of tuning
voltage at the frequency of interest at 1.5GHz. This phase shifter is then fabricated 15 times to provide
an individual phase shifter for the required sources in the active cloak.
166
Appendix A. A Bridged-T Phase Shifter
Cdc Rrf
Cdc
Rrf
Cvar
Rrf
Cvar
Cdc
Cvar
Cdc
Cvar
(a)
(b)
(c)
Magnitude (dB)
0
−10
−20
−30
−40
0
2
4
6
8
10
12
14
16
18
12
14
16
18
DC Bias(V)
Angle (deg)
600
400
200
0
0
2
4
6
8
10
DC Bias(V)
(d)
(e)
Figure A.3: (a) A schematic of the final design of the bridged-T phase shifter in microstrip. The lumped
loadings are also shown in the drawing. (b) The simulated |S11 | as a function of both frequency and bias
voltage. (c) The simulated |S21 | as a function of both frequency and bias voltage. (b) The simulated
phase of S21 as a function of both frequency and bias voltage. (e) The simulated S-parameters as a
function of bias voltage at 1.48GHz (the measured frequency of the cloak).
167
Appendix A. A Bridged-T Phase Shifter
(a)
(b)
(c)
Magnitude (dB)
0
−10
−20
−30
0
2
4
6
8
10
12
14
16
18
12
14
16
18
DC Bias(V)
Angle (deg)
600
400
200
0
−200
0
2
4
6
8
10
DC Bias(V)
(d)
(e)
Figure A.4: (a) A picture of the fabricated phase shifter. (b) The measured |S11 | as a function of both
frequency and bias voltage. (c) The measured |S21 | as a function of both frequency and bias voltage.
(d) The measured phase of S21 as a function of both frequency and bias voltage. (e) The measured
S-parameters as a function of bias voltage at 1.48GHz (the measured frequency of the cloak).
APPENDIX
B
Floquet Analysis Of A Periodic Magnetic Current Sheet
Following the analysis in Chapter 6 where the Floquet modes scattered by an induced and periodic
surface current density J were found, a similar analysis is presented here for a periodic magnetic surface
current density M for completeness sake. Because of the duality of the problem the fields being solved
for are the scattered magnetic field as given by the boundary condition (6.4). The geometry of the
N =1
N =0
M (x)
N =1
N =0
kin
Z(x)
N = −1
z
y
N = −1
x
W
Figure B.1: The Floquet mode model of a Huygens surface. Since the impedance function Zm is
periodic the problem is analyzed in a periodic environment. Here, given an incident plane wave and
the impedance, Zm of the surface, the induced current M(x) can be decomposed into Floquet modes
as shown. The scattered fields are also decomposed into 2P − 1 Floquet modes as shown and their
coefficients are solved for.
problem is shown in Fig. B.1 where the impedance surface is placed again in a 2D periodic waveguide of
168
Appendix B. Floquet Analysis Of A Periodic Magnetic Current Sheet
169
width W . This periodic waveguide only supports propagation of plane waves with a discrete spectrum,
P∞
given by p=−∞ e−jkx,p x e−jky,p y where the kx,p and ky,p are defined as usual in Chapter 6.
The surface is again illuminated by a TE-polarized plane wave with a magnetic field tangential to
the surface given by Hinc =
−Eo cos ϕi −jk sin ϕi x −jk cos ϕi
e
e
x̂.
η
The magnetic surface current density is
given by
P
−1
X
Mx,y = δ(y)
Ap e−jkx,p x x̂.
(B.1)
p=−(P −1)
These induced magnetic surface currents radiate the scattered field created by the impedance surface.
By duality the Green’s function for a magnetic surface current in a periodic waveguide takes on the same
form as before [140, 141],
Ḡm (x, y|x0 , y 0 ) =
1
2jW
0
∞
X
e∓jky,m (y−y ) −jkx,m (x−x0 )
e
Ī
ky,m
m=−∞
(B.2)
where Ī is the identity matrix. With this Green’s function, the field radiated by the magnetic current
density M are given by
F = εo
εo
F=−
ẑ
2jW
Z
∞
−∞
Z
Z
Ḡm (x, y|x0 , y 0 )M(x0 , y 0 )dx0 dy 0 ,
(B.3)
S
W
0
0
∞
X
e∓jky,m (y−y ) −jkx,m (x−x0 )
e
δ(y)
ky,m
m=−∞
P
−1
X
Ap e−jkx,p x dx0 dy 0 ,
(B.4)
p=−(P −1)
where F is the electric vector potential. Using the orthogonality of the complex exponentials, the vector
potential reduces to,
F=−
εo
2j
P
−1
X
p=−(P −1)
e∓jky,p y
Ap e−jkx,p x x̂,
ky,p
(B.5)
where the ∓ indicates forward propagating waves for y > 0 and backwards propagating waves for y < 0.
The tangential magnetic field Hx can be found from the vector potential and is given by,
Hx,s
ωεo
=−
2
2
kx,p
1− 2
ko
!
−jω
Hx = 2
k o
P
−1
X
p=−(P −1)
∂2
k +
∂x2
2
Fx ,
(B.6)
e∓jky,p y
Ap e−jkx,p x
ky,p
(B.7)
The total field then is simply Hx,tot = Hx,inc + Hx,s . With the total field and the surface currents each
defined as a sum of Floquet modes, they can be inserted into the boundary condition imposed by the
impedance surface given in (6.2) which results in,
−Eo cos ϕi −jk sin ϕi x −jk cos ϕi ωεo
e
e
−
η
2
Hx,tot (x, y = 0) = Ym (x)M (x, y = 0),
! P −1
2
X
kx,p
e∓jky,p y
1− 2
Ap e−jkx,p x
ko
ky,p
p=−(P −1)
= Ym (x)
P
−1
X
p=−(P −1)
Ap e−jkx,p x
(B.8)
(B.9)
170
Appendix B. Floquet Analysis Of A Periodic Magnetic Current Sheet
Again the magnetic impedance of the Huygens surface Zm = Ym−1 was found in Chapter 6 and can be
expressed as
πx
Y (x) =
πx
ωµo ej W + e−j W
πx
πx
2ky,1 ej W − e−j W
(B.10)
This expression for the admittance is inserted into (B.9) which after rearranging results in,
P
−1
X
p=−(P −1)
"
ωεo
2ky,p
πx
−Eo cos ϕi −j (kx,0 x− πx
W ) − E e−j (kx,0 x+ W ) =
e
o
η
!
2
kx,p
πx
πx
1 − 2 Ap e−j (kx,p x− W ) − e−j (kx,p x+ W ) +
ko
πx
πx
k cos ϕo
Ap e−j (kx,p x− W ) + e−j (kx,p x+ W ) .
2ηko
(B.11)
πx
Again the admittance term has shifted every Floquet plane wave by a factor of e±j W . The resultant
wave vectors after such a shift can now be expressed as,
0
kx,2p∓1 = ko sin ϕi +
π(2p ∓ 1)
.
W
(B.12)
Substituting this into (6.31) gives
1
2ωµo
P
−1
X
p=−(P −1)
−Eo cos ϕi −jkx,−1 x
e
− Eo e−jkx,1 x = (B.13)
η
h
i
0
0
0
0
ky,p Ap e−jkx,2p−1 x − e−jkx,2p+1 x + k cos ϕo Ap e−jkx,2p−1 x + e−jkx,2p+1 x .
To solve for the 2P − 1 unknown coefficients, Ap , of this Floquet expansion, (B.13) is again expanded
into 2P − 1 equations using the inner product defined in (6.34). The resulting system of equation takes
the form
1
ky,−(P −1) + ko cos ϕo A−(P −1)
2ωµo
1
ko cos ϕo − ky,−(P −1) A−(P −1) +
ky,−(P −2) + ko cos ϕo A−(P −2)
2ωµo
..
.
cos ϕi Eo
1
1
−
=
(ko cos ϕo − ky,−1 ) A−1 +
(ko cos ϕo + ky,0 ) A0
η
2ωµo
2ωµo
cos ϕi Eo
1
1
−
=
(ko cos ϕo − ky,0 ) A0 +
(ko cos ϕo + ky,1 ) A1
η
2ωµo
2ωµo
1
1
0=
(ko cos ϕo − ky,1 ) A1 +
(ko cos ϕo + ky,2 ) A2
2ωµo
2ωµo
1
1
0=
(ko cos ϕo − ky,2 ) A2 +
(ko cos ϕo + ky,3 ) A3
2ωµo
2ωµo
..
.
1
1
0=
(ko cos ϕo − ky,P −2 ) AP −2 +
(ko cos ϕo + ky,P −1 ) AP −1
2ωµo
2ωµo
0=
0=
1
2ωµo
(B.14)
By inspection it can be determined again that the coefficients Ap = 0 for p 6= {0, 1}. Thus the only
Appendix B. Floquet Analysis Of A Periodic Magnetic Current Sheet
171
non-zero coefficients are A0 and A1 . With the Floquet expansion solved for, the electric field can also
be found from the electric vector potential in (B.5) and is given by,
E=
−1
1 ∂Fx
∇×F=
.
εo
εo ∂y
(B.15)
From here it can easily be determined that when A0 and A1 are substituted into the scattered electric
fields, the amplitude of the scattered field is the same as found in Chapter 6. This set of equations in
(B.14) can also be used to numerically solve for the fields in the case where the incident field differs from
the designed case as also done in Chapter 6.
APPENDIX
C
A Frequency Domain TLM/MOM Solver
In Section 8.3 a TLM solver was used to model a tensor Huygens surface supporting TE and TM
polarizations in a two-dimensional domain. The TLM solver is based on the one described in [149] which
solved for the voltages in a TE-polarized grid at a single frequency in a two-dimensional domain. The
termination of the TLM grid was done by using the method-of-moments to allow for the voltages and
currents on the periphery to radiate to infinity. Here, further details are given regarding how this method
can be extended to support two polarizations in a two-dimensional domain.
The TLM solver, solves for the voltages and currents in a transmission-line grid which map one-toone to the electric and magnetic field respectively in free space. The TLM unit cell for two-dimensional
free-space supporting both a TE and TM polarization is shown in Fig. 8.6. It consists of two, twodimensional transmission-line sections, a shunt-node section (TE) and a series-node section (TM). The
TE and TM grids are completely decoupled from one another and each cell is characterized by an 8 × 8
matrix given by,
H=
"
Z
0
0
Y
#
,
(C.1)
which is block diagonal, where 0 is a 4 × 4 matrix of zeros. The matrices Z and Y are 4 × 4 matrices
which are the impedance matrix of the series-node grid and the admittance matrix of the shunt-node
grid and are given by,
−1
1
1
YI+ 1
,
Z=
2
Z
−1
1
1
Y=
ZI + 1
.
2
Y
(C.2)
(C.3)
Here 1 and I are a matrix of ones and the identity matrix respectively while Z = jk0 dη and Y = jk0 d/η
are the inductance and capacitance of free-space respectively for a given unit cell size of d. The shuntnode grid is characterized by the voltages at its terminals while the series node grid is characterized by
172
173
Appendix C. A Frequency Domain TLM/MOM Solver
its currents. These unit cells can be stitched together into a global matrix. This is given by,

TM
Vper
 TM
 Vint

 ITE
 per
ITE
int


Z11
 
  Z21
=
  B
  11
B21
Z12
A11
Z22
A21
B12
Y11
B22
Y21
A12

ITM
per
  TM

A22 
  Iint

TE
Y12  
 Vper
TE
Vint
Y22






(C.4)
It can be seen that the voltages and currents in the overall matrix are annotated based on whether they
are on the perimeter or in the interior of the grid as shown in Fig. 8.6. The A and B block matrices
detail the coupling between the TE and TM waves and are zero for free space. As stated, the voltages
and currents on the perimeter can be radiated to the far-field using the method of moments. Here
two integral equations are solved, one for the far-field voltages in the TE grid and one for the far-field
currents in the TM grid, which are given by [82],
Z
1 TE
TE
Vper + jk0 η
Iper
G(k0 |ρ − ρ0 |)dρ
2
per
Z
TE
(k0 |ρ − ρ0 |)
T E ∂G
−
Vper
dρ,
∂n0
per
Z
1 T M jk0
= Iper
+
V T M G(k0 |ρ − ρ0 |)dρ
2
η per per
Z
TM
(k0 |ρ − ρ0 |)
T M ∂G
−
Iper
dρ,
∂n0
per
V TE =
IT M
(C.5)
(C.6)
where GT E (k0 |ρ − ρ0 |) and GT M (k0 |ρ − ρ0 |) are the free space greens functions for the TE and TM modes
in two-dimensions. Following [149], the method of moments discretization of these equations gives,
TE
TE
TE TE
Vmom
= MTE
1 Iper + M2 Vper
(C.7)
TM TM
ITM
Vper + MTM
ITM
mom = M1
2
per
(C.8)
TE
TM
where MTE
and MTM
are the matrices discretizing the integral equations in (C.5),(C.6).
1 , M2 , M1
2
TM
These equations can be solved for ITE
per and Vper which can then be substituted into the left-hand side
(C.4). This gives the global matrix equation of the hybrid TLM/MOM method as shown in (C.9)

(MTM
)−1 ITM
mom
1

TM

Vint

 (MTE )−1 VTE

mom
1
ITE
int


 
 
=
 
 
Z11 + MTM
1
−1
MTM
2
Z12
A11
Z21
Z22
A21
B11
B12
Y11 + MTE
1
B21
B22
Y21
A12
−1
MTE
2

ITM
per
  TM

A22 
  Iint

TE
Y12  
 Vper
TE
Vint
Y22






(C.9)
The sources which excite the grid are either interior point sources or perimeter sources such as plane
waves and Gaussian beams which can be generated by substituting the required perimeter voltages and
currents into (C.9). Current sources excite the TE grid and voltage sources excite the TM grid. Note
that these sources are also weighted by the method of moment weights as to terminate the domain
properly. A combined current/voltage source can create the desired polarization of the incident wave.
The tensor Huygens surface can be included in the global matrix equation by replacing the free-space
174
Appendix C. A Frequency Domain TLM/MOM Solver
unit cells with an 8 × 8 matrix describing each tensor Huygens surface unit cell. Because the tensor
Huygens unit cell is a four-port circuit, only four ports of the 8-port unit cell are connected, with the
remaining left open as shown in Fig. 8.6 (those transverse to the direction of propagation). To fill in
the relevant non-zero entries of the 8 × 8 circuit, the entries of the 4 × 4 impedance matrix given in
(8.23) are used to describe the tensor Huygens surface unit cell. However, this impedance matrix needs
to be rearranged in a different form where the TE ports are related by admittances, the TM-ports by
impedances and the cross-coupling between TE and TM ports (and vice versa) by unit-less parameters
(V/V and I/I respectively). This can be rewritten as the following matrix

Vy,1

 Vy,2

 I
 z,1
−Iz,2


Zyy
 
  Zyy
=
  B
  zy
Bzy
Zyy
Ayz
Zyy
Ayz
Bzy
Yzz
Bzy
Yzz
Ayz

Iy,1


Ayz 
  −Iy,2

Yzz  
 Vz,1
Vz,2
Yzz






(C.10)
where the entries of this matrix can be found by algebraically rearranging the 4 × 4 impedance matrix
and its inverse admittance matrix. These entries can be inserted into the 8 × 8 matrix describing the
unit cell and subsequently into the global matrix of (C.9).
It is noted that the only non-zero parameters of the Aij and Bij block matrices in (C.9) arise from
the tensor Huygens surface and induce the cross-coupling between the TE and TM waves which would
otherwise be independent. Inverting this matrix gives the voltages on the TE grid and the currents on
the TM grid.
When plotting the fields, the field in free-space is plotted as a function of the voltage (electric field) so
that the polarization of the wave can be determined. When inverting (C.9) the solved quantities are the
voltage in the TE grid and the current in the TM grid. Therefore the spatial derivative of the currents
on the TM grid must be taken to find the actual voltages on the TM grid in the x and y directions. This
is done by applying Ampere-Maxwell’s equation to get,
1 ∂I T M
,
jωε0 ∂y
1 ∂I T M
Vy = −
.
jωε0 ∂x
Vx =
(C.11)
(C.12)
The above allows for the voltages in the TE grid and the voltages in the TM grid to be plotted as
demonstrated in Section. 8.3 when tracing out the polarization of the field.
Finally, it is noted that the far-fields can be found if necessary, by following a similar procedure given
in [149] where the voltages and currents on the perimeter can be used to find the corresponding far-field
quantities.
APPENDIX
D
S-parameter Conversion
In Chapters 8 and 9, four-port S-parameters are used to understand the transmission and reflection of TE
and TM waves off of various configurations of tensor impedance surfaces. As part of this analysis, various
transformation of S-parameter matrices are undertaken and the mechanics of these transformations are
summarized here.
D.1
Converting Linearly-Polarized S-parameters To Circularly
Polarized S-parameters
In Chapters 8 and 9 the use of circularly-polarized S-parameters is relevant to understanding chiral
polarization effects such as polarization rotation or circular polarization selectivity. Given that the
default basis, defined in this thesis is a linearly polarized TE/TM basis, a conversion from these two
different bases can be found
The incoming and outgoing linearly polarized and circularly polarized waves are defined for a fourport network as given by the port numbering defined in Chapter 8. From here the conversion from the
incoming linearly polarized waves to incoming circularly polarized waves is given by,

1


1 0
√ 
2
1
0
0
1
0
1
−j
0
j
0

 

aLH
aT1 M
1
  T M   LH 

 

j 
 a2  = a2  ,



TE 
RH 
0   a3  a3 
−j
aT4 E
aRH
4
0
Ca aL = aCP .
(D.1)
(D.2)
Likewise, the conversion for outgoing linearly polarized waves to outgoing circularly polarized waves is
175
176
Appendix D. S-parameter Conversion
defined by,

1

1 0
√ 
2
1
0
0
j
1
0
0
−j
1

 

bLH
bT1 M
1
  T M   LH 

 

−j 
 b2  = b2  ,



RH 
TE 
0   b3  b3 
bRH
bT4 E
j
4
0
0
Cb bL = bCP .
(D.3)
(D.4)
The definition of the linearly and circularly polarized S-parameters in terms of these incoming and
outgoing waves is given by,
bL = SL aL ,
(D.5)
bCP = SCP aCP .
(D.6)
From here, (D.2) and (D.4) can be substituted into (D.5) which is given by,
Cb −1 bCP = SL Ca −1 aCP ,
bCP = Cb SL Ca
−1
(D.7)
aCP .
(D.8)
Thus it can easily be seen that the circularly-polarized S-parameters can be defined in terms of the
linearly-polarized S-parameters,
SCP = Cb SL Ca −1 .
(D.9)
The opposite can also easily be found from (D.9).
D.2
Rotating S-parameters
In Chapters 8 and 9, tensor impedances are partially defined in terms of a rotation angle which ultimately
maps to a physical rotation of the crossed-dipoles and crossed-loops. The tensor impedance of these
crossed loops and dipoles is characterized by a 4-port Z-parameter/Y-parameter matrix as defined in
Chapter 8 which can be easily converted to S-parameters. When designing or analyzing these kinds of
structures it is often ideal to examine their S/Z/Y-parameters in a rotated basis where the resulting
impedance tensor is diagonal or vice versa. To do this this the S-parameter matrix must be rotated. For
the waves incident on each port their rotation is given by,

cos θ

 0

 sin θ

0
0
cos θ
0
sin θ
− sin θ
0
cos θ
0

  
aθ1
aT1 M
  T M   θ
  

− sin θ
 a2  = a2  ,


θ
TE 
0   a3  
a3 
cos θ
aT4 E
aθ4
0
Rθ a = aθ .
(D.10)
(D.11)
177
Appendix D. S-parameter Conversion
Likewise for the outgoing waves the rotation is given by,

cos θ

 0

 sin θ

0
0
cos θ
− sin θ
0
0
cos θ
sin θ
0
  

bθ1
bT1 M
  T M   θ
  

− sin θ
 b2  = b2  ,


θ
TE 
0   b3  
b3 
bθ4
bT4 E
cos θ
0
Rθ b = bθ .
(D.12)
(D.13)
With the S-parameters defined to be b = Sa, (D.11) and (D.13) can be substituted into this definition
which results in the following,
Rθ −1 bθ = SRθ −1 aθ ,
bθ = Rθ SRθ
−1
aθ .
(D.14)
(D.15)
Thus it can easily be seen that the rotated S-parameters can be defined in terms of the TE/TM Sparameters,
Sθ = Rθ SRθ −1 .
(D.16)
The opposite can also easily be found from (D.16) where the rotated S-parameters can be rotated back
to a TE/TM basis.
D.3
Transfer Matrices and Cascading S-parameters
To determine the S-matrix of the overall transmitarray, the S-parameters of the individual layers and
spacers can be converted into 4 × 4 transfer matrices. The transfer matrix is defined to be,
 
 
b2
a1
 
 
a2 
 b1 
  = T ,
b 
a 
 4
 3
b3
a4
(D.17)
where ai and bi are the incoming and outgoing waves at ports 1-4, with the numbering of the ports given
in Fig. 9.2. These transfer matrices can be multiplied together to model the whole stack of admittance
sheets and dielectric layers as given by,
Tarray = TY,1 TTL TY,2 ...TTL TY,N−1 TTL TY,N ,
(D.18)
for an N layer transmitarray. The reason that these matrices can be multiplied together is that the
input waves to the T-matrix have the same direction as the output waves of the T-matrix. This overall
transfer matrix can be converted back to an S-parameter matrix to find the reflected and transmitted
fields off of the transmitarray. A procedure for converting between S-parameters and transfer matrices
is given in [140, Appendix C].
For the interconnecting transmission-lines of length d and characteristic impedance η1 and propagation constant k the TE and TM modes are decoupled as stated earlier. Thus the S-parameter matrix is
178
Appendix D. S-parameter Conversion
simply given below by,
STL

j sin ko d(η1 −η 2 /η1 )
2η cos ko d+j(η1 +η 2 /η1 ) sin ko d

 2η cos ko d+j(η 2η+η2 /η ) sin ko d
1
1
=

0

2η
2η cos ko d+j(η1 +η 2 /η1 ) sin ko d
j sin ko d(η1 −η 2 /η1 )
2η cos ko d+j(η1 +η 2 /η1 ) sin ko d
0
D.3.1
0

0
0




2η
2
2η cos ko d+j(η1 +η /η1 ) sin ko d 
0
0
j sin ko d(η1 −η 2 /η1 )
2η cos ko d+j(η1 +η 2 /η1 ) sin ko d
0
2η
2η cos ko d+j(η1 +η 2 /η1 ) sin ko d
j sin ko d(η1 −η 2 /η1 )
2η cos ko d+j(η1 +η 2 /η1 ) sin ko d
(D.19)
The S-parameters Of Cascaded Tensor Huygens Surfaces
In Section 8.5, two tensor Huygens surfaces are cascaded back-to-back as shown in Fig. 8.12. The Sparameters for this structure can be found by converting the S-parameter matrices into transfer matrices
as discussed above. However, for the specific examples discussed in Section 8.5 one or both of the tensor
Huygens surfaces are reflectionless in the ideal case and thus the overall S-parameters can be easily
calculated for this ideal scenario without resorting to transfer matrices.
To see this, first note that the spacer layer of length d between the two Huygens surfaces has no
overall effect on the S-parameters apart from some arbitrary phase shift that is accumulated by all the
S-parameters and can be disregarded. It is also assumed that the second surface has no cross-coupling
terms, i.e. S41 = S23 = S32 = S14 = 0. For the case where two reflectionless surfaces are cascaded,
the overall S-parameters can be easily calculated. The S-parameters for the transmitted coefficients are
then given to be,
1
2
1
2
b1,T M = S12
S12
a2,T M + S14
S34
a4,T E ,
(D.20)
1
2
1
2
b2,T M = S21
S21
a1,T M + S23
S21
a3,T E ,
(D.21)
1
2
S34
S34
a4,T E ,
(D.22)
1
2
1
2
b4,T E = S41
S43
a1,T M + S43
S43
a3,T E ,
(D.23)
b3,T E =
1
2
S32
S12
a2,T M
+
where the superscript indicates the S-parameters of either the first surface or the second surface. Remember that the reflection coefficients are zero for both the individual surfaces and the overall structure.
A similar method can be carried out for the case when the first surface is not reflectionless as in the
case of the CPSS discussed in Section 8.5. The same assumption is made about the second surface, in
that it is reflectionless with zero cross-coupling. This results in the following equations for the cascaded
S-parameters,
1
1
2
1
1
2
b1,T M = S11
a1,T M + S12
S12
a2,T M + S13
a3,T E + S14
S34
a4,T E ,
b2,T M =
1
2
S21
S21
a1,T M
b3,T E =
1
S31
a1,T M
b4,T E =
1
2
S41
S43
a1,T M
+
+
1
2
2
S22
S21
S12
a2,T M
1
2
S32
S12
a2,T M
+
+
+
1
2
S23
S21
a3,T E
1
S33
a3,T E
1
2
2
S42
S43
S12
a2,T M
+
+
+
1
2
2
+S42
S21
S34
a4,T E ,
1
2
S34
S34
a4,T E ,
1
2
S43
S43
a3,T E
+
1
2
2
S44
S43
S34
a4,T E ,
(D.24)
(D.25)
(D.26)
(D.27)
From a synthesis perspective these equations can be used to decompose the desired chiral S-parameters
of the overall structure into two constitutive surfaces with anisotropic but non-chiral behaviour.
APPENDIX
E
Quasi-Optical Measurements
In Chapters 8 and 9 the fabricated polarization controlling surfaces are characterized in a 4-port quasioptical setup. Here some of the basic details of the quasi-optical setup used in these measurements
are reviewed. While having 4-ports in a quasi-optical setup is new, the actual details regarding the
quasi-optical components of the measurement are the same as in a 2-port setup. A good overview of
quasi-optical measurements can be found in [209] and the specific setup used here is an extension of
the quasi-optical setup used in [221]. The main things to highlight about the quasi-optical setup are
the properties of the Gaussian beam generated by the horn antenna, the characteristics of the dielectric
lenses and their relative configuration.
E.1
Horn Antenna
From the configuration of the quasi-optical system shown in Fig. 8.18, the horn antennas used in the
measurement setup are conical horn antennas (Cernex CCA081120M-01). A conical horn is chosen
because it has a high coupling efficiency to a fundamental Gaussian beam. For the specific horn antenna
under consideration a back-of-the-envelope calculation can lead to an estimate of the parameters of the
fundamental Gaussian beam generated by the horn antenna. A schematic of the horn antenna is shown
in Fig. E.1a.
Following [209], the two main things to determine are the radius of curvature of the Gaussian beam
at the aperture of the horn and the waist of the beam at the aperture of the horn as well. For a conical
horn, the radius of curvature is given by R =
r
sin θ
where r is the radius of the aperture of the horn
antenna and θ is the angle shown in Fig. E.1a. The waist at the horn aperture is defined simply by
w = 0.32d where d is the diameter of the horn antenna as also shown in Fig. E.1a. From these definitions
the properties of the Gaussian beam can be found from standard textbook calculations [153].
179
180
Appendix E. Quasi-Optical Measurements
Lens
Circular Horn
r d
θ
f1
y
f2
y
t
z
x
z
x
(a)
(b)
Lens
DUT
Circular Horn
f1
f2
y
t
z
x
(c)
Figure E.1: (a) The parameters of the conical horn used to determine the fundamental Gaussian beam
radiated by the horn. (b) The focal lengths of the dielectric lens. (c) The horn and lens and DUT
placed together. The horn is placed relative to the lens such that the focal length of the lens is where
the minimum of the waist in the Gaussian beam would be.
E.2
Dielectric Lenses
The dielectric lenses are biconvex lenses made of rexolite (n=1.59) with surfaces defined by a hyperbolic
profile given by
y=
p
(n2 − 1)x2 + 2(n − 1)f x
(E.1)
where f is the focal length of the lens. The lens is asymmetric with the focal length given by f1 =
186.8mm on one side and f2 = 289.8mm on the other side as shown in Fig. E.1b. The lens itself has a
thickness of t = 45mm. The focal length of these lenses was chosen for the quasi-optical measurement
setup in [221] and is treated as a constant here. To use these lenses in conjunction with the horn
antennas, their ABCD matrix must be defined [153]. For a lens with a hyperbolic curvature on either
side, the ABCD matrix for the entire lens is given by,
"
A
B
C
D
#
lens
=
"
A
B
C
D
#
h1
"
A
B
C
D
# "
t
A
B
C
D
#
,
(E.2)
h2
where the matrix is defined by the ABCD matrix for the hyperbolic surface with focal length f1 , the
ABCD matrix for the thickness of the lens and the ABCD matrix for the hyperbolic surface with focal
181
Appendix E. Quasi-Optical Measurements
length f2 . These are given by,
"
A
B
C
D
"
A
"
E.3
C
A
C
#
=
h1
B
D
#
B
D
#
"
1
0
−1
nf1
1
n
=
t
h2
=
"
"
#
,
(E.3)
,
(E.4)
#
(E.5)
#
1
t
0
1
1
0
1
f2
n
Overall Configuration
Finally, the configuration of the horn and lens is chosen such that the horn is placed at the focal length,
f1 , of the lens. The location of the minimum in the output waist and its size can be found through
standard quasi-optical calculations. This is a sanity check to confirm that the output waist is indeed at
f2 away from the lens as shown in Fig. E.1c. From the ABCD matrix given in (E.2),
dout =
(Af1 + B)(Cf1 + D) + ACzo2 )
,
((Cf1 + D)2 + C 2 zo2 )
wo
;
wout = p
(Cf1 + D)2 + C 2 Zo2
(E.6)
(E.7)
where zo is the Rayleigh range, and wo is the minimum waist which are both parameters of the input
Gaussian beam. It can also be confirmed that the Gaussian beam is confined to the aperture of the
DUT as well.
APPENDIX
F
Determining The Number Of Layers In A Tensor Impedance Transmitarray
3
1
Y1
a
Y2
b
c
Y3
d
e
f
4
Γf,Yf
2
Γa,Ya
Γc,Yc Γd,Yd
Γb,Yb
Γe,Ye
Figure F.1: An N = 3 tensor impedance transmitarray. As shown below, this network cannot be
reflectionless if γ is different from layer to layer.
In Section 9.2 it was stated N = 4 layers are needed to construct a chiral tensor impedance transmitarray that is reflectionless. Here further details are given as to how this can be determined. To help
make this case analogies to scalar impedance transmitarrays are used as necessary.
When designing a transmitarray, the variables or degrees of freedom are the impedance surfaces of
each layer, which are used to control the reflection and transmission coefficients. For a scalar impedance
transmitarray enough degrees of freedom are needed to set the complex valued reflection to zero (S11 = 0)
and to control the transmission phase (S21 = eiφ ) [138, 170]. Another way of looking at the design
problem is as a matching network [161], where the impedance at the output is taken to be Y = Yo and
the impedance of each surface is chosen such that the input impedance is also Y = Yo . From either
perspective, it is found that for a scalar transmitarray N = 3 impedance surfaces are needed to make a
reflectionless transmitarray.
For a tensor impedance transmitarray, the variables and reflection/transmission coefficients all increase proportionately. Each tensor admittance surface has three degrees of freedom. Meanwhile, the
complex reflection coefficient matrix now has six degrees of freedom while the the transmission parameters have another three (the relative phase and amplitude as well as the absolute phase). Thus
182
Appendix F. Determining The Number Of Layers In A Tensor Impedance Transmitarray183
intuitively, it would seem as if three layers would be sufficient1 . For a tensor impedance transmitarray
that implements chirality, it is implicitly assumed that for the admittance tensor at each layer, γ varies.
The problem then is to figure out how many layers are needed to create a reflectionless transmitarray
for this structure.
It is clear that having either N = 1 or N = 2 layers is insufficient. For an N = 3 transmitarray,
shown in Fig. F.1 it can be shown why this design cannot be reflectionless if γ changes from layer to
layer. From Fig. F.1, it is assumed that the values for Y1 and Y3 are given with the assumption that
γ1 6= γ3 implying that Y1 6= Y3 . It is also assumed that Ya = Yf = Yo . Using (9.3) and (9.5) the
complex input admittance can be found at points c and d in the transmitarray shown in Fig. F.1. These
are given to be
Yc = R1 (Gc,diag + jBc,diag )R1 −1 ,
(F.1)
Yd = R3 (Gd,diag + jBd,diag )R3 −1 ,
(F.2)
where R1 and R3 are the rotation matrices corresponding to Y1 and Y3 as defined in (8.8). It can then
be seen that the tensor admittance for the second layer is found to be Y2 = Yc − Yd . However since
γ1 6= γ3 the real part for Y2 is non-zero implying that a lossless admittance tensor can’t be found for
Y2 . Thus N = 3 layers are insufficient.
When there are N = 4 layers there are enough degrees of freedom to enforce a reflectionless transmitarray and to manipulate the transmission coefficients as desired while allowing γ to change from layer
to layer. From a matching point of view it can be seen that when N = 4, two layers can be used to take
the admittance matrix, which is Yo at the output, to a complex value inside the transmitarray as shown
using (9.6). Then, the remaining two layers are used to bring this complex admittance matrix back to
Yo at the input using (9.6) again. This is the basis for the semi-analytical procedure in Chapter 9. By
having (9.6), the design procedure is able to converge on reflectionless solutions.
Of course, N = 4 is simply the minimum number of layers needed to construct a reflectionless chiral
transmitarray, more layers can be used as they add extra degrees of freedom which may help other
parameters such as the bandwidth.
There is also the case of the CPSS which only required N = 3 layers as opposed to N = 4. This can
be reasoned using the same example above with similar logic. Here however, it can be shown that an
N = 3 transmitarray with different values of γ can support reflections. From a matching perspective,
a similar argument can be made as above, where taking the the input admittance at point f , which is
Yf = Yo in Fig. F.1, and using Y1 , Y2 and Y3 to make Ya 6= Yo at point a. This requires six degrees of
freedom. With three layers there are also enough degrees of freedom to control the transmission through
the structure to realize the functionality of the CPSS.
1 In fact if one looks at three layer tensor impedance transmitarrays that are symmetric (Y = Y ), reflectionless
1
3
structures can be designed that allow for the control of the relative phase, relative amplitude and absolute phase of the
transmitted field. However chiral effects would not be possible.
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