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Experimental and computational studies of electromagnetic cloaking at microwaves

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EXPERIMENTAL AND COMPUTATIONAL
STUDIES OF ELECTROMAGNETIC
CLOAKING AT MICROWAVES
By:
Xiaohui Wang
A DISSERTATION
Submitted in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
In Electrical Engineering
MICHIGAN TECHNOLOGICAL UNIVERSITY
2014
© 2014 Xiaohui Wang
UMI Number: 3622833
All rights reserved
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UMI 3622833
Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author.
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P.O. Box 1346
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This dissertation has been approved in partial fulfillment of the requirements for the
Degree of DOCTOR OF PHILOSOPHY in Electrical Engineering.
Department of Electrical and Computer Engineering
Dissertation Advisor:
Elena Semouchkina
Committee Member:
Durdu Guney
Committee Member:
Miguel Levy
Committee Member:
Warren Perger
Department Chair:
Daniel R. Fuhrman
Content
Content .................................................................................................................................................. iii
Preface .................................................................................................................................................... v
Acknowledgements............................................................................................................................... vi
Abstract ................................................................................................................................................ vii
1. Introduction ....................................................................................................................................... 1
1.1 Background of invisibility cloak ............................................................................................... 1
1.1.1 Transformation Optics based invisibility cloak ...................................................................... 1
1.1.2 Metamaterials in the design of the transmission cloak ........................................................... 4
1.2 Other cloaking approaches ........................................................................................................ 4
1.3 Problems of the existing cloak designs ..................................................................................... 5
1.4 Objectives and contributions ..................................................................................................... 6
2. Development of experimental facilities for characterizing resonator structures at microwave
frequencies ............................................................................................................................................. 9
2.1 Permittivity measurements by using the microstrip line method .............................................. 9
2.2 Complex permittivity measurements by the split-post method ............................................... 11
2.3 Fabrication of the fixture and establishing of the Hakki-Coleman technique for
characterization of dielectric resonators ........................................................................................ 12
2.4 Assembly and establishing of the waveguide technique for the characterization of dielectric
resonators and their arrays............................................................................................................. 13
2.5 Design and implementation of the parallel plate waveguide chamber for characterization of
multi-resonator structures .............................................................................................................. 14
2.5.1 Electromagnetic field mapping ............................................................................................ 15
2.5.2 Determination of the spectra of the total scattering cross-width .......................................... 15
3. Designing and implementing microwave cloaks by using all-dielectric metamaterials based on
transformation optics .......................................................................................................................... 18
3.1 Introduction ............................................................................................................................. 18
3.2 TO-based invisibility cloak design .......................................................................................... 18
3.2.1 General coordinate transformation ....................................................................................... 18
3.2.2 Cylindrical coordinate transformation.................................................................................. 19
3.2.3 Effective permeability of metamaterials composed of identical resonators ......................... 20
3.3 Selection of dielectric resonators ............................................................................................ 21
3.4 Spoke-like microwave cloak ................................................................................................... 22
3.4.1 Cloak Design ........................................................................................................................ 22
3.4.2 Fabrication and experiment .................................................................................................. 24
3.4.3 Simulation results on EM field analysis ............................................................................... 25
3.5 Impact of the effective permittivity of cloaking medium ........................................................ 27
3.6 Microwave cloak utilizing constant inter-resonator distances within concentric arrays ......... 29
3.6.1 Cloak design ......................................................................................................................... 29
3.6.2 Cloak implementation .......................................................................................................... 30
3.6.3 Simulation and experimental results .................................................................................... 31
3.7 Conclusions ............................................................................................................................. 33
4. Cylindrical cloaking by using multilayer dielectric coating ........................................................ 35
4.1 Introduction ................................................................................................................................... 35
4.2 Scattering from a cylindrical multilayer structure ................................................................... 35
4.3 Genetic algorithm based optimization of the cloak design...................................................... 38
4.4 Cloak comparison with TO based design ................................................................................ 40
4.5 Conclusion............................................................................................................................... 43
5. Spherical cloaking using multilayer shells of ordinary dielectrics.............................................. 44
5.1 Introduction ............................................................................................................................. 44
iii
5.2 Theoretical analysis of scattering from a spherical multilayer target ...................................... 45
5.3 Genetic algorithm based optimization of the cloak design...................................................... 47
5.4 Full-wave simulations of the cloak performance .................................................................... 49
5.5 Loss dependence of the cloak performance............................................................................. 51
5.6 Conclusion............................................................................................................................... 53
6. Cloaking of electrically large objects by ENZ material ............................................................... 54
6.1 Introduction ............................................................................................................................. 54
6.2 Theoretic analysis .................................................................................................................... 55
6.3 Analysis and discussion ........................................................................................................... 57
6.4 Simulation results of wave propagation through ENZ material cloaking a metal target ......... 58
6.5 Conclusions ............................................................................................................................. 61
7. Summary and future work ............................................................................................................. 62
7.1 Summary ................................................................................................................................. 62
7.2 Future work ............................................................................................................................. 63
References ............................................................................................................................................ 65
iv
Preface
This dissertation presents my research work in pursuing the Ph.D degree in Electrical
Engineering at Michigan Tech University. This dissertation includes previously published
articles in Chapter 3-5. All the published articles are collaborated with my supervisor Dr.
Elena Semouchkina.
Chapter 3 contains two articles published in IEEE International Symposium on Antenna
and propagation (APS/URSI) and IEEE Microwave and Wireless Components Letters. As
the first author of both papers, I mainly contributed on the methodology investigation,
experiments and draft writing. Fang Chen, as the second author provided most of the
simulation results. Steven Hook, as the third author in the APS/URSI article helped on the
prototype fabrication. As the last author, Dr. Elena Semouchkina took charge of the
revisions and final submission.
Chapter 4 contains one article previously published in Applied Physics Letters. As the
first author, I generated the idea and conducted the derivations and simulations. My
supervisor, Dr. Elena Semouchkina as the second author provided invaluable revisions
and advices on choosing the objective journal.
Chapter 5 contains one article previously published in AIP Advances. As the first author, I
generated the idea and conducted the mathematic derivations and programming. The
simulation results were obtained by the second author Fang Chen using CST. My
supervisor, Dr. Elena Semouchkina as the third author provided invaluable revisions.
v
Acknowledgements
First, I would like to give my sincerely thanks to my supervisor, Prof. Elena
Semouchkina for her continuous support during my four years’ study at Michigan
Technological University. Her supervision on my research conduct, paper writing, and
academic activities provided unique contributions on my way to accomplish this Ph.D.
Second, I want to give my great thanks to my committee members Dr. Durdu Guney, Dr.
Miguel Levy, Dr. Warren Perger and Dr. Kim Fook Lee for willing to serve on my
dissertation committee, spending time to read my thesis and giving me valuable
comments.
Third, I also want to thank the Electrical and Computer Engineering Department, from
the department chair to all the staff. Their well-organized work made my education go so
smoothly that I could focus more on my own research topics.
Big thanks will go to group members in our research lab: Dr. George Semouchkin,
Haolun Zhang, Sheng Mao, Michael Brush, Arash Hosseinzadeh and Ran Duan, who
accompanied me in this unforgettable journey.
The final thanks belong to my family members, for their unconditional love and support.
Special thanks go to my wife Fang Chen and my son Eric, who brought me amazing luck
even when he was not yet born.
vi
Abstract
An invisibility cloak is a device that can hide the target by enclosing it from the incident
radiation. This intriguing device has attracted a lot of attention since it was first
implemented at a microwave frequency in 2006. However, the problems of existing cloak
designs prevent them from being widely applied in practice. In this dissertation, we try to
remove or alleviate the three constraints for practical applications imposed by loosy
cloaking media, high implementation complexity, and small size of hidden objects
compared to the incident wavelength.
To facilitate cloaking design and experimental characterization, several devices and
relevant techniques for measuring the complex permittivity of dielectric materials at
microwave frequencies are developed. In particular, a unique parallel plate waveguide
chamber has been set up to automatically map the electromagnetic (EM) field distribution
for wave propagation through the resonator arrays and cloaking structures. The total
scattering cross section of the cloaking structures was derived based on the measured
scattering field by using this apparatus.
To overcome the adverse effects of lossy cloaking media, microwave cloaks composed of
identical dielectric resonators made of low loss ceramic materials are designed and
implemented. The effective permeability dispersion was provided by tailoring dielectric
resonator filling fractions. The cloak performances had been verified by full-wave
simulation of true multi-resonator structures and experimental measurements of the
fabricated prototypes.
With the aim to reduce the implementation complexity caused by metamaterials
employment for cloaking, we proposed to design 2-D cylindrical cloaks and 3-D
spherical cloaks by using multi-layer ordinary dielectric material (εr>1) coating. Genetic
algorithm was employed to optimize the dielectric profiles of the cloaking shells to
provide the minimum scattering cross sections of the cloaked targets. The designed
cloaks can be easily scaled to various operating frequencies. The simulation results show
that the multi-layer cylindrical cloak essentially outperforms the similarly sized
metamaterials-based cloak designed by using the transformation optics-based reduced
vii
parameters. For the designed spherical cloak, the simulated scattering pattern shows that
the total scattering cross section is greatly reduced. In addition, the scattering in specific
directions could be significantly reduced. It is shown that the cloaking efficiency for
larger targets could be improved by employing lossy materials in the shell.
At last, we propose to hide a target inside a waveguide structure filled with only epsilon
near zero materials, which are easy to implement in practice. The cloaking efficiency of
this method, which was found to increase for large targets, has been confirmed both
theoretically and by simulations.
viii
1. Introduction
1.1 Background of invisibility cloak
An invisibility cloak is a device that can hide the target by enclosing it. When the
electromagnetic wave hits the cloaked target, it does not travel through the cloaked region
and follows undisturbed along the original trajectory after passing the cloaked target as if
the target does not exist (Figure 1.1). This intriguing device has attracted a lot of attention
since it was first implemented for operation at microwave frequency in 2006 [1]
following two other pioneering findings, i.e. the appearance of the– transformation optics
(TO) and metamaterials.
Figure1.1 Schematic of the transmission invisibility cloak
(From[2]. Reprinted with permission from AAAS)
1.1.1 Transformation Optics based invisibility cloak
TO has provided the theoretical support for designing the first invisibility cloaks since it
paved a way to arbitrary redirect the electromagnetic (EM) waves [2]. Owing to the forminvariant property of the Maxwell’s equations in different coordinate systems, TO
exemplifies the EM identity between a physical space and the transformed or virtual
space. In particular, the equivalence of these two spaces for the wave propagation could
be accomplished by setting anisotropic and inhomogeneous materials with the prescribed
properties in the transformed space according to a specific mapping function. Based on
TO, two types of invisibility cloaks have been proposed, i.e. a transmission cloak and a
reflection cloak.
1

Transmission cloaks
A transmission invisibility cloak is a device that can hide the target in free space. As
shown in Figure 1.1, there is no reflection in the backward propagation direction or any
shadow in the forward propagation direction experienced by the travelling wave.
Transmission cloaks of canonical geometries, such as cylindrical or spherical ones,
present a special interest because these symmetric geometries reduce the anisotropic
requirements for the material. Specifically the effective parameter tensors of the cloaking
medium prescribed by TO will be diagonal matrices, and thus, the implementation
complexity will be reduced. One apparent feature of TO-based transmission cloaks is the
need of providing the effective relative permittivity or permeability values to be less than
one in the cloaking medium to support superluminal phase velocities along curved paths
around the object. Since natural materials do not provide such values, resonant
metamaterials seemed to be the suitable candidates for the cloak media.
The first cylindrical invisibility cloak composed of metallic split ring resonators (SRRs)
operating at microwave frequency was designed by Schurig et al. [1]. Close-packed SRRs
of different sizes were arranged in concentric layers to provide radial dispersion of the
effective permeability of the cloak medium at TE wave incidence. The invisibility effect
was narrow-band due to the causality constraints of metamaterials [3]. In addition, for the
cloak medium with the loss tangent on the level worse than 10-2, the so-called delay-loss
constraint [4] limited the size of the target being cloaked by one wavelength. Other
restrictions were added at scaling the SRRs forming the cloak down to operate at optical
frequencies due to the magnetic response saturation [5].
Employment of metamaterials comprised of dielectric resonators (DRs) was expected to
provide much lower loss and better scalability. A terahertz cloak designed by using DRs
of different sizes was the first attempt to explore an all-dielectric route for cloaking[6].
However, DRs of different sizes inevitably increased the implementation complexity at
decreasing the resonator size. A new approach to utilize identical DRs in a spoke-type
arrangement was proposed for designing an infrared invisibility cloak [7], where the
control of air fractions was used to provide spatial dispersion of the effective permeability
2
in the cloak medium. The designs in [6, 7] only aimed at the wave front reconstruction by
providing the required effective permeability dispersion, while the prescribed effective
permittivity of the cloak medium was not matched to the prescribed by TO values. This
affected the efficiency of reducing the scattering cross section (SCS) of the cloaked
object according to the theoretical analysis of the cylindrical cloaks scattering [8, 9]. To
decrease the scattering, a nonlinear function was proposed to map the physical space to
virtual space in the cloak design [10].

A unidirectional TO cloak
The described above transmission cloaks were designed to operate for waves incident
from any direction. To remove the anisotropic requirement to the cloaking medium, a
one-directional cloak was proposed based on TO approach that utilized homogeneous
metamaterials [11]. Later on, it was implemented and characterized experimentally at
microwave frequencies [12].

Reflection cloaks
Figure 1.2 Under-carpet cloak: (a) PEC ground without target, (b) with target, and (c) with cloaked target.
In addition to transmission cloaks hiding targets in free space, the so called under-carpet
or ground-plane cloaks have been proposed to hide a bump on the ground plane of a
perfect electrical conductor (PEC) [13]. As shown in Figure 1.2, when the cloaked bump
is illuminated by an EM wave, it reflects the wave as if there is just the PEC ground plane
without the bump; therefore, such cloak was also called a reflection cloak. The reflection
cloak does not require materials with the effective relative permittivity/permeability
values below than one, because the cloak does not need to support superluminal phase
3
velocity of the wave traveling inside the cloak. Therefore, the under-carpet cloak could be
realized by using non-resonant metamaterials not limited to the causality constraints and
providing a wide operation bandwidth [14-17]. However, the reflection cloaks only work
in a very rare scenario where a PEC ground plane is introduced as the background. Since
practical applications of reflection cloaks are very limited, this work was focused only on
the transmission cloaks.
1.1.2 Metamaterials in the design of the transmission cloak
The described previously TO-based transmission cloaks require the material parameters
of the cloaking shell medium to be anisotropic and spatially dispersed. To provide the
superluminal phenomenon, the effective permittivity (εr) or effective permeability (μr)
values less than one are further required in cloak medium. Metamaterials that support
Lorentz-type resonance could provide both the desired parameter values below one and
the dispersion, since their properties could be modified by a proper choice of the type and
the size of constituent resonators. Materials with (εr<1) can be provided by most of the
naturally existing noble metals near their electric plasma frequency [18]. Metamaterials
with (μr<1) were first proposed by Pendry using the split-ring resonator (SRR) medium
[19]. Later on, the double negative material or left handed material was constructed by a
combined SRR/metal wire medium [20]. In conclusion, metamaterials with flexible
constitutive parameters provide solid substance support for the invisibility cloak designs.
1.2 Other cloaking approaches
Except for TO, some other techniques have been proposed to reduce the scattering of the
target. In particular, hard surfaces were used to reduce the forward scattering of 2-D
masts with cross sections elongated in the wave incidence direction [21]. The hard
surface technique can reduce the scattering in a relatively broad band, but only for one
direction of wave incidence. It was also proposed that a cylinder with the diameter
smaller than the wavelength can be hidden by a set of annular metal layers with
thicknesses smoothly increasing toward the hidden object in the center for the incident
wave with E-field parallel to the axis of the cylinder [22].
Another popular approach for cloaking is based on scattering cancellation [23-25]. In
4
particular, it was shown that scattering from a dielectric target could be cancelled by
scattering from the shell made of a plasmonic material [23]. The above cancellation
techniques, however, targeted the dominant scattering mode and, therefore, hiding objects
much smaller than incident wavelengths, while for larger objects higher order scattering
modes deteriorated the cloaking effect. To enhance the quality of scattering suppression,
multi-layer metamaterials cloaking shells were proposed, which utilized secondary
reflections from the boundaries of layers [26, 27]. Most of the proposed up to date multilayer cloaks, however, employ layers of plasmonic or other complex artificial materials
that are difficult to implement in practice.
Matched zero index material (MZIM), metamaterials with both relative permittivity εr
and the relative permeability μr equal zero were also employed to design cloaks [28, 29].
It was proposed that objects made of a perfect electric conductor (PEC) [28] or dielectric
rods with the properly designated radius and permittivity [29] could be hided inside a 2-D
waveguide structure filled with MZIM. Further studies showed that using only epsilon
near zero (ENZ) material in the above structure could also result in total transmission if
the geometric size and material property of the dielectric rods are properly designed [30].
Recently, the level set-based topology optimization method was introduced to design a
unidirectional cloak [31]. Cloaking shell designed by this method is optimized in discrete
format of two kinds of materials: air and one specific ordinary dielectric material. Due to
the ease of fabrication, this unidirectional cloak was also experimentally demonstrated
[32]. The latest improved level set-based topology optimization cloak implemented by
using a polymer was able to work for a finite number of directions by employing some
symmetric structures in the cloaking shell [33].
1.3 Problems of the existing cloak designs
In summary, there are several serious problems that need to be solved on the way to
provide the invisibility in practice.

High loss and limitation on scalability
Metamaterials composed of metallic resonant elements usually have high ohmic losses.
As indicated above, for the cloak medium with the loss tangent on the level worse than
5
10-2, the so-called delay-loss constraint limits the size of the target being cloaked to the
wavelength [4]. For practical applications, however, the size of cloaked objects is desired
to be at least several wavelengths. This predicted drawback substantially limits the
application merits of the cloaks. In addition, other restrictions are added at scaling the
SRR-based cloaks for operation at optical frequencies, i.e. the magnetic response
saturation [5].

Implementation complexity
Transmission cloaks designed based on TO and dominant scattering mode cancellation
approaches require metamaterials or plasmonic material as the cloaking medium. The
artificial metamaterials, however, are difficult to implement in practice. Another
complexity of metamaterials employment is the accurate control of the effective
parameters dispersion in the cloak design. Based on the effective medium theory, the
effective parameters of metamaterials are obtained assuming the medium is infinite and
periodic. However, metamaterials employed in the cloaks are inevitably finite structures
and the periodicity of metamaterials is destroyed when spatially dispersed material
parameters are required

Relatively small objects to hide
The cloaks designed by using the dominant scattering mode cancellation [23-25]
approach can only hide objects much smaller than incident wavelengths, while for larger
objects, higher order scattering modes deteriorate the cloaking effect. TO-based cloaks
designed by metamaterials with metallic inclusions cannot hide large target because of
the delay-loss constraints. The other reported cloak designs [6, 7, 22, 26, 27] also did not
demonstrate the capability to cloak electrically large targets, while significant practical
applications are indeed expected for cloaking object substantially larger than the
illumination wavelengths.
1.4 Objectives and contributions
The objective of this dissertation is to find the ways to solve or relief currently existing
challenges of the invisibility realization by using different approaches to make practical
cloaking applications possible.
6
Metamaterials composed of dielectric resonators will be employed in the TO-based
cloaks design. For this purpose experimental techniques will be developed and relevant
devices will be assembled to characterize the relative permittivity of dielectric materials
at microwaves. Finally, a parallel plate waveguide chamber will be designed and
assembled for characterizing the resonator arrays and the entire cloak structures.
Programs related to the measurement automation, data collection and processing will also
be designed.
To reduce the loss of metamaterials medium and enhance the scalability of the cloak,
identical DRs will be used to design a low scattering microwave cloak. The proposed
cloak designs will be simulated as true multi-resonator structures to assess their
performance, their prototypes will be fabricated and complete experimental
characterization of the cloak will be performed inside the designed parallel plate
waveguide chamber.
To target the reduction of the implementation complexity, a new cloaking approach using
multi-layer ordinary dielectric coatings instead of metamaterials, based on the
cancellation of wave scattering from cloak layers will be developed. This work includes:
theoretical calculations on deriving the scattering coefficients of multi-layer cylindrical
and spherical structures; designing 2D cylindrical and 3D spherical cloaks from ordinary
dielectric layers using the Genetic Algorithm (GA) based optimization procedure.
To substantially increase the size of the hidden objects, a new type of cloak based on the
electromagnetic tunneling effect will be developed. Epsilon near zero (ENZ) materials
will be employed to design this cloak. The proposed cloak is inclined to hide larger
objects: the larger object, the better cloaking efficiency it can achieve.
The rest of this dissertation is organized as follow: Chapter 2 describes the experimental
facilities developed in the lab, which includes relative permittivity measurement devices
and a unique parallel plate waveguide chamber for recording wave transmission through
cloaking structures and resonator array. Chapter 3 is devoted to the development of the
TO-based microwave cloak designed by using identical dielectric resonators. Full wave
simulations of true multi-resonator structures and measurement results have proved the
7
cloaking efficiency. In Chapter 4, a cylindrical cloak composed of multi-layer ordinary
dielectric materials (εr>1) is proposed to hide target with diameters exceeding the
wavelength of incident wave. The dielectric profile of the cloak shell is optimized by GA
to provide the minimal scattering cross section. Chapter 5 extends the idea of Chapter 4
to spherical cloaking by using 3D multi-layer dielectric layers. Chapter 6 reports the
development of a cloak from an epsilon near zero (ENZ) material. This cloak shows an
intriguing character that better cloaking efficiency can be achieved when the size of the
PEC target is increasing. Chapter 7 concludes this dissertation and outlines some
perspective future work.
8
2. Development of experimental facilities for
characterizing resonator structures at microwave
frequencies
Dielectric resonators are basic elements of the dielectric metamaterials for the cloak
design. According to [34], the resonant frequency of the DR depends on its geometry and
material permittivity. Therefore, the permittivity of dielectric resonators at microwave
frequencies and their resonant frequencies should be characterized. This chapter describes
the efforts on the development of several techniques for measuring the complex
permittivity of dielectric materials at microwave frequencies and for the characterization
of microwave resonators and resonator arrays in the new Microwave Characterization
Lab of MTU ECE Department.
Furthermore, the techniques for the measurement of EM wave transmission through big
resonator arrays and cloaking structures have been established. In particular, a unique
parallel plate waveguide chamber has been designed and implemented to automatically
map the EM field distribution for the resonator arrays and cloaking structures. The
technique for measuring the total scattering cross-section (TSCW) of the structures by
using this apparatus has also been developed.
2.1 Permittivity measurements by using the microstrip line
method
Figure 2.1 The schematic of the microstrip line
The geometry of a typical microstrip line, which is widely used in microwave circuits, is
shown in Figure 2.1. It is known that the permittivity of the dielectric substrate defines
the characteristic impedance and the propagation constant of a microstrip line. Because
9
the dielectric substrate is not fully clamped by the strip conductor and the ground plane,
part of the fields from the microstrip conductor leaks into the air. Therefore, the effective
permittivity of the microstrip line is somewhat less than the real permittivity of the
dielectric substrate separating the conductor from ground plane. The expressions for the
relative effective permittivity εeff and the propagation constant β are[35]:

ε r + 1 ε r − 1 
ε eff =
2
+
h
1 + 12 

W
2 

−
1
2
2
 W 
+ 0.04 1 − 
h 


β = ω ε 0 µ0ε eff ,
(2.1)
(2.2)
where, ‘εr’ is the dielectric constant of the substrate, ‘h’ and ‘W’ are the height of the
dielectric substrate and the width of the strip conductor, respectively.
To measure the permittivity, the dielectric material under study should serve as the
substrate of the microstrip line, which can be fabricated at the MTU Microfabrication
Lab. We have assembled a set-up consisting of the vector network analyzer Agilent
N5230A and the ICM microstrip test fixture to measure the phase spectrum of the
transmission coefficient S21. The steps to obtain the material permittivity are listed below:
1. Calibrate the microstrip test fixture, which is shown in Figure 2.2.
Figure 2.2 ICM microstrip test fixture
2. Record the dimensions of the microstrip line, i.e. the microstrip width W and
length l, and the thickness h of the substrate.
3. Measure the phase delay φ of the transmission spectrum S21, then the effective
relative dielectric constant can be calculated by:
10
ε eff
 φc 
=

 2π fl 
2
(2.3)
where, ‘ c ’ is the speed of light, f is frequency.
4. Substitute equation (2.3) into equation (2.1), then relative permittivity of the
substrate is:
1
−
2

2
h


 W 

2ε eff − 1 + 1 + 12  + 0.04 1 − 

W
h 



εr =
1
−
2

h 2

 W
1 + 1 + 12  + 0.04 1 −  

W
h 



(2.4)
By using this method, only the real part of the relative permittivity can be measured,
since we derived the relative permittivity from the propagation constant β, which is only
related to the real part of the relative permittivity.
2.2 Complex permittivity measurements by the split-post
method
The split-post dielectric resonator (SPDR) provides an accurate technique for the
measurement of the complex permittivity of low loss dielectric materials, thin films and
wideband gap semiconductors that are difficult to measure by other techniques [36]. As
shown in Figure 2.3, the thin film sample is put inside the gap between the split dielectric
resonators, a pair of magnetic loop probes located in the diametric direction of the
dielectric resonator work as the transmitter and receivers. SPDR typically operates at the
TE01δ mode that restricts the electric field component to the azimuth direction so that the
electric field remains continuous on the dielectric interfaces. The real part of the relative
permittivity of the sample is determined from the resonance frequency shift due to the
sample insertion in the gap of the split post. The loss tangent can be found from the Q
factors of an empty cavity and a cavity with the sample, respectively. The expressions for
the real relative permittivity and loss tangent according to [36] are:
11
ε r' = 1 +
tan δ =
f0 − f s
hf 0 Kε (ε r' , h)
Q −1 − QDR −1 − Qc −1
pes
(2.5)
(2.6)
where: h is the thickness of the sample, Kε is a function of ε r' and h that tabulates all the
values for a given SPDR fixture, Q is the unloaded Q-factor of the resonant fixture
containing the dielectric sample, QDR is the Q-factor of the dielectric resonator, Qc is the
Q-factor of the resonant fixture containing the sample that depends on the metal losses,
and pes is the electric energy filling factor of the sample.
Figure 2.3 Schematic diagram of a SPDR fixture[37]. (©2012 IEEE)
Such fixture for the MTU Lab is being developed and constructed in collaboration with
the Penn State University[37, 38]. The limitation of the split post method is that the
permittivity of pre-measured samples is needed to interpolate the measurement result.
Hence the accuracy of the measured data depends on the results of the measurements of
preliminary samples.
2.3 Fabrication of the fixture and establishing of the HakkiColeman technique for characterization of dielectric
resonators
The Hakki-Coleman fixture [39, 40] has been assembled in the MWC Lab to measure the
resonant frequency of dielectric resonators. As shown in Figure 2.4, the fixture consists of
four parts: two poles to fix the transmitter and receiver, a metallic cavity formed by two
copper plates to accommodate the DRs under test, a screw jack that is supporting the
12
metallic cavity and a triangle rail to support the screw jack.
Figure 2.4 Assembled Hakki-Coleman fixtures in the Microwave Characterization Lab
To measure the resonant frequency of the dielectric resonator, it should be placed in the
center of the metallic cavity. A pair of electric/magnetic probes located in the vicinity of
the dielectric resonator work as the transmitter and receiver. At the resonance the received
signal increases substantially because the receiver probe detects the strong resonance
fields near the dielectric resonator. By measuring the transmission spectrum S21, the
resonant frequencies of the dielectric resonator can be detected.
2.4 Assembly and establishing of the waveguide technique for
the characterization of dielectric resonators and their arrays
Figure 2.5 Waveguide WR137and a DR array embedded into the foam
In order to characterize resonance properties of DR arrays, a waveguide technique has
been established, when resonators are placed inside a standard waveguide. The
waveguide WR137 (3.48cm*1.58cm) working in the TE10 mode was found suitable for
characterizing the resonance frequencies of single DRs and DR arrays in the microwave
range of interest. In order to locate the DRs accurately in the center of the waveguide
WR137, a foam substrate with the permittivity close to that of air is used to support the
13
DRs, as shown in Figure 2.5. The transmission spectrum S21 of the resonator-loaded
WR137 is measured by the network analyzer N5230A. The resonant frequencies of single
DRs and DR arrays can be then directly detected from the measured transmission
spectrum S21.
2.5 Design and implementation of the parallel plate waveguide
chamber for characterization of multi-resonator structures
A unique parallel-plate waveguide chamber (Figure 2.6) has been designed and
assembled in the MWC lab to map the field distribution of 2-D multi-resonator structures
including the entire cloaking structures. In addition, TSCW of the structures can be
determined from the field mapping data. The parallel plate waveguide consists of two
parallel aluminum plates. The distance between the two plates is adjusted to 12 cm so that
only TEM mode could propagate inside the parallel plate waveguide below the cutoff
frequency 12.5 GHz. The upper plate is fixed on the frame, while the lower one is
mounted to the computer-controlled X-Y translation stage driven by two step motors. The
transmitter and receiver is a WR90 waveguide adapter mounted on the front edge of the
lower plate and an electrical probe mounted on the center of the upper plate, respectively.
A ring microwave absorber with saw-toothed pattern and high attenuation rate sticking to
the lower plate serves to reduce the reflection from the edges.
Figure 2.6 Photograph of the parallel plate waveguide chamber
When the field mapping chamber is operating, the upper plate is kept stationary while the
lower plate with the target is moving with respect to the upper plate along with the
14
translation stage. The movement of the lower plate is driven by two step motors. In
addition, the data communication between the network analyzer and the computer is
supported by the USB-3488A GPIB card. Both step motors and GPIB card have been
programmed to obtain automatic measurement. A custom Labview program has been
developed to synchronize the movement of the translation stage and the data collection at
each step of the movement. The transmission spectrum S21 data are stored as complex
values in matrices which are later processed by using Matlab.
2.5.1 Electromagnetic field mapping
Figure 2.7 shows electrical field mapping result of a rectangular region inside the empty
waveguide chamber. The field map is plotted by the real part of the complex S21 matrix
for this region.
0
H
5
Y (cm)
K
10
15
20
0
5
10
X (cm)
15
Figure 2.7 Field mapping result of a rectangular region of empty parallel plate waveguide
2.5.2 Determination of the spectra of the total scattering cross-width
The EM field distribution is a powerful tool for the assessment of the cloak performance.
However, the TSCS can bring further quantitative information of the cloaking band. The
TSCS is usually referred to the total scattering cross area (TSCA) for the 3-D or total
scattering cross width (TSCW) for the 2-D case and is defined by the ratio of the total
scattering power and the power density of the incident wave:
=
TSCS
∫
2π
0
Es ( ρ 0 , θ )
Ein
15
2
2
ρ0 ⋅ dθ
(2.7)
where: ‘Es’ is the scattered electric field, ‘Ein’ is the incident electric field, and ‘ρ0’ is the
radial distance that is in the far field range
Therefore, TSCS is a metric that can judge the cloaking effect efficiency quantitatively
because it accounts for the omnidirectionally scattered field, including reflections and
shadowing. Mathematically, the scattered field is the relative difference between the total
field and incident field Es=Etot-Ein. The experimentally measured TSCW of an invisibility
cloak was first reported in [41]. Here, a similar method is employed to measure and
calculate the TSCS. In particular, the scattered fields in a circle that encompasses the
cloak were expanded as the summation of a series of Bessel functions. The TSCS can
then be expressed in terms of the Bessel amplitudes. The 2-D TSCS then can be
calculated by
=
TSCS
2
k0 Einc
∑( A
∞
2
m=0
m
2
+ Bm
2
) (1 + δ
m0
)
(2.8)
where, Am and B m are the amplitudes of the mth order Bessel function and δm0 is the
Kronecker delta function.
Scattering Cross Section (m)
0.25
Measured
Calculated
0.2
0.15
0.1
0.05
7
7.5
8
8.5
Frequency (GHz)
9
9.5
Figure 2.8 The measured and calculated SCS spectrum of a metallic cylinder
Figure 2.8 shows the measured TSCS of a metallic cylinder with the diameter 55 mm and
height 11 mm by using the experimental apparatus as Figure 2.6. For comparison, the
16
theoretically calculated TSCS of 2-D infinite long cylinder with the same diameter is also
given. The discrepancy between the measured TSCS and the calculated one is caused by
the gap between the top of the cylinder and the upper plate, which is necessary and
inevitable for the relative movement between the top plate and bottom plate of parallelplate waveguide chamber. However, this gap will destroy the periodicity of the metallic
cylinder in z-direction and therefore causes the discrepancy.
17
3. Designing and implementing microwave cloaks
by using all-dielectric metamaterials based on
transformation optics•
3.1 Introduction
This chapter describes the efforts on designing and implementing the microwave
cylindrical cloak by using identical dielectric resonators based on TO approach. Two
cloak designs employing various DR filling fractions to provide the effective
permeability dispersion have been proposed based on utilizing the reduced set of TOcloak parameters for TE wave incidence. The cloak performances have been investigated
by full-wave simulation of true multi-resonator structures and experimental measurement
of the fabricated prototypes. The first design, i.e. a spoke-like DR arrangement having
the same number of resonators in each concentric layer was found to provide for the
wave front reconstruction, however, exhibited high total scattering cross width (TSCW).
The second cloak design with fixed inter-resonator distance in each layer demonstrated a
clear cloaking effect, i.e. both the wave front reconstruction and reduction of the TSCW.
3.2 TO-based invisibility cloak design
3.2.1 General coordinate transformation
The general coordinate transformation for solving Maxwell’s equation between two
coordinate systems was first presented in [44], however, the effective parameters were
given in obscure format. Yan, et al. presented the coordinate transformation from
Cartesian space (x,y,z) to an arbitrary curved space (q1,q2,q3) in the matrix format[45].
The tensor format effective permeability µ̂ and permittivity εˆ in a curved space are given
below:
=
εˆ det
=
( g ) ( g T ) ε g −1 µˆ det ( g ) ( g T ) µ g −1
•
(3.1)
The material contained in this chapter was previously published in IEEE AP-S/URSI
[42]
X.
Wang, F. Chen, S. Hook, and E. Semouchkina, "Microwave cloaking by all-dielectric metamaterials," in
Antennas and Propagation (APSURSI), 2011 IEEE International Symposium on, 2011, pp. 2876-2878. and
IEEE Microwave and Wireless Components Letters [43] X. Wang, F. Chen, and E. Semouchkina,
"Implementation of Low Scattering Microwave Cloaking by All-Dielectric Metamaterials,"2013. (©IEEE)
18
where, ‘g’ is the metric tensor of the curved space,
 ∂f1 ∂f 2 ∂f3 


 ∂q1 ∂q1 ∂q1 
 ∂f
∂f 2 ∂f3 
g= 1 =
, x
 ∂q2 ∂q2 ∂q2 
 ∂f1 ∂f 2 ∂f3 


 ∂q3 ∂q3 ∂q3 
f=
f=
f3 (q1 , q2 , q3 )
1 ( q1 , q2 , q3 ), y
2 ( q1 , q2 , q3 ), z
‘ ε ’ and ‘ µ ’ are the effective permittivity and permeability in Cartesian space,
respectively.
3.2.2 Cylindrical coordinate transformation
Figure 3.1 shows the schematic of the cylindrical coordinate transformation, which is a
special case of the general coordinate transformation. The electromagnetic field inside a
physical cylindrical air region r < b can be compressed into a virtual annular region
a<r<b, so that an object could be hidden in the region r<a. To make these two regions
equivalent for wave propagation, the material parameters in the annular region were
given as in [1, 46]:
r−a
ε=
µ=
r
r
r
r
ε=
µ=
θ
θ
r−a
2
 b  r−a
ε=
µ=
z
z


b−a r
(3.2)
As seen from equation (3.2), all material parameters are anisotropic and dispersed with
the radius. Such material parameters dispersion is very difficult to implement in practice.
Figure 3.1 Coordinate transformation of cylindrical space
To ease the implementation, the reduced parameters have been proposed for linearly
polarized incident wave, for example, TE wave with electric field polarized along the
cylinder axis [46]. In this case, the Maxwell’s equations in the cylindrical coordinate
system are:
19
jωε z Ez
=
1  ∂ ( rHθ ) ∂H r 
−


r  ∂r
∂θ 
jωµr H r =
(3.3)
1 ∂ ( Ez )
r ∂θ
(3.4)
∂Ez
∂r
(3.5)
jωµθ Hθ = −
Only three material parameters εr, μr and μθ are relevant in equations (3.3-3.5). To
simplify the cloaking material parameters in equation (3.2), the permeability parameters
(both μr and μθ) can be normalized by μθ, while the permittivity component εr can be
multiplied by μθ to keep the refraction indices in radial and azimuthal directions
unchanged [47]:
nr
=
=
µθ ε z , nθ
µr ε z
(3.6)
Therefore, the reduced parameters for the cloak design under TE wave incidence are:
2
2
 b 
r−a
=
ε z =
 µr =
 µθ 1
b−a
 r 
(3.7)
Equation (3.7) shows that only the μr is spatially dispersed, while the other two
parameters are constant. The reduced cloak parameters hold the same refraction index as
the full cloak parameters, thus the cloaks designed by the reduced parameters equation
(3.7) have the same wave front propagation as the cloaks designed by the full parameters
equation (3.2). The reduced cloak parameters have successfully resulted in some practical
cloak realization [1, 6, 7, 48]. It should be noted that the simplicity of the reduced
parameters is at the cost of impedance mismatch that increases the scattering crosssection of the cloaked target.
3.2.3 Effective permeability of metamaterials composed of identical
resonators
As shown in [19, 20], the effective permeability of a medium composed of magnetically
resonating elements can be expressed by the relation:
20
f ω2
µreff =
1− 2
=
1 − f µrres
2
ω − ωres
+ iγω
(3.8)
where f is the resonance filling factor, which represents the ratio of the volumes of the
resonator (Vres) and the metamaterials unit cell (Vuc) that includes air fraction, ωres is the
angular resonant frequency of the resonators and γ is the dissipation factor. Equation
(3.8) shows that the dispersion of the effective permeability in the cloak can be provided
by tailoring f that has been used at the development of the design proposed here.
3.3 Selection of dielectric resonators
As discussed in Chapter 1, the target of this work is to employ low-loss dielectric
resonators in the design of the cloak for operation at microwave frequencies. A solid
solution made of (ZrTiO4/SnTiO4) ceramics was chosen as the material for the dielectric
resonators since it has high permittivity of 37 and very low loss of 2*10-4. Higher
permittivity values are required for DRs to be employed in the cloak for two reasons: 1)
to reduce the resonant frequency for a given shape of the DR to satisfy the subwavelength requirement for the effective homogeneous medium; 2) to provide better field
confinement inside the resonators at the magnetic resonance excitation in order to limit
the coupling between DRs.
0
-5
-10
S21 (dB)
-15
-20
-25
-30
-35
measurement
simulation
-40
-45
8
8.1
8.2
8.3
8.4
8.5
Frequency (GHz)
8.6
8.7
8.8
Figure 3.2 Transmission spectrum of a single dielectric resonator. Inset: H-field at f=8.34 GHz in the
diametrical cross-section of the DR
As it will be shown below, cylindrical DRs with the diameter of 6.06 mm and the height
21
of 3.06 mm were found to resonate at the desired frequency of 8.34 GHz. The DR aspect
ratio corresponding to the diameter twice as large as the height was chosen since it can
provide high Q factor and support magnetic moments along the DR axis in a wide range
of wave incidence angles [7, 49].
The electromagnetic (EM) response of a single DR has been investigated both by
simulations and experimentally. The transmission spectrum and field distributions were
simulated by using the CST Microwave Studio commercial software package. In
experiments, the resonators were embedded into a foam substrate and then put in the
center of a 2-inch waveguide WR137 section with their axis parallel to the magnetic (H)
field of the TE10 waveguide mode. Figure 3.2 shows the measured and simulated
transmission spectra with the above model. The inset H-field distribution inside the
resonator proves that the TE01δ resonance mode is excited at the resonant dip of the
transmission spectrum. The obtained TE01δ mode has the field distribution similar to that
of a magnetic dipole and therefore, DRs can successfully play the role of SRRs in the
cloak designs.
3.4 Spoke-like microwave cloak
3.4.1 Cloak Design
The design method for spoke-like cloak based on tailoring resonator filling fraction was
first reported by Semouchkina et al. in [7]. The cloak was designed by increasing the air
fraction from inner to the outer layer, while maintaining the same quantity of resonators in
each of concentric arrays. Therefore, all the DRs were arranged in spokes to provide the
required effective permeability dispersion as described in [7, 42].
Suppose there are several concentric layers in the cloak and each layer has the same
number N of dielectric resonators, as shown in Fig.3.3. The effective permeability
dispersion is provided by employing various inter-layer distances. On this assumption, the
resonator filling fraction fi of the ith layer can be found as:
fi =
N ⋅ Vres
2π riδ i h
22
(3.9)
where, Vres is the volume of the DR, ri is the radius of the ith layer and δi is the thickness of
the ith layer in the r-direction and h is the height of the cloak shell in z-direction.
δi
ri
δ1
r1
Figure 3.3 Geometric of spoke-like cloak
In order to design the cloak, the radius r1 and thickness δ1 of the first layer are supposed to
be known and the loss of the dielectric resonator is ignored. According to the reduced
parameters, the effective permeability of each layer should satisfy:
(µ )
eff
r
i
 ri − r1 
f ω2
1− 2 i 2 =
=


ω − ωres  ri 
2
(3.10)
Since the effective permeability of the first layer should be zero, then:
f ω2
=
1− 2 1 2 =
0
1
ω − ωres
(µ )
eff
r
(3.11)
Then the operating frequency of the cloak could be found as:
ω=
ωres
F1
, F1 = 1 − f1
(3.12)
where, F1 is the air volume fraction of the first layer.
By solving the Equation (3.10)-(3.12), the effective permeability of the ith layer can be
expressed in terms of the products δ·r for the first and the ith layer:
23
(µ )
eff
r
i
r δ  ri − r1 
=
1− 1 1 =


riδ i  ri 
2
(3.13)
Taking r1 = 40.1mm, δ1 = 7.5mm and N=18, the radii of the other cloak shells can be
found recursively from Equation (3.13). Table 3.1 presents the radii and the effective
permeability for each cloak layer. It should be noted that in this design only the effective
permeability dispersion was provided, while the required effective permittivity was not
exactly matched to Equation (3.1).
Table 3.1 The radius and the effective permeability for the spoke-like cloak
Ri /mm
μeff
40.1
0
53.6
0.0634
65.0
0.1467
75.5
0.2198
85.5
0.2819
3.4.2 Fabrication and experiment
The fabricated spoke-like cloak prototype made of identical DRs is shown in Figure 3.4. A
styrofoam substrate with the height of 14 mm was used to support the cloak. The cloak
consists of 5 concentric layers of DRs and each concentric layer is composed of 18
resonators at an equalized angle interval [42]. The target to be hidden is an aluminum
cylinder with the diameter of 5.5 cm, which corresponds to 1.5 wavelengths at the
frequency 8 GHz.
Figure 3.4 Fabricated spoke-like cloak with 18 DRs in each layer
To evaluate the performance of the cloak, the field maps for the cloaked target have been
measured inside the designed parallel plate waveguide chamber at different frequencies.
24
Figure 3.5 shows the field mapping results of the spoke-like cloak at frequencies 7.5 GHz,
8.3 GHz and 9.1 GHz. The field maps at 7.5 GHz and 9.1 GHz reveal wave front
reconstruction in contrast to the field map at 8.3GHz, where a shadow is observed.
H
K
8.3 GHz
7.5 GHz
9.1 GHz
Figure 3.5 E- field mapping result of the spoke-like cloak at different frequencies
3.4.3 Simulation results on EM field analysis
(a)
(b)
H
(c)
K
Figure 3.6 H field distribution at different frequencies: (a) f= 7.5 GHz, (b) f=8.3 GHz and (b) f=9.1 GHz.
Full wave simulations of the multi-resonator 3D cloak structure have been performed by
using CST Microwave Studio. The simulated field maps of the cloak were plotted at the
25
same frequencies as in the experiment. As shown in Figure 3.6, the wave front
reconstruction was clearly seen at the low frequency 7.5 GHz and high frequency 9.1 GHz
(Figure 3.6a and 3.6c), while at the frequencies 8.3 GHz a typical “shadow” of the object
was observed (Figure 3.6b).
SCW (m)
0.6
0.55
0.5
spoke-like cloak
bare cylinder
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
7
7.5
8
f (GHz)
8.5
9
9.5
Figure 3.7 Simulated total scattering cross-width (TSCW) of the spoke-like cloak with the target in
comparison with that of bare cylinder
The experimental and simulated field mapping results show that the spoke-like cloak
designed only by using the effective permeability dispersion can provide the wave front
reconstruction, however, this cloak did not exhibit TSCW reduction. As shown in Figure
3.7, the simulated TSCW spectrum of the cloaked cylinder is about three times higher than
the TSCW of the bare cylinder. Two reasons may contribute to the higher TSCW. First,
the inter-resonator distance in the outer layer of the cloak is about 3 cm, which makes the
unit cell size to be beyond the sub-wavelength dimensions, though the resonator size is
sub-wavelength at 9 GHz. This could possibly destroy the effective parameters of the
formed medium. Another factor is the not exact matching of the effective permittivity
value to the prescribed one. The effective permittivity extracted for the spoke-like design
from the simulated S-parameters was found to be around 1.2, which is less than the
required value 3.2. The mismatch of the effective permittivity may also result in
impedance mismatch between the air and cloak media increasing the reflection effect.
26
3.5 Impact of the effective permittivity of cloaking medium
In order to outline the ways for further optimization of the cloak design to increase its
efficiency, this section investigates the influence of the effective permittivity on both
wave front reconstruction and TSCW. The cloak model in Section 3.4 is modified that all
the concentric DR layers are replaced by effective medium layers with material
parameters corresponding to equation (3.1), while the geometry parameters of layers are
reserved. Figure 3.8a depicts the total E-field distribution with reconstructed wave front,
though there is circular scattering, as shown in Figure 3.8b. The TSCW spectrum of the
cloaked target has also been simulated by using the COMSOL Multiphysics 4.2
commercial software package as shown in Figure 3.9. The results prove that this cloaked
target has about 2/3 TSCW of the bare target, which is around 0.13m at 7~9.5GHz.
Figure 3.8 and Figure 3.9 show that a cylindrical cloak designed by using the reduced set
of parameters of equation (3.1) can have decreased TSCW and acceptable wave front
reconstruction.
(a)
(b)
Figure 3.8 E- field distribution of a 2D invisibility cloak designed by using reduced parameters at f= 8.0
GHz (a) the total field distribution (b) the scattering field distribution
27
SCW (m)
0.2
0.18
0.16
0.14
0.12
5-layer cloak with effective paramter
bare cylinder
0.1
0.08
0.06
0.04
0.02
0
7
7.5
8
f (GHz)
8.5
9
9.5
Figure 3.9 TSCW of the cloak designed by effective medium layers model in Section 3.4
To analyze the influence of the effective permittivity, the same cloak designed with
reduced effective parameters has also been investigated, except that the effective
permittivity was not matched, as shown below:
1.2 ≠ (
ε=
z
b 2
)
b−a
(3.7)
Figures 3.10 (a)-(b) shows that the wave front is distorted due to the strong scattering
particularly in the forward propagation direction, and the TSCW of the cloak with
unmatched permittivity surrounding the target is about 3 times higher than that of the
target, as seen in Figure 3.11.
k
(a)
(b)
Figure 3.10 E- field distribution of the cloak with the effective permittivity unmatched at f= 8.0 GHz (a) the
total field distribution (b) the scattering field distribution
28
Figure 3.11 Scattering cross width of the cloak with unmatched effective permittivity
In conclusion, matching of the constant value of the effective permittivity in the cloak
design plays an important role for both the wave front reconstruction and the decrease of
TSCW. These factors will be accounted for in the next design.
3.6 Microwave cloak utilizing
distances within concentric arrays
constant
inter-resonator
In this part, a cylindrical cloak design with constant inter-resonator distance inside each
layer is proposed with the purpose to provide decreased TSCW in addition to wave front
reconstruction. In difference from the spoke-like cloak, the inter-resonator distances in
the new design were fixed (d=10 mm) along circumferences in all layers, while interlayer distances were gradually increased from inner to outer shells. By this way, the main
drawback of the spoke-type microwave design in section 3.4 i.e., improper size of unit
cells in outer shells was avoided. The denser arrangement of DRs in this design also
increased the effective permittivity to make it closer to the value prescribed by Equation
(3.1). The number of layers was limited to four to avoid violation of the sub-wavelength
requirement for the radial dimensions of unit cells.
3.6.1 Cloak design
On the condition that the inter-resonator distance inside each layer is fixed, by solving
joint Equations (3.1) and (3.2), the effective permeability of the ith shell of the cloak can
be expressed in terms of the thickness δ1 of the first shell and the thicknesses δi of the ith
29
shell:
2


( µ )i =  ri −r r1  = 1 − δδ1
i
 i 
eff
r
(3.8)
Taking the radius of the first shell r1 to be 35 mm and its thickness δ1 to be 8 mm, the
radii of the other cloak shells can be found recursively from equation (3.8). The radius of
the first shell r1 and the thickness δ1 were chosen to operate with objects exceeding the
wavelength in air, while still providing for the sub-wavelength requirements for the unit
cells. Table I presents the radii, the effective permeability and permittivity values and the
number of resonators for each shell of the cloak. It should be noted that due to the applied
calculation procedure, the presented values of permeability exactly corresponded to the
prescribed ones given in equation (3.1). The permittivity values calculated by using
mixing formulas were less than the prescribed by equation (3.1) value of 4.95, however,
they were higher than the value of 1.7 in Section 3.4.
Table 3.2. Parameters of the cloak design
ith shell
ri (mm)
N
μeff
εeff
1
35
22
0
2.525
2
43.15
27
0.036
2.469
3
51.77
33
0.105
2.357
4
61.13
40
0.182
2.230
3.6.2 Cloak implementation
Z
Y X
h
Figure 3.12 The photograph of the cloak and its structural parameters
30
Figure 3.12 shows the fabricated prototype of cloak utilizing identical DRs and a
styrofoam substrate with the height h=12 mm. This prototype was fabricated to
correspond to parameters listed in Table 3.2. The object to be hidden was an aluminum
cylinder with the diameter of 55 mm, which corresponded to about 1.67 wavelengths at
the frequency of 9 GHz in free space.
3.6.3 Simulation and experimental results
V/m
Y
(a)
Z
X
(b)
Figure 3.13. E-field distributions of the (a) bare target and (b) cloaked target at 9.0 GHz. (In order to
provide comparison, the electric field intensity is clamped in the interval [-1, 1])
The performance of the cloak has been evaluated by using the COMSOL Multiphysics
software. In order to save the CPU memory and computational time, only a quarter of the
cloak has been modeled due to the structure symmetry with respect to both z=0 and y=0
planes, at which PEC/PMC boundary conditions have been applied. Figure 3.13
compares the E-field distributions for the wave propagating through the uncloaked bare
target and through the cloaked target in the plane z=0. The plane wave having a unit
magnitude and E-field polarized in the z direction is incident from the –x direction.
Strong E-field distortions are clearly observed in the field map of the bare target, i.e.,
reflections in the backward direction and a shadow in the forward direction. In
comparison, the shadow is significantly mitigated in the field map of the cloaked target
31
demonstrating a fair wave front reconstruction at 9.0 GHz.
In addition to verification of the wave front reconstruction, the efficiency of the cloak
was quantitatively evaluated by determining the TSCW, which is defined by the total
energy scattered from all directions by an object normalized to the incident energy
density:
=
σ
∫
2π
0
Ezs
2
2
Ezi r ⋅ dφ
(3.9)
=
Ezi E0 exp(− jkx) is E-field of the incident plane wave polarized in z direction and
where
Ezs is the scattered field.
For a PEC cylinder of radius ‘a’ illuminated by TE incident wave, the TSCW can be
derived by substituting the scattered field into (3.9):
s
z
=
E
E0
∞
∑
n = −∞
j − n (−1) J n (ka ) H n(2) (ka ) H n(2) (kr )e jnφ
(3.10)
4
Simulated
Measured
Normalized TSCW
3.5
3
2.5
2
1.5
1
0.5
8.5
8.6
8.7
9.1 9.2
9
8.8 8.9
Frequency (GHz)
9.3
9.4
9.5
Figure 3.14 The simulated (solid curve) and measured (dashed-dotted curve) TSCW spectra normalized to
the TSCW of the bare cylinder
Figure 3.14 presents the TSCW spectra of the cloaked target normalized by the TSCW of
the bare target obtained, correspondingly, from simulations and measurements. The
ripples appearing in the measured TSCW spectrum as shown in Figure 2.8 can be
32
removed by normalization. When the normalized TSCW is bigger than one, the cloaking
effect is not provided, while the cloaking band corresponds to the normalized TSCW
smaller than one. In experiments, the network analyzer N5230A was used to measure
phase-sensitive transmission S21 through the cloaked target, which was placed inside the
parallel plate waveguide chamber described in Chapter 2. After expanding the measured
scattering field in Bessel functions, the TSCW spectrum could be then calculated by the
summation of Bessel function amplitudes of different orders. The experimental and
simulated TSCW spectra are in good agreement and demonstrate the cloaking bandwidth
extending from 8.95 GHz to 9.15 GHz, where the normalized TSCW is less than one.
Based on air fractions controlled permeability dispersion in the cloak medium, a cloak
has been implemented by using four layers of equidistant identical dielectric resonators.
Both wave front reconstruction and the TSCW spectrum for the developed design have
confirmed an acceptable cloaking effect, which, for the first time, was simulated by using
the true model of the multi-resonator cloak structure. The respective data reported in
literature so far used cloak models composed of material layers (shells) with prescribed
parameters. The experimentally measured TSCW spectrum has demonstrated very good
agreement with the simulated one. Compared to other cloaking techniques [22-25], the
proposed cloak can also conceal bigger objects with the size exceeding the wavelength.
3.7 Conclusions
This chapter reports the progress on the development of microwave cloaks utilizing
identical DRs based on tailoring the resonator filling fractions. Two kinds of microwave
cloak have been proposed based on different mechanisms of tailoring the resonator filling
fraction. The spoke-like cloak was found only to provide acceptable wave front
reconstruction, while the TSCW was deteriorated due to the large unit cell size in the
outer layer and the mismatched impedance, which is resulted from the ignored effective
permittivity prescribed by reduced TO. This ignored effective permittivity, though as
constant, actually relates to the refraction index and impedance, which govern the wave
front reconstruction and TSCW reduction, respectively. Taking into account of the
effective permittivity, we proposed another cloak having fixed inter-resonator distance
inside each concentric cloak layer. The simulation and experiment results show that this
33
cloak can provide relative better wave front reconstruction and TSCW reduction.
In conclusion, the performance of invisibility cloak based on TO approach is relatively
limited. In particularly, the TSCW reduction, although comparable to the current
literature, actually is only 25%. In the following two chapters, we proposed cloak designs
by using ordinary dielectric based on other approaches that surpass TO-based cloaks.
34
4. Cylindrical cloaking by using multilayer
dielectric coating•
4.1 Introduction
Most of the proposed multi-layer cloaks to date employ layers of plasmonic or other
complex artificial materials that are difficult to implement in practice. Meanwhile, in
older studies of the radar detection problem, it was revealed that coating of an infinite
conducting cylinder by a dielectric layer with properly chosen permittivity and thickness
could cause significant decrease in backscattering [51]. Even better results for
backscattering were observed at a multilayer dielectric coating [52]. Recently, a two-layer
dielectric shell was also found efficient for decreasing the total scattering cross width
(TSCW) of a cylindrical metallic object [53], although the diameter of the object was
much smaller than the radiation wavelength.
In this chapter we investigate an attractive opportunity to design a multi-layer cylindrical
cloaking shell from ordinary dielectric materials (εr>1) for hiding objects with diameters
exceeding the wavelengths of incident illumination. In addition, in difference from most
of works on cloaking, we targeted an essential decrease of the shell thickness compared
to the diameter of the object. Illumination by TM mode was considered (with magnetic
field directed along the axis of the cylindrical target). We aimed to minimize the TSCW
of the cloaked target and employed the Genetic Algorithm (GA) for optimizing the
dielectric profile of the multilayer shell.
4.2 Scattering from a cylindrical multilayer structure
Scattering from the target covered by a multi-layer shell (layers numbered from 1 to M in
Figure 4.1) was analyzed for the incident wave described =
by H zi H 0 exp( − jk0 x) ,
where H0 is the magnetic field magnitude and k0 is the wave number in free space. The
radii of the outer boundaries of the layers are denoted by rm (m=1, 2…M) (Figure4.1).
The target region and free space are numbered as layer 0 and layer M+1, respectively, for
•
The material contained in this chapter was previously published in Applied Physics Letters [50] X. Wang
and E. Semouchkina, "A route for efficient non-resonance cloaking by using multilayer dielectric coating,"
Applied Physics Letters, vol. 102, pp. 113506-113506-4, 2013. (©2013 AIP)
35
the notation consistence. The material of the target was taken to be a perfect electric
conductor (PEC), however, an object made of any material covered by a thin PEC layer
could be taken instead.
Figure4.1. Schematic of the multi-layer cylindrical cloak with the target inside
The magnetic field Hzm in the mth cloak layer for TM illumination could be found from
the solution of the Helmholtz equation analogous to that used for the electric field at
consideration of TE case [26]:
1 ∂  r ∂H zm  1 ∂  1 ∂H zm  2
0

+

 + k0 µ zm H zm =
r ∂r  ε φ m ∂r  r 2 ∂φ  ε rm ∂φ 
(4.1)
The presence of only one z-component of the permeability tensor in equation (4.1) makes
the TM case advantageous for implementation since it allows for avoiding magnetic
materials in the shell. After separation of variables, the general solution of equation (4.1)
can be expressed as:
H zm =
∞
∑a
n = −∞
mn
 JVmn (km r ) + R m ( m −1) n HV(2)
(km r )  ⋅ exp( jnφ )
mn
36
(4.2)
where ‘n’ is the order of azimuthal harmonic function, amn is unknown coefficient,
km = k0 µ zmε φ m is the wave number in the mth layer, and Vmn = n ε φ m ε ρ m is the order
of the Bessel function. J(x) is the Bessel function of the first kind, while H (2) ( x) is the
Hankel function of the second kind representing the outward (in the +r direction)
traveling cylindrical wave with time dependence exp(jωt), and R m ( m −1) n is the total
outward scattering coefficient at the interface between mth layer and (m-1)th layer. The
total scattering wave at the interface r = rm-1 consists of two components: the wave
resulting from partial reflection of the inward propagating wave and controlled by the
coefficient Rm(m-1) and an additional wave coming from the region r < rm-1 and governed
by the term, which takes into account the wave totally scattered at the interface r = rm-2:
−1
T( m −1) m ⋅ R( m −1)( m − 2) ⋅ (1 − R( m −1) m ⋅ R( m −1)( m − 2) ) ⋅ Tm ( m −1) , where Tm(m-1) is transmission coefficient of
inward propagating wave at the above interface, while R(m-1)m and Tm-1)m are direct
reflection and transmission coefficients for outward propagating wave at the same
interface. Therefore, R m ( m −1) n at any interface can be defined by the given below
expression (where the index ‘n’ is omitted for shorter notation) [54]:
−1
Rm ( m=
Rm ( m−1) + T( m−1) m ⋅ R( m−1)( m− 2) ⋅ (1 − R( m−1) m ⋅ R( m−1)( m− 2) ) ⋅ Tm ( m−1)
−1)
(4.3)
The direct reflection and transmission coefficients have been derived by matching the
tangential E-field and H-field components at the interfaces:
j ′j−1 − ηm −1 / ηm j−′1 j
h′h−1 − ηm −1 / ηm h−′1h


=
=
R
R
(
1)
(
m
m
m
−
−
1)
m


ηm−1 / ηm j−′1h − j−1h′
ηm−1 / ηm j−′1h − j−1h′




2i
1
2i
1
T
T
⋅
⋅
m ( m −1) =
( m −1) m =
π km rm−1 ηm−1 / ηm j−′1h − j−1h′ 
π km−1rm−1 j−′1h − ηm / ηm−1 j−1h′

where, j = JV (km rm−1 ) , j ′ = JV′ (km rm −1 ) , j−1 = JV
nm
n ( m−1)
nm
(km −1rm −1 ) ,
(4.4)
j−′1 = JV′n ( m−1) (km −1rm −1 ) ,
(2)
′ (km −1rm −1 ) , and η m is the
h = HV(2)
(km rm −1 ) , h ' = HV(2)′ (km rm −1 ) , h−1 = HVn ( m−1) (km −1rm −1 ) , h−′1 = HV(2)
nm
n ( m−1)
nm
wave impedance in the mth layer.
With account for equation (4.4), equation (4.3) could be employed for deriving
37
recursively the total scattering coefficient for the outer surface of the shell, however,
initialization of the recursive procedure demanded knowing of the scattering coefficient
R10 at the interface between the PEC target and the first layer of the shell. For the case of
TM wave incidence this coefficient could be defined by the equation [55]:
R10 n = − J n' (k1a ) / H n(2)' (k1a )
(4.5)
After the total scattering coefficient at the outer cloak surface for each order of
cylindrical wave R( M +1) Mn is found, the scattered field of the multilayer cloak could be
∞
(2)
s
jnφ
−n
determined by using the expression H z = H 0 ∑ j R( M +1) Mn H n (k0 r )e . In the far field
n = −∞
approximation we employed the asymptotic formula of Hankel function, therefore, the
TSCW of the cloak σ could be derived as:
σ
=
∫
2π
0
H zs
2
2
r=
⋅ dφ
2
π k0
H0
2π
∞
∫ ∑
0
n = −∞
2
R( M +1) Mn e jnφ dφ
(4.6)
4.3 Genetic algorithm based optimization of the cloak design
The obtained expression provided an opportunity for quantitative characterization of the
cloak performance and was further used to define the fitness function (1/σ) in the cloak
design optimization procedure based on GA. The goal of this procedure was to search for
the relative permittivity values exceeding or equal to 1 for each dielectric layer, which
would provide the minimal possible value of σ. The efficient elitist strategy with
tournament selection [56] has been employed for optimization. Taking into account the
trade-off between implementation simplicity and optimization diversity, an eight-layer
cloak of equi-thick layers has been explored here. At the given value of target radius ‘a’
equal to either 0.75λ, or 1.0λ the thickness of each of 8 layers was controlled by the
optimal value of ‘b’.
In the GA optimization procedure, the shell parameters (εi, b) were coded by binary digits
and the search domain for ‘εi’ was chosen between 1 and 128(27). The search domain of
the outer radius ‘b’ was defined by the range between 1.05a and 2a that provided for
diverse options for the layer thicknesses. To produce next generations of the cloak
38
parameters, the probabilities of crossover and mutation were set to be 0.8 and 0.1,
respectively. The results of the cloak parameter optimization for two chosen targets with
a=0.75λ and a=1.0λ are presented in Table 4.1 and in Figure 4.2.
Table 4.1. Optimized dielectric profile and the outer radius of the cloak
a/λ
ε1
ε2
ε3
ε4
0.75 1.0 65.0 65.0 17.0
1.0
1.0
1.0
ε5
ε6
ε7
3.0
3.0
4.0
ε8
b/λ
58.0 0.8695
65.0 65.0 25.0 6.0 50.0 65.0 1.1125
As seen from the table and Figure 4.2, radial dispersion of the cloak permittivity in the
optimized shell cannot be described by a simple monotonic dependence for both target
sizes. One of noticeable features is the need to have an air gap between the target and the
cloak. It is interesting to note that in the earlier work devoted to the solution of the
backscattering problem for conducting cylinders coated by dielectric shells it was also
found useful to introduce an air gap with width up to the wavelength [57]. Another
specific feature of the optimized dielectric profiles is their U-shape (Figure 4.2). In
addition, it is worth noting equal permittivity values in some neighboring layers that
could further simplify the cloak implementation.
Dielectric Constant
70
a=0.75λ
a=1.0λ
60
50
40
30
20
10
1
1
2
3
4
5
6
Number of Layer
7
8
Figure 4.2. Optimized radial dielectric profiles of the multi-layer shells for two targets with different radii.
39
An important result of optimization is that the optimized shell thickness is only 16% of
the target radius at a=0.75λ and 11.25% at a=1.0λ, respectively. As known, the thickness
of cloaks designed by using the TO approach was usually comparable to ‘a’, i.e. was 8-9
times bigger than that of the proposed cloak. Among other designs only the mantle cloak
had comparable shell thickness [25].
The optimized values of shell parameters in Table 4.1 were used for calculating the
TSCW for cloaked and bare targets by using equation (4.6). For the target with the radius
of 0.75 λ the TSCW was found to be only 1.24λ, i.e. 50% of the TSCW value for the bare
target (2.48λ), while for the target with the radius of 1.0 λ the TSCW of the cloaked target
(2.15λ) was found to be 62% of the TSCW value for the bare target (3.47λ). It should be
noted that earlier works on backscattering from PEC targets coated by multi-layer shells
also did not provide perfect scattering cancellation and mentioned some restrictions in
scattering reduction [51, 52].
4.4 Cloak comparison with TO based design
To additionally ensure advantages of the optimized multi-layer dielectric cloaks beyond
their material simplicity, the performance of one of them was compared with that of the
cloak having the same dimensions (a=0.75λ, b=0.8695λ) and the same thicknesses of 8
layers (0.015λ) but designed by using the TO approach, in particular, possessing with
radial dispersion of the effective permittivity and values of other parameters satisfying
the given below relations derived for TM wave incidence [48]:
µ z =1, εθ =[b /(b − a)]2 , ε r =[b /(b − a)]2 [(r − a) / r ]2
(4.7)
Although a TO-based cloak with such thin layers as those obtained at optimization of the
multi-layer dielectric cloak (0.015λ) could not be implemented using conventional
metamaterials composed of resonance elements with dimensions of about 0.1λ, there is
no fundamental restriction for the theoretical comparison of two cloaks. Providing for the
parameters of the layers μz=1, εθ=52.92 and εri<1 (i=1…8) prescribed by equation (4.7)
would demand in practice meta- or plasmonic materials. Since the above mediums are the
resonance ones, their operation is frequency dependent. However, for the desired
comparison of two cloaks these specifics could be ignored.
40
The multi-layer dielectric cloak could perform at any frequency if the dielectric profile
presented in Table 4.1 is provided. However, the geometric parameters should be varied
accordingly for various operation frequencies since they have been normalized with
respect to the wavelength of incident radiation.
Figure 4.3 Snap-shots of Hz distributions for fwork=8 GHz at TM wave incidence on: (a) bare metallic
cylinder with the radius 0.75λwork; (b) the same target cloaked by the shell with material parameters
prescribed by equation (4.7); (c) the same target cloaked by the multi-layer dielectric shell; (d) zoomed-in
view of (c).
Taking into account the above considerations, the two cloak performances could be
compared at arbitrary frequency fwork, while the dimensions of the multi-layer dielectric
cloak should be determined by using λwork= c/ fwork. The presented below results have
been obtained for fwork=8 GHz, i.e. for the cloaks with the dimensions defined by
λwork=3.75cm. The software COMSOL Multiphysics was used to simulate wave incidence
on multi-layer dielectric and TO-based cloaks. Snap-shots of Hz field distributions of two
cloaks with the same target are compared in Figure 4.3. As seen from Figure 4.3a, strong
distortions of the wave front and a “shadow’ behind the target are clearly observed in the
latter case. In contrast, both cloaks provide decreased reflections and essentially less
41
expressed shadows, however, the multi-layer dielectric cloak performance is obviously
better.
The above observations were also justified by TSCW comparison for two cloaks obtained
by using COMSOL-based simulations of integrated far-fields of scattered waves. The
obtained TSCW values for the bare target and for the target cloaked by either the TO
based cloak or the multi-layer dielectric cloak were 9.56 cm (2.55λwork), 7.60 cm
(2.026λwork) and 4.80 cm (1.28λwork), respectively. From the obtained values it is obvious
that the TO-based metamaterials cloak essentially concedes to the simple multi-layer
dielectric cloak. It is worth mentioning that the presented above TSCW data for the bare
and dielectric coated targets are consistent with the respective TSCW values found by
using the analytical equation (4.6) (2.48λ and 1.24λ, respectively).
The presented in Figure 4.3d zoomed-in image of H-field shows that the performance of
the multi-layer dielectric cloak is accompanied by trapping of a part of incident
electromagnetic energy inside the shell and transferring it along the shell circumference
as a kind of azimuthal surface wave. The observed by-passing of the target could possibly
contribute in reducing the TSCW.
The TSCW spectra for targets with the radii of 0.75λwork and 1.0λwork were obtained by
integrating the simulated far-fields. The obtained TSCW spectra for frequency swept from
6 GHz to 10 GHz for the bare and cloaked targets are presented in Figure 4.4. As seen
from the figure, the cloaking effect (the drop of the TSCW below the value characteristic
for the bare target) is observed near 8.0 GHz in both cases, but for the larger target the
TSCW drop does not exceed 38%. The latter result agrees with noticed in other works
trend toward decreasing the cloaking effect for larger targets due to enhancement of the
higher order scattering modes [23]. However, in the presented cloak designed to provide
maximal suppression of total scattering from the cloaked target, the changes in
contributions from various scattering modes in responses from larger targets should
produce smaller effect on cloaking than that in cloaks aimed on suppression of only the
dominant mode. It is also worth noting that the cloaking effect bandwidth in Figure 4.4 is
relatively wide (more than 5%) for both targets. Since no optimization of this parameter
42
was performed in the proposed design, there is a potential for increasing it further in future
work.
22
20
TSCW (cm)
18
16
Bare target (a=0.75λ)
Cloaked target (a=0.75λ)
Bare target (a=1.0λ)
Cloaked target (a=1.0λ)
14
12
10
8
6
4
6
6.5
7
8.5
8
7.5
Frequency (GHz)
9
9.5
10
Figure 4.4. The TSCW spectra obtained by integrating the simulated far-fields for the cloaked and bare
targets of two different sizes. Dimensional parameters of the multi-layer dielectric cloaks in both cases were
fixed for the wavelength corresponding to fwork =8 GHz.
4.5 Conclusion
In conclusion, we have developed an approach for designing multi-layer dielectric cloaks
that could be easily scaled to various frequency ranges. In particular, it was demonstrated
that a 50% TSCW reduction could be achieved for a metal target with the radius of 0.75λ,
when it’s covered by a shell composed from only few dielectric layers each of 0.015λ
thick. It was shown that the multi-layer dielectric cloak essentially outperformed the
similarly sized metamaterials cloak designed according to the TO requirements. The
design procedure developed in this work could be used for further advancing the cloak
parameters and for adjusting it to practical needs.
43
5. Spherical cloaking using multilayer shells of
ordinary dielectrics•
5.1 Introduction
Implementation of spherical cloaks, whose performance is independent on the
polarization of incident radiation, presents a special interest for both the theory of
cloaking and practical applications. The first approach for spherical cloak design based
on the transformation optics (TO) and demanded the effective parameters of the cloak
media to be spatially dispersed and anisotropic [2]. While for TO-based cylindrical cloaks
under either transverse electric (TE) or transverse magnetic (TM) illumination a set of
reduced effective medium parameters could be employed to alleviate the material
complexity [1, 46, 48], similar simplification is not possible for spherical cloaks, since
decomposition of any incident plane wave includes both radial TE and TM components.
In search for an opportunity to fulfill the requirements for the cloak medium, segmenting
an anisotropic material into isotropic layers with sub-wavelength thicknesses has been
proposed [59]. In particular, a design of a spherical cloak composed of multilayer
isotropic metamaterials and based on linear TO was presented [60]. Even better results
for the total scattering cross section (TSCS) reduction of a cloaked sphere were obtained
when non-linear TO was applied [61]. However, the above cloak designs [60, 61]
required too many layers of metamaterials with different parameters that prevented
implementation of such cloaks. As means to decrease the number of isotropic layers in
the cloak shell, the particle swarm optimization technique was employed to design a
spherical cloak [62] .
An alternative spherical cloaking approach was dominant scattering mode cancellation
under the quasi-static conditions [23-25]. Such incorporated plasmonic metamaterials in
the cloak medium and its efficiency was found to depend on the target material and size.
The target size also had to be small compared to the wavelength, as, otherwise, the higher
order scattering modes became dominant so that the suppression of the dominant mode
•
The material contained in this chapter was previously published in AIP Advances [58] X. Wang, F.
Chen, and E. Semouchkina, "Spherical cloaking using multilayer shells of ordinary dielectrics," AIP
Advances, vol. 3, p. 112111, 2013. (©2013 AIP)
44
lost sense. All listed above designs of spherical cloaks [2, 23-25, 60, 61] employed
metamaterials or plasmonic materials, whose electromagnetic properties were frequency
dependent and most of which were difficult to implement practically.
In Chapter 4, we demonstrated that, instead of metamaterials, ordinary dielectric
materials can be used in a multi-layer cylindrical shell for providing efficient cloaking of
cylindrical objects [50]. It should be noted that an opportunity to conceal a spherical
target by using a single layer dielectric shell was also considered in [63]; however, the
designed cloak could perform only for a dielectric target. In this work, we extend our
approach in Chapter 4 to a much more challenging case of cloaking spherical metal
targets by 3D multi-layer dielectric shells.
5.2 Theoretical analysis of scattering from a spherical
multilayer target
Figure 5.1 Schematic of a multi-layer spherical cloak with a PEC core inside
First, the scattering of a monochromatic plane wave by a spherical metal target covered
with a multilayer dielectric coating is considered. Figure 5.1 shows the schematic of the
cloaked target, which is a perfectly electric conducting (PEC) sphere with the radius ‘a’.
The radii of the outer boundaries of the cloak layers are denoted by rm (m=1, 2…M). The
target region and free space are considered as layer 0 and layer M+1, respectively. Based
on the standard Mie expansion, the field scattered by the spherical target covered with a
multi-layer shell at the outer surface of the cloak can be represented by the sums of
45
TM
spherical harmonics TMr and TEr of different orders ‘n’ with coefficients R( M +1) Mn and
R(TE
M +1) Mn respectively. The TSCS of the cloaked target QS can be expressed by using the
formula for a spherical particle [64]:
λ2
Qs =
2π
∑ ( 2n + 1) ( R
∞
2
TM
( M +1) Mn
n =1
+ R(TEM +1) Mn
2
)
(5.1)
TM
TE
The coefficients R( M +1) Mn and R( M +1) Mn at the outer layer of the cloak can been derived
recursively by using the equation for spherical multi-layered media from [54] (in which
index ‘n’ is omitted for shorter notations):
−1
Rm ( m=
Rm ( m−1) + T( m−1) m ⋅ R( m−1)( m− 2) ⋅ (1 − R( m−1) m ⋅ R( m−1)( m− 2) ) ⋅ Tm ( m−1)
−1)
(5.2)
where, Rm ( m−1) is the total scattering coefficient at the spherical surface with r = rm-1; Rm ( m −1)
and Tm ( m −1) are, respectively, the direct reflection and transmission coefficients for the
inward propagating wave at r = rm-1; R( m −1) m and T( m−1) m are the direct reflection and
transmission coefficients for the outward propagating wave at r = rm-1; and R( m−1)( m−2) is the
total scattering coefficient at the spherical surface r = rm-2. The direct reflection and
transmission coefficients for both TMr and TEr spherical waves have been determined by
matching the E-field and H-field at the interfaces of layers to satisfy the boundary
conditions:
 TM
j j ′ − ηm −1 / ηm j−1′ j
 Rm ( m −1) = −1
ηm−1 / ηm j−1′h − j−1h′


i
 TmTM( m −1) =

ηm−1 / ηm j−1′h − j−1h′

h−1h′ − ηm −1 / ηm h−1′h
 TM
=
R
−
(
1)
m
m

ηm −1 / ηm j−1′h − j−1h′

 TM
i
 T( m −1) m =
hj−1′ − ηm / ηm −1 j−1h′

46
 TE
j j ′ − ηm / ηm −1 j−1′ j
 Rm ( m −1) = −1
ηm / ηm −1 j−1′h − j−1h′


i
 TmTE( m −1) =

ηm / ηm −1 j−1′h − j−1h′

h−1h′ − ηm / ηm −1h−1′h
 TE
 R( m −1) m =
ηm / ηm −1 j−1′h − j−1h′

 TE
i
 T( m −1) m =
j−1′h − ηm −1 / ηm j−1h′

(5.3)
where, j = Jˆn (km rm−1 ) , j ′ = Jˆn′ (km rm−1 ) , j−1 = Jˆn (km −1rm −1 ) , j−′1 = Jˆn′ (km−1rm−1 ) , h = Hˆ n(2) (km rm−1 ) ,
, Ĵ (x ) is the spherical Bessel function
k
η
Hˆ ( 2 ) (x )
and
is the second kind of the spherical Hankel function; m and m are the wave
(2)
h ' = Hˆ n(2)′ (km rm −1 ) h−1 = Hˆ n (km −1rm −1 )
,
,
h−′1 = Hˆ n(2)′ (km −1rm −1 )
number and the impedance of the mth layer, respectively.
To initialize the recursive procedure based on equation (5.2), one needs to know the
scattering coefficients R10TMn and R10TEn at the interface between the target and the first layer
of the shell. These two coefficients could be obtained similarly to obtaining scattering
coefficients of a PEC sphere in free space:
R10TMn =
− Jˆn′ (k1a) / Hˆ n(2)′ (k1a), R10TEn =
− Jˆn (k1a) / Hˆ n(2) (k1a)
(5.4)
The set of equation (5.1-5.4) allows one to determine the value of QS for the multi-layer
cloak, which characterizes the cloak performance, and can be used as an objective
function for the optimization procedure described below.
5.3 Genetic algorithm based optimization of the cloak design
The Genetic Algorithm (GA) has been further applied to determine the optimal profiles of
the dielectric permittivity in the shell for providing minimal values of QS. In order to
develop doable cloak designs, all cloak layers were considered as formed from ordinary
nonmagnetic materials with the relative permittivity values either exceeding or equal to 1.
The search domain for the relative permittivity values ‘εi’ (where index ‘i’ counts the
layers outwards from the target), was taken between 1 and 129. Materials with thus high
permittivity can be easily found on the market, in particular, among the BST composites
based on barium/strontium titanates. As known, these materials can demonstrate the
relative permittivity up to 600 [65] and even higher at microwaves, while in THz range
they have been used in dielectric resonators with the relative permittivity of 200[6]. It is
important to mention that the BST-based materials are compatible with the film
technologies, so that it is common to deposit BST films with the characteristic for
coatings thicknesses of several hundred microns [66]. In order to reduce the number of
optimization variables, all layers of the shell are presumed to have equal thicknesses
47
defined by the chosen outer radius ‘b’ of the cloak and by the number of layers. It is
worth mentioning that this assumption does not prevent optimization of the thickness of
the cloak layers. As it could be seen from Table 5.1 below, which presents the optimized
parameters of the cloak layers, some neighboring coating layers are prescribed the same
permittivity values, so that they could be considered as a single layer with a larger
thickness. Therefore, instead of making the thickness of layers variable at the
optimization, the overall number of layers in the shell could be increased, if the quality of
the obtained results looks insufficient. In this work, the domain for ‘b’ was taken between
1.05a and 2a and an eight-layer shell has been designed to restrict the fabrication
complexity. Since the convergence of QS calculated by using equation (5.1) for the Mie
expansion series is known to deteriorate when the size of the scatter exceeds the
wavelength of incident radiation, our consideration was limited by targets with the radii
‘a’ equal to 0.5λ and 0.2λ, respectively.
Table 5.1 Optimized Dielectric profiles and the Outer Radii of Cloaks
a/λ
ε1
ε2
ε3
ε4
0.5 1.0 1.0 129.0 129.0
0.2 1.0 1.0 129.0
65.0
ε5
ε6
2.0
1.75
ε7
ε8
72.9 97.0
b/λ
0.58
129.0 129.0 12.9 79.7 0.335
In GA optimization, the shell parameters [εi, b] are coded by binary digits and the
efficient elitist strategy with tournament selection has been employed [56]. To produce
each next generation of the cloak parameters, the probabilities of crossover and mutation
were set to be 0.8 and 0.1, respectively. The results of the cloak parameter optimization
for two chosen targets are presented in Table5.1.
As seen from Table 5.1, the radial dependences of the permittivity in the shell media for
both targets are non-monotonous. Another noticeable feature is an air gap between the
target and the shell. It should be noted that similar features were characteristic for the 2D
cylindrical cloak in Chapter 4.
48
The parameters in Table 5.1 were employed to calculate the TSCS (QS) values for the
cloaked targets by using equation (5.1-5.4). For the target with a=0.5λ QS was found to be
0.72λ2, i.e. 42.4% of the value for the bare target (1.70λ2), while for the target with a=
0.2λ, QS appeared to be 0.047λ2, i.e. only 16.4% of the value for the bare target (0.287λ2).
Better cloaking efficiency for the smaller target is apparently related to fewer scattering
multipoles that need to be suppressed, compared to the case of the larger target.
5.4 Full-wave simulations of the cloak performance
(a)
(b)
(c)
(d)
Figure 5.2 EX distribution in YZ cross-section for: (a) and (c) bare PEC targets with a=0.5 λ and a=0.2 λ; (b)
and (d) respective cloaked targets at 6.0 GHz. The incident wave has E-field polarized in X-direction and
propagates in Z-direction (For better comparison, the electric field intensity is clamped in the interval [-1,
1])
The performances of the designed cloaks have been investigated by full-wave simulations
using the commercial software CST Microwave Studio. Since the dimensions of the
target and the shell in the cloak design (Table 5.1) are normalized by the wavelength of
incident radiation, the cloak performance could be evaluated for any operating frequency.
The near field distribution in Figure 5.2 is obtained for the cloak operation at 6 GHz, i.e.
at the wavelength of 5 cm in free space. As seen from the figure, while bare targets
strongly distort propagating waves, cloaking of both targets substantially reduces back
scattering and provides wave front reconstruction for waves passing the cloaked objects,
which is especially efficient in the case of the smaller target.
In addition, the calculated above QS values were verified by their comparison with the
49
simulated by using the CST solver values of the far-field TSCS. As seen from Figure 5.3,
the simulated TSCS spectra for the cloaked targets of both sizes demonstrated deep drops
at around 6 GHz: down to 50% of the TSCS of the bare larger target and down to 20% of
the TSCS of the bare smaller target, respectively. The minimal TSCS values for the
cloaked targets were found to be equal to 18.5cm2 and 1.2 cm2, respectively, and were in
agreement with the respective theoretical values of 0.72λ2 and 0.047λ2 derived above.
80
Cloaked target (a=0.2λ)
Bare target (a=0.2λ)
Cloaked target (a=0.5λ)
Bare target (a=0.5λ)
70
Total SCS(cm 2)
60
50
40
30
20
10
0
5.8
5.9
6
6.1
Freqency(GHz)
6.2
Figure5.3. TSCS spectra for the cloaked and bare targets of two different sizes, with radius a=0.2λ (1.0 cm)
and a=0.5λ (2.5 cm) respectively. Cloaks are designed to work at f=6GHz.
It should be noted that the TSCS defined by equation (5.1) characterizes the integral 3D
effect of the power scattered in all directions. However, for practical purposes,
minimizing scattering in a specific direction, i.e. making the object invisible under a
specific angle of view, can present a special interest. Fig. 4 presents the simulated
scattering patterns of the cloaked targets with the radii a=0.5λ (2.5 cm) and a=0.2λ (1.0
cm) in the E-k and H-k planes, respectively. As seen in the figure, the backscattering
cross section (BSCS) at θ=180o is significantly reduced, i.e. by 4 dB when the target with
the radius a=0.5λ is cloaked, and by 18 dB for the cloaked target with the radius a=0.2λ.
There can also be found specific directions, at which scattering can be considered as
practically eliminated. So the largest reduction by 30 dB of the SCS of the cloaked target
with the radius a=0.5λ occurs at the elevation angle θ=102o and the azimuth angle φ=90o
50
or 270o, while for the cloaked target with the radius a=0.2λ the largest reduction of SCS
by 35 dB is observed at the elevation angle θ=75o and the azimuth angle φ=0o or 180o.
(a)
o
o
90 (φ=0 )
o
120
-15
o
o
o
90o(φ=90 )
120
30o
-25
150
(b)
60o
-30
150o
-35dBsm
θ =0
θ =180
o
o
θ =0
o
o
60
(c)
o
-20
(d)120
o
60
-40
o
150
o
o
o
90
120
30
120
90 (φ=180 )
o (φ=0o)
o
o
150
o
120o
o(φ=270o)
90
o
90o(φ=90 )
-25
o
θ =0
150o
30o
120o
o(φ=180o)
60o
30o
-45dBsm
-60dBsm
o
60
-35
o
150
30o
θ =180
o
30
θ =180
30o
o
Bare Sphere
Cloaked Sphere
-50dBsm
o
o
150
o
60
-10
o
o
θ =180
θ =0
o
o
30
150
o
60o
120
o
o
60o
90 (φ=270 )
90
Figure 5.4. Scattering patterns of cloaked targets: (a) with a=0.5λ in E-k plane, (b) with a=0.5λ in H-k plane,
(c) with a=0.2λ in E-k plane, and (d) with a=0.2λ in H-k plane. Cloaks are designed to work at f=6GHz.
5.5 Loss dependence of the cloak performance
An interesting opportunity to enhance the scattering cancellation effect by using lossy
materials in the cloak layers has also been explored. For this purpose, the loss
dependences of both the TSCS and BSCS were examined by introducing different loss
tangents ( tan δ = ε ′′ / ε , where ε ′′ is the imaginary part of the dielectric constant and ε is
the dielectric constant given in Table 5.1) to calculate the wave number and the
impedance in equation (5.3). The analytical expression used here for the BSCS of a
51
spherical structure has been derived as:
λ2
σ=
4π
∞
∑ ( −1) ( 2n + 1) ( R
n
TE
( M +1) Mn
n =1
− R
TM
( M +1) Mn
)
2
(5.5)
When the loss tangent of the cloak materials is zero, the BSCS for the cloaked target with
a=0.5λ calculated from equation (5.5) is 0.2953λ2, i.e. 49.7% of the value for the bare
target (0.5941λ2), while for the cloaked target with a=0.2λ the BSCS is 0.0064λ2, i.e. only
1.8% of the value for the bare target (0.3459λ2). At the design frequency of 6 GHz, the
calculated BSCS values are -31.3 dBsm, -28.3 dBsm, -48.0 dBsm and -30.6dBsm,
respectively, for the cases of: cloaked target with a=0.5λ, bare target with a=0.5λ, cloaked
target with a=0.2λ and bare target with a=0.2λ. These numbers agree well with the data
obtained from the simulated scattering patterns presented in Figure 5.4.
TSCS
BSCS
0.02
0.04 0.06
tan δ
SCS / λ2
SCS/ λ2
1.4
1.2
1
0.8
0.6
0.4
0.2
(a) 00
0.08
0.1
0.35
0.3
0.25
0.2
0.15
0.1
0.05
(b)00
TSCS
BSCS
0.02
0.04 0.06
tan δ
0.08
0.1
Figure 5.5. Loss dependence of the TSCS and BSCS for the cloaked targets with the radii: (a) a=0.5λ and
(b) a=0.2λ.
Figure 5.5 shows the TSCS and BSCS dependences on the loss tangent of the cloak
materials with parameters listed in Table 5.1. It is interesting to find out that employment
of lossy materials in the cloak medium can improve the performance of the cloak for a
larger target: the TSCS for the target with a=0.5λ has a minimum when the loss tangent is
around 0.01 and the BSCS has a minimum at the loss tangent of 0.09. For the cloaked
target with a=0.2λ, however, both the TSCS and BSCS were found to increase
monotonically with the loss tangent increase.
52
5.6 Conclusion
In conclusion, by using the Mie scattering theory and the GA based optimization
procedure, we have developed an approach for designing 3D spherical invisibility cloaks
employing multilayer shells of ordinary dielectrics. The cloaks can be easily scaled for
operation at any frequency. Full-wave field simulations demonstrated that the cloaks
designed by using this approach provided efficient back scattering reduction and wave
front reconstruction for the cloaked targets. Theoretical calculations verified by full-wave
simulations have shown that the designed cloak reduced the TSCS and BSCS by 50% for
the target with the radius of 0.5 λ, while the TSCS for the target with the radius of 0.2λ
was reduced by more than 80% and BSCS by 98%. In addition, the specific directions, at
which the scattering was practically eliminated, have been detected for both targets. At
last we have demonstrated that proper loss values of the cloaking shell materials could
improve the cloaking efficiency for larger targets.
53
6. Cloaking of electrically large objects by ENZ
material
6.1 Introduction
Materials with the refraction index equal to zero are called zero index material (ZIM). An
intriguing property of ZIM is that the phase velocity is infinite when wave propagates
through it. Since the refraction index is related to the relative permittivity εr and the
relative permeability μr, ZIM can be achieved when any of these parameters is
approaching zero. When both permittivity and permeability are approaching zero, a
matched zero index material (MZIM) is achieved since, in addition, the wave impedance
is preserved as that in free space. The wave propagation and scattering inside MZIM had
been first analyzed by Ziolkowski [67]. As a MZIM is very complicated to implement,
ZIM resulting from the single zero constitutive parameter draws more interest.
Silveirinha and Engheta proposed that the electromagnetic (EM) wave can tunnel through
very narrow channels filled with epsilon near zero (ENZ) material [68, 69], although
intrusively the intrinsic mismatched impedance can degrade the transmission efficiency.
Later on, this electromagnetic tunneling was experimentally demonstrated by using ENZ
material composed of complementary split ring resonators [70].
The idea of designing an invisibility cloak by employing ZIM material is originates from
the EM tunneling property through ZIM. J. Hao proposed that a 2-D perfect electric
conductor (PEC)/perfect magnetic conductor (PMC) object can be hidden inside the
MZIM clapped by two parallel PEC/PMC plates of the waveguide, where EM wave
tunnels through this clapped structure without seeing the object [28]. Except for
PEC/PMC object, dielectric rods with properly designated radius and permittivity can
also be hidden by MZIM [29]. A further study shows that ENZ material, which does not
depend on the permeability, can also result in total transmission if the geometric size and
material property of the dielectric rods are properly designed [30]. By introducing more
complexity and freedom, it was also shown that full transmission can be achieve by ENZ
material with the embedded PMC defect coated by properly designed dielectric shell[71].
In this chapter, we first analyze the transmission coefficient of a PEC target embedded in
54
ENZ material that is clapped by waveguide structure. By interpreting the closed
analytical formula of transmission coefficient, we find that better cloaking effect can be
achieved when the size of the PEC target is increasing, in other words, the channel of the
ENZ material becomes narrower along the propagation direction. This intriguing
phenomenon, although contrary to people’s intuition, is further proved by simulations.
6.2 Theoretic analysis
The schematic of the wave propagation through ENZ material embedding a target is
given in Figure 6.1. The whole structure is clapped by a PEC parallel plate waveguide
with the height d. The whole system has four regions: region (0) and region (3) are filled
with air; region (1) extending from x=0 to x=a is filled with ENZ material; region (2) is
filled with ENZ material with the embedded target inside.
Figure 6.1 The model of wave propagation through the epsilon near zero (ENZ) material clapped by a PEC
parallel plate waveguide.
To facilitate the analysis, only TEZ incident wave is considered in the following. In
accordance with Figure 6.1, the TEZ incident wave propagates in the +x direction and can
be described by the relations below:

 H in = H 0i e − jkx zˆ
 
− jkx
 Ein = η0 H 0i e yˆ
(6.1)
where the time factor exp(jωt) is omitted. The field in each region can be expressed
theoretically by using the following considerations. When the incident wave hits the
boundary between air and ENZ material at x=0, the wave is reflected. Suppose that the
55
reflection coefficient for the magnetic field at the interface x=0 is R, thus, the field in the
region (0) can be expressed as:

 H 0 [ H 0i e − jkx + RH 0i e jkx ]zˆ
=
 0
− jkx
=
− Rη0 H 0i e jkx ] yˆ
 E [η0 H 0i e
(6.2)
Let’s further suppose that the transmission coefficient at x=a for the incident wave is T,
thus, the field in the region (3) can be expressed as:

 H 3 = TH 0i e − jk ( x − a ) zˆ
 3
− jk ( x − a )
yˆ
 E = Tη0 H 0i e
(6.3)
So far, the fields in region (0) and region (3) have been expressed with the two unknowns
R and T. To cloak the target by ENZ material placed inside the above proposed
waveguide structure, we need to provide for total transmission of the wave in the Region
(3) from Region (0), which means that the transmission coefficient T should be as close
as possible to one.
To express the transmission coefficient T in a closed form, Ampere’s law and Fraday’s
law are applied successively for region (1) filled with the ENZ material. From the
Ampere’s law,


∇ × H1 = − jωε1r ε 0 E1
(6.4)


where H1 and E1 are the magnetic field and the electric field inside region (1), ε1r is the
relative permittivity of ENZ material. Since region (1) is filled with ENZ material (ε1r≈0),

the right side of equation (6.4) is zero. This means that the magnetic field H1 is constant
everywhere inside region (1). By matching the H-field at the boundary x=0 and x=a from
equation (6.2) and (6.3), we obtain:
1+ R =
T
(6.5)
From the Faraday’s law:
1 
1 
∂
E
⋅
dl
=
−
B
⋅ dS1
∫ S1
∂t ∫∫
56
(6.6)
where, S1 is the surface of the boundary of region (1), which includes the boundary at
x=0, the boundary at x=a, the top and the bottom PEC plates, and also the outer area S2 of
the target. By calculating the two sides of equation (6.6), we obtain:
1 
2(T − 1)η0 H 0i d − 
E
−iωµ0 µ1r H 0iTA1
∫ ⋅ dl =
S2
(6.7)
where, μ1r is the relative permeability of ENZ material, A1 is the area of the ENZ material
1 
−
E
and ∫ S2 ⋅ dl is the result of E-field integration along the outer surface of the target.
Solving equation (6.7), we obtain:
1 

⋅ dl 
E
1 + ∫ s2

T=

2η0 H 0i d 


 iπµ1r A1 
1 +

λ0 d 

(6.8)
6.3 Analysis and discussion
The transmission coefficient derived in equation (6.8) can vary in dependence on the size
of the embedded target and the relative permeability of the ENZ material. In the simple
case when µ1r = 0 , the ENZ material becomes MZIM, equation (6.8) reduces to
 
1 + ∫ E1 ⋅ dl 2η0 H 0i d , the transmission coefficient solely depends on the E-field
s2
integration along the outer surface of the target, and it will be equal to one when there is
no target or the embedded target has PEC boundary. This is the case that had been studied
in [28]. In this section, we focus on the cloaking by using ENZ material ( µ1r ≠ 0 ), while
considering a PEC target, since any target can be covered by a thin PEC shell. At these
1 
0 , and equation (6.8) reduces to:
assumptions, ∫ S2 E ⋅ dl =
 iπµ1r A1 
=
T 1 1 +

λ0 d 

(6.9)
The transmission coefficient depends on the operating frequency and the geometry of the
ENZ material. In particular, the lower is the frequency and the smaller is the ratio A1 /d,
the higher is the transmission. At the fixed frequency and the waveguide height d, a
smaller A1 /d, means smaller A1 that in turn means that the blue area in Fig. 6.1 filled with
57
ENZ material should be as small as possible. The simplest way to make A1 smaller is to
shorten the length of the ENZ material; however, this will also reduce the size of the
target embedded inside the ENZ material.
Another way to reduce A1 is to increase the size of the target and, correspondingly, to
squeeze the area of ENZ material. Since the outer shape of ENZ material region is
rectangular, a conformal rectangular target should squeeze the area of ENZ material as
much as possible. The simulation results presented in the following section show that a
large rectangular PEC target will increase the transmission and therefore improve the
cloaking efficiency. It should be noted that the size of the rectangular target cannot be so
big that the target boundaries are infinitely close to the waveguide walls, since the gap
between the target and the waveguide should accommodate at least one unit cell of ENZ
material.
6.4 Simulation results of wave propagation through ENZ
material cloaking a metal target
ENZ
A/m
(a)
Y (cm)
Target
Z
X(cm)
(b)
Figure 6.2 Hz field distribution of wave propagation through ENZ material with PEC targets of different
cross-sections: (a) small circular and (b) large square at f=10 GHz.
In order to verify the cloak performance, we conducted the simulation by using the
58
COMSOL Multiphysics software. The effective permittivity of the cloak medium
introduced into the simulation has been described by the Drude-Lorentz dispersion
model:
ω p2
ε1r = 1 −
ω ( ω − iΓ ω )
(6.10)
where ω p is the angular plasma frequency and Γ is the damping factor. In this simulation,
the plasma frequency is set to be 10 GHz and the damping factor Γ is set to be 10-5.
Figure 6.2 illustrates wave propagating through ENZ material with embedded targets of
two different shapes and sizes. The outer shape of the ENZ material is a square with the
side length of 15cm (5λp at the plasma frequency 10GHz). The smaller target is circular
with the diameter of 9 cm (3λp), while the larger target is a square with the side length of
14.4 cm (4.8λp). The presented results clearly prove the conclusion of the theoretical
analysis that the larger is the target, the higher transmission is provided by the cloak.
Figure 6.3. Transmission spectrum of the embedded larger square target with the side length of 14.4 cm.
The transmission spectrum of the larger square target with the side length of 14.4 cm
embedded inside the ENZ material is given in Figure 6.3. Although the transmission
coefficient is about 0.625 at the plasma frequency, the field map in Figure 6.2b shows
that the wave exactly restores its front after passing through the cloaked target. It is
interesting to note that above the plasma frequency, there are several transmission peaks
that demonstrate even higher transmission efficiency than that at the plasma frequency. In
59
particular, the transmission coefficients are 0.976 and 0.919, respectively, at 10.04 GHz
and 10.12 GHz, i.e. just to the right of the plasma frequency. The Hz field distributions in
Figure 6.4 show that the wave fronts are distorted at these frequencies. It could be
suggested that this high transmission is caused by the Fabry-Perot resonances inside the
two horizontal channels.
(a)
(b)
Figure 6.4 Hz field distribution of wave propagation through ENZ material at the transmission peaks at (a)
f=10.04 GHz and (b) f=10.12 GHz.
0.4
0.2
0.1
0
-8
(b)
0.5
Hz (A/m)
Hz (A/m)
0.3
1
(a)
0
-0.5
-1
-6
6
4
2
0
-2
-4
Distance along the channel (cm)
-8
8
-6
6
4
2
0
-2
-4
Distance along the channel (cm)
8
Figure 6.5 Dependence of Hz field on the distance along the horizontal channel of ENZ at (a) f=10.04 GHz
and (b) f=10.12 GHz.
60
The dependencies of magnetic field on the distance along the horizontal channel at
frequencies 10.04 GHz and 10.12 GHz presented in Figure 6.5 that correspond to the first
and the second order Fabry-Perot resonances, respectively, do confirm this assumption.
Consequently, the non-uniform wave fronts behind the target in Figure 6.4 result from the
superposition of waves originating from two sources, i.e. from the exits of the two
horizontal channels.
6.5 Conclusions
The wave transmission through the epsilon near zero (ENZ) material with the embedded
metal target has been investigated. By analyzing the theoretically derived transmission
coefficient, we propose a cloaking method that is inclined to hide larger objects using
only an ENZ material. The simulation results have verified the cloaking effect. ENZ
materials for operation at microwave frequencies can be implemented, for instance, by
using an array of metal wires [68, 72], an array of complementary SRRs [70] or an array
of dielectric rods.
61
7. Summary and future work
7.1 Summary
In Chapter 1, we first reviewed different approaches to invisibility cloaking at microwave
frequencies and formulated our research objectives based on the analysis of the problems
of the existing cloak designs, i.e. lossy cloaking medium, high implementation
complexity, and small hidden target size compared to the incident wavelength.
To replace the lossy metamaterials cloaking medium, we proposed to design microwave
cloaks by using dielectric resonators composed of low loss ceramic materials. In Chapter
2, we reported the development of several facilities for measuring the complex
permittivity of dielectric materials at microwave frequencies. In particular, a unique
parallel plate waveguide chamber has been set up to automatically map the EM field
distribution for the resonator arrays and cloaking structures. The technique for measuring
the TSCW of 2-D cloaking structures by using this apparatus has also been developed.
Chapter 3 described the efforts on designing and implementing the TO- based microwave
cylindrical cloaks by using identical dielectric resonators, which have better scalability
and lower loss than metal SRRs. The effective permeability dispersion prescribed by the
reduced TO parameters for TE wave incidence was provided by tailoring DR filling
fractions. The cloak performances had been verified by full-wave simulation of true
multi-resonator structures and experimental measurement of the fabricated prototypes. It
proved that the designed cloak having fixed inter-resonator distance in each layer
demonstrated a clear cloaking effect, i.e. both the wave front reconstruction and the
reduction of the TSCS. The results of this Chapter have been published in two conference
papers presented at the international IEEE Antenna and Propagation Symposium (AP-S)
and in the journal paper-in the IEEE Microwave and Wireless Components Letters.
Aiming to reduce the implementation complexity of the cloak medium and to increase the
operation bandwidth of the cloak, we proposed to design 2-D cylindrical cloaks by using
multi-layer ordinary dielectric material (εr>1) coating in Chapter 4. Genetic algorithm
was employed to optimize the dielectric profile of the cloaking shell, in order to minimize
the TSCW of the cloaked target. The designed cloak with much thinner cloaking shell
62
compared to the size of the target can be easily scaled to various operating frequencies. It
was also shown that the multi-layer dielectric cloak essentially outperformed the
similarly sized TO-based metamaterials cloak. The results of this Chapter have been
published in two conference papers presented at the international IEEE Antenna and
Propagation Symposium (AP-S), and at the “Metamaterials 2013”, 7th International
Congress on Advanced Electromagnetic Materials in Microwaves and Optics, and in the
journal paper in Applied Physics Letters.
In Chapter 5, we extended the multi-layer dielectric coating method to 3-D spherical
cloak design. The specific directions, at which the scattering could be practically
eliminated, were detected from the scattering patterns of cloaked targets. The influence of
the target size and the dielectric material loss on the cloaking efficiency was analyzed. It
was shown that the cloaking efficiency for larger targets could be improved by employing
lossy materials in the shell. The results in this Chapter have been published in the journal
paper in the American Institute of Physics (AIP) Advances.
In Chapter 6, we theoretically analyzed the transmission coefficient of target embedded
inside ENZ material clapped by a waveguide and proposed a cloaking method by using
only ENZ material, which is easy to implement in practice. The cloaking efficiency of
this method, which is inclined to larger targets, has been confirmed both theoretically and
in simulations. The paper based on the results of this Chapter is currently in preparation.
In conclusion, the cloak designs proposed in Chapters 3-5 can be readily implemented
and are easily scalable to other operation frequencies. The cloak design proposed in
Chapter 6 using only ENZ material is also easy to implement in practice compared to
MZIM, in addition, its cloaking efficiency increases for larger targets.
7.2 Future work
The ideas and works on microwave cloaks presented in this dissertation also inspired
some possible future works.
The experimental facilities and measurement techniques developed in the Microwave
Characterization Lab have the capability of measuring the complex relative permittivity
of dielectric materials, mapping the field and characterizing the scattering cross width of
63
2-D targets, which might be PEC targets or any periodic artificial medium. One possible
work that can be conducted with the designed parallel plate waveguide is experimentally
characterizing the photonic crystal composed of dielectric rods. This kind of photonic
crystals can be used to develop ENZ materials, and the experimental demonstration of the
cloak proposed in Chapter 6 can be then performed.
One of the advantages of cloak designed by identical dielectric resonators is the
possibility to scale it for applications at higher frequencies. The work is currently
ongoing on fabricating dielectric resonators for the infrared range from glass materials
and by the Si-based technology in collaboration with other universities. Once the
fabrication work is done, the microwave cloak can be redesigned and implemented for
the infrared range operation with the help of collaborators.
The multilayer cylindrical and spherical cloaks proposed in this dissertation can be
fabricated in future from the ceramic materials. The theoretical derivations of the
scattering from multi-layer cylindrical structures and spherical structures can be used to
reduce the bistatic scattering cross section in the desired direction by employing the
genetic algorithm optimization. To enhance the scattering reduction, materials with
proper loss can be introduced into the optimization procedure.
64
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