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Design of guided mode resonant filters at microwave and millimeter wave frequencies using genetic algorithms

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THE CATHOLIC UNIVERSITY OF AMERICA
Design o f Guided Mode Resonant Filters at Microwave and Millimeter Wave
Frequencies Using Genetic Algorithms
A DISSERTATION
Submitted to the Faculty o f the
Department of Mechanical Engineering
School o f Engineering
O f The Catholic University o f America
In Partial Fulfillment o f the Requirements
For the Degree
Doctor o f Philosophy
©
Copyright
All Rights Reserved
By
Joseph R. Krycia.
Washington, D. C.
2005
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UMI Number: 3169861
Copyright 2005 by
Krycia, Joseph R.
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Design o f Guided Mode Resonant Filters at Microwave and Millimeter Wave
Frequencies Using Genetic Algorithms
Joseph R. Krycia, Ph.D.
Director: Mark S. Mirotznik, Ph.D.
For decades engineers have exploited the structural properties o f composites to
develop material systems with desirable mechanical properties. Only recently
have investigators applied similar concepts towards the development o f composite
materials with "designer" electromagnetic properties. These designer materials
are called metamaterials and are generally defined as a composite o f two or more
materials producing an "effective" electromagnetic material with desirable
properties. Examples o f metamaterials includes the very popular research topics
o f photonic crystals and left handed materials. One application o f metamaterials
that is o f particular interest is the design of frequency selective materials or
surfaces (FSS). An FSS is in general, a single material or material structure that
transmits (or reflects) only specific frequencies. Most FSS are designed using a
periodic array of metallic patterns (e.g. dipoles, crosses, patches ...). Analytical as
well as numerical algorithms have been developed for producing FSS structures
based on the size and shape o f the metallic arrays. For very narrow band
applications, however, losses in the metal surfaces can be significant enough to
necessitate the search for a different FSS design methodology. Moreover, for
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
applications that have stringent out-of-band signature requirements it may be
necessary to reduce the amount of metal in the FSS structure. As a result, an all
dielectric FSS is o f interest. Fortunately, a design approach has been developed
using the concepts o f metamaterials to produce all dielectric FSS structures. This
approach, called the Guided Mode Resonant Filter (GMRF) method, constructs a
periodic pattern o f wavelength scale dielectrics o f different permittivity.
The overall goal o f this project was to study the application o f GMRF at lower
frequencies (i.e. microwave and millimeter wave hands) and large scale
applications (e.g. radomes) requiring in some cases 2D GMRF. To this end, the
following specific goals were accomplished; (1) Rigorous ID and 2D analysis
codes were developed using the rigorous coupled wave (RCW) method. (2)
Design algorithms based on genetic algorithms were developed for the synthesis
o f ID and 2D GMRF transmission filters. (3) Specific designs requiring only a
single low dielectric material and air were demonstrated. (4) Specific designs
with relatively low dielectric constant materials were demonstrated. (5) Finite
element analysis was used to validate ID and 2D GMRF designs. (6)
Experimental test samples were fabricated at both the microwave and millimeter
wave frequency band.
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This dissertation by Joseph R. Krycia fulfills the dissertation requirements for the
doctoral degree in Mechanical Engineering approved by Mark S. Mirotznik,
Ph.D., as Director, and by John J. McCoy, Sc.D., Steven J. Russell, Ph.D., and
John A. Judge, Ph.D. as Readers.
Mark S. Mirotznik, PhD, Director
John J. McCoy, Sc.D., Reader
ii
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To my beautiful and loving wife Kristin Krycia, without her endless
patience through this long journey I would not have finished.
iii
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C on ten ts
List o f Figures
vi
1 Introduction
1.0.1 M otivation.................................................................................
.......................................
1.0.2 Problem Statement
.
1
1
4
1.0.3
1.0.4
Literature R eview ..................
New C ontributions
.............................................
5
8
1.0.5
Overview of this dissertation
9
..........................................
2 A nalytical M ethods: R igorous C oupled W ave (R C W ) A nalysis
2.1 Overview of RCW T h eo ry
.
2.2 Geometry of 2D Grating Stack..............................................................
2.3 Fields in the Incident and Exit regions . ..........
2.3.1 Definition of Fields in the Incident Region
.................
2.3.2 Definition of Fields in the Exit Region
....................
2.4 Fields in the Grating and Homogeneous la y ers..............
2.5 Boundary C onditions
.................
2.6 Formulation of Matrix Equations ................
2.7 Code Development and Validation. ............................
11
11
. 12
14
14
16
17
22
25
29
3 A nalytical M ethods:
Itera tive D esign o f G uided M ode R eso­
nant F ilters
30
3.1 Design Approach Considerations and Assumptions . . . . . . . . . . .
32
3.2 Iterative Design Approach . ..............................
34
3.2.1 Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2.2 Direct Search
39
.............. .............................................. ... .
3.2.3 GMRF Design Algorithm
.............................
iv
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40
4
5
A n aly tical R esu lts
4.1 One-Dimensional and Two-Dimensional GMRF Designs at Microwave
...............
and Millimeter Wave Frequencies
4.1.1 E xam ple#!: One-dimensional GMRF Band pass Filter Design
at 14GHz' . .....................
4.1.2 Example #2: One-dimensional GMRF Band pass Filter Design
at 38 G H z ..................
4.1.3 Example #3: Two-dimensional GMRF Band pass Filter De­
sign at 36 GHz and 42GHz . . . . . . . . . . . . . . . . . . . .
45
E x p e rim e n ta l M eth o d s
5.1 T est. Article Fabrication
.............
5.2 Description of Measurements
............................
5.2.1 University of Delaware Millimeter Wave M easurement............
5.2.2 SPAWAR Systems Center Dual Anechoic Chamber Spatially
69
69
71
73
Averaged Transmission Microwave Measurement . . . . . . . .
6
E x p e rim e n ta l R e su lts
48
47
53
63
76
81
6.1
Experimental Results: Microwave Guided Mode Resonant Filter at 14
G H z ........................................
Experimental Results: Millimeter Wave Guided Mode Resonant Filter
82
6.2
at 36 GHz
85
...............
7 D iscussion a n d C onclusions
89
7.0.1
7.0.2
7.0.3
Significant Accomplishments
........................................
Theoretical Discussion . . . . . . . . . . . . . . . . . . . . . .
Experimental D iscussion.................
90
91
93
7.0.4
7.0.5
Future Work
....................
C onclusions..........................................
94
96
B ibliography
98
v
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L ist o f Figures
2.1
Generic two-dimensional grating structure ..................................... ... .
13
3.1
3.2
3.3
GMRF design and evaluation process . . . . . . . . . . . . . . . . . .
Generation of children in next generation for genetic algorithm . . . .
Mesh pattern for direct search .................................................................
31
38
41
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
RCW predcition of 14 GHz design performance ..................................
Objective function evaluation for 14GHz design ............
Dimensioned schematic of 14 GHz design ............................................
Comparison of design with HFSS at 14 GHz .....................................
36 GHz Design . . ..................
Objective function evalaution by iteration
..................
Dimensioned schematic of 36 GHz design
......................
Comparison of Design with HFSS predicition at 36G H z...............
Loss sensitivity comparison........................................... ... . ..................
Incident angle sensitivity comparison
........................
Two Dimensional dual pass band design
...............
Dimensioned schematic of two-dimensional 36GHz and 39GHz filter .
Comparison of 2 dimensional dual band design with HFSS . . . . . .
49
50
51
52
55
56
57
58
60
62
65
66
67
..................................
Conceptual experimental setup
Schematic of University of Delaware experiemtal setup . . . . . . . .
University of Delaware millimeter wave meaurement s y s t e m ............
Schematic of SPAWAR measurement facility
..................
Inside SPAWAR chamber looking at the back of the horn toward the
test sample
...............
5.6 Inside SPAWAR chamber showing the aperature for tranmission through
the GMRF
..................
72
74
75
77
5.1
5.2
5.3
5.4
5.5
6.1
Comparison of the predicted and measure data for the 14 GHz guided
mode resonant filter.
...................
vi
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78
79
83
6.2 Test article within aperature surrounded by radar absorbing materia!
6.3 Comparison, of the measured vs the predicted data for the 36 GHz
guided mode resonant filter . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Image on the left shows the amplitude across the lens, the image on
the right shows the phase
...............
vii
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84
86
87
A cknow ledgem ents
There are so many pepole I wish to thank for their support and encouragement.
First, I would like to thanks my advisor and friend, Dr. Mark Mirotznik for not only
his invaluable guidance, but for Ms persistent faith in my abilities. I would also like
to thank Dr. John Meloling and David Hurdsman at SPAWAR Systems Center and
Dr. Dennis Prather for their expertise in measurement systems.
I would like to thank my parents for stressing the importance of a good education.
Most of all, I would like to thank my family. My wife Kristin who has been with me
through this entire journey, my daughter Erin who joined us at the very beginning,
my son Nichloas who joined us during the journey, and my daughter Juliette who
came to us near the end. I can not express enough gratitude for their scarifices of
time and the encouragement they gave me. Thank You.
viii
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C hapter 1
In trod u ction
1.0.1
M o tiv atio n
For decades engineers have exploited the structural properties of composites to
develop material systems with desirable mechanical properties. [1] Only recently have
investigators applied similar concepts towards the development of composite materi­
als with “designer” electromagnetic properties. In fact, this idea of ’'engineering" the
electromagnetic properties of composite structures has become so popular that the
research community has coined the term "metamaterials" to refer to such material
systems. A metamaterial is generally defined as a composite of two or more materials
producing an “effective” electromagnetic material with desirable properties. This rel­
atively recent interest in metamaterials is primarily due to advances in both efficient
computer design codes and the availability of state-of-the-art micro-fabrication facili-
1
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2
ties. Examples of metamaterials includes the very popular research topics of photonic
crystals and left handed materials. These "man made" composites exhibit very un­
usual electromagnetic properties not achievable with naturally found materials. One
application .of metamaferiais that is of particular interest is the design of frequency
selective materials or surfaces (FSS). [2]An FSS is in general, a single material or
material structure that transmits (or reflects) only specific frequencies.
Most FSS
are designed using a periodic array of metallic patterns (e.g. dipoles, crosses, patches
. . . ) . Analytical as well as numerical algorithms have been developed for producing
FSS structures based on the size and shape of the metallic arrays. For very narrow
band applications, however, losses in the metal surfaces can be significant enough to
necessitate the search for a different FSS design methodology. Moreover, for applica­
tions that have stringent out-of-band signature requirements it may be necessary to
reduce the amount of metal in the FSS structure. As a result, an all dielectric FSS
is of interest. Fortunately, a design approach has been developed using the concepts
of metamaterials to produce all dielectric FSS structures.
This approach, called
the Guided Mode Resonant Filter (GMRF) method, constructs a periodic pattern of
wavelength scale dielectrics of different permittivity. Due to the periodic nature of
the grating structure the diffracted reflection and transmission orders are a result of
the interactions of the incident energy and the resonant modes of the grating. At the
resonant frequency the grating structure can exhibit very strong transmission and
reflection coefficients approaching unity for a narrow frequency band. This combina­
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tion of th e efficient reflection and transmission properties coupled with the frequency
selectivity can provide very efficient pass band and stop band filters. Additionally,
since the GMRF design does not employ any metallic structures it can be designed
with good out-of-band signature control as well as a minimal amount of loss. Since
their inception nearly all applications of GMRF have focused on optical and infrared
wavelength scale devices. In fact, GMRF have been commercially employed as effi­
cient dielectric ,mirrors in laser systems and other optical components. Additionally,
most investigators have concentrated on one-dimensional GMRF since they can pre­
cisely control the state of polarization in their applications and thus do not require
complicated 2D designs. The overall goal of this project was to study the application
of GMRF at lower frequencies (i.e. microwave and millimeter wave bands) and large
scale applications (e.g. radomes) requiring in some cases 2D GMRF. To this end, the
following specific goals were accomplished;
•
Rigorous ID and 2D analysis codes were developed using the rigorous cou­
pled wave (RCW) method.
«
Design algorithms based on genetic algorithms were developed for the syn­
thesis of ID and 2D GMRF' transmission filters.
•
Specific designs requiring only a single low dielectric material and air were
demonstrated. By using only a single dielectric material, other than air, the fabri­
cation is considerably simplified.
•
Specific designs with relatively low dielectric constant materials were demon­
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strated. This is important since for applications requiring large areas at microwave
and millimeter wave frequencies it Is difficult to find inexpensive high dielectric con­
stant materials.
•
Finite element analysis was used to validate ID and 2D GMRF designs.
•
Experimental test samples were fabricated at both the microwave and mil­
limeter wave frequency bands.
•
Experimental validation was conducted for both microwave and millimeter
wave test samples.
The results, presented in detail later in this thesis, will show that the combination
of rigorous coupled wave code and a well thought out iterative design algorithm can be
used to efficiently synthesize GMRF at microwave and millimeter wave frequencies.
I will also show that these designs can be fabricated using common materials and
validated both numerically and experimentally.
1.0.2 Problem Statem ent
All-dielectric narrow-band frequency selective surfaces (FSS) are useful for a num­
ber of applications of interest to security and defense. The design of such structures
can be challenging and for most applications requires the development of rigorous
computational electromagnetic analysis and synthesis algorithms.
The goal of this
work was to develop a methodology for the practical design of all-dielectric narrow
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5
band FSS at microwave and millimeter wave frequencies. To this end, I developed a
methodology for the design of ID and 2D all-dielectric FSS structures based on the
concept of guided mode resonant filters (GMRF), Specifically, I developed an imple­
mentation of the Rigorous Coupled Wave (RCW) method in MATLAB that can be
used for multiple arbitrary layers in one or two dimensions.
I also developed and
applied an optimization techniques based on a genetic algorithm and direct search to
find candidate designs.
I numerically validated the design algorithm using a finite
element model. Samples designs were fabricated and experimentally measured as a
further validation.
1.0.3
Literature R eview
A large amount of research has been conducted in frequency selective surfaces
(FSS). At microwave and millimeter wave frequencies this research had been focused
on the development of metallic elements that create the desired FSS behavior. [2]
Although the periodic metal patch FSS technology is maturing to the point in which
there are fielded applications, there are a couple of shortfalls with the metallic element
approach. The losses in the metal patches can often create, a wider than desired pass
band, consequently very narrow pass bands can be difficult to achieve.
Also, the
out of band characteristics of the metallic structure can create undesirable scattering
from the surface at frequencies removed from the desired pass band.
Specifically
with finite sized metallic FSS arrays, there are grating lobes at higher frequencies
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and propagation of surface wav® at lower frequencies that contribute to the out of
band scattering. [3]
An alternative to the traditional metallic element FSS, is an all dielectric option
called a .guided mode resonant filter (GMRF). Rather than relying upon.the periodic
metallic patterns, GMRFs rely upon a periodic modulation in the dielectric constant
of the materials.
When the period of this modulation is near the wavelength, the
resonant modes of the grating couple with the diffracted and transmitted inodes of
the grating. [4] [5] This resonant behavior is strongly dependent upon the frequency
and consequently it can be used to design very efficient narrow band filters known as
GMRF. [6] [7]
GMRF theory has been developed to show their theoretical ability to produce high
efficiency reflection and transmission filters. [8] [9] (10) [11] [12] [13] [14] [15] [18] [17] [18] [19] [20]
Wang et. ai. had shown that the frequency response of these filters can be con­
trolled by varying the geometry of the grating structure. Lemach et, al. and others
also showed that by dictating the geometric configuration of the filter, including pe­
riod, fill factor, material properties, and number of layers, the spectral response can
be controlled.[21] [22] [23] [24][25] [26] The early GMRF work was all done for one­
dimensional gratings until Peng and Morris further developed the theory to two di­
mensional gratings and theoretically investigated the scattering properties and later
foliwed by others. [46] [28] [29] Applications of GMRFs were considered for optical
wavelengths with a primary focus upon using GMRF to replace homogeneous layers
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7
thin film reflection filters. The advantages of the GMRFs was an increases frequency
selectivity through the inherent efficient.and narrow filter properties.[30][31][32][33]
Further work has been done to show the effectiveness of cascaded reflection filters.
Magnusson et. al. showed the effect of cascading filters gave even lower side bands
and even narrower reflection bands. [34] Norton et. a l conducted experimental work
that investigated the effect of varying one of the grating feature dielectric constants
and the resultant effect upon reflection efficiency and frequency. [35] It should be
noted, however, th at the vast majority of this work has been done at optical frequen­
cies with few studies aimed at the millimeter and microwave frequency bands. The
exception to this is a study done by Tibuleac et. al. that conducted experiments on
multilayer gratings with high dielectric contrasts at microwave frequencies. [36] To the
best of my knowledge no prior studies have been conducted on the design, fabrication
or testing of GMRF at millimeter wave frequencies.
For the analysis of GMRF, a number of effective analytical -method have been de­
veloped. Beyond typical numerical techniques such as method-of-moments (MOM),
finite element methods (FEM), and the finite-difference time-domain (FDTD), the
rigorous coupled wave (RCW) algorithm has been particularly useful for the solu­
tion of periodic planar structures such as GMRF.[37] [38] [39] It is considered rigorous
because if directly applies Maxwell’s equations with no .approximations unlike the
numerical techniques.
RCW has been extended to three dimensions to analyze
many classes of gratings th at include GMRF. [40] [41] [42] [43]
Methods of increas-
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8
lag the stability of the method and increase its efficiency have been researched and
validated. [44][45] [42] [47] [48] [49]. These previous RCW algorithms, particularly the
work of Noponen et. al. [40] were extended to multiple dielectric layers in this study
and applied towards the analysis of GMRF'.
While a reasonable .amount of. work has been reported on the development of rigor­
ous analysis algorithms little research has been published on efficient iterative design
algorithms. Most of the reported GMRF designs seem to have been .found through
a "trial and error" design approach. The only reported use of modem optimization
algorithms towards the design of GMRF was presented by Zufada et. al.[50]. Thus,
I believe the work presented here on the use of a genetic algorithm/pattern search
approach towards GMRF design is quite novel and fills a hole in the literature.
1.0.4
N ew Contributions
As mentioned earlier the principal goals of this study were to (1) analyze the elec­
tromagnetic characteristics of GMRFs at microwave and millimeter wave frequencies,
(2) develop a methodology to predict and design practical GMRFs, and (3) to develop
and apply an experimental protocol to validate the analytical and numerical models
applied in this study.
Of specific interest in this research is the development of filters with single
grating layers made from a low contrast dielectric. The advantage of a low contrast
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9
dielectric is th at at the frequencies of interest high dielectric materials are difficult to
find and work with in addition to being prohibitively expensive for practical designs.
Major new work I present in this dissertation is:
Novel iterative design algorithm for GMEF design based on a genetic algo­
rithm and direct pattern search
Two measurement techniques for GMRF characterization at millimeter wave
and microwave frequencies was applied
Low dielectric contrast GMRF designs were obtained
Experimental data for both microwave and millimeter wave GMEF designs
was obtained and used to validate designs
1.0,5
Overview of this dissertation
The organization of this dissertation is as follows. In Chapter 2, I discuss the
theoretical models used to analyze the transmission properties of GMRFs.
The
primary method used was a rigorous coupled wave (RCW) technique. This method
was used for both design and analysis of the GMRFs. A full three dimensional RCW
method was developed to analyze two-dimensional gratings of finite thickness and a
two-dimensional method was developed for the analysis and design of one-dimensional
gratings.
design.
Chapter 3 addresses the iterative design methods developed for GMRF
The iterative design scheme uses both a genetic algorithm and a pattern
search method to find a design solution. Is Chapter 4 ,1 present analytical results for
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both microwave and millimeter GMRF designs. This chapter also provides numerical
validation of these designs using the finite element method. In Chapter 5, I present
the experimental methods. This includes a description of two distinct experimental
methods used to characterize samples.
Namely, a spatially averaged measurement
method th at was used for the microwave experiments and a collimated lens system
that was used for the millimeter wave experiments. In Chapter 6 ,1 present the results
and analysis of three experiments.
Finally, in Chapter 6, I present a discussion of
the results, concluding remarks and a discussion of future research directions based
on the results of this work.
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C hapter 2
A n alytical M ethods: R igorous
C oupled W ave (R C W ) A n alysis
In this chapter I present the analytical method used to analyze GMRF. I give
a detailed derivation of the Rigorous Coupled Wave analysis applied to a multilayer
planar grating problem.
I .also provide the formulation of the matrix equations I
used for encoding in MATLAB.
2,1
O verview o f R C W T h e o ry
Rigorous Coupled Wave.Analysis (RCWA) is a rigorous numerical .method for
solving Maxwell’s equations for planar periodic structures.
The term rigorous.is
used to denote th at no hypothesis or simplification of Maxwell’s equations axe used.
11
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1*2
In this dissertation, I developed RCWA codes for both one- and two-dimensional
rectangular binary gratings that includes multiple grating/homogenous layers.
G e o m e try o f 2D G ra tin g S ta c k
2.2
An example of a generic structure is shown in Figure 2.1. The generic two di­
mensional grating structure in Figure 2.1 contains an incident region, grating layer,
a single homogeneous layer and exit region. It should be noted that in general the
code can actually handle an arbitrary number of layers sandwiched between the inci­
dent and exit regions. However, for the designs developed during this study only a
single grating and homogenous layer were needed. In the grating region, where 0 <
z < h, the periodicity of the structure is given by A x and Ay in the x and y direction
respectively. In this grating region, the permittivity is expressed as a doubly periodic
function e(x,y).
The incident and exit regions are semi-infinite isotropic dielectric
media with permittivity of ei (z < 0) and
€4
(z>h2). The unit polarization vector
of the incident wave is denoted in terms of 6 , 6 , and ip. Here 9 denotes the elevation
angle,
0
denotes the azimuthal angle, and ip denotes the angle between the E field
vector and the incident plane defining the polarization. For my analysis, the incident
and exit regions were set to free space.
As an overview, in RCWA the fields inside all interior layers are expanded in
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13
z=oI J ■.Tl .'it ill J ! t U I >ij
Z=h.
;Jl
£4
i-tii'iii
r." E-it ttxtkm
Figure 2.1: Generic two-dimensional grating structure
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terms of a sum of spatial harmonics. This expansions can be expressed and solved
as an eigenvalue problem for all of the harmonic orders. At the interfaces between
each interior layer and at material interface separating the incident and exit regions
(denoted in the Figure), electromagnetic boundary conditions are enforced.
Once
the all of the electric and. magnetic fields axe matched, a set of linear simultaneous
equations result. The solution of. those equations is used to calculate the fields within
the incident, exit and all interior regions. Based on those solutions the reflected and
transmitted fields for all diffractive orders can be determined.
It should be noted
that since the goal of a GMRF device is to act as a narrow band filter that the
generation of propagating diffractive orders, other than the zeroth order, is normally
undesirable.
Thus all the grating structures are normally subwavelength and all
reflective and transmitted orders are evanescent with the exception of the zeroth
order.
In the next section the fields within each of the regions (i.e. incident, exit
and grating layers) are defined. I then define the boundary conditions th at must be
satisfied at the interface.
2.3
F ield s in th e In c id e n t and E x it region s
2.3.1
Definition of Fields in the Incident Region
The incident field is assumed to be a monochromatic plane wave with unit ampli­
tude, a free-space wavelength of Xa, and arbitrary polarization. Mathematically this
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15
is written as
Emc(x, y, z) = «exp(—s k inc • r*)
(2.1)
The time dependence exp(iut) is assumed throughout this thesis and omitted
everywhere.
The wave vector in the incident region, k inc, is given by
k inc — k()ti2 ^siu 9 cos
"1“ sin 0 sin tpy ~f~cos 0 z) — oY,.x + (30y + r0z
where n* is the index of refraction of the incident layer and kQ =
(2.2)
The
polarization vector u may be written in terms of direction cosines as
u — (cos ip cos 9 cos <p —sin ip sin o)x -f (cos ip cos 9 sin <p+ sinip cos <p)y — (cos ip sin 6)z
(2.3)
The unit polarization vector of the incident wave is shown in Figure 2.1 in terms
of 8, p, and ip. The electric field in the incident region is the sum of the incident plane
wave and all reflected orders including those that may be evanescent. Thus the total
electric and magnetic field distributions within the incident region can be written in
the form of Rayleigh expansions where the coefficient R mn denotes the reflection
coefficient of the mn-th reflected orders
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
E j
— E in c 4~ ^
(
R-rrm
J k Ijnn " T j
(2.4)
m = —oo r = —oo
oo
oo
— E in c 4" ^ ^
^ ^ ^?mn®XP [ J (^ira® 4~ kynV
j
(2.5)
m = —oo ra=—oo
If/ =
— Aync x J ^ rec w/i0
J
■
oo
^
oo
(2.8)
^
^
^ ^ ^ i,mn ^ R jn n CXp | j k j,TOn * T J
(2.7j
’X^ ° m = —oo r = —oo
The wave vectors for the incident fields axe given by the following
kxm
=a m = a 0 +
(2.8)
**» = A » = & + ( ^ )
(2J)
kz,mn
= t'rrm = \ji™A ) 2 - G?m - /3^
*£mn
= rVr», = ~ J \ / + 0m~ (»l*3o) if
if
(bi&0)2 >
+ 0m
(n l ^ f < ° 4 + 0L
(2.10)
(2-U )
where Ax and Ay are the grating periods in the x and y directions respectively as
shown in Figure 2.1.
2.3.2
Definition of F ie ld s in the Exit Region
The electric and magnetic fields in the exit region is the sum of the mn-th trans­
mitted orders, T m n propagating out from the final grating layer. Similar to 2.4 it
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
17
can be expressed in terms of a Rayleigh expansion:
OO
~Ee
=
OO
^ T mne x p ^ ~ j k E^m n - ( y - h2z ) ^
(2.12)
m = —oo n = —oo
oo
oo
5 ] T mn exp { —j [k^nX + kyny + k f mn (z - h2)]}
= £
m = —oo n ——oo
^
oo
oo
H
where
E
/1 2
(2.13)
^
~ ~
*y ^ ^ k
OJHo■ m = —o o » = —oo
E ,m n
T rn„eXp ( j k
E ,m n
' { T"
h^Z) ^
(2.14)
is the total thickness of the grating and homogeneous layers. The wave
vectors for the exit fields are given by the following
OLm
f 2Ttm\
OLq + |
J
(2.15)
vyn — f i n ~ fio +
( ? )
<“
Kmn = tmn = y («4&o)2 ~ d 2m ~ f i n if («***) >
kf,mn =
tOT.„ = -
j
y
- (n4fco) if
+ fin
( n A ) 2 < ^ 4 + dn
>
(2.17)
(2.18)
In my numerical implementation of 2.1 and 2.12 the infinite series is truncated
such that there are Lx orders in the x direction and Ly orders in the y direction for
a total of Lt — Lx Ly orders.
2.4
F ield s in th e G ratin g a n d H o m o g en eo u s layers
Inside the region 0 < z < h% shown in 2.1 all of the grating and homogeneous
layers are contained. These fields are defined by Maxwell’s curl equations
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18
V x E
=
iojji0H
(2.19)
V x ff
-
-iw e E
(2.20)
When these equations are projected upon a Cartesian coordinate system, they
lead to a set of six coupled equations
dEz
dEx
dz
dEy
dx
dHz
dy
dHx
dz
dHv
dx
dE y
dz
dE z
dx
dEx
dy
dHy
dz
dH z
dx
dHx
dy
(2.21)
ibJIlHy
(2.22)
iwfj,Hz
(2.23)
—iojeEx
(2.24)
—iuieEy
(2.25)
—iuieEz
(2.26)
To eliminate the z component we can solve 2.26 and 2.23 for E z and H z respec­
tively.
*■ - - « - ( £ - £ )
Substituting Equations 2.27 and 2.28into Equations 2.21, 2.22, 2.24, and 2.25
results in the following set of four coupled partial differential equations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In deriving Equations 2.29, 2.30, 2.31, and 2.32 it was assumed that the material
properties (e and y) can vary only in the x and y direction within any planar layer
(i.e. uniform in z). If it is also assumed that the material layers are non-magnetic
(i.e. y = y Q) then the equations above simplify to
5
-sr
.
j o « -'( 5 - f ) )
-
m
dz
" »
uiy0dy Vdx
,3.36,
dy J
In theECW formulation the materialproperties within thegrating are assumed
to vary periodically in the x and y directionswith periodsof A*
and Ay respectively.
Due to the grating’s periodic structure the material properties can be expanded into
a Fourier series as given by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
p~ oo
« x ,v )
q—oo
= < . £ £
fp,,e)2' £ V 2'* »
(2.37)
p ~ —oo q= ~ o o
1
«*-*> -
i
q—oo
•vt—
q=oo
"i.—^
* 3 T 5 . E
p = —o o q = —oo
(2.38)
In addition, due to the presence of a doubly periodic medium the field distributions
within the grating layer can be expanded, via Floquet, into a sum of harmonic inodes
given by
—
E x(x, y. z) —
E v(x, y, z)
—
H j x , y, z)
_
Hy{x, y , z)
—
m—oo n=oo
V
V F
J (amx+Pny+jz}
/
/ ^ juarmnc
yri=—oo n=~-oo
m=oo n=oo
EymneP(-amX+^nV+lz^
77?==—oc n=—OO
m=oo n=oo
Hxmnerta™x+®nV+'yz'>
i7i=—oon=—
oo
n=oo
Hvmn^
^
m=—oc n=—oo
(2.39)
(2.40)
(2.41)
371=00
(2.42)
Substitution of■Equations 2.39, 2.40, 2.41, and 2.42 into Equations. 2.33, 2.34,
2.35, and 2.36 performing all partial derivatives results in the following set of coupled
linear equations.
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21
9= L g
P = Ia s
SjJ€(f ) ,E xmn
=
^ “)
k H y mn
S,m -p,n-q (p tp^ypq
0 q E Xpq)
(2.43)
~~ ftqH xpq)
(2.44)
|3=—Ij£ £^——£jy
p=Lx
■UJCajEymn
= —fe H
^ “]
^ n — Qn
p= -Lx
p=Lx
UJflolHxmn =
-A:2 E
Cro-p.M.-g {°hp-^ypq
«=-£»
q—Lv
E Cm—
jmx~-§
"FOcm M
p » - P » ) (2.45)
•p=x—Lx q——Ly
p~£*x
E
E w
p~~Lx g=—Ly
- qExpq "t" Pn ipt-mEymn
0 n-Exmn)
(2-46)
Equations 2,43,2.44, 2.45, and 2.48 .are of the form of an eigenvalue problem. Note
that the infinite summation of modes has been truncated in Equations 2.47, 2.48, 2.49,
and 2.50 to 2Lt modes where L t = L xLy. The solution of the eigenvalue problem
results in the propagation constants
amplitudes E ^ a , Eymni,
eigenmode I.
-jl
(i.e. eigenvalues) and corresponding complex
and Hymrd (i.e. eigenvectors), for each distinct
Once determined the fields within the grating layers are calculated as
a superposition of each mode as
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22
2 Lt.
E x(x ,y ,z)
£
m=i>*
{Ae>
B ie-3
E
E xrmd^ amlB+^ v)- (2.47)
£
?72.—~~Lx, ft—
.2 L t
E y(x,y, z) =
m=L-r,
£ M
£
1=1
~
k
*° 1£= 1
2Lt
Hy{x, y, z)
,
£
(2.48)
m = ~Lx n= ~Lv
j m =Lx n=L v
£
£
Hxmnlej(amX+0^ A 9 )
2Lt
Hx(x,y, z)
n—
!%
fm
+ B ie ~ j'>l<'z ~h'!
Ate ^ z
+ B i e - ^ z~k)
1 m - ~ L* n=—Ly
7Tl~L/x.
—*-*lj
£ £
m = —L z n = —L,.
The only unknowns in Equations 2.47, 2.48, 2.49, and 2.50 are the complex coeffi­
cients At and B[. These coefficients are determined by applying boundary conditions
at the interface between each layer. This is described in the next section.
2.5
B oundary C on d ition s
Maxwell’s, equations, require that at the boundary between any two dielectric ma­
terials the tangential component of electric .and magnetic be continuous.
For the
geometry described in 2.1 the boundary condition is written mathematically at the
grating boundary (z=0) as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
23
2Lt
■F R x m n
—
'y
~F B i fsx p ^ iy g h i)jE & fjfg n x
( 2 .5 1 )
~F B t
( 2 .5 2 )
e=i
2Lt
U yS fj^ S fiQ -j- R y rn n
=
g h i j j E ym ,ni
1=1
(J3q V,z
=
TQoUy ) 8 m$)8 raO
&n R gm n + F m nR ym n
, 2Lt
/■ftp
k \ i€„, ^ Ai *' e=i
{fo o U x — a o u z ) 8 m odno
( 2 .5 3 )
(2.54)
^m nR xm n
&m R zm n
( 2 .5 5 )
21,
=
fci / — y ^[A f - B i e x p ( ijehi))H ymni
V e° *=i
(2.56)
A t z = hi, interface separating the grating region and homogenous region,
the equations describing the boundary conditions are as follows (homogeneous layer
goes from hi to h^) [note that the appended subscript h denotes values from the
homogeneous layer]
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24
(2.57)
y^A |?exp(i 7 ^ i ) + B t e x p (ijt (hi - h2)}Exmra
G_-i
K.----A.
2L t
=
J ^ [A a exp(i7a h 1) + B ih e x p ( - t j eh(hi - h2))j£a™ntt
8_1
K
- X
2 Lt
'^T{Aeexp(i'y£hi) -f B ee x p (ije(hx - h2)]Emini
(2.58)
(2.59)
G_1
2Lt
=
T i A ehexp (i'ythhi) 4- B the x p i- i') th{hi - h2))]Eymnlh
(2.60)
8_"i1
£—
2L t
]T [A f
j?_-i
£—A
(2.61)
exp(*7A) - B £e x p (ije(hi - h%)]Hxmni
21,
= J 2 { A th e x p (ijmhi) - S a e x p (-’i7&(h1 - h2))\Hxmjdh
f _IJ.
2
^ [ A ^ e x p f ^ h i) - B ie x p ( ijg(hi - h2)]Hyrrml
(2.62)
(2.63)
/?,_—
_-jA
K
2L t
(2.64)
- Bgh ex.p(-E feh(hi fc=i
At z = h2, interface separating the homogenous region from the exit region, they
are as follows
2L t
y ^ )lAihexp(i'Yihh2) + B ih]Ex,mnih = Txmn
e=i
(2.65)
^
■&=!
2Lt
(2.66)
fc\j
V
j
k%f
exp(yyift^2) A B tfjjE pm nlh
=
T-yrrvn
6Xp{yygfjh^j)
-flxroraife — 0 nh^'z^n
^mnT'-ytrm
(2.67)
exp (ij(hh2)
Hynrnih == ^rtvnXxmri
^mh^'zmn
(2.68)
£=i
2Lf
€n0 i=i
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25
Note for any additional grating and/or homogenous layers, the boundary condi­
tions will take the form of the z = hi boundary conditions.
2.6
F o rm u la tio n o f M a trix E q u a tio n s
By applying the 12 boundary conditions described in the following section (4
for each of the 3 material interfaces) to the electric and magnetic field distributions
defined for each of the 4 regions (i.e. incident, homogenous, grating and exit regions)
a large system of linear equations can be constructed.
This system of equations
when cast as a matrix problem can then be numerically solved for the unknown field
The final matrix equation will have form
distribution in each of the four regions.
shown below.
C2
0
0
P
Cs c 4
0
0
Q
Cr
2 -W i
Di
d
Ds
d4
d5
-ir3
-W 2
A
0
-W 4
B
0
Dq -W 5 - W 9
Dr Dg
-W 7 -W 8
1
T J
0
0
0
0
Ft
f
2
0
0
0
f
3
f
4
0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.69)
where, C i, C 2 , Cs and CAjare {2Lt x Li) submatrices derived from applying bound­
ary conditions 2.51 to 2.55, D \ —D$ and W \ —W% are {Lt x Li) submatrices derived
from applying boundary conditions 2.57 to 2.63 and F ^ F ^ , F% and i^ a re {2Lt x L t)
submatrices derived from applying boundary conditions 2.65 to 2.68. Also A, B , R
and 5 are (2Lt x 1) vectors representing the unknown coefficients of each of the spa­
tial harmonics in the grating and homogenous layers respectively and P and Q are
(Lt x 1 } excitation vectors. The values for each of the submatrices given in Equation
2.69 is provided below.
Applying boundary conditions 2.51 and 2.55 with the substitution of Equations
2.47 and 2.50 results in
Clnl
rj
$ n O cm
f coIj f ^ o Uxnl---------Arol
m + J
i'mti
■n'
G^nl
ki i — Hxni —
C3nl
k JIK
— Hyrd + | r,
1 (c
r„
^
Eynl
(2.70)
f'nm/
Exnl
( r™”+ if e )
aW
2l
jp
E"
exp(i7#/ii)
, Pna iri rp
Z-'xnl "i
& yn l
(2.71)
(2.72)
'
C*4ni
k\ j
Hyrd "F j T-nm
OZ
Fxal
(3n Oim
Eyni exp(*7 ^ 1 ) (2.73)
Applying boundary conditions 2.57 and 2.63 with the substitution of Equations
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27
2.47 and 2.50 results in
Di„i
= E m lexp(i'yehi)
D 2ni
= E xnie x p ( - i^ £(hi
D 3ni
= Eyni exp(«7|fti)
(2.74)
h2)}
e x p ( - ^ (fei - fe2))
(2.75)
(2.76)
(2.77)
D 5ni
= k ^ H ^ e M iJ e h i)
(2-78)
Deni
= - k xf ^ H xrde K p (-il l (hl - /*,))
V ^0
(2.79)
Dm;
= k xf ^ H ynlexp(i'yeh l )
(2.80)
V
Aw
= —k y ~ ^ H ynlexp(—i'yi (hi —h2))
(2.81)
W Xni
= E„d exp{i^th t)
(2.82)
W 2rd
= E xn(&q>{-i'ye (hi - h^)}
(2.83)
W3nl
= Dy„iexp(i7£.fe1)
(2.84)
W 4ni
= D jp u e x p f-^ C ^ i-/la ))
(2.85)
Wsni-
= k ^ H mdexp(i'yth l)
(2.86)
W'eni
= - k x [ ^ H xru exp (—i'yt(h1 - h 2))
V £e
(2.87)
WWi
= k J ^ H snlexpii-fth)
(2.88)
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28
Applying boundary conditions 2.85 and 2.88 with the substitution of Equations
2.47 and 2.-50 results in
Fo tj
|
Hxnl + —
J
o
&nm
Flnl —
Finl =
— k i f —
H zn i +
V €e
fin \
fijZ
m £^ x n l,
.
i
'■nm.
F$nl ~
.
k i l — Hynl — (4
1 «nm
e0
\
Flnl —
-Jfe4
ynl
(t
I hn m
\
j
\
i
,
\
x?
I -&xnl
/
/
&xnl
p
Jvnwi
Z / Eynl
B2
+j ,• n J e-Ljyn
tllTO/
W
B„a„ 271
exp(i7£h2)
(2.90)
(2.91)
,
^
*-*yn l
^nm
Bnotm
.
E'ynl
exp(?>/fh2)
(2.92)
(2.93)
Applying boundary conditions 2.512.55 and 2.652.68 with the substitution of
Equations 2.4 and 2.12 results in
=
a^ - n x - ^2r0 +
o£\
( 2r'
% + Bquz JnS
i aoBa
-Uy - a Quz
(2.94)
(2.95)
After substitution of Equations 2.70 through 2.95 into Equation 2.69 and solving
the resultant matrix equation the reflected and transmitted diffracted orders can be
determined by
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23
2 Lt
R xm n — T , ' A
(2.96)
i= \
2Lt
Ryrnn
—
Txmn ~
-f- B ( GXp{i,y £ h i)^ E p mni
*=i
2Lt
Y ^ iR e v x p iiJ ih ) + Se)Esmnl
e=i
Uy$mQ$nQ
(2.97)
(2.98)
2L t
Tyum =
2.7
exp(n'i^2) + s e)Eymni
(2.99)
C ode D evelop m en t and V alidation
I implemented this numerical formulation of RCWA in MATLAB.
The code
I developed resulted in two separate MATLAB .m files, also known as MATLAB
scripts.
I had one file for the one dimensional GMRJFs and a second for the two
dimensional GMRFs.
The two dimensional .m file was a direct implementation
of the numerical development I outlined above.
The one dimensional .m file was
a simplification that eliminated either the x or y components dependent upon the
polarization of the incident field.
Once I had the developed code, I used published GMKF results to verify the
code. [46] [51] Both my one dimensional and two dimensional codes did a very good
job of duplicating published results of GMEFs.
In the next chapter I will describe the iterative design algorithm developed during
the course of this project.
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C hapter 3
A n alytical M ethods:
Itera tiv e
D esign o f G uided M ode R esonant
F ilters
In this chapter I will present the design methodology developed for the synthesis
of GMRFs.
Overall the design process is an iterative approach in which the ROW
algorithm, presented in Chapter 2, is used to analyze various test structures until
a desired response is found.
Figure 3.1 shows the design process flow graphically.
As an overview, an initial design was selected based upon wavelengths .and design
constraints.
The initial design was used as input to the optimization algorithm.
This chapter describes the design assumptions, iterative design approach, and grating
designs including predicted performance.
The output of the design algorithm was
30
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31
IVaiacg
! forGMRF
r ----
y-j
'TffiSiVi.-
2W‘V :
vxii A'idc?;
^jxari &V<vr»L
Beslggt AlgaritiHti
u
1afc:*Crr.fim»v.
M«asaraiiejsS
X s im e ru a }
&
£ x |i « r i c i e > i t s f !
V'aiTjlHBon
Step *L Evaluate «bjeirtiv*fimctie»wi& RCW
mode! for iuisial value
Step #2: Apply genetic algorithm using RCW
model to evaluate each child in the objective
function to begin mixuntizatiou
Step #& Apply direct search method to refiae
andfm aike uaramizatuni of objective function
Step M i Ensure fabrication constraints are met,
i on e "with n-/v c o n s tr a in t.
if not return t
littU i
Anai^<w
Figure 3.1: GMRF design and evaluation process
numerically evaluated not only with the RCW developed for this research, but also
against a commercially available finite element code. The fabrication, experimental
measurement and analysis will be covered in following chapters.
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32
3.1
D esign A p p ro ach C o n sid e ra tio n s a n d A ssu m p ­
tio n s
Once the.RCW code was verified, I developed a methodology to design .a GMRF
with a desired pass band response. The RCW7' code allowed me to change both the
geometric and material property parameters of the GMRF including (1) grating pe­
riod, (2) grating fill factor, (3) grating thickness, (4) homogeneous layer thickness, (5)
number of layers either grating or homogeneous, (6) grating layer(s) material prop­
erties, and (7) homogeneous material layer(s) properties. By appropriately changing
these seven properties, I could completely control the response characteristics of the
GMRF. Although the RCW code can handle the complete range of parameters and
calculate the response, real world experimental constraints had to be considered such
as fabrication capabilities, available materials, and measurement methods and capa­
bilities.
The fabrication method used was a computer numerically controlled (CNC)
industrial router. The CNC router provided the capability to precisely fabricate the
lamellar grating structure within excellent tolerances, approximately 1 mil.
With
this fabrication method comes constraints on the ratio of the fill factor to grating
thickness.
If the fill factor was too high ( i.e. only a small amount of material
in width to be machined away) and the grating was thick the grating could not be
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
33
fabricated with this method. This is due to the constraint on the cutting bit profile
that the total depth of cut could be no more than 3 times the width of the bit used.
There is no machining constraint on the homogeneous layer other than a. reasonable
thickness for handling, measurement, .and cost.
The material selected for this experiment was REXOLITE®. REXOL1TE® is
a thermoset rigid plastic produced by cross linking polystyrene with divinylbenzene.
The resultant translucent plastic has several excellent properties for this experiment.
It has a dielectric constant of 2.53 from virtually 0 GHz to 500 GHz with a very low
loss factor. The low loss factor matches my material assumptions in the RCW code
for a near lossless material and the dielectric of 2.53 large enough to design reasonably
good GMRF. Additionally, this rigid plastic has excellent machining properties that
allow the use of the CNC router with standard carbide tipped tools for fabrication.
The final design constraint to consider is the measurement techniques and capabil­
ities. Two measurement methods at different frequencies were used to experimentally
validate the designs. One technique was a collimated lens system for millimeter wave
(32-50GHz) at the University of Delaware and the other was a sliding phase elimina­
tion system for microwave (6-18Ghz) at SPAWAR Systems Center in San Diego. The
details of these experimental systems are described in Chapter 4.
The constraints
of these systems limited the frequency bands that the experimental GMRFs had to
operate in.
Given the constraints of fabrication, material, and measurement I designed two
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
34
one-dimensional GMRF test articles, one at microwave frequencies and one at mil­
limeter wave frequencies for fabrication.
I also designed a two dimensional GMRF
at millimeter wave frequencies in a material other than REXOLITE®.
As de­
scribed later in this document, it was found to be very difficult to design a good
two-dimensional GMRF in a low dielectric contrast material such as REXOLITE®.
As a consequence, a design, was shown and validated numerically using the FEM for
a material with a higher dielectric constraint than REXOLITE®, however it was not
fabricated due to the expense of the high dielectric materials. For the designs that
were fabricated I ensured that the aspect ratio of the profile fell within the fabrication
constraints and that the overall layer thickness was a reasonable thickness that could
be readily purchased.
3.2
Iterative D esign A pproach
In Chapter 2 ,1 presented a method based on RCW for analyzing the transmission
of a GMRF given an incident plane wave. This a straight forward function evaluation
problem.
The inverse problem, however, is of greater significance for filter design.
The inverse problem is to find the GMRF parameters that produce, as closely as pos­
sible, the desired filter characteristics. This inverse problem can be easily approached
as an iterative optimization calculation illustrated in Figure 3.1.
Put simply, given
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35
some desired output, mathematically described by an objective or cost function, I
iteratively analyze test structures using the RCW algorithm until an acceptable de­
sign is found. Clearly, the brute force method, of testing all possible combinations of
design parameters (i.e. all grating periods, fill factors, ...) is not practical due to the
infinite number of perturbations.
Thus, a more intelligent optimization algorithm
must be identified and used. I found th at a combination of two techniques provided
good designs in a short period of time. These were a genetic algorithm coupled with
a direct search method. The genetic algorithm is run initially to find a subspace in
the entire design space in which a good solution exits. The direct search is then used
to refine the genetic algorithm search until it locates the final design.
In the next
sections the details are these algorithms are presented.
3.2.1
G enetic Algorithm
The genetic algorithm is a method for solving optimization problems that is based
on natural selection. This is the process that drives biological evolution. The genetic
algorithm repeatedly modifies a population of individual evaluations of an objective
function. At each step or generation, the genetic algorithm selects individuals from
the current population to be parents and integrates their most attractive features to
produce children for the next generation. Over successive generations, the population
"evolves" towards an optimal solution. The initial population is randomly generated,
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36
while the succeeding generations evolve from the parent population. Each succeeding
generation is referred to as the children of the previous generation. [52]. Occasionally,
a parameter is randomly varied, referred to as a mutation, to prevent the algorithm
from being, stuck in a local minimum.
The following outlines ,the genetic algorithm process: The algorithm is initiated
by creating a random initial population of test samples. A single test sample, for
example, might be a vector of design parameters that includes the grating period, fill
factor and grating thickness. Each of these design parameters is a single number in
the test vector and is called a gene in genetic algorithm terminology. The algorithm
then evolves a sequence of new populations of test samples, or generations. At each
step, the algorithm uses the individuals in the current generation to create the next
generation. To create the new generation, the algorithm perforins the following steps:
(1) scores each member of the current population by evaluating its objective function
value, (2) scales the objective function evaluations to convert them into a more usable
range of fitness values, (3) selects parents based on their fitness scores, (4) generates
children from the parents. Children are produced either by making random changes
to a single parent - mutation - or by combining the vector entries of a pair of parents
- crossover, and (5.) replaces the current population with the children to form the
next generation. The algorithm stops when one of the stopping criteria is met, which
include a stable best fitness value and/or exceeds the maximum number of allowable
generations.
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To create the next generation at each step, the genetic algorithm uses the current
population to create the children that make up the next generation. The algorithm
selects a group of individuals in the current, population, called parents, who contribute
their genes - the entries of their vectors - to their children. The algorithm selects
individuals that have better fitness values as parents. The genetic algorithm creates
three types of children for the next generation: (1) Elite children are the individuals
in the current generation with the brat fitness values. These individuals automatically
survive to the next generation, (2) Crossover children are created by combining the
vectors of a pair of parents and (3) Mutation children are created by introducing
random changes, or mutations, to a single parent.
Crossover children are created
by combining pairs of parents in the current population. At each coordinate of the
child vector, the default crossover function randomly selects an entry, or gene, at the
same coordinate from one of the two parents and assigns it to the child. Mutation
children are created by randomly changing the genes of individual parents by adding
a random vector from a Gaussian distribution to the parent.
Figure 3.2shows the children of the initial population, that is, the population at
the second generation, and indicates whether they are elite, crossover, or mutation
children.
My implementation of the genetic algorithm is sometimes referred to as a microgenetic algorithm. The micro prefix is added because the implementation I selected
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38
Current Generation
Next Generation
<
! Crossover Child
j Mutation Child /
Figure 3.2: Generation of children in next generation for genetic algorithm
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39
has both a reduced population size and limited number of generations. 1 implemented
the algorithm to quickly evolve the solution to an area in the domain where a design
solution was likely to exist,
I then passed that final elite child to a direct search
methods to rigorously search the design space for the best solution.
3.2.2
Direct Search
Direct search is a method for solving optimization problems that does not require
any information about the gradient of the objective function, it is strictly a numerical
evaluation of the objective function at a discrete set of points. As opposed to more
traditional optimization methods that use information about the gradient or higher
derivatives to search for an optimal point, a direct search algorithm searches a set
of points around the current point, looking for one where the value of the objective
function is lower than the value at the current point. A direct search is useful to solve
problems for which the objective function is not differentiable, or even continuous.
The special class of direct search method I used is a pattern search algorithm. The
pattern search algorithm computes a sequence of points that get closer and closer
to the optimal point. At each step, the algorithm searches a set of points, called a
mesh, around the current point - the best point computed at the previous step of
the algorithm. The algorithm forms the mesh by adding the current point to a scalar
multiple of a fixed set of vectors called a pattern. If the algorithm finds a point in
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40
the mesh that improves the objective function at the current point, the new point
becomes the current point at the next step of the algorithm.
Each set of points, the mesh, is called a poll.
The points of the mesh for the
current poll are selected by adding and subtracting the current mesh size to the set
of variables that were the best point of the previous poll, see Figure 3.3. If the poll
was successful, i.e. one of the new points had a better fitness value for the objective
function, the new point becomes the center of the next mesh and the mesh size is
expanded. If the poll is unsuccessful, the current best point is retained and the mesh
size is reduced. This algorithm can be stopped in several ways including: mesh size
below a threshold, a set maximum number of polls, and a threshold value of change for
2 successful polls. I choose to incorporate both the mesh size criteria and maximum
number of polls. The maximum number of polls ensured that the computational time
would not become unreasonably long, while the mesh size stopped the algorithm when
the parameter changes were much smaller than the constraints previously identified.
3.2.3
G M RF D esign Algorithm
The complete algorithm I used combined these two optimization techniques, ge­
netic algorithm and direct search. The genetic algorithm is used to initiate the search
and was given the geometric constraints of the problem along with the algorithm con­
straints of a maximum number of generations. The best fitness values variables were
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41
n
u
Poll Point
initial Point
Poll Point
Pol! Point
Poll Point
J “L
Figure 3.3: Mesh pattern for direct search
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42
then passed to the pattern search algorithm. The best fit results of the pattern search
algorithm was then compared to the desired filter properties.
The final piece of the design optimization problem was to develop a reliable objec­
tive function. The objective function for the transmission filter had to combine the
following criteria of good narrow band pass characteristics at the desired frequency
with good band stop characteristics away from the band pass frequency.
The objec­
tive function was constructed to use the RCW formulation to calculate the transmis­
sion characteristics of a grating structure for a given frequency band. The frequency
band was centered on the desired pass band frequency denoted as P(A)with a band­
width of j3w.. At the pass band frequencies the goal was to maximize transmission
while minimizing the transmission throughout the out-of-band portion of the fre­
quency range, denoted as £(Aj) for the lower side band and S(XU) for the upper side
band. This lead to an objective function th at calculated one transmission coefficient
at the deign pass band frequency and one transmission coefficient at each of the out
of band frequencies.
(3.1)
Where A is the design center frequency, j3w is the bandwidth of the pass band, A*
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43
is the lower bound of the lower stop band stop band, and Xu is the upper bound of
the upper stop band.
Depending upon the bandwidth of the stop band regions above and below the
pass band, this can be many frequency points.
Given the nature of guided mode
resonant filter’s characteristic of a narrow sharp passband, dense sampling in the
stop band is required to ensure there is not a undesirable second pass band within
the design range. This need to balance the requirement of one band pass evaluation
point with many, typically numbering in the hundreds, stop band evaluation points
leads to weighting the passband evaluation such that it has an equivalent impact on
the objective function as the many stop band points.
A ^ p.
(3.2)
The value of the weighting coefficient was determined through numerical experi­
mentation
An initial guess was to have a value for W equal to the total number of
lower and upper stop band points.
This value gave equal weight to the pass band
frequency relative to all of the stop band frequency. The equal weighting sometimes
lead to a solution where the stop band was acceptable but the desired pass band was
weak to nearly nonexistent.
After conducting a series of numerical experiments, I
found that a value for W that was 25% greater than the total number of stop band
frequency points yielded consistently good results.
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44
la the next chapter I present analytical results for the design of GMRF using the
genetic algorithm presented in this chapter.
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C hapter 4
A n alytical R esu lts
In this chapter I present several analytical results of GMRF designed at microwave
and millimeter wave frequencies using the iterative design algorithm described in the
previous chapter. In addition to the RCW predictions I provide numerical validation
by comparing the results to a commercial finite element program HFSS. I will follow
this chapter with experimental results that provide further validation.
45
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46
4.1
O ne-D im ensional a n d T w o -D im en sio n al G M R F
D esig n s a t M icrow ave a n d M illim eter W ave
Frequencies
Gratings were designed for both microwave and millimeter wave frequencies. These
particular designs were selected to account for the previously identified fabrication
and material property constraints.
The designs were achieved by using the de­
sign algorithm as detailed in the previous chapter, including both genetic and direct
search algorithms. Two designs were done as one dimensional gratings and one de­
sign was done as a two dimensional grating. One-dimensional GMRF designs were
achieved at both microwave frequencies and millimeter wave frequencies while the
two-dimensional design was accomplished at millimeter wave frequencies.
For all of my designs, I validate them with a separate numerical technique, namely,
a commercially available Finite Element Method (FEM) code marketed by ANSOFT
as High Frequency Structural Simulator (HFSS). FEM is a well understood numer­
ical technique that can be used to conduct rigorous electromagnetic analysis.
In
fact, the ANSOFT HFSS package is an industry leading standard for high frequency
electromagnetic simulation codes. It is a frequency domain technique and therefore
must be run at each individual frequency. One of the inputs that drive the accuracy
is the size of elements used in the FEM mesh.
The resonant structure of GMRFs
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47
have characteristically rapidly changing fields within the structure which drive a small
element size in the mesh. The small element size is needed to accurately capture the
effects of those rapidly changing fields. 'This in turn creates a significantly long run
time for the analysis - more elements require more memory and CPU time to solve.
Convergence of the HFSS results was ensured by refining the mesh until the GMRF
performance curves converged.
In the next several sections I will provide three different GMRF designs that serve
as examples of the analysis and design methods developed during the course of this
project.
4.1.1
E xam ple # 1 : O ne-dim ensional G M R F B an d p ass F il­
te r Design at 14GHz
In this section, I present the design of a one-dimensional GMRF’ with a passband
at 14GHz to demonstrate a design a microwave frequencies. This design is done is
REXOLITE with few other constraints on the design. I allowed the period, fill factor,
and both grating and homogeneous layer depths to vary.
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48
D esign analysis for exam ple # 1
The objective of this example was to design a one-dimensional GMR transmission
filter with a. center passband of 14 GHz and a fractional bandwidth of less than 1% .
The initial design parameters chosen for this design were a grating period A = 2cm,
fill factor =0.5, and a grating thickness of each layer was 1cm. The design range was
given as: 1 < A < 4, 0.2 < fill factor < 0.8, and both the grating and homogeneous
layer thicknesses 0.75 < A* < 3.
The stop band was calculated at 125 frequency
points above and 125 frequency points below the stop band frequency.
Using the
objective function 3.2 the weighting factor for the stop band frequency value was 325,
or slightly more than 125% of the total number of stop band frequency points. I then
ran through the algorithm as described in Chapter 3. As shown in Figure 4.1 an
good design was achieved with a clear pass band at 14GHz and fractional bandwidth
of much less than 1%.
The initial guess was a rather inefficient design. This can be seen in Figure
4.2. The genetic algorithm portion of the design process produced a substantially
better design while the direct search algorithm further refined the design.
The
initial guess shown as the top point on the y axis produced a value for the objective
function of approximately 480. Upon completion of the genetic algorithm (shown in
the graph as the drop from the initial point on the y axis to the next point at iteration
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49
Transmission
Designed 14 GHz P ass Band
1.35
1.4
Frequency
1.45
x 10
Figure 4.1: RCW predcition of 14 GHz design performance
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50
Best Function Vitos: 235.1797
§00?
m
*23
« m >
c
©
130
200
0
100
200
300
€30
iteration
Figure 4.2: Objective function evaluation for 14GHz design
#1), the objective value functions was reduced to approximately 245.
The direct
search algorithm was then used to refine the design to a final value of 235.
design was used to fabricate a test article.
This
Final values from the design algorithm
were: A = 1.5, fill £actor=0.5, the thickness of the grating hi = 1.8cm, and thickness
of the homogeneous layer h2 = 0.85cm.
A schematic of the output design is shown Figure 4.3. It shows one-dimensional
grating view with the dimensions used, for development of both the numerical valida­
tion model and the experimental test article.
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51
Example#! one-dimensional 14 GHz
0.75
0,75
—
■siiii
2.65
i
i
Profile view
ABiM mmmmsfgreim eemiimisiem'
Figure- 4.3: Dimensioned schematic of 14 GHz design
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52
HFSS Validation of 14 GHz P ass Band
w
so
£m
as
-6 j-
-10
j — * — D esign Prediction
I • - • Q- - H FSS Prediction
-12
1.25
1.3
1.35
1.4
Frequency
1.45
1.5
1.55
Figure 4.4: Comparison of design with HFSS at 14 GHz
H F S S v alid atio n for exam ple # 1
The results from this 14 GHz design as predicted by the RCW formulation were
numerically validated by comparison to HPSS. As shown in Figure 4.4 the HFSS
prediction clearly is in agreement with the RCW prediction. It should be noted, that
when the predicted curve in Figure 4.4 is compared with the curve in Figure 4.1, there
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53
is a noticeable difference in the magnitude of the passband of the RCW prediction.
The reason for this difference is that the two RCW predictions are done at different
frequency points. Fewer frequency points were predicted and plotted with HFSS than
the RCW. This reduced set of points was used because the run time requirements
of HFSS would rapidly become excessive with more points.
Even with a reduced
frequency point set, here 40 HFSS frequency points vs. 1001 RCW frequency points,
the HFSS would run overnight to produce the result while the RCW would run in
approximately 5 minutes. With this numerical validation, I found the 14GHz design
to be acceptable for fabrication.
4.1.2
E xam ple # 2 : O ne-dim ensional G M R F B an d p ass F il­
te r D esign at 36 GHz
This section presents the design of a transmission filter with a center passband of
36 GHz to demonstrate a millimeter wave filter. I put additional design constraints
on the thickness of the material to better conform to economically available material.
D esign analysis for ex am ple # 2
With 36 GHz as the center band frequency the initial values used were a A = 7mm
and a fill factor =0.5. For the thickness of the grating and homogeneous layers I added
additional constraints.
The motive for these additional constraints were two fold.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
One was to apply additional constraints due to material availability and fabrication
constraints.
A total thickness of 19mm was used because this was the thickness
of material available off the shelf.
A grating depth of 6mm was used because this
corresponded to the maximum depth of the smallest groove that could be removed by
the CNC milling machine - the smallest bit available is 2mm and so the constraint on
the depth was placed at 3 times the bit diameter, for a 6 mm maximum depth at the
smallest feature size. The second motive was to test the design algorithm to see if
an acceptable design could be found with only the period and fill factor as variables.
The design range was given as: 5 < A < 9 and 0.2 < fil l factor < 0.8.
The stop
band was calculated at 125 frequency points above and 125 frequency points below
the stop band frequency. Using the objective function 3.2 the weighting factor for the
stop band frequency value was 325, or slightly more than 125% of the total number
of stop band frequency points.
I then ran through the algorithm as identified in
Chapter 3. The result is shown in Figure 4.5. The result was an efficient response
with a nearly symmetrical line shape below -2db.
My initial guess for the 36GHz design was not much better than the 14GHz design
with an initial value of nearly 382.
In this case, however, the genetic algorithm
reached its maximum number of generations before a significantly better design had
been achieved, see Figure 4.6.
The direct search algorithm took over and did an
excellent job of refining the design as seen by reducing the value of the objective
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55
Designed 36 GHz P ass Band
‘
"T
Transmission
-5 -
-10 h
-15
-20
-
-25 L3.4
3.45
3.5
3.55
3.8
3.85
Frequency
3.?
3 .7 5
3.8
x1G1°
Figure 4.5: 36 GHz Design
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56
390
r
Objective Function Evaluation
380 ^
370
-
360
-
350
-
340
-
330
-
320 s-
10
20
30
40
50
Iteration
Figure 4.6: Objective function evalaution by iteration
function to 327.5.
Final values from the design algorithm were:
A = 7.8, fill
factor=0.7, the thickness of the grating hi = 6mm, and thickness of the homogeneous
layer
= 13mm.
A schematic of the output design is shown Figure 4.7. It shows one-dimensional
grating, view with the dimensions used for development of both the numerical valida­
tion model and the experimental test article.
H FSS validation for ex am ple # 2
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57
Example #2 one-dimensional 36 GHz
5.45
•2.35
\
\
-— 5 , 4 5 - 4 - 2.35—
i
mm
mm
■ ill
■aBBi
mam
■SB!
mmm.
I
j
IBB
mSm
IS S rt’l
■Bi
iiiiigii
—
f
8
t
j
1
19
Ml
13
Pkii mew
Profile view
ANBimmmmis m e m mtSimetm's
Figure 4.7: Dimensioned schematic of 38 GHz design
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58
Transmission
HFSS Validation of 36 GHz P ass Band
-10 r
-12
-14
h
-16 — * — Design Prediction j
HFSS Prediction i
-18
-20
3.4
3.45
3 .5
3.55
3 .6
3.65
3.7
3.75
Frequency
Figure 4.8: Comparison of Design with. HFSS predication at 3SGHz
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59
Figure 4.8 shows the HFSS prediction has reasonably good agreement with the
design value for the 36 GHz design.
The location and magnitude of the passband
were nearly identical. As with the 14 GHz design, I found this numerical validation
to be acceptable and proceeded to fabricate the 36 GHz design.
S en sitiv ity o f D esign to M aterial Loss
For this design, I also conducted a numerical study with ECW on the effects of
material loss. The objective was to investigate if even a small amount of material loss
would have a significant effect on the filter’s performance. To this end, I varied the
loss tangent of the material from 0 to 0.01. The loss tangent is defined as the ration
of the imaginary component of the complex permittivity to the real component (i.e.
loss tangent = A-). The results are shown in Figure 4.9. As can be seen in the data,
even small amounts of material loss can have a noticeable impact on the magnitude
of the filter performance while the location of the passband remains substantially
unchanged.
The pass band drops off rapidly with increasing loss.
A very slight
widening of the pass band can also be seen, but that effect is small compared to the
impact on the magnitude. This effect is not too surprising. Since the GMRF response
is a very narrow band resonance it would be very sensitive to any material loss. It
should be noted that the loss tangent of REXOLITE is known to be approximately
0.006.
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80
L oss Sensitivity Comparison
Loss Tan 0.01
-7.6 8 db
c -10
0
1
E
sro
Loss Tan 0 .0 0 6
-9.4 5 db
Loss Tan 0 .0 0 3
-12.18 db
t— -15
No Loss
-20
No Loss
-20.76 db
L oss Tan 0 ,0 0 3
—
L oss Tan 0 .0 0 6
• — - L oss Tan 0.01
-25
3 .55
3.6
3 .65
Frequency
3 .7
Figure 4.9: Loss sensitivity comparison
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3 .75
61
S en sitiv ity o f D esign to Sm all Changes in Incident A ngle
For this design, I also conducted a numerical study with RCW on the effects of
small changes in the incident angle measured.
For all the designs it was assumed
th at a plane wave is normally incident on the GMRF. The objective of this numerical
experiment was to see if even small changes of incident angle off normal would have
an impact on the filter’s.performance. To this end, I varied the incident angle from
0 (normal incidence) to 1.5 degrees off normal. The results are shown in Figure 4.10.
As can be seen in the plot, the effect of incident angle is significantly more profound
than the effect of material loss previously shown.
The first striking difference is
the separation of the single passband into 2 pass bands, one higher and one lower
in frequency. This is seen on the 0.5 degree incident data, where a decrease in the
magnitude can also be seen for both of the new pass bands. As the incident angle
increases, the two pass bands move farther way from the original pass band.
Clearly this example illustrates that GMRF that are designed for a given incident
angle are unlikely to work at other incident angles.
It also, however, provides a
potential means of producing good calibration targets for applications that require
precise system alignments.
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82
Angie Sensitivity Comparison
s
0 D©Qf©©
0.5 Degree
1 Degree
1.5 Degree
0.5 Degree
15.1 db
36.79 GHz
1.5 Degrees
-17.4 db
36.04 GHz
1.5 Degrees
18.63 db
37.11 GHz
1 Degree
-19,4 db
36.21 GHz
0 Degree
-23.8 db
36.44 GHz
0.5 Degree
-22.3 db
36.36 GHz
.............. .................. -
J
1
j
I
1 Degree
-21.5 db
36.95 GHz
-25
x 10
Figure 4.10: Incident angle sensitivity comparison
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63
4,1.3
E xam ple # 3 '. T w o-dim ensional G M R F B an d p ass F il­
te r D esign a t 36 G H z and 42G H z
In this section I will present the design analysis and numerical validation of a
two-dimensional filter design with two band pass frequencies.
The purpose of the
two-dimensional filter is to have a polarization independent design and to show that
multiple band passes are possible. It was determined that to achieve a good 2D design
th at a material with a higher dielectric constant than REXOLITE was needed.
D esign analysis for exam ple # 3
The objective of the two-dimensional design was to produce a transmission filter
with center passbands of 36 GHz and 42 GHz and fractional bandwidths of less than
1% at each passband.
Initial attempts at this design employed the same material
as the one dimensional designs (i.e. REXOLITE) but it was found that for these
structures that a dielectric contrast ratio higher than 2.5:1 was needed for an ac­
ceptable design.
As a result, the dielectric constant of the material was included
as an additional variable to the design algorithm. The initial values of this design
were chosen as, periods in the x and y directions Az = Av = 6mm, fill factors in
the x and y directions Sx —ffy—Q.S, initial dielectric constant e = 5.0, and a grating
thickness of each layer was restricted to the same value as the 36 GHz and 40 GHz
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64
designs.
The grating depth of 6mm was still a valid constraint and the total thick­
ness was carried as still a standard stock quantity. The design range was given as:
4 < AXty < 9, 0.2 < fill factor < 0.8, and both the grating and homogeneous layer
dielectric 4 < e < 7. I then ran through the algorithm as described in Chapter 3 and
the results are shown in Figure 4.11. The design has a deep pass band at 36 GHz
and a second good passband at 39 GHz. Both pass bands have notable side lobes
higher in frequency from the pass band.
A schematic of the output design is shown Figure 4.12.
It shows the two-
dimensional grating view with the dimensions used for development of both the nu­
merical validation model.
H FSS v alid atio n for exam ple # 3
The comparison of the 14 GHz design as predicted by the RCW formulation
with the HFSS prediction shows reasonably good agreement as shown In Figure 4.13.
There is a good match in the relative magnitudes and locations of both of the pass
bands.
In this Chapter I presented three examples illustrating of the design of GMRF
using the iterative design algorithm presented in Chapter 3.
The results were val-
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65
Designed 2 dimensional dual p assband of 36 GHz and 39 GHz
-10
Transmission
-1 5
-20
-2 5 b
-30
-35
-40
3.4
3.5
3.6
3.8
Frequency
3.7
3.9
4.1
x 10
Figure 4.11: Two Dimensional dual pass band design
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10
88
Example #3 two-dimensional
36 & 39 GHz
-
i«- 4.S- *
.
"H2.3j—
:llllllll 1I
lM ^
Wmm |
-*|2.5|—
LJ LJ
U
--------------------------------
125
4.8
■2.3-
----- ------------------------------
1.25
□ j
.
19
LJ
LJ
13
1
i 1i P i
j
.
view
Profile view
AH MmmMmtM m e m miliimieiem
Figure 4.12: Dimensioned schematic of two-dimensional 36GHz and 39GHz filter
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87
Designed 2 dimensional dual passband of 36 GHz and 39 GHz
-10
$
£ -20
i/3
C
H
E
-25 -
-30
Design Prediction j
-X- • HFSS Prediction !
-35 j-
-40
3.4
3.5
3.6
3.7
3.8
Frequency
3.9
4
4.1
Figure 4.13: Comparison of 2 dimensional dual band design with HFSS
R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission.
idated using an independent numerical algorithm (FEM). Additionally, for one of
the designs numerical experiments were conducted to determine its sensitivity to ma­
terial less and incident angle.
The results of these experiments indicate that the
performance of GMRF are indeed very sensitive to both material noise and incident
angle. In the next chapter I will present the experimental methods th at were used
to. fabricate and characterize the examples presented in this chapter.
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C hapter 5
E xp erim en tal M eth od s
In this chapter I will discuss the experimental methods used during the course
of this project.
Specifically, I will provide a description of the methods used to
fabricate test articles as well as provide a detailed description of two different mea­
surement techniques used to characterize the samples. Specifically, a collimating lens
measurement technique for the millimeter wave designs, located at The University of
Delaware, and a spatial averaging technique for the microwave designs, located at
Space Warfare .(SPAWAR) Systems Center in San Diego, CA .
5.1
T e st A rticle F a b ric a tio n
In this section I will describe the materials and methods used to fabricated
89
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70
test samples.
All test articles were fabricated out of REXOLITE®, a thermoset
plastic material with both excellent machining qualities and good dielectric proper­
ties.
REXOLITE® is easily worked with standard carbide tipped machine tools.
For my fabrication I used a computer numerically controlled (CMC) router at The
Catholic Universfty of America. The CMC router was able to mill the gratings into
REXOLITE® with an accuracy of 1 mil (“0.025 mm). REXOLITE’s® dielectric
properties are very consistent and well known. It has a dielectric constant of er =2.53
and a loss tangent of approximately 0.0066. These values are consistent across both
of the frequency bands of interest in this dissertation. The REXOLITE® was pur­
chased in 12”xl2" squares which were of sufficient size for both measurement systems.
The CNC router used to fabricate the test articles was a computer controlled three
axis machine that had a 1 mil tolerance in fabrication. The geometry of the grating
was inputted into the computer using a program called BOBCAD. Within this soft­
ware, the two dimensional structure of the grating was drawn using the BOBCAD
interface.
Using the geometric information and machine movement speeds deter­
mined by the REXOLITE®, the software automatically generated the machine code
used to drive the CNC router.
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71
5.2
D escrip tion o f M easurem ents
Experiments were conducted at both microwave and millimeter wave frequen­
cies. The.microwave frequency measurements were conducted at the Space Warfare
(SPAWAR) Systems Center in San Diego, CA and the millimeter wave frequency
measurements at the University of Delaware. Department of Electrical Engineering.
The two measurement approaches were different in both their operational frequency
range as well as the details in their construction. Conceptually, however, both mear
surement techniques are similar. The basic idea is that the transmission through a
sample can be measured by simply placing a sample between two horns on either side
of the test article. One horn is used to transmit energy and the other horn is used as
a detector. A vector network analyzer is used to both supply energy to the transmit
side and detect energy on the detector side. This basic concept is illustrated in the
figure below Figure 5.1.
However, the actual experiment proves to be much more complex than described
in Figure 5.1. The resonance phenomena in the GMRF depends upon a flat phase
front, i.e. a plane wave, incident upon the material.
Typically available horns
have a curved wave front at reasonable horn separation distances. In addition, the
conceptually simple measurement set up is complicated by the finite size of the panel
and interactions from within the room. The finite panel size introduces edge effects
while the room can introduce additional scattering from other objects.
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The two
72
HSPNetwwk
Art*fys«r
D
C
i
T5*¥t«*»'l¥towa
/
Receiving
horn
Transmitting
t i :
\
horn
x- REXOLITE
Test sample
Figure 5.1:. Conceptual experimental setup
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73
measurement techniques take separate approaches to eliminating the edge effects and
the other scattering from non-test article objects. The following two sub-sections will
describe in detail the different measurement apparatus used as well as the calibration
procedures.
5.2.1
U niversity o f Delaware M illim eter Wave M easurem ent
The millimeter wave measurements were conducted at the University of Delaware.
The experimental setup, shown in Figure 5.3, integrated a vector network analyzer
(HP 8530) with two millimeter wave horns configured to radiate through a pair of
microwave lenses.
GHz.
This system works well with frequencies ranging from 30-110
The test article was placed between the lenses where the beam was to have
the maximum degree of collimation. The desired result is a nearly flat phase front
beam impinging on the center of the test article. In this manner, the curved wave
front was converted to a flat wave front by the lens and the beam width of the
wave front was smaller than the test article. This achieved two of the goals of the
measurement system: (1) a nearly incident plane wave, and (2) elimination of edge
effects and scattering from within the room. The network analyzer was swept through
the desired frequency range obtaining transmission, measurements at each frequency.
In addition to measurements taken with the test article in place I also conducted
measurement with no sample (i.e. free-space) to use as a calibration sample. Since
the free-space measurement should, in theory, provide a transmission coefficient of
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74
N«te*Ariaiyier
x
Horns with
/" " Collimating Jens
1
k\!
Test Sample
Figure 5.2: Schematic of University of Delaware experiemtal setup
1.0 for all frequency it can be used to determine correction factors. Errors that are
corrected using this type of calibration are; (1) finite beam widths of the horns, (2)
frequency dependency of the horns, cables and connectors and (3) any polarization
mismatch between the two horns.
The Figure 5.3 shows the setup with a sample
placed between the collimating lenses.
5.2.2
SPAW AR S ystem s C en ter D u al A necholc Cham ber Spa-
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75
Network sna=> t3s"
Test Article
Receive Horn with
Coiiimating iens
i ransmit Horn with ji
Coilimatmg lens ■
Figure 5.3: University of Delaware millimeter wave meaurement system
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76
tially A veraged Transm ission M icrow ave M easurem en t
A second set of measurements were conducted at microwave frequencies at the
3PAWAR Systems Center. This system employs the Dual Aaechoic Chamber Spa­
tially Averaged Transmission Measurement technique.
This method has several
significant differences from the UDEL system. The most important difference is the
ability to translate the transmit and receive horns closer to and farther away from the
test article and the anechoic chambers under computer control. By doing so spatial
averaging can be used to remove a number of noise sources and other artifacts from
the measurement. See Figure 5.4.
The measurement process is controlled using a PC running LabView. Lab View is
a data acquisition software package that interfaces with the HP 85 IOC Network An­
alyzer and the stepper motor th at moves the transmit horn. Both the transmit and
receive horns are placed within a tapered anechoic chamber to minimize scattering
from anything other than the te s t.article which is placed between the horns.
One
complete measurement is achieved by making separate measurements of the back­
ground, an empty test fixture, and the test article. Each background and test article
measurement actually consists of multiple measurements and several distances to take
advantage of spatial averaging.
The receive horn is then moved and another mea­
surement is taken. This cycle is repeated until the desired number of measurements
has been taken, normally 49. When the measurement cycle has been completed for
each background and test article, the data is processed. The processing consists of
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77
1
%
l
.1
1
HPtM nmit
A t«Jp*r
o
c
T» ¥ ^ C ¥ ,
direct path
a
/
!
/
transmitting
horn
f \
receiver
translation
■receiving
hero
indirect patii
Figure 5.4: Schematic of SPAWAR measurement facility
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78
Figure 5.5: Inside SPAWAR chamber looking at the back of the horn toward the test
sample
dividing the test article measurements by the background measurements, scaling the
data to the empty aperture. The scaling is done for every frequency at each position
at which a measurement was made. The results of these divisions is then averaged,
i.e. spatial averaging. [53]
The advantage of this technique is to eliminate the scattering due to edges of the
test article and other objects by averaging out the phase variations. This elimination
occurs because the scaled direct transmission path energy measured through the filter
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79
Figure 5.6: Inside SPAWAR chamber showing the aperature for tranmission through
the GMRF
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80
is unchanged and each distance, however, the indirect path energy, such as that from
the edges, changes with the distance.
The change in distance due to the indirect
path has a corresponding phase shift that is different for every distance measured.
When the results are averaged over a sufficient number of distances, the effects of
the edges and other non-direct path scattering are effectively, averaged out.
is illustrated graphically in Figure 5.4.
This
The result is a measurement that contains
transmission data from the direct path only effectively eliminating the edge effects of
the aperture. [54] [53]
In the next chapter I will describe some experimental results obtained using the
experimental methods described in this chapter.
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C hapter 6
E xp erim en tal R esu lts
In this chapter I present results from my experimental measurements. I compare
the these experimental results to the analytical predictions designs from the RCW
model and HFSS. These comparisons were conducted for both of the one dimensional
GMRF designs.
Unfortunately, I was not able to fabricate the two-dimensional
design for experimental validation. Materials for this experiment were prohibitively
expensive .and difficult to specify the exact dielectric constant required.
Therefore
the numerical validation with HFSS is the filial validation for the two-dimensional
design.
6.1
E x p e rim e n ta l R e su lts: M icrow ave G uided M o d e
81
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82
R esonant F ilter a t 14 G H z
The 14 GHz microwave design was experimentally measured using the previously
described dual anechoic chamber spatially averaged apparatus at SPAWAR Systems
Center in San Diego, GA. A comparison of the measure results with the predicted
design are shown in Figure 6.1.
In this prediction the material loss (tangent =
0.0006) of REXOLITE® was included. As shown in Chapter 4 even small amounts
have material loss can have a significant impact on the performance of the GMRF
at its resonance frequency. As shown in the figure the RCW prediction are in good
agreement with the measurement. The pass band is centered at 14 GHz for both the
prediction and the experiment and the magnitudes are quite similar. The oscillations
seen in the measured data are due most likely to noise in the measurement system.
A broadening of the passband can also be seen in the measurement.
The widening of the measured transmission notch can be attributed, I believe,
to the finite aperture that was actually illuminated in the measurement- Figure 6.2
shows the illuminated aperture window with the test article in place. The theoretical
predictions assumed that both the incident plane wave and the filter grating were
infinite in two dimensions. Obviously neither the test article nor the plane wave are
infinite in the experiment, but the plane wave is significantly larger than the actual
illuminated area of the filter grating. In this case there are approximately 14 periods
illuminated from corner to comer in the aperture. As can be seen in Figure 6.2, the
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33
14 GHz Measured vs Predicted
SPAWAR Systems Center
I. f i n * I . .
r'jUvtAlu
fii*
V II® |
18
i
%n i
?• I®.-*.
*
a
5
■Measured
■Predicted (w/ loss)
-10
1.2
1.25
1.3
1.35
1.4
Frequency
1.45
1.5
1.55
1.6
x 10
10
Figure 6.1: Comparison of the predicted and measure data for the 14 GHz guided
mode resonant filter..
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84
Ifi
Figure 6.2: Test article within aperature surrounded by radar absorbing material
aperture cover effectively eliminates the edges of the test article, but sacrifices the
.size of the illuminated area. The spatial averaging techniques help reduce the effects
of the edges of the aperture, however, the number of periods illuminated still is quite
finite.
The limited illuminated area has been shown to impact the performance of the
GMEF. Bendickson et. ah and Boye and Kostuk investigated both the effects of
finite grating sizes of GMRFs and finite incident beams. [55] [56] Both papers showed
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85
that when the illuminated area of the grating is finite the response both the width
and efficiency of the pass band are effected.
Specifically they found the efficiency
decreased as the size of the illuminated area decreased. In addition to the decreased
efficiency, the pass band broadened in frequency. This is seen, in the data of the 14
GHz design measurement with the noted decrease in efficiency and the broadening of
the passband in the low frequency.
6.2
E xp erim en tal R esu lts: M illim eter W ave G uided
M ode R esonant F ilter at 36 G H z
The 36 GHz design was experimentally measured in the millimeter wave labo­
ratory at the University Of Delaware in Newark, DE using the collimated beam system
previously described. A comparison of the measured data with the predicted data is
shown in Figure 6.3. Since the designed GMRF actually had an additional passband
at a frequency that was also measured, I extended the frequencies range in the plot
for comparison. As with the 14 GHz design, I added the actual REXOLITE® loss
to the prediction. The design pass band as measured was slightly lower in frequency,
but the second pass band at 34 GHz also had a good match. Measurement data in
this system had some significant noise that could not be eliminated. For the main
pass band the magnitude of the measured data was reduced by approximately 2 dB
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88
36 GHz Measured vs Predicted
University of Delaware
.§
.1
E
S
t-5
-10
-12
Measured
predicted (wI toss)
-14
-16
3.3
3.4
3.5
i
|
3.6
Frequency
3.7
3.8
x 10™
Figure 6.3: Comparison of the measured vs the predicted data for the 36 GHz guided
mode resonant filter
and was shifted down in frequency 0.17 GHz. For the second pass band, the magni­
tude was reduced by 1.5 dB and was shifted down in frequency 0.08 GHz. Given the
large amount of noise in the system and the previously seen sensitivity in incident
angle, I consider this a reasonably good match.
The frequency shifts of the measurement may be traced to the experimental
setup. The millimeter wave experimental setup used a collimating lens to produce a
plane wave incident upon the test article. The collimating lens system was charac­
terized at Delaware using a near field probe scanned, over the lens to measure both
amplitude and phase. To replicate a plane wave, the amplitude and phase should be
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87
Amplitude
Phase
ta f
-100 -75 -50 -25 0 25 50 75 100 125
xjmmS
..
■
1??25-1£» -75 -50 -25
0
25 50 75 100 125
x{mm)
Figure 8.4: Image on the left shows the amplitude across the lens, the image on the
fight shows the phase
constant across the lens. Figure 6.4 shows the amplitude and phase characteristics
of the collimating lens.
Examining the images there is a constant amplitude as expected across the lens.
However, there is a variation in phase across the lens system. This variation, or phase
.tilt, varies nearly 380 degrees across the lens.
This tilt in phase can be thought of
as caused by a plane wave at a non-normal angle of incidence.
The impact of
this on the measurement can have.substantial effects, as demonstrated in Chapter
4,
including a shift in the pass band center frequency and/or a decrease in the amplitude
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88
of the passband. For the 36GHz design there appears to be a combination of these
two phenomena.
Additionally, much the same as the 14GHz design, there were a
finite number of periods illuminated that lead to a decrease in the amplitude of the
passband notch. The phase tilt of the lens can have the impact of moving the center
frequency. This effect was seen in the sensitivity data presented earlier.
However,
even considering these experimental artifacts, the comparison between experiment
and theory were reasonable good and, I believe, serve as conclusive validation of the
GMRF design algorithm developed in this project.
In the next chapter I will present a brief discussion and conclusion of the results
presented here as well as some comments on potential fixture work.
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C hapter 7
D iscu ssion and C onclusions
The principal objective of this project was to develop a practical method of de­
signing guided mode resonant filters at microwave and millimeter wave frequencies.
From a theoretical standpoint I developed rigorous analysis code based on the rigor­
ous coupled wave method. I also developed an iterative design algorithm for GMRF'
synthesis using a combination of genetic algorithms and a direct search. These iter­
ative design algorithms allowed the design of guided mode resonant filters that met
performance requirements within design constraints. Several examples using the al­
gorithms developed in this project were presented. These examples were numerically
validated using a commercially available finite element package (HFSS). The exper­
imental research focused on developing methods for fabricating and characterizing
GMRFs for transmission passband applications at microwave and millimeter wave
frequencies.
I demonstrated that GMRF at both microwave and millimeter wave
89
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90
frequencies could be fabricated and characterized using two different measurement
techniques.
Although the two techniques were at different frequencies, 1 was able
to compare and contrast, them for their suitability for the measurement of transmis­
sion guided mode resonant filters.
In the next sections I will briefly outline the
significant accomplishments and in the following two sections I will provide a brief
discussion/summary of the major finding of this project from both a theoretical and
experimental perspective.
7.0.1
Significant Accomplishments
In the course of this dissertation I had several significant accomplishments:
1. I implemented a three-dimensional RCW code in MATLAB that can ana­
lyze one and two-dimensional GMRF with an arbitrary number of layers in an arbi­
trary order.
2. I developed a design algorithm for GMRF using genetic algorithms and
direct search methods.
3. I developed three unique GMRF designs. One microwave design in one
dimension and two millimeter wave designs, a one-dimensional design and a dual band
pass two dimensional design.
4. I used two different measurement techniques for GMRF, the dual anechoic
chamber spatial averaging system and a collimating lens system.
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91
5. I experimentally validated the two one dimensional designs, one in each of
the different measurement systems.
6. I demonstrated simple filter designs of single grating layers with low con­
trast dielectrics (2.5:1).
7- I designed and measured a GMRF at millimeter wave, no literature exists
in this frequency regime.
7.0.2
T h eo retical D iscussion
With the use of REXOLITE® and air as the two contrasting media in the grating
structure I was able to demonstrate a simple guided mode resonant filter using low
dielectric contrast materials. This simple design consisted of using a single grating
layer with a single homogeneous layer for a 2 layer structure. My unique contribution
was to use these low contrast materials for applications in both the microwave and
millimeter wave frequency regimes.
There has only been a few published papers
of research at microwave frequencies and none at millimeter wave frequencies. My
research showed that REXOLITE® with a dielectric of 2.53 and air make an excellent
material system for a grating. The advantage of a guided mode resonant filter with
a modulated air and REXOLITE® grating layer on top of a homogeneous layer is
that it is a quite straightforward to fabricate.
The ability to use the a combination optimization algorithm of genetic algorithm
and pattern search techniques demonstrated the ability to solve the backward design
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92
problem for guided mode resonant filters in a reasonable amount of time. I used this
algorithm to design three filters, one at microwave frequencies and two at millimeter
wave frequencies. Thus algorithm was developed to find designs that were not only
two layer designs, but also fell within the material fabrication constraints identified.
The design algorithm provided the capability to use stock material sizes and standard
machine tools and practices to build custom guided mode resonant transmission fil­
ters. The microwave design at 14 GHz demonstrated the ability to use the developed
design algorithm for a transmission filter that met all of the assumption and con­
straint criteria. The 2 millimeter designs at 36 GHz and the two-dimensional design
demonstrated the robustness and versatility of the design algorithm to tailor the filter
performance to the required needs. All of the designs were in good agreement with
the HFSS predicted performance which provide a numerical validation of the designs.
The 36 GHz design illustrated the sensitivities of the GMRF design to material
loss and incident angle. The effect of changing the incident angle even small amounts
was quite profound. The center of the passband was shown to split into two nearby
passbands that were shifted both up and down in frequency.
increased as the incident angle increased.
And the separation
Adding loss made the magnitude of the
pass band rapidly decrease to the point in which the filter was marginally effective.
7.0.3
E x p erim en tal Discussion
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The initial experimental setup seemed quite simple but turned out to be more
challenging than anticipated.
My original plan was to use a focused lens system
for the microwave measurements but that proved to be a failure. The problem was
the illuminated area from the focused lens was only approximately 1.5 inches at the
frequency of interest. This illuminated too small of an area to measure the filter re­
sponse because too few periods of the structure were illuminated. Additional research
showed there were two options, (1) a another more conventional collimating beam ap­
proach and (2) a novel spatial averaging approach. The University of Delaware had
an appropriate collimating beam system at millimeter wave that I was able to use.
This system proved to be a good experimental setup for measuring GMRFs. One of
the few difficulties was aligning the test sample perfectly perpendicular to the uniform
phase front of the collimated beam. The characteristic tilt to the phase front need
to be accounted for in the physical orientation of the test articles.
The SPAWAR dual anechoic chamber with spatial averaging was an excellent
method for measuring the characteristics of the GMRF. This technique eliminated
all of the problems including edge effects and background scattering while only slightly
reducing the illuminated area. This reduced area results in a decreased measurement
of the amplitude of the pass band filter.
Overall the experimental data validated the predicted performance of the designed
GMRFs.
The dual chamber spatial averaging technique was demonstrated to an
excellent new measurement technique to compliment the more traditional collimating
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94
lens approach I used for the millimeter wave measurements.
For both measurement techniques, the effects of finite size had an impact relative
to the efficiency.
This effect was apparent at 14 GHz and very pronounced in the
40 GHz data. There was .a slight degrade in the 36 GHz data but it was not nearly
as significant. It was understandable to see this effect at 14 GHz because given the
wavelength, the illuminated are was rather small, on the order of 6 wavelengths across.
One hypothesis to explain the large efficiency decrease in the 40 GHz measurement is
the very narrow nature of the predicted filter. This predicted response may be much
more sensitive to a finite illuminated area than the more symmetric responses seen
in the 14 GHz and 36 GHz design.
7.0.4
Future Work
This work provides a good foundation for application of GMRFs at microwave and
millimeter wave frequency bands. Further research with GMRF can fall into two cat­
egories: (1) advanced designs and design issues, and (2) experimental characterization
improvements.
Advanced designs with two-dimensional GMRF could include different shaped
gratings. My design in this dissertation used periodic square elements for the grat­
ing.
Round elements could be used to produce a GMRF with a response that is
independent of the polarization orientation of the incident field.
Also, a two di­
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95
mensional design could be developed that has different frequency response dependent
upon the polarization and could achieve this using rectangular elements.
Finally,
the rectangular elements could be converted to ovals to introduce some polarization
tolerance to the filter.
For the experimental measurements, my data showed that the spatially averaging
techniques is an excellent method to measure these filters. Extending this technique
to millimeter wave would be beneficial. Another option for the millimeter wave mea­
surement is to carefully analyze and adjust the collimating lens system to eliminate
the measurable phase tilt and reduce the noise in the system to a more acceptable
level.
One GMRF application of a is to serve as a calibration standard for other mea­
surement systems. Given the distinct frequency response that is directly dependent
upon incident angle, the measured output of a well characterized GMRF could be
used to align a test article within a collimating lens system.
Additionally, a well
characterized polarization dependent GMRF could be used to find the uniform phase
plane of the system by examining the location and amplitudes of the transmission
peaks and nulls. This frequency movement of the pass band with changes in the
incident angle were shown in the 36GHz design.
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96
C onclusions
7.0.5
The goals of my research were: (1) to analyze the electromagnetic characteris­
tics of GMRFs, (2) define a methodology to predict and design practical GMRFs
for fabrication, and (3) develop and apply an experimental protocol to validate the
analytical and numerical models applied in this study. In this dissertation I showed
the following:
1. A RCW method for one and two dimensional gratings with an arbitrary
number of layers. The RCW I implement in MATLAB was validated both numeri­
cally and experimentally as an excellent method to predict GMRF performance.
2. I developed a novel design techniques using a genetic algorithm and direct
search method for GMRFs. I was able to put arbitrary requirements on a GMRF
and develop designs with this algorithm that were validated numerically and with
experimentation.
3.
I conducted two different types of measurements of GMRFs.
At mi­
crowave I showed the use of a unique spatial averaging techniques to experimentally
characterize the filters. And at millimeter wave I showed to utility of a collimating
lens approach to the measurement.
4. I combined the analysis and design techniques with the demonstrated
experimental protocol to show a validated design technique for GMRFs.
Additionally, I contributed to the field of GMRFs by designing and developing
efficient GMRF with minimal number of layers out of low contrast dielectric materials.
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The unique combined genetic algorithm and direct search design method has shown
that a simple design of one grating layer design utilizing only 2 different dielectric
materials such as the 14 GHz design exists and can be designed. The two dimensional
design I developed had 2 designed band passes with both of them nearly symmetrical
in shape and were efficient. I added experimental data at microwave and millimeter
frequencies for transmission filters.
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B ibliography
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Publishing Company, USA, 1998
[2] Munk, B., "Frequency Selective Surfaces: Theory and Design", John Wiley and
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[3] Munk, B., "Finite Antenna Arrays and FSS", John Wiley and Sons, New York,
2003
[4] Hessel, A., Oliner, A. A., "A New Theory of Wood’s Anomalies on Optical Grat­
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[5] Lord Rayleigh, "On the dynamical theory of gratings", Proc. Royal Soc., (Lon­
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[6] Wang, S., Magnusson, R., "Theory and applications of guided-mode resonant
filters", Applied Optics, vol. 32, no. 14, May 1993
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[7] Fehrembach, A., Maystre, D., Aentenac, A., "Phenomenological theory of filter­
ing by resonant dielectric gratings”, J. Opt. Soc. Am. A, vol. 19, no. 6, June
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[8] Wang, S., Magnusson, R., "Multilayer waveguide-grating filters", Applied Optics,
vol. 34, no. 14, pp 2414-20, May 1995
[9] Jacob, D., Dunn, S. C~, Moharam, M., "Flat-top narrow-band spectral response
obtained from cascaded resonant grating reflection filters”, Applied Optics, vol.
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[10] Santos, J., Bernardo, L., "Antireflection structures with use of multilevel sub­
wavelength zero-order gratings", Applied Optics, vol. 36, no. 34, pp 8935-8, De­
cember 1997
[11] Jacob, D., Dunn, S. C., Moharam, M., "Normally incident resonant grating
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