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A MODE-MATCHING APPROACH TO DETERMINE
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DOUBLY—PERIODIC ARRAY OF APERTURES IN A
THICK CONDUCTING SCREEN

presented by

DERIK CLAYTON LOVE

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2/05 c:/CIFIC/DateDue.indd-p. 15

A MODE-MATCHING APPROACH TO DETERMINE
THE SHIELDING EFFECTIVENESS OF A
DOUBLY—PERIODIC ARRAY OF APERTURES IN A
THICK CONDUCTING SCREEN

By

Derik Clayton Love

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2004

ABSTRACT

A MODE-MATCHING APPROACH TO DETERMINE THE
SHIELDING EFFECTIVENESS OF A DOUBLY—PERIODIC ARRAY
OF APERTURES IN A THICK CONDUCTING SCREEN

By

Derik Clayton Love

The transmission of electromagnetic waves through apertures in a conducting
screen is a problem that has been examined many times before. Several techniques
have been used when the apertures are periodically arranged, and computational
approaches have allowed for the analysis of complex aperture shapes. However, past
literature is typically concerned with screens whose thickness is comparable to or
smaller than the aperture dimensions (i.e. thin screens). Further, the usual focus is
on transmission within a narrow band of frequencies.

The shielding properties of planar, periodic structures have been considered in
prior efforts. For a thick conducting screen of apertures, one approach for estimating
the shielding effectiveness is to treat the screen as an array of cylindrical waveguides.
This is referred to as the waveguide below cut—off principle. The result is dependent
on the attenuation constant of the aperture and the aperture length. This technique
is limited by the fact that it was developed to describe the attenuation of waves
propagating in an opening whose length is at least five times its width. In addition,
this approach is only relevant when the frequencies of interest are below the cut-off
frequency of the dominant waveguide mode.

This dissertation uses mode-matching to determine the shielding effectiveness of a
doubly-periodic conducting screen of apertures whose thickness can be several times
the aperture size. This is accomplished by modeling the screen as an array of cylin-

drical waveguides. This study considers rectangular and circular apertures, and the

fields within them are represented using waveguide modal fields. The reflected wave
above the screen and the transmitted wave below the screen are found by applying
Floquet’s Theorem, thereby exploiting the doubly-periodic nature of the screen of
apertures. After enforcing boundary conditions and building a system of linear equa-
tions, the system is then truncated to produce a matrix equation which is solved
using standard techniques. The shielding effectiveness of the screen is determined by
comparing the transmitted power to the incident power carried by a plane wave. It
is clear that as the thickness of the screen increases, the transmitted power is greatly
reduced at frequencies below the cut-off frequency of the dominant waveguide mode.
However, increasing the thickness also increases the attenuation of the higher-order
waveguide modes, leading to non-convergent solutions to the matrix equation. By
selectively eliminating higher-order modes from consideration, meaningful solutions
are found. Results also show the effect of increasing the number of Floquet modes,
varying the incidence angle, and changing the incident plane wave polarization.

The mode-matching results for rectangular apertures are very similar to data
obtained by applying the waveguide below cut-off principle. However, the mode-
matching approach can be used in cases where the frequencies of interest are above
the cut-off frequncy of the dominant waveguide mode, when higher-order modes will
begin to propagate. Comparisons are also made to previously published data using
the mode-matching approach. The data curves are in strong agreement in each com-
parison. However, it should be noted that the previously published data considers the
principal Floquet mode as the only propagating mode. That approach is inconsistent
with the definition of the propagation constant for Floquet waves. Experimental data
using commercial-grade aluminum honeycomb is also presented as another compari-
son for the mode-matching results. In each case, the curves are in good agreement in

describing the transition from strong shield to weak shield.

For my two boys, Andrew and Malcolm

iv

ACKNOWLEDGMENTS

I want to begin by acknowledging all of the guidance and support that has been
provided to me by various individuals during my graduate studies here at Michigan
State. I have never felt less than a tremendous amount of pride in the support that I
have received within the EM research group, within the ECE department, and across
the campus. I also want to thank all of the family, friends, colleagues and others that
encouraged me to pursue a doctorate in order to achieve my career goals. In fact,
a portion of my degree should go to my wife Amena. I can’t imagine that I would
have made it through this process without her supporting me in every way that she
possibly could. Being a husband, father, and graduate student is busy work, but I
could not have asked for a more supportive Spouse.

To my guidance committee members, Dr. Edward Rothwell, Dr. Dennis Nyquist,
Dr. Leo Kempel, and Dr. Byron Drachman, I want to thank you for doing just that:
guiding me. I am looking forward to a long and productive career as a researcher
and possibly as a teacher, and I know that my interest in both has been advanced
by my experiences in your classrooms, office hour sessions, etc. Before arriving at
MSU, I was told by a former co—worker that studying electromagnetics at Michigan
State is special because of the people, and I completely agree. I also want to thank
the faculty and staff of the ECE department, in particular Dr. Percy Pierre and Dr.
Barbara O’Kelly. Their intial interest in me is what attracted me to MSU, and their
constant pledge of financial and career development support is what kept me here

and helped me through. NASA Goddard Space Flight Center has been instrumental

in my progress due to their financial support of my doctoral studies. Finally, I would
like to recognize both Eric Pulley from Benecor, Inc. of Wichita, KS and Luke Young
from Plascore, Inc. of Zeeland, MI. My experimental work could not have been done
without them graciously agreeing to provide samples of aluminum honeycomb at little

to no cost.

vi

TABLE OF CONTENTS

LIST OF TABLES ................................. ix

LIST OF FIGURES ................................ x
CHAPTER 1

Introduction ..................................... 1
CHAPTER 2

Theoretical Formulation .............................. 4

2.1 Floquet Waves .............................. 4

2.2 Waveguide Fields ............................. 11

2.2.1 Modal Fields ........................... 11

2.2.2 Mode Emctions for Rectangular Waveguides .......... 16

2.2.3 Mode Functions for Circular Waveguides ............ 18

2.3 Enforcement of Boundary Conditions .................. 19

2.4 System of Linear Equations ....................... 21

2.5 Shielding Effectiveness .......................... 24
CHAPTER 3

Numerical Results ................................. 30

3.1 Rectangular Apertures .......................... 30

3.1.1 Calculations ............................ 30

3.1.1.1 Computing Slmn and 52m" ............... 30

3.1.1.2 Computing fpe and fph ................. 32 3

3.1.1.3 Computing Sp ...................... 34

3.1.1.4 Computing Plpmn, ngmn, lemn, and Qgpmn ..... 36

3.1.2 Discussion of Results ....................... 40

3.2 Circular Apertures ............................ 43

3.2.1 Calculations ............................ 44

3.2.1.1 Computing 51m" and ngn ............... 44

3.2.1.2 Computing fpe and fph ................. 46

3.2.1.3 Computing 5,, ...................... 48

3.2.1.4 Computing Plpmn, ngmn, lemn, and Qgpmn ..... 52

3.2.2 Discussion of Results ....................... 61
CHAPTER 4

Experimental Results ................................ 84

4.1 Setup .................................... 84

4.1.1 Sample Materials ......................... 84

4.1.2 Equipment ............................. 84

vii

4.2 Procedure ................................. 86

4.2.1 Low Frequency Measurements .................. 86
4.2.2 High Frequency Measurements .................. 86
4.3 Calculations ................................ 87
4.4 Discussion of Results ........................... 87
CHAPTER 5
Discussion of Mode Selection ........................... 94
CHAPTER 6
Conclusions ..................................... 109
APPENDIX A
Derivation of 2-D Fourier Series Representation of a Periodic Function in Skewed
Coordinates ..................................... 112
BIBLIOGRAPHY ................................. 119

viii

LIST OF TABLES

Table 3.1 Number of modes used to calculate the curves in Figure 3.3. . . . 62

Table 3.2 Cutoff frequencies for modes of square apertures. ......... 62

ix

Figure 2.1

Figure 2.2

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7

Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

LIST OF FIGURES

Doubly-periodic conducting screen of apertures with general cross-
sectional shape .............................

Unit cell for a doubly-periodic conducting screen of apertures with
general cross-sectional shape. ....................

Doubly-periodic conducting screen of apertures with rectangular
cross-section. .............................

Unit cell for a doubly-periodic conducting screen of apertures with
rectangular cross-section ........................

Comparison of transmission coefficient for screens of different thick-
nesses for normally incident TM polarized plane wave. ......

Comparison of transmission coefficient versus frequency for various
incidence angles when t=5.5 mm for TM polarized incident plane
wave. .................................

Comparison of transmission coefficient versus frequency for various
incidence angles when t=5.5 mm for TE polarized incident plane
wave. .................................

Comparison at t=4.4 mm between using 882 Floquet modes vs.
1922 Floquet modes. .........................

Comparison at t=4.4 mm between using 60 waveguide modes vs.
2 waveguide modes ...........................

Comparison at t=1.1 mm between mode-matching approach and
waveguide below cutoff formula ....................

Comparison at t=2.2 mm between mode-matching approach and
waveguide below cutoff formula ....................

Comparison at t=4.4 mm between mode-matching approach and
waveguide below cutoff formula ....................

Comparison at t=7.7 mm between mode-matching approach and
waveguide below cutoff formula ....................

Comparison at t=18.0 mm between mode-matching approach and
waveguide below cutoff formula ....................

Comparison at t=26.0 mm between mode-matching approach and
waveguide below cutoff formula ....................

Comparison at t=40.0 mm between mode-matching approach and
waveguide below cutoff formula ....................

28

64

68

69

70

71

72

73

74

75

76

Figure 3.15

Figure 3.16

Figure 3.17
Figure 3.18

Figure 3.19

Figure 3.20
Figure 3.21
Figure 4.1
Figure 4.2
Figure 4.3
Figure 4.4
Figure 4.5
Figure 5.1
Figure 5.2
Figure 5.3

Figure 5.4

Comparison between proposed mode-matching formulation and
mode-matching results of Chen using 0, = 20° near transmission
null (Wood’s anomaly). Assumes only (0,0) Floquet mode is pI‘Op—
agating. ................................

Comparison between proposed mode-matching formulation and
mode-matching results of Chen using 0,- = 80° near transmission
null (Wood’s anomaly). Assumes only (0,0) Floquet mode is prop-
agating. ................................

Comparison between proposed mode-matching formulation and
mode-matching results from Widenberg et a1 using 6,- : 0° .....

Comparison between proposed mode-matching formulation and
mode-matching results from Widenberg et a1 using 0,: = 30°. . . .

Comparison between proposed mode-matching formulation and
mode-matching results from Widenberg et a1 using 0,- = 60° near
transmission null (Wood’s anomaly). Assumes only (0,0) Floquet
mode is propagating ..........................

Doubly-periodic conducting screen of apertures with circular cross-
section. ................................

Hexagonal unit cell for a doubly-periodic conducting screen of aper-
tures with circular cross-section ....................

Equipment arrangement for taking low frequency measurements
(2-18 GHz) ..............................

Styrofoam board with window used to hold aluminum honeycomb
samples. The dark grey region is covered with foil tape .......

Equipment arrangement for taking high frequency measurements
(20—40 GHz) ..............................

Comparison of numerical and experimental results for rectangular
apertures ...............................

Comparison of numerical and experimental results for hexagonal
apertures ...............................

Condition number versus frequency for several values of thickness
when using 60 waveguide modes and 882 Floquet modes ......

Plot of shielding effectiveness for t=4.4 mm when varying the num-
ber of waveguide modes used. ....................

Plot of condition number for t=4.4 mm when varying the number
of waveguide modes used. ......................

Plot of shielding effectiveness for t=6.6 mm when varying the num-
ber of waveguide modes used. ....................

xi

77

78

79

80

81

82

83

90

91

93

97

98

100

Figure 5.5

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11

Figure 5.12

Figure A.1
Figure A.2

Plot of condition number for t=6.6 mm when varying the number
of waveguide modes used. ......................

Plot of shielding effectiveness for t=7.7 mm when varying the num-
ber of waveguide modes used. ....................

Plot of condition number for t=7.7 mm when varying the number
of waveguide modes used. ......................

Plot of shielding effectiveness for t=8.8 mm when varying the num-
ber of waveguide modes used. ....................

Plot of shielding effectiveness for t=9.9 mm when varying the num-
ber of waveguide modes used. ....................

Plot of shielding effectiveness for t=18.0 mm when varying the
number of waveguide modes used ...................

Plot of shielding effectiveness for t=26.0 mm when varying the
number of waveguide modes used ...................

Plot of shielding effectiveness for t=40.0 mm when varying the
number of waveguide modes used ...................

Doubly-periodic direct lattice ....................

Alternate description of doubly-periodic lattice ..........

xii

102

103

104

105

106

107

108
116
117

CHAPTER 1

INTRODUCTION

The transmission of electromagnetic waves through apertures in a conducting screen
is a problem that has been examined many times before [1H4]. Several techniques
have been used when the apertures are periodically arranged [5]-[12], and computa-
tional approaches have allowed for the analysis of complex aperture shapes [13]-[19].
However, past literature is typically concerned with screens whose thickness is com-
parable to or smaller than the aperture dimensions (i.e. thin screens). Further, the
usual focus is on transmission within a narrow band of frequencies.

The shielding properties of planar, periodic structures have been considered in
prior efforts [20]-[22]. For a thick conducting screen of apertures, one approach for
estimating the shielding effectiveness is to treat the screen as an array of cylindrical
waveguides. This is referred to as the waveguide below cut-off principle [23]. The
result is dependent on the attenuation constant of the aperture and the aperture
length. This technique is limited by the fact that it was deveIOped to describe the
attenuation of waves propagating in an opening whose length is at least five times its
width [24]. In addition, this approach is only relevant when the frequencies of interest
are below the cut-off frequency of the dominant waveguide mode.

This dissertation uses mode-matching to determine the shielding effectiveness of a
doubly-periodic conducting screen of apertures whose thickness can be several times
the aperture size. This is accomplished by modelling the screen as an array of cylin-
drical waveguides. This study considers rectangular and circular apertures, and the
fields within them are represented using waveguide modal fields. The reflected wave
above the screen and the transmitted wave below the screen are found by applying

Floquet’s Theorem, thereby exploiting the doubly-periodic nature of the screen of

apertures. After enforcing boundary conditions and building a system of linear equa-
tions, the system is then truncated to produce a matrix equation which is solved
using standard techniques. The shielding effectiveness of the screen is determined by
comparing the transmitted power to the incident power carried by a plane wave. It
is clear that as the thickness of the screen increases, the transmitted power is greatly
reduced at frequencies below the cut-off frequency of the dominant waveguide mode.
However, increasing the thickness also increases the attenuation of the higher—order
waveguide modes, leading to non-convergent solutions to the matrix equation. By
selectively eliminating higher-order modes from consideration, meaningful solutions
are found. Results also show the effect of increasing the number of Floquet modes,
varying the incidence angle, and changing the incident plane wave polarization.

The mode-matching results for rectangular apertures are very similar to data
obtained by applying the waveguide below cut-off principle. However, the mode-
matching approach can be used in cases where the frequencies of interest are above
the cut-off frequncy of the dominant waveguide mode, when higher-order modes will
begin to propagate. Comparisons are also made to previously published data using
the mode-matching approach. The data curves are in strong agreement in each com-
parison. However, it should be noted that the previously published data considers the
principal Floquet mode as the only propagating mode. That approach is inconsistent
with the definition of the prOpagation constant for Floquet waves. Experimental data
using commercial-grade aluminum honeycomb is also presented as another compari-
son for the mode-matching results. In each case, the curves are in good agreement in
describing the transition from strong shield to weak shield.

This dissertation describes all of the theoretical, numerical, and experimental
investigations involved in this study. Chapter 2 outlines the theory, including deriva-
tions of fields produced both by the Floquet waves and within the apertures. Also

included is a detailed description of the enforcement of boundary conditions on the

electric and magnetic fields, and a full explanation of how the boundary conditions
were used to create a system of linear equations. Finally, the computation of shielding
effectiveness will be shown. Chapter 3 outlines calculations and numerical results for
both rectangular and circular apertures, including computation of the integral expres-
sions that are described in Chapter 2. The experimental set-up and results are dis-
cussed in Chapter 4. An open set-up using horn antennas and a network analyzer was
used to conduct shielding measurements on samples of aluminum honeycomb. The
results confirmed the general behavior versus frequency that was expected. Chapter
5 discusses the considerations that were made in order to obtain the numerical results
in Chapter 3. This involves the truncation of the matrix equation in order to solve
for the unknowns. Most important was the observation that as the thickness of the
screen increased, the number of waveguide modes had to be reduced in order to get
convergent solutions to the matrix equation. The remaining chapters include conclu-
sions, references, and appendices for techniques that were critical to the development
of the theory.

Mode-matching applied to thin screens is not a new technique, and neither is com-
puting and/or measuring shielding effectiveness. However, using mode-matching to
determine the shielding effectiveness of thick screens across a wide range of frequen-
cies is a new direction. To the author’s knowledge, the comparison of mode-matching
numerical results to experimental data for aluminum honeycomb has not been done
until now. Also, the use of styrofoam boards covered in foil tape to analyze the hon-
eycomb samples is a new approach, but similar to using large metal plates with an

aperture to analyze composite materials.

CHAPTER 2

THEORETICAL FORMULATION

2. 1 Floquet Waves

The layout of a screen of apertures is shown in Figure 2.1. It is considered to be of
infinite extent in the x and y directions and have thickness t in the z direction. The
screen contains apertures that are arranged in a doubly-periodic fashion, with the first
axis of periodicity being the x-axis and the second making an angle (150 with respect
to the x-axis. From Figure 2.1, the screen has a periodicity of c along the x-axis
and d along the skewed axis. By representing the screen as an array of cylindrical
waveguides, each element of the array is represented by a cell and the center element
is regarded as the unit cell. The unit cell (S) is composed of two regions, the aperture
region (Q) and the conducting region (S —Q), as shown in Figure 2.2 (Note: Although
the figure shows a rectangular unit cell, there are other possible choices for the unit
cell geometry; the rectangular cell is used as an example to graphically indicate S and
Q). A plane wave is incident on the screen with wave vector It:i = :izkj, + 9k; — 2kg,
where

k; = k1 sin 0,- cos (23,-,
k; = k] sin 01‘ SlIl $15
k: 2 k1 cos 0,,

k1: (AM/(M161),

and 61 and #1 are the permittivity and permeability, respectively, in region I (z > 0).
Similarly, 62 and #2 are the permittivity and permeability in region 11 (—t < z < 0),

and 63 and ug are the permittivity and permeability in region 111 (z < —t). The

azimuthal angle (1),: is measured from the positive x-axis toward the positive y-axis,
while the zenith angle 9,- is measured from the positive z-axis toward the negative
z-axis.

If regions I, II, and III are assumed source-free, the fields in each region can be
expressed using a TE/TM decomposition of the fields [25]. Furthermore, Hertzian
potential functions can be used to determine the fields while still maintaining the

wave nature of the solution [25]. The Hertzian potentials that are used are

He(:r,y,z) = 2He(:r,y,z) (TMz Case),

I'Ih(:z:,y, z) = 2Hh(a:,y, z) (TEz Case).

Due to the longitudinal nature of the potentials, their satisfaction of the wave equa-
tion simplifies to only requiring the scalar component to satisfy the scalar Helmholtz
equation [25], or

(V2 + k2)He,, = 0.

The fields due to the potentials are

 

3H8 , 82 . .
E = V; 62 + Z (a + k2) He 4']pr X VtI-Ih, (2.1)
H 2
H = Vt%z—h' 'I' 2 (% + [62) H}, — jtdéé X VtIIe, (2.2)
where
A 3
Vt — V — 2E,
k2 = w2uc,

so that It? 2 w2ulel and kg = w2u3c3.

Because of the doubly-periodic nature of the array of apertures and the plane

wave excitation, the potentials in regions I and III must obey Floquet’s Theorem,
such that

Ileh(a:,y,z) = e—J'ke'e I e‘Jkei/nfha y,z 2).

Using the results from Appendix A, HZ, can be expanded in a Fourier series such that

   

     

 

 

Heh(x y,Z Z) = Z Hehmn(z j27rm ‘7“ L2” CSC ¢Oy + ‘7_ 272m COt $03].
171, 712—“)
Using this, He], can be rewritten as
n..<e,y, e) = e—jkixe ”ii/115m v.2 z)
= if unmet-1121’" .e.> — (2%" eseee— ——coteo + my
m,n=—oo
: Z I_I¢3hmn(z Z)6 _‘jam$e— jflmny (23)
m, n——oo
where
2 .
a... = 7"” + k; (2.4)
c
2 2m
fimn— — 1:; csc 450 — —:— cot $0 + 19;, (2.5)

and the indices m and n are the Floquet mode indices. Substituting Heh into the

Helmholtz equation leads to

2 2 2
(V2 + mud, (3— + a_ + a_ + k2) He,

81:2 By? 82 2
oo 2 . .
= Z (—a?,, - fimn + k2 + 823) Hehmn(z)e—]amxe—Jflm"y
= 0. (2.6)

If a function an(z) is defined such that

52
an(z) = (413,, — 63",, + k2 + 5—3) Hehmn(z),
(2.6) can be rewritten as
Z an(z e—jamxe—jflmny : 0. (2.7)

m ,n—-oo

The orthogonality of the exponentials can be used to simplify (2.7) by removing the
infinite summations. This is done by multiplying (2.7) by another pair of exponential

expressions and integrating over the unit cell region over which the Floquet waves are

defined. This leads to

[ejaaxejflaby [ Z an Z)8 ("jamxe—jfimny] d3 : O (2.8)

S m, n=—oo

Interchanging the order of summation and integration, (2.8) becomes

2 an(z )[f ej(aa — am):z:ej(fiab — 5mn)yd3] = 0, (2.9)
m, 712-00 3

The orthogonality of the exponentials allows the integral in (2.9) to be evaluated such
that

/8j(aa _ am)$ej(flab “7 ,an)yds —__—_ Aséaméb‘na
S

where A, is the area of the unit cell, and 6 is the Kronecker delta function [26].

Rewriting (2.9) once more leads to

As 2 (Sam6anmn(Z) : Asan(Z) : 0

m,n=—oo

With A, being a constant and assumed not equal to zero, it follows that

2

an(z) 2 (“air _ 181277.11 + k2 + 1‘) Hehmn(3) = 0i

822
or
32
(F3711: + 6—22‘) Hehmn(z) = 0, (2.10)
where
F2 = k2 — a2 — 2 .

The solution to the ordinary differential equation in (2.10) is

Hehmfl(z) : Cl’g-Irmne-_‘7I‘Trmz + aghmne+JPmnzi

which represents waves either propagating or evanescent in the +z and -z directions.

The potential expression in (2.3) can now be rewritten as

oo _. _. . 3)
Ileh(:1:,y,z)= Z aghmne jamxe Jflmnye'I'JI-‘Snnz

m,n=—oo

when 2 < —t, and

0° —' _° _' (1)
Heh($,y,2)= Z ajhmne Jamilie Jfimnye JanZ

m,n=—oo

when 2 > 0, where

12 2 2 2
Final = k1_ am _ Ian’

F92=%—a;—fi

mn mn’

and k1 = CIA/(#161) and k3 = w‘/(,Ll.3€3) represent the wave number in regions I and

+
ehmn

ehmn

III, respectively. Also, the coefficients a and a represent the complex Floquet

wave coefficients in regions I and III, respectively. Two observations are made here.

The first is that the sign of FEM must be chosen such that

§R{M} >0 and 3{M} <0

to ensure that propagating F loquet modes propagate away from the screen and evanes-
cent Floquet waves in each region decay away from the screen. The second observation

is that the potential expressions can be rewritten as

00 — — '1' -r + '1‘“) z
Ugh: Z aehmne J "m 6 J m" (2.11)
m,n=-oo
when 2 < —t, and
00 — '1' -r — T“) z
Ugh: Z agihmne J "m c J "m (2.12)
m,n=—oo

when 2 > 0, where Tmn = iam + gfimn and r = 5:2: + 32y + 22.

The Floquet wave fields will be matched to fields within a waveguide whose lon-
gitudinal axis is the z-axis, and thus only the transverse components of the electric
and magnetic fields will be needed. The transverse electric field is taken from (2.1)
as

He . -
Et 2 V??? + jwpz x th'lh, (2.13)

while the transverse magnetic field is taken from (2.2) as

all),

Ht = Vt 52

 

— jwei x th'Ie. (2.14)

With complete expressions for the Hertzian potentials in regions I and III, the electric

and magnetic field expressions for both regions can be determined. Substituting (2.11)

and (2.12) into (2.13) leads to the electric field, which is

(1)

(1) . .
00 quJngZe-]Tmn ' 1‘.

E53) : Z:

m,n:—oo

(1)
:FagfnnI‘gngn — wp(1)a$nn(rmn x 2)

(3)

 

 

Similarly, the magnetic field can be found by substituting (2.11) and (2.12) into (2.14),

which leads to

(1)

(1) 00 (1) . (3) .
H53) =2 Z $ahimnl‘rgl1'mn + wquagfimhmn x z) eTJanze_]Tm" ' r.
m,n:—oo (3)

 

 

In cases where signs, subscripts, and/or superscripts are stacked, the upper signs,
subscripts and superscripts correspond to region I (z > 0), while the lower signs,
subscripts and superscripts correspond to region III (2 < —t). Making some substi-
tutions for constants and taking :2 x H t, the transverse electric and magnetic fields

can be expressed as

00

Et = Z [:FAjfnn jinn _ AiitmnRIEnn 7 (2'15)
00 (1) (1)
2 x H. = 2: scAinYh‘3’Rl-f... + Ainw31Ri. , (2.16)

where
(1)
143:1"! : ainrgg T7117! V AS,

:l: :l: /
Ahmn : (up?) Cl’hmnTTnn A8,
3

(1)
- (3)
lerznn : 7A-mnlerrEn : R2mne:F]I-‘mnz3
83
Ri : (f-mn X ‘2)an : len32F‘7anza

lmn

10

(1) <1)
Ri : 1 e;jr§,?,lze—jrmn.r= Rmnezpjréilz,

m" «A:

 

 

1 .
R... = ”97m" "3
me
len : (ff-mu X 2)Iimna (217)
R2mn : 'f-mnanna (218)
.. Tmn
7Imn : a
Tmn

Tmn : ITmnl = V 03,; + 53mm

and the wave admittances are given by

 

(3) _ (3)
Y8 (1) ’
(3)

mu

(1)

(1) (3)
1453’ _ m" .
wM1)

(3)

2.2 Waveguide Fields

2.2.1 Modal Fields

To determine the fields within the screen, a modal expansion will be performed using

Hertzian potentials. The Hertzian potentials used for this expansion are

Hep(:c,y, z) = 2H8p(a:,y, 2) (TM, Case),

th(:1:,y, z) = illhpcr, y, z) (TEZ Case),

11

where p is an index to represent a particular waveguide mode. Due to their longitu-

dinal nature, these potentials must satisfy the scalar Helmholtz equation, or

(V2 + k§)He,,p = 0, (2.19)

where kg = (22222262, and 62 and #2 represent the permittivity and permeability, re-
spectively, in the aperture regions within the screen (region II). The waveguide modal

electric and magnetic fields in terms of the Hertzian potentials are

 

311,, 62 . .
Ep = Vt 62 +2 g + kg Hep + JUJ/JQZ X Vtth1 (2.20)
2
Hp th L161:p+ 2(66— 2 +k2) th— jwfgi X VtHep- (2.21)

The separation of variables technique can be used to express the solutions to (2.19)
as [25]

Hep : wrap“): y)e¥jkzz,
Hm, = who, new”,

where the mode functions ([26,, and 21),”, must satisfy
vfwehp + kzweh, = 0, (2.22)

and
k3 = k3 — k3. (2.23)

To complete the satisfaction of the Helmholtz equation, Z (2) = eszkzz satisfies

62Z

2 _
07 +kZ—0.

12

As was the case with the Floquet wave derivation, the fields of interest within
the waveguide are the fields transverse to the direction of wave travel, or the fields
transverse to the longitudinal axis of the waveguide, which is the z-axis. Therefore,

(2.20) and (2.21) can be used to obtain the transverse fields as

 

 

8H6 , A

Etp 1' Vt 82p + mez X thhp) (2.24)
an

H,,, = v, a :1" — 32.2622 x thep. (2.25)

For the TM, case, the potential Hep can be used in (2.24) and (2.25) to express the

modal transverse field components as

E, = acetpezfjkzz, (2.26)
Htp : _}/ep(2 X etp)€:ijzZ, (2.27)

where
etp($a y) = jkzvtwepcra 3!), (2'28)

and the complex wave admittance is

 

A modal expansion of the total transverse electric field within the waveguide in terms

of sinusoidal functions and complex coefficients AP and 3,, leads to

E? = 21/1,, sin(kzz) + B,, cos(kzz)]Etp (2.29)
p

13

where

 

Etp _ 8t? a
fpe
fpe = fetp ' etp d3, (2.30)
9

and fpe is used to make the modal fields orthonormal. For a modal expansion of the

transverse magnetic field (2 x H g), (2.26) and (2.27) can be used to reason that
. T 8 A T
chgEt 2 BER: x Ht ]. (2.31)

This is accomplished by taking the cross product of 2 and H tp, and then taking a

derivative with respect to 2 such that

(796; (2 x Htp) = — (—Yep(2 x 2 x tp)e:‘tjkz )

 

= :lzjwfzethTjkzz

: jWCQEtp.
Substituting (2.29) into (2.31) leads to

jwézEzw = jLUCQ Z[AP SinUCzZ) + Bp COS(kzz)]EtP
p
,LUEQ .

: ZJk—kz[Ap31n(kzz) + Bp COS(kzZ)]Etp
p Z
1 .k2 -

= — ZJ—kz[Ap sm(kzz) + Bp COS(kzZ)]Etp
772 p k2

a .
= 5;[ZXHtTl

14

where 772 represents the intrinsic impedance in region II. The solution for 2 x H f is
A T 1 .k2 .
z x H, = —E ij—[Ap cos(kzz) — Bp Sln(kzZ)]Etp. (2.32)
p Z

For TE modes, a similar approach can be used to determine the fields. th can

be used in (2.20) and (2.21) to express the modal transverse field components as

Htp : ¥htpe¥jkzz, (2.33)
E”, = z,,,(2 x htp)e:ijzz, (2.34)

where
htp($a y) :jkzvtwhp($ay)1 (235)

and the complex wave impedance is

 

The modal expansion for the total transverse electric field is
E? = {[0, sin(kzz) + Dpcos(k.z))E,,, (2.36)
p

where
_ th(2 X htp)

 

Etp — a
V fph
fph = [th(2 X htp) ' th<£ x htp) d8, (2-37)
{2

Cp and Dp are complex coefficients for the TB modes, and fp), is used to make the

fields orthonormal. To obtain the magnetic field expansion, it can be shown using

15

(2.33) and (2.34) that

6

Jam (:2 x H?) = ——(E‘f). (2.38)

82

This is accomplished by taking the cross product of 2 and H tp, and then multiplying

by jam such that

jwpg (2 X Htp)

Substituting (2.36) into (2.38) leads to

6
5;

The solution for 2 x H tT is

2xH3‘

T

E)

t

— (2 x (3.532))

raw/12¢ x h...) eszkzz

8 .
5; 2,10,, sm(kzz) + DP cos(kzz)]Etp
Z kz[Cp cos(kzz) — DP sin(kzz)]Etp
p

jUJ/Lg (2 x HE).

:—J— 2 kz[C'p cos(kzz) — Dp sin(kzz)]Etp
Z kz[Cp cos(kzz) — DP sin(kzz)]Etp

:1 :j—k—Z[Cp cos(kzz) — DP sin(kzZ)]Etp- (2-39)
772 p k2

2.2.2 Mode Functions for Rectangular Waveguides

For the case involving rectangular apertures, the screen is modeled as an array of

rectangular waveguides. Many textbooks have analyzed the rectangular waveguide,

16

including [25]-[27]. Some details of determining the mode functions are repeated here.

Consider a rectangular aperture defined such that —% S a: S g and —g g y g g,
and filled with a material with permittivity 62 and permeability [1.2. To properly
represent the fields within the aperture, the mode functions $8,, and 1pm, must satisfy
(2.22) and (2.23). In addition, each mode function must also satisfy the appropriate
boundary condition. For TM modes, that is the homogeneous Dirichlet boundary

condition, or

$61) = wepctvy) : 0: 117,3] 6 P,

where I‘ is the contour defining the boundary of the waveguide. Also, the index p
refers to the pt” mode, whether it is a TM or TE mode. The well-known result is

that 1126,, can be represented as

([28,, = sin [191C (:1: + 3)] sin [Icy (y + 3)] , (2.40)

where

and

k§=k§—k§=k§+k§.

Just to clarify, the index p refers to the 1)“ mode combination of the indices m and
n for TM and TE modes.

For TE modes the requirement is satisfaction of the homogeneous Neumann

17

boundary condition, or

air/”1p _ at()pr _
an — an ($,y)—0, $,yEF,

 

 

where n is the variable for the normal direction to I‘. The well-known result is that

112;”, can be represented as

b
7,0,”, = cos [k3, (x + 3)] cos [Icy (y + 5)] , (2.41)
where
k. = M, m = 0,1,2,3,...
a.
kyzfl, n=0,1,2,3,...
b
and

k§=k§—k§=k§+k§.

Also, m and 72. cannot be simultaneously equal to zero for TB modes.
2.2.3 Mode Functions for Circular Waveguides

For the case involving circular apertures, the screen is modeled as an array of circular
waveguides. Many textbooks have analyzed the circular waveguide, including [28] [27].
Some details of determining the mode functions are repeated here.

Consider a circular aperture defined such that 0 S r S a, and filled with a material
with permittivity 62 and permeability [1.2. To properly represent the fields within the
aperture, the mode functions 1126p and 1,12,”, must satisfy (2.22) and (2.23). In addition,
each mode function must also satisfy the appropriate boundary condition. To satisfy

the homogeneous Dirichlet boundary condition for TM modes, the well-known result

18

is to define 2128p as

 

([43,, 2 JC (X: 7*) [AC cos(c¢) + BC sin(cd>)], (2.42)
where xcd is the d“ zero of the ct" order Bessel function of the first kind. Also,

42:43—14:43.

 

kr : Xcd
a

To satisfy the homogeneous Neumann boundary condition for TB modes, the

well-known result is to define whp as

whp = JC (2%”) [AC cos(c¢) + BC sin(c¢)], (2.43)

where x’cd is the d“ zero of the derivative of the ct" order Bessel function of the first

 

kind. Also,
kgmn : kg _ kgmn : kfmnz?
kl : XIX!
r a °

2.3 Enforcement of Boundary Conditions

In order to relate the Floquet wave and waveguide coefficients, boundary conditions
are enforced at the upper and lower surfaces of the screen, which correspond to
z = 0 and z = —t, respectively. This is accomplished by taking the transverse field
expressions and equating them at the boundaries. Using (2.15), (2.29), and (2.36),

the continuity of transverse electric field within the aperture region leads to

hmn

E; + ' {—21ijan — 4+ len] = ZGpEtp, r e o (2.44)
m,n P

19

when 2 = 0 and

_. _ (3) _ (3lt
Z [AemnRzmne jrmnt _ Ahmanmne jrmn

m,n

= Z [—Fp sin(kzt) + G, cos(kzt)] Etp, r e 9 (2,45)
12

when 2 = —t. F], and G,p are used here to represent the waveguide coefficients for
both TM and TE modes, given the similarity between (2.29) and (2.36). It is assumed
that the transverse electric field outside of the aperture but within the unit cell will
go to zero at z = -t due to the presence of a perfect electrical conductor in that
space. The quantity E; represents the transverse electric field in the plane 2 = 0 due

to the incident plane wave, and it is defined as

E1: E3(§Iex + gey)e_jk0($ cos (bisin 0,- + ysin ¢,sin 61-), (2.46)

where e...c and 8,, are used to describe the transverse field components and E3 is the
incident electric field amplitude. Using (2.16), (2.32), and (2.39), the continuity of

transverse magnetic field within the aperture region leads to

877171

2 x H; + Z [A+ 14.0%,,” + AhmnY(1)R1mn] = E [32312,] , r e o

 

p 772
(2.47)
when 2 = 0 and
- (3) . (3)
Z AemnY€(3) R2mne_Jant _ fimnYh(3)R1mn€_]ant
=2 {—FP cos( (k t) + Gp sin(k t)] Etp, 1- E Q (2.48)
p

20

when 2 = —t. The quantity 2 x H 2 represents the transverse part of the incident

magnetic field in the plane 2 = 0, and it is defined as

2 x H: ___ (3)111 _ ihy)-§—ée—jk0(x cos (bisin 6.- + ysin (i). sin 6,), (2.49)
1

where hm and hy describe the transverse field components and 171 is the intrinsic

impedance in region I. Also, up is defined such that

.192
Up = Jig—
for TM modes and
.kz
VP — J};

for TE modes.

2.4 System of Linear Equations

The F loquet waves are orthogonal such that
[len ' Rpm'n'ds : 611’6mm’6nn’a
s

and 6 is the Kronecker delta function [26]. Having enforced the boundary conditions,
the orthogonality of the Floquet waves can be used on (2.44) and (2.45) to solve for
the Floquet wave coefficients. Multiplying (2.44) by Rfmw and integrating over S
gives

Aim 2 [E1 - R'fmnds — 20,, [13,, - R‘fmnds. (2.50)
s 1’ n
Multiplying (2.44) by R‘Sm’n’ and integrating over S gives
Ari-rm = /E: ' Rgmnds — Zap/EU) ° R’Emnds' (251)
s P n

21

Multiplying (2.45) by Ring”, and integrating over S gives

- 3
Ah—mn = ejrinzlt Z [Fp sin(kzt) — Gp cos(kzt)] lEtp - R’fmnds. (2.52)
P n
Multiplying (2.45) by R3“, and integrating over S gives
- 3
24;," = ejrinllt 2 [—F,, sin(kzt) + Gp cos(kzt)] [Etp - Rgmnds. (2.53)
p n

The modal waveguide fields are orthogonal [28][29] such that
[Etp ' EthdS = 6ppl.
n

In fact, they are orthonormal due to the normalization that is applied by (2.30)
and (2.37). The waveguide field orthogonality can be used with (2.47) and (2.48) to
obtain additional expressions involving the Floquet wave coefficients and waveguide

coefficients. Multiplying (2.47) by Etp: and integrating over 9 gives
— 2 / E.p-[A;tnnve<1>R2mn + Army/gnaw] ds = fiFp+/[2 x H§]-E.,,ds. (2.54)
m,nQ 772 Q
Multiplying (2.48) by Etpr and integrating over 9 gives

emn

' 3
Z / e-JrlnlztEtp. [Afman’len —- A" Yengmn] ds
m,nQ

= ;£[Fp cos(kzt) + G,, Sin(kzt)l- (2-55)
2

Making some substitutions, (2.50)-(2.55) can be rewritten as

Afil-mn : Slmn _ ZGPPIpmna (2.56)
P

22

Ajr-nn : 32m?! _ Z GpP2pmm (2.57)
P

_ ‘ (3
Ahmn = ejrmilt 2le sin(kzt) — G'p cos(kzt)]P1pmn, (2.58)
p

_ __ jF(3)t _ .

Aemn — e m" :1 PP sm(kzt) + G,D cos(kzt)]ngmn, (2.59)

p
U
77—:FP : — ZlA:mnYe(l)Q2pmn + AtmnYIfI)lemnl — Sp, (2-60)
m,n

_‘ (3) _ _
Z 6 erntlAhmnYh(3)Q1Pmn _ Aemn3/e(3)Q2Pmnl : glFP COSUCzt) + GP Sin(kzt)la
2

(2.61)
where

Slmn — fE; ' Rimnds, (2.62)
S

S2mn — /E; ' R2mndsa (2.63)
S

5,, = /(2 x H;) Etpds, (2.64)

0

Plpmn — [Etp Rimnds, (2.65)
0

P2,"... _ f E.,, Rgmnds, (2.66)
0

lemn : [Etp ' lendsa (2.67)
0

Q2pmn : [Etp ' R2mnd3' (2.68)

Eliminating the Floquet coefficients by substituting (2.56) and (2.57) into (2.60), and

(2.58) and (2.59) into (2.61), and making some additional substitutions, leads to

Fp — Z: Gil—{pi 1‘ jp,

23

(2.69)

FPWPC + Gpwg — Z[F,~Tp,- — GiUpi] = 0, (2.70)

where

JP 2 —‘Sp — Z[Y;(1)S2an2pmn — Y/fl)Slan1pmnla

m,n

Kpi Z ZlYIEI)Plianlpmn + n(l)P2ian2pmnla
m,n

sz' Z Sin(kzt) Z[n(3)P2ian2pmn + Ye(l)Plian1pmn]a

m,n

Upi : cos(kzt) Z[Ye(3) P2ian2pmn + K(I)Plianlpmnla

- 772
J =—J,
:0 up?
- 772
Kiz—Ki,
:2 Vp :0

u
Wcz—acos kzt,
p 772 (P)

s V -
WP :2 l sm(kzt).
772

Expressions (2.69) and (2.70) can be used to construct a square matrix equation of
the form Ax = b, where the unknowns are the waveguide coefficients. Once the
waveguide coefficients are known, (2.56)-(2.59) can be used to compute the Floquet

coefficients.

2.5 Shielding Effectiveness

Shielding effectiveness is defined as the ratio of power carried by the Floquet waves
in region III to the power carried by the incident plane wave in region I. Computing
the power carried by the Floquet waves requires determining the power transmitted
through the unit cell. This is computed by integrating the time-average Poynting
vector,

1 , .
P=S/§R(Ex H )-zds. (2.71)

24

Using vector identities and passing the dot product inside of the 3% expression, (2.71)
can be rewritten in terms of the transverse components of the electric and magnetic

fields, leading to

=—/-2—32 (2 x H*))ds. (2.72)

Recalling (2.15) and the complex conjugate of (2.16) for the fields in region 111

gives
00
Et 2 Z [Ac—mnR‘Z—mn — hman—mn] ’
m,n=—oo
0° (3) 3
z x H; = Z [— .4;qu prq + A;,,‘;Y‘ >*R.;;;,].
p,q=-oo
Writing out the dot product in (2.72) gives
A a: 00 —* It —* — 3 *
Et ' (Z X Ht) 2 Z [AemnAepq- mn Rque( YB) + AhmnAhpq lmn Rlpqh ( )

m,,np,q=—oo

(3)* — —* — —ar 3*
Aemn Ahqu'Zmn Rlpqh _AhmnAepq 1mn R2pq}/e( ) ] ' (273)

Substituting (2.73) into (2.72) leads to four integral expressions. The integrals can

be evaluated using the orthogonality properties of the Floquet waves, giving

. 3 _- 3* ' (3) _ 3)*
[R2mn’ 7;.qu : eJPSane JFian/Rzmn , 13;?qu : (3]an Ze JIM“ nz5mp5nq,
S

lpq lpq

' 3 t 3)
S/Rl—mn' 12““ d3_ — eijnz e_.7F$nh 2 f len' R“ d3_ — ejrmnze_j11(h*z6mp6nq,
S

(3) z

__ _* _. (3)*z
/R2mn ' Rlpqu— — ejI-‘mn 6 ijn zS/R2mn ' lpqd :0,
S

(3) z

_ —¢ _ F _ F(3)*Z * _
/ Rm, .122,qu _ 6] mu 6 J / R1,... - 122,,qu _ 0.
S S

25

I l

Equation (2.72) can now be rewritten to reflect transmitted power such that

00 .

1 * ‘ 3 _ 3 t
Ptrans = —2§R Z (lA_ l2Ye(3)* + lAhmnIZYhm) )ejrgnZIZe ‘71“:an ' (274)

emn
m,n=-oo

For prOpagating F loquet modes,
F532 = k; — a?" — 6,2,", > 0, F312, is real and positive,
while for evanescent Floquet modes,
F53? 2 kg — a; — 5,2,," < 0, r513, is imaginary and negative.

The Floquet wave admitttances Ye”) and Yhm are defined as

 

(3) (.063
Ye (3) ’
mn
Y<3> F533.
h 01/13

and w, 63, and #3 are always real and positive. Therefore, for evanescent modes, Ye“)
and Yhm are imaginary, so these modes provide no contribution to the series, and

(2.74) can be rewritten as

1 _ _
Ptrans : _"2— [Z (lAemnl23/e(3) + lAhmnl2Yh(3))] a (275)

m,n

where m and n correspond to the indices of propagating modes. The power carried
by the incident plane wave is determined by substituting (2.46) and (2.49) into (2.72),

leading to

Rnc : —/
5

1‘2

/ [(eyhx—erhy) 0 ds
S 7h

 

§R[E§- (2 x H;'*)] ds = —

 

NIH
NIH

26

_ —A3 COS 6i

'2
E' . 2.76
2m 0 < >

Shielding effectiveness (SE) is computed based on the ratio of transmitted power to
incident power so that

SE”; = ~1010g10 (

Prans
‘ ) . (2.77)

Pine
The transmission coefficient (T) is the negative of the shielding effectiveness in dB

such that

 

(2.78)

Prans
TdB = —SEdB = 1010g10 ( t ) .

P inc

27

>'~<1

 

W m
i r x

Figure 2.1. Doubly-periodic conducting screen of apertures with general cross-sec-
tional shape.

28

 

 

 

 

Figure 2.2. Unit cell for a doubly-periodic conducting screen of apertures with general
cross-sectional shape.

29

 

CHAPTER 3

NUMERICAL RESULTS

3.1 Rectangular Apertures

The layout of the screen for the rectangular aperture case is shown in Figure 3.1,
where the skew angle of the array, (120, the x-periodicity c and the skewed periodicity
d are shown. Figure 3.2 shows the unit cell, where the aperture dimensions are a and

b, and the unit cell dimensions are u and v.

3.1.1 Calculations

In order to compute the shielding effectiveness of a screen of rectangular apertures,
a total of 14 integral calculations must be made. Four of them, (2.30), (2.37), (2.62),
and (2.63), refer to TM or TE modes, or neither. By contrast, (2.64), (2.65), (2.66),
(2.67), and (2.68) account for the other ten because each must be calculated for both
TM and TE modes within the aperture. However, some similarities between the
different formulas will lead to some redundancy in the calculations.

3.1.1.1 Computing Slmn and 52".”

Substituting (2.17) and (2.46) into (2.62) leads to

Slmn 2 [E2 ' lends
S

6:3,an — eyam E6
”a?” + 6,2,", \/A—s
X [fa-714160 cos at: sin 9. - am)e-J'y(ko sin r157 sin 97 — fimnlds
S

 

 

= (eff—3.11;?)é’i—EFGVG) <3“

30

 

where

sin [3:(ko cos ()5,- sin 0,- — am)]
F =

(:5) (k0 cos ab,- sin 0,- - am)

 

‘)

__ sin [y(k0 sin ()5,- sin 0,- —- flmn)]
G(y) _ (k0 sin ¢,— sin 0, — 577m)

 

Equation (3.1) is true for all combinations of the Floquet mode indices m and 11 except

for three. If m = n = 0, (2.62) becomes

 

51m, = 875” ‘ €me 133%. (3.2)
7013.. + [33,...

If m = 0 only, (2.62) is expressed as

Slmn : (63.6mm _ eyam) 2EOU’G (B) . (33)

\/a3n+fifm W” 2

And if n = 0 and (150 = 90°, then (2.62) is found to be

Slmn = (ezflmn _ eyam) 2EOUF (E) . (34)

i/afnflfm WW 2

Using similar reasoning, substituting (2.18) and (2.46) into (2.63) leads to

 

 

 

S2771" : /E: ' REmnds
3

exam + eyflmn E5

2 («aw/33,... ) W17

X [e—j$(ko cos d).- sin 0.- — am)e-J'y(ko sin <fn- sin 01-— 67”,.)(13

3

(67355213.?) 35—14969 ““5”

 

 

31

Equation (3.5) is also true for all combinations of the F loquet mode indices m and n

except for three. If m = n = O, (2.63) becomes

 

52"... 2 (exam + eyfim") Egm. (3.6)

V0371 + 337m

If m = 0 only, (2.63) is expressed as

 

 

_ exam + eyfimn 2E3u (3)
S2mn — ( \/agn + ’63”! ) M G 2 - (3.7)
And if n = O and $0 = 90°, then (2.63) is found to be
_ exam + eyfimn 2E3v (u)
S2mn '— ( “0112" + 1812"” ) m F 2 . (3.8)

3.1.1.2 Computing fpe and fph.

To evaluate fpe, the normalization integral for TM modes in the aperture, (2.40) is

substituted into (2.28), leading to

 

kz(x+%>ismlky(v+%>l
wlkxnsflwslwéfll

which is then substituted into (2.30) such that

etp = jkz (ikr cos

0

'- 7
= —k§ / k: cos2 [km (m + g” sin2 ky (y + 3) ds
9 _ -
493/19: sin2 [km (a: + g)] cos2 16,, (y + %)l ds
0 . .

= _kg (kg + kg) ”1” (3.9)

 

 

32

 

This result is found by using basic substitution to evaluate the integrals and discarding
any terms that involve the sine of an integer multiple of 77, which is always zero.
To evaluate fph, the normalization integral for TB modes in the aperture, (2.41)

is substituted into (2.35), leading to

th (73 X htp) : .719th? (fig?! cos [km (£13 + 3.)] Sin [kg (y + 3)]
_ 32k: sin [km (a: + 3)] cos [kg (y + g”),

which is then substituted into (2.37) such that

 

fph : /Zh,,(2 x hip) - th(2 x htp)ds '
n
= —w2,u2 / k: cos2 [km (a: + g)] sin2 ka (y + g) ds
9 ..

—w2,u2 / k: sin2 [kgc (x + E» cos2 lky (y + g) ds
L
n .

99
4 .

 

 

= —w2p2(k§+k3) (3.10)
This result is also found by using basic substitution to evaluate the integrals and
discarding any terms that involve the sine of an integer multiple of 7r, which is always
zero. Equation (3.10) is valid for all TE modes in the aperture except for those

involving either km 2 O or 19,, = 0. If k,D = 0, fph becomes

f,,, = / th(2 x h,,,) - th(2 x h.,)ds
Q

-w2u2 / k: sin2 [16,, (y + 3)] ds
9

2ab
313'

= —w2u2k (3.11)

33

If Icy = O, fp), becomes

f,,. = / th(2 x h,,,) . th(2 x h,,,)ds
fl

: —w2p2/k: sin2 [k1, (a: + 3)] d3
9

= —w2u2k332§. (3.12)

3.1.1.3 Computing Sp

Using (2.26), (2.28), (2.40), and (2.49) in (2.64) and making some substitutions gives

53’“ = /(2 x H;') - Etpds
Q

. a b
= Gogeuflevyy 3111 [km (a: + 5)] cos [Icy (y + 5)] ds
_ “93$ ’0 y 9)] ' 9
Huge ey cos [km (a:+2 S111 [kg (y+2)] ds
— GG(E)G 9-HH(9)H 9 (313)

for TM modes, where

Go = —]szO hxky,
771V fpe
H0 2 Mhykx,

771\/f—pe

_ —kxeu$$[1 — 2sin2(k$a:)] + kme‘umx

 

 

 

v evyyu —— 2sin2(k y)] — v e—vyy
G = y y y , 3.15
111(1)) : uxe [1 28m ($33)] uxe ’ (3.16)

u§+k§

34

—kyevyy[1 — 25in2(kyy)] + [rye—”31y

 

112(9) = 113+ k3 , (3.17)
ux = —jko cos Q5,- sin 6,, (3.18)
'03, = —jk0 sin (b,- sin 61'. (3.19)

There are a couple of special situations that should be noted for making these cal-
culations. If either um = 0 or k1, = 0, the effect on (3.14) and (3.16) can be found

directly from those expressions. However, if um = km 2 0, (3.13) becomes

32’“ = —H0aH2 (g) . (3.20)
Similarly, if vy 2 kg = O, (3.13) simplifies to
a
33"“ = GOG1 (5) b. (3.21)

The effect of either 12,, = 0 or kg = 0 can be found by directly evaluating (3.15) and
(3.17).
For TE modes, using (2.34), (2.35), (2.41), and (2.49) in (2.64) and making some

substitutions gives

33E = [(2xH§)-E,,ds
Q

r .
= —G0/eu1‘xevyy sin [km (:16 + g» cos kyn (y + g) ds

9 l .

u :1: v y a - F b 1
—H0/e-Tey cos kat x+§ smkan y+-2- ds

9 .

—GOG1 (g) 02 ('3‘) — HoHl (g) H2 (2;) , (3.22)

where 0(5), 02(3)), H1(3:), H2(y), ux, and 2),, are defined by (3.14), (3.15), (3.16),

 

 

35

(3.17), (3.18), and (3.19) respectively, just as they were for the TM case. The change

is in the definition of Go and H0, where

Go = J——————"‘Z"”E°h.k.,
"Iprh
'kzZ .-
H02] ”BO/2,3,,

Tin/f7);

for TB modes. Similar to the TM case, if um = km 2 0, (3.22) becomes

b
SEE Z —H0aH2 (é) .
If 21,, 2 kg = 0, (3.22) simplifies to

5;”? = 43001 (521-) b.

3.1.1.4 Computing Plpmn, ngmn, lemn, and Qgpmn

(3.23)

(3.24)

In order to evaluate Plpmn, ngmn, lemn, and Qgpmn, the same computations that

were carried out previously for SI, are repeated. In fact, the formulas to be shown will

include the functions 01(23), G2(y), H1(:c), and H2(y), which are defined by (3.14),

(3.15), (3.16), and (3.17), respectively.

To calculate Plpmn for TM modes, substituting (2.4), (2.5), (2.17), (2.26), (2.28),

and (2.40) into (2.65) gives

1pmn

PTM = f E", ~R‘1'mnds
Q

. b ,
= —Go/eu1$evyy sin [km (:1: + 525)] cos ky (y + 2)
n L ..

+Ho/euxxevyy cos [k1, (a: + 3)] sin ky (y + 3)]
n L- .1

 

36

 

d3

d3

 

= m(,).2(g)_..(g).(g)

where
jkzkyam

G = ,
° «Wig/a3 + 32,...
ijkI/an
H0 2 ,
mam + 3.2....

”at : Jam,

 

vy : jfimn-

If u,c 2 km, 2 0, (3.25) becomes

TM
Plpmn

= HoaH2 (g) .

If 1),, = kg“ = 0, (3.25) simplifies to

PTM

1pmn

= —G001 (g) b.

(3.26)

(3.27)

(3.23)

(3.29)

For TE modes, Plpmn is found by substituting (2.4), (2.5), (2.17), (2.34), (2.35), and

(2.41) into (2.65), giving

P575111 : [Etp ' Rimnds
Q

= GO/euflevyy sin [1:3, (as + g)] cos [kg (y + 3)] ds
{2
u :1: v a - b
+H0/e x e 313’ cos [km (:1: + 2)] sm [Icy (y + 5)] ds

0

= GOG1 (g) 02 (g) + HOH1 (g) H2 (3) ,

37

(3.30)

 

where
j k2 ka: am Z hp

\/—A3(/fph\/03n + 33.;
H0 2 jkzkyflmnzhp
«713/3». M. + 53,...
If at = 16;; = 0, (3.30) becomes

0

b
1317,;an = HoaH2 (5) . (3.31)
If 12,, 2 kg = 0, (3.30) simplifies to
P55... = 0001 (g) b. (3.32)

To calculate ngmn for TM modes, substituting (2.4), (2.5), (2.18), (2.26), (2.28),
and (2.40) into (2.66) gives

1);):an : [Em ' Rgmnds
n
__ u a: v y - a b
—— GO/e 3‘ e y sm [19,610+ -)] cos [kg (y+—)] ds
9 2 2
+H0/euxxevyy cos [kI (a: + g» sin [Icy (y + 3)] ds

0
= G001 (g) 02 (g) + mm (‘23) H2 ('3') , (3.33)

where
= jkzkyfimn
V A3 V fpe a?" + (8721111,

jkzkzam

x/A—.(/f;(/a3n+ 3...’

 

0

 

H0:

38

and (3.39) and (3.40) still apply. If um 2 .163C 2 0, (3.33) becomes

1327,34,, = Home!2 (g) . (3.34)
If vy = Icy = O, (3.33) simplifies to

P5X" = (1001 (g) b. (3.35)

When calculating ngmn for TE modes, substituting (2.4), (2.5), (2.18), (2.27),
(2.35), and (2.41) into (2.66) gives

P” = / E,p . R’fmnds
Q

2pmn
= uxsz: v y - 9)] 9
Goa/e e 1’ sm [163(3) + 2 cos [16,, (31+ 2)] (18

+H0 / euxxevyy cos [3, (a: + g)] sin ]k, (y + 3)] ds
9

 

 

a b a b
= 0001 (5) 02 (5) + HoHl (5) H2 (2) , (3.36)
where
G jkzkzamth
0 = a
m" fph V (1727; + fig";
H0 : jkzkyfimnzhp

mM(/a3n + 33.3

If um = km = 0, (3.30) becomes

b

39

If vy = Icy = 0, (3.30) simplifies to
P223, = GOG1 (g) b. (3.38)

In order to compute lemn and Qgpmn for TM and TE modes, the formulas are
almost identical to what has been shown for Plpmn and ngnm. The only difference is

that (3.39) and (3.40) are defined such that
um = —jam, (3.39)

’Uy : _jfimna (3.40)

which are the complex conjugates of the versions used in computing Plpmn and ngm.

Otherwise, the formulas and constants are exactly the same.
3.1.2 Discussion of Results

Figure 3.3 shows a plot of transmission coefficient versus frequency, where each curve
represents a different value of screen thickness. The transmission coefficient is the
negative of the shielding effectiveness in dB. The aperture is square shaped with di-
mensions a = b = 3.6 mm, while the unit cell is also square shaped with dimensions
11 = v = 3.6254 mm. The difference accounts for the thickness of the conducting
region. These aperture dimensions were chosen because they are typical of the di-
mensions of aluminum honeycomb, samples of which were used for an experimental
comparison to the numerical data. The excitation is a normally incident plane wave
with magnetic field perpendicular to the y — 2 plane, and the array of apertures is
unskewed, i.e. (60 = 90°. Because the array is unskewed, the periodicity and the
unit cell dimensions along each direction are the same, i.e. c = u and d = v. It is
clear that increasing the thickness of the screen decreases the transmission of power

through the screen at the lower frequencies. It is also important to note that once

40

 

the frequency reaches approximately 41 GHz, the shielding effectiveness approaches
0 dB irrespective of the screen thickness. This is expected given that the aperture
size in this case is 3.6 mm, which corresponds to a cut-off frequency of 41.64 GHz for
the dominant TElo and TE” modes of a square waveguide. For a normally incident
plane wave with electric field perpendicular to the x— 2 plane, the results are virtually
identical. To compute the curves in Figure 3.3, 882 Floquet modes were used. 60
waveguide modes (mode indices 3 5) were used to compute the curves for the 1.1 mm,
2.2 mm, and 4.4 mm curves. For the larger thickness values, the curves for the shield-
ing effectiveness did not provide meaningful data when using 60 waveguide modes.
The trends that were observed in increasing from 1.1 mm to 2.2 mm to 4.4 mm were
not apparent. However, using less waveguide modes did produce useful results that
showed a continuing trend toward better shielding performance at lower frequencies
when thickness was increased. The reasoning behind the use of less waveguide modes
for higher thickness values is explained in Chapter 5, including the techniques that
were used to evaluate problems with high thickness values. The number of waveguide
modes used to compute the other curves in Figure 3.3 are shown in Table 3.1. All 60
waveguide modes and their corresponding cut-off frequencies are shown in Table 3.2.

Figure 3.4 shows the impact of changing the incidence angle 0,- for a TM-polarized,
incident plane wave with a screen thickness of 5.5 mm. Results for 15° and 30° were
computed, but are omitted due to their similarity to the 0° case. The dips in the
higher frequency region are due to forced resonances that occur just prior to the
onset of grating lobes. These points are known as Wood’s anomalies [5] [9], and they
occur at the point in frequency where the separation between the apertures is about
a wavelength. For the TE—polarized incident plane wave case, the curves in Figure
3.5 show more modest differences for the change in incidence angle. However, this
case does indicate a downward shift in the resonant frequency as 6,- is increased, as

was noted in [5]. Here, the resonant frequency is considered the point at which full

41

 

transmission is achieved. This is somewhat different from the T M-incidence case,
where the increase in 0,- has no effect on the resonant frequency. Similar to the TM—
incidence plot, results for 15° and 30° were omitted due to their similarity to the 0°
case.

In getting numerical results for the rectangular aperture case, the choice for the
number of Floquet modes was (2 x 10 + 1)2 x 2 = 882, where each mode index (m
and n) ranges from —10 to +10 and there are TM and TE sets of modes. This was
the number of modes used in [19], and it represents the starting point for this study.
Some consideration was given to the idea that using more Floquet modes, while more
time-consuming for computations, might provide more accurate data. Figure 3.6
shows the comparison between using 882 Floquet modes and 1922 Floquet modes
while holding the number of waveguide modes at 60. No appreciable difference was
seen between the two cases when t=4.4 mm, and all curves were computed using
882 Floquet modes. A similar consideration was made with regard to the number of
waveguide modes being used. Increasing the thickness beyond 6.6 mm prevents the
use of 60 waveguide modes, but Figure 3.7 shows that just using the two dominant
modes (TEm and TEm) 0f the square waveguide gives virtually the same curves as
using 60 modes when t=4.4 mm. However, it should be noted that if the apertures
were larger, higher-order modes would contribute at a lower point in frequency.

Figure 3.8 - Figure 3.14 show comparisons between mode-matching results and
data using the waveguide below cut-off formula at different thicknesses. The

waveguide below cut-off formula for transmission coefficient is given by

T,”3 = zolog10 (em) (3.41)

f 2
a=wm,|(7€) —1. (3.42)

42

where

The quantity d is the length of the aperture, and fc is the cut-off frequency of the
dominant rectangular waveguide mode, which in this case is given by fc = 41.64 GHz.
As the thickness of the screen is increased to five times the aperture width and larger,
the agreement between the curves gets stronger. In fact, the waveguide below cutoff
formula was developed to describe the attenuation of waves propagating in an opening
whose length is at least five times its width [24]. The mode-matching approach is
not limited to a certain ratio between aperture width and aperture length/screen
thickness, making it more flexible for making shielding effectiveness calculations. In
addition, the mode-matching approach can be used for frequencies above the cut-off
frequency of the dominant waveguide mode.

Figure 3.15 and Figure 3.16 show mode-matching results using the proposed ap-
proach in comparison to past published results for mode-matching from [10]. Also,
Figure 3.17, Figure 3.18, and Figure 3.19 compare results to published data from [19].
In each figure, both sets of data are very closely related. However, it should be noted
that in [10] and [19], only the (m = 0,n = 0) Floquet mode was considered to be
propagating. The curves are in very good agreement if that consideration is taken into
account. However, if the frequencies of interest are high enough, other Floquet modes
will become propagating and should therefore be included in the analysis [21][30]. In
fact, the transmission nulls in Figure 3.15, Figure 3.16, and Figure 3.19 are due to the
Wood’s anomaly mentioned earlier. And in each case, the next propagating Floquet

mode occurs very close in frequency to the location of the Wood’s anomaly.

3.2 Circular Apertures

The layout of the screen for the circular aperture case is shown in Figure 3.20, where
the skew angle of the array, (150, the x-periodicity c and the skewed periodicity d are
shown. Figure 3.21 shows the unit cell, where the circular aperture radius is a, and

the hexagonal unit cell dimension is u.

43

3.2.1 Calculations

In order to compute the shielding effectiveness of a screen of circular apertures, a
total of 14 integral calculations must be made. Four of them, (2.30), (2.37), (2.62),
and (2.63), refer to TM or TE modes, or neither. By contrast, (2.64), (2.65), (2.66),
(2.67), and (2.68) account for the other ten because each must be calculated for both
TM and TE modes within the aperture. In addition, due to the mode functions used
for the circular aperture case, calculations must be done seperately for even and odd
modes. However, some similarities between the different formulas will lead to some

redundancy in the calculations.
3.2.1.1 Computing Slmn and 82"",

Substituting (2.17) and (2.46) into (2.62) leads to

Slmn : [Ef fmnds
S

 

 

_ exfimn — eyam E6
«a; + (33.... Wis
x fe—ijco cos ¢,- sin 6,- — am)e-jy(k0 sin ¢,sin 0,- — flmnlds
s

 

 

 

 

_ exflmn — eyam 4E5
" ( m ) m [F1(u) + F2(u)l (343)
where
_jp 2“ jp 2n
F1(u) = e p 3’75 (ej(%+px)u_1) ———ep 3’72? (1_e—j(%—px)u),
p, (7% + p.) M75 - pm) 3 44
—jp 2“ jp 2" ( ° )
F201): 8 p 3’73 (ej(%_px)u_1) ——§E_—:7—3—(1—e—j(%+pf)u),
py (7% _ pm) py(\/§ + pm) (3 45)

px 2 k0 cos ¢,—sin 6’, — am,

44

py 2 k0 sin (b,- sin 0,- — 5",".

This result is obtained by forming linear equations for the unit cell boundaries and

integrating over the limits. The equations are

( ) —1 2n
yl :8 fix fl,
( )_ _L + _23
yz :1: fix fl,
(1‘) — -1— _ 2—
313 «gm fl,
( ) —1 2H.
y4 I \/§ «3,

and F1(:c) and F2(2:) are expressed as

0 312(1)
171(27): /e_]p:rx [ / e_]pyydy] dx,

_“ 311(17)

u 314(3)
F2(:r)= fe-Jpzar[ / e_]pyydy]d:r.

0 y3($)
Equation (3.43) is true for almost all combinations of the Floquet mode indices. If

m = n = 0, (3.43) becomes

_ 62:,an — eyam i
Slmn — ( m ) EO\/Z;' (346)

If the mode indices are such that pg 2 Pxfi, (3.43) still applies with (3.44) and (3.45)

 

simplifying to

 

 

. 2u ' 2u
F1(u)= e )(e]( +pm)u_1) '6

V3
py (% +pa:

45

2n

' 2a
—pr75 _. p e-pr75
F2(u) = e p (e “7% + pflu — 1)+j—-——u. (3.48)
p, (7% + pm) pg

Using similar reasoning, substituting (2.18) and (2.46) into (2.63) leads to

 

S2mn : [Ef ' Rgmnds
S

_ exam + eyfimn E},
«a; + 5,2,... «A.
x [e—jsrUco cos qbisin 6, — am)e—jy(k0 sin qfi, sin 9,- — fimn)ds
S

exam + eyfimn

_ ( 02 +_5_2 ):€A§8[F1(u)+F2(u)] (3-49)

with F1(u) and F2(u) given by (3.44) and (3.45), respectively, for most Floquet mode

 

 

index combinations. If m = n = 0, (3.49) becomes

exam + e m, ,
52m, = ( 2 31‘: )Em/As. (3.50)
v am + 1617171

If the mode indices are such that pg 2 p3, J3, (3.49) still applies with (3.47) and (3.48)

 

defining F1(u) and F2(u), respectively.
3.2.1.2 Computing fm3 and fph

To evaluate fpe, the normalization integral for TM modes in the aperture, (2.42) is

substituted into (2.28), leading to

a, = jk. (ragga?) cos(ca) — ¢3§Jc(krr) sin(c¢)) , (3.51)

for even TM modes, and

em = jkz (feréflcrr) sin(cq§) + 43;J6(kr7‘) cos(c¢)) , (3.52)

46

for odd TM modes. Substituting either (3.51) or (3.52) into (2.30) gives

 

n
: —kfk3 [[Jé(krr)]2[cos(c¢)]2ds — [9302/—12—[Jc(krr)]2[sin(c¢)]2ds
Q Q T
: —7rk§k3/T[J£(krr)]2dr — 7rk302/l[Jc(krr)]2dr, (3.53)
0 0 T -
aslongascsdéO. Ifc=0,
fpe = [em ' etp d8
0 V
= 4.31.3 / [.1kade
n

= —2nk§k3 / r[J;(k.r)]2dr (3-54)

0

for even TM modes, and fpe = 0 for odd TM modes. The integrals in (3.53) and
(3.54) can be computed in closed form, or they can be determined numerically.
To evaluate fph, the normalization integral for TE modes in the aperture, (2.43)

is substituted into (2.35), leading to
2,, (2 x h,,,) = 3'15.th (3k.Jg(k;r) cos(cqb) + 1=§J6(k;r) sin(c¢)) , (3.55)
for even TE mdoes, and
Z»... (23 x h,,,) = jkthp (3k.J;(k;r) sin(c¢) — achucy) cos(c¢)), (3.55)
for odd TE modes. Substituting either (3.55) or (3.56) into (2.37) gives
f,,, = / 2,,(2 x h,,) . th(2 x h,,)ds
o

47

= —(kzk:.th)2[[J£(k;r)]2[cos(c¢)]2ds — c2(kthp)2/ :—2[Jc(k;r)]2[sin(c¢)]2ds
:2

Q
a

= —7r(kzk;th)2/T[Jé(k;r)]2dr —- 7rc2(kthp)2/-71:[Jc(k;r)]2dr (3.57)

0

aslongascaé0.1fc=0,

fph = [ZIP(2 X htp) - th(23 x htp)ds
n

—(k.k;zhp>2 / [J;(k;r)12ds
fl

= —7r(kzk;th)2 / r[J;(k;r)]2dr (3.58)

for even TE modes, and fph = 0 for odd TE modes. The integrals in (3.57) and (3.58)

can be computed in closed form, or they can be determined numerically.
3.2.1.3 Computing 5,,

Using (2.26), (2.28), (2.42), and (2.49) in (2.64) and making some substitutions gives

5,?“ = [(2 x H;‘) . Etpds
Q

a 21r
= +G0/1‘Jé(krr) [/ cos(c¢)sin(¢)e_jkor sin 6,-cos(¢ _ (qufiJ dr

0 0

a 21r
—G1/Jc(k,r) [/ sin(c¢)cos(¢)e—jkor sin 9,-cos(¢ _ ¢‘)d¢] dr
0

0

a 27r
—02/7‘J£(krr) [/ cos(c¢)cos(d))e_jk0r sin 6,cos(q’> — ¢i)d¢] dr

0 0
a 27r
—G3/Jc(krr) [/ sin(c¢>)sin(¢)e—jkor sin O‘COSW _ ¢i)d¢] dr
0 0
:- + [GOG4 — 0206) / rJ;(k,r)Jc,1(kor sin 0,)dr
0

— [0005 + 0207] / rJ;(k,r)Jc_1(k0r sin 0,)dr
0

48

— [0105 + G367] / Jc(k,r).]c+1(kor sin 0,)dr
0

— [GIG4 — G305] / Jc(k,.r)Jc_1(k0r sin 0,)d7‘ (3.59)
o
for even TM modes, and

53‘“ = [(2 x H;) - Etpds

o
a 21r
= +GO/rJé(krr) [/ sin(c¢)sin(¢)e—jkor sin 0,cos((b — ¢i)d¢] dr
0 o
a 21r
+G1/Jc(k,r) [/ cos(c¢)cos(¢)e_jkor sin 0,cos(¢ _ ¢i)d¢:l dr
0 o

a 21r
—GgfrJ£(k,.r) [/ sin(c¢)cos(¢)e_jkor sin 6,-cos((b _ ¢i)d¢] dr

0 0

+G3 [6 Jc(krr) [7cos(cq5) sin(¢)e—jkor sin 0" COS<¢ _ ¢i)dq§] dr
0 o
= — [COGS + 0204] [arJé(k,r)Jc+1(kor sin 0,)d7‘
o
'l' [GOG7 —' 0205] jTJé(kTT)Jc-1(k0T sin 9i)d7‘
0
+[G1G5 + G3G4] [0 Jc(k,.'r)Jc+1(k0r sin 0,)dr
. o

+ [GIG7 — 0305] / Jc(k,.r)Jc_1(korsin 6,)dr (3.60)
0

for odd TM modes, where
jkzkrEf,

 

GO : hm,
771V fpe
GI = JkZCEO ha?)

01%

49

 

376.6235

 

 

 

Gz = h ,
771V fpe y
jkzcEf,
G3 = h ,
771V fpe y
G4 = 7rj°+1 sin[(c +1)(,75,~], (3.61)
G5 = m“ sin[(c — 1)¢,], (3.62)
G6 = 7rj°+1cos[(c + 1)¢.-], (3.63)
G, = 7rj°71cos[(c -— 1)(b,~]. (3.64)
(3.59) and (3.60) are valid as long as c ¢ 0. If c = 0,
SE“ = [(2 x H;) -Et,,ds
o
a 21r . .
= +00 / rJ{,(k.r) ]/ sin(¢)e—Jko’"Sln 92C03<¢ ‘ ¢i)d¢] dr
0 0
a 27r
-02/TJ6(k,-T) '/cos(¢>)e_]kor 81” 61°03“) _ 050MB] dr
0 o
= j27r[Go sin (b,- — G2 cos 65,-] [TJ6(k,.r)J1(k0r sin 6,)dr (3.65)
0

for even TM modes, and S; M = 0 for odd TM modes. The integrals in (3.59), (3.60),

and (3.65) must be determined numerically.

For TE modes, using (2.34), (2.35), (2.43), and (2.49) in (2.64) and making some

substitutions gives

5;“? = [(2ng).E,,,ds
Q
27r

= +00 / Jc(k,’,r) [ / sin(c¢) gamma—31““Sin 92' COW ‘ ¢ild¢ dr
0

0

50

a 27r
+01/rJ£(k,'.r) [/ cos(c¢)cos(¢)e_jkor sin O‘COS(¢ — ¢i)d¢] dr

0 0

0

—Gg/a.]c(k:_r) [7sin(c¢)cos(¢)e—jkor sin 6,-cos(¢ _ ¢i)d¢] dr
0
+Gg/arJé(k;r) [7cos(cq§)sin(¢)e_jkor sin 0‘C05(¢ _ ¢i)d¢] dr
0 o
+ [G1G6 "l' G364]/0TJ£(I€TT)JC+1(IC0T sin 01)d7‘
0
+ [6'le — 0305]/arJé(krr)Jc_1(k0r sin 0,)dr
o

— [GOG6 + 020.] / Jc(k,r)JC+1(k0r sin 0,)dr
0

+ [0007 — 0205] / Jc(k,r)Jc_1(k0rsin 6,)dr (3.66)
0

for even TE modes, and

TE
SP

[(2 x H;') - Etpds
Q

a 21r
—G0/rJé(k,.r) [/ cos(c¢)sin(¢)e_jkor sin O‘COSW — ¢i)dq§] dr

0 0

a 21r
+G1/Jc(krr) [/ sin(c¢)cos(¢)e—jk0r sin 0,cos(¢ _ ¢i)d¢] dr
0

0

a 21r
+G2/1'J,’,(k,r) [/ cos(c¢)cos(¢)e_jkor sin Q‘COSM — dado] dr

0 0

a 2n
+03/Jc(krr) [/ sin(c¢)sin(¢)e_jkor sin 6,-cos(¢ — ¢‘)d¢] dr
0 o

+[G'1G'4 — G3G6] [TJé(krr)Jc+1(k0r sin 0,)d7‘
o

+[0105 + G307] / rJ;(k,r)J,_1(k0r sin 0,)dr
0

51

— [GOG4 — G2G6] / Jc(krr)Jc+1(kor sin 0,)d7‘
0

+ [0005 + G207] / Jc(k,r)Jc_1(k0r sin 6,)dr (3.67)
0

for odd TE modes, where
_ jkthpCEa

771m
_J'kthpkf-Eé
— 771m
_ jkthchf]
— 771m
= jkthpkiEii h

771m y,

and G4,G5,G6, and G7 are defined by (3.61), (3.62), (3.63), and (3.64), respectively.

GO hm)

GI hr:
02 hy,

G3

(3.66) and (3.67) are valid as long as c ¢ 0. If c = 0,

s,” = [(2 x H;‘) . Etpds

n
a ‘21r 1,

= +G1/rJé(k,'.r) /cos(¢)e_jkor sin 0,cos(¢ — (mdoj dr
0 -0

 

a. '21r
+G3 / rJ,’,(/s;.r) / sin(¢)e—jk°TSin 92m“? — $066] dr

0 -0
a.

= j27r[G'1 cos gb, + 03 Sin 65,-] /rJ6(k,'.r)J1(k0r sin 0,)dr (3.68)

0

for even TE modes, and SEE 2: 0 for odd TE modes. The integrals in (3.66), (3.67),

and (3.68) must be determined numerically.
3.2.1.4 Computing Plpmn, ngmn, CAM", and Qgpmn

In order to evaluate Plpmn, P2pmm lemn, and Qgpmn, the same computations that

were carried out previously for 8,, are repeated.

52

To calculate Plpmn for TM modes, using (2.4), (2.5), (2.17), (2.26), (2.28), and
(2.42) in (2.65) gives

 

 

P17131511: = [Em ' Rfmnds

o

a "21r '1
= +G2/rJé(krr) fcos(c¢)cos(¢>)ejr<m" cos(d) _ ”m")d¢ dr

0 -0 .
a '21r

—G0/7‘Jé(krr) fcos(c¢)sin(¢)ejr<m" cos(qb — 14"")qu dr
0 .0 .
a. ’21r

+G3 / Jc(krr) fsin(c¢) sin(¢)ejr<m" cos(¢ — l’7'1")qu] dr
0 .o
a '21r

+G1/Jc(krr) fsin(c¢) cos(¢)ejr<m" cos(qb _ ”m")d¢] dr
0 -0

 

+ [0207 + G065] jTJé(krT)Jc_1(CmnT)dT
0

+ [026:6 — G004] iTJé<krT)Jc+1(<mnT)dT
0

— [G3G5 — G1G4] j Jc(krT)Jc+1(CmnT)dT
0

+ [G3G7 + G105] [a Jc(krT)Jc—1(Cmnr)dr (3.69)
0

for even TM modes, and

TM _
Plpmn '—

”211'

 

-0
" 271’

 

/sin(c¢>) COS(¢)ejTCmn COSW — an)d¢

 

]/ sin(c¢) Sin(¢)ej7‘§mn 005015 — anldqil
0 .
21r

53

dr

d7‘

—03/Jc(krr) [/ cos(c¢)sin(¢)ejr<mn 005” .— Vm")d¢ dr
0 0

a 21r
—G1/Jc(k,.r) [/ cos(c¢)cos(¢)ejr<mn cos(qb _ ”m")d¢ dr
0

0
a

: + [G205 — 0007]/7J(:(krT)Jc—1(Cmnr)dr

0
a

+ [G204 + G006] frJé(krr)Jc+1(Cmnr)dr

0

— [0304 + 0.06] / Jc(k,r)Jc+1(Cmnr)dr
0

+ [G305 — 0.07] / J.(k.r)J._i(<m.r)dr (3.70)
0

for odd TM modes, where
jkzkram

: i/Aicmnfl’

jkzcam

Z «2.60.03;

2 jkzkrflmn
mcmnfl’

_ j 1620an

_ mcmnfl’

G4 = 7rj°+1 sin[(c + 1)Vm,,], (3.71)

 

0

 

1

 

2

 

3

G5 = 7rj°"1sin[(c — 1)Vm,,], (3.72)
G6 = 7rj°+1cos[(c +1)um,,], (3.73)
G7 = 7rj°“1cos[(c — 1)an], (3.74)

(mu : V (13,, + 53mm

[an

an = arctan .
Om

 

54

 

(3.69) and (3.70) are valid as long as c 75 0. If c = 0,

TM __ t
Plpmn _ [Etp ' lend8
9
a

”211'
'2 +02/7‘J6(k,r) [cos(¢)ejr<m" cos(qS — Um")d¢] dr
0 .0

 

a " 271'
—(;O / rJ6(k,r) / sin(¢)efl€mn 00505 — an)d¢] dr

0 .0
a

= j27r[Go sin(1/m,,) — G2 cos(umn)]/rJ6(k,r)J1(k0rsin 6,)dr (3.75)
0

for even TM modes, and PTM = 0 for odd TM modes. The integrals in (3.69),

1pmn
(3.70), and (3.75) must be determined numerically.
For TE modes, using (2.4), (2.5), (2.17), (2.34), (2.35), and (2.43) in (2.65) gives

1pmn

P” = f 13,, - R’fmnds
Q

”211' _
2' —GO/1‘J£(k;r) foos(c¢)sin(¢)ejr<m" cos(gb _ ”m")d¢] dr

0 -0

 

a '21r
—6'2/rJé(k;r) fcos(c¢)cos(¢)ejr<m" COSW _ ”m")d¢] dr

0 .0
" 21r

+G1/Jc(k,',r) lsin(c¢)cos(¢)ejrcm" cos((b _ ”m")dq§ dr
0 L0 .

a '21r

—G3/Jc(k;.7') fsin(c¢)sin(¢)ejrcmn COSW _ VWqufl dr
0 .

.0

 

 

= + [0005 - G2G7] [TJé(k,-T)JC_1(CmnT)dT
0

— [0004 + Gian] / rJ;(k.r)J..i(<m.r)dr
0

+ [010. + 0306] / Jc(k.r)J..i(<m.r)dr
0

55

+ [0105 - G307] / Jc(k,r)Jc_1(§mnr)dr (3.76)
0

for even TE modes, and

TE _ at
Plpmn _ [Etp ' Izlmn.dS
Q
a

 

'21r
= —GO/r.]é(k:.r) /sin(c¢)sin(¢)ej7‘<mn COSW _ VWJdgb] dr
0 .0
a ’21r
—02/rJé(k,’.r) fsin(c<f>) cos(¢)ejr<m" COS” — anldqs] dr
0 .0

a ['21r .
~01 / Jc(k:.r) fcos(c¢) cos(¢)efl<mn cos(qb — ”m")dqb] dr
0

I I
O

 

.0

+03 / J.(k;r) 766s(c¢) sin(6)ej7"<mn COW — an)d¢] dr
0
= — [0007 + G265] i TJ£(krT)JC_.1(CmnT)dT

0

+ [GoGs — G204] jTJ£(k7-T)Jc+1(<mn1')d7‘
0

— [GIGG — G3G4] j Jc(krT)Jc+1(CmnT)dT
0

— [0107 + G305] j Jc(krT)Jc_1(CmnT)dT (3.77)
0

for odd TE modes, where
G0 : jkzk;thIan
V AsCmnV fph

l = jkz thcfimn
V AsCmnV fph

_ jkzkitham
V AsCmnV fph,

 

 

 

2

56

_ jkz thcam

V AsCmnV fph ,
and G4, G5,G'6, and G7 are defined by (3.71), (3.72), (3.73), and (3.74), respectively.

 

3

(3.76) and (3.77) are valid as long as c # 0. If c = 0,

P” = f 13,, - Rimnds
Q

1pmn

a ’27r
= —GO/TJ6(k;.T) fsin(¢)ejr<"m COS” _ ”m")d¢] dr

0 L0

 

a '21r
—G2/1‘J6(k;r) fcos(¢)ejr<m" COSM’ _ ”7"")dcb] dr

0 ..0

= —j27r[Gosin(1/mn) +chos(1/mn)]frJ6(k,'.T)J1(CmnT)dT (3-78)
0

for even TE modes, and Pfin '2 0 for odd TE modes. The integrals in (3.76), (3.77),
and (3.78) must be determined numerically.

To calculate ngmn for TM modes, using (2.4), (2.5), (2.18), (2.26), (2.28), and

(2.42) in (2.66) gives

132,111)?an : [Etp ' 1?";mndS
Q

a ”271'
= +G2/rJé(krr) /cos(c¢)cos(¢)ejr<mn COSW — an)d¢] dr
0 L0
0 ”211'
+G0/rJé(krr) fcos(cq’>) sin(¢)ejr<m" cos(qb _ ”m”)dq§] dr
0 -0

 

a ’21r
—G1/Jc(krr) fsin(c¢)cos(¢)ejr<m" COS(¢ _ ”m")d¢] dr
0

.0

 

a '21r
+03 / Jc(k,r) l/ sin(c¢)sin(¢)ej’"4mnc°s(¢ — an)d¢] dr
0

0

z — [G0G5 — G207]frJé(krr)JC_1(Cmn7‘)dT
0

57

+ [G2G5 + GoG4] / TJé(krT)JC+1(CmnT)dT
0

— [6104 + 0306] / Jc(k,r)Jc+1(Cmnr)dr
0

— [0,05 — G307] / Jc(k,.r)Jc_1(Cm,,r)dr (3.79)
0

for even TM modes, and

P2297247: = [Etp ' ‘Il‘gmnd8
Q

a ['27r
2: +02 frJéUcJ) [sin(c¢) cos(¢)ejr<m" (”SW — Um")d¢] dr
0 .0

 

a [‘21r

+G0/rJé(krr) fsin(c¢) sin(¢)ejr<m" COS(¢ — Um")d¢] dr
0 .0

a ' . ]

+01 / J,(k.r) / cos(cns) cos(¢)e17‘Cmn COS(¢— ”mums dr
0

-0 ..
' 21r 1

-03/Jc(krr) fcos(c¢)sin(¢)ejr<mn COSM) — ”m")dqb dr
0 .

L0

 

 

= + [0205 + 0007] / rJ;(k.r)Jc_1(gm,r)dr
0
+ [0204 - (3005] / rJ;(k,r)Jc,1(gm,,r)dr
0
+[01G6 - G304] / Jc(krT)Jc+1(CmnT)dT ’
0

+ [0107 + G3G5] / Jc(k,.r)Jc_1((mnr)dr (3.80)
0

for odd TM modes, where

 

jkzkrfimn
0 = a
V AsCmnV fpe

1 _ J' 1920an

— mcmnm’

 

58

j k, krozm

G = ,
2 «Kennfi;

jkzcam

V AsCmnV fpe,
(3.79) and (3.80) are valid as long as c 75 0. If c = 0,

 

 

3

P2751211; — [Em ' ligmndS
Q

a '21r "
= +00 / "J6(k.r) / cos(¢)ej’"4m"°°°(° ‘ ”m")d¢ d"
0 -0 -
a '21r . j
+G2/7‘J6(kr7') [sin(¢)e-7TC"‘" cos(qS _ an)d¢, dr
0 -0 .1

 

 

j27r[G2 cos(z/mn) + Co sin(1/,,,,,)] [TJ6(k,r)J1(Cm,,r)dr (3.81)
0

for even TM modes, and PTM = 0 for odd TM modes. The integrals in (3.79),

2pmn

(3.80), and (3.81) must be determined numerically.
For TE modes, using (2.4), (2.5), (2.18), (2.27), (2.35), and (2.43) in (2.66) gives

P27115111 : [Etp . Rflmnds
Q

a ’27r -
= +G2/rJé(k,'.r) [/ cos(c¢)cos(¢)ejr€m" COS(¢ _ an)d¢ dr

0 0
a 21r

—Go/rJé(k:.r) [/ cos(c¢)sin(¢)ejr<m" COS“) _ an)d¢ dr

0 0

 

 

a '21r
+G1/Jc(k;r) fsin(c¢) cos(¢)ejr<m" COS(¢ _ ”m")d¢] dr
0

-0

 

a 21r
+03 / J,(k;r) / sin(c¢) sin(6)ei’“<mn COW — an)d¢] dr
0

.0

= + [G207 + G005][TJé(krT)Jc—1(Cmnr)dr
0

59

+ [G206 '_ G004] /TJ£(krT)Jc+1(Cmnr)dT
o

+ [G104 “— G3G6l/Jc(krT)Jc+1(Cmn7')dT
0

+ [0105 + G3G7] / Jc(k,.r)Jc_1(Cm,,r)dr (3.82)
0

for even TE modes, and

P27917321: : [Etp ' Rfmnds
Q

a "21r
: +G'2/7'Jé(k;r) /sin(cq§)cos(¢)ejr€m" COS(¢ — ”m")d¢] dr

0

I r
O

 

a 21r

—Go/rJé(k;r) fsin(c¢)sin(¢)ejr<m" COSW _ ”m")d¢] dr
0 .0

'27r 7

—GI/Jc(k:.r) foos(c¢)cos(¢)ejr<mn COS(¢_ ”m")d¢ dr
0 -

-0
" 27r

—G3 / Jc(k:.r) / cos(cqb) sin(¢)ej"<mn 008(4) — ”mn)d¢[ dr
0 .

b0

 

 

= + [G205 — 0007] iTJ£(krT)JC_1(CmnT)dT

0

+ [G204 + GoGs] jTJ£(k,-T)Jc+1(cmn7‘)d7'
0

— [GIGS + G3G4l ] Jc(krT)Jc+l(CmnT)dT
0

—' [0107 — G305] [0 Jc(krT)Jc_1(CmnT)d7‘ (3.83)
0

for odd TE modes, where

 

jkszthOfm
0 = a
mcmnm

jkz thcam

: mcmwfi

 

1

60

= .775szthan
V AsCmnV fph

: jkz thcflmn
V AsCmn V fph

and G4,G5,Gs,and G7 are defined by (3.71), (3.72), (3.73), and (3.74), respectively.

 

2

 

3

(3.82) and (3.83) are valid as long as c yé 0. If c = 0,
P2351” : [Etp ' Rzmnds
n

a '21r
= +G2/7‘J6(,€;T) [cos(¢)ej7~<mn cos(qS _ ”m")d¢] dr

0 -0

 

a "211'
—Go/7‘J6(k;r) fsin(¢)ejr€m" COSW _ an)d¢] dr

0 -0
a

2: j27r[G2 cos(umn) — Go sin(1/mn)l frJé(k;T)J1(CmnT)dT (3-84)
0

for even TE modes, and PTE = 0 for odd TE modes. The integrals in (3.82), (3.83),

2pmn
and (3.84) must be determined numerically.

In order to compute lemn and Q2pmn for TM and TE modes, the formulas are
identical to what has been shown for Plpmn and BMW. This is true despite the fact
that len and R2mn are used to compute lemn and szmn instead of RIM and
R3,...-

3.2.2 Discussion of Results

At this time, numerical results for the circular aperture case are not available.

61

Table 3.1. Number of modes used to calculate the curves in Figure 3.3.

 

 

 

 

 

 

 

 

 

 

Thickness (mm) Modes Used

1.1 60

2.2 60

4.4 60

7.7 30

8.8 30

9.9 30

18.0 6
26.0 4
40.0 2

 

 

 

 

Table 3.2. Cutoff frequencies for modes of square apertures.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Modes Cut-off frequency fc (GHz)

TE10,TE01 41.64

TM11,TE11 58.89

TE02, TE20 83.28
TM12,TM21,TE12,TE21 93.11
TM22, TE22 117.77

TE03, TE30 124.91
TM13,TM31,TE13,TE31 131.67
TM23, TM32, TE23, TE32 150.13
TE04, TE40 166.55
TM14,TM41,TE14,TE41 171.68
TM33, TM33 176.66

TM24, TM42, TE24, TE42 186.21
TM34, TM43, TE34, TE43, TE05, TE50 208.19
- TM15,TM51,TE15, TE51 212.31
TM25, TM52, TE25, TE52 224.23

T M44, T E44 235.54
TM35, TM53,TE35, TE53 242.79
TM45, TM54, TE45, TE54 266.62
T M55, TE55 294.43

 

62

>‘<

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>X

 

 

 

 

 

5-
CL\‘

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.1. Doubly-periodic conducting screen of apertures with rectangular cross—

section.

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ 11 >

 

 

 

Figure 3.2. Unit cell for a doubly-periodic conducting screen of apertures with rec-
tangular cross-section.

64

 

    
  

 

 

 

 

 

 

0 "'"”"”‘"""""""””"“'"”“”'°”““'””""“""'"""'"'”"”"_"
$-20 .4.-----_-_- ‘_<““‘- A“::;'::‘::::25v"'-::::::::: ........ ". I, t.1'1m
E ..-::::::;::== iii +1.2.2mm
O ..:::::::2 5555
g .40 . .."-.----------:::,:::==' ............................. . '1 - -B-t-4.4rnm
5 ° +t-7.7mm
C _. ‘
§ -60 i» -------------------- . ------------------------------ x ,_8_8mm
u .1223'
E ”J; -e-t-9.9mm
'80 - "-‘ ................................... -
E --t-1B.0mm
'—
-l-—t-26.0mm
-100 J ........................
--t-40.0mm
'120 ‘r If I I 1 4‘
0.0 10.0 20.0 30 O 40.0 50.0

Frequency (GHz) .

Figure 3.3. Comparison of transmission coefficient for screens of different thicknesses
for normally incident TM polarized plane wave.

65

20 —5

 

vb
o
1

 

 

 

Transmission coefflclent (dB)

 

 

 

 

I - —e- theta=0 degrees ]

_’ —+— theta=45 degrees I

-80 - ...................................................... +theta=60degrees

-e- theta=75 degrees [

l

-100 ~ ---------------------------------------------------- - --------------------------------------------------- [

'1 20 r I I J.
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (6"!)

Figure 3.4. Comparison of transmission coefficient versus frequency for various inci—
dence angles when t=5.5 mm for TM polarized incident plane wave.

66

20

 

    
 

 

 

 

 

o s .....................................................................................
g -20 . .............................................................................................
E
.9
2
E .40 _. ...........................................................................
1:
-§ .60 4 ---------------------------------------------------------------------------------------
“E + theta=0 degrees
a -80 -. ____________________________________________________________ +theta=45 degrees ___________
'- +theta=60 degrees ,
+theta=75 degrees
-100 - ............ . ........ , ................................................................. , _______
420 . r . .
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 3.5. Comparison of transmission coefficient versus frequency for various inci-
dence angles when t=5.5 mm for TE polarized incident plane wave.

67

 

 

 

 

L ,"l I -9- 60 waveguide modes, 882 Floquet modes 1

l 1

-E- 60 waveguide modes, 1922 Floquet modes ‘

 

Transmlsslon (:00!!!ch (dB)

 

 

.80 .
-100 ~
-120 r v v - '
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 3.6. Comparison at t=4.4 mm between using 882 Floquet modes vs. 1922
Floquet modes.

68

20

 

 

 

 

-60 _ .'_ ______________________________ —B— 60 waveguide modes, 882 Floquet modes J ________ i

 

—e— 2 waveguide modes, 882 Floquet modes J

 

Transmission Coefficient (dB)

-100 w --------------------- 7 ---------------------------- 7 ------------------------------------------- :

 

-120 r x

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GI-12)

 

Figure 3.7. Comparison at t=4.4 mm between using 60 waveguide modes vs. 2
waveguide modes.

69

 

 

 

 

 

 

 

 

 

8
E
E
.2
o
340 q_ ______ J +Results from mode matching
c ' approach
2
3-50 «-
E -a- Results from using "waveguide
g -60 J ______________ - __________________________________ J below cutoff" formula ____________
E
.—
-7o
-80 ~r -------------------------------------------------------------------------------------------------------- i
i
i
'90 r I T I i
0.0 10.0 20 0 30.0 40.0 50.0

Frequency (GI-l2)

Figure 3.8. Comparison at t=1.1 mm between mode-matching approach and
waveguide below cutoff formula.

70

8
l

 

 

  

 

 

 

 

 

0" """""""""""""""""""""""""""""""""""""" “MAuvmé

-1O ---------------------------------- J
a J :::::::::::::::::::::::::::::::::::::':_?;7.—." J
“EL-20 4 ---------------------------------------------------------------------------------------------------
g
o -30 4 . ,. _, _ ‘
g + Results from mode *7
o .40 fl ___________________________________________________ matching approach ' ,,,,,,,,,,
r.-
% _50 J _______________________________________________________ -a— Results from using _____________
.2 ”waveguide below cutoff"
E formula J
5-60 -J+ --------------------------------------------------------------------------------------------------------
.—

-70 ,,,,,,

-80 ,

-90 I . T 7

0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 3.9. Comparison at t=2.2 mm between mode-matching approach and
waveguide below cutoff formula.

71

 

_I.
o

 

    

 

 

 

 

0 _. ..............................................................................................

-10 s ................................................................................................
i'n‘ a
3-20« -------------------------------------------------------------- ----------------------- J
E . g .1 . - - - l
g -30
2 '40 q """""""""""""""""""""""""" + Results from mode if V
.2 matching approach
3.: -50 - ---------------------------------------------------------------- [
.5, -a- Results from using - L
t: -50 J ., .................................................... "waveguide below cutoff" % ............. J
S tormula '

-70 4 ---------------------------------------------------------------------------------------------------------

-80. ......... - ..................................................................................

'90 1 —1 r r ‘J

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GHz)

Figure 3.10. Comparison at t=4.4 mm between mode-matching approach and
waveguide below cutoff formula.

72

 

    
  

 

 

 

 

 

o J ————————————————————————————————————————————
-10 _ ....................................................................................
a
B -20 ------------------------
E
o -30 . .....................
r:
2 L
-50 ,,-.-" memm -. —~~-——~~*~-~-~~~-- . ........ g
'2 ___=:=::===J. + Results from mode 5
2 |c:::::::::::::::::::::== === . ,- ' matChfng appmaCh J
E -60 'J" - --------------------------------------
l- .
-70 fl ___________ ______________________________________________ -a-Results from using _
"waveguide below cutoff"
formula
-90 . . , , .
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GI-l2)

Figure 3.11. Comparison at t=7.7 mm between mode-matching approach and
waveguide below cutoff formula.

73

 

 

    

 

 

 

 

 

O J ---------------------------------------------------------------------------------------
A -25
in
E
E -50 ..............................................................................................
2
2
E -75 J --------------------------------------------------------------
5 -1oo «J ------------------------------------------------------------------------------------------
g . . +Results from mode ;
g -125 J ' ' ----------------- matching approach ---------- f
8 l ""3““ '''''
'- -150 .J —a-Results from using ..
"waveguide below cutoff" J
formula
-175 -J ................................................................................ ~. ...................
-200 . T I r
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 3.12. Comparison at t=18.0 mm between mode—matching approach and
waveguide below cutoff formula.

74

 

   
    

 

 

  

 

 

 

o .. —————————————————————————————————————————————————————————————————————————————————
A -25 ------------------------------- J ------------------------------------------------------------------------
g ;+ Results from mode matching
‘5 -50., ,. _ .. ., __J approach
%
i—a- Results from using

g '75 ‘ """""""""""""""" "waveguide below cutoff" """"""""""""""""""
0 formula
.5 -100 -4 .....................................................................................................
.3
s -125 e
r:
S

-150 ~ -----------------------------------------------------------------

-175 +~-«----o---—-----------~------- --- " -------------------------------------

'200 '_‘ ---::::: :::: r I I 1

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GI-l2)

Figure 3.13. Comparison at t=26.0 mm between mode-matching approach and
waveguide below cutoff formula.

75

 

 

 

 

 

 

 

 

0 a ........................................................................................
-25

ii
3
one -50 ........................................................................................................
5 .. _W a
g 75 + Results from mode
3 ' matching approach
0
.5 400 -- ------------------------- -a— Results from using ----------------- ' ---------
3 ”waveguide below cutoff" ‘
5 125 - _________________________ formula ________________
c -
E
p.

-150 a ..........................................................................

_175 - ......................................... , ................... ,1

-200 , . 1 ‘ r '

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GI-l2)

Figure 3.14. Comparison at t=40.0 mm between mode-matching approach and
waveguide below cutoff formula.

76

 

2,0 w Wm “WM- mg -m---

.0
o
J

 

iv
o

.43
O
L

  

Transmission coefficient (dB)
sh
O

 

 

........................................................

—theta=20 degrees

—9— theta=20 degrees (Chen)

..............................................

 

 

 

-3J0 ......................................................
-10.0 r If i r . J
14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0

Frequency (GHz)

Figure 3.15. Comparison between proposed mode-matching formulation and mode—
matching results of Chen using 0,- = 20° near transmission null (Wood’s anomaly).

Assumes only (0,0) Floquet mode is propagating.

77

2.0 -. mm---“ fl

 

 

 

0.0 4.-----“ .--- ._--.

-2.0 '4

-4.0 ~

 

 

 

—theta=80 d rees
.5_0 - ............................................ eg 3 ............. -- --

o theta=80 degrees (Chen)

Transmission coeiiicient (dB)

 

 

 

 

 

 

 

-8.0 - ............................................................................ 7
'10.0 T i I 4 i T I
14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0
Frequency (GHz)

Figure 3.16. Comparison between pr0posed mode-matching formulation and mode—
matching results of Chen using 0,- : 80° near transmission null (Wood’s anomaly).
Assumes only (0,0) Floquet mode is propagating.

78

 

 

 

 

 

 

 

 

 

—theta=0 degrees

 

 

o theta=0 degrees (Widenberg)

-6 . #__ Anfm____-_-_—_

 

Transmission coefficient (dB)

 

 

 

 

 

'10 1rTIT Y 1 I : V: TT T rTI T i I I I r. T T T T l’ i i i T r'

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0
Frequency (GI-i2)

Figure 3.17. Comparison between proposed mode-matching formulation and mode—
matching results from Widenberg et 3.1 using 0,- : 0°.

79

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 ..
A C
in
B
E -2
.9
0 1
g
g 1 —iheta=30 degrees ; \
.2 j o theta=30 degrees (Widenberg) ‘
E
.—

-8 —4

-10 T T T A f T T T T - ' ‘ T T T T T 1 t T T T T F I v. T T T T T l T T T . ,
8.0 8.5 9.0 9.5 10.5 11.0 11.5 12.0

10.0
Frequency (GHz)

Figure 3.18. Comparison between proposed mode-matching formulation and mode—
matching results from Widenberg et al using 0,- = 30°.

80

 

 

 

.4?__

 

—theta=60 degrees

0 iheta=60 degrees (Widenberg) /
-6 «L . .

 

 

 

—~<

Transmission coefficient (dB)

 

 

 

 

 

'10' i T T T T T T T T ; 1 - T T T T T T T T T 1 = , T T T l T T T T T T T ‘ T

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0
Frequency (GHz)

Figure 3.19. Comparison between proposed mode-matching formulation and mode—
matching results from Widenberg et a1 using 0,- = 60° near transmission null (Wood’s
anomaly). Assumes only (0,0) F loquet mode is prOpagating.

81

 

 

 

 

V
X

 

 

 

 

/
\

 

 

 

 

 

Figure 3.20. Doubly-periodic conducting screen of apertures with circular cross-sec-
tion.

82

 

 

Figure 3.21. Hexagonal unit cell for a doubly-periodic conducting screen of apertures
with circular cross-section.

83

CHAPTER 4

EXPERIMENTAL RESULTS

4.1 Setup

4.1 .1 Sample Materials

As a means of comparing the numerical results with measured data, an experiment
was set up in order to test the mode-matching approach against actual measure-
ments to determine how well it can predict shielding performance [31]. Samples of
commercial-grade aluminum honeycomb were used in the experiment. The honey-
comb was available with rectangular apertures and hexagonal apertures. The rectan-
gular aperture sample, which was supplied by Benecor, Inc. of Wichita, KS, is about
2 ft. in length and width, and approximately 1 / 2 inch in thickness. The apertures
are square with about 1 / 4 inch in length and width. The hexagonal aperture sample,
which was supplied by Plascore, Inc. of Zeeland, MI, is about 1 ft. in length and
width while the thicknesses is approximately 13 / 16 in. The aperture cell size is about

3 / 4 in. The foil thickness for all of the samples is approximately 0.003 in. (3 mils).
4.1.2 Equipment

An HP851OC Network Analyzer was used to perform frequency domain transmission
measurements. For samples whose apertures have a dominant mode cut-off frequency
in the range of 2—18 GHz, the Michigan State University arch range was used along
with American Electronic Laboratories H-1498 horn antennas for transmitting and
receiving. Figure 4.1 shows a sketch of the low frequency measurement set-up, in-
cluding the arch range, the network analyzer, the antennas, and the sample being
tested. The arch range, designed and built by the Georgia Tech Research Institute, is
a metallic structure 20 ft. in diameter and 4 ft. high. The receiving and transmitting

antennas are connected to the network analyzer using coaxial cables, and lenses are

84

used to collimate the transmit and receive beams. The sample was mounted into
an empty window in a styrofoam board, and the rest of the board was covered with
aluminum tape (see Figure 4.2 for a description). The board is 4 ft. high, 4 ft. wide,
and 1 in. thick. This approach is similar to the one mentioned in [32] to measure
the shielding effectiveness of composite material samples. Mounting the sample in
a board allowed for reducing the effect of a large beam spot size at low frequencies.
The board was then placed onto a styrofoam mount between the transmit and receive
antennas.

For samples whose apertures had a dominant mode cut-off frequency in the range
of 20-40 GHz, a benchtop set-up was used to conduct the experiment. See Figure 4.3
for a sketch of the high frequency measurement set-up. The 8510C network analyzer
was connected to two EMCO 3116 horn antennas with K-type connector cables. Just
as with the arch range set-up, the samples were placed into windows in 4’ x 4’ x 1”
styrofoam boards, and the rest of the board was covered with aluminum tape. A
styrofoam mount was used to position the board containing the sample. The cables
that were used with the high frequency horns were about 3 ft. long, so the transmit
and receive antennas were about 2 ft. from each other, with the sample in the middle.
For horn antennas to be operating in the far-zone, the wave must travel a distance
larger than 2 = 2D2//\0, where D is the largest dimension of the antenna and A0 is
the free—space wavelength at the frequency of interest. For the 2-18 GHz range, the
distance 2D2//\0 ranges from 0.7 ft at 2 GHz to 6.34 ft at 18 GHz. The distance
between the horn antenna and the sample is approximately 10 ft, so the far-zone
requirement is achieved for the low-frequency case. For the 20-40 GHz range, the
distance 2D2/A0 ranges from 4.8 ft at 20 GHz to 9.6 ft at 40 GHz. These values are
both larger than the distance between the antennas and the sample, so the set-up
does not meet the far-zone requirement. However, despite the fact that the horns are

not in the far-zone, the agreement between numerical and experimental results is still

85

pretty good.

4.2 Procedure

4.2.1 Low Frequency Measurements

For all measurements, a two-level calibration was conducted. The first level involved
calibrating the network analyzer using the appropriate standards. For the low fre-
quency measurements, the standards were applied to the ends of the 2.4 mm connec-
tion cables. From the ends of the 2.4 mm connection cables, connector adapters were
used to go from 2.4 mm to 3.5 mm to SMA. Coaxial cables with SMA connectors
were used to link the 8510C with the transmit and receive antennas, both of which
use SMA connectors. The second level of calibration involved taking two transmission
measurements in addition to the sample measurement. The first is an empty mea-
surement, which in this case is a measurement with the window in the board being
empty (no sample). The second is a noise measurement, where a board of the same
size as the one with the window is used to perform a transmission measurement. This
board is completely covered with aluminum tape. The noise measurement includes
the effects of diffraction of the transmitted wave around the edges of the measure-
ment board. The use of the three measurements will be explained in the calculations

section.
4.2.2 High Frequency Measurements

For all measurements, a two-level calibration was conducted. The first level involved
calibrating the network analyzer and cables using the appropriate standards. From
the ends of the 2.4 mm connection cables, connector adapters were used to go from
2.4 mm to 3.5 mm to K. Cables with K-type connectors were used to link the 8510C
with the transmit and receive antennas, both of which use K-type connectors. And
due to the compatibility of 3.5 mm connectors and K-type connectors, 3.5 mm cali-

bration standards were used to calibrate to the ends of the K-type connector cables.

86

As was the case with the low frequency experiment, the second level of calibration
involved taking empty and noise measurements in addition to the sample measure-

ment.

4.3 Calculations

To determine the tranmission coefficient at a particular frequency, the three measure-

ments that are used are

Ssampge — Measured transmission S-parameter for sample in window,

Sum-88 — Measured transmission S-parameter for blocking plate,
Sammy — Measured transmission S-parameter for empty window,

where 512 or 5'21 can be used depending on the set-up. Using those three values, the

transmission coefficient is defined as

Ssample — Snoise

 

ng = —SEdB = 2010g10 (4.1)

 

 

Sempty _ Snoise

It should be noted that because of the use of S-parameter measurements in (4.1),
2010g10 is used instead of 1010g10 because S-parameters are analogous to voltage and
current measurements. By contrast, (2.77) and (2.78) use IOlog10 because power

quantities are involved.

4.4 Discussion of Results

Figure 4.4 shows a comparison between numerical and experimental data for a screen
of rectangular apertures (Benecor sample). The apertures are square shaped with
a width of 1/4 in. (approximately 6 mm). The screen thickness is about 1/2 in.

(approximately 13 mm). Overall, there is pretty good agreement in terms of the

87

transition from low transmission (below ~20 dB) to full transmission. This case, as
with all others in this section, is for normal incidence.

Figure 4.5 shows a comparison between numerical and experimental data for a
screen of hexagonal apertures (Plascore sample). The apertures have a width of
3 / 4 in. (approximately 19 mm) between parallel sides. The screen thickness is about
13/ 16 in. (approximately 21 mm). Square apertures of the same width are used to
model the hexagonal apeture sample, and the results are in good agreement with the
measured data. It is anticipated that once results are available, the use of circular
apertures may prove to be even closer to the measured result. It should also be noted
that this data was obtained despite large losses in measured power due to attenuation
in the coaxial cables.

In both cases, the measured data goes above the 0 dB mark, which at first does not
seem reasonable. However, the contribution to transmission due to diffraction around
the edges of the screen could lead to that kind of behavior. In general, the measured
and numerical data are in good agreement. It is also worth noting that in the context
of a shielding application, where the frequency range of operation would be well below
the point where full transmission occurs, the effects due to edge diffraction are not

expected to be as significant.

88

Tranhsmitting Receiving
orn <1 horn
antenna antenna

Lens

Styrofoam board
with window

 

 

 

HP 851 00
Network Analyzer

 

 

 

 

 

Figure 4.1. Equipment arrangement for taking low frequency measurements
(2—18 GHz)

89

 

Figure 4.2. Styrofoam board with window used to hold aluminum honeycomb sam-
ples. The dark grey region is covered with foil tape.

90

Transmitting Receiving

 

 

 

horn T— horn
antenna antenna
Styrofoam board
with window
HP 85100

 

 

 

 

Network Analyzer

 

 

 

Figure 4.3. Equipment arrangement for taking high frequency measurements
(20—40 GHz)

91

 

 

20

 

 
   
 

—Benecor sample (numerical)

3 l
-Benecor sample (measured) : l

 

Transmission Coefficient (d8)
r'o
o

 

 

'40 T T T
24.1 26.6 29.1 31.6 34.2
Frequency (6112)

Figure 4.4. Comparison of numerical and experimental results for rectangular aper-
tures

92

 

 

 

 

20

 

   
 

—Plascore sample (numerical)

-20 . --------------------------- _— Plascore sample (measured)

 

 

Transmission Coefficient (dB)

 

 

-40

T 1*

6.0 8.0 10.0 12.0 14.0 16.0 18.0
Frequency (GHz)

Figure 4.5. Comparison of numerical and experimental results for hexagonal apertures

93

CHAPTER 5

DISCUSSION OF MODE SELECTION

One of the major obstacles in this research was determining why the shielding calcula-
tion does not produce useful results for 60 waveguide modes when the screen thickness
exceedes 6.6 mm. Originally, the matrix equation was solved using Gaussian Elim-
ination [33]. When the convergence problems arose, Singular Value Decomposition
(SVD) [33] was used in order to determine if the increase in thickness caused the
square matrix in the matrix equation to become badly conditioned. This was accom-
plished by computing the condition number of the square matrix versus frequency for
all of the thickness values of interest. According to [33], the matrix is badly condi-
tioned once the condition number of the matrix exceeds the accuracy of the computer,
which is about 1015. Figure 5.1 shows a plot of the condition number versus frequency
for a variety of thickness values, where 60 waveguide modes and 882 Floquet modes
were used in each case. There is no absolute threshold where having a condition
number lower than that point leads to meaningful solutions. However, at the highest
thickness values, the condition number is well past the values that produce useful
results. Further, increasing the thickness of the screen does make the condition num-
ber higher when using 60 waveguide modes. An attempt was made to use SVD in
order to discard small singular values in order to improve matrix conditioning, but
the solutions were not any better than before. To deal with the problem, the condi-
tion number of the square matrix was reduced by considering less waveguide modes
for very thick screens. Modes were removed in order of highest cut-off frequency.
The concern was that a significant amount of accuracy would be lost by taking that
approach. However, Figure 5.2 shows that for the case of the 4.4 mm thick screen,

the use of 60 waveguide modes provides virtually the same result as the use of several

94

other combinations of waveguide modes, including using only the 2 dominant modes.
This would seem to suggest that the use of 60 modes is unnecessary. However, it
should be stated that the need to disregard higher-order modes arises from the fact
that the increase in thickness attenuates those modes so severely that they need to be
removed from consideration. Figure 5.3 shows that even with 60 waveguide modes,
the condition number never exceeds 1015 for the 4.4 mm thickness case. As was stated
before, 1015 appears to be a value such that if the condition number is well below that
value, the shielding effectiveness calculations will converge. If the condition number
far exceeds that value, the calculations generally do not converge.

For the case of the 6.6 mm and 7.7 mm thick screens, a point should be made.
Figure 5.4 indicates that the shielding effectiveness converges for all choices of number
of waveguide modes used when the thickness is 6.6 mm. Also, Figure 5.6 shows that
the shielding effectiveness results do not converge for the 7.7 mm case when using 60
or 40 waveguide modes, but they do converge for 30 or less. However, an interesting
observation is that Figure 5.7 shows a condition number at or above 1020 for the
7.7 mm thickness when using 60 modes, while 40 modes leads to condition numbers
between 1013 and 1015. Meanwhile, Figure 5.5 shows condition numbers of between
1016 and 1019 for the 6.6 mm thickness when using 60 modes. So, despite the fact that
the 7.7 mm case with 40 modes has a lower condition number than the 6.6 mm case
with 60 modes, the former does not produce practical data while the latter does. This
reemphasizes the point that while the condition number does give some indication of
how the thickness is affecting the solution of the matrix equation, an absolute point
where having a lower or higher condition number guarantees a convergent result was
not found.

More examples of the effect of reducing the number of waveguide modes on the
convergence of the shielding computation are shown in Figure 5.8 - Figure 5.12 for

thicknesses of 8.8 mm, 9.9 mm, 18.0 mm, 26.0 mm, and 40.0 mm. The vertical scales

95

were adjusted to include the most information possible, but some curves were not
included. For example, the 60 waveguide mode case for 9.9 mm thickness is not

included in Figure 5.9 because it deviated too much from the curves shown in the

figure.

96

 

1.0E+100

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0E+90 ~ »
1.0E+80 4
1.0E+70 A Q ;¢;i‘.’1m‘a ‘
E +t=2.2mm
g 1.0E+60 ‘ __t=44mm
:
g 1.0E+50 1 x—i=7.7mm
E " +t=8.8mm
g 1.0E+40 . +t=9.9mm
O
+t=18.0mm
1.0E+30 — !
:+t=26.0mm
1.0E+20 « W. :4. five—a; ‘- . +t=40.0mm
k M
1.0E+10 ‘Qn... A
4""""'i-9-e-24.4.::::::::::::::::::::;::::::::::: “““““““““““““““““““““““ “‘H‘W‘ """
1.0E+00 I r . .
0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GHz)

Figure 5.1. Condition number versus frequency for several values of thickness when
using 60 waveguide modes and 882 Floquet modes.

97

 

20

   

 

 
   

 

 

 

 

 

l
0 _
3.20,
E
.2 .
o .
g + 60 waveguide modes 9
5 -60 4* ___________________________________________________________ +40 waveguide modes ___________ {
g + 30 waveguide modes .
a .
22wav demode ‘
5'30 ______________________________________________________________ + egur S ___________ 1a
'- +14 waveguide modes
100 -°- 6 waveguide modes
+ 2 waveguide modes
'120 T r T r l
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 5.2. Plot of shielding effectiveness for t=4.4 mm when varying the number of
waveguide modes used.

98

 

1.0E+25

 

 

 

 

 

 

l
1.0E+20 4
. :60 modes
“g 1.0E+15 4 +40 modes
2 +30 modes
5 +22 modes
2 .
+ 14 modes
‘5’ 1.0E+10 - ‘
o 3 +6 modes
M—ef—g2gmodes
1.0E+05 ,
10E+00 4 ~‘:;:;_..__,,::::;::::==:::::::::: ----- - Y a
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GI-i2)

Figure 5.3. Plot of condition number for t=4.4 mm when varying the number of
waveguide modes used.

99

 

20

 
 
 
  
 

 

0 .4 ............................................................................................

3 -20 ~ ----------------------------------------------------------------------------------------------------
E

.2 .
g
E -40 .. - - - --------------------------------------------------------------------------- g
5 —a— 60 waveguide modes l
g '60 d """""""""""""""""""""""""""""""""""""""""""""" + 40 waveguide modes """"" t
E, —e— 30 waveguide modes

5 '80 """""""""""""""""""""""""""""""""""""""" + 22 waveguide modes """""""

+ 14 waveguide modes ;
-100~ -----------

 

 

 

 

 

+ 6 waveguide modes 5

-e— 2 waveguide modes l

’1 20 : 1 r r J.
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 5.4. Plot of shielding effectiveness for t=6.6 mm when varying the number of
waveguide modes used.

100

 

1 .OE+25

 

 

1.0E+20 1
‘ ‘ " +60 modes
+40 modes
1.0E+15 ~ +30 modes

\\ -- 2 modes
- AM ' +14 modes
\N... vvvvv M + 6 modes

1.0E+10 .. ______________ -_ - -
k “““““““““““““““““““““““ +2 modes
1.0E+05 \‘mhy

........

...........

.....................................
"""'VVTTrr vvvvvvvvvvvvv

 

 

Condition number

 

 

 

1 .0E+00 1W

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (Gt-i2)

Figure 5.5. Plot of condition number for t=6.6 mm when varying the number of
waveguide modes used.

101

 

20

 

  

   

 

o _ .................................................................................................... E
8
E .....................................................................................................
E
.2
.2 _____________________________________
E . . -a-60 waveguide modes 3 1
§ -60 4-- 3:. E +40waveguide modes E9?
__ i '. ‘ - , ‘ ‘
e . _ +30wavegu1de modes l
a _ E
g -30 ................................................................. +22 waveguide modes E.
'- +14 waveguide modes
_100 . __________ . ___________________________________________________________ E +6 waveguide modes .
: +2 waveguide modes .
'120 T T T T E
0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GI-12)

Figure 5.6. Plot of shielding effectiveness for t=7.7 mm when varying the number of

waveguide modes used.

102

 

1.0E+25

 
  

.
- ,-- _ _v4
5;.- .4 {:3 .‘.'. ._.j.‘
u“.- _. _-
q ..4 1‘ v Q . ":"-
1.0E+20 w».- .;_._,,._.,.:=.:.-
...~_- _-,- L- --

i-B— 60 modesi

+ 40 modes

1.05.45 «7 ..
+30 modes

 

 

E. """" l+14 modes

1.0E+10 k“ ____________________________________________________________

‘4‘
-------------------
......................................
------------------------------
__________________________

Condition number

'—+—6modes
+2modesj

105105 “WE

 

,o..
I ‘~. .....
I

~
‘.~.

......

....

~
“.“--
-,_..

  

.
.
a 1
._ . .
..........
...........
.............
~--.__ .........

- ‘_
.- ,-
~-.- -.- -0 _--
--._-::-u..._,--- ....

 

 

 

0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GHz)

1.0E+00

Figure 5.7. Plot of condition number for t=7.7 mm when varying the number of
waveguide modes used.

103

 

O)
O

 

 

 

 

 

4O -4 ........................................................................................................
A 20 ~ ---------------------------------------------------------------------------------------------------------
m
3
"' 0 ‘ """""""""""""""""" ."'r~" n " -~ _:“ " n _ r" -. ., n " n 2,-. -'
5 :‘ r:, '1‘ .. c: ._ H L“ A .. __“u u h 5' 1.. ._ U u ".4 r n. E h
o '3-1 r u u U L'J U U L”: ' H
3 -2o U a
I
5 .40 q ...................................................................................................
n
g , . _____ .v .1
.5, -60 - ................................................... 4 +40 waveguide modes ?
§ + 30 waveguide modes
I'-
-80 ~ ------------------------------------------------------- f + 22 waveguide modes
+ 14 waveguide modes ,
-100 --------- - w ~ +6waveguidemodes
1 -e- 2 waveguide modes J;
-120 Y r r 1 at
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GHz)

Figure 5.8. Plot of shielding effectiveness for t=8.8 mm when varying the number of
waveguide modes used.

104

 

O)

O
i
.i

b
0
ji

N
O

.__-

O

.......................................................................................

 

.....................................................................................................

is
o

   

 

   
  
  

-------- -i + 40 waveguide modes
5 + 30 waveguide modes :;

Transmission Coefficient (dB)
é: vb
O O

 

 

 

.30. --------------------------------------------------------- +22waveguidemodes;--~——---§

i +14 waveguide modes

400» -------------------------------------------------------------- +6waveguidemodes

+2 waveguide modes

420 i . . r i
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (GI-l2)

Figure 5.9. Plot of shielding effectiveness for t=9.9 mm when varying the number of
waveguide modes used.

105

 

 

 

 

 

 

 

 

 

30
0 r
a -30 .
E
.2
e -60 .. ..............................................................................
g . - __
5-90. we“ ....................................
:3
E ‘ _ +12waveguide modes
3120 ~ g ----------------- + 10 waveguide modes
.—
' V -x— 6 waveguide modes
-150 ........... l .......................................... +4waveguidemodes
f 7'7 -e- 2 waveguide modes
-180 r r r .
0.0 10.0 20.0 30.0 40.0 50.0
Frequency (GI-12)

Figure 5.10. Plot of shielding effectiveness for t=18.0 mm when varying the number
of waveguide modes used.

106

 

30 7 i

Transmission Coefficient (dB)
8

 

 

 

 

 

 

 

-120 ~ » . -
_ ‘ ‘ .. .9. 6 waveguide Rimes
-150 ._ 11‘. ~ 4‘ --------------- +4 waveguide modes
- . —a— 2 waveguide modes
-1eo . r .- — . . . .
0.0 10.0 20.0 30.0 40.0 50.0

Frequency (3"!)

Figure 5.11. Plot of shielding effectiveness for t=26.0 mm when varying the number
of waveguide modes used.

107

 

60 w‘fi

...............................................................................................

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a ..........................................................
B
E ‘ ............. - M ..............................................
0 I
s i
8 .
o .........................................................................
c
.2
3
_E. -90 "i ...........................................
m
E -120 +4waveguide modes ......................................................
+2 waveguide modes
-150 «i- ----------------------------------------------------------------- , --------------------------------
’180 I —r‘ 1 r
0.0 10.0 20.0 30.0 40.0 ' 50.0

Frequency (GHz)

Figure 5.12. Plot of shielding effectiveness for t=40.0 mm when varying the number
of waveguide modes used.

108

CHAPTER 6

CONCLUSIONS

A mode-matching approach for computing the shielding effectiveness of a doubly-
periodic array of apertures in a thick conducting screen has been presented. The
effect of choosing thicknesses much larger than the aperture is the expected result
of lower transmission of power. Mode-matching has shown some agreement with the
waveguide below cutoff formula and strong agreement with other published results
using mode-matching. The technique does suffer from an inability to yield meaning-
ful data when the thickness is increased to a point where inaccurate solutions are
produced. The use of Singular Value Decomposition led to the conclusion that the
condition number is well beyond the accuracy of the computer when the thickness
is increased a great deal. To produce useful and accurate solutions, the number of
waveguide modes included in the analysis is reduced, which also reduces the condition
number.

This dissertation considers mode-matching applied to apertures that are rectan-
gular and circular in shape. Future work will consider hexagonal and other shapes
for aperture arrays in thick screens. The shielding prediction is more flexible than
the waveguide below cutoff approach, and the result is more exact. Other benefits
include observance of the Wood’s anomaly, a strong reliance on the well-known prin-
ciples of waveguide theory, and practical application to the prediction of shielding
performance of aluminum honeycomb and other doubly-periodic structures.

The measurements performed as part of this study were successful in confirming
some of the results that were predicted numerically. Future work will be done in order
to measure the effects of changing the incidence angle, predicting the occurrence and

overall effect of the Wood’s anomaly, determining the impact of using smaller and

109

larger sample sizes, and using other measurement techniques.

110

APPENDICES

111

 

APPENDIX A

DERIVATION OF 2-D FOURIER SERIES REPRESENTATION OF A
PERIODIC FUNCTION IN SKEWED COORDINATES

Following the method of [34], a Fourier Series expansion is applied to a periodic poten-
tial function in a skewed coordinate system. Figure A.1 shows a lattice configuration
with periodicity in two directions, one along the y-axis and the other along a direction

at an angle 0 with respect to the y-axis. If r is defined as

1‘ =lld1+l2d2,

where 11 and Z2 are integers, and d1 and d2 are vectors describing the direct lattice,

the reciprocal lattice can be defined by the vectors b1 and b2, and they must obey

bi'dk=6ik7 Z,k=1,2

or
b1 .1. £12
and
()2 _L d1.
Letting

d1 = dull? + dlyga
d2 = d2x5? + dzygi
b1 :; b19353 + blyga

b2 = b2xi + b2yga

112

and

 

dlzr: dly
D 2 row vectors, (A.1)
(12:1: d2y
blx b2:
B 2 column vectors, (A2)
bly b2y
then
DB = dlxbla: + dlybly dlzb2x + dlyb2y (A3)
.. d2xblx + d2yb1y d2xb2x + d2yb2y
_ d. -b. all . b2
. d2 - b. at - b2
1 0
0 1
Therefore,
B = D“,

and any point r in the x-y plane can be described using coordinates (51,62) such that

7‘ = 51611 + 52612

where
51 = 7'51, (AA)

52 = 7' ' b2- (A.5)

A periodic function F (:r, y) in the direct lattice has the same value at the points

r = (:12, y) and r’ = r +l1d1 +12d2, where ll and 12 are integers. The Fourier Series

113

 

Representation of F (x, y) in skewed coordinates is obtained by using
F(x, y) = f (51, £2) = Z ewe—27W ("151 + "62), (A.6)

where m and n are integers, cm" are the Fourier series coefficients, and f has a period

of unity in (51,52) coordinates. Substituting (A.4) and (AS) into (A.6) leads to

17(3), y) = Z Cmne-27l'j[m(b1 ' 7') + ”(b2 ° 7)] (A7)
01‘
F($, y) = E: C e—27i'j[h - 1‘],
where
h = mbl ‘l' nbg.

Since dxdy = deéldfg, where 3,; is the unit cell area, the Fourier coefficients cm are

found using

1 ' .
Cmn = ELF($,y)e+2m[h rldxdy.

Using Figure A.2, which was adopted from [19], (A.1) can be rewritten as

all 0
D = (A.8)

d2 COS Q50 d2 SlIl Q50

Substituting the elements from (A.8) into (A.3) leads to

1
d1b1x+0= 1 =>blz = —,
d1

(111)254—0201?be =0,
. —1
d2 cos dob” + (12 SH] 460ny = 0 => bly = — cot Q50,

d1

114

d2 COS ¢ong + (12 sin (250be = 1 fi b2y :

d2 8111 (150
Rewriting b1 and b2 gives
1 - 1
b1 d—IIB — a: C01} Q5031,
b - i csc (b
2 — d2 0y:
and using those results in (A.3) leads to
' 1
DB 2 d1 0 3;
d2 cos c150 d2 sin (150 all cot qio Ell; csc (150
F
1 0
O 1

 

L

(A.7) can now be expressed as

Fe, y) = z c...e-2vrjim<b1 -r> + nu» - r>1

m,n

: Z cmne—27rj [m ((%i — 31; cot 4303)) ~r) + n ((2%; csc $03)) -r)

l 1

: Zcmne
mn

115

 

—27rj [fin—2: — g; cot qioy + 5’; C80 (150.71]

1
— d—2 csc $0.

0

 

"<

 

 

 

   

 

 

 

 

 

 

 

 

 

Figure A.1. Doubly-periodic direct lattice

116

’X

4%

 

 

 

 

 

 

 

 

., a»)

Figure A.2. Alternate description of doubly-periodic lattice

117

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118

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