Wave Scattering Theory: A Series Approach Based on the Fourier Transformation

Wave Scattering Theor:

Springer Berlin Heidelberg New York Barcelona Hongkong London Milan Paris Singapore Tokyo

Hyo J. Eorn

Wave Scattering Theory A Series Approach Based on the Fourier Transformation

With 62 Figures

t Springer

Professor Hyo J. Eom

Korea Advanced Institute of Science and Technology Department of Electrical Engineering 373-1, Kusong-dong, Yusong-gu 305-701 Teajon I Korea

ISBN-13:978-3-642-63995-1 Springer-Verlag Berlin Heidelberg New)brk

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Die Deutsche Bibliothek - CW-Einheitsaufnahme Eom, Hyo J.: Wave scattering theory: a series approach based on the Fourier transformation 1 Hyo J. Eom. - Berlin; Heidelberg; New York; Barcelona; Hongkong ; London; Milan; Paris; Singapore; Tokyo: Springer, 2001

ISBN-13:978-3-642-63995-1 e-ISBN-13:978-3-642-59487-8 DOl: 10.1007/978-3-642-59487-8

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Preface

The Fourier transform technique has been widely used in electrical engineering, which covers signal processing, communication, system control, electromagnetics, and optics. The Fourier transform-technique is particularly useful in electromagnetics and optics since it provides a convenient mathematical representation for wave scattering, diffraction, and propagation. Thus the Fourier transform technique has been long applied to the wave scattering problems that are often encountered in microwave antenna, radiation, diffraction, and electromagnetic interference. In order to u~derstand wave scattering in general, it is necessary to solve the wave equation subject to the prescribed boundary conditions. The purpose of this monograph is to present rigorous solutions to the boundary-value problems by solving the wave equation based on the Fourier transform. In this monograph the technique of separation of variables is used to solve the wave equation for canonical scattering geometries such as conducting waveguide structures and rectangular/circular apertures. The Fourier transform, mode-matching, and residue calculus techniques are applied to obtain simple, analytic, and rapidly-convergent series solutions. The residue calculus technique is particularly instrumental in converting the solutions into series representations that are efficient and amenable to numerical analysis. We next summarize the steps of analysis method for the scattering problems considered in this book.

1. Divide the scattering domain into closed and open regions. 2. Represent the scattered fields in the closed and open regions in terms of

the Fourier series and transform, respectively. 3. Enforce the boundary conditions on the field continuities between the

open and closed regions. 4. Apply the mode-matching technique to obtain the simultaneous equa

tions for the Fourier series modal coefficients. 5. Utilize the residue calculus to represent the scattered field in fast conver-

gent series.

This monograph discusses time-harmonic wave scattering problems and a time factor exp( -iwt) is suppressed throughout the analysis. In each section, a set of simultaneous equations for the Fourier series coefficients is boxed. A set of the boxed simultaneous equations is the rigorous final formulation and

VI

must be numerically evaluated to further investigate the wave scattering behaviors. This book contains 9 chapters. In Chapter 1 electromagnetic scattering from rectangular grooves in a conducting plane is considered. In Chapter 2 electromagnetic wave radiation from multiple parallel-plate waveguide with an infinite flange is analyzed. In Chapter 3 electromagnetic, electrostatic, and magnetostatic penetrations into slits in a conducting plane are considered. In Chapter 4 electromagnetic wave guidance by a certain class of waveguides and couplers is considered. In Chapter 5 electromagnetic and acoustic wave scattering from junctions in rectangular waveguides is analyzed. In Chapter 6 wave scattering from rectangular apertures in a plane is studied. In Chapter 7 wave scattering from circular apertures in a plane is examined using the Hankel transform. In Chapter 8 wave scattering from an annular aperture in a conducting plane is considered. In Chapter 9 electromagnetic wave radiation from circumferential apertures on a circular cylinder is analyzed. All the work presented in this monograph was performed from 1992 through 2000 at the Korea Advanced Institute of Science and Technology (KAIST). My sincere thanks go to my former graduate students at KAIST (T. J. Park, K. H. Park, S. H. Kang, J. H. Lee, J. W. Lee, Y. C. Noh, K. H. Jun, Y. S. Kim, H. H. Park, S. B. Park, K.W. Lee, J. S. Seo, J. G. Lee, S. H. Min, J. K. Park, H. S. Lee, J. Y. Kwon, and Y. H. Cho), who carried out the tedious problem formulations under my guidance. My thanks also go to my wife and son for their patience with me while I was working on this monograph. Any comments and suggestions from readers to improve the monograph would be gratefully received.

Taejon, Korea Hyo J. Eom

Contents

Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. XI

Transform Definitions ........................................ XII

1. Rectangular Grooves in a Plane .......................... 1 1.1 EM Scattering from a Rectangular Groove in a Conducting

Plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . 1 1.1.1 TE Scattering [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 TM Scattering [7] ................................ 3 1.1.3 Appendix....................................... 4

1.2 EM Scattering from Multiple Grooves in a Conducting Plane 6 1.2.1 TE Scattering [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 TM Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Appendix....................................... 10

1.3 EM Scattering from Grooves in a Dielectric-Covered Ground Plane.. . . ...... .... .......... .... .. .... .... .. .... ..... 11 1.3.1 TE Scattering [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 1.3.2 TM Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13 1.3.3 Appendix....................................... 15

1.4 EM Scattering from Rectangular Grooves in a Parallel-Plate Waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.4.1 TE Scattering [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17 1.4.2 TM Scattering [17] ............................... 19

1.5 EM Scattering from Double Grooves in Parallel Plates [18] .. 21 1.5.1 TE Scattering ............... : . . . . . . . . . . . . . . . . . . .. 21 1.5.2 TM Scattering. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 24 1.5.3 Appendix....................................... 26

1.6 Water Wave Scattering from Rectangular Grooves in a Plane 29 References for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

2. Flanged Parallel-Plate Waveguide Array. .......... ....... 35 2.1 EM Radiation from a Flanged Parallel-Plate Waveguide. . . .. 35

2.1.1 TE Radiation [8,9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36 2.1.2 TM Radiation [10]. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. 36

VIII Contents

2.2 EM Radiation from a Parallel-Plate Waveguide into a Dielectric Slab ....................................... 37 2.2.1 TE Radiation ................................... 37 2.2.2 TM Radiation [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 42

2.3 TE Scattering from a Parallel-Plate Waveguide Array [18] ... 47 2.4 EM Radiation from Obliquely-Flanged Parallel Plates.. .. . .. 48 2.5 EM Radiation from Parallel Plates with a Window [24].. . . .. 51 References for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53

3. Slits in a Plane.. . . .. . . .. . . . . .. . . . . . . . . .. . . . . .. . . .. . . . . . .. 57 3.1 Electrostatic Potential Distribution Through a Slit in a Plane

[1] .................................................... 57 3.2 Electrostatic Potential Distribution due to a Potential Across

a Slit [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.3 EM Scattering from a Slit in a Conducting Plane .......... 63

3.3.1 TE Scattering [10] ...... : . . . . . . . . . . . . . . . . . . . . . . . .. 64 3.3.2 TM Scattering [11] ............................... 66

3.4 Magnetostatic Potential Distribution Through Slits in a Plane 67 3.5 EM Scattering from Slits in a Conducting Plane [13] . . . . . . .. 70

3.5.1 TE Scattering... ...... .... ...... .......... ....... 71 3.5.2 TM Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72

3.6 EM Scattering from Slits in a Parallel-Plate Waveguide. . . .. 74 3.6.1 TE Scattering [26] .. . . . .. . . . . . . . . .. . . . . . . . . . . . . . .. 74 3.6.2 TM Scattering [27] ............................... 77

3.7 EM Scattering from Slits in a Rectangular Cavity .......... 78 3.7.1 TM Scattering [29] ............................... 78 3.7.2 TE Scattering [30] . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . .. 80

3.8 EM Scattering from Slits in Parallel-Conducting Planes [31] . 82 References for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85

4. Waveguides and Couplers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87 4.1 Inset Dielectric Guide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87 4.2 Groove Guide [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90

4.2.1 TM Propagation........ .......... ...... ......... 91 4.2.2 TE Propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 93

4.3 Multiple Groove Guide [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95 4.4 Corrugated Coaxial Line [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 4.5 Coaxial Line with a Gap [13] ............................. 100 4.6 Coaxial Line with a Cavity [16] .......................... 102 4.7 Corrugated Circular Cylinder [18] ........................ 105 4.8 Parallel-Plate Double Slit Directional Coupler [23] .......... 111 4.9 Parallel-Plate Multiple Slit Directional Coupler [29] ......... 115 References for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 118

Contents IX

5. Junctions in Parallel-Plate/Rectangular Waveguide ...... 121 5.1 T-Junction in a Parallel-Plate Waveguide .................. 121

5.1.1 H-Plane T-Junction [4] ........................... 122 5.1.2 E-Plane T-Junction [5] ........................... 123

5.2 E-Plane T-Junction in a Rectangular Waveguide [6] ......... 125 5.3 H-Plane Double Junction [8] ............................. 129 5.4 H-Plane Double Bend [9] ................................ 133 5.5 Acoustic Double Junction in a Rectangular Waveguide [11] .. 135 5.6 Acoustic Hybrid Junction in a Rectangular Waveguide [15] .. 140

5.6.1 Hard-Surface Hybrid Junction ..................... 140 5.6.2 Soft-Surface Hybrid Junction ...................... 144 5.6.3 Appendix ....................................... 145

References for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147

6. Rectangular Apertures in a Plane ........................ 149 6.1 Static Potential Through a Rectangular Aperture in a Plane. 149

6.1.1 Electrostatic Distribution [3] ....................... 149 6.1.2 Magnetostatic Distribution ........................ 151

6.2 Acoustic Scattering from a Rectangular Aperture in a Hard Plane [7] .............................................. 153

6.3 Electrostatic Potential Through Rectangular Apertures in a Plane [9] .............................................. 155

6.4 Magnetostatic Potential Through Rectangular Apertures in a Plane [10] ............................................ 157

6.5 EM Scattering from Rectangular Apertures in a Conducting Plane [11] ............................................. 160

6.6 EM Scattering from Rectangular Apertures in a Rectangular Cavity [18] ............................................ 164

References for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170

7. Circular Apertures in a Plane ............................ 173 7.1 Static Potential Through a Circular Aperture in a Plane .... 173

7.1.1 Electrostatic Distribution [1] ....................... 173 7.1.2 Magnetostatic Distribution [4] ..................... 175

7.2 Acoustic Scattering from a Circular Aperture in a Hard Plane [6] .................................................... 176

7.3 EM Scattering from a Circular Aperture in a Conducting Plane179 7.4 Acoustic Radiation from a Flanged Circular Cylinder [15] ... 185 7.5 Acoustic Scattering from Circular Apertures in a Hard Plane

[17] ................................................... 187 7.6 Acoustic Radiation from Circular Cylinders in a Hard Plane

[19] ................................................... 193 References for Chapter 7 ..................................... 196

X Contents

8. Annular Aperture in a Plane ............................. 199 8.1 Static Potential Through an Annular Aperture in a Plane ... 199

8.1.1 Electrostatic Distribution [4,5] ..................... 199 8.1.2 Magnetostatic Distribution [4] ..................... 202

8.2 EM Radiation from a Coaxial Line into a Parallel-Plate Waveguide [6] .......................................... 204

8.3 EM Radiation from a Coaxial Line into a Dielectric Slab [10] 208 8.4 EM Radiation from a Monopole into a Parallel-Plate

Waveguide [17] ......................................... 213 References for Chapter 8 ..................................... 216

9. Circumferential Apertures on a Circular Cylinder ........ 219 9.1 EM Radiation from an Aperture on a Shorted Coaxial Line [1]219

9.1.1 Field Analysis ................................... 219 9.1.2 Appendix ....................................... 222

9.2 EM Radiation frolll Apertures on a Shorted Coaxial Line [3] 223 9.3 EM Radiation from Apertures on a Coaxial Line [4] ........ 225 9.4 EM Radiation from Apertures on a Coaxial Line with a Cover

[6] .................................................... 228 9.5 EM Radiation from Apertures on a Circular Cylinder [10] ... 231

9.5.1 TE Radiation .................................... 232 9.5.2 TM Radiation ................................... 235

References for Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

A. Appendix ................................................. 239 A.1 Vector Potentials and Field Representations ............... 239

Index ......................................................... 243

Notations

EM: electromagnetic eil1(_l)m _ e-il1

Fm (",) = ",2 _ (m7r/2)2

H~1)(",),H~2)(",): nth order Hankel functions of the first and second kin

i = v'-I Im{··· }: imaginary part of { ... }

I n (,,,), Nn (",): nth order Bessel functions of the first and second kinds

PEe: perfect electric conductor

(1',1>,2) : unit vectors in cylindrical coordinates

Re{-··}: real part of { ... }

(x, fj, 2) : unit vectors in rectangular coordinates

c5mn : Kronecker delta

co = 2 Cm = 1 (m = 1,2,3, ... )

w: angular frequency ( ... )*: complex conjugate of ( ... )

Transform Definitions

Note that J(() is called a transform of f(x) and J(() is called an inverse transform of f(x). The respective transform pairs are as follows:

Fourier transform

J(() = i: f(x)ei(x dx

f(x) = ~ 100 J(Oe-i(x d( 27f -00

J((,11) = i:i: f(x,y)exp(i(x+il1y)dxdy

1 100 100 -f(x, y) =:t;2 -00 -00 f((, 11) exp( -i(x - il1Y) d(dl1

Fourier cosine transform

J(() = 100 f(x) cos (x dx

f(x) = - J(()cos(xd( 2100

7f 0

Fourier sine transform

J(() = 1000 f(x) sin (x dx

2100 -f(x) = - f(() sin (x d(

7f 0

1. Rectangular Grooves in a Plane

1.1 EM Scattering from a Rectangular Groove in a Conducting Plane

z

Incident wave Scattered wave

Fig. 1.1. A rectangular groove in a conducting plane

A rectangular groove in a conducting plane is a canonical structure in electromagnetic scattering study. Electromagnetic wave scattering from a groove in a conducting plane was considered in [1-5] due to its practical applications in optics and microwaves. Practical applications, for instance, include a design of optical-video disks and a radar-cross-section estimation. In the next two subsections we will consider TE (transverse electric to the y-axis) and TM (transverse magnetic to the y-axis) scattering from a two-dimensional rectangular groove in a perfectly-conducting plane.

H. J. Eom, Wave Scattering Theory© Springer-Verlag Berlin Heidelberg 2001

2 1. Rectangular Grooves in a Plane

1.1.1 TE Scattering [6]

Consider TE scattering from a rectangular groove in a perfectly-conducting plane. In region (I) (z > 0) a uniform plane wave Et(x, z) is assumed to be incident on a rectangular groove in a perfectly-conducting plane. Region (II) (-d < z < 0 and -a < x < a) is an infinitely-long rectangular groove engraved in the y-direction. The wavenumbers in regions (I) and (II) are ko (= W.jILofo) and k (= w..(ii€), respectively, where IL = ILrILo and f = frfO.

The total E-field in region (I) is a sum of the incident, specularly-reflected, and scattered fields

E~(x, z) = exp(ikxx - ikzz)

E;(x, z) = - exp(ikxx + ikzz)

E;(x,z) = 21 ['Xl E;(()exp(-i(x+i~oz)d( 1f i-oo

(1)

(2)

(3)

where kx = ko sin (), kz = ko cos (), and ~o = Jk5 - (2. We include the reflected field E;(x, z) in the total field expression, for convenience, although its inclusion is unnecessary. The total transmitted field in region (II) is represented in terms of the modal coefficient Cm

00

Et(x, z) = L Cm sinam(x + a) sin em (z + d) (4) m=l

where am = m1fj(2a) and em = Jk2 - a~. To determine the unknown coefficient Cm , we enforce the boundary con

ditions on the tangential E- and H-field continuities. The tangential E-field continuity at z = 0 is

Ixl < a Ixl >a.

Taking the Fourier transform of (5) yields 00

E;(() = L Cm sin(emd)ama2 Fm((a) . m=l

(5)

(6)

The tangential H-field continuity along (-a < x < a) at z = 0 is written as

2ikzeikzx _ roo i2~o E;(()e-i(X d( i-oo 1f

~ cm~m. ( ) ( ) = L...J --- smam x + a cos emd . m=l ILr

(7)

We multiply (7) by sinan(x + a) and integrate from -a to a to obtain

1.1 EM Scattering from a Rectangular Groove in a Conducting Plane 3

(8)

where

AI(ko) = i: a2Fm«a)Fn(-(a)lI:od( . (9)

It is convenient to transform Al (ko) into a numerically-efficient form by performing a contour integration. The result is available in Subsect. 1.1.3 to give

(10)

Note that Al (ko) is a numerically-efficient integral, which vanishes in highfrequency limit; thus for koa -+ 00,

y'k'5 - a;' Al (ko) -+ 271" 2 8mn .

aam (11)

Substituting (11) into (8) yields an approximate high-frequency solution, which agrees with that in the Kirchhoff approximation.

The far-zone scattered field at distance r from the origin is shown to be

E~«(J8,(J) = J2~rexp(ikor-i7l"/4)COS(J8E~(-kosin(Js). (12)

1.1.2 TM Scattering [7]

Consider a TM wave impinging on a rectangular groove in a conducting plane. In region (I) the total H-field is a sum of the incident, reflected, and scattered fields as

H;(x, z) = exp(ik.,x - ikzz)

H;(x, z) = exp(ik.,x + ikzz)

H;(x, z) = -21 roo if;«) exp( -i(x + ill:oz) d( . 71" i-oo

In region (II) the total transmitted field is 00

H~(x, z) = L em cosam(x + a) cos~m(z + d) . m=O

(13) (14)

(15)

(16)

Applying the Fourier transform to the tangential E-field, E.,(x,O), continuity along the x-axis yields


(17)

The tangential H-field continuity along the boundary (-a < x < a and z = 0) requires

00

= L Cm cosam(x + a) cos(emd) . (18) m=O

Multiplying (18) by cosan(x + a) and integrating from -a to a, we obtain

2ikza2 Fn(kza) • 00

= ~ L cmema2 sin (em d) ill (ko) - CnlLCn cos (en d) 7r€r m=O

(19)

where

ill(ko) = i: a2 Fm ((a)Fn (-(a)(2/bol d( . (20)

Performing the residue calculus, we get

() 27r€n - ( ) ill ko = t5mn - ill ko aJk3 -a~

(21)

where ilt{ko) -+ 27r€n/(aJk3 - a~)t5mn in high-frequency limit (koa -+ 00). The far-zone scattered field at distance r is

H;((}s,(i) = J ;:r exp(ikor - i7r /4)cos (}Ji;( -ko sin(}s) . (22)

1.1.3 Appendix

Consider

Al(ko) = i: a2Fm((a)Fn(-(a)/bod(. (23)

When m + n is odd, Al (ko) = O. When m + n is even, Al (ko) is rewritten as

Al(ko) = 100 2 1 - (-I)nexp(i2(a)/bo d(. (24) -00 ((2 - a~)((2 - a~Ja2

Integrating along the deformed contour r l , n, r3 and r4 in the upper-half plane, we get

Al (ko) = 27rJk32 - a~ t5mn - .11 (ko) aam

(25)


where Sl = C 0~51) (0.5i)I-1.5, h = (a - l)i, t2 = (-a - l)i, t3 = ({J - l)i,

t4 = (-{J - l)i, and

A(t) = (_1)11l't1- O.5 exp(pt)erfc( vPt) 1-1

+21- 1y'7rpO.5-1 ~)2l- 2r - 3)!!( -2pW . (30) r=O

Note that erfc(·) denotes the complementary error function and p = 2koa. Consider

(31)

An evaluation of ,01 (ko) is similar to that of Al (ko) discussed earlier. When m + n is odd, 'o1(ko) = O. When m + n is even,

where

J = (Xl -4i(-1)n exp[2koa(i-v)](1+iv)2 dv 1 10 (koa)2[(l + iv)2 - a2][(1 + iv)2 - {J2]VV( -2i + v)

J2 = roo 4i(1 + iv)2 dv 10 (koa)2[(1 + iv)2 - a2][(1 + iV)2 - {J2]VV( -2i + v)

-4i ( a . 1 {J. 1 f.I) = - sm - a + sm - iJ (koa)2(a2 - {J2) v'1- a 2 V1- {J2

a = am/ko and {J = an/ko.

1.2 EM Scattering from Multiple Grooves in a Conducting Plane

(32)

(33)

(34)

(35)

Electromagnetic wave scattering from multiple rectangular grooves in a conducting plane has been considered extensively in [9-11] due to optical diffraction grating and polarizer applications. Most previous studies have dealt with electromagnetic wave scattering from infinite periodic rectangular grooves by utilizing Floquet's theorem. In the next two subsections we will study TE and TM scattering from finite rectangular grooves engraved in a perfectlyconducting plane without recourse to Floquet's theorem. The present section is an extension of Sect. 1.1, which discusses scattering from a single groove in a conducting plane.

1.2 EM Scattering from Multiple Grooves in a Conducting Plane 7

Incident wave

~ .... ..... • ... . - /

1=-L1

PEe

z

Scattered wave

~:??Fr X ..~

1/=1 I=L2 1 1 Region (II)

1+----+1.1 I

Fig. 1.3. Multiple rectangular grooves in a conducting plane


Consider a uniform plane wave E;(x, z) incident upon a perfectly conducting plane with multiple rectangular grooves (width 2a, depth d, and period T). Regions (I) and (II) denote the air and a groove medium whose wavenumbers are ko = wJ/1of.o = 21f/>" and k = w..jJiE (/1 = /1o/1r and f. = f.Of. r ), respectively. The total E-field in region (I) is composed of the incident, specularlyreflected, and scattered fields as


E~(x,z) = -exp(ikxx+ikzz)

E;(x,z) = -21 {ex; E;(()exp(-i(x+i~oz)d( 1f Lex;

(1)

(2)

(3)

where kx = ko sin (1, kz = ko cos (1, and ~o = Jk'5 - (2. The total transmitted field inside the lth groove of region (II) is

ex; Et(x, z) = L c~ sin am(x + a -IT) sin~m(z + d) (4)

m=1

where am = m1f/(2a) and ~m = Jk2 - a;". The tangential E-field continuity at z = 0 for integer l (-L1 :S l :S L 2 ) is

given by

E;(x, O) = {~t(x, 0), Ix -lTI < a Ix -lTI > a.

(5)


Applying the Fourier transform to (5) gives

L2 00 E~(() = L L c~ei(IT sin(~md)ama2 Fm((a) . (6)

1=-L1 m=l

The tangential H-field continuity along (IT - a) < x < (IT + a) at z = 0 is given by

2ikzeikzx - /00 i211:0 E~(()e-i(X d( -00 7r

00 I~ = - L Cm m sin am (x + a -IT) cos(~md) . (7)

m=l J.lr

We substitute (6) into (7), multiply (7) by sinan(x + a - rT), and integrate from (rT - a) to (rT + a) to get

(8)

where

A2(ko) = i: a2 Fm((a)Fn( -(a) 11:0 exp [i(l- r)(T] d( . (9)

Using the residue calculus, we evaluate A2(ko) in Subsect. 1.2.3

y'k5 - a~ -A2(ko) = 27r 2 OmnOlr - A2(ko)

aam (10)

where A2(ko) is a branch-cut integral in numerically-efficient form. It is also possible to transform A2(ko) into a series whose nth term is on the order of (koa)0.5-n; hence, .12(ko) vanishes for large koa.

The far-zone scattered field at r is

E~(()s, ()) = J 2~r exp(ikor - i7r /4) cos()sE~( -ko sin()s) . (11)

1.2.2 TM Scattering

Consider scattering of a uniform plane wave H~(x, z) incident on a perfect conducting plane with a finite number ofrectangular grooves (width 2a, depth d, and period T). Regions (I) and (II), respectively, denote the air and rectangular grooves, which lie in parallel with the y-direction. In region (I) the total H-field consists of the incident, reflected, and scattered components

1.1 EM Scattering from a Rectangular Groove in a Conducting Plane 5

1m ( s) Branch cut

-a n Re(s)

Branch cut

Fig. 1.2. Contour path in the (-plane

where the first term is a residue contribution at ( = ±am when m = nand the second term results from integration along the branch cut r3 and r4 associated with the branch point ko. Assuming ( = ko + ikov, we obtain a branch cut contribution

(26)

where h and I2 are in numerically-efficient integral forms

100 -4i(-1)nexp[2koa(i-v)]v'v(-2i+v) d h = v

o (koa)2[(l + iv)2 - a 2][(1 + iv)2 - ,82] (27)

100 4iv'v( -2i + v) d h= v o (koa)2[(1 + iv)2 - a 2][(1 + iv)2 - ,82]

-4i (~ . -1 ~. -1 ) = (koa)2(a2 _ ,82) - a sm a + ,8 sm,8 (28)

a = am/ko and,8 = an/ko. It is also possible to transform h into asymptotic series [8]

h = _ 2 exp(2ikoa) (_l)n (koa)2(a2 - ,82)

00

L Sz{ [A(h) - A(t2)]/a - [A(t3) - A(t4)]/,8} Z=1

(29)

1.2 EM Scattering from Multiple Grooves in a Conducting Plane 9

H;(x, z) = exp(ik",x - ikzz) (12)

H;(x, z) = exp(ik",x + ikzz) (13)

H;(x, z) = 21 roo ii;(() exp( -i(x + i~oz) d( (14) 7r i-oo

where k", = ko sinO, kz = ko cosO, and ~o = ..jk'5 - (2. Inside the lth groove of region (II) the total transmitted field is

00

H~(x,z) = L c~ cosam(x + a -IT) cos~m(z + d) (15) m=O

where am = m7r/(2a) and ~m = ..jk2 - a~. Applying the Fourier transform to the tangential E-field continuity along

the x-axis (z = 0) yields for every integer l

£2 00 (

ii;(() = L L C~~m sin(~md)-a2 Fm((a)ei(IT . (16) 1=-£1 m=O ~O€r

The tangential H-field continuity along (IT-a) < x < (IT+a) at z = 0 gives

2eik"", + 1=~1 ~ d~!m sin(~md) i: a2~:€~(a) (exp[-i((x -IT)] d(

00

= L c~ cosam(x + a -IT) cos(~md) . (17) m=O

We multiply (17) by cos an(x + a - rT) and integrate with respect to x from (rT - a) to (rT + a) to obtain

_2ik",eik"rT a2 Fn(k",a) • £2 00

= -~ L L C~~ma2 sin(~md)n2(ko) + c~acn cos(~nd) (18) 7r€r 1=-£1 m=O

where

n2(ko) = i: a2 Fm((a)Fn( _(a)(2 exp[i(l - r)(T]~ol d( . (19)

It is possible to transform n2(ko) into a numerically efficient form based on a residue integral technique as shown in Subsect. 1.2.3. The result is

() 27rcn -n 2 ko = 8mn8'r - n2(ko) .

avk5 - a~ (20)

The far-zone scattered field at r is

H;(Os,O) = exp(ikor - i7r /4)V 2~r cosOsii;( -ko sin Os) . (21)


1.2.3 Appendix

Consider

A2(ko) = I: a2 Fm((a)Fn( -(a) "'0 exp[i(l- r)(T] d( (22)

in the complex (-plane as shown in Fig. 1.2 of Subsect. 1.1.3. Integrating along the deformed contour r 1 , n, r3 , and r4 in the upper-half plane, we obtain

A2(ko) = 2: Jk5 - a~OmnOZr - A2(ko) (23) aam

where the first term is a residue contribution at ( = ±am when m = nand l = r, and the second term A2(ko) is due to integration along the branch cut r3 and n. When l = r, A2(ko) degenerates into Al (ko) as considered in Subsect. 1.1.3. When l =j:. r, we get

A2(ko) = 2{[( _l)m+n + l]h(qT) - (_l)m h(qT + 2a)

-( _l)n h(qT - 2a)}

where q = l - rand

h(c) = { exp(i(lclh/k5 - (2 d( ir4 a2((2 - a~)((2 - a;)

(24)

(25)

a = am/ko and (3 = an/ko. By letting ( = ko +ikov, we obtain a numericallyefficient integral

rOO iexp[kolcl(i - v)]Jv( -2i + v) h(c) = io (koa)2[(1 + iv)2 - a 2][(1 + iv)2 _ (32] dv .

The evaluation of Jl2(ko) is similar to that of A2(ko), leading to

21fcn - ( ) Jl2(ko) = omnOZr - Jl2 ko .

aJk5 - a~

When l = r, D2(ko) degenerates into Dl(ko) given in Subsect. 1.1.3. When l =j:. r,

D2(ko) = 2{[( _l)m+n + 1]J3(qT) - (_l)m h(qT + 2a)

-( -1)nJ3(qT - 2a)}

where q = l - rand

J3 (c) =

(26)

(27)

(28)

roo i(l + iv)2 exp[kolcl(i - v)] dv (29) io (koa)2Jv( -2i + v)[(l + iv)2 - a 2][(1 + iv)2 - (32]

a = am/ko and (3 = an/ko.

1.3 EM Scattering from Grooves in a Dielectric-Covered Ground Plane 11

1.3 EM Scattering from Grooves in a Dielectric-Covered Ground Plane

,"- 1 z

t x Region (I) E:1

Fig. 1.4. Rectangular grooves in a dielectric-covered ground plane

Surface wave scattering from multiple grooves in a dielectric-covered ground plane was studied for Bragg reflector and leaky wave antenna applications [13] . A use of multiple grooves in the leaky wave antenna can improve the antenna radiation characteristics in terms of bandwidth, gain, and beamwidth. In the next two subsections we will analyze a problem of TE and TM surface wave scattering from grooves in a dielectric-covered ground plane.


A surface wave, which is transverse electric (TE) to the x-axis, is incident on periodic rectangular grooves. Regions (I), (II), and (III), respectively, denote the air (wavenumber; kl = wy1i€1), a dielectric slab (wavenumber; k2 = w,jii72 = 21f/)..) , and an Nnumber of grooves (wavenumber; k3 = wJli€3). In region (I) the total E-field has the incident and scattered fields

(1)

(2)


where 1\,1 = Jki - (2 and kZ1 = Jk;, - ki. The E-field in region (II) similarly has the incident and scattered components

E~II (x, z) = sin kz2 (z + b)eik~x (3)

E;I(x,z) = 2~ i: [EiI«()eiI<2Z +E~I«()e-iI<2Z] e-i(xd( (4)

where 1\,2 = Jk~ - (2, kZ2 = Jk~ - k;" and the characteristic equation

tan(kz2 b) = _kkz2 determines kx. In region (III) (IT - a < x < IT + a and z1

-d - b < z < -b: l = 0,1, ... ,N - 1), the total E-field is a sum of discrete modes

00

m=1

where am = m7f/(2a), (m = 1,2,3, ... ), and ~m = Jk~ - a~. The tangential E-field and H-field continuities at z = ° yield

E~I «() = (1\,2 - 1\,1) EiI «() . 1\,2 + 1\,1

The tangential E-field continuity along (IT - a l = 0,1, ... ,N - 1) is

< x < IT + a, z

E;I (x, -b) = { ~~II (x, -b), Ix -lTI < a Ix -lTI > a.

Applying the Fourier transform to (7) yields

EII «() = ~ f: cl am sin(~md)(1\,2 + 1\,1)ei(IT a2 Fm«(a) . + 1=0 m=1 m 2 [1\,2 cos(1\,2b) - il\,1 sin(1\,2b)]

(5)

(6)

-b:

(7)

(8)

The tangential H-field continuity along (rT - a < x < rT + a and z = -b: r = 0,1, ... ,N - 1) requires

H!II (x, -b) + H;I (x, -b) = H;II (x, -b) . (9)

We substitute (6) and (8) into (9), multiply (9) by sinan(x + a - rT), and integrate from (rT - a) to (rT + a) to obtain

(10)

where

A4(kt) = 100 1\,2 [1\,1 - ~1\,2tan(1\,2b)] -00 1\,2 -11\,1 tan(1\,2 b)

·a2 Fm«(a)Fn( -(a) exp[i(l - r)T] d( . (11)


Using a contour integral technique as shown in Subsect. 1.1.3, it is possible to transform (11) into a numerically-efficient form.

We represent the transmitted and reflected fields at x = ±oo in regions (II) and (I) as

E;I (±oo, z) = K± sin kZ2(Z + b)e±ik.x

E;(±oo,z) = K±sin(kz2 b)exp(±ikxx - kzlZ)

where

(12)

(13)

~l ~ iam sin(~md)kz1kz2 (. ) 2 ( ) () K± = 6 !::::t Cm kx(1 + kz1b) exp "TlkxlT a Fm "Tkxa. 14

The time-averaged incident, transmitted, and reflected powers (Pi, Pt, and

/ 12 / I 12 kx (l+kz1b) Pr ) are Pt Pi = 11 + K+ ,Pr Pi = K_ ,and Pi = 4wJl kzl . The

far-zone scattered field at x = r sin Os and z = r cos Os is

EyS(r, Os) = J k1 cosOs exp(ik1r - i1r/4) 21Tr

~ ~ I am sin(~md)/1;2ei(IT a2 Fm«(a) I . 6 !::::t Cm [/1;2 cos(/1;2b) - i/1;l sin(/1;2b)] (=-kl sinO •. (15)

1.3.2 TM Scattering

We will consider TM wave scattering from grooves that are periodically engraved in a dielectric-covered ground plane. In region (I) the total H-field consists of the incident and scattered fields

H;I (x, z) = exp(ikxx - kZ1Z)

H; (x, z) = 2~ i: if; «() exp( -i(x + i/1;l Z ) d(

(16)

(17)

where /1;1 = Jki - (2 and kzl = Jki - ki. In region (II) the total H-field has the incident and the scattered fields

H ill ( ) _ coskz2(z + b) ik.x y x, Z - (k b) e cos z2

(18)

H;I(X,z) = 21 100 [if~?«()ei"2Z +if~I«()e-i"2Z] e-i(xd( 1T -00

(19)

where /1;2 = Jk~ - (2,kz2 = Jk~ - k~, and tan(kz2 b) = kkz1f 2. In region z2fl

(III) the total transmitted H-field is

00

H;Il(X,Z) = L c~ cosam(x+a-1T) cos~m(z+b+d) (20) m=O


where am = m1f /(2a), (m = 0,1,2, ... ), and ~m = Jk~ - a;". The tangential E-field and H-field continuities at z = ° give

ii~I(() = (£1~2 - £2~1) ii!/(() . (21) £1~2 + £2~1

The tangential E-field continuity at z = -b is given by

EII(x -b) = {EiII(X' -b), Ix -lTI < a (22) x' 0, Ix - lTI > a .

Taking the Fourier transform of (22) gives

N-l 00 ·(IT 2 iiII (() = " " l £2~m sin(~md)((£1~2 + £2~1)el a Fm((a)

+ ~ ~ cm 2£3~2 [£2~1 cos(~2b) - i£1~2 sin(~2b)]· (23)

We multiply the tangential H-field continuity along (rT - a < x < rT + a and z = -b: l = 0,1, ... ,N - 1) by cos an(x + a - rT), and integrate from (rT - a) to (rT + a) to get

kxa2 Fn(kxa) exp(ikxrT) cos(kz2b)

~ ~ l [£2~m sin(~md)] 2 ( ) • r = ~ ~ cm 21f£3 a {l4 kl + lCncnacos(~nd) (24)

where

100 (2 [£1~2 - i£2~1 tan(~2b)] a2 Fm((a)Fn ( -(a) di (25) {l4(kt} = ."

-00 ~2 [£2~1 - i£1~2 tan(~2b)] exp[i((r -l)T] .

It is expedient to transform {l4 (k1) into a numerically-efficient integral based on the contour integral technique. An analytic evaluation of {l4(k1) is summarized in Subsect. 1.3.3.

are The transmitted and reflected fields at x = ±oo in regions (II) and (I)

H II(± )=K COSkz2(Z+b) ±ikzx y 00, z ± (k b) e cos z2

H:(±oo, z) = K± exp(±ikxx - kzlZ)

(26)

(27)

where N-l 00

K±= L L c~ l=O m=O

[ =F£~£2kzlk~2~m sin(~md)a2 Fm(=Fkxa) eXP(=FikxlT)] (28) . £3 cos(kz2b) [£1£2(k~1 + k~2) + kzlb(£~k~2 + £~k~l)]

The transmission and reflection coefficients are T (= Pt! Pi) = 11 + K+ 12 and {! (= Pr / Pi) = IK _12. The far-zone scattered field at x = r sin 08 and z = rcosOs is


1.3.3 Appendix

-a n

(n=m)

1m (~)

Branch cut

Branch cut

Fig. 1.5. Deformed contour path in the (-plane

When l = T, n4 (k1 ) is rewritten as

Re(~)

n4 (kt} = 100 [40111:2 - ~f211:1 tan(1I:2b)] (2 a2 Fm((a)Fn( -(a)d(. (30) -00 40211:1 - 140111:2 tan(1I:2b) 11:2

When m + n is odd, n4 (kt} = 0 by odd symmetry. When m + n is even, n4 (kt} is

{]4(k1 ) =

100 [40111:2 - if211:1 tan(1I:2b)] 2(2 [1- (_1)mei2<a] -00 40211:1 - i€111:2 tan(1I:2b) a211:2((2 - a~)((2 - a~J de·

(31)

In view of Fig. 1.5, the integrand has a pair of branch points at the zeros of 11:1 and two single poles at ( = ±am when m = n. Also it has a finite number of single poles at ( = ±kx (surface wave poles) that are solutions to


the equation t:2K1 = it:1K2 tan(K2b). The integral contour can be deformed in the upper-half plane by using the residue calculus, yielding

f.?4(k1) =

8rl {27r€n a [t:1K2m - it:2 K1m tan(K2mb)] 8 a2 K2m t:2K1m - it:1K2m tan(K2m b) nm

- r fa«()d( - r fb«()d( + L irs ir4 kz

47rkzkz1(t:~k~2 +t:~k~1) [1- (_I)me2ikza] } (32)

[t:1t:2(k~1 + k~2) + kz1b(t:~k~2 + t:~k~1)] (k; - a~)(k; - a~J

where 8nm : Kronecker delta, K1m = y'k~ - a~, K2m = y'k~ - a~, kz : zeros of [t:2K1 - it:1K2 tan(1I:2b)], kZ1 = y'k; - k~, and kz2 = y'k~ - k;. Furthermore we note that

[t:1K2 +it:2K1 tan(K2b)] 2(2 [1- (_I)mei2(a]

fa ( =-( ) t:2K1 + it:1K2 tan(K2b) K2«(2 - a~)((2 - a;)

() [t:1K2 - it:2K1 tan(K2b)] 2(2 [1- (_I)mei2(a] fb ( = t:2K1 - it:1 K2 tan(K2b) K2«(2 - a~)«(2 - a;) .

When l ¥- r, f.?4(kd is

(33)

(34)

f.?4(k1) = ['Xl [t:1K2 - ~t:211:1 tan(K2b)] (2a2 Fm(.(a)Fn ( -(a) d(. (35) i -00 t:2K1 - 1t:1K2 tan(K2b) K2 e1«r-I)T

The integrand has a pair of branch points at the zeros of K1 and a finite number of single poles at ( = ±kz that are solutions to the equation t:2K1 = it:1K2 tan(K2b). The integral contour depends on the sign of (l - r) as

f.? (k) {upper half plane when l > r (36) 4 1 lower half plane when l < r .

Using the residue calculus, we obtain

f.?4(k1) =

1

'"' 27rkzkzdt:~k~2 + t:~k~1)a4 Fm(kza)Fn( -kza) L..J [t:1t:2(k2 + k2 ) + k 1b(t:2k2 + t:2k2 )] eik.,(r-I)T kz z1 z2 z 1 z2 2 z1

- r ff«()d( - r ff«()d( when l > r irs ir4

'"' 27rkzkz1(t:~k~2 + t:~k~1)a4 Fm( -kza)Fn(kza) L..J [t:1t:2(k2 + k2 ) + k 1b(t:2k2 + t:2k2 )] eikz{l-r)T k., z1 z2 z 1 z2 2 z1

+ irs f~«()d( + ir4 ff!«()d( when l < r

(37)

where kz: zeros of [t:2K1 - it:1K2 tan(K2b)], kZ1 = y'k; - ki, and kZ2 = y'k~ - k;.


Pi • Incident

•

Region (II)

Region (I) ko,llo,to

z

b --•• Pt

Transmitted

~x

Fig. 1.6. Rectangular grooves in a parallel-plate waveguide

E~ (() = _ei2 l<ob Et (() . (4)

The continuity of tangential E-field at z = ° is ES( 0) = {Et(x,O), Ix-tTI<a

y x, 0, Ix - tTl> a . (5)

Taking the Fourier transform of (5) yields

L2 00

Et(()(l - ei2 l<Ob) = L L c!nei(IT sin(~md)ama2 Fm((a) . (6) 1=-L1 m=l

We multiply the tangential H-field continuity along (tT - a) < x < (IT + a) at z = ° by sin an (x + a - rT) and integrate with respect to x from (rT - a) to (rT + a) to obtain

kzsan exp(ikxsrT)a2 Fn(kxsa) L2 00 r ~

- an L L c!nama2 sin(~md)A5(ko) = cn na cos(~nd) (7) 2n ~r

1=-L1 m=l

where

A5(ko) = i: cot(r.:ob)a2 Fm((a)Fn( -(a)r.:o exp[i(t - r)(T] d( . (8)

Performing the residue calculus, we transform (8) into a rapidly-convergent series

1.4 EM Scattering from Rectangular Grooves in a Parallel-Plate Waveguide

A5(ko) = 2: Jk'5 - a~ cot (Jk'5 - a~b) 8mn8'r + A5(ko) aam

- . ~ 27r/i;5Al I A5(ko) = -1 L...J b«(2 _ a2 )«(2 _ a2)(a2

v=1 m n (=..jk5-(V7r/b)2

Al = [( _1)m+n + 1] exp(i(li - rlT)

-( _1)m exp [i(l(i - r)T + 2al]

-( _1)n exp [i(l(i - r)T - 2al]

The scattered field at x = ±oo is

E~(±oo, z) = L K'± sin(kzvz)e±ikzvx v

where v: integer, 1 ~ v ~ kob/7r, kzv = v7r/b, kxv = Jk'5 - k~v' and

The time-averaged incident, reflected, and transmitted powers are

19

(9)

(10)

(11)

(12)

(13)

R - kxsb (14) • - 4WIlo

Pr = ~ LkxvlK~12 (15) Wllo v

Pt = ~ {kxs[1 + 2Re(K~J + IKtI2] + L kxvlKtl2} (16) Wllo vI's


An electromagnetic wave, which is transverse magnetic (TM) to the x-axis, is incident on an N number of grooves. The total H-field in region (I) is composed of the incident and scattered fields

H;(x, z) = cos(kzsz)eikzsx (17)

1 100 -.( H~(x, z) = -2 COS/i;o(z - b)H;«()e-1 x d(

7r -00

(18)

where 0 ~ S < kob/7r, s: integer, kzs = s7r/b, kxs = Jk5 - k~s' and /i;o = Jk5 - (2. In region (II) (iT - a < x < iT + a and -d < z < 0: i = -L1 , •.• ,L2 ) the H-field is

00

H~(x, z) = L c!n cosam(x -iT + a) cos~m(z + d) (19) m=O

1.4 EM Scattering from Rectangular Grooves in a Parallel-Plate Waveguide 17

Note that

ft(() = - [t1K2 + ~t2K1 tan(K2 b)] fc(() (38) t2K1 + 1t1K2 tan(K2b)

ff(() = [t1K2 - ~t2K1 tan(K2 b)] fc(() (39) t2K1 - 1t1K2 tan(K2b)

f:-(() = - [t1K2 + ~t2K1 tan(K2 b)] fc(() (40) t2K1 + 1t1K2 tan(K2b)

ff(() = [t1K2 - ~t2K1 tan(K2 b)] fc(() (41) t2K1 - 1t1K2 tan(K2b)

fc(() = (2a4Fm((a)Fn(-(a) (42) K2 ei«(r-l)T

1.4 EM Scattering from Rectangular Grooves in a Parallel-Plate Waveguide

Corrugated waveguide structures may find many practical applications for microwave filter, polarizer, and antenna [15). In the next two subsections we will consider TE and TM scattering from a finite rectangular grooves periodically engraved in a perfectly-conducting parallel-plate waveguide. The present section analysis is applicable to scattering from a rectangular waveguide with multiple grooves, which is a typical microwave filter component.

1.4.1 TE Scattering [16)

Consider a TE wave Et(x, z) incident on rectangular grooves (width: 2a, depth: d, period: T, and total number of grooves: N) in a parallel-plate waveguide. Regions (I) and (II) denote a parallel-plate waveguide interior and grooves with wavenumbers ko (= wJ/-loto = 21f/>") and k (= w..jiiE, /-l = /-lo/-lr, and t = totr), respectively. The total E-field in region (I) has the incident and scattered components

(1)

(2)

where kzs = s1f/b, s: integer, kxs = Jk5 - k;s' and KO = Jk5 - (2. The E-field inside the lth groove is

00

E~(x, z) = L c!n sinam(x + a -IT) sin~m(z + d) (3) m=l

where am = m1f/(2a) and ~m = Jk 2 - a;". The continuity of tangential E-field at z = b gives


where am = m7r/(2a) and em = Jk2 - a~. The tangential E-field continuity at z = 0 yields

jiS(() = ~ ~ c' [ifoemSin(emd)] [(a2Fm~(a)ei"T] (20) y L...J L...J m f KO SlllKOb

1=-L1 m=O

The tangential H-field continuity at the aperture of grooves (rT - a < x < rT + a,z = 0: r = -£1' ... ,£2) gives

-ikxsa2 Fn(kxsa) exp(ikxsrT)

(21)

where

J15 (ko) = i: cot(Kob)a2 Fm((a)Fn ( -(a)Ki)I(2 exp[i(l - r)(Tl d( .(22)

Using the residue calculus, we transform J15 (ko) into

where

J1 (k ) - 27r€n 8nm8rl - ti (k ) 5 0 - _ Ik2 _ 2 (_ Ik2 _ 2 b) 5 0 ay 0 am tan y 0 am

- () ~ i27rkxw AI J15 ko = L...J b(k2 _ 2) (k2 _ 2) 2 w=o €w xw am xw an a

Al = [1 + (_I)m+n] exp(ikxwll- rlT)

-( _1)m exp[ikxwl(l- r)T + 2all

-(-I)nexp[ikxwl(l- r)T - 2all.

We evaluate the total scattered fields at x = ±oo as

H;(±oo, z) = L K; cos(kzvz)e±ik .. vx v

where 0 ~ v < kob/7r, v: integer, and L2 00

K;= L LC~ 1=-L1 m=O

=Ffoem sin(emd)a2 Fm(=Fkxva) exp(=FikxvlT) f€v b

The transmission and reflection coefficients are

T = Pt/Pi = 11 + K:12 + + L€vkxvlKtl2 €s X8 vi-s

(! = Pr/Pi = -k1 L€v kxvI K ;12 €s xs v

where 0 ~ v < (kob/7r), v: integer, and Pi = kxs€sb/(4wfO).

(23)

(24)

(25)

(26)

(27)

(28)

(29)

1.5 EM Scattering from Double Grooves in Parallel Plates [18] 21

1.5 EM Scattering from Double Grooves in Parallel Plates [18]

x

[=0 l=1 l=2. . . . . . . . l=l \ Fig. 1.7. Double rectangular grooves III a conducting paral el-plate waveguide

Corrugated waveguides are of fundamental interest in microwaves for their practical filter application. Electromagnetic wave scattering from rectangular grooves in a parallel-plate waveguide was studied in Sect. 1.4. In this section we will investigate scattering from double rectangular grooves in a metallic parallel-plate waveguide. A multiple groove scattering analysis given in this section is in continuation of the discussion given in Sect. 1.4.

1.5.1 TE Scattering

Consider a TE wave E~(x, z) incident on a finite number of double rectangular grooves in a conducting parallel-plate waveguide. Regions (I), (II), and (III) denote a parallel-plate waveguide interior, lower grooves, and upper grooves, respectively. The wavenumbers in regions (I), (II) , and (III) are ko (= W";f.lOfO), k2 (= kO";f.l2f2), and k3 (= kO";f.l3f3), respectively. In region (I) the incident and scattered fields are

E~(x, z) = exp(ikxsx) sin(kzsz) (1)

E;(x,z) = 21 roo [Et(()eiKZ + E~(()e-iKZ] e-i(xd( . (2) 7r J-oo

where kzs = s7r/h, 1::; S < koh/7r, s: integer, kxs = Vk5 -k;s' and /'i, = Vk5 - (2. In the lth groove (0 ::; l ::; L1 ) of region (II) the field is


00

E~I(X,Z) = L b~sinam(x+a-tTdsin~m[z+dl(t)] (3) m=l

m 1f r-=---,,..--where am = ~ and ~m = y'k~ - a;'. In the rth groove (0 ~ r ~ L2 ) of

region (III) the field is 00

n=l

h - n1f d - y'k2 2 were gm - 2g an TJn - 3 - gw

The Ey field continuity at z = 0 requires

- a + tTl < X < a + tTl otherwise


L1 00

E~(() + E:(() = L L b~ sin[~mdl(t)]F:n(() 1=0 m=l

where

Similarly from the Ey field continuity at z = h, we obtain

L2 00

(5)

(6)

(7)

E+(()eil<h + E:(()e-il<h = - L L c~ sin[TJnd2(r)]G~(() (8) r=O n=l

where r gnei«rT2H) [( -l)nei(g - e-i(g]

Gn (() = (2 2 . - gn

(9)

The Hx field continuity at z = 0 for (-a + tTl < X < a + lTd is

kzs exp(ikxsx) + ;1f i: Ii [E~(() - E:(()] e-i(xd(

1 00

= - L ~mb~ sinam(x + a -lTI) cos[~mdl(l)l· f.l2 m=l

(10)

Substituting E.+(() and E~(() of (6) and (8) into (10), multiplying (10) by sinap(x + a - UTI), and integrating over (-a + UTI < X < a + UTI), we get


kzsF;:(kxs )

= t f {sin[~mdl (l)]Il~!n + !!...- COS[~mdl (l)]8u[8pm } b!n [=0 m=l /-L2 L2 00

+ L L {sin[1Jnd2(r)]I2~~} c~ r=On=l

where

Il~!n = :7r i: tan~l\:h) F:r,(()F;: ( -() d(

12~~ = 21 roo . ~ h) G~(()F;:( -() d( . 7r 1-00 SIn I\:

Similarly from the Hx field continuity at z = h,

L1 00

kZ8 cos(kzsh)G~ (kxs ) = L L {sin[~mdi (l)]I3~!n} b!n 1=0 m=l

where

13~!n = 2~ i: sin~l\:h) F:r, (()G~ ( -() d(

14~~ = 2~ i: tan~l\:h)G~(()G~(-()d(.

(11)

(12)

(13)

(14)

(15)

(16)

It is convenient to transform II~!n, 12~~' 13~!n, and 14~~ into rapidly-convergent series via the residue calculus. The results are summarized in Subsect. 1.5.3 Appendix.

The scattered field at x = ±oo is given by

E;(±oo, z) = L K~ sin(kzuz) exp(±ikxux) (17) u

(18)

Let the time-averaged incident, transmitted, and reflected powers be denoted by Pi, Pt , and Pr , then


T = Pt/Pi = 11 +K;-1 2 + f- Lkxv lKtl 2

xs vi's

/ . kxsh where 1 ::; v < koh 7r, v: mteger, and Pi = -4w .

Po

1.5.2 TM Scattering

In region (I) the incident and scattered fields are

H;(x, z) = exp(ikxsx) cos(kzsz)

H;(x,z) = 21 roo [H~()eiI<Z +H~()e-iI<Z] e-i(xd( 7r 1-00

(19)

(20)

(21)

(22)

where kzs = 87r/h, ° ::; s < ko h/7r , s: integer, kxs = .Jk~ - k~s' and", = .Jk~ - (2. In the lth groove (0 ::; l ::; L1) of region (II) the scattered field is

00

HtI(x,z) = L b!n cosam(x + a -lT1) cos~m[z + d1(l)] (23) m=O

m 7r ,-:-n--;:-

where am = ~ and ~m = y'k~ - a~. In the rth groove (0 ::; r ::; L 2) of

region (III) the scattered field is 00

HtIl (x, z) = L c~ cosgn(x + 9 - rT2 - 8) COS7Jn[z - h - d2(r)] (24) n=O

h - n7r _ . /k2 2 were gm - 2g and 7Jn - V 3 - gn·

The Ex field continuity at z = ° requires

E!(x,O) + E;(x, O) = {Eo,iI(x,O), - a + lT1 < X < a + lT1 (25) otherwise.


• L1 00

H~() - H~() = _1 L L b~~m sin[~md1(l)]F:n() E2'" 1=0 m=O

(26)

where 1 i(ei(lTJ [e-i(a - (_I)mei(a]

Fm() = (2 _ a~· (27)

Similarly from the Ex field continuity at z = h,

L2 00

H~()eil<h - H~()e-il<h = ~ L L c~7Jnsin[7Jnd2(r)]G~() (28) 1E3'" r=O n=O


where r _ i(ei«rT2+<l) [e-i(y - (_I)nei(y]

Gn (() - (2 2 . -gn

(29)

The Hy field continuity at z = 0 for (-a + ITI < X < a + ITI ) is

exp(ikxsx) + 2~ i: [Ht(() + H~(()]e-i(xd( 00

= L b!n cos am (x + a - ITI ) cOS[~mdl (l)] . (30) m=O

Substituting Ht(() and H~(() of (26) and (28) into (30), multiplying (30) by cos apex + a - UTI), and integrating over (-a + UTI < X < a + uTt), we get

where co = 2, em = 1 (m = 1,2,3, ... ), 8ul is the Kronecker delta, and

II~!n = 2~ i: li:ta~(li:h)F:n(()F;(-()d( 12~~ = 21 100

• 1( h) G~(()F;( -() d( . 7r -00 Ii: sm Ii:

Similarly from the Hy field continuity at z = h, we get

(31)

(32)

(33)

(34)

(35)

(36)

Rapidly-convergent series forms for II~' 12;';", 13~z.,., and 14~';" are available in Subsect. 1.5.3.


The scattered field at x = ±oo is

H;(±oo, z) = L K!: cos(kzuz ) exp(±ikxux) (37) u

(38)

Let Pi, Pr , and Pt denote the time-averaged incident, reflected, and transmitted powers, respectively. Then the transmission (T) and reflection ((!) coefficients are

T = PdPi = 11 + K';1 2 + + LEvkxv lK:12 (39) Es xs v#s

(40)

where 0 ~ v < koh/7r, v: integer, and Pi = kxsEsh/(4wfo).

1.5.3 Appendix

• TE wave Utilizing the technique of contour integration, we evaluate I1;:/n, I2~~' I3~~' and I 4;;; analytically in the complex (-plane.

Il~~ = Almbul bpm + Jl~~

{ 27riPl (() k=ap + 27riQl (() k=9n ,

I2~~ = X 1(() + 2 .f{(()SI(() - h(()s~(() I m ~(() ,

(=a p

{ 27riP2 (() k=am + 27riQ2 (() k=9q ,

I3~~ = X 2(() + 2 .f~(()S2(() - h(()s~(() m s~(()

(=a m

where

AIm = aJk5 - a;' cot ( Jk5 - a;,h)

J ul ~ • 2 WI (()

ap:l gn

ap = gn

am :lgq

, am =gq

Ipm = - L...J wmap'" h(((2 _ a2 )((2 _ a2) 0=1 m p (=Jk"5-(arr/h)2

(41)

(42)

(43)

(44)

(45)

(46)


Wl(() = [(_l)m+p + 1]ei(II-UIT1 _ (_1)mei(IU-u)Tl+2al

_(_1)pei(I(I-ulTt -2al (47)

h(() = IWP9n[(_1)n+Pei(lrT2-uTl+g-aHI

_( _1)Pei(lrT2-uTl-g-aHI

_( _1)nei(lrT2-uTl +g+aHI + ei(lrT2-UT1-g+aHI]

SI(() = 211"sin(l\;h)(( + ap )2

~ il\;h(() I X 1(() = - L.. h ( 1),8( 2 a2)( 2 2)

,8=1 (- (- P ( - 9n (=.jk5-(,81r/h)2

P2(() = h(() 211" sin(l\;h)(( + am )((2 - 9~)

h(() Q2(() = 211"sin(l\;h)(( + 9q)((2 - a~)

h(() = l\;am9q [( _1)m+qei(llTt-vT2+a-g-ol

_( _1)mei(IITl-vT2+a+g-ol

_ ( -l)q ei(IIT1 -vT2-a-g-ol + ei(IIT1 -VT2-a+g-ol]

S2(() = 211" sin(l\;h)(( + am )2

X 2 (() = - f: il\;h(() I ,8=1 h((-1),8((2 - a~)((2 - 9~) (=Jk5-(,81r/h)2

W2(() = [(_l)q+n + 1]ei(lr-VIT2 _ (_1)nei(l(r-v)T2+2g1

_( _1)QeiC1 (r-v)T2-2gl .

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)


• TM wave Similarly we evaluate 11;/n, 12;~' 13~'m, and 14;:;' analytically in the complex (-plane.

Wl() = [(_I)m+p + l]ei(II-U 1Tl _ (_I)mei(I(I-u)Tl+2al

_( _1)Pei(I(I-u)Tl-2a l

!I(() = [(_It+Pei(lrT2-uTl+g-aHI _ (_I)Pei(lrT2-uTl-g-aHI

(67)

(68)

(69)

_( _1)nei(lrT2-uTl +g+aHI + ei(lrT2-UTl-g+aHI] (70)

I (1")_{(2!I((), p¥O (71) JOl .. - !I (), p=O

s (() - {27rI\;Sin(l\;h)(+ap )2, p¥O (72) 1 - 27rl\;sin(l\;h), p = 0

00 i(!I() X 1 (() = - L c: h( 1)13((2 a2)((2 2) (73)

13=0 13 - - P - gn (=.jk"5-(i31r / h)2

1.6 Water Wave Scattering from Rectangular Grooves in a Plane 29

12«(,) = [(_1)m+qei<llTI-vT2+a-9-ol _ (_1)mei<llTI-vT2+a+9-ol

_( _1)Qei<IITl-vT2-a-9-ol + ei<IITI-VT2-a+9-ol]

102«(,) = {(,2 12«('), m f 0 12«(,), m = 0

s «(,) _ { 21l"~ sin(~h)«(, + am )2, m f 0 2 - 21l"~sin(~h), m = 0

00 i('h«(') I X 2 «(,) = - L c: h( 1)!3«(,2 a2 )«(,2 2)

!3=0 !3 - - m - gQ <=..jk5-(!31C / h )2

W2«(,) = [(_l)Q+n +1]ei<lr-vIT2 _ (_1)nei<l(r-v)T2+291

_( _1)Qei<l(r-v)T2-29j .

(76)

(77)

(78)

(79)

(80)

(81)

(82)

1.6 Water Wave Scattering from Rectangular Grooves in a Plane

A modeling of water flow over obstacles is of importance due to its practical applications in many fluid-mechanic related areas. In this section we will consider a problem of small-wateewave scattering from a horizontal water bed consisting of an N number of rectangular grooves [19,20]. An incident wave velocity potential is given by pi(X,y) = Ai coshko(y+h)eikox. It is convenient to use an approximate linear modeling approach [21] for small-water wave scattering. For small waves, a dispersion relation w2 = gko tanh(koh) satisfies the surface boundary condition

2A;i( ) 8p i(x,y) 0 -w ~ x,y + g 8y = at y = 0 (1)

where g and ko are the gravitational acceleration and the wavenumber, respectively. In region (I) (-h < y < 0) the scattered wave is


z y=O

Incident <1>1

Region (I) (Water) Scattered <1>5 Scattered <1>5

h

Bed surface (Reflected) (Transmitted)

x=(N-1J!'-~ _ ~=(N-1)T +a

Region (II)

Fig. 1.8. Rectangular grooves on a horizontal water bed

1 100 . pS(x , y) = 27f _00[Acosh((y+h)+Bsinh((y+h))e-1(Xd(. (2)

Inside the lth groove of region (II) (Ix - lTI < a and -(h + d) < y < -h: l = 0,1,2, . . . ,N - 1) the water wave is

00

pt(x, y) = L C;, cosam(x + a -IT) cosham(y + h + d) (3) m=O

where am = m7f/(2a). The surface boundary condition at y = °

gives

2.,J;.s( ) 8pS(x, y) I 0 -w '£' x, Y + g 8 = y y=O

A = [g( - w2 tanh((h)] B . w2 - g(tanh((h)

The boundary condition at a bed surface (y = -h)

8pS(x,y) = {~Pt(X'Y) 8y 8y'

is rewritten as

Ix -lTI > a

Ix -lTI < a

(4)

(5)

(6)

1.6 Water Wave Scattering from Rectangular Grooves in a Plane 31

~ 100 B(e-i(Xd( 271" -00

{ 0, Ix - lTI > a

= ~ C!nam cosam(x + a -IT) sinh(amd) , Ix -lTI < a. (7)

Applying the Fourier transform to (7) yields N-I 00

B = - L L iC!nam sinh(amd)Fm(a)a2ei(IT . (8) 1=0 m=O

Multiplying the boundary condition, qii(x, -h) + qi8(X, -h) = qit(x, -h) for IX-lTI < a, by cosan(x+a-rT), and integrating from (rT-a) to (rT+a), we obtain

N-I 00

-ikoAia2 Fn(koa)eikorT + 2~ L L C!nam sinh(amd)I 1=0 m=O

where

1=100 [g( - w2 tanh(h)] -00 w2 - g(tanh(h)

-(a4 Fm(a)Fn ( -(a) exp[i«(l - r)T] d( .

U sing the residue calculus, we transform I into

1= -471"ik5E (ko) + 271" f (vE(i(v)[g(v - w2 tan(v h)] sinh(2koh) + 2koh 9 v=1 (vhsec2(vh) + tan(v h)

(9)

(10)

{

471"a( 2 -2 9 - w h)8mn81r. m = 0

+ w (11) 2a7l" [gam -w2tanh(amh)]s: J:

with

2 h( h) UmnUIr. otherwise am w - gam tan am

Al E() = ((2 _ a~)((2 _ a;)

Al = [(_I)m+n + 11exp[i(l(l- r)Tll

-( _1)m exp[i(I(1 - r)T + 2aIJ

-( -It exp[i(I(1 - r)T - 2all

(12)

(13)

and the positive real numbers (v (v = 1,2, ... ) are determined by the characteristic equation w2 =-g(v tan((vh).

It is possible to obtain the scattered field in terms of rapidly-convergent series using the residue calculus. For x > (N - I)T + a,


N-l 00

q;S(X,y) = L L C!namsinh(amd)

For x < -a,

1=0 m=O

.{ 2[w2 sinh(koY) + gko cosh(koY)] g[2koh + sinh(2koh)]

· cosh(koh)a2 Fm( -koa) exp( -ikolT + ikox)

f 2[w2 sin((vY) + g(v cos((vY)] + v=l g[2(v h + sin(2(v h)]

· cos((vh)a2 Fm( -i(va) exp((vlT - (vx) } .

N-l 00

q;8(X,y) = L L C!namsinh(amd) 1=0 m=O

.{ _ 2[w2 sinh(koY) + gko cosh(koy)] g[2koh + sinh(2koh)]

· cosh(koh)a2 Fm(koa) exp(ikolT - ikox)

_ f 2[w2 sin((vY) + g(v cos((vy)] v=l g[2(vh + sin(2(v h)]

· cos((vh)a2 Fm(i(va) exp( -(vlT + (vx) } .

References for Chapter 1

(14)

(15)

1. J. M. Jin and J. L. Volakis, "TM scattering by an inhomogeneously filled aperture in a thick conducting plane," lEE Proceedings, pt. H, vol. 137, no. 3, pp. 153-159, June 1990.

2. T. B. A. Senior, K. Sarabandi, and J. R. Natzke, "Scattering by a narrow gap," IEEE Trans. Antennas Propagat., vol. 38, no. 7, pp. 1102-1110, July 1990.

3. S. K. Jeng, "Scattering from a cavity-backed slit on a ground plane- TE case," IEEE Trans. Antennas Propagat., pp. 1523-1529, Oct. 1990.

4. S. K. Jeng and S. T. Tzeng, "Scattering from a cavity-backed slit on a ground plane- TM case," IEEE Trans. Antennas Propagat., vol. 39, no. 3, pp. 661-663, May 1991.

5. K. Yoshidomi, "Scattering of an electromagnetic beam wave by rectangular grooves on a perfect conductor," TI'ans. Inst. Electron. Commun. Eng. Jpn., vol. E. 67, no. 8, pp. 447-448, Aug. 1984.

6. T. J. Park, H. J. Eom, and K. Yoshitomi, "An analytic solution for transverse magnetic scattering from a rectangular channel in a conducting plane," J. Appl. Phys., vol. 73, no. 7, pp. 3571-3573, 1. April 1993.

References for Chapter 1 33

7. T. J. Park, H. J. Eom, and K. Yoshitomi, "An analysis of transverse electric scattering from a rectangular channel in a conducting plane," Radio Sci., vol. 28, no. 5, pp. 663-673, Sept.-Oct. 1993.

8. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, vol. I Elementary Functions, pp. 105-325, Gordon and Breach, New York, 1986.

9. R. Petit, Electromagnetic Theory of Gratings, vol. X of Topics in Current Physics, New York, Springer-Verlag, 1980.

10. J. W. Heath and E. V. Jull, "Perfectly blazed reflection grating with rectangular grooves," J. Opt. Soc. Am., vol. 66, pp. 772-775, 1976.

11. Y. L. Kok, "Boundary-value solution to electromagnetic scattering by a rectangular groove in a ground plane," J. Opt. Soc. Am. A., vol. 9, pp. 302-311, 1992.

12. T. J. Park, H. J. Eom, and K. Yoshitomi, "Analysis of TM scattering from finite rectangular grooves in a conducting plane," J. Opt. Soc. Am. A, vol. 10, no. 5, pp. 905-911, May 1993.

13. K. Uchida, "Numerical analysis of surface-wave scattering by finite periodic notches in a ground plane," IEEE Trans. Microwave Theory Tech., vol. 35, no. 5, pp. 481-486, May 1987.

14. K. H. Park, H. J. Eom, and T. J. Park, "Surface wave scattering from a notch in a dielectric-covered ground plane: TE-mode analysis," IEEE Trans. Antennas Propagat., vol. 42, no. 2, pp. 286-288, Feb. 1994.

15. I. L. Verbitskii, "Dispersion relation for comb-type slow-wave structures," IEEE Trans. Microwave Theory Tech., vol. 28, no. 1, pp. 48-50, Jan. 1980.

16. J. H. Lee, H. J. Eom, J. W. Lee, and K. Yoshitomi, "Transverse electric mode scattering from a rectangular grooves in parallel-plate," Radio Sci., vol. 29, no. 5, pp. 1215-1218, Sept.-Oct. 1994.

17. K. H. Park, H. J. Eom, and K. Uchida, "TM-scattering from notches in a parallel-plate waveguide," IEICE Trans. Commun., vol. E79-B, no.2, pp. 202-204, Feb. 1996.

18. S. B. Park, Electromagnetic Scattering from a Parallel-Plate with Double Rectangular Corrugations, Master Thesis, Department of Electrical Engineering, Korea Advanced Institute of Science and Technology, Taejon, Korea, Dec. 1998.

19. J. J. Lee and R. M. Ayer, "Wave propagation over a rectangular trench," J. Fluid Mech., vol. 110, pp. 335-347, 1981.

20. J. W. Miles, "On surface-wave diffraction by a trench," J. Fluid Mech., vol. 115, pp. 315-325, 1982.

21. G. D. Crapper, Introduction to Water Waves, pp. 154-157, Ellis Horwood Limited,1984.

2. Flanged Parallel-Plate Waveguide Array

2.1 EM Radiation from a Flanged Parallel-Plate Waveguide

z

Transmitted 1 Region (I) ko

~~..,.,.,~~""";i-------

1 1 Incident Reflected

Region (II) k

Fig. 2.1. A flanged parallel-plate waveguide

Electromagnetic wave scattering from a conducting double wedge and a flanged parallel-plate waveguide was considered in [1-7]. A flanged parallelplate waveguide is one of the basic radiating structures used for various aperture antennas. A study of radiation from a flanged parallel-plate waveguide is applicable to radiation from a flanged conducting rectangular waveguide that is a practical radiating element. In the next two subsections we will formulate TE and TM radiation from a flanged parallel-plate waveguide.


36 2. Flanged Parallel-Plate Waveguide Array

2.1.1 TE Radiation [8,9]

Consider a TE wave Et (x, z) radiating from a flanged parallel-plate waveguide with width 2a. The incident and reflected E-fields inside a parallel-plate waveguide (region (II) with wavenumber k) are

Et(x, z) = sin ap(x + a) exp(i~pz) (1) 00

E;(x, z) = I>m sinam(x + a) exp( -i~mz) (2) m

where am = m7r /(2a) and ~m = Jk2 - a;'. Note that m = 1,3,5, ... for odd p and m = 2,4,6, ... for even p. In region (I) (z > 0 with wavenumber ko) the transmitted E-field is

1 100 ~ Et(x, z) = -2 Et(() exp( -i(x + ilioz) d( 7r -00

(3)

where 1i0 = J k5 - (2. Applying the Fourier transform to the tangential E-field continuity at

z = 0 yields

(4) m

Multiplying the tangential H-field continuity at (-a < x < a and z = 0) by sinan(x + a) and integrating over (-a < x < a), we get

(5)

where Al(ko) is given in Subsect. 1.1.3. The far-zone transmitted field at distance r from the origin is

t f§ko . . Ey (Os) = - exp(Ikor + I37r /4) cos Os 7rr

. ~(8 ) [am COS(koaSinOs )] L...J mp + Cm (k . 0)2 2 m 0 SIn s - am

(6)

2.1.2 TM Radiation [10]

Assume that a TM wave Ht(x, z) radiates from the aperture of a parallelplate waveguide (region (II) with wavenumber k). The incident and reflected H-fields within a parallel-plate waveguide are

Ht(x, z) = cos ap(x + a) exp(i~pz) (7) 00

H;(x, z) = L Cm cos am(x + a) exp( -i~mz) . (8) m=O

2.2 EM Radiation from a Parallel-Plate Waveguide into a Dielectric Slab 37

In region (I) (wavenumber ko) the transmitted H-field is

1 {'XJ_ H~(x, z) = 2; Loo H~(() exp( -i(x + iKoz) d( (9)

where KO = y'k'5 - (2. The tangential E-field continuity at z = 0 yields

00

Koii~(() = -i(~pa2 Fp((a) + L i(~mcma2 Fm((a) . (10) m=O

The tangential H-field continuity along the aperture (-a < x < a and z = 0) gives

00

L (8mp - cm)~mafll (ko) = 27f(8np + cn)cn (11) m=O

where fll (ko) is given in Subsect. 1.1.3. When the operating frequency approaches infinity (koa ~ 00), fll(ko) ~ 27fcn/(ay'k'5 - a;;J8mn , thereby leading (11) to a solution in the Kirchhoff approximation.

The far-zone transmitted field at r is

H~(Bs) = J 2~r exp(ikor + i7f/4) sinBs

00

. L(8mp-cm)~ma2Fm(-koasinBs). (12) m=O

2.2 EM Radiation from a Parallel-Plate Waveguide into a Dielectric Slab

Electromagnetic wave radiation from a parallel-plate waveguide and a rectangular waveguide into a dielectric slab was studied in [11-16]. A study of radiation from a parallel-plate waveguide into a dielectric slab is useful for the design of radome and microwave permittivity sensors. In the next two subsections we will investigate TE and TM wave radiation from a flanged parallel-plate waveguide into a displaced dielectric slab.

2.2.1 TE Radiation

Consider a problem of radiation from a flanged parallel-plate waveguide into a dielectric slab. A wave transverse electric (TE) to the z-axis, E~(x, z), is incident on a dielectric slab from inside a parallel-plate waveguide. Regions (I), (II), (III), and (IV), respectively, denote the air (wavenumber: kl wVJiEl = 27f/>" and >..: wavelength), a dielectric slab (wavenumber: k2 =


Region (I) k1 -

1 1 PEe

Incident

Region (IV) k4

Fig. 2.2. A flanged parallel-plate waveguide radiating into a dielectric slab

w"fiii2), a background medium (wavenumber: k3 = w..jiif3), and an aperture (wavenumber: k4 = w..jji€.i). In region (I) the total E-field is

E~(x,z) = 21 100 EI(()exp(-i(x+i";1z)d( (1)

7r - 00

where ";1 = Jkr - (2. In regions (II) and (III) each E-field is represented as

E~I (x, z) = 2~ i: [EtI(()eiI<2 Z + EiI(()e-iI<2 Z ] e-i(x d( (2)

E~II (x, z) = ~ i: [EtII(()eiI<3 Z + EiII(()e-iI<3 Z ] e-i(x d( (3)

where ";2 = Jk~ - (2 and ";3 = Jk~ - (2. In region (IV) (-a < x < a) the total incident and reflected fields are

00

E~(x, z) = L em sinam(x + a)e-i~mz m=l

where ~m = Jkl- a~ and am = m7r/(2a).

(4)

(5)

The tangential field continuities at z = d2 = d1 + band z = d1 give, respectively,


Eill(() = ei21<3 d l

[ (~3 - ~2)(~2 + ~1)e2iI<2dl + (~3 + ~2)(~2 - ~t}e2iI<2d2]

. (~3 + ~2)(~2 + ~1)e2iI<2dl + (~3 - ~2)(~2 - ~1)e2iI<2d2 -+ ·EllI (()

_ (a l ) -+ = a2 EIII (()

- -+ = aEIII (() . (7)

The tangential E-field continuity at z = 0 gives

-+ ( a l ) _ ~ EIII (() 1 + a2 - aKp((a) + !::t cmaKm((a) (8)

where

(9)

Substituting (8) into the tangential H-field continuity at the aperture (-a < x < a and z = 0) gives

(10)

where

(11)

(12)

It is expedient to transform (12) into a numerically-efficient integral based on the residue calculus. Let's consider a complex (-plane in Fig. 2.3. For analytic convenience, £2 is assumed to have a small positive imaginary part. Integrating along the deformed contour TI , n, T3 , and T4 in the upper-half plane, we obtain

Inm = h m 8nm + lnm + rnm

where 8nm is the Kronecker delta and

(13)


-a n

(n=m)

1m (~)

Branch cut

Branch cut


hm = 27faJk~ - a~ ( Ct2 - Ctl) I Ct2 + Ctl (=a m

lnm = L 27fiaman(Ct2 - Ctd[1 - (_I)m ei2k.a] 1

k. (Ct2 + Ctd'((2 - a~)((2 - a;) (=k.

rnm = - 1= dvf(v)aman

2i[1 - (_I)mei2ak1e-2ak1Vh/k~ - ki(1 + iV)2

krl(1 + iv)2 - (am/k1 )2][(1 + iv)2 - (an/kI)2]

Re(~)

f ( v) = _ ( Ct2 - Ctl ) I + ( Ct2 - Ctl ) I Ct2 + Ctl 1<1 -+-1<1 ,(=k1 +ik1 v Ct2 + Ctl (=k1 +ik1 V

(14)

(15)

(16)

(17)

and (.)' denotes differentiation with respect to (. Note that hm and lnm are the residue contributions at ( = ±am and kx, respectively, where kx is a zero of (Ct2 + Ctl), and rnm is a branch-cut integration along nand r4 associated with a branch-point at ( = k1 .

The far-zone field E~ (r, 0) and the total radiated power (Ps) normalized by the incident one (Pi) are


(18)

(19)

where

A = { [- i cosO cos(k1d1 cosO) V(f2/f1) - sin2 0

V(f2/fl) - sin2 0 l. / - cosO sin(k1d1cosO) sin(k1by (f2/fl)-sin20)

+ exp( -ik,d, cos 9) cos (k,b,/(,,/ ,,) - sin' 9) } -, (20)

The reflection and transmission coefficients associated with the surface waves in regions (I), (II), and (III) are

(21)

(22)

72 = PII/Pi

= ~ {IKc 12b IKc 12 [sin(2kz2d2) - sin(2kz2ddl IK S 12b 2~pa II + II 2kz2 + II

_ IKS 12 [sin(2kz2d2) - sin(2kz2ddl } II 2kz2

4~ kXk (cos kz2 d2 - cos kz2ddRe [KlI(KiI)* + KiI(KII )*1 (23) pa z2


73 = PIlI/Pi

= 2~;a (IKIIII2d1 - IKIIII2 2L3 sinh 2kz3d1 + IKIIII 2d1

+ IKIIII2-kl sinh2kz3 d1) 2 z3

2~ kXk [1 + cosh(2kz3dt}] pa z3

·Re [iKIII(K1II)* - iK1II(KIII )*] (24)

where kZ1 = Jki - ki, kZ2 = Jk~ - ki, kZ3 Jki - k~, and the power conservation requires (J + 71 + 72 + 73 + f2 = 1. We further note that

KI = -2iAexp[i(kz2 - ikzt}d2]kz2 (25)

KII = _2iAeikz2d2 [kz2 cos(kz2d2) + kZ1 sin(kz2 d2)] (26)

Kh = _2iAeikz2d2 [kz2 sin(kz2 d2) - kZ1 cos(kz2d2)] (27)

KI II = 2Be -kz3 dl kZ3 . [ikz1 (ei2kz2dl - ei2kz2d2) + kZ2(ei2kz2dl + ei2kz2d2)] (28)

A = { [aKp((a) + f; cmaKm((a)] (a2eil<3 dl + a1 e- il<3 dl) ekz3dl}

{ [eil<2d1 (1\:2 + I\:t) + exp(i21\:2d2 - i1\:2d1)(1\:2 - I\:t}]

. [(a2 + at}e-il<3 dl]' }-11_

C,--k~

B= [aKp((a) + %;1 cmaKm((a)]

C,=-km

where (.), denotes differentiation with respect to (.

2.2.2 TM Radiation [17]

(30)

(31)

Consider a flanged parallel-plate waveguide radiating a TM wave into a dielectric slab as discussed in Subsect. 2.2.1. Regions (I), (II), (III), and (IV),


respectively, denote a half-space (wavenumber: kl = W../J-tEOEI = 27f / )..1), a dielectric slab (wavenumber: k2 = W..jJ-tEOE2 = 27f/)..2), a background medium (wavenumber: k3 = W..jJ-tEOE3 = 27f/)..3), and an aperture (wavenumber: k4 = W..jJ-tEOE4 = 27f/)..4). A wave transverse magnetic (TM) to the z-axis, H~(x, z), impinges on a dielectric slab from inside a parallel-plate waveguide. In region (I) the H-field is

H: (x, z) = -/; i: HI(() exp( -i(x + i#l:IZ) d( (32)

where #1:1 = v'k~ - (2. In region (II) the total H-field is

H:I (x, z) = -/; i: [HiI(()e i I<2 Z + Hil(()e- i I<2 z ] e-i(x d( (33)

where #1:2 = v'k~ - (2. In region (III) the total H-field is

H:II (x, z) = -/; i: [HiII(()ei I<3 Z + Hill (()e-iI<3 Z ] e-i(x d( (34)

where #1:3 = v' k~ - (2. In region (IV) (-a < x < a) the incident and reflected fields are

00

H;(x, z) = L em cosam(x + a)e-i~",z m=O

where ~m = v'k~ - a~ and am = m7f/(2a).

(35)

(36)

The tangential E-field and H-field continuities at z = d2 = d1 + b yield

H- (() = ei21<2d2 (El#1:2 - E2#1:1 ) H+ (() . (37) II El#1:2 + E2#1:1 II

Similarly the tangential E-field and H-field continuities at z = d1 yield

HilI(() = [(E2#1:3 - E3#1:2) (El #1:2 + E2#1:t}e2il<2 d l

+(E2#1:3 + E3#1:2) (El#1:2 - E2#1:t}e2iI<2d2 ]

• [(E2#1:3 + E3#1:2) (El #1:2 + E2#1:t}e2i1<2d1

+(E2#1:3 - E3#1:2)(El#1:2 - E2#1:1)e2iI<2d2 ] -1 ei21<3 d l HiII(()

( a 1 ) - + == a2 HIII (()

- -+ = aHIII (() . (38)

The tangential E-field continuity at z = 0 yields


(39)

where

(40)

From the tangential H-field continuity along the aperture (-a < x < a and z = 0), we obtain

( 41)

where

(42)

When m + n is odd, Jnm = O. When m + n is even, Jnm is rewritten as

Jnm = /00 2 (a2 + ( 1 ) (2 [1 - (-1 )mei2(a) -00 a2 - a1 ((2 - a;,)((2 - a;)f\;3 d( .

(43)

Performing a contour integration in view of Fig. 2.3, we obtain

(44)

We note that hm and lnm are the residue contributions at ( = ±am and kx, respectively, where kx is a zero of (a2 - ad, and rnm is a branch-cut integration associated with a branch-point at ( = k1 . They are given as

hm = 27fa/ Vk~ - a;' (a2 + ( 1 ) I a2 - a1 (=a",

47fi(2(a2 + ad[1 - (_1)m e i2k X a) 1

lnm = L (a2 _ ad'((2 _ a2 )((2 - a2)f\;3 _ kx m n (-kx

rnm = 100 dV{ 2i[1- (_1)m e i2ak1e-2ak1V)(1 + iv)2 f(v)}

.{ kd(1 + iv)2 - (am /kd 2)[(1 + iv)2 - (an /kI)2)

.Vk~ - ki(1 + iv)2 } -1

f (v) = _ (a2 + a1 ) I + (a2 + a1 ) I a2 - a1 1"<1->-1"<1 ,(=k1 +ik1 V a2 - a1 (=k 1 +ik1 v

The far-zone field H; (r, 0) and the normalized radiated power are

(45)

(46)

(47)

(48)


(49)

(50)

where

Ls(() = ~ [~pKp(() - f Cm~mKm(()l f3 (51) ~ m~ 4

11 = { [-if3f1 c! 0 cos(k1d1 cos 0) - f22 co; 0 sin(k1d1 cos 0)]

. sin(k1bj3) + exp( -ik1d1 COSO)f3f2 COS(k1b(3)} -1 (52)

and (3 = V(f2/fd - sin2 O. The reflected (Pr) and transmitted powers (PI, PII , and PII I) associated

with the surface waves in regions (I), (II), and (III) are

{} = Pr/Pi

(53)

(54)

T2 = PII/Pi

= kxf4 {IKII I2 b + IKIII2 [sin(2kz2 d2) - sin(2kz2 ddl 2f2~pCpa 2kz2

IK s 12b _ IK S 12 [sin(2kz2 d2) - sin(2kz2 ddl } + II II 2kz2

kxf4 ( ) - 4~ k cos kz2 d2 - cos kz2 d1

pf2cp z2a

·Re [KlI(Kh)* + Kh(K1I )*] (55)

46 20 Flanged Parallel-Plate Waveguide Array

73 = PIlI/Pi

= kt4 (IKIllI2d1 - IKIlll2 2k1 sinh 2kz3d1 2£3 pcpa z3

+lKhIl2d1 + IKIll1 2-k1 sinh2kz3 d1) 2 z3

~ kx£~ [2 + 2 cosh(2kz3dt)l 4 p£3cp z3a

oRe [iKIll(KIll )* - iKIll(KIll )*] (56)

where kZ1 = Jk;, - kr, kZ2 = Jk~ - k;" kZ3 = Jk;, - k~, and the power conservation requires a + 71 + 72 + 73 + [! = 1. Note that

KI = -2iAexp[i(kz2 - ikzt)d2l£lkz2 (57) KII = _2iAeikz2d2 [£lkz2 cos(kz2d2) + £2kz1 sin(kz2 d2)l ekz3dl (58)

Kh = _2iAeikz2d2 [Etkz2 sin(kz2 d2) - £2kz1 cos(kz2d2)l ekz3dl (59)

KIll = 2Be-kz3d, £2kz3 [i£2kz1 (ei2kz2dl - ei2kz2d2)

+£lkz2(ei2kz2dl +ei2kz2d2)] (60)

KIll = 2Be-kz3dl£3kz2 [£1 kZ2 (ei2kz2dl _ ei2kz2d2)

+i£2kz1(ei2kz2dl +ei2kz2d2)] (61)

A = {(£3/£4) [~pKp(() - Io cm~mKm(()l o (a2ei l<3 d l + a1 e -iI<3dl) }

o {1\:3[ei I<2d1 (£11\:2 + £21\:t) + ei21<2d2-iI<2d, (£11\:2 - £2l\:t)l

o [(a2 - at)e- i I<3 d,]' }-~ (--k.

(62)

B= [~pKp(() - foCm~mKm(()l (£3/£4)

(63)

(=-k.

where the symbol (-)' denotes differentiation with respect to (0

2.3 TE Scattering frOID a Parallel-Plate Waveguide Array [18] 47

2.3 TE Scattering from a Parallel-Plate Waveguide Array [18]

z


Region (I)

Fig. 2.4. A parallel-plate waveguide array

Electromagnetic wave radiation from a parallel-plate waveguide array has been extensively studied in [19-21] due to its practical applications in array antenna design. In this section we will consider TE wave scattering from a flanged parallel-plate waveguide array when a uniform plane wave E~(x, z) is incident on a waveguide array (width 2a, period T, and array number N). This section is a continuation of Sect. 2.1 where radiation from a single parallel-plate waveguide was considered. Regions (I) and (II) denote the air and a waveguide interior with wavenumbers ko = WJf.Lofo = 27r/J... and k = w-.,fiIE (f.L = f.Lof.Lr and f = fOf r ), respectively. In region (I) the E-field has the incident , reflected, and scattered waves

(1)

(2)

(3)

where kx = ko sinO, kz = ko cosO, and "-0 = y'k5 - (2. The transmitted E-field inside the lth waveguide of region (II) is


00

Et(x, z) = L c~ sin am(x + a _IT)e-i~mZ (4) m=l

where am = m7r/(2a) and em = Jk2 - a~. Applying the Fourier transform to the tangential E-field continuity, we

obtain L2 00

E;(() = L L c~ama2 Fm((a)ei<IT . (5) 1=-L1 m=l

Multiplying the tangential H-field continuity at (IT - a) < x < (IT + a) and z = 0 by sin an (x + a - rT), and integrating with respect to x from (rT - a) to (rT + a), we obtain

2ikz an exp (ikx rT)a2 Fn(kxa) • L2 00 r C Ian '"' '"' I 2 ). cn .. na = 2 L...J L...J cmama A2(ko + 1-- .

7r 1=-L1 m=l J.Lr (6)

where the expression for A2(ko) is given in Subsect. 1.2.3. The far-zone scattered field at distance r from the origin is

E;(Os,O) = J 2~r exp(ikor - i7r /4) cos OsE; ( -ko sin Os) . (7)

The transmission coefficient, which is a ratio of the transmitted power to the incident one over the apertures, is

1 L2 <>./4a

T = - '"' '"' Icl 12e a/J.L* . 2wJ.Lo L...J L...J m m r 1=-L1 m=l

2.4 EM Radiation from Obliquely-Flanged Parallel Plates

(8)

Electromagnetic radiation from a flanged parallel-plate waveguide is an important subject due to its flush-mounted antenna applications. Radiation from a right-angled and infinitely-flanged parallel-plate waveguide has been extensively studied in Sects. 2.1 through 2.3. Radiation from an obliquelyflanged parallel-plate waveguide is of some theoretical interest, but its radiation study is relatively very little [22,23]. In this section we intend to study radiation from an obliquely-flanged parallel-plate waveguide. Consider a TE (transverse electric to the z-axis) wave radiating from an obliquely-flanged parallel-plate waveguide into a conducting plane. When a conducting plane is removed, the scattering geometry becomes a half-space radiation problem. The wavenumber inside the waveguide is k (= 27r / A = w..jji€ and A: wavelength). In region (I) the total incident and reflected E-fields are

2.4 EM Radiation from Obliquely-Flanged Parallel Plates 49

{} -

- -0- ---- ---- -- --'"l.- - - ,/ ----0-- r~----~- r--- ~ ··....,· ...... 11 ... _ .............

(1) (Xl

E;(x, z) = L Em sin(amx)e-ikmZ (2) m=l

where ap = p1f/a and kp = Jk2 - a~. In region (III) the transmitted E-field

is

E;II (u, v) = ~ {(Xl [E+(()e iKV + E_(()e- iKV ] e-i(u d( 21f J-(Xl

where", = Jp - (2. We apply the second Green's formula

J (E BG _ G BEy) dt = 0 lc y Bv Bv

(3)

(4)

to the contour OAB with the auxiliary functions satisfying the Helmholtz equation in region (I) as

G; = sin(aqx)e±ikqz, q = 1,2, .. . . (5)

The fields on the line OB are assumed on the interval (-9,0) as

EyloB = n~(Xl Cnexp (i2~nu) (6)

BEy I ~ (.21fn ) Bv = ~ Dnexp l-U .

OB n=-(Xl 9 (7)


As a result of the application of Green's formula, we obtain

m=l 00

= L [Cni ( aq sin ax~ + kqgq cos a) - Dngq] Yq (X~) (8) n=-oo

00

= L [Cni(aqSinax;;--kqgqcosa)-Dn9q]Yq(X;;-) (9) n=-(X)

1 - (-I)qe- ig( where Yq(() = 2 (2 ' gq =

gq -q7r ± g' Xn

27rn . - ± kqsma, and q =

9 1,2,3 ....

The Ey continuity at v = b yields

E_(() = _E+(()ei2 I<b • (10)

The Ey continuity at v = 0 between regions (II) and (III) yields 00

E+(() + E_(() = -i L CnS;;--(() (11) n=-(X)

1 - e±ig( where S~ (() = ( / . The Hu continuity at v = 0 is given as

+ 27rn 9

~ (27rn) 1 100 [- -]. n~oo Dn exp igu = 27r -00 i,.,; E+(() - E_(() e-1(ud(. (12)

We multiply (12) by exp ( -i 2:m U) and perform integration with respect

to U from -g to 0 to get

where

00 Cn Dm=- L -Jmn

n=-oo 9 (13)

Jmn = -J-l°O ,.,;cot(,.,;b)S;!;(()S;;--(() d( . (14) 7r -00

Note that

.00 2b;(I-eig(t) (f+(27r/g)2mn Jmn = -1 t; bet [(f - (27rm/g)2] [(;- (27rn/g)2]

+g8mnVk2 - (27rn/g)2cot [bVk2 - (27rn/g)2] (15)

2.5 EM Radiation from Parallel Plates with a Window [24] 51

where bt = t7f/b and (t = Vk2 - (t7f/b)2. When b goes to 00, it is expedient to transform (14) into a numerically-efficient radiation integral based on the residue calculus. The result is given by

Jmn = _-.!.k2 vv(v - 2i) 2· 100

7f 0

[1- exp[igk(1 + iv)]] [k2(1 + iv)2 + (27f/g)2mn] ~~--~~~--~~~~--~~~~~~~dv [k2(1 + iv)2 - (27fm/g)2] [k2(1 + iv)2 - (27fn/g)2]

+6mn 9 j k2 - (27fm/ g)2 .

The scattered field at u = ±oo in region (III) is

E~II (±oo, v) = L Kf sin bt(v - b)e±i(,u t

where

K± _ 00 bt (1 - e±i(,9)

t - n~oo en b(t( -1)t(=f(t + 27fn/ g)

(16)

(17)

(18)

o < t, and t: integer. The transmission (71 and 72) and reflection (0) coefficients are

(19)

(20)

(21)

where 0 < t < kb/7f, t: integer, 0 < m < ka/7f, and m: integer. When b goes to 00, the far-zone scattered field in region (III) is

E~II (r, 0) = J 2:kr exp(ikr - i7f /4)k cos 0 E+( -k sin 0) . (22)

2.5 EM Radiation from Parallel Plates with a Window [24]

A use of a window in an open-ended waveguide can reduce an undesirable reflection when an open-ended waveguide is used as a radiating element in phased array antennas. The effect of a window in an open-ended waveguide on radiation characteristics has been studied theoretically in [25,26]. In this section we will investigate a TM wave radiating from a flanged parallel-plate waveguide with a window. In region (I) (z > 0) an incident field H~(x, z) impinges on a slit (width: 2a and depth: d) in a thick perfectly conducting


H' _ y

x

Z _-----(: >---f'~.....L...----__

Region (III)

Fig. 2.6. A flanged parallel-plate waveguide with a window

plane. Region (II) (-d < Z < 0 and -a < x < a) denotes a slit and region (III) denotes a half-space (z < -d). The wavenumber in regions (I), (II), and (III) is k = w..jJif. = 27r/>". The fields in regions (I) (Ht and H;), (II) (H~), and (III) (H~) are represented as

H~(x,z) = coswp(x+w)exp(-i1Jpz)

m=O 00

H;(x, z) = L (bm cos~mz + em sin~mz) cosam(x + a) m=O

Ht(x, z) = 21 roo Ht(() exp [-i(x - ill;(z + d)] d( 7r J- oo

(1)

(2)

(3)

(4)

where Wm = m7r/(2w), 1Jm = y'k2 - w~, am = m7r/(2a), ~m = y'k2 - a~, and II; = y'p - (2.

The tangential E-field continuity at -w < x < wand z = 0 yields


The tangential H-field continuity at -a < x < a and z = 0 gives

~(d 1.:) [(-l)nSinWm(a+W)+sinwm(a-w)] 6 m+ump 2 2 Wm m=O wm -an

= bnacn .

The tangential field continuities at z = -d give

(bn cos end - Cn sin end)acn • 00

= ..:... ~ em(bm sinemd + Cm cosemd)a2 il1(k) 21f m=O

where il1(k) is

il1(k) = i: a2Fm«(a)Fn(-(a)(21\;-ld(.

An explicit evaluation of ill (k) is available in Subsect. 1.1.3. The far-zone transmitted field at distance r is

Ht(r,O) = ei(kr-7r/4JV k sinO y 21fr

00

(5)

(6)

(7)

(8)

. ~ [bm sin (em d) + Cm cos(emd)]ema2 Fm( -kasinO) (9) m=O

where 0 = sin-1 (x/r) and r = y'x2 + (z + d)2. The transmission coefficient T, a ratio of the time-averaged power transmitted through a slit to that incident on a slit, is given by

T = _Im{_a_ f cmem[JbmJ2 sin emd(cos emd)* cp'f/pw m=O

+b:ncmJ cosemdJ2 - bmc:nJ sinemdJ2

_JCm J2 cosemd(sinemd)*] } . (10)


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14. W. D. Burnside, R. C. Rudduck, L. L. Tsai, and J. E. Jones, "Reflection coefficient of a TEM mode symmetric parallel-plate waveguide illuminating a dielectric layer," Radio Sci., vol. 4, no. 6, pp. 545-556, June 1969.

15. Y. Sugio, T. Makimoto, and T. Tsugawa, "Two dimensional-analysis for gain enhancement of dielectric loaded antenna with a ground plane," IEICE Trans. Commun. (in Japanese), vol. J73-B-II, no. 8, pp. 405-412, Aug. 1990.

16. G. B. Gentili, G. Manara, G. Pelosi, and R. Tiberio, "Radiation of open-ended waveguides into stratified media," Microwave Opt. Technol. Lett., vol. 4, no. 10, pp. 401-403, Sept. 1991.

17. J. W. Lee, H. J. Eom, and J. H. Lee, "TM-wave radiation from flanged parallelplate into a dielectric slab," lEE Proceedings-H Antennas Propagat., vol. 143, no. 3, pp. 207-210, 1996.

18. T. J. Park and H. J. Eom, "TE-scattering and reception by a parallel-plate waveguide array," IEEE Trans. Antennas Propagat., vol. 42, no. 8, pp. 862-865, June 1994.

19. S. W. Lee, "Radiation from an infinite aperiodic array of parallel-plate waveguides," IEEE Trans. Antennas Propagat., vol. 15, pp. 598-606, Sept. 1967.

20. C. P. Wu, "Analysis of finite parallel-plate waveguide arrays," IEEE Trans. Antennas Propagat., vol. 18, pp. 328-334, May 1970.

21. G. F. Vanblaricum and R. Mittra, "A modified residue-calculus technique for solving a class of boundary value problems-part II: waveguide phased arrays,


modulated surfaces, and diffraction gratings," IEEE 171ms. Microwave Theory Tech., vol. 17, no. 6, pp. 310-319, June 1969.

22. R. C. Rudduck and L. L. Tsai, "Aperture reflection coefficient of TEM and TEol mode parallel-plate waveguides," IEEE Trans. Antennas Propagat., vol. 16, no. 1, pp. 83-89, Jan. 1968.

23. C. Thompson, "Radiation from a tapered open end of a waveguide with an infinite flange," Radio Eng. Electron. Phys., vol. 22, no. 1, pp. 23-30, Jan. 1977.

24. H. J. Eom and T. J. Park "Radiation from a parallel-plate fed slit in a thick conducting screen," IEICE 1rans. Commun., vol. E78-c, no. 8, pp. 1131-1133, Aug. 1995.

25. R.W. Scharstein, "Two numerical solutions for the parallel plate-fed slot antenna," IEEE Trans. Antennas Propagat., vol. 37, no. 11, pp. 1415-1425, Nov. 1989.

26. B.N. Das, A. Chakraborty, and S. Gupta, "Analysis of waveguide-fed thick radiating rectangular windows in a ground plane," lEE Proceedings-H, vol. 138, no. 2, pp. 142-146, Apr. 1991

3. Slits in a Plane

3.1 Electrostatic Potential Distribution Through a Slit in a Plane [1]

z

PEe

v Region (I)

<1>1 (x,z)

0 x

Region (II) <1>11 (x,z)

Region (III) I. . 1 1 28 1

<1>111 (x,z)

Fig. 3.1. A thick slit near a conducting plane at potential V

Electromagnetic interference (EMI) problems often require an estimation of field strength penetrating into a slit in a conducting plane. Many analytic and numerical approaches have been used to predict the effect of field leakage on electric circuit and system performance. For instance, a low-frequency field penetration into a slit is approximately modeled in terms of the electrostatic potential that is governed by Laplace's equation [2]. In this section we will analyze an electrostatic potential distribution through a slit in a thick perfectly-conducting plane. Consider a thick and perfect conducting slit at


58 3. Slits in a Plane

zero potential placed near a conducting plane at potential V. The electrostatic potentials in regions (I), (II), and (III) are represented as

pI(x,z) = ~1OO [~+(()e-<z +~-(()e<z] e-i<xd( (1) 211" -00

00

pIl (x, z) = L [bm cosham(z + d) + em sinham(z + d)] m=l . sinam(x + a)

pIlI (x, z) = - pIlI (() exp[l(I(z + d) - i(x] d( 1 100 -

211" -00

where am = m1l"/(2a) and m = 1,3,5, .... The boundary conditions on the field continuities require

pI(x, h) = V

pI (x, 0) = { ~,Il (x, 0),

pIlI(x,_d) = {tI(X,-d),

Ixl < a Ixl > a

Ixl < a Ixl > a.

Applying the Fourier transform to (4) through (6) yields

(2)

(3)

(4)

(5)

(6)

~+(()e-<h +~-(()e<h = 211"V15(() (7) 00

~+(() + ~-(() = L (bm coshamd + em sinhamd) ama2 Fm((a) (8) m=l

00

~IIl (() = L bmama2 Fm((a) (9) m=l

where 8 (() is the Dirac delta. We substitute ~+ (() and ~- (() into the boundary condition

OpI (x, z) I = OpIl (x, z) I ' oz z=o oz z=o

Ixl < a (10)

multiply (10) by sinan(x + a), and integrate with respect to x from -a to a to get

V[I-(-I)n] 00 •

ha = L (bm cosh amd + em smh amd)I n m=l

+ana(bn sinh and + en cosh and) . (11)

Similarly from the boundary condition

OpIl I (x, z) I = OpIl (x, z) I ' OZ z=-d OZ z=-d

Ixl < a (12)

3.2 Electrostatic Potential Distribution due to a Potential Across a Slit [3] 59

we obtain 00

L bmJ = anacn m=l

where I and J are

1 roo ( I = 27r i-oo tanh(hFm«(a)Fn(-(a)amana4d(

1 roo J = 27r i-oo 1(IFm«(a)Fn(-(a)amana4 d(.

Performing the residue calculus, we get the rapidly-convergent series ana -

I = h h 8mn - I tan an

J = ana8mn - J where

(13)

(14)

(15)

(16)

(17)

l= L vaman e 00 [2 1 + -2av ]

1=1 h (v2 + a;')(v2 + a~J v=17r/h' m + n = even (18)

{ 2 (4~m~n 2 ) [In(an/am)+ci(m7r)-ci(n7r)], m + n = even

_ 7r an am J=

am [2 - n7rsi(n7r)], m = n . 7ran

Note that ci(·) and si(·) are the cosine and sine integrals defined as

. 100 sint Sl(X) = - -dt x t . 100 cost Cl(X) = - -dt . x t

For m + n = odd, I = J = O.

3.2 Electrostatic Potential Distribution due to a Potential Across a Slit [3]

(19)

(20)

(21)

Section 3.1 discusses an electrostatic potential distribution through a slit when a charged perfectly-conducting plane is placed nearby a slit. In this section we will consider an electrostatic potential distribution through a slit when a potential V is applied across a slit with thickness d and width 2a. In regions (I) (z > 0) and (III) (z < -d) the scattered potentials are written as


z

Region (I)

<1>[ (x,z) y

0 Region (II)

<1>11 (x,z)

----------Region (III) 28

<1>111 (x,z)

Fig. 3.2. A thick sli t with a potential difference V

z z

III

3-a ~-tx':~-Ea '/ X 3-a --i---Ea '/ X

, 0 Region (II) 0 , 0 Region (II) V

---~-- ---~--<l> III (x,-d) 0

(a) (b) 1'0. ".0. fill '==ljUIVGLu::aU,t jJJ.UUlt:l11 U~t:U Ul1 I"ut;:: ;:;upt:::1j.JU;:; ..... lUll }-Il"Ull,;l}.Jlt:

1 ('Xl_ pI(x,z) = 271" J-

oo pI(()exp(-I(lz - i(x)d( (1)

1 roo -pIlI(x,z) = 271" J-oo

pI(()exp[l(I(Z+d) -i(x]d(. (2)

In (1) and (2) we note that pI (x , 0) = pIlI (x, -d). Based on the superposition principle, the original problem in Fig. 3.2 may be decomposed into two different cases (a) and (b) as shown in Fig. 3.3. Hence the total potential in

3.2 Electrostatic Potential Distribution due to a Potential Across a Slit [3] 61

region (II) is a sum of two solutions to problems (a) and (b) as

q;lI (x, z) = ~l bm cosh [am (z + ~) ] sin am (x + a)

. k1r(x + a) 4V 00 smh d brz

- -:;- L . br2a sin d k=l ksmh--

d m7r

where am = -, m = 1,2,3, ... , and k = 1,3,5, .... 2a

(3)

Due to a symmetry of the problem geometry, pI (x, 0) = pIlI (x, -d) and

8pI (x, z) I 8pI II (x, z) I Th" d' h 1 h b d 8z z=O =- 8z z=-d' IS m lcates t at on y t e oun ary

conditions at z = 0 need to be enforced as

{ 0,

pI (x, 0) = pIl (x, 0), V,

8pI (x, z) I 8pIl (x, z) I 8z z=o 8z z=o'


x <-a Ixl < a x>a

Ixl < a.

;PI(C) = 1=1 bmcosh (a;d) Em(() + Vei(a [7rl5(C) - i~] where o(C) is the Dirac delta and

~ am [( -1)m e i(a - e-i(a]

=m(C) = (2 2 . -am

Rewriting (5) gives

-~ 100 ;PI (c)I(le-i(xd( 27r -00

= ~1 ambm sinh (a;d) sinam(x + a)

. h k7r(x + a) 4V 00 sm d

-d" L . k7r2a k=1 smh-

d

(4)

(5)

(6)

(7)

(8)

We multiply (8) by sin an(x + a) and integrate with respect to x from -a to a to get


. (and) 4V 00 an cosn7r =aanbnsmh 2 +d~(k7r/d)2+a;·

Substituting (6) into (9) gives

~ roo ei(a~Sn( -()d( 27r ) -00 I(

- %;1 bm cosh (a~d) 2~ i: 1(ISm(()Sn( -()d(

. (and) 4V 00 an cosn7r = aanbnsmh 2 + d ~ (k7r/d)2 + a; .

Using the residue calculus, we evaluate

~ roo ei(a~Sn( -()d( = _( _1)n [V + V Si(n7r)] 27r } -00 I( 2 7r

(9)

(10)

(11)

(12)

where 8nm is the Kronecker delta and J is given by (19) in Sect. 3.1. Substituting (11) and (12) into (10), we obtain a matrix equation

(13)

where B is a column vector of bn and the matrix elements are

'I/J - 8 _ exp[(am - an)d/2] + exp[-(am + an)d/2] J nm - nm 2aan

(14)

"In = (_1)n+I exp( -an d/2) aan

[V V. 4V ~ an 1 . 2" + -;sI(n7r) + d ~ (k7r/d)2 + a; (15)

It is of practical interest to represent the scattered potential and the surface charge distribution in fast convergent series. Substituting (6) into (1) and performing a contour integration, we get a series form

00

= L ambm cosh amd 7r 2

m=I

·1m {( _1)m K(am, a - x + iz) - K(am, -(a + x) + iz)}

V V (a-x) +- - -arctan -- , 2 7r Z

x <-a (16)

3.3 EM Scattering from a Slit in a Conducting Plane 63

. 1m {( -1) m K (am' a - x + iz) - K (am, a + x + iz)}

+bm cosh a;d exp( -amz) sinam(a + x)

v V (a-x) +- - - arctan -- , 2 7f Z

V V (x -a) +- + -arctan -- , 2 7f Z

x>a

(17)

(18)

where K(f3, j.t) = 1 [ci(f3j.t) sin(f3j.t) - si(f3j.t) cos(f3j.t)]. The surface charge den

sity on a slit is

Ips(x, z)1

:tl am: m cosh ( am ~) [( -1) m P( am, a - x) - P (am, - a - x)]

V 1 +---, 7fa-x f ambmcosh [am (z +~)] 4V f sin(k7fzjd)

-d sinh(k7f2ajd) ' k=1,3

x < -a, z = °

- d < z < 0, x = -a

(19)

where P(f3, j.t) = ci(f3j.t) cos(f3j.t) + si(f3j.t) sin(f3j.t) and E is the medium permittivity.

3.3 EM Scattering from a Slit in a Conducting Plane

Electromagnetic wave scattering from a slit in a conducting plane is a canonical problem [4-9] often encountered in electromagnetic interference and compatibility-related area. In the next two subsections we will consider TE and TM scattering from a two-dimensional slit in a perfectly-conducting plane. Since the slit axis (y) is chosen to be perpendicular to the plane of incidence, the scattering analysis becomes a two-dimensional problem. An analytical formulation given in this section is similar to that in Sect. 1.1.


z


Region (I) Air -a

Region (III)

Transmitted wave

Fig. 3.4. A slit in a thick conducting plane

3.3.1 TE Scattering [10)

In region (I) (z > 0) a field E~(x,z) is incident on a slit (width: 2a and depth: d) in a thick perfectly conducting plane. Regions (II) (-d < z < 0 and -a < x < a) and (III) (z < -d) are a slit and a lossless half-space, respectively. The wavenumbers in regions (I), (II), and (III) are ko, kl' and k2' respectively, where ko = wJJ-Lo€o, kl = WJJ-Ll€l, and k2 = WJJ-L2€2. The E-field in region (I) has the incident, reflected, and scattered waves


E;(x, z) = - exp(ikxx + ikzz)

E~(x, z) = 21 100 E~«() exp( -i(x + i/\,oz) d( 7r -00

(1)

(2)

(3)

where kx = kosinO, kz = kocosO, and /\'0 = Jk'5 - (2. In region (II) the total E-field is

00

Et(x, z) = L (bm cos~mz + em sin~mz) sinam(x + a) (4) m=l

where am = m7r/(2a) and ~m = Jk; - (2 . In region (III) the transmitted field is

Et(x, z) = 2~ i: Et«() exp[-i(x - i/\'2(z + d)) d( (5)

3.3 EM Scattering from a Slit in a Conducting Plane 65

where K.2 = v'k~ - (2. Applying the Fourier transform to the tangential E-field continuity along

the x-axis (z = 0) yields 00

E;(() = L bmama2 Fm((a) . (6) m=l

The tangential H-field continuity along (-a < x < a) at z = 0 yields

2ikzan 2 D (k ) _ ian ~ b 2A (k ) _ cn~na a L'n xa - 2 ~ mama 1 0

J.to 7rJ.to m=l J.t1 (7)

where the explicit expression for Al (ko) is available in Subsect. 1.1.3. Similarly the tangential E-field continuity at z = -d gives

00

Et(() = L (bm cos~md - Cm sin~md)ama2 Fm((a) . m=l

The tangential H-field continuity along (-a < x < a) and z = -d gives

i J.t2 (bn sin ~nd + Cn cos ~nd)~na J.tl

00

= ~; L ama2(bm cos~md - Cm sin~md)Al(k2) m=l

where

A1 (k2) = i: a2Fm((a)Fn(-(a)K.2d(.

(8)

(9)

(10)

The far-zone scattered and transmitted fields at distances TI and T2 are

E;(Os,{}) = V2ko exp(ikoTI-i7r/4)cosOsE;(-kosinOs) (11) 7rTl

Et(Ot,O) = V2k2 exp(ik2T2 -i7r/4)cosOtEt(-k2sinOt). (12) 7rT2

The transmission (reflection) coefficient is a ratio of the transmitted (reflected) power to that incident on a slit. The transmission coefficient T and the reflection coefficient (! are

T = Im{ 2kJ.to f ~~ [lbml2 cos~md(sin~md)* + bmc~1 cos~mdl2 OJ.tl m=l

-b~cml sin~mdl2 -lcml2 sin~md(cos~md)*] } (13)

{! = :ko Im{ f ~bmc:n~~} . (14) o m=l J.tl



Consider an incident H~(x, z) impinging on a slit in a conducting plane. The wavenumbers in regions (I), (II), and (III) are ko = WVf.-LO€O = 27r/A, kl = WVf.-LOf.-Lrl€O€rl, and k2 = WVf.-LOf.-Lr2€O€r2, respectively. In region (I) the field consists of the incident, reflected, and scattered components as

H;(x, z) = exp(ikxx - ikzz)

H;(x, z) = exp(ikxx + ikzz)

H;(x, z) = 2~ i: ii;(() exp( -i(x + i/\;oz) d( .

In region (II) the H-field is

=

(15)

(16)

(17)

H;(x,z) = L (bmcosemz+cmsinemz)cosam(x+a). (18) m=O

In region (III) the total transmitted H-field is

H;(x,z) = 21 (= ii;(()exp[-i(x-i/\;2(z+d)]d(. 7r J-=

(19)

Applying the Fourier transform to the tangential E-field continuity along the x-axis yields

ii;(() = - f emcm (a2 Fm((a) . (20) m=O /\;O€rl

The tangential H-field continuity along (-a < x < a) at z = 0 gives

. = -2ikxa2 Fn(kxa) = -2 1 L ema2Cm{)1(ko) + bna£n .

7r€rl m=O (21)

The explicit expression for {)l (ko) is given in Subsect. 1.1.3. The tangential E-field continuity at z = -d similarly gives

= ii:(() = ~ L em(bm sinemd + Cm cos em d) (a2 Fm((a) . (22)

€rl/\;2 m=O

The tangential H-field continuity along (-a < x < a) at z = -d gives

(bn cos end - Cn sin end)acn • 00

= 2l€r2 L ema2 (bm sinemd + Cm coSemd){)1(k2) 7r€rl m=O

where

(23)

(24)

3.4 Magnetostatic Potential Distribution Through Slits in a Plane 67

The far-zone scattered and transmitted fields at distances rl and r2 are

H~(()s,()) = V2ko exp(ikorl - i'll'/4)cos()Ji~(-kosin()s) (25) 'lITl

H~(()t,()) = -V2k2 exp(ik2r2 -i1l'/4)cos()Ji~(-k2sin()t). (26) 1I'r 2

The transmission coefficient T is

T = -Im{ v!-- f €m~m [lbml2 sin~md(cos~md)* Ofrl m=O

+b;;'cml cos~mdl2 - bmc;;'1 sin~mdl2

-lcml2 cos~md(sin~md)*] } . (27)

3.4 Magnetostatic Potential Distribution Through Slits in a Plane

H

z

! Region (I) PEe . ~ y' .

~~E*~:~.- ~.~~~*~~~!r x T Region (II)

1=-1 1=0 1=1

Region (III)

Fig. 3.5. Multiple slits in a thick conducting plane

A study of magnetostatic potential distribution through apertures in a conducting plane is of interest in EMI/EMC-related problems. In the present section we will study a problem of magnetostatic potential distribution through


multiple slits in a thick conducting plane. Consider an incident magnetostatic potential ~i(X, z) impinging on a finite number of slits (thickness: d, width: 2a, period: T, total number of slits: N) in a thick conducting plane. Regions (II) (-d < Z < 0 and Ix -lTI < a) and (III) (z < -d) denote a slit interior and a half-space, respectively. In region (I) (z > 0) the total magnetostatic potential consists of the incident and scattered potentials

~i,,(x,z) = x - AI, Ix -lTI < T/2, 1 = 0,±1,±2,... (1)

~B(X, z) = ~ ('X) ¥B«()e-i(ze-I(lz d( (2) 27r J-oo

The magnetostatic potential for a multiply-connected domain may be a multivalued function of position; hence, it is necessary to introduce a cut in a domain to make the magnetostatic potential a single-valued function [12]. This implies that a constant A' is yet to be determined later when the boundary conditions are matched. In region (II) of the lth slit the magnetostatic potential is

00

~d,l(x, z) = ci + L [b~ sinham(z + d) + c~ cosham(z + d)] m=l

. cos am (x + a -IT) (3)

where am = m7r/(2a) and m = 1,2,3, .... In region (III) the total transmitted potential is

1 (OO_ ~t(x, z) = 27r J -00 ~t«() exp[-i(x + 1(I(z + d)] d( . (4)

The boundary condition on the continuity of normal derivative of the potential at z = 0 requires

:z [~i,,(x, z) + ~B(X, z)t=o

= { ! [~d,l(x,z)t=o' Ix -lTI < a

0, Ix - lTI > a .

(5)


I(I¥B«() L2 00

= i( L L (b~ coshamd + c~ sinhamd)ei(IT a2amFm«(a) . (6) I=-Ll m=l

The boundary condition on the continuity of the potential across the slits requires

Ix -lTI < a. (7)

3.4 Magnetostatic Potential Distribution Through Slits in a Plane 69

Multiplying (7) by cosan(x + a - rT) (n = 1,2,3, ... ) and integrating with respect to x from (rT - a) to (rT + a), we get

(_1)n -1 L2 00

-'-----'---- = L L (b~ coshamd + c~ sinhamd) I an 1=-L 1 m=l

+ana (b~ sinh and + c~ cosh and)

where

1= - el«(l-r)TI(la4amanFm((a)Fn( -(a) d( . 1 100 •

21T -00

Using the contour integration, we evaluate

I = ana OmnOlr + I .

When l = r,

{ (4~man 2 ) [In(an/am)+ci(m1T)-ci(n1T)]' m + n = even

_ 21T an - am 1= -

am [2 - (n1T)si(n1T)], m = n 1Tan

where ci(·) and si(·) are the cosine and sine integrals, respectively. When l =I- r,

1= aman {[(_1)m+n + 1]Imn(qT) 21T

-(-1)mlmn(qT + 2a) - (-1)nlmn(qT - 2a)}

where q = l- rand

100 1(lei(lcl Imn(c) = -00 ((2 _ a;,)((2 _ a~) d( .

Utilizing the residue calculus, we obtain

Imn(c)

( 2 2 2) [-ci(amc) cos(amc)+ci(anc) cos(anc) an -am

-si(amc) sin(amc)+si(anc) sin(anc)], m =I- n

(8)

(9)

(10)

(11)

(12)

(13)

(14)

It is necessary to obtain another set of simultaneous equations for cb by integrating directly (7) with respect to x from (IT-a) to (IT+a). The result is


2ac~ = 2alT - 2aA' £2 00

+ L L (b~coshamd+c~sinhamd) J (15) 1=-£1 m=l

where 1 roo ei(a - e-i(a

J = 27f i-oo 1(1 a2 amFm((a) d( . (16)

Similarly from the boundary conditions at z = -d

8 t _{:z[4>d"(X,Z)L=_d' 8z [4> (x,Z)L=_d-

0,

Ix -lTI < a (17)

Ix -lTI > a

Ix -lTI < a (18)

we get

£2 00

L L b~I = anac~, n = 1,2,3 ... 1=-£1 m=l

£2 00

2ac~ = - L L b~ J . (19) 1=-£1 m=l

The magnetic polarizability of the lth slit at z = -d is

['T+a 8

tPl == L x- [4>d,l(x z)] dx IT-a 8z ' z=-d

= 4aL f b~ 7f m m=l

where L is a slit length in the y-direction.

(20)

3.5 EM Scattering from Slits in a Conducting Plane [13]

Electromagnetic scattering from a strip-grating was studied in [14-18] for microwave-optical filter and polarizer applications. A study of electromagnetic scattering from multiple slits is also useful to solve the field-leakage problems in electromagnetic interference. In the next two subsections we will study TE and TM scattering from multiple slits in a thick and perfectlyconducting plane. A scattering analysis given in this section is an extension of the single-slit case considered in Sect. 3.3.

3.5 EM Scattering from Slits in a Conducting Plane [13] 71

Incident wave

Region (II) k2 , (:2

Region (III) k3 , (:3

z

Scattered wave

Transmitted wave

Fig. 3.6. Multiple slits in a thick conducting plane

3.5.1 TE Scattering

Consider an incident wave Et(x, z) impinging on a perfect conducting plane with multiple slits (width: 2a, depth: d, period: T, and total number: N = L1 + L2 + 1). Regions (I), (II), and (III) are the air, slits, and a lower half-space where their respective wavenumbers are k1(= W..jJ.L1f1 = 27r/)..) , k2(= W..jJ.L2f2) , and k3(= W..jJ.L3f3). The E-fields in regions (I) (incident Et, reflected E;, and scattered E~), (II) (Et), and (III) (transmitted E;) are assumed to be

00

m=l

(1)

(2)

(3)

(5)

where kx = k1 sinO, kz = k1 cosO, 1\;1 = Jki - (2, am = m7r/(2a) , ~m = Jk~ - aTn, and 1\;3 = Jk~ - (2.

Applying the Fourier transform to the tangential E-field continuity condition at z = 0 gives


L2 00

EZ«() = L L b!neiC;IT ama2 Fm«(a) . (6) 1=-L1 m=1

The tangential H-field continuity over (IT - a) < x < (IT + a) (l = 0, ±1, ±2, ... ) at z = 0 yields

Note that the expression for A2 (k1 ) is available in Subsect. 1.2.3 as

A2(kt} = i: a2 Fm«(a)Fn ( -(a) 1\;1 exp [i(l- T)(T] d( .

Similarly the tangential E- and H-field continuities at z = -d yield

/"3 (b~ sin end + c~ cos en d) en a J.L2

L2 00

(7)

(8)

= ~; L L am(b!ncosemd-c!nsinemd)a2A2(k3). (9) 1=-L1 m=1

The far-zone scattered fields at distances T1 and T2 are

EZ(Os,{}) = V2k1 exp(iklTl-i7r/4)cosOsEZ(-klsinOs) (10) 7rTI

E;(Ot,O) = V 2k3 exp(ik3T2 -i7r/4)cosOt 7rT2

L2 00

. L L [b!n cos(emd) - c!n sin(emd)]

·am exp( -ik3 sin OtlT)a2 Fm (-k3a sin Ot) .

The transmission coefficient (r) is shown to be

r = .!!2..~Im { f f ~b!nc~e~} 2k1 N 1=-L1 m=1 J.L2

where N = Ll +L2 + 1.

3.5.2 TM Scattering

(11)

(12)

Asume that a TM wave H~(x, z) is incident on multiple thick slits in a perfectly-conducting plane. The H-fields in regions (I), (II), and (III) are

3.5 EM Scattering from Slits in a Conducting Plane [13]

00

m=l

1 100 -HZ(x, z) = -2 HZ(() exp[-i(x - i1£3(Z + d)] d(

7r -00

where kx = k1 sinO and kz = k1 cosO. The tangential E- and H-field continuities at z = 0 yield

-i2kxeik.rT a2 Fn(kxa) . L2 00

1101 '"' '"' 1 1 2 r = -2- ~ ~ ~mcma D2(k1) + bnacn 7rf2 1=-L1 m=O

where the explicit expression for D2 (kd is given in Subsect. 1.2.2 as

73

(13)

(14)

(15)

(17)

(18)

D2(kd = i: a2 Fm((a)Fn( _(a)(2 exp[i(l - r)(T]1£11 d( . (19)

The tangential E- and H-field continuities at z = -d also yield

(b~ cos ~nd - c~ sin ~nd)acn . L2 00

=~ '"' '"'~m(b!nsin~md+c!ncos~md)a2D2(k3) (20) 27rf2 ~ ~

1=-L1 m=O

where

D2(k3) = i: a2 Fm((a)Fn( _(a)(2 exp[i(l - r)(T]1£31 d( . (21)

The far-zone fields are

H;(Os,O) = J k1 exp(ik1r1 -i7r/4)sinOs 27rr1

L2 00

.101 L L ~mc!nexp(-ik1sinOsIT)a2Fm(k1asinOs)(22) 102 1=-L1 m=O

HZ(Ot,0) = J2k3 exp(ik3r2 - i7r/4) sinOt 103 7rr2 102

L2 00

. L L ~m[b!n sin(~md) + c!n cos(~md)] 1=-L1 m=O

(23)


The transmission coefficient (T) is

3.6 EM Scattering from Slits in a Parallel-Plate Waveguide

z

P t - (E,H);

Region (I) k1, " 1

_"" ........ ~_ z:~.~.~.~:::::~ x

z=-b-d I , ' . IT I T . 8+

Region (III) Air k3 , "3

8 (E,H)y

Ps .l"lg. li.7. IVIUltlpie slits ill a parallel-plate wavegUide

(24)

Electromagnetic wave radiation and scattering from a slitted parallel-plate waveguide was studied in [19-25] for aperture array antenna applications. A slitted waveguide is an important array antenna element that enables us to realize the narrOw beamwidth and wideband radiation characteristics. In this section we will consider TE and TM scattering from a finite number of slits in a parallel-plate waveguide. An analytical formulation in this section is similar to the discussion in Sect. 3.5.


Consider a parallel-plate waveguide with N slits of width 2a and depth d. Regions (I) , (II), and (III) respectively denote a parallel-plate waveguide interior (wavenumber k1 = Wy'J.L1Ed, an N(= L1 + L2 + 1) number of slits

3.6 EM Scattering from Slits in a Parallel-Plate Waveguide 75

(k2 = Wy'J.L2f2), and a lower half-space (k3 = WVJ.L3f3 = 27r/>"). Two incident TE waves, E~1 (x, z) and E~2(X, z), are assumed to excite regions (I) and (III), respectively. The total E-field in region (I) has the incident and scattered fields

Et1(X,Z) = A1eik.sxsinkzs(z+b)

E~(x, z) = ~ ('Xl E~(() sin(1\;1 z)e- i(x d( 7r 1-00

(1)

(2)

where 0 < 8 < k1b/7r (8: integer), kzs = 87r/b, kxs = Jkr - k;s' and 1\;1 = Jkr - (2. In region (II) (IT - a < x < IT + a and -d - b < z < -b: I = - L 1 , ... ,L2 ) the E-field is

00

m=1 (3)

where am = m7r/(2a) and ~m = Jk~ - a~. In region (III) the total E-field is composed of

E~2(X,Z) = A2exp[ikxx+ikz(z+b+d)] (4)

E~(x, z) = -A2 exp [ikxx - ikAz + b + d)] (5)

E~II (x, z) = ~ roo E~II (() exp[-i(x - il\;3(z + b + d)] d( (6) 27r 1-00

where kx = k3 sinO, kz = k3 cosO, and 1\;3 = Jk~ - (2. Applying the Fourier transform to the tangential E-field continuity at

z = -b gives

L2 00 -2isin(1\;1b)E~(() = L L b~amei(IT a2 Fm(a() . (7)

1=-L1 m=1

Multiplying the tangential H-field continuity at the slit apertures (rT - a < x < rT + a and z = -b: r = -L1' ... ,L2 ) by sinan(x + a - rT) and integrating, we get

(8)

where

A5(kd = i: 1\;1 cot(l\; l b)a2 Fm((a)Fn ( -(a) exp[i(1 - r)(T] d( (9)


and a fast-convergent series for A5(kd is available in Subsect. 1.4.1. Applying the Fourier transform to the tangential E-field continuity at Z = -b - d gives

E:II(() L2 00

= L L [b~ cos(~md) - c~ sin(~md)]ama2ei(IT Fm((a) . (10) 1=-L1 m=l

The tangential H-field continuity at the slit apertures (rT - a < x < rT + a and Z = -b - d: r = -£1"" ,£2) gives

2kz A 2 • 2 ---an exp(lkxrT)a Fn(kxa)

W/13

1 L2 00

+-2 - L L aman [b~ cos(~md) - c~ sin(~md)] a2 A2 (k3) 7r/13 1=-L1 m=1

= J.-a~n[b~ sin(~nd) + c~ cos(~nd)] (11) /12

where A2 (k3 ) is given in Subsect. 1.2.3 as

A2 (k3) = i: a2 Fm((a)Fn ( -(a)K3 exp riel - r)(T] d( . (12)

The total scattered field at x = ±oo is

E:(±oo,z) = LK;=sinkzv(z + b)exp(±ikxvx) (13) v

where 0 < v < k1 bj7r, v: integer, kzv = v7rjb, kxv = v'kr - k~v' and

L2 00

K;= = iLL b~amkzv exp(=fikxvlT)a2 Fm(=fkxva)j(kxvb) . (14) 1=-L1 m=1

When Al = 1 and A2 = (Pi2 =) 0, the time-averaged incident, reflected, transmitted, and radiated (scattered into region (III)) powers are, respectively

(15)

(16)

(17)

3.6 EM Scattering from Slits in a Parallel-Plate Waveguide 77

Ps = f: f Re{ ~~;,. a l=-Ll m=1 /12

·[b~ cos(';md) - c~ sin(';md)][b~ sin(';md) + c~ cos(';md)] * } (18)

where 0 < v < (k1b/'IT), v: integer, and Pr + Pt + Ps = Pi1 . When Al = (Pi1 =) 0 and A2 = 1, then

P i2 = aNk3

W/13

Pr = ~ Lkxv lK;;-12 W/11 v

_ b " 1 +1 2 Pt - 4-- L....J kxv Kv . W/11 v

The far-zone scattered field at distance r is

III ( ) [k; (. . 'IT) ~III(') Ey r,Bs = v~exp lk3r-14 cosBsEy k3smBs ·


(19)

(20)

(21)

(22)

Assume that two incident TM waves propagate in regions (I) and (III). In region (I) the total field consists of the incident and scattered components

H~1 (x, z) = Al eikzsx cos kzs(z + b)

H;(x,z) =! {'XJ if;(() COs(fLlz)e-i(x d( . 'IT} -00

In region (II) the field is 00

H;I (x, z) = L cosam(x + a -IT) m=O

·[b~ cos';m(Z + b) + c~ sin';m(z + b)] .

In region (III) the incident, reflected, and scattered fields are

H~2(X, z) = A2 exp[ikxx + ikz(z + b + d)]

H;(x, z) = A2 exp[ikxx - ikz(z + b + d)]

1 100 ~ H;II (x, z) = -2 H;II (() exp[-i(x - ifL3(Z + b + d)] d( . 'IT -00

The tangential E- and H-field continuities at z = -b yield

-iA1kxs exp(ikxsrT)a2 Fn(kxsa)

L2 00 ~ = - L L c~ ~1 m a2 !?5(k1) + b~Ena

l=-L1 m=O 'ITt2

(23)

(24)

(25)

(26)

(27)

(28)

(29)


where

f}5(kt}

= i: ~11(2 cot(~lb)a2 Fm(a)Fn( -(a) exp[i(1 - r)(T] d( . (30)

A fast convergent series representation for f}5(kt} is given in Subsect. 1.4.2. The tangential E- and H-field continuities at z = -b - d yield

-i2A2kx exp(ikxrT)a2 Fn(kxa)

_ f: f: faem[b~ sin(em~) + em cos(emd)] a2 f}2(ka)

1=-L1 m=O 2111"102

= €na[b~ COS(end) - C~ sin(end)] (31)

where f}2(ka) is given in Sect. 1.2 as

f}2(ka) = i: a2 Fm(a)Fn( _(a)(2 exp[i(1 - r)(T]~31 d( . (32)

3.7 EM Scattering from Slits in a Rectangular Cavity

A study of electromagnetic wave penetration into a cavity with multiple slits has been of importance in electromagnetic wave interference and compatibility problems [28]. In this section we will study an electromagnetic field penetration into a two-dimensional rectangular cavity with multiple slits. Scattering analyses for the TM and TE wave incidences are, respectively, presented in the next two subsections.


A TM (transverse magnetic to the wave propagation direction) uniform plane wave is obliquely incident on a cavity. In region (I) the incident and reflected H-fields are

H~(x, y) = - exp(ikxx + ikyy)

H;(x, y) = - exp(ikxx - ikyy)

(1)

(2)

where kx = ko sin (I, ky = ko cos (I, and the free-space wavenumber is ko = W.jJ.LOfO = 211"/ A. The scattered H-field in region (I) is given by

H;(x, y) = -/; i: H;() exp( -i(x + i~oY) d( (3)

where ~o = .,Jk5 - (2. In region (II) (Ix -ITI < a and 0 < y < d), which is a slit interior. the H-field is

3.7 EM Scattering from Slits in a Rectangular Cavity 79

x

II)

Region (I)

2aTI -I I - I

- Y

Incidence Region (III)

Fig. 3.8. M slits in a rectangular cavity

00

Hi! (x, y) = L [c!n cos~m(Y - d) + d!n sin~m(Y - d)] m=O

· cosam(x-IT+a) (4)

where am = m7r/(2a) and ~m = Jkg - a?,.. In region (III) (Ixl < a and d < Y < d + h), which is a cavity interior, the H-field is

00

Hill (x, y) = L eq cos I'q (y - d - h) cosaq(x + a) (5) q=O

where a q = q7r/(2a) and I'q = Vkg - a~ . The boundary conditions on Ex and Hz continuities at y = 0 require,

respectively

E~(x, 0) + E~(x, 0) + E;(x, 0) = { ~!! (x, 0),

and

Ix -ITI < a otherwise

(6)


Ix -lTI < a. (7)

Applying the Fourier transform to (6) yields a representation for il;«() in terms of the modal coefficients em and d~. We then substitute il;«() into (7), multiply (7) by cosan(x-rT+a), and perform integration from (rT -a) to (rT + a) to get

2aikxFn (kx a )eikz rT

• L2 00

-;: L L em [c~ sin(emd) + d~ cos(emd)] (}2(ko) 1=-L1 m=O

= en [c~ cos(end) - ~ sin(end)] , n = 0,1,2, ... (8)

where

(9)

The analytic evaluation of (}2{ko) is available in Subsect. 1.2.3. In addition, the boundary conditions of E- and H-field continuities at y = d yield

where

1 L2 00 00 yl yr ~ ~ (: d' ~ mq nq r

ao: L..J L..J <"m m L..J tan{ h) = encn 1=-L1 m=O q=O '"(q '"(q

0:2 - a2 q m

{

O:q

rfnq = . {sin[O:q(a - 0: -IT)] + (_I)m sin[O:q{a + 0: + IT)]}

2a, m = q = O.


(10)

(11)

Assume that an electromagnetic TE wave is obliquely incident on thick multiple slits in a rectangular cavity. In region (I) the incident and reflected E-fields are represented as

E~{x, y) = -Zo exp{ikxx + ikyY)

E~{x,y) = Zoexp{ikxx - ikyY)

(12)

(13)

where kx = kosinB, ky = kocosB, ko = w-./J.LO€O = 27r/>' is the free-space wavenumber, and Zo is the intrinsic impedance in free space. The scattered E-field in region (I) is given by

E!{x, y) = 2~ i: E!{() exp{ -i(x + il\;oy)d( (14)

3.7 EM Scattering from Slits in a Rectangular Cavity 81

where "'0 = y'k5 - (2. In region (II) (Ix -lTI < a and 0 < y < d) the E-field inside the slit is

00

m=l

. sin am(x -IT + a) (15)

where am = m1f/(2a) and ~m = y'k5 - a~. In region (III) (Ixl < 0: and d < y < d + h) the E-field is

00

E;II(x,y) = Leqsin'}'q(y-d-h)sino:q(x+o:) q=l

where O:q = q1f/(20:) and '}'q = Jk5 - o:~. The boundary conditions on Ez and Hx at y = 0 require

and

Ix -lTI < a otherwise

(16)

(17)

Ix - lTI < a . (18)

Applying the Fourier transform to (17), substituting E;(() into (18), multiplying (18) by sinan(x - rT + a), and performing integration from (rT - a) to (rT + a), we get

where

A2(ko) = i: a2"'oFm((a)Fn( -(a) exp[i((l - r)T]d( . (20)

The explicit evaluation of A2(ko) is given in Subsect. 1.2.3. Similarly from the boundary conditions at y = d, we obtain

where

1 L2 00 00 yl yr _ '"' '"' c1 '"' '}'q mq nq = -~ndr ao: ~ ~ m~ tan('V h) n

1=-L1 m=l q=l Iq

yl = am mq 0:2 _ a2

q m

(21)

. {sin[O:q(a - 0: -IT)] + (_I)m sin[O:q(a + 0: + IT)]} . (22)


3.8 EM Scattering from Slits in Parallel-Conducting Planes [31]

z Incident Field Scattered Field

Transmitted Field It'ig. 3.9. Multiple slits m paraJlel-conQucting planes

Electromagnetic wave mutual coupling between slits in two parallel conducting screens was studied in [32-34]. In this section we will investigate electromagnetic wave scattering from multiple slits in parallel conducting planes.

3.8 EM Scattering from Slits in Parallel-Conducting Planes [31] 83

The present section is an extension of Sec. 3.3 where electromagnetic scattering from a single slit was discussed. In region (I) (air, z > (30 = 0) a TM wave Ht(x, z) is incident on a slit (width: 2ao and depth: dfJ) in a thick perfectly conducting plane. Region (II) (_(31 - dl < z < _(31, ci - al < x < a l + ai, and relative permittivity ErII) and region (IV) (_(31+1 - dl +1 < z < _(31+1, al+1 - al+1 < x < al+1 + a1+1, and relative permittivity ErIV) denote the lth and (l+l)th slits, respectively. Region (III) (_(31+1 < z < -(3I-dl and relative permittivity Er I II) denotes a dielectric slab bounded by the parallel conducting planes. Regions (I) and (V) are half-spaces that situate above and below the parallel-conducting planes, respectively. In region (I) (wavenumber k = WYJ.tEo) the total H-field is assumed to have the incident, reflected, and scattered components

H;(x, z) = exp(ikxx - ikzz)

H;(x, z) = exp(ikxx + ikzz)

H;(x, z) = 21 100 ii;(() exp( -i(x + i~z) d( 7r -00

(1)

(2)

(3)

where kx = ksinO, kz = kcosO, and ~ = Jk2 - (2. In regions (II), (III), (IV), and (V) the total H-fields are given as

00

HtI(x,z) = L [b~cos~~(z+(3I) +c~sin~~(z+(3I)] m=O

(4)

1 100 -HtII (x, z) = 27r -00 {HtII+(() exp[i~(z + (31 + dl )]

+iitII-(() exp[-i~(z + (31 + dl)] }e-i(X d( (5)

00

HtV (x, z) = L [b~1 COS~!;;1 (z + (3/+1) + C~1 sin ~!;;1 (z + (31+1)] m=O

. cos a~1 (x + al+1 - a1+1) (6)

H~ (x, z) = 2~ i: iiv (() exp [-i(x - i~(z + (3L + dL)] d( (7)

where a~ = m7r/(2al), ~~ = Jk2 - (a~)2, and ~ = Jk2 _ (2. The boundary condition On the tangential E- and H-field continuities at

z = 0 yields

(8)


where

(9)

Note that a in fl2(k) needs to be replaced by aO that is a half of the slit width in region (I). The evaluation of fl2(k) is available in Subsect. 1.2.3 as

[fl (k)) - 21fcn 8 - r:n (k)] (10) 2 a--+aO - aOJk2 _ (a~)2 mn L'<2 a--+aO

where 8mn is the Kronecker delta, co = 2, and Cn = 1 (n = 1,2,3, ... ). Similarly the boundary conditions at z = _(31 - d1 and _(31+1 give, respectively,

21f(b1 cos c-l d1 _ c1 sin c-1 d1)a1c n ~n n ~n n

(11)

(12)

where

h = 100 (2(a1)2 Fm((a1)(al)2 Fn( -Cal) d( -00 r;, tan r;,((31 + d1 _ (31+1) (13) _ 100 (2 (a1+1 )2 Fm((aI+1 )(al)2 Fn( _Cal) exp[i((oH1 - oJ))

h - -00 r;, sin r;,(f3l + dl _ (3l+1) d( (14) _ 100 (2(al)2Fm((al)(al+1)2Fn(-(al+1)exp[i((o:l - 0:1+1)) Is - -00 r;, sin r;,((31 + d1 _ (31+1) d( (15)

100 (2 (a1+1 )2 Fm ((aI+1 )(aI+1)2 Fn( _(aI+1) 14 = l I I d( . (16)

-00 r;,tanr;,((3 + d - (3 +1)

It is possible to transform (13) through (16) into numerically-efficient series based on the residue calculus. The boundary conditions on the tangential Eand H-field continuities at z = _(31+1 - d1+1 yield

21f(bL cos c-L dL _ cL sin c-L dL)aL c n ~n n ~n n • 00

= 1fr V L ~~(b~ sin~~dL + c~ cos~~dL)(aL)2 [D2(k))a--+aL (17) frIV m=O


where

(18)


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29. H. H. Park and H. J. Eom, "Electromagnetic penetration into 2D multiple slotted rectangular cavity: TE-wave," Electron. Lett., vol. 35, no. 1, pp. 31-32, 1999.

30. H. H. Park and H. J. Eom, "Electromagnetic penetration into 2D multiple slotted rectangular cavity: TM-wave," IEEE Trans. Antennas Propagat., vol. 48, no. 2, pp. 331-333, Feb. 2000.

31. J. W. Lee and H. J. Eom, "TM scattering from slits in thick parallel conducting screens," IEEE Trans. Antennas Propagat., vol. 46, no. 7, pp. 1117-1119, July 1998.

32. L. R. Alldredge, "Diffraction of microwaves by tandem slits," IRE Trans. Antennas Propagat., pp. 640-649, Oct. 1956.

33. Y. E. Elmoazzen and L. Shafai, "Mutual coupling between parallel-plate waveguides," IEEE Trans. Microwave Theory Tech., vol. 21, no. 12, pp. 825-833, Dec. 1973.

34. Y. Leviatan, "Electromagnetic coupling between two half-space regions separated by two slot-perforated parallel conducting screens," IEEE Trans. Microwave Theory Tech., vol. 36, no. 1, pp. 44-51, Jan. 1978.

4. Waveguides and Couplers

4.1 Inset Dielectric Guide

Fig. 4.1. Inset dielectric guide

The inset dielectric guide (IDG) is a dielectric-filled rectangular groove and its waveguiding characteristics have been extensively analyzed in [1-3]. In this section we will analyze an electromagnetic wave guided by the inset dielectric guide and obtain its dispersion relation. Region (I) (-d < y < 0 and Ixl < a) is filled with a dielectric material of permittivity f and region (II) (y > 0) is the air of fO. Due to a dielectric discontinuity at y = 0 between regions (I) and (II) , a hybrid wave, which is a combination of the TE and TM waves, is assumed to propagate along the z-direction. The assumption of a hybrid wave propagation is necessary to satisfy the boundary conditions on the field continuities at y = O. In region (I) the total E- and H-field z-components are


88 4. Waveguides and Couplers

00

E; (x, y, z) = ~=>k sin ak(x + a) sin O'k(Y + d)eii3z (1) k=l

00

H;(x,y,z) = L qmcosam(x+a)cosO'm(y+d)eii3z (2) m=O

where am = m7r/(2a), O'm = vkr - a;'" - (P, and k1 = wy'iiOE = W..j/-tOEOEr. In region (II) the fields are given as

1 roo -E;I (x, y, z) = 27r } -00 Ez(() exp {i[-(x + ry(()y + ,8z]} d( (3)

1 roo -H;I (x, y, z) = 27r } -00 Hz(() exp {i[-(x + ry(()y + ,8z]} d( (4)

where ry(() = Vk~ - (2 _,82 and k2 = w..j/-toEo = 27r/>'. The transverse components of E and H fields are immediate from

E I,II( ) _ i (,8 EI,!I HI,II) t x, y, z - k 2 _,82 V t z + w /-t V t X z

1,2 (5)

H I,II( ) _ i (,8 HI,!I EI,!I) t x, y, z - k2 _,82 Vt z - WEI,II Vt x z

1,2 (6)

where the subscript t means taking the component transverse to the zdirection.

The Ez and Ex field continuities at y = 0 yield 00

EzC() = LPk sin(O'k d)a2akFk((a) k=l

where

Am = ,8am(k~ - kI) sin(O'm d) iw/-t(kr - ,82)

B _ O'm (k~ - ,82) sin(O'm d) m - i(kr - ,82)

(7)

(8)

(9)

(10)

Multiplying the Hz and Hx field continuities at Ixl < a and y = 0 by cosan(x + a) and sinal(x + a), respectively, and integrating with respect to x from -a to a, we obtain

(11)

4.1 Inset Dielectric Guide 89

(12)

where

(13)

(14)

Note Jmn = a2 flt (Jk~ -/32) and Imn = a2amanAl (Jk~ - {32) where the expressions for {II and Al are available in Subsect. 1.1.3. For m + n is odd, {II (Jk~ - {32)= Al (Jk~ - {32) = O. For m + n is even,

Al ( Jk~ - {32) = 21fJki a~f2 - a;' 8mn - Al ( Jk~ - {32) (15)

{II ( Jk~ - {32) = aJk~ ~~~ _ a;' 8mn - {It ( Jk~ - {32) . (16)

In order to determine {3 we form a determinant

(17)

The dispersion relation (17) is given by a product of even and odd mode determinants where tJil , tJi2 , tJi3 , and tJi4 elements are

(18)

(19)

(20)


[{3Bmalacm (k~_{32) ] 'l/J4,lm = 1J(am) + ki _ {32 {3am cos(amd)a 81m

a2[ !?1]mn-tlm{3Bmal 21f

(21)

Note that the indices m and n associated with !?l and Al in (IS) through (21) need to be changed according to the indices for 'l/JI ,2,3,4.

A dominant-mode approximate solution, 'l/J2,OO = 0, is given by

. / h/k2 - {32 _ V k~ - {32 + iVkr - {32 cot(aod) - 421fa J = 0 (22)

where - Si J = ---,=:=;;;==~

Jk~ - {32

-4i roo [eXp(i2Jk~ - {32a) exp( -2Jk~ - {32 aV)] dv. (23) Jo Jk~ - {32(1 + iv)2Jv(v - 2i)

4.2 Groove Guide [4]

y

PEe

Region (I) PEe

Fig. 4.2. Groove guide

The groove guide is known to have a low-loss high-power capacity above 100 GHz and experimental and theoretical studies were performed in [5-7]. In this

4.2 Groove Guide [4] 91

section we will analyze its guiding characteristics and derive the dispersion relations for the TE and TM waves. The groove guide interior consisting of regions (I), (II), and (III) is filled with a homogeneous dielectric medium with permittivity Eo This means that the groove guide can support either a TE or TM wave propagation. The E-field, E(x, y, z), propagating along the z-direction may be represented as E(x, y)eif3z • In regions (I), (II), and (III), their respective Ez and Hz fields are

00

E;(x,y) = L Pm sinam(x + a) sinam(y + d) (1) m=l

00

H;(x,y) = L qm cosam(x + a) cosam(y + d) (2) m=O

E;I(x,y) = 2~ i: (E:eiKY +E;e-iKY ) e-i~Xd( (3)

H;I (x, y) = 2~ i: (ii:eiKY + ii;e-iKY ) e-i~x d( (4)

00

(5) m=l

00

H;ll(x,y) = L smcosam(x+a)cosam(y-b-d) (6) m=O

where am = mn/(2a), am =Jk2 - a;' - /32 , K, = Jk2 - /32 - (2, and the wavenumber is k = w.,[iif. = 2n / A. The remaining field components are immediately available from (5) and (6) in Sect. 4.1.

4.2.1 TM Propagation

Consider a TM wave propagation with Ez "# 0 and Hz = O. The boundary condition E;I (x, 0) = E; (x, 0) is written as

2.. roo (E+ + E-) e-i(x d( 2n i-oo Z Z

= { f,pm sin am(x + a) sin (am d) ,

0,

Applying the Fourier transform to (7) yields 00

Ixl < a

Ixl > a.

E: + E; = L Pmam sin (am d) a2 Fm((a) . m=l

Similarly the condition E;ll (x, b) = E;I (x, b) gives

(7)

(8)


00

E5;e i l<b + E5;e- i l<b = - L rmam sin(amd)a2 Fm((a) . (9) m=1

The condition HI'! (x, 0) = H; (x, 0) for Ixl < a and the condition H;II (x, b) = HP(x,b) for Ixl < a give, respectively

00

L Pmam cos(amd) sinam(x + a) m=1

(10)

00

L rmam cos(amd) sin am(x + a) m=1

= - K, [-Acsc(K,b) + Bcot(K,b)] e-'(x d( 1 /00 .

27f -00

(11)

where 00

A = L Pmam sin(amd)a2 Fm((a) (12) m=1

00

B = - L rmam sin(amd)a2 Fm((a) . (13) m=1

We multiply (10) and (11) by sin an(x + a) and integrate from -a to a to obtain

%;1 {pm [~:p cos(amd)8mn + Sin(amd)Jr]

+rm sin(amd)I2 } = 0 (14)

%;1 {pm sin(amd)lz

+rm [~cOS(amd)8mn + Sin(amd)Jr] } = 0 (15)

where

(16)

(17)

Using the residue calculus, we transform II and lz into fast convergent series

4.2 Groove Guide [4] 93

{ 4aam cot(amb) . ~ (/ )

II = (mn)2 8mn -1 ~ Umn vn b , m + n = even

0, m+n = odd

(18)

12 = (mn)2 umn 1 ~ mn vn ,m + n = even (19) { 4aam csc(amb) ~ _ . ~(-I)VU ( /b)

v=1 0, m+n=odd

where

(20)

A dispersion relation is obtained by setting a determinant of simultaneous equations for Pm and r m to zero

I:~ :~ 1= ° (21)

. where the elements of WI and 1}/2 for m + n = even are

'l/Jl,mn = ~:p [cos(amd) + sin(amd) cot(amb)]8mn

00

-isin(amd) L Umn(vn/b) (22) v=1

4aam . ( ) ( ) 'l/J2,mn = (mn)2 sm amd csc amb 8mn

00

-isin(amd) L(-I)VUmn(vn/b) . (23) v=1

In low-frequency limit, a dominant-mode (m = 1) approximate solution is

'l/Jl,ll - 'l/J2,1l = ° . (24)

4.2.2 TE Propagation

We consider a TE wave propagation by assuming Ez = ° and Hz :f:. 0. By applying the Fourier transform to the condition E:1(x,0) = E:(x,O), we get

-+ --_(~ . 2 Hz - Hz - - ~ qmam sm(amd)a Fm((a) .

K, m=O

(25)

Similarly from the condition E:II (x, b) = E:I (x, b), we get


jj+eil<b _ jj-e-il<b = _f ~ S a sin(a d)a2p ((a) (26) Z Z K L...J m m m m .

m=O

From the condition H;I (x, 0) = H; (x, 0) for Ixl < a and the condition H;II (x, b) = H;I (x, b) for Ixl < a, we obtain

where

~o { qm [aca: cos(amd)omn + Sin(amd)13]

+sm sin (am d) 14 } = 0

~O { qm sin(amd)14

+sm [~cOS(amd)Omn+sin(amd)h]} =0

1 100 ((a)2 h = - cot(tl:b)--Fm((a)Fn( -(a) d( 2n -00 tl:

1 100 ((a)2 14 = - csc(tl:b)--Fm((a)Fn( -(a) d( . 2n -00 tl:

Note that 13 and 14 are represented in fast convergent series

{

Cm cot (am b) . ~ 1 (j ) I Omn - 1 L..J - V mn vn b, m + n = even

3 = aam v=O Cv

0, m+n = odd

{cmcsC(amb)A _.~(-1)vlT ( jb)

I Umn 1 L..J v mn vn ,m + n = even 4 = aam v=O Cv

0, m+n = odd

where

To determine f3, we form a determinant for m + n = even

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

4.3 Multiple Groove Guide [8] 95

where

'ljJ3 mn = cm [cos(amd) + sin(amd) cot(amb)]omn , aam

-isin(amd) f= -.!..Vmn(mr/b) v=O Cv

(35)

'ljJ4 mn = cm sin(amd) csc(amb)omn , aam

00 (-l)V -isin(amd) L --Vmn(mr/b) .

v=O Cv

(36)

A dominant-mode solution is approximately given by

'ljJ3,OO - 'ljJ4,OO = 0 . (37)

4.3 Multiple Groove Guide [8]

y

.,.,.,.""- _x

A multiple groove guide is a potential high-power coupling structure that consists of a finite number of parallel rectangular groove guides. A doublegroove guide has been analyzed in [9] to assess its utility as a waveguide or a power coupler. In this section we will investigate the guiding and coupling characteristics of a multiple groove guide by deriving its dispersion relation. A theoretical analysis given in this section is an extension of the single groove case discussed in Sect. 4.2. Consider a multiple rectangular groove guide (N: number of groove guides) where the TE wave propagates along the z-direction with H(x , y,z) = H(x , y)eit3z . In regions (I) (-d < y < 0), (II) (0 < y < b), and (III) (b < y < b + d), the Hz components are


N-l 00

H;(x,y) = L L q~ cosam(x - nT) cos em (y + d) n=O m=O

·[u(X - nT) - u(x - nT - 2a)]

H;I (x, y) = 2~ i: (ii:ei1)Y + ii;e-i1)Y) e-i'Xd(

N-l 00

H;II(X,y) = L L s~ cosam(x - nT) cosem(Y - b - d) n=O m=O

·[u(x - nT) - u(x - nT - 2a)]

(1)

(2)

(3)

where am = m7r/(2a), em = Jk2 - a;' - {32, T/ = Jk2 - (2 - {32, k = 27r/AO, and u(·) is a unit step function.

Applying the Fourier transform to the Ex continuity at y = 0 yields

N-l 00

ii: - ii; = - L L ~q~em sin(emd)G~(() n=O m=O IT/

(4)

where

Gn (I") = i([1 - (_I)mei '2a] i'nT m .. 1"2 2 e . .. -am

(5)

Similarly the Ex continuity at y = b gives

N-l 00

ii:ei1)b - ii;e-i1)b = L L ~s~em sin(emd)G~(() . (6) n=O m=O IT/

We multiply the Hz continuity at y = 0 by cosal(x-pT), (P = 0, ... ,N -1), and integrate over (PT < x < pT + 2a) to get

N-l 00

L L {q~[em sin(emd)Il + acos(emd)OmIOnpcm] n=O m=O

+s~em sin(emd)I2 } = 0 (7)

where

It = 2-100 cot(T/b) Gn (()GP(-()d( 27r -00 T/ m I

(8)

12 = 2-100 csc(T/b) Gn (()GP ( -()d( . 27r -00 T/ m I

(9)

Note that II and 12 are transformed into rapidly-convergent series

4.4 Corrugated Coaxial Line (10) 97

I - cm8ml8np _ ~ f: (vAl (10) 1 - a ~m tan(~mb) b v=O cv(; - a~)(; - an

I = cm8ml8np -~f:(-I)V (vAl (11) 2 a ~m sin(~mb) b v=O cv(; - a~)(; - an

Al = [( _1)m+l + 1] exp(i(vln - piT)

-( _1)m exp [i(vl(n - p)T + 2al]

-( _1)1 exp [i(vl(n - p)T - 2al] (12)

where (v = Jk2 - (V1r/b)2 - 132. Similarly the Ex and Hz continuities at y = b between regions (II) and (III) yield

N-l 00

L L {q;::'~m sin(~md)I2 n=O m=O

+s;::'[~m sin(~md)Ir + acoS(~md)8mI8npcm]} = O.

A dispersion relation is formed from (7) and (13) as

I:~ :~ 1= 0

where the elements of l[1l and l[12 are

'lj;~~l = ~m sin(~md)Il + acos(~md)8mI8npcm 'lj;;::nl = ~m sin(~md)I2 .

(13)

(14)

(15)

(16)

When N = 1 (single-groove case), (14) reduces to (34) in Subsect. 4.2.2. When N = 2 (double-groove case), (14) reduces to a dominant-mode solution (m=O)

'lj;~?OO - 'lj;~?oo = ±('lj;}?oo - 'lj;~?oo) (17)

where ± sign corresponds to the TE12 and TEll waves, respectively.

4.4 Corrugated Coaxial Line [10]

A study of electromagnetic wave scattering from a corrugated coaxial line is of interest due to practical applications for microwave filter design [11,12]. In this section we will analyze scattering from multiple grooves in the inner conductor of a coaxial line. An incident TEM wave propagates along a coaxial line whose inner conductor has an N number of grooves. In region (I) (b < r < a) the H-field consists of the incident and scattered fields

(1)

(2)


,

o d d+w 2d+w N(d+w)-w

Region (II) t:2

Fig. 4.4. Corrugated coaxial line

where Ii = Jkr - (2, 1/1 = J l-to/El, R(lir) = Jo (lir) No (lia) - No (Iir)Jo (lia) , , dR(lir)

kl = Wy'l-toEl = Wy'l-tOErlEo, and R (lir) = d(lir) . Note that (r, </J, z) are

circular cylindrical coordinates, and Jo(-) and No(-) represent the Oth order Bessel functions of the first and second kinds, respectively. In region (II) (c < r < b) with permittivity E2 the H-field is

N-l <Xl n

H¢II(r, z) = iWE2 L L Pm m(limr) cos am[z - s(n)) n=O m=O lim

·{u[z - s(n))- u[z - d - s(n))}

where

Ro(limr) = { JO(limr)No(limc) - No (limr) Jo (limc), JO(lim r),

s(n) = n(d + w)

C¥=O c=O

(3)

(4)

(5)

lim = Jk~ - a~, k2 = Wy'l-tOEr2EO, am = m7rld, u(·) is a unit step function, N is the number of grooves, and P~ is an unknown coefficient associated with the mth mode in the (n + l)th groove.

The tangential E-field continuity is

E (b z) = {EZII(b, z), s(n) <.z < d + s(n), n = 0,1, ... ,N - 1 (6) zl , 0, otherwIse.


4.4 Corrugated Coaxial Line [10] 99

where n -i([( _l)me i(d - 1] .

Gm (() = (2 _ a;' exp[l(s(n)] .

The H¢ field continuity at r = b over the groove apertures requires

H~J(b, z) + H¢J(b, z) = H¢IJ(b, z) .

(8)

(9)

We multiply (9) by cos adz - s(p)] (where 1= 0,1,2, ... andp = 0,1,2, ... ,N-1) and integrate from s(p) to s(p) + d to obtain

(10)

(11)

Using the residue calculus, it is possible to transform I into rapidly-convergent series that are efficient for numerical computation. The result is

1- cmd R'(r;,b) r5 r5 I - 2 r;,R(r;,b) ml np

(=a=

00 2i([1 _ (_l)me i (d] I - ~ b[l- J6(r;,b)/J6(r;,a)]((2 - a;,)((2 - an (=(j

ik1 [1- (_1)meik1d]

bIn (b/a)(ki - a;,)(ki - an' n = p and m + I is even

= 0, n = p and m + I is odd

= _ f i([(-l)mei(d -l][(-l)le-i(d -l)exp[i([s(n) - s(p)))

j=l b[l - J1;(r;,b)/ J1;(r;,a))((2 - a;') ((2 - an

ik1 [( _1)meik1d - 1][( _1)le- ik1d - 1) exp[ik1 [s(n) - s(p)))

2bln(b/a)(ki - a;,)(ki - af) n>p

(12)

(13)

(14)


_ ~ i([( _1)mei(d - 1][( _1)le- i(d - 1] exp[i([s(n) - s(p)]] I - ~ b[l- J6(",b)/J6(",a)]((2 - a;") ((2 - an

1-1 (=-(j

ikd(-I)me-ik1 d _1][(_I)leik1d -1]exp[-ikds(n) - s(P)]]

2bln (b/a)(kr - a;")(kr - an n<p.

(15)

Note that (j is a root of the characteristic equation R(",b) = O. The first term in (12) represents a contribution of mode-scattering in region (II), and the second and third terms in (12) account for the effects of higher and TEM mode-scattering in a coaxial line, respectively. The field reflection coefficient r at z = 0 and the transmission coefficient T at z = (N - 1)(d + w) + d= N (d + w) - ware

H¢/ _ r= --. = -Lo H¢/

(16)

H~/ + H¢/ + T = . = 1 +Lo

H¢/ (17)

N-l 00 n R ( b)'k [1 (l)m 'fik1d] L ± - '"' '"' Pm 0 "'m 1 1 - - e [ 'k ( )]

o - =F ~ ~ 2ln (b/a)(k2 _ a2 ) exp =Fl IS n . n~m~ 1 m

(18)

4.5 Coaxial Line with a Gap [13]

r

··· ··· ·R .. ·· • .... (II)····· ·· .. ..... eglon· · .. ... . .............. ... .. ' .. .. . ,', .. .... .

-z ::::::: :: ~i :~ : ~ri~p:: ::: :::: """"""""""""""",,,,,,,,,,,,,,,,, ...... ..... .. .... ..

4.5 Coaxial Line with a Gap [13] 101

A study of TEM wave scattering from a coaxial line terminated by a gap is of practical interest in microwave circuit design [14,15]. In this section we will analyze scattering from a coaxial line terminated by a gap, where the inner conductor of a coaxial line is partially removed and replaced by a dielectric medium with permittivity 102 = t:Ot:r2' A theoretical analysis shown in this section is similar to that discussed in Sect. 4.4. An incident TEM

iklZ

wave, H~ = _e -, propagates from left inside a coaxial line. In region (I) the 'TIl r

scattered H-field is

e-ik1Z 2100 1 _ H¢I(r, z) = -- + iWt:l - -EI(()R'(K,r) cos((z)d(

'TIlr rr 0 K, (1)

where K, = v'k~ - (2, kl = w.,[iifl = 2rr/>..1, 'TIl = v'J.t/t:l, R(K,r) = Jo(K,r)No(K,a) - No(K,r)Jo(M), and R'O = dR(·)/d(·). Note that JoO and NoO are the Oth order Bessel and Neumann functions, respectively. In region (II) the scattered H-field is

H¢[[(r, z) = iwt:2 f: _l-R~(K,mr) cos(amz) m=O K,m

(2)

where Ro(K,mr) = PmJO(K,mr), mo = dRoO/d(·), K,m = v'k~ - a~, k2 = w".fii€2 = 2rr/>"2, and am = mrr/d.

The Ez-field continuity at r = b is

E (b ) _ { Ez[[(b, z), zI ,z - 0,

-d<z<O otherwise.

Applying the Fourier cosine transform to (3) gives

EI(() = ~o ~~:~)b) Em(()

where

;:;' (() = (-l)m(sin((d) ~m ((2 _ a~) .

The H¢-field continuity at r = b for (-d < z < 0) is

eik1Z H¢I(b,z) + -b = H¢[[(b,z).

'TIl

(3)

(4)

(5)

(6)

Multiplying (6) by cos(asz) (where s = 0,1,2, ... ) and integrating with respect to z from -d to 0, we obtain

(7)

where 8ms is the Kronecker delta and


We transform 1ms into a fast-convergent series

I - cmdR'(r;,b) 8 I ms - 2r;,R(r;,b) ms (=a m

i( _1)m+s k1 (1 - e2ikl d)

2bln (b/a)(kr - a;,,)(kr - a;)

(X) i( _1)m+s((1 _ e2i(d) I

- ~ b[1 - JJ(r;,b)!JJ(r;,a))((2 - a;") ((2 - a;) (=(n

where (n is given by R(r;,b) k=(n = O. The scattered field for (z < -d) in region (I) is

H¢[(r,z) = (1 + Lo)e-ik,Z _ I: Ln(()R'(r;,r)ei(ZI 771 r n=l (=-(n

where

Lo = k1sin(k1d) I: (-1) mRo(r;,mb) In(b/a) m=O kr - a;"

2WE1 sin((d) ~ (-1)m Ro(r;,mb) Ln(() = bR'(r;,b)[1- JJ(r;,b)/JJ(r;,a)) ~o (2 - a;" .

The field reflection coefficient r at z = -d is

r = - H~[ I = -(1 + Lo)ei2k1 d .

H¢ z=-d

4.6 Coaxial Line with a Cavity [16]

(8)

(9)

(10)

(11)

(12)

(13)

A coaxial line-radial line junction has been widely used as an antenna feeder or a power combiner and was studied in [17) to obtain its equivalent circuit representation. A coaxial line with a material-filled cavity is useful as a microwave sensor to estimate the material permittivity but its theoretical investigation is very little. In this section we will investigate TEM-wave scattering from an infinitely-long coaxial line with a dielectric-filled circular cylindrical cavity. The geometry of a scattering problem is shown in Fig. 4.6. A theoretical analysis given in this section is similar to the discussions in Sects. 4.4 and 4.5. In region (I) (a < r < b and permittivity: 100101'1) the total E-field is assumed to have incident and scattered components

4.6 Coaxial Line with a Cavity [16] 103

z c

Z=01

Incidence .l'lg. -',l.U. vOctXIaI llne wnn a cavny

. e ik1Z

E~l(r, z) = --r

Ezl(r, z) = 2~ i: El(()R(f\;r)e-i(zd(

Erl(r, z) = ;: i: El(()i(R'(f\;r)f\;-l e-i(Zd(

H¢I(r, z) = iWfOfr121 roo El(()R'(f\;r)f\;-l e-i(Zd( 1f J-oo

(1)

(2)

(3)

(4)

where R(f\;r) = JO(f\;r)No(f\;a) - No(f\;r)Jo(f\;a), R'(·) = dR(·)/dO, k1 = Wy'J.Lofofr1 = 21f/)..1: wavenumber, f\; = .jkr - (2, and JoO and NoO denote the Oth order Bessel and Neumann functions, respectively. In region (II) (b < r < c, - d < z < 0, and permittivity: fOf r 2) the scattered fields are


00

EZH(r, z) = L PmQ("'mr) cos(amz) (5) m=O

00

ErH(r, z) = - L PmamQ'("'mr)",~l sin(amz) (6) m=O

00

H"'H(r, z) = iW€O€r2 L PmQ'("'mr)",~l cos(amz) (7) m=O

where Q("'mr) = JO("'mr)No("'mc) - NO("'mr)JO("'mc), "'m = v'k~ - a~, k2 = wVt-tO€O€r2, and am = m7r/d. If a cavity radius c is chosen to be infinite

(00), Q("'mr) is given by H~1)("'mr). Enforcing the boundary conditions on the Ez-field continuity at r = b

and the H",-field continuity at r = band (-d < z < 0), we get

where

(9)

1 - dcmR'(",b) 8 I Gm(-kt} ms - 2",R(",b) ms (=am bIn (b/a)(ki - a;)

00 2Gm(-() I - ~ b[l- JJ(",b)/JJ(",a)]((2 - a;) (=("

(10)

Note that 8ms is the Kronecker delta, Cm = 2 (m=O), and 1 (m=I,2, ... ). The field reflection (F) and transmission (T) coefficients in region (I) are

shown to be

F = E~I(r, z) = _ f: PmQ("'mb) Gm(kl)e-2iklZ E~I(r,z) m=O 2In(b/a)

(11)

T = E~I(r, z) = 1 + f: PmQ("'mb) Gm( -kt} . E~I(r, z) m=O 2In(b/a)

(12)

It is also interesting to consider scattering from a shorted coaxial line with a cavity as shown in Fig. 4.7. When the coaxial line is shorted at z = 0, the field reflection coefficient F is given as

4.7 Corrugated Circular Cylinder [18] 105

z c

-..----------- z=o ~"'1~-----t-- r

9

d

PEe

r = E;I(r,z) E~I(r, z)

{ ~ PmQ("'mb)k1

= - 1 + ~O In(b/a)(kr - a;")

. [( _l)m sin(kl (d + g)) - sin(klg)] }e-2iklZ .

4.7 Corrugated Circular Cylinder [18]

(13)

A study of electromagnetic wave propagation within a conducting corrugated circular cylinder is important for the design of antenna feed horn and gyrotron. The radiation characteristics from an open-ended corrugated circular waveguide should have low sidelobes, low cross polarization levels, axial beam symmetry, and low attenuation [19-22] for antenna feed application. In


Fig. 4.8. Corrugated conducting circular cylinder

this section we will analyze scattering from a corrugated circular cylindrical waveguide that supports the propagation of a hybrid wave (a combination of the TE and TM waves). This implies that the boundary conditions need to be enforced in terms of the TE and TM waves simultaneously. An incident TEll wave propagates within a perfectly conducting cylindrical waveguide that is filled with a dielectric medium with permittivity to and permeability f.L . For simplicity we introduce the normalized fields E(t') and H(t') that are given by E(t) = VJiE(t'), H(t) = -i"fiH(t'), and t = ,jJi£t'. Maxwell's equations for the normalized fields E(t') and H(t') are therefore written in time domain as

\7 X E(t') = i aH(t') at'

\7 x H(t') = i aE(t') . at'

(1)

(2)

It is possible to represent the fields in terms of the Hertz vector potentials as

E = \7 x \7 x IItm +w\7 x IIte == E tm + E te

H = \7 x \7 x IIte + w\7 x IItm == H te + H tm

(3)

(4)

where IItm and IIte are the electric (TM) and magnetic (TE) Hertz vector potentials, respectively. In region (I) (r < a) the scattered field is represented by the Hertz vector potentials


II:m = zcos¢> i: lltm()J1(~r)ei(Zd( (5)

II:e = zsin¢> i: llte()h(~r)ei(zd( (6)

where ~ = J w2 - (2 and J1 (.) is the first-order Bessel function. In region (II) (r > a and Sl < Z < Sl +d, corrugation region) the fields in the lth groove are

00

IItm = -z L EnlPnl(~nr) cos ¢>cos n; (z - st) n=O

00

IIte = Z L HnlQnl (~nr) sin ¢> sin n; (z - Sl) n=1

where

Pnl(~nr) = J1(~nr)N1(~nbl) - J1(~nbl)N1(~nr)

Qnl(~nr) = J1(~nr)N{(~nbl) - J{(~nbl)N1(~nr)

(7)

(8)

(9) (10)

~n = Jw 2 - (mrjd)2, and b1 is a radius at the lth corrugation. The incident field takes the form of

(11)

where (3 = J w2 - (3~, (3c = xi j a, and xi is the first root of derivative of the Bessel function J1 (.).

The tangential Ez continuity at r = a requires

(Ei + E!m + E!e) . Z

= { (E~~ + E~~) . z, 0,

Sl < Z < Sl+1 (l = 1,2,3, ... ) otherwise

(12)

where a symbol (.) denotes the dot product of two vectors. Applying the Fourier transform to (12) and solving for lltm(), we get

where

The tangential Et/> continuity at r = a requires

I I' (Ei + E tm + Ete) . ¢

= { (E~~ + Em . ~, 0,

Sl < Z < SI+1 (l = 1,2,3, ... ) otherwise.

(14)

(15)


Applying the Fourier transform to (15), sUbstituting fItm(c,) into (15), and solving for fIte ( (), we obtain

fIte (C,) IiW J{ (lia) =

- 2~ LEnl Pnl~na) [(';)2 - (C,:n r] exp(-iC,sdEn(C,) n,l

- 2~ L HnlWlinQ~l(lina) n; exp( -iC,sl)En(() . n,l

The tangential Hz continuity at r = a in the qth groove

(Hi + H~m + H~e) . Z = (H~!n + H~~) . Z

is rewritten as

(3~Jl((3ca)eif3z + I: iite (()J1 (lia)ei(ZdC,

= L HnlQnl(kna)li;' sin n; (z - Sl) . n,l

(16)

(17)

(18)

Multiplying (18) by sin ":t7f (z - Sq) and integrating from Sq to Sq + d with

respect to z gives

(19)

where

(20)

(21)

(22)

is rewritten as


,,[ mrQnl(K,na) I()] n11"( ) = L...J Hnl d r + EnlK,nWPnl K,na cos d Z - Sl . n,l

(23)

Multiplying (23) by cos ":t11" (z - Sq) and integrating from Sq to Sq + d with

respect to z, we get

(24)

where

13 = 21 100 (:~:~K,a~ exp[i(Sq - sl)]Em(-()En«()d( (25) 11" -00 K, 1 K,a

14 = 21 100 (2;\(K,a; exp[i(Sq - sl)]Em( -()En«()d( . (26) 11" -00 K, 1 K,a

It is possible to evaluate the integrals, 11 , 12 , h, and 14 as fast-convergent series by modifying them in the complex (-plane. In view of (24) in Subsect. 1.1.3, the rewritten integrals contain poles due to the Bessel functions and poles due to Em and En when m = nand q = l. Evaluating the pole contributions with a residue calculus, we obtain

I = _." (jJ1(Xj) A(rl.) 1 1L...J J"( ') '»

j a 1 Xj

(27)

(28)

(29)


where

A(() = Al [(2 _ (m1l"/d)2] [(2 _ (n1l"/d)2]

Al = [1 + (_1)m+n) exp(i(ISq - sd)

-(-l)n exp(i(ISq - Sl- dl)

-(-l)m exp(i(ISq - Sl + dl) .

(30)

(31)

(32)

We note that (j = Jw2 - (Xj/a)2, (j = JW2 - (xj/a)2, and Xj and xj are

the jth roots of J1 (-) and Jf(.), respectively. We evaluate the scattered E-field when z -+ ±oo as

E = L GJej exp(±i(jz) + L Cjej exp(±i(jz) (33) j j

The reflected TE wave power at z = -00 is

h

P: = ~ L IGJ 12w (j(Xj2 -l)J;(xj) j=1

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

4.8 Parallel-Plate Double Slit Directional Coupler [23] 111

where JI is the largest number that satisfies the condition w2 - (xj/a)2 > O. The reflected TM wave power at z = -00 is similarly given as

h P;" = ~ L ICjI2W(jX]J~2(Xj)

j=l

(42)

where 12 is the largest number satisfying w2 - (Xj / a)2 > O. The total reflected power is given by (p; + p;n).

4.8 Parallel-Plate Double Slit Directional Coupler [23]

z

P1 - - 0: -82 Z=-d1 p-~ P+83 - P2

~ ~~.w:L-==r _t~ Port 3 Region (III) Port 4

Fig. 4.9. Parallel-plate double slit directional coupler

Electromagnetic wave coupling by slits between rectangular waveguides has been considered in [24-28] for directional coupler applications. In this section we will examine TE (transverse electric to the x-axis) wave coupling through a double slit between two parallel-plate waveguides. The present study is directly applicable to a practical directional coupler that has coupling elements in the common narrow wall of rectangular waveguides. In region (I) (-d1 < z < 0) the total E-field consists of the incident and scattered components


E;(x,z) = eikzsx sin(kzsz)

E~(x,z) = ~lOO E~(()sin"'lze-i(Xd( 27f -00

(1)

(2)

h ki dl _ S7f _ V 2 2 - V 2 2 were 0 < S < --, kzs - -d ' kxs -ki - kzs ' and", -ki - ( . In 7f I

region (II) (-d2 < z < -dl , - 0: - a2 < x < -0: + a2) the E-field is 00

m=1

m7f . / 2 2 . ( ) ( where a2m = -2 and 6m = V k2 - a2m · In regIOn III -d2 < z < a2

-dl , (3 - a3 < x < (3 + a3) the E-field is 00

E~II (x, z) = L [em cos(6mz) + 1m sin(6mz)) sina3m(x - (3 + a3) (4) m=1

m7f ~~~-

where a3m = -2 and 6m = Vk~ - a~m· In region (IV) ( -d3 < Z < -d2 a3

) the transmitted E-field is

E~v (x, z) = ~ [: E~v (() sin "'4(Z + d3)e-i(x d( (5)

where "'4 = Jk~ - (2 . The tangential E-field continuity at z = -di is given by

! E[I(x, -dt}, -0: - a2 < x < -0: + a2

E;(x, -dl ) + E~(x, -dl ) = E[II (x, -dt}, (3 - a3 < x < (3 + a3 (6)

0, otherwise.

Taking the Fourier transform of (6) gives

- sin("'ddE~(() 00

= e-i(a L [bm cOS(6mdl) - em sin(6mdd)a2ma~Fm((a2) m=l

00

+ei (,6 L [em cos(~3mdd - 1m sin(6mdl))a3ma~Fm((a3) . (7) m=1

The tangential H-field continuity between regions (I) and (II) for -0: - a2 < x < -0: + a2 is

(8)

Multiplying (8) by sin a2n(X+0:+a2) and integrating from -0:-a2 to -0:+a2, we get

4.8 Parallel-Plate Double Slit Directional Coupler [23] 113

27fkzs (-I)Sa2na~F(kxsa2)e-ikZBO< 00

= L [bm COS(6mdl) - em sin(6mdl)]a2ma2nItmn m=l

00

m=l

where

lImn = i: ibl cot(ibldda~Fm((a2)a~Fn( -(a2)d( (10)

l2mn = i: ibl cot(ibldl)a~Fm((a3)a~Fn( -(a2)ei<:(0<+,B)d( . (11)

It is possible to transform lImn and l2mn into fast-convergent series based on the technique of contour integration. Similar evaluations are available in Sects. 1.5 and 5.3. The tangential E and H-field continuities between regions (I) and (III) give

00

= L [bm cOS(6mdl) - em sin(6mdd]a2m a2nl 3mn m=1

00

+ L [em cOS(6mdl) - 1m sin(6mdl)]a3ma2nl 4mn m=1

(12)

where

l3mn = i: ibi cot(ibldl)a~Fm((a2)a~Fn( -(a3)e- i<:(0<+,B)d( (13)

l4mn = i: ibi cot(ibldda~Fm((a3)a~Fn(-(a3)d( . (14)

The tangential E and H-field continuities between regions (II) and (IV) give

27f6n[bn sin(6nd2) + en cOS(6nd2)]a2 00

= L [bm cOS(6md2) - em sin(6md2)]a2ma2nl 5mn m=1

00

+ L [em cOS(6md2) - 1m sin(6md2)]a3ma2nl 6mn (15) m=l

where


Ismn = i: ~4 cot ~4(d3 - d2)a~Fm((a2)a~Fn( -(a2)d( (16)

hmn = i: ~4 cot ~4(d3 - d2)a~Fm((a3)a~Fn( -(a2)ei«a+,8)d(. (17)

The tangential field continuities between regions (III) and (IV) give

21f6n[en sin(6nd2) + in cOS(6nd2)]a3 00

= L [bm cOS(6md2) - Cm cos(6md2)]a2ma3nI7mn m=l

00

+ L [em cOS(6md2) - im sin(6md2)]a3ma3nIsmn (18) m=l

where

hmn = i: ~4 cot ~4(d3 - d2)a~Fm((a2)a~Fn( -(a3)e- i«a+,8)d( (19)

Ismn = i: ~4 cot ~4(d3 - d2)a~Fm((a3)a~Fn( -(a3)d( . (20)

The scattered field at x = ±oo in region (I) is

E1(±oo z) - " L± sin(k z)e±ik.,vx Y ,- L...J v zv

v

00

L; = L ia2m[bm cos(6mdd - Cm sin(6mdd] m=l

a~kzve±ik.,va . d k (k d) Fm(=t=kxva2)

1 xv cos zv 1 00

+ L ia3m[em cOS(6mdl) - im sin(6mdl)] m=l a~kzve'fik.,v,8

. d k (k d) Fm(=t=kxva3) . 1 xv cos zv 1

In region (IV) the scattered field at x = ±oo is

E~V(±oo,z) = LK~sin(kz1/(z+d3»e±ik.,'1X 1/

(21)

(22)

(23)

where 1 ::; 'T} < k4(d3 - d2)/1f ('T} = integer), kZ1/ = 'T}1f/(d3 - d2), kX1/ = Jk~ - k~1/' and

4.9 Parallel-Plate Multiple Slit Directional Coupler [29] 115

00

K~ = - L ia2m[bm cOS(6md2) - Cm sin(6md2)] m=1

00

m=1

(d3 - d2) cos(kz1)(d3 - d2))kx1) .

The reflection and transmission coefficients are given as

PI 1" _ 2 (} = p. = k L..- kxvlLv I

1, XS v

= Pi + P2 = 11 L+1 2 _1" k IL+1 2 72 P. + s + k L..- xv V

t xs vies

= P3 = (d3 - d2 ) "k IK-1 2 73 p. k d L..- X1) 1)

t xs I 1)

(24)

(25)

(26)

(27)

(28)

4.9 Parallel-Plate Multiple Slit Directional Coupler [29]

In Sect. 4.8 we examined TE wave coupling through a double slit between two parallel-plate waveguides. In this section we will consider TM (transverse magnetic to the z-axis) wave scattering from multiple slits in between two parallel-plate waveguides. A TM wave scattering analysis in this section is similar to the TE case in Sect. 4.8. A TM wave propagating along the zdirection is incident from port 1 onto multiple slits. In region (I) (0 < x < dd the total H-field consists of the incident and scattered field components

H~(x, z) = eikzsz cos kxs(x - dd

H~ (x, z) = ~ roo ii~ (() cos K(X - dde-i(z d( 21f Loo

(1)

(2)

where 0 ::; S < kdI/1f, kxs = S1fld l , kzs = Jk2 - k~s' K = Jk2 - (2, and k is the wavenumber. In region (II) (-d2 < X < 0 and Tl < z < Tl + al: l = 1,2, ... ,N) the total H-field is

00

H~I (x, z) = L [b~ cos(6mx) + c~ sin(~lmx)J cosalm(Z - T1) (3) m=O


X=d1 - - -P1 Pi Region <I) P2

Port 1 Port 2 y z

J~L~ ___ x=o

X=-d2

a1 a2 ••• - T1 12 P3 Region (III)

Port 3

I~~I TN -P4

Port 4 X=-d3

ler

where aZm = m7f/az and ~Zm = Jp - arm· In region (III) (-d3 < X < -d2) the transmitted H-field is

HtII(X,z) = 2~ i: iltII(()COSK,(X + d3)e-i(zd( . (4)

Between regions (I) and (II) the tangential field continuities give

(5) Noo Ze br

= L L cm<"Zm J(dd _ nCnar 27f 2

Z=l m=O

where co = 2, cz = 1 (l = 1,2,3, ... ), and

J(dd _ roo (2[1_ (-l)ne- i(a r J[l- (_1)mei(a1] exp[i((Tz - Tr)] - i-oo K, tan(K,d1) ((2 - a;m)((2 - a~n) d( .

(6)

Utilizing the technique of contour integration, we obtain

4.9 Parallel-Plate Multiple Slit Directional Coupler [29] 117

J(d ) = 7rcnalrSnmrSrl 1 y'k2 _ arm tan( y'k2 - armdd

00 A - L i27rkzw d (P 21 )(k2 2 ) (7)

w=O Cw 1 zw - aim zw - arn

A1 = exp(ikzwlTI - Trl) + (-l)m+nexp(ikzwlal - ar + TI - Trl)

-( _l)m exp(ikzwlTI - Tr + atl)

-( _l)n exp(ikzwlTI - Tr - arl) . (8)

Similarly the tangential field continuities at x = -d2 yield

7r [b~ cOS(~rnd2) - c~ sin(~rnd2)] Cnar N 00

+ L L [b!n6m sin(6md2) + c!n6m Cos(~lmd2)] J(d3 - d2) = 0 (9) 1=1 m=O

where

(10)

The total transmitted and reflected fields at z = ±oo in region (I) are

v

In region (III) the scattered field at z = ±oo is

H[II(x,±oo) = LK~coskx1](x+d2)e±ikzryZ 1]

(11)

(12)

(13)

where 0 :s 'f} < k(d3 - d2)/7r ('f} = integer), kX1] = 'f}7r/(d3 - d2), kZ1] =

Jp - k;'1]' and

K± = t f exp(=t=ikz1]TI ) - (-l)mexp[=t=ikz1](Tt +al)] 1] ± 1=1 m=O c1](d3 - d2)(k;1] - arm)

. [b!n6m sin(6md2) + C!n~lm COS(6md2)] (14)

The reflection and transmission coefficients are


(15)

(16)

(17)

(18)

where Pi is the incident power at port 1, 0 :::; v < kdl/rr, and 0 :::; 'T} < k(d3 - d2 )/rr.

It is possible to apply the present theory to a directional coupler consisting of wide-waH-slit rectangular waveguides with a dimension a x b with a > b. Our theory with a TMo incident wave (8 = 0) is approximately applicable to a rectangular waveguide directional coupler with a TElO incident wave by replacing k in (5) and (9) with Jk2 - (rr/b)2.


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5. Junctions in Parallel-Plate/Rectangular Waveguide

5.1 T-Junction in a Parallel-Plate Waveguide

z

1 Region (II) Pt2

PEe

Fig. 5.1. T-junction in a parallel-plate waveguide

The T -junction in a rectangular waveguide is one of basic power-coupling components used in microwave circuit applications. The power-combining characteristics of the T-junction in a rectangular waveguide have been extensively studied in [1-3]. In the next two subsections we will consider scattering from the H-plane and E-plane T-junctions in a parallel-plate waveguide. A scattering study of the T-junction in a parallel-plate waveguide is directly applicable to the T-junction in a rectangular waveguide.


122 5. Junctions in Parallel-Plate/Rectangular Waveguide

5.1.1 H-Plane T-Junction [4]

Assume that a TE wave E~(x, z) is incident on the parallel-plate waveguide Tjunction in H-plane (wavenumber k). Scattering from the H-plane T-junction in a parallel-plate waveguide is equivalent to that from the H-plane T-junction in a rectangular waveguide with the lowest mode (TElO wave) excitation. The total E-field in region (I) (-b < z < 0) has the incident and scattered waves

E~(x, z) = exp(ikxsx) sin kzs(z + b) (1)

E~(x,z) = ~/oo E~(()sin(J>;z)e-i(xd( (2) 21f -00

where kzs = s1flb, (s = 1,2,3, ... ), kxs = Jk2 - k;s, and J>; = Jk2 - (2. In region (II) (-a < x < a and z < -b) the total transmitted field is

00

E~I(x,z) = L cmsinam(x+a)exp(-i~mz) (3) m=l

where am = m1fI(2a) and ~m = Jk2 - a;". Applying the Fourier transform to the tangential E-field continuity at

z = -b yields 00

E~(() sin( -J>;b) = L cmam exp(i~mb)a2 Fm((a) . (4) m=l

The tangential H-field continuity at (-a < x < a and z = -b) is written as

kzs exp(ikxsx)

_ f: Cm am eX;;i~mb) 100 J>; cot (J>;b)a2 Fm((a)e-i(x d( m=l -00

00

= -i L cm~m exp(i~mb) sin am(x + a) . (5) m=l

Multiplying (5) by sinan(x + a) and integrating with respect to x from -a to a, we get

kzs ana2 Fn(kxsa)

_ Loo aman exp(i~mb) 2A (k) . C (. C b) - Cm a 3 - lCn<"na exp l<"n

21f m=l

where

A3(k) = i: J>;cot(J>;b) a2Fm((a)Fn(-(a)d(.

For m + n = odd, A3(k)=0. For m + n = even,

(6)

(7)

5.1 T-Junction in a Parallel-Plate Waveguide 123

A ( ) 27fem 3 k = 2 (e b/nm aam tan m

00 i47f(I7f/b)2 [1- (-I)mexp (i2Jk2 - (17f/b)2 a)]

- ~ Jk2 - (17f /b)2a2b [e;. - (17f /b)2] [e; - (I7f /b)2] (8)

The total transmitted and reflected fields at x = ±oo in region (I) are

E:(±oo, z) = L K,7= sin kzv(z + b) exp(±ikxvx) (9) v

where 1 ::::; v < kb/7f, v: integer (1,2,3, ... ), kzv = v7f/b, kxv = Jk2 - k~v' and

The reflection ({!) and transmission (7"1 and 7"2) coefficients are

7"1 = Ptl / Pi = 11 + K;12 + ,}- L kxv lKtl2 xs v#s

where 1 ::::; m < 2ak/7f, 1::::; v < kb/7f, and m and v are integers.

5.1.2 E-Plane T-Junction [5]

(10)

(11)

(12)

(13)

Consider the E-plane T-junction in a parallel-plate waveguide when a TM wave H~(x, z) is incident on the junction. In region (I) (-b < z < 0) the total H-field has the incident and scattered waves

H;(x, z) = eik",.x cos kzs(z + b)

H: (x, z) = 21 roo if: (() cos(l\;z)e-i(x d( 7f 1-00

(14)

(15)

where kzs = S7f Ib, s: integer (0,1,2, ... ), kxs Jk2 - k~s' and I\; =

Jk2 - (2. In region (II) (-a < x < a and z < -b) the transmitted field is

00 H:1(x,z) = L cmcosam(x+a)e-i~",z (16)

m==O

where am = m7f/(2a) and em = Jk2 - a~. The tangential E-field continuity at z = -b yields


00

H~ «()Ksin(Kb) = L -cm~meie",b(a2 Fm«(a) . (17) m=O

Multiplying the tangential H-field continuity along (-a < x < a and Z = -b) by cosan(x + a) and integrating with respect to x from -a to a, we obtain

where

For m + n = odd, D3(k)=0. For m + n = even,

( ) 27r€m D3 k = ~ma tan(~mb) 8nm

(18)

(19)

00 -i47rJk2 - (l7r/b)2 [1- (-I)mexp (i2Jk2 - (l7r/b)2 a)]

+ ~ €lba2[~;' - (l7r/b)2][~; _ (l7r/b)2] . (20)

The total transmitted and reflected fields at x = ±oo in region (I) are

H~(±oo,z) = LK;=coskzv(z+b)exp(±ikxvx) (21) v

where 0 :::; v < kb/7r, v: integer, kzv = v7r/b, kxv = Jk2 - k~v' and

00 • (: ie b K ± - ~ =fICm<"me '" 2 D (k ) v - L..J b a I'm =f xv a .

m=O €v

The transmission (71 and 72) and reflection (e) coefficients are

71 = Ptl/Pi = 11 + K:1 2 + + L€vk xv lKtl2 €s XB vi's

(22)

(23)

(24)

(25)

where 0 :::; m < 2ak/7r, 0 :::; v < kb/7r, m and v: integers, and the incident power is Pi = kxs€sb/(4wf.)

5.2 E-Plane T-Junction in a Rectangular Waveguide [6] 125

z

x

Incident

Fig. 5.2. E-plane T-junction in a rectangular waveguide

5.2 E-Plane T-Junction in a Rectangular Waveguide [6]

A study of electromagnetic wave scattering from the E-plane T-junction in a rectangular waveguide has been performed in [7]. In this section we will revisit the scattering problem of the E-plane T-junction in a rectangular waveguide. The TElO wave is incident on the T-junction filled with a medium with wavenumber k (= 27f I >. = w"fiif). The incident TElO wave is given as

. k ·k H' (x, y) = ~ cos(k y)e1 zX (1)

x IWJ.1 y

where ky = 7f/d and kx = Jp - k~. The E-plane T-junction in a rectangular

waveguide supports the propagation of a hybrid wave that is a combination of the TE and TM waves. In region (I) (-b < z < 0 and 0 < y < d) the scattered field consists of the TE (transverse electric to the x-direction) and TM (transverse magnetic to the x-direction) waves

(2)

(3)

where liz = Jk2 - k~ - (2. In region (II) (-a < x < a, Z < -b, and 0 < y < d) assuming the longitudinal direction is z, we express the z-components as


CXl

H;I (x, y, Z) = L Cn cos kn(x + a) cos(kyy)e-iKnZ n=O

CXl

E;I (x, y, z) = L dn sin kn(x + a) sin(kyy)e-iKnZ n=l

where kn = mr/(2a) and K,n = .Jk2 - k; - k~. Applying the Fourier transform to the Ex (x, y, -b) continuity

i: E~ (x, y, -b)ei'Xdx = iaa E~I (x, y, -b)ei'Xdx

yields

b(() = a2(

sin(K,zb)

(4)

(5)

(6)

. [~ cnwJ.tkyeiKnb Fn((a) + ~ dnknK,neiKnb Fn((a)] (7) L...J k2 _ K,2 L...J k2 - K,2 n=O n n=l n

Similarly applying the Fourier transform to the Ey continuity at z = -b gives

a = a2 [ CXl cn(k2k; - (2k; - (2k~)eiKnbFn((a) (() K,z sin(K,zb) ~ k2 - K,;

_ ~ dnk2knkyK,neiKnbFn((a)] L...J wII.(k2 - K,2) . n=l r n

(8)

Substituting a(() and b(() into the H-field continuities i I _ II Hx(x,y)+Hx(x,y,-b)-Hx (x,y,-b) (9) i I _ II Hy(x, y) + Hy (x, y, -b) - Hy (x, y, -b) (10)

taking iaa (9) sin kt(x + a)dx (t = 1,2, ... ), and i: (10) coskt(x + a)dx (t =

0,1, ... ), we obtain

(11)

5.2 E-Plane T-Junction in a Rectangular Waveguide [6] 127

(12)

(13)

(14)

(15)

(16)

(17)

It is possible to evaluate the integrals II through h using the residue calculus. The results are summarized below. For n + t = odd, h = 12 = 13 = 14 = 15 = O. For n + t = even,

I - Pn 8 _ 41fi ~ E((v) t ...J. 0 and n...J. 0 (18) 1 - k2 nt 4b L...J 1"' f f

n a v=o cv,>v

(19)

(20)

n=t=O

otherwise (21)


I = PnEn"';' 8 _ 27rikky coth(kyb) E(k) 5 (k2 _ k;;,) nt a4

_ 47ri ~ (v (V7r /b)2 E( (V) (22) a4 b ~ (k2 - (;)

[1- (_1)nei2(a] where E(() = ((2 _ k;;)((2 _ kl)' (v =Jk2 - k~ - (v7r/b)2, and

Pn = 27rC~t("'nb). a "'n

The scattered fields for x :::; -a and x 2 a are

H:(x,y,z) = LKtcos(kyy)cos(klz)e±i(IX (23) I

E:(x, y, z) = L J1± sin(kyY) sin(k/z)e±i(lx (24) I

where

where 0 :::; l < b/7rJP - k~, l: integer, 0 :::; n < 2a/7rJk2 - k~, and n:

integer.

5.3 H-Plane Double Junction [8] 129

Fig. 5.3. H-plane double junction

5.3 H-Plane Double Junction [8]

In this section we will consider a problem of electromagnetic wave scattering from two junctions in the H-plane waveguide where two junctions are either short- or open-circuited. The two-junction scattering analysis given in this section is an extension of the single H-plane T-junction discussed in Subsect. 5.1.1. A TE wave Et(x , z ), which is transverse electric (TE) to the x-axis , is incident on the junctions with wavenumber k (=27r / >..=w..Jii€). In region (I) (-b < z < 0) the total E-field is assumed to have the incident and scattered components Et (x ,z) and E~(x,z) as

E~(x , z ) = eikzsx sin kzs (z + b) (1)

E;(x, z ) = 2~ i: [E~+(()eiI<Z + E~- (()e-i l<z ] e-i(x d( (2)

where kzs = s7r/b, s: integer, kxs = Jk2 - k;s' and r;, = Jk2 - (2. In region (II) (-a < x < a and -q - b < z < -b) the transmitted field is

00

E;l (x, z ) = L em sin am(x + a)Yq(z) (3) m=l

where am = m7r/(2a) , ~m = Jk2 - a~ , and


y _ {e-i~=Z, open-circuited (q -+ 00) q(z) - sin~m(z + b + q), short-circuited

Y;(z) = dYq(z)jdz .

(4)

(5)

In region (III) (0: - d < x < 0: + d and 0 < z < p) the transmitted field is represented as

00

E~II (x, z) = I: dm sin bm(x - 0: + d)Xp(z) m=l

where bm = m7rj(2d), 17m = Jk2 - b'fn, and

X ( ) _ {ei1)m Z , open-circuited (p -+ 00) p z - sin 17m (z - p), short-circuited

X;(z) = dXp(z)jdz .

(6)

(7)

(8)

Applying the Fourier transforms to the tangential E-field continuities at z = -b and 0 yields respectively

00

(9) m=l

00

E~+(() + E~-(() = I: dmXp(0)bmei(G:d2 Fm((d) . (10) m=l

Substituting (9) and (10) into the tangential H-field continuities along (-a < x < a and z = -b) and along (0: - d < x < 0: + d and z = 0) gives explicitly

00

(11) m=l

kzseik'8x cos(kzsb)

-f cmYq(-b) ~m foo ~( b)a2 Fm((a)e-i(x d( m=l 7r -00 SIn K,

00 b fOO + I: dmXp(O) 2: _ K,Cot(K,b)ei(G:d2 Fm((d)e-i(x d( m=1 00

00

= I: dm sin bm(x - 0: + d)X;(O) . (12) m=l

5.3 H-Plane Double Junction [8] 131

Multiplying the H-field continuity at z = -b, (11), by sinan(x + a) and integrating with respect to x from -a to a, we get

-f dmXp(O) b;;n 12 + cnaY;( -b)8nm m=l

(13)

where

(14)

(15)

Utilizing the technique of contour integration, we transform the integrals into fast-convergent series

1 _ {O, 1 - hm8nm + rnm ,

h = gm8nm L1 + fnm

where

n+m = odd n+m = even

L1 = { 1, 0,

d = a and a = ° otherwise

(16)

(17)

(18)

h _ 2na~m ( ) m - a~ tan(~mb) 19

BmTJm gm = b~ sin(TJm b) (20)

Bm = -n [Id - al exp(i(ld - aD

-( -1)mld + al exp(i(ld + al)] L:=am (21) 00

rnm = L2cos(ln)TI;I; [1- (-1)mexp (i2Jk2 - (In/b)2a)] (22) 1=1 00

1=1

-( _1)m exp(i(ld + a + al) - (_1)n exp(i(la - a - dl)

+ exp(i(la + a - dl)] I (=Jp -(lrr /b)2

T _ -i2n(ln/b)2 uv - Jk2 - (In/b)2b[u~ - (In/b)2][v; - (In/b)2]cos(ln)

u,v = ~,TJ.

(23)

(24)


Multiplying the H-field continuity at Z = 0, (12), by sinbn(x - a + d) and integrating from (a - d) to (a + d) similarly, we obtain

kzs cos(kzsb) bneiku Otd2 Fn(kxsd) =

where

100 /'i,ei(Ot

13 = -=---( b)a2Fm«(a)~Fn(-(d)d( -00 SIn /'i,

14 = i: /'i,cot(/'i,b)d2Fm«(d)~Fn(-(d)d(. Note that

where 00

n+m = even n+m = odd

(25)

(26)

(27)

(28)

(29)

Znm = L2cos(lrr)T'1'1 [1- (-I)mexp (i2Jk2 - (lrr/b)2d)] (30) 1=1 00

tnm = LTe'1 [( _1)m+n exp(i(l- d + a - al) 1=1

-( _1)m exp(i(ld + a - al)

-( _1)n exp(i(l- a - a - dl)

+ exp(i(1 - a - a + dl)] I . (31) (=Vk2-(l1r/b)2

The transmitted and reflected fields at x = ±oo in region (I) are

E~(±oo, z) = L L; sin kzv(z + b)e±ik~vx (32) v

where 1 ~ v < kb/rr, v: integer (1, 2,3, ... ), kzv = vrr/b, kxv = Jk2 - k~v' and

00

± '" . () amkzv 2 ( ) Lv = L...J lCmYq -b Tba Fm =t=kxva m=1 xv

00 b k e=Fik~v Ot - ~ idmXp(O) ~x:~(-I)V d2Fm(±kxvd). (33)

The transmission (71, 72, and 73) and reflection (e) coefficients are

5.4 H-Plane Double Bend [9] 133

(34)

(35)

(36)

(37)

where 1:::; "( < 2ak/rr, 1:::; 0" < 2dk/rr , 1:::; v < kb/rr , and ,,(,O",v are integers.

5.4 H-Plane Double Bend [9]

z

Fig. 5.4. H-plane double bend

Wave scattering from a rectangular waveguide double bend has been considered in [10) and applied to the problem of quantum waveguide structures. A TE wave E~(x, z) is incident on a right-angled double bend with the wavenumber k = w.,fiii. In region (I) (0 < x and -b < z < 0) the field has the incident, reflected, and scattered components as


E~(x, z) = e-ik2SX sin kzs(z + b)

E; (x, z) = _eikzsx sin kzs (z + b)

E~(x,z) = 3. r>o E~(()sinl\;zsin(xd( 7r 10

(1)

(2)

(3)

where kzs = s7rjb, s: integer, kxs = Jk2 - k;s' and I\; = Jk2 - (2. In region (II) (0 < x < a and -c < z < -b) the field is

00 E~I (x, z) = L sin(amx) (am cos~mz + 13m sin~mz) (4)

m=l

where am = m7rja and ~m = Jk2 - a;;". In region (III) (x' < 0, x' = x - a, and -d < z < -c) the field is

E~II (x, z) = 3. [00 E~II (() sin I\;(z + d) sin (x' d( . (5) 7r 10

We apply the Fourier sine transform to the E-field continuities to represent Et (() and EtII (() in terms of am and 13m. The field continuities at z = -b and -c yield

(6)

where the matrix elements are

'l/Jrf' = onm ~~n sin(~nb) + 2~ cos(~mb)I(b) (7)

'l/JrY' = onm ~~n cos(~nb) - 2~ sin(~mb)I(b) (8)

'I/J~f' = onm ~(-1)n~n sin(~nc) - 2~ (_1)m cos(~mc)I(d - c) (9)

'I/J~Y' = onm ~(-1)n~n cos(~nc) + 2~ (_1)m cos(~mc)I(d - c) (10)

n kzsan[( _l)ne-ikzs a - (_l)neikzsa] 'Y = k2 _ a2 (11)

zs n

00 -i27r(17r ju)2( -l)m+naman [1 - exp (i2Jk2 - (l7r ju)2a)] I (u) = L --~====;::====;==;:::;;::-:-:-;:--=---:-:-~~--:-:----:--:--::--~

l=1 Jp - (l7rjU)2U[~;;" - (17rju)2][~; - (17rju)2]

+onm7ra~m cot(~mu) . (12)

The reflected field for x > 0 and the transmitted field for x' < 0 are respectively

5.5 Acoustic Double Junction in a Rectangular Waveguide [11] 135

P

E~II (-00, Z) = L L; sin k';-v(z + d)e-ik';vx' (14) q

where k"tv = p7T/b, k;v = q7T/(d - c), k;=v = Vk2 - (k;:v)2, and

L+ = f 2k"tv( -1)m+Pam sin(ktva) (am cos~mb -;Jm sin~mb) (15) P m=l ktvb[(ktv)2 - a;"]

L- = f 2k.;-v( -1)qam sin(k;va) (am cos~mc -;Jm sin~mc) (16) q m=l k;v(d - c)[(k;v)2 - a;"]

The transmission (T) and reflection (Q) coefficients are

T = P, /P = '"' k;v(d - c) IL-1 2 t , ~ k b q

q xs

Q = Pr/Pi = 11- L;-1 2 + L ~tv IL:12

Pi's xs

where 1 -:; p < kb/7T , p: integer, 1 -:; q < k(d - C)/7T, and q: integer.

5.5 Acoustic Double Junction in a Rectangular Waveguide [11]

z t 2g 't3 i--!

y

8

(17)

(18)

x


A study of acoustic wave scattering from a T-junction and a right-angled bend in a rectangular waveguide has been performed in [12-14]. In this section we will study acoustic wave scattering from two junctions in a hard surface rectangular waveguide. In region (I) (0 < z < h) an incident field (velocity potential) .pi (x, y, z) propagates along the x-direction and impinges on two junctions where the wavenumber is k (= 2rr / >.., >..: wavelength). In region (I) the total field consists of the incident and scattered potentials

.pi(X,y,z) =cos(bpy)coshl(z-h)ei{3p/x (1)

.pI (x, y, z) = 2~ I: [A() cos Kp(Z - h) + .8() COS(KpZ)]

(2)

where bp = prr/b, hi = lrr/h, /3pl = Jk2 - b~ - hi, and Kp = Jk2 - (2 - b~. In region (II) (-a < x < a and Z < 0) the transmitted field is

00

.pIl(x,y,z) = L Cmcosam(x + a)cos(bpy)e-iempz (3) m=O

where am = mrr / (2a) and ~mp = J k2 - a~ - b~. In region (III) (d - g < x < d + g and h < z) the transmitted field is

00

.pIlI (x, y, z) = L Dm cosgm(x - d + g) cos(bpy)eiT/mpZ m=O

where gm = mrr/(2g) and'T/mp = Jk2 - g~ - b~. The velocity continuity condition at z = 0 requires

8 [.pi (x, y, z) +.pI (x, y, Z)]I 8z z=O

_ { 8.pIl(x,y,z) I, Ixl < a, 0 < y < b - 8z z=O

0, otherwise.

Taking the Fourier transform of (5) results in

2( 00

A() = - ~ ( h) L ~mpCmFm(a) . Kpsln Kp m=O

The velocity continuity condition at z = h similarly yields

_ _g2(ei(d 00 .

B() = . ( h) L 'T/mpDmFm(g)elT/mph . Kpsln Kp m=O

The pressure continuity condition at z = 0 . I II

.p~(x, y, 0) +.p (x, y, 0) =.p (x, y, 0), Ixl < a, 0 < y < b

(4)

(5)

(6)

(7)

(8)

5.5 Acoustic Double Junction in a Rectangular Waveguide [11] 137

is written as

(_I)le i /1p1x + ~ ('JO [A(() cos(Kph) + B(()] e-i(x d( 2n J-oo

00

= L Cm cosam(x + a) . m=O

Similarly from the pressure continuity condition at z = h, we obtain

e i/1p 1x + ~ foo [A(() + B(() cos(Kph)] e-i(x d( 2n -00

00

= L Dm cosgm(x - d + g) exp(i7Jmph) . m=O

(9)

(10)

We substitute A(() and B(() into (9), multiply (9) by cosan(x + a) (n = 0,1,2,3, ... ), and perform integration from -a to a to get

• 00

ia,8pl(-I)I+1Fn(,8pla) = ;: L [a2~mpCmftnm(a) m=O

+l7JmpDmeirt""ph 12nm] + Cm€n 8nm (11)

where

(12)

(13)

v=o v = 1,2,3, ....

(14)

Note that ftnm(a) and 12nm are transformed into rapidly-convergent series

I () - { 0, n + m = odd (15) Inm a - Pm(a)8nm + qnm(a), n + m = even

j2niP( () I (=an

+ 2niQ (() 1(=9"" ' an :j:. gm

12nm = X(() + (16) 2ni f'(()s(() - f(()s'(() I

--2 -2 2(1') , an = gm €n a 9 s ., (=a n

where 8nm is the Kronecker delta, 0' denotes differentiation, and

( ) _ 2n€m cot(~mph) Pm a - at

a <"mp (17)

(18)


f(() = {h((), h((),

an = gm I:- 0 an = gm = 0

h(() = (2[(-1)m+nexp(i(ld-a+gl)

-( _1)m exp(i(ld + a + gl)

-( _1)n exp(i(ld - a - gl)

+ exp(i(ld + a - gl)]

h(() = [exp(i(ld - a + gl) - exp(i(ld + a + gl)

- exp(i(ld - a - gl) + exp(i(ld + a - gl)]

an = gm I:- 0 an = gm = 0

(19)

(20)

(21)

(22)

(23)

(24)

X(() = _ 27ri ~ (25) a2g2 h ~

h(() I (€v(-1)V[~;p - (V7r/h)2][1J~p - (v7r/h)2] (=Jk2_(V7r/h)2-b~ .

We similarly multiply (10) by cosgn(x-d+g) (n = 0,1,2,3, ... ) and perform integration to obtain

. (3 i{3p/ d D ((3 ) - ig ~ [ 2 to C [. -lg pie rn pig - 27r ~ a c.,mp m 2mn

+l1JmpDmei'f/,",ph hnm (g)] + ei'f/,",ph Dmcn8nm . (26)

In low-frequency approximation, the lowest wave (m = n = 0) becomes dominant; thus (11) and (26) with p = l = 0 lead to

n '¢4"/1 - '¢2"/2 1..10 ~ ....:.....:....:..=.....----'-::....:.:=:...

'¢1'¢4 - '¢2'¢3 (27)

D '¢1 "/2 - '¢3"/1 o ~ ....:.....:....:..=.....--.:,-::....:.:=:... '¢1'¢4 - '¢2'¢3

(28)

where

5.5 Acoustic Double Junction in a Rectangular Waveguide [11]

-ikh 00 Po ( /") 'l/J1 = 2i e + 2k9:.. ""' 0 -a..,v ei(v a

sin(kh) h f;:o cv(v

'l/J2 = eikh [- f~(O) + ~ ~ h((v) 1 2asin(kh) ah f;:o cv(-I)vQ

'l/J3 = - fHO) + ~ ~ h((v) 2gsin(kh) gh f;:o cv( -1)V(~

'l/J4 = eikh 2i e + 2k!L L Fo( -g(v) ei (v9 [ -ikh 00 1

sin(kh) h v=o cv(v

'/'1 = -iakFo(ka)

'/'2 = -igkeikdFo(kg)

(v = y'k2 - (V7r/h)2 .

The scattered fields for (x ~ -a) and (x 2: d + g) are given as

pI (x, y, Z)

= { ~ Kt cos(bpy) cos hu(z - h)eif3~uX,

L K;;- cos(bpY) cos hu(z - h)e-lf3pux, u

x2:d+g

x ~-a

139

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

o ~ u and u: integer. The transmission (71, 72, and 73) and reflection (g) coefficients are

71 = 11 + Ktl2 + L cu/3p ulK tl 2 (38) u=/-l cd3pl

2g ""' 2 73 = -h /3 L....J IDnl cn'T}np

Cl pi n

g = L cu/3puI K;;-12 u cl/3pl

where 0 ~ u < (h/Tr)Jk2 - b~, u: integer, 0 ~ m <

integer, 0 ~ n < (2g/,rr)Jk2 - b~, and n: integer.

(39)

(40)

(41)


5.6 Acoustic Hybrid Junction in a Rectangular Waveguide [15]

z

Port 3

w a

Port 1

'. a I rig. 0.0. ACOUStiC nYOflU JunctIOn

x

The electromagnetic hybrid junction in a rectangular waveguide, also known as a magic T, is a microwave passive component (duplexer) to decouple powers between the E-plane and H-plane arms [16]. In this section we will consider a similar hybrid junction structure for acoustic wave scattering and investigate its transmission and reflection coefficients. Next two subsections present the scattering analyses for an acoustic hybrid junction in a rectangular waveguide consisting of hard and soft surfaces, respectively.

5.6.1 Hard-Surface Hybrid Junction

Consider a hybrid junction in a hard-surface rectangular waveguide where the wavenumber is k. In region (I) (0 < x < a and 0 < z < b) the incident fields (velocity potentials), <Pt( x, y, z) and <P~ (x, y, z), propagate along the ±y directions. Then the total field in region (I) consists of the incident and scattered components

<pi(x, y,z) = COS(AIX)cos(BlZk,61Y (1)

<p;(x, y,z) = cos(A2x)cos(B2 z)e-i,62Y (2)

1 foo 00 [ <pI (x, y, z) = 2 L Am(() cos(h":mx) cos(bm z) 1f -00 m=O

+Bm(() cos(amx) COs(1Jmz)] e-i(Yd( (3)

5.6 Acoustic Hybrid Junction in a Rectangular Waveguide [15] 141

where A1 = a1T / a, B1 = j.t1T /b, /31 = Jk2 - Ai - Bi, A2 = e1T / a, B2 = V1T /b, /32 = Jk2_A~-B~ (a,j.t,e,v = 0,1,2, ... ), am = m1T/a, bm = m1T/b, K-m = Jk2 - <? - b;" and "1m = Jk2 - (2 - a;'. In region (II) (0 < x < a, -w/2 < y < w/2, and b < z) the incident and transmitted fields through port 3 are

4'~(x, y, z) = COS(A3X) cos W3(y + w/2)e-i,8a z (4) 00 00

4'II(x,y,z) = LLapqcos(apx)cosWq(y+w/2)ei~pqZ p=Oq=O

(5)

where A3 = </J1T/a, W3 = T1r/w, /33 = Jk2_A~-Wi (</J,T = 0,1,2, ... ),

Wq = Q1T/W, and epq = Jk2 - a~ - w~. In region (III) (a < x, - g/2 < y < g /2, and 0 < z < b) the incident and transmitted fields through port 4 are

4'~(x,y,z) =cosG4(y+g/2)cos(B4z)e-i,84x (6) 00 00

4'III (x, y, z) = L L bpq cosgp(Y + g/2) cos(bqz)ehpqx (7) p=Oq=O

where G4 = 'ljJ1T/g, B4 = X1T/b, 134 = Jk2 - G~ - B~ ('IjJ,x = 0,1,2, ... ),

gp = p1T/g, and "(pq = Jk2 - g~ - b~. The velocity continuity condition at z = b requires

8[4'1 (x, y, z) + 4'~(x, y, z) + 4'1 (x, y, Z)ll 8z z=b

= { 8[4'~(x, y, z) ~ 4'II (x, y, Z)llz=b' 0 < X < a, Iyl < w/2 (8)

0, otherwise.

Taking the Fourier transform of (8) yields 00

- L Bm(()"Im cos(amx) sin("Imb) m=O

00 00

+ 2: 2: apqepq ( cos(apx)E:(()ei~pqb (9) p=Oq=O

where

~w (_1)qei(w/2 - e- i(w/2

=q(()= (2_W~ (10)

We multiply (9) by cos(amx) and perform integration with respect to x from Otoatoget


Bm(() = ~(33 5 W (()e- i ,8sb TJ¢ sm(TJ¢b) r

-f am ~~mq 5w(()ei~mqb . q=O q TJm sm(TJm b) q

(11)

Similarly we apply the Fourier transform to the velocity continuity condition at x = a, multiply by cos(bmz), and perform integration from 0 to b to get

Am(() = ~(34 59 (()e- i ,84a IbX sm(lbxa) 'if;

-f b m ~'Ypm 59(()ei'Ypma . p=o p Ibm sm(lbma) p

(12)

Note that the pressure continuity condition at Z = b, 0 < x < a, and Iyl < W /2 requires

. . I . U 4il(x, y, b) + 4i;(x, y, b) + 4i (x, y, b) = 4iHx, y, b) + 4i (x, y, b) . (13)

Re-expressing (13) explicitly yields

COS(A1X)( _1)l'ei ,81Y + COS(A2X) ( -lte-i,82Y

1 roo 00

+ 27f L 2)Am(() COS(lbmX) ( _1)m 00 m=O

+Bm(() cos(amx) Cos(TJmb)]e-i(Yd(

= COS(A3 X) cos W3 (y + i) e-i,8sb

00 00

+ L L apq cos(apx) cosWq (y + i) ei~pqb . p=Oq=O

We substitute (11) and (12) into (14), and perform

(14)

i: loa (14) cos(asx) cos Wt (y + i) dxdy, (8 = 0,1, ... and t = 0,1, ... ), to 2

obtain

- i( -1)1'~(31Cct8cts5f((3d + i( -It ~(32c"8,,s5f( -(32)

a s: -i,8sb (. (3 [baw W s:) + "2 u<!>se 1cs 3 l<!>rt - 2 c<!>CrUrt

+ i(-I)X+s(34e-i,84a[~~'if;t 00 00

"" [ a s: i~ b (. (: [baw W s: ) = L..J L..J apq"2csupse pq l<"pq 1pqt + 2ctUqt p=Oq=O

+ b i(-I)q+s'Y ei'Ypqa 19w ] pq pq 2sqpt (15)

where 8ms is the Kronecker delta and

5.6 Acoustic Hybrid Junction in a Rectangular Waveguide [15] 143

{ 2, Cs = 1,

8=0 8 = 1,2,3, ....

(16)

The pressure continuity condition at x = a, Iyl < g/2, and 0 < z < b requires iiI i III P1 (a,y,z)+P2 (a,y,z)+p (a,y,z) =P4 (a,y,z)+P (a,y,z). (17)

We substitute (11) and (12) into (17), and perform

I t rb (17) cos(bhz) cosgn (y + ~) dzdy, (h = 0,1, ... and n = 0,1, ... ) to -t Jo

get

- i( _1)Cl! ~(31cI-'8I-'h5~({31) + i( -1)11 ~(32cv8vh5~( -(32)

+ i(-I)Hh{33e- i(33br;:/hTn

b ~ -i(34a (. (3 Iabg 9 ~) + "2uxhe lCh 4 lx'l/m - "2cxc,pU,pn

(18)

where

00 (2 cot (bVk2 - (2 - a2) Ifa~ = ~ 1 p 5 W (()5f(-() d(

pq 271" _ Jk2 1"2 2 q 00 -." - ap

(19)

19w _ ~ 100 (2 ;:;g(l") ;:;W(_I") dl" 2sqpt - 2 {32 _ 1"2 ~p ." ~t ." .". 71" -00 sq ."

(20)

In Subsect. 5.6.3 Appendix, we transform It;~ and Ig~Pt into rapidlyconvergent series.

The scattered fields for (y ~ -g/2) and (y ~ g/2) are obtained with a residue calculus as

pI (x, y ~ g/2, z) = L L Ktm cos(aux) cos(bmz)ei.BumY (21) u=Om=O

pI (x, Y ~ -g/2, z) = L L K;;m cos(aux) cos(bmz)e-i.BumY (22) u=Om=O

(23)


5.6.2 Soft-Surface Hybrid Junction

Consider acoustic scattering from a soft-surface hybrid junction. In region (I) the total field consists of the incident and scattered components

4>1 (x, y, Z) = sin(Alx) sin(B1z)ei ,81Y (24) 4>; (x, y, z) = sin(A2x) sin(B2z)e-i,82Y (25)

1 (Xl 00 [_

4>I(x,y,z) = 271" J-oo"I; Am«()sin(l\:mx)sin(bmz)

+Bm«() sin(amx) Sin(l1mZ)] e-it;;Yd( (26)

where (1, j.L, (!, v = 1,2,3, .... In regions (II) and (III) the incident and transmitted fields take the forms of

4>; (x, y, z) = sin(A3x) sin W3(y + w/2)e-i,8gZ (27) 00 00

4>II (x, y, z) = L L apq sin(apx) sin wq(y + w/2)eiepqz (28) p=lq=l

4>~(x, y, z) = sin G4(y + g/2) sin(B4z)e-i,84x (29) 00 00

4>III (x, y, z) = L L bpq singp(Y + g/2) sin(bqzk'YpqX (30) p=lq=l

where </J,T,'I/J, x = 1,2,3, .... From the pressure continuity conditions at z = b and x = a, we get

Bm«() = . W3 5 W «()e-i,8gb + fam . Wq 5 W «()eie",qb (31) sm(l1¢b) r q=l q sm(l1mb) q

Am«() = . G4 59 «()e- i ,84a + f b m . gp 5 9«()ei 'Ypm a • (32) sm(l\:xa) t/J p=l P sm(l\:ma) p

The velocity continuity conditions at z = b and x = a, respectively, give

Bl (-1)1' ~Wt8as5;V«(3d + B2( -1)'" ~Wt8e85;V( -(32)

a A -i,8gb (UT Ibaw . W f.I A ) +"2u¢se Yf3Wt 3¢rt + l"2,u3 Urt

+( I) X+s B G -i,84aI9w - as 4 4 Wte 4sXt/Jt 00 00

= L L [apq ~8pseiepqb ( -WqWtIg;~ + i ~~pq8qt ) p=lq=l

b ( l)q+s b i'YpqaI9w] - pq - as qgpwte 4sqpt

(33)

5.6 Acoustic Hybrid Junction in a Rectangular Waveguide [15) 145

Al (_1)0: ~9n8IthS~({3I) + A 2( -I)" ~9n8vhS~( -(32)

+( _1)<P+h A3bh9nW3e-i,6ab r:::hrn

b ~ -i,64a (G f ab9 .9{3 ~ ) +"2uxhe 49n 3X'lm + 1"2 4Utf;n 00 00

= LL [ - apq(-I)p+hapbh9nWqeiepqb1:;'~qn p=lq=1

+b b ~ i-ypqa ( fab9 +' 9 ~) ] pq"2Uqhe -9p9n 3qpn IfypqUpn

where q f; 0, t f; 0, h f; 0, n f; 0, and

(34)

1 100 Jk2 - (2 - a2

1;;~ = -2 ( J p ) s:' (()S;" ( -() d( (35) 7r -00 tan b k 2 - (2 - a~

19W - ~ 100 1 :::;9(1") :::;W(_I") dl" (36) 4sqpt - 2 {32 _ 1"2 ~p ." ~t ." .". 7r -00 sq ."

The rapidly-convergent series expressions for 1~~~ and 1f~pt are available in Subsect. 5.6.3 Appendix.

The scattered fields for (y :::; -9/2) and (y ~ 9/2) are

cpI(x,y ~ 9/2,z) = L L Ktm sin(aux) sin(bm z)ei ,6umY (37) u=1 m=1

u=1 m=1

_ ~ (b i')'pm a + -i(34a8 8 ) i9pauS$(1={3um) (39) ~ pme e ptf; mx a{3um(-I)u

o < u, 0 < m, and u, m: integers.

5.6.3 Appendix

Consider

00 (2 cot (bo /k2 _ (2 - a2 )

1fa~=~1 V p SW(()S;"(-()d( (40) pq 27r -00 Jk2 - (2 - a~ q

f9W _ ~ 100 (2 :::;9(1") :::;W(_I") dl" (41) 2sqpt - 2 {32 _ 1"2 ~p ." ~t ." .". 7r -00 sq ."


The integral Jf;;t contains simple poles at ( = ±wq and an infinite number of

poles corresponding to sin (bJk2 - (2 - a~) = O. The integral J~~Pt contains

simple poles at ( = ±/3sq and at ( = ±gp, ±Wt (double poles when gp = wt). U sing the residue calculus, we obtain the series representations as

Jbaw WCq cot(bJk2 - w~ - a~) J:

Ipqt = Uqt 2 ik2 - w2 - a2 V q p

• 00

-~ L {([( _1)q+t - (_I)qei(\w\ - (_I)tei(\w\ + I]} v=o

. {cv((2 - W~)((2 - wi)} -1 1(=v'k2-a~-b~ (42)

{ iP, «() k=,. + iQ, «() k=w" gp"l Wt

J~~Pt = X 1(() + ~f{(()s(() - h(()s'(() I (43)

cp S2(() (=gp' gp = Wt .

Similarly we get

wJk2 -w2 -a2 baw q p 8

J3pqt = ( qt 2w~ tan bVk2 - w~ - a~)

• 00

1 L( 2 2 2) -I) k - ( - ap v=1

. [(_I)q+t - (_I)Qe i(w - (_I)te i(w + 1]

. [(((2 _ W~)((2 - wi)r1 1(=v'k2-a;-b~ (44)

{iP,«()k=,. +iQ,«()k=w" gp"l Wt

Jt~Pt = X 2(() + . fH()s(() - h(()s'(() I (45)

1 S2(() (=gp' gp = Wt

where (-)' denotes differentiation and

-if·(() I j = 1,2 (46) X j (() = (/3s + ()((2 ~ g2)((2 _ w2) _ ' q p t (-f38q

/i(() Pj (() = (f3';q - (2)(( + gp)((2 - wF) (47)

/i(() Qj(() = (f3';q - (2)(( + Wt}((2 - g~) (48)

h(() =


gp = Wt # 0

gp = Wt = 0

(2 [( _l)p+t exp(i(lg - wl/2) - (-l)P exp(i(lg + wl/2)

-( _l)t exp(i(l- 9 - wl/2) + exp(i(1 - 9 + WI/2)],

gp = Wt # 0

exp(i(lg - wl/2) - exp(i(lg + wl/2)

- exp(i(1 - 9 - wl/2) + exp(i(1 - 9 + wl/2), gp = Wt = 0

(49)

(50)

h(() = [( _l)p+t exp(i(lg - wl/2) - (-l)P exp(i(lg + wl/2)

-( _l)t exp(i(1 - 9 - wl/2) + exp(i(1 - 9 + wl/2)] . (51)


1. N. Marcuvitz, Waveguide Handbook, Radiation Laboratory Series, vol. 10, New York: McGraw-Hill, 1951.

2. X. P. Liang, K. A. Zaki, and A. E. Atia, "A rigorous three plane mode-matching technique for characterizing waveguide T-junctions and its application in multiplexer design," IEEE Trans. Microwave Theory Tech., vol. 39, no. 12, pp. 2138-2147, Dec. 1991.

3. M. Koshiba and M. Suzuki, "Application of the boundary-element method to waveguide discontinuities," IEEE Trans. Microwave Theory Tech., vol. 34, no. 2, pp. 301-307, Feb. 1986.

4. K. H. Park and H. J. Eom, "An analytic series solution for H-plane waveguide T-junction," IEEE Microwave Guided Wave Lett., vol. 3, no. 4, pp. 104-106, April 1993.

5. K. H. Park, H. J. Eom, and Y. Yamaguchi "An analytic series solution for Eplane T-junction in parallel-plate waveguide," IEEE Trans. Microwave Theory Tech., vol. 42, no. 2, pp. 356-358, Feb. 1994.

6. H. J. Eom, K. H. Park, J. Y. Kwon and Y. Yamaguchi "Fourier transform analysis for E-plane T-junction in a rectangular waveguide," Microwave Opt. Technol. Lett., vol. 26, no. 1, pp. 34-37, July 2000.

7. F. Arndt, I. Ahrens, U. Papziner, U. Wiechmann, and R. Wilkeit, "Optimized Eplane T-junction series power dividers," IEEE Trans. Microwave Theory Tech., vol. 35, no. 11, pp. 1052-1059, Nov. 1987.

8. J. W. Lee and H. J. Eom, "TE-mode scattering from two junctions in H-plane waveguide," IEEE Trans. Microwave Theory Tech., vol. 42, no. 4, pp. 601-606, April 1994.

9. J. W. Lee, H. J. Eom and K. Uchida, "Right-angle H-plane waveguide double bend," IEICE Trans. Commun., vol. 77, no. 12, 1994.


10. A. Weisshaar, J. Lary, S. M. Goodnick, and V. K. Tripathi, "Analysis and modeling of quantum waveguide structures and devices," J. Appl. Phys., vol. 70, no. 1, pp. 355-366, July 1991.

11. J. Y. Kwon, H. H. Park, and H. J. Eom, "Acoustic scattering from two junctions in a rectangular waveguide," J. Acoust. Soc. Am., vol. 103, pp. 1209-1212, 1998.

12. J. W. Miles, "The diffraction of sound due to right-angled joints in rectangular tubes," J. Acoust. Soc. Am., vol. 19, no. 4, pp. 572-579, July 1947.

13. J. C. Bruggeman, "The propagation of low-frequency sound in a twodimensional duct system with T joints and right angle bends: Theory and experiment," J. Acoust. Soc. Am., vol. 82, no. 3, pp. 1045-1051, Sept. 1987.

14. C. Thompson, "Linear inviscid wave propagation in a waveguide having a single boundary discontinuity: Part II: Application," J. Acoust. Soc. Am., vol. 75, no. 2, pp. 356-362, Feb. 1984.

15. J. Y. Kwon and H. J. Eom, "Acoustic hybrid junction in a rectangular waveguide," J. Acoust. Soc. Am., vol. 107, no. 4, pp. 1868-1873, April 2000.

16. T. Sieverding and F. Arndt, "Modal analysis of the magic tee," IEEE Microwave Guided Wave Lett., vol. 3, no. 5, pp. 150-152, May 1993.

6. Rectangular Apertures in a Plane

6.1 Static Potential Through a Rectangular Aperture in a Plane

A low-frequency field penetration into an aperture is often modeled in terms of the electrostatic or magnetostatic potential that is governed by Laplace's equation. A static potential distribution through a thin rectangular aperture in a conducting plane was studied in [1,2] using approximate methods. In the next two subsections, we will consider electrostatic and magnetostatic potential distributions through a thick rectangular aperture and derive their polarizabilities.

6.1.1 Electrostatic Distribution [3]

z

x R~Clion (111\

Fig. 6.1. A rectangular aperture in a conducting plane


150 6. Rectangular Apertures in a Plane

In this section we will investigate an electrostatic potential distribution through a rectangular aperture in a thick conducting plane. In region (I) (z > 0) an incident potential pi(x, y, z) impinges on a rectangular aperture in a perfectly-conducting thick plane at zero potential. In region (I) the electrostatic potential has the incident (primary) and scattered (perturbed) potentials

pi(x, y, z) = z (1) 1 {'XJ ('XJ_

pS(x,y,z) = (27r)2 l-ool-oo pS((,ry)

. exp ( - i( x - iryy - J (2 + ry2 z) d( dry . (2)

The electrostatic potential in region (II) (-d < z < 0, Ixl < a, and Iyl < b) is

00 00

pd(x,y,z) = L L[cmnsinhkmn(z+d) + dmncoshkmn(z + d)] m=l n=l . sin am (x + a) sin bn (y + b) (3)

where am = m7r/(2a) , bn = n7r/(2b), and kmn = Ja;, + b~. The transmitted electrostatic potential in region (III) (z < -d) is

1 roo roo _ pt(x,y,z) = (27r)2 l-ool-oo pt((,ry)

·exp [-i(x - iryy + J(2 + ry2(Z + d)] d(dry. (4)

The boundary condition on the field continuity at z = ° requires

pi(x, y, 0) + pS (x, y, 0) = { PO,d(X, y, 0), Ixl < a, Iyl < b (5) otherwise.

Applying the Fourier transform to (5) yields 00 00

m=l n=l ·ambn(ab)2 Fm ((a)Fn (ryb) . (6)

The boundary condition for Ixl < a and Iyl < b

O[pi(x, y, z) + pS(x, y, z)]1 = opd(x, y, z) I oz z=o oz z=o

(7)

is rewritten as

00 00

= L L kmn[cmn cosh(kmnd) + dmn sinh(kmnd)] m=l n=l

(8)

6.1 Static Potential Through a Rectangular Aperture in a Plane 151

We multiply (8) by sin ap(x + a) sin bq(y + b) (p, q = 1,2,3, ... ) and integrate over the aperture to obtain

where

'Y - apbq (ab)3 ~ ~ (2'71-)2 L...J L...J

m=l n=l ambn[cmn sinh(kmnd) + dmn cosh(kmnd)]I

= kpq[cpq cosh(kpqd) + dpq sinh(kpqd)] (9)

'Y = 4[1- (-I)P][I- (-I)q] (10) pq7r2

1= i: i: )(2 + TJ2 Fm((a)Fp( -(a)Fn(TJb)Fq( -TJb) d(dTJ . (11)

Also the boundary conditions at z = -d yield

The electric polarizability is shown to be

X(z) == 2 ibb i: pd(x, y, z) dxdy

00 00

= 2ab L L'Ymn m=l n=l

'[Cmn sinh kmn(z + d) + dmn cosh kmn(z + d)] .

6.1.2 Magnetostatic Distribution

(12)

(13)

A problem of magnetostatic potential distribution through various apertures has been studied extensively [4-6]. An incident magnetostatic potential pi(x, y, z) impinges on a rectangular aperture (thickness: d and area: 2a by 2b) in a thick conducting plane. Regions (II) (-d < z < 0, Ixl < a, and Iyl < b) and (III) (z < -d) denote an aperture interior and a half-space, respectively. In region (I) the total magnetostatic potential is a sum of the incident and scattered potentials

pi (x, y, z) = x cos (}i + y sin (}i (14)

1 100 100 ~ p8(X, y, z) = (27r)2 -00 -00 p8((, TJ)

. exp (-i(X - iTJY - )(2 + TJ2 z) d(dTJ . (15)

In region (II) the magnetostatic potential is


00 00

pd(X,y,Z) = L L[cmnsinhkmn(z + d) + dmncoshkmn(z + d)] m=On=O . cos am(x + a) cos bn(y + b) (16)

where am = m7f/(2a), bn = n7f/(2b), and kmn = Ja'Tn + b~. In region (III) the total transmitted magnetostatic potential is

1 roo roo pt(x,y,z) = (27f)2 J- oo J- oo ;pt(('TJ)

·exp [-i(x - iTJY + J(2 + TJ2 (z + d)] d(dTJ. (17)

The boundary condition on the continuity of normal derivative of the potential at Z = ° is

:Z [pi (x, y, z) + pS (x, y, Z) L=o

= { :z [pd(x, y, z)L=o' Ixl < a, Iyl < b

0, otherwise.

(18)

Applying the Fourier transform to (18) yields 00 00

;PS(('TJ) = L Lkmn[cmncosh(kmnd)+dmnsinh(kmnd)] m=On=O

.(ab)2 (TJ Fm((a)Fn(TJb) . J(2 + TJ2

(19)

The boundary condition on the continuity of the potential across the aperture is

Ixl < a, Iyl < b . (20)

We multiply (20) by cosap(x + a)cosbq(y + b) (p,q = 0,1,2,3, ... ), and perform integration to get

(ab)4 00 00 •

ry - (27f)2 fo ~ kmn [cmn cosh(kmnd) + dmn smh(kmnd)] I

= f [cpq sinh(kpqd) + dpq cosh(kpqd)] (21)

where

j2b cos B. [( -l)P - 1] • 2 ' ap

ry= . [(-l)q -1]

2asmBi b2 ' q

(p # 0, q = 0)

(22)

(p = 0, q # 0)

6.2 Acoustic Scattering from a Rectangular Aperture in a Hard Plane [7]

{ 4ab,

E = ab, 2ab,

(p = 0, q = 0) (p # 0, q # 0) (p # 0, q = 0) or (p = 0, q # 0) .

Also the boundary conditions at Z = -d give

153

(23)

(24)

(25)

The longitudinal magnetic polarizability 7/Jx and the transverse magnetic polarizability 7/Jy are given by

7/Jx == - ibb i: x :z [pd(x, y, z)] dxdy

=2bf 1-(-1)m m=l am

. [cma cosham(z + d) + dma sinham(z + d)] (26)

7/Jy == - ibb i: y :z [pd(x, y, z)] dxdy

00 1- (_1)n =2aL b [cancoshbn(z+d)+dansinhbn(z+d)] (27)

n=l n

6.2 Acoustic Scattering from a Rectangular Aperture in a Hard Plane [7]

A study of acoustic wave scattering from a rectangular aperture in a thick hard plane was performed in [8] using an approximate technique based on the radiation impedance. In this section we will revisit the problem of acoustic wave scattering from a rectangular aperture in a thick hard plane. In region (I) (z > 0) a field (velocity potential) pi(x, y, z) is incident on a rectangular aperture with wavenumber k( = 21f /).. and )..: wavelength). In region (I) the total field has the incident, reflected, and scattered waves

pi(X, y, z) = exp(ikxx + ikyy - ikzz)

pT(X, y, z) = exp(ikxx + ikyy + ikzz)

1 roo roo -p8(X,y,Z) = (21f)2 1-00 1-00 p8(('1])

. exp( -i(x - i1]y + i~z) d(d1]

(1)

(2)

(3)


z

x Region (III)

Fig. 6.2. A rectangular aperture in a hard plane

where kx = k cos </>sin B, ky = k sin </> sin B, kz = k cos B, and K, = Jk2 - (2 _1]2. In region (II) (-d < z < 0, Ixl :s a, and Iyl :s b) the field is

00 00

m=On=O . cosam(x + a) cosbn(y + b) (4)

where am = m7r/(2a), bn = mr/(2b), and ~mn = Jk2 - a~ - b;'. In region (III) (z < -d) the transmitted field is

q/(x,y,z) = (2!)2 1:1: ;jt((,1])

. exp[-i(x - i1]y - iK,(z + d)] d(d1] . (5)

The velocity continuity condition at z = ° requires

8[<Jii (x, y, z) + <Jir(x, y, z) + <Jis (x, y, Z)]I 8z z=O

_ { 8<Jid (x, y, z) I ' - 8z z=O

0,

Ixl < a, Iyl < b (6) otherwise.


;jS((,1]) = -i(ab)2 (: f f m=On=O

~mn[cmn sin(~mnd) - dmn cos(~mnd)]Fm((a)Fn(1]b) . (7)

6.3 Electrostatic Potential Through Rectangular Apertures in a Plane [9] 155

The pressure continuity condition at z = 0 for Ixl < a and Iyl < b is

pi(x, y, 0) + pr(x, y, 0) + pS(x, y, 0) = pd(x, y, 0) . (8)

We substitute j;S«(, "7) into (8), mUltiply (8) by cos apex + a) cos bq(y + b) (p, q = 0,1,2,3, . .. ), and perform integration to obtain

i(ab)3 00 00 •

'Y + (27f )2 ~ ~ emn[cmn sm(emnd) - dmn cos(emnd)]I

= cpcq[cpq cos(epqd) + dpq sin(epqd)] (9)

where

'Y = -2abkxkyFp(kxa)Fq(kyb)

roo ro «("7)2 I = i-oo Loo -",-Fm «(a)Fp (-(a)Fn ("7b)Fq (-"7b) d(d"7.

The boundary conditions at z = -d similarly yield

i(ab)3 00 00

~ ~ ~ emndmnI = CpCqCpq .

The reflection coefficient (J and the transmission coefficient Tare

{

00 00

(J = 4~z Im ~~Cpcqe;q[lcpqI2sin*(epqd)cos(epqd)

+c;qdpql sin(epqdW - cpqd;ql cos(epqd) 12

-ldpq l2 sin(epqd) cos*(epqd)] }

6.3 Electrostatic Potential Through Rectangular Apertures in a Plane [9]

(10)

(11)

(12)

(13)

(14)

An electrostatic potential penetration into a single rectangular aperture in a conducting plane was considered in Sect. 6.1. In this section we will study a potential distribution through multiple rectangular apertures by extending the theoretical analysis given in Sect. 6.1. An incident electrostatic potential pi(X, y, z) impinges on rectangular apertures in a perfectly-conducting thick plane at zero potential. In region (I) (z > 0) the potential consists of the incident and scattered components as


z

Fig. 6.3. Multiple rectangular apertures in a conducting plane

pi(X,y,Z) = z (1)

1 100 100 -pS(x,y,z) = 47r2 - 00 - 00 pS((,TJ)

. exp (-i(X - iTJY - J (2 + TJ2 z) d(dTJ . (2)

In region (II) (-d < z < 0, Ix - lTal < a, Iy - uTbl < b: l = 0, . .. ,MI ,

U = 0, . .. ,Mu, and MI and Mu are integers) the total potential is 00 00

pd(x, y, z) = L L[c~n sinh kmn(z + d) + d~n cosh kmn(z + d)] m=l n=l

(3)

where am = m7r/(2a), bn = n7r/(2b), and kmn = Ja~ + b';. The transmitted potential in region (III) (z < -d) is

pt(x,y,z) = 4~2 i:i: ¥t((,TJ)

. exp [-i(X - iTJY + J (2 + TJ2(Z + d)] d(dTJ . (4)

From the boundary conditions on the field continuities at z = 0, we obtain

6.4 Magnetostatic Potential Through Rectangular Apertures in a Plane [10] 157

(ab)3 M, Mu 00 00

'Ypq - 4rr2 apbq L L L L ambn 1=0 u=O m=l n=l

. [c~n sinh(kmnd) + d~n cosh(kmnd)] I

= kpq [c;~ cosh(kpqd) + ~~ sinh(kpqd)] (5)

wherep= 1,2, ... , q = 1,2, ... , r = 0,1, ... ,MI, v = 0,1, ... ,Mu, and

4[1- (-I)P][I- (-I)q] 'Y - ~--~~~~~~~ pq - pqrr2

1= i: i: ../(2 + rpFm«(a)Fp(-(a)Fn(T/b)Fq(-T/b)

. exp[i((l- r)Ta + iT/(u - v)n] d(dT/ .

The boundary conditions at z = -d similarly yield

( b)3 M, Mu 00 00

:rr2 apbq L L L L ambnd~nI = kpqc;~ . 1=0 u=o m=l n=l

The total electric polarizability is

M, Mu /.UTb+b llTa+a X(z) == 2 L L pd(x, y, z) dxdy

1=0 u=o un-b 1Ta-a M, Mu 00 00

= 2ab LL L L 1=0 u=o m=l n=l

'Ymn [c~n sinh kmn (z + d) + d~n cosh kmn (z + d)]

6.4 Magnetostatic Potential Through Rectangular Apertures in a Plane [10]

(6)

(7)

(8)

(9)

The magnetic polarizability is a useful concept for approximately estimating the shielding effectiveness of an enclosure with apertures in low-frequency regime. In this section we will derive the magnetic polarizability of multiple rectangular apertures in a thick conducting plane. A magnetostatic potential analysis in this section is similar to the electrostatic case considered in Sect. 6.3. Consider an incident magnetostatic potential pi(x, y, z) impinging on multiple rectangular apertures in a thick conducting plane. In region (I) (z > 0) the total magnetostatic potential is a sum of the incident and scattered potentials. The incident potential is

pi(x,y,z) = x cos (Ji + ysin(Ji - AI,u (1)

for Ix - ITal < a, Iy - unl < b, I = 0, ... ,MI, and u = 0, ... , Mu. In the multiply-connected structure as considered in this section, the magnetostatic


Reg~on(lII)

potential is a multi-valued function. In order to make the magnetostatic potential a single-valued function, a constant Al,u needs to be determined later in matching the boundary conditions. The scattered potential is

1 100 100 -p8(X,y,Z) = 471"2 -00 -00 p8((,1])

. exp ( -i(x - i1]y - J (2 + 1]2 Z) d(d1] . (2)

In each aperture of region (II) the potential take the form of 00 00

pd(x,y,z) = L L [c!;:nsinhkmn(z+d)+d!;:ncoshkmn(z+d)] m=On=O

(3)

where am = m7l"/(2a), bn = n7l"/(2b) , and kmn = Ja;' + b;. Note that Ta and n denote the periods of apertures in the x- and y-directions, respectively. In region (III) (z < -d) the transmitted potential is

1 100 100 -pt(x,y,z) = 471"2 -00 -00 pt((,1])

. exp [-i(x - i1]y + J(2 + 1]2(Z + d)] d(d1] . (4)

The boundary conditions at Z = 0 yield

6.5 EM Scattering from Rectangular Apertures in a Conducting Plane [11] 159

(ab)4 MI M" 00 00

'Ypq - 41f2 LL L L 1=0 u=O m=O n=O

kmn [c~n cosh(kmnd) + d~n sinh(kmnd)] I

= E [c;~ sinh(kpqd) + d;~ cosh(kpqd)] ab (5)

where p = 0,1,2, ... , q = 0,1,2, ... , r = 0,1, ... ,MI , V = 0,1, ... ,Mu , and

'Ypq = {

4ab(rTa cos (h + VTb sinBi - AI,U), (p = 0, q = 0)

2bcosBd(-1)P-1l/a~, (P:lO, q=O) 2asin Bd(-1)q -1l/b~, (p = 0, q:l 0)

0, (p :I 0, q:l 0)

(6)

(7)

Note that E is 4 for (p = 0, q = 0), 1 for (p:l 0, q :I 0), and 2 for others. Similarly the boundary conditions at z = -d give

(8)

We solve (5) and (8) for the modal coefficients c~n and d~n when (p:l 0) or (q :I 0), and then subsequently determine the constant terms d~~ and AI,u.

The total longitudinal ('lj;x) and transverse ( 'lj;y) magnetic poiarizabilities are

(9)

Ml M" 00 1- (_1)n 1/Jy =2aLLL bn

1=0 u=On=l

. [c~':, cosh bn (z + d) + d~':, sinh bn (z + d) ] (10)


Z Region (I)

E t, H t EI, H t Fig. 6.5. Multiple rectangular apertures in a conducting plane

6.5 EM Scattering from Rectangular Apertures in a Conducting Plane [11]

A study of electromagnetic penetration into an aperture in a conducting plane is important in EMI/EMC-related problems [12-17]. In this section we will analyze electromagnetic wave scattering from multiple rectangular apertures in a perfectly-conducting thick plane. Consider multiple rectangular apertures with area 2a x 2b and thickness d where Ta and Tb are the periods in the x- and y-directions, respectively. Assume that an electromagnetic wave is obliquely incident on multiple rectangular apertures. The incident and reflected fields in region (I) take the forms of

Ei = Zo{(usin¢+vcos¢cosO)x - (ucos¢ - v sin ¢cos O)fj

+vsinOi}exp(ikxx+ikyy-ikzz) (1)

ET = - Zo{( u sin ¢ + v cos ¢ cos O)x - (u cos ¢ - v sin ¢ cos O)fj

-v sin Oi} exp(ikxx + ikyy + ikzz) (2)

where X, fj, and i are the unit vectors, kx = kocos¢sinO, ky = kosin¢sinO, kz = ko cosO, ko = WJl-tOfO = 2n/).. is the wavenumber, and Zo = Vl-to/fo is the free-space intrinsic impedance. A polarization state of the incident field is determined by the choice of u and v. It is convenient to represent the fields in terms of the electric and magnetic vector potentials. The relations between the electric and magnetic vector potentials and the fields are given in

6.5 EM Scattering from Rectangular Apertures in a Conducting Plane [11) 161

Appendix A.l. The scattered ES_ and jfs-fields in region (I) are obtained from the z-components of the scattered electric and magnetic vector potentials, F:(x,y,z) and A~(x,y,z)

1 100 100 -F:(x,y,z) = 47r2 -00 -00 F:((,'T/)

· exp( -i(x - i'T/y + iK.z)d(d'T/ (3)

1 100 100 -A~(x,y,z)= 47r2 -00 -00 A~((,'T/)

· exp( -i(x - i1]y + iK.z)d(d1] (4)

where K. = .jk~ - (2 - 1]2. In region (II) (lth and kth aperture in the x and y directions) the vector potentials F!(x, y, z) and A~(x, y, z) within the rectangular aperture are

00 00

Ff(x,y,z) = L L [c~ncosemn(z+d) +d~nsinemn(z+d)] m=On=O · cos am(x -lTa + a) cos bn(y - kTb + b) (5)

00 00

A~(x, y, z) = L L [~n cosemn(z + d) + ~n sinemn(z + d)] m=ln=l · sinam(x -lTa + a) sin bn(y - kTb + b) (6)

where (m,n):f. (0,0), emn = .jkr - a~ - b;, kl = WJ/-LIEl, am = m7rj(2a), and bn = n7r j(2b). In region (III) the transmitted vector potentials F;(x, y, z) and A;(x, y, z) are

F;(x, y, z) = 4~2 i: i: F;((, 1])

·exp[-i(x - i1]y - iK.(z + d)]d(d1] (7)

A;(x,y,z) = 4~2 i: i: A;((,1])

· exp[-i(x - i1]y - iK.(z + d)]d(d1] . (8)

The Ex,y(x, y, 0) field continuity is given by

E!,y(x, y, 0) + E;,y(x, y, 0) + E;,y(x, y, 0)

_ { E~,y(x, y, 0), Ix - lTal < a and Iy - knl < b - 0, otherwise. (9)

Applying the Fourier transform to (9) and solving for F:((,1]) and A~((,1]), we obtain

( )2 Ml Mk 00 00 (2 2) F:((,1])=- a~ EOLLLL(1]~m++;n

1 1=0 k=O m=O n=O ( 1]

. [C~n cos(emnd) + d~n sin(emnd)] ·Fm ((a)Fn (1]b) exp(i(lTa + i1]kn) (10)


( b)2 M, Mk 00 00 (2 2 b2 (2) 1~((,1]) = a :JLO€O t;~l;~ a~~ +-1]2)K

. [C~n cos(emnd) + d~n sin(emnd)]

·Fm ((a)Fn (1]b) exp(i(lTa + i1]kn)

.( b)2 M, Mk 00 00 b (: +1 a JLo€o LL L L am n<"mn

JLI €1 1=0 k=O m=1 n=l K

. [~n sin(emnd) - ~n cos(emnd)]

·Fm ((a)Fn (1]b) exp(i(lTa + i1]kn) . (11)

The boundary condition on the Hx,y(x, y, 0) field continuity requires

H;,y(x, y, 0) + H;,y(x, y, 0) + H:,y(x, y, 0)

= H;,y(x, y, 0), Ix -lTal < a and Iy - knl < b . (12)

We substitute 1:((,1]) and ff':((,1]) into (12), multiply (12) by cosap(xrTa + a) sin bq(y - WTb + b) or sinap(x - rTa + a) cosbq(y - WTb + b), and perform integration over the area of aperture to obtain

(b)3b M, Mk 00 00

1'+ a47r2 q LL L L 1=0 k=O m=O n=O

[c~n cos(emnd) + d~n sin(emnd)]

. [W€O b~I2 _ (a~ + b~,) It] €l WJLO€1

_ i(ab)3€ob q ~ ~ ~ ~ 47r2JLI€1 L...., L...., L...., L....,

1=0 k=O m=l n=l

[c~n sin(emnd) - ~n cos(emnd)] ambnemnI2

= icpbq (c;~ sine epqd) - tt;~ cos( epqd)] epq WJLI€1

- cpap [C;~ cos(epqd) + a;~ sin (epq d)] , JLI

P = 0,1, . .. and q = 1,2, ... (13)

6.5 EM Scattering from Rectangular Apertures in a Conducting Plane [11] 163

where

+ i(ab)3 fOap ~ ~ ~ ~ 41[2J.tlfl L...J L...J L...J L...J

1=0 k=O m=l n=l

[C!:;n sin(~mnd) - ~n Cos(~mnd)] ambn~mnIa

= icqap [c;~ sin(~pqd) - rr;,~ cos(~pqd)] ~pq WJ.tlfl

+ cqbq [c;~ cos(~pqd) + a;~ sin(~pqd)] , J.tl

P = 1,2, . .. and q = 0, 1, ...

'Y = 2iab(u sin ¢ cos 0 + v cos ¢)kxbq

·Fp(kxa)Fq(kyb) exp(ikxrTa + ikywTb)

'Y = 2iab(ucos¢cosO - vsin¢)kyap

·Fp(kxa)Fq(kyb) exp(ikxrTa + ikywTb)

II = i: i: ((~)2 L((,T})

· exp[i((l- r)Ta + iT}(k - w)Tbl d(dT}

12 = i: i: ~ L((, T})

· exp[i((l- r)Ta + iT}(k - w)nl d(dT}

['Xl (CO 2

13 = l-col-co : L(("T})

· exp[i(,(l - r)Ta + iT}(k - w)nl d(dT}

(14)

(15)

(16)

(17)

(18)

(19)

with L((" T}) = Fm((,a)Fp( -(a)Fn(T}b)Fq( -T}b). In high-frequency limit for a single aperture case, 11 ,12, and 13 have dominant contributions from the poles of L((, T}) at m = p and n = q when l = rand k = w, leading

41[2 1 41[2 1 41[2 1 to h -t (ab)3 ~pq CpCq, 12 -t (ab)3 b~~pq cp, and 13 -t (ab)3 a~~pq Cq. The

boundary conditions on the Ex,y and Hx,y continuities at z = -d similarly yield


(20)

( b) 3 Ml Mk 00 00 [ (2 b2 )] a a p"""" Ik WEo 2 I _ am + n I 4 2 L..J L..J L..J L..J Cmn am 3 I

7r 1=0 k=O m=O n=O EI W/-LOEI

i(ab)3 Eoap ~ ~ ~ ~ (}k bel 4 2 L..J L..J L..J L..J mn am n .. mn 3

7r /-LIEI 1=0 k=O m=l n=l

_ i€qa p .J1"W C _ €qb q -rw - u_ .. pq C,

W/-LIEI pq /-LI pq

P = 1,2, . .. and q = 0,1, .... (21)

When region (III) is filled with a perfect conducting medium (PEe), the field representations for F; (x, y, z) and A! (x, y, z) are unnecessary. This implies that we need to match the boundary conditions at z = 0 only, thus leading to (13) and (14) with clk = (j}k = 0 mn mn .

The transmission coefficient T of an aperture, a ratio of the total transmitted power to that incident on an aperture, is shown to be

6.6 EM Scattering from Rectangular Apertures in a Rectangular Cavity [18]

(22)

A subject of electromagnetic wave penetration into a cavity with apertures is a canonical problem in electromagnetic compatibility and interference. Various approaches [19-23] have been used to estimate the shielding effectiveness of a rectangular enclosure with apertures. The present section provides an analysis for electromagnetic wave penetration into a three-dimensional cavity with multiple rectangular apertures in a conducting plane. An electromagnetic wave is normally incident on the cavity-backed multiple rectangular

6.6 EM Scattering from Rectangular Apertures in a Rectangular Cavity [18] 165

y

········Ta···················· ····················2"8······1

Db··· "'OD4b

00··· "'OOIib 00'" : x

00···0···00

00"'0"'00 .................................................... ............ I" -I

20:

(a ) Front view

y

Re

h

Ey

HxL-. z c £ZI

ICI Region (III)

Region (I) ~ (b) Side view

Fig. 6.6. Multiple rectangular apertures in a rectangular cavity

apertures with periods Ta and Tb in the x- and y-directions, respectively. The total field in region (I) (z > 0) consists of the incident, reflected, and scattered waves


(1)

(2)

where the wavenumber is ko = WJP,OEO = 2rr/).. and Zo = y'P,O/EO is the free-space intrinsic impedance. The scattered ES-and HS-fields in region (I) are given in terms of the z-components of the electric and magnetic vector potentials, F;(x,y,z) and A~(x,y,z)

1 100 100 -F:(x,y,z) = 4rr2 -00 -00 F:((,'f/)

· exp( -i(x - i'f/y + il\;z)d(d'f/ (3)

1 100 100 -A:(x, y, z) = 4rr2 -00 -00 A:((, 'f/)

· exp( -i(x - i'f/y + il\;z)d(d'f/ (4)

where I\; = y'k3 - (2 - 'f/2. In region (II) (lth and kth aperture in the x and y directions) the vector potentials Ff(x, y, z) and A~(x, y, z) are

00 00

Ff(x,y,z) = L L [c!:;ncoS~mn(z+d) +d!:;nsin~mn(z+d)] m=On=O

(5) 00 00

A~(x,y,z) = L L [c!!ncoS~mn(z+d) +d!!nsin~mn(z+d)] m=l n=l · sin am (x -ITa + a) sinbn(y - kTb + b) (6)

where (m, n) :I (0,0), am = mrr/(2a), bn = nrr/(2b), and ~mn = y'k3 - a;' - b;. In region (III) of a cavity interior (Ixl < a, Iyl < (3, and -d - h < z < -d) the corresponding vector potentials F;(x, y, z) and A;(x, y, z) are

00 00

F~(x,y,z) = LLegjsin')'gj(z+d+h) g=Oj=O · cos a g (x + a) cos (3j (y + (3) (7) 00 00

A~(x,y,z) = LLegjcos')'gj(z+d+h) g=lj=l · sinag(x + a) sin (3j (y + (3) (8)

where (g,j) :I (0,0), ag = grr /(2a), (3j = jrr /(2(3), and ')'gj = Jk3 - a~ - (3J.

The boundary conditions on the Ex,y and Hx,y continuities at z = 0 require respectively

E!,y(x, y, 0) + E;,y(x, y, 0) + E;,y(x, y, 0)

= { E;,y(x, y, 0), Ix -ITal < a and Iy - kTbl < b 0, otherwise

(9)

6.6 EM Scattering from Rectangular Apertures in a Rectangular Cavity (18) 167

and

H;,y(x, Y, 0) + H;,y(x, Y, 0) + H;,y(x, Y, 0) = H;,y(x, Y, 0), (10)

Ix -lTal < a and Iy - kTbl < b.

Applying the Fourier transform to (9) and solving for F:((,,,,) and A~((,,,,), we get

M, Mk 00 00

F:((,,,,) = -(ab)2LL L L 1=0 k=O m=O n=O

(",(a~ + b;) [ Ik (C d) d1k . (C d)] · (2 2 Cmn cos <"mn + mn SIn <"mn

+",

·Fm ((a)Fn (",b) exp(i(lTa + i",kn) (11) M, Mk 00 00

A=((,,,,) = (ab) 2w/toLL L L 1=0 k=O m=O n=O

(a~",2 - b;(2) [ Ik (C d) d1k . (C d)] • ((2 + ",2)K, cmn COS <"mn + mn sm <"mn

·Fm ((a)Fn (",b) exp(i(lTa + i",kTb )

M, Mk 00 00 b C

+i(ab)2 L L L L am n<"mn 1=0 k=O m=l n=I K,

· [~nsin(~mnd) - ~ncos(~mnd)] ·Fm((a)Fn(",b) exp(i(lTa + i",kTb) . (12)

We substitute (11) and (12) into (10), multiply (10) by cosap(x - rTa + a) sinbq(y - WTb + b) or sinap(x - rTa + a) cosbq(y - wn + b), and perform integration over the area of apertures to get

(b)3b M, Mk 00 00

a41f2 qLLLL 1=0 k=O m=O n=O

. [c~n cos(~mnd) + d~n sin(~mnd)] [Wb~J2 - (a~ + b;) It] w/to€o

.( b)3b M, Mk 00 00

I4:2JLoq ~?;~]; . [~n sin(~mnd) - ~n Cos(~mnd)] ambn~mnI2

= icpbq [c;~ sin(~pqd) - d;~ cos(~pqd)] ~pq wjLo€o

- cpap [C;~ cos(~pqd) + d:';~ sin(~pqd)] , /to

p = 0,1, ... and q = 1,2, ... (13)


(a!~32ap ff f f 1=0 k=O m=O n=O

. [C~n cos(~mnd) + d~n sin(~mnd)J [wa;'J3 - (a~ + b;,) h] W/-Lofo

+i~a:t:p ff f f: /-L 1=0 k=O m=1 n=1

. [C~n sin(~mnd) - ~n Cos(~mnd)J ambn~mnh - s;~

= icqap [c;~ sin(~pqd) - d;~ cos(~pqd)J ~pq W/-Lofo

+ cqbq [c;~ cos(~pqd) + d;~ sin(~pqd)J ' /-Lo

p = 1,2, ... and q = 0, 1, ...

where 8mp is the Kronecker delta, co = 2, C1 = C2 = ... = 1, and

rw _ {8[1 - (-I)P]/(p1T), Spq - 0,

q=o otherwise.

(14)

(15)

Note that h, hand h are given by (17), (18), and (19) in Sect. 6.5. The boundary conditions on the Ex,y and Hx,y continuities at z = -d for Ixl < CY and Iyl < f3 require

E~,y(x, y, -d)

= {E~,y(x,y, -d), 0, otherwise

(16)

and

H~,y(x, y, -d) = H:,y(x, y, -d), Ix -iTal < a, Iy - kTbl < b. (17)

Performing !~ i: {Ex continuity in (16)} cos CYg' (x + CY) sin f3j l (y + (3)dxdy

and !~ i: {Ey continuity in (16)} sin CYg' (x + CY) cos f3j' (y + (3)dxdy, and

solving for egj and egj, we get

1 ~ ~ ~ ~ (bnf3jY:/;n + amCYgY:/;n) Ik egj = CYf3 L...J L...J L...J L...J sin('" ·h)(c .cy2 + c (./2) Cmn

1=0 k=O m=O n=O Ig) ) 9 gl-') . M/ Mk 00 00

+ W/-L~CYf3 t; t; 1=1 ~ (amf3jY:/;n - bnCYgY:/;n) ~ ;P (18) sin("(gjh)(cjCY~ + cgf3J) mn mn'

g = 0,1,2, ... , j = 0,1,2, ... , (g,j):j:. (0,0)

6.6 EM Scattering from Rectangular Apertures in a Rectangular Cavity [18] 169

• Ml Mk 00 00 (b ylk (3 y-Ik ) _ . = lWp'O "'"'"'"' "'"'"'"' nag mn - am j mn Ik egJ {3 ~~ ~ ~ .' ( ·h)( 2 (32) Cmn

a 1=0 k=O m=O n=O "(gJ sm "(gJ ag + j

Ml Mk 00 00 ( ylk b (3 ylk ) _ ~ "'"'"'"' "'"'"'"' amag mn + n j mn c (jk (19)

{3 ~~ ~ ~ .' ( 'h)( 2 (32) <"mn mn' a 1=0 k=O m=l n=l "(gJ sm "(gJ ag + j

where

ylk = mn

9 = 1,2,3, ... , j = 1,2,3, ...

({3;~~~) {sin[{3j(b - (3 - kTb )]

+( _1)n sin[{3j(b + {3 + kn)]} , g=m=O

(a~ - a~)~~J _ b~) {sin[ag(a - a -iTa)]

+( _1)m sin[ag(a + a + iTa)]} . {sin[{3j(b - (3 - kTb)] + (_I)n sin[{3j(b + {3 + kn)]},

otherwise

( ;bam2 ) {sin[ag(a - a -iTa)] a g -am

+( _1)m sin[ag(a + a + iTa)]), j=n=O

(a~ - a~)~;J _ b~) { sin[ag(a - a - iTa)]

+( _1)m sin[ag(a + a + iTa)]} . {sin[{3j(b - (3 - kTb)] + (_I)n sin[{3j(b + {3 + kn)]},

otherwise.

(20)

(21)

We substitute (18) and (19) into (17), multiply by cos ap(X-rTa +a) sin bq(ywn + b) or sinap(x - rTa + a) cosbq(y - wn + b), and perform integration over the area of apertures to get

(22)


(23)

where

00 00 (b (3 ylk ylk ) Q __ 1_ '"' '"' (3., .yrw n j mn + amag mn

- Wf..LoEo 66 J gJ pq tan(, ·h)(a2 + c; (32) g=o J=1 gJ 9 9 J

00 00 (b ylk (3 ylk ) + '"' '"' yrw nag mn - am j mn W 6 6 ag pq , . tan(, ·h)(a2 + (32)

g=1 j=1 gJ gJ 9 J

(24)

00 00 ( (3 ylk b ylk) Q __ 1_ '"' '"' (3., .yrw am j mn - nag mn - Wf..LoEo 66 J gJ pq tan(, ·h)(a2 + C; (32)

g=o J=1 gJ 9 9 J

00 00 ( ylk b (3 ylk ) +w'"' '"' a yrw amag mn + n j mn 66 9 pq , . tanh ·h)(a2 + (32) g=1 j=1 gJ gJ 9 J

(25)

1 00 00 (b (3 ylk + y-lk ) L L - n j mn amag mn p - __ a , . yrw -'-----''---'c:::.:,::.,.--;---:::-''----~:..c. - Wf..LoEo . 9 gJ pq tan(, ·h)(c; ·a2 + (32)

g=1 J=O gJ J 9 J

00 00 (b ylk (3 y-lk ) -W '"' '"' (3. yrw nag mn - am j mn

66 J pq , . tan(, ·h)(a2 + (32) g=1 j=1 gJ gJ 9 J

(26)

00 00 ( (3 ylk b ylk) P __ 1_ '"' '"' a .yrw am j mn - nag mn - Wf..LoEo 66 g,gJ pq tanh ·h)(c; ·a2 + (32)

g=1 J=o gJ J 9 J

00 00 ( ylk b (3 ylk ) _ LL(3.yrw amag mn + n j mn

W g=1 j=1 J pq ,gj tanhgjh)(a~ + (3]) (27)


1. N. A. McDonald, "Polynomial approximations for the electric polarizabilities of some small apertures," IEEE Trans. Microwave Theory Tech., vol. 33, no. 11, pp. 1146-1149, November 1985.

2. S. B. Cohn, "The electric polarizability of aperture of arbitrary shape," Proc. I.R.E., vol. 40, pp. 1069-1071, 1952.

3. H. H. Park and H. J. Eom, "Electrostatic potential distribution through a rectangular aperture in a thick conducting plane," IEEE Trans. Microwave Theory Tech., vol. 44, no. 10, pp. 1745-1747, Oct. 1996.

4. S. B. Cohn, "Determination of aperture parameters by electrolytic-tank measurements," Proc. I.R.E., vol. 39, pp. 1416-1421, 1951.


5. R. L. Gluckstern and R. K. Cooper, "Electric polarizability and magnetic susceptibility of small holes in a thin screen," IEEE Trans. Microwave Theory Tech., vol. 38, no. 2, pp. 186-192, Feb. 1990.

6. L. K. Warne and K. C. Chen, "Relation between equivalent antenna radius and transverse line dipole moments of a narrow slot aperture having depth," IEEE Trans. Electromagn. Compat., vol. 30, no. 3, pp. 364-370, Aug. 1988.

7. H. H. Park and H. J. Eom, "Acoustic scattering from a rectangular aperture in a thick hard screen," J. Acoust. Soc. Am., vol. 101, no. 1, pp. 595-598, Jan. 1997.

8. A. Sauter, Jr. and W. W. Soroka, "Sound transmission through rectangular slots of finite depth between reverberant rooms," J. Acoust. Soc. Am., vol. 47, pp. 5-11, 1970.

9. H. H. Park and H. J. Eom, "Electrostatic potential distribution through thick multiple rectangular apertures," Electron. Lett., vol. 34, no. 15, pp. 1500-1501, 23rd, July 1998.

10. J. G. Lee and H. J. Eom, "Magnetic polarisability ofthick multiple rectangular apertures," Electron. Lett., vol. 35, no. 21, pp. 1850-1851, Oct. 1999.

11. H. H. Park and H. J. Eom, "Electromagnetic scattering from multiple rectangular apertures in a thick conducting screen," IEEE Trans. Antennas Propagat., vol. 47, no. 6, pp. 1056-1060, June 1999.

12. A. EI-Hajj and K. Kabalan, "Characteristic modes of a rectangular aperture in a perfectly conducting plane," IEEE Trans. Antennas Propagat., vol. 42, no. 10, pp. 1447-1450, Oct. 1994.

13. K. Barkeshli and J. L. Volakis, "Electromagnetic scattering from an aperture formed by a rectangular cavity recessed in a ground plane," J. Electromagn. Waves Appl., vol. 5, no. 7, pp. 715-735, 1991.

14. J. M. Jin and J. L. Volakis, "Electromagnetic scattering by the transmission through a three-dimensional slot in a thick conducting plane," IEEE Trans. Antennas Propagat., vol. 39, no. 4, pp. 1544-1550, Apr. 1991.

15. C. C. Chen, "Transmission of microwave through perforated flat plates of finite thickness," IEEE Trans. Microwave Theory Tech., vol. 21, no. 1, pp. 1-6, Jan. 1973.

16. T. K. Sarkar, M. F. Costa, C-L. I, and R. F. Harrington, "Electromagnetic transmission through mesh covered apertures and arrays of apertures in a conducting screen," IEEE Trans. Antennas Propagat., vol. 32, no. 9, pp. 908-913, Sept. 1984.

17. T. Andersson, "Moment-method calculations on apertures using basis singular functions," IEEE Trans. Antennas Propagat., vol. 41, no. 12, pp. 1709-1716, Dec. 1993.

18. H. H. Park and H. J. Eom, "Electromagnetic penetration into a rectangular cavity with multiple rectangular apertures in a conducting plane," IEEE Trans. Electromagn. Compat., vol. 42, no. 3, pp. 303-307, Aug. 2000.

19. T. Wang, R. F. Harrington, and J. R. Mautz, "Electromagnetic scattering from and transmission through arbitrary apertures in conducting bodies," IEEE Trans. Antennas Propagat., vol. 38, no. 11, pp. 1805-1814, Nov. 1990.

20. K. P. Ma, M. Li, J. L. Drewniak, T. H. Hu~ing, and T. P. Van Doren, "Comparison of FDTD algorithms for subcellular modeling of slots in shielding enclosures," IEEE Trans. Electromagn. Compat., vol. 39, no. 2, pp. 147-155, May 1997.

21. H. Kogure, H. Nakano, K. Koshiji, and E. Shu, "Analysis of electromagnetic field inside equipment housing with an aperture," IEICE Trans. Commun., vol. E80-B, no. 11, pp. 1620-1623, Nov. 1997.


22. B. W. Kim, Y. C. Chung, and T. K. Kang, "Analysis of electromagnetic penetration through apertures of shielded enclosure using finite element method," 14th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, vol. II, pp. 795-798, Mar. 1998.

23. M. P. Robinson, T. M. Benson, C. Christopoulos, J. F. Dawson, M. D. Ganley, A. C. Marvin, S. J. Porter, and D. W. P.Thomas, "Analytical formulation for the shielding effectiveness of enclosures with apertures," IEEE 1)oCLns. Electromagn. Compat., vol. 40, no. 3, pp. 240-248, Aug. 1998.

7. Circular Apertures in a Plane

7.1 Static Potential Through a Circular Aperture in a Plane

z I <l>S Region (I)

Region (III)

Fig. 1.1. A circular aperture in a conducting plane

7.1.1 Electrostatic Distribution [1]

A potential penetration into a circular aperture in a thick conducting plane has been studied in [2-3] due to its application in low-frequency field leakage problems. In this section we will study an electrostatic potential penetration into a circular aperture in a thick perfectly-conducting plane. In region (I) (z > 0) an incident potential pi(r, z ) impinges on a circular aperture (radius:


174 7. Circular Apertures in a Plane

a and depth: d) in a thick perfectly-conducting plane at zero potential. Regions (II) (-d < z < 0 and r < a) and (III) (z < -d) represent a circular aperture and a half-space, respectively. The total electrostatic potential in region (I) has the incident and scattered potential components as

4.>i(r, z) = z (1)

4.>8(r, z) = 100 ~8(()Jo((r)e-'Z(d( . (2)

The total electrostatic potential in region (II) is 00

n=l where kn is determined by the characteristic equation Jo(kna) = O. The total transmitted electrostatic potential in region (III) is

4.>t(r, z) = 100 ~t(()Jo((r)e'(Z+d)(d( . (4)

The boundary condition on the field continuity at z = 0 requires

4.>i(r, 0) + 4.>8(r, 0) = {04.>,d(r, 0), r < a (5) r > a.

Applying the Hankel transform to (5) yields

~8(() = ~(bnSinhknd+cnCOShknd) [-akn72(~~1(kna)] (6)

The boundary condition at z = 0 is given as

:z [4.>i(r, z) + 4.>8(r, z)t=o = :z [4.>d(r, z)L=o' r < a . (7)

Substituting ~8(() into (7), multiplying (7) by Jo(kpr)r, and integrating with respect to r from 0 to a, we get

00

: - L(bnsinhknd+cncoshknd)a2knkpJl(kna)I p n=l

a2 = 2"kpJ1 (kpa)(bp cosh kpd + cp sinh kpd) (8)

where

[00 J6((a)(2 I = Jo ((2 _ k~)((2 _ k~) d( . (9)

The additional boundary conditions at z = -d give

00 1 L cnknJ1(kna)I = 2bpJl(kpa) . n=l

(10)

7.1 Static Potential Through a Circular Aperture in a Plane 175

The electric polarizability is shown to be

X(z) == 27r loa pd(r, z)rdr

00

= 27ra L [bn sinh kn(z + d) + en cosh kn(z + d)] J1 (kna)jkn . (11) n=l

7.1.2 Magnetostatic Distribution [4]

A behavior of magnetostatic potential penetration into a circular aperture was studied in [2,3,5]. Consider an incident magnetostatic potential pi(x, y, z) impinging on a circular aperture (radius: a and depth: d) in a thick conducting plane. Regions (II) (-d < z < 0 and r < a) and (III) (z < -d) are a circular aperture and a half-space, respectively. In region (I) (z > 0) the total magnetostatic potential consists of the incident and scattered potentials

pi(r,cp,z) = x = rcoscp

pS(r,cp,z) = coscp 1000 ~S(()J1((r)e-(Z(d(.

The total magnetostatic potential in region (II) is

pd(r,cp,z) 00

(12)

(13)

= cos cp L [bn sinh kn(z + d) + en cosh kn(z + d)] J1(knr) (14) n=l

where the constant kn is determined by : [J1 (knr)]r-a = O. The total trans-ur -

mit ted magnetostatic potential in region (III) is

(15)

The continuity of normal derivative of the potential at z = 0 requires

:z [pi (r, cp, z) + pS (r, cp, z) L=o

= { :z [pd(r, cp, z)L=o' r < a

0, r > a.

(16)


_ 1 00

pS (() = _"( L kn (bn cosh knd + en sinh knd) n=l

. [J1(kna)[J1((a) - a(Jo((a)]] (2 - k~ . (17)


The continuity of the potential across the aperture is

4ii(r, 4>, 0) + 4iS(r, 4>, 0) = 4id(r, 4>, 0), r < a. (18)

Substituting j;S(() into (18), multiplying (18) by J1 (kpr)r (p = 1,2,3, ... ), and integrating with respect to r from 0 to a, we get

00

n=l

(19)

where

I = roo [Jl((a) - a(Jo((a)J2 d( 10 ((2 - k;)((2 - k~) .

(20)

Similarly the boundary conditions at Z = -d yield

(21)

The magnetic polarizability of a circular aperture is

00

= tra2 L [bn coshkn(z + d) + en sinhkn(z + d))J2(kna) . (22) n=l

7.2 Acoustic Scattering from a Circular Aperture in a Hard Plane [6]

Acoustic and scalar wave scattering from a circular aperture in a hard plane has been studied in [7-9). In this section we will consider acoustic scattering from a circular aperture in a thick hard plane. In region (I) (z > 0) a field (velocity potential) 4ii(r, 4>, z) is incident on a circular aperture (radius: a and depth: d) in a thick hard plane. Regions (II) (-d < Z < 0 and r < a) and (III) (z < -d) denote a circular aperture and a half-space, respectively, where the wavenumber is k (= 2tr/A and A: wavelength). The total field in region (I) consists of the incident, reflected, and scattered waves

7.2 Acoustic Scattering from a Circular Aperture in a Hard Plane [6] 177

z

Region (I)

Region (III)

Fig. 7.2. A circular aperture in a hard plane

pi (r, rp, z) = exp(ikx sin () - ikz cos ()) 00

= exp(-ikz cos ()) L imJm(krsin())eim¢

m=-oo

pT(r,rp,z) = exp(ikxsin()+ikzcos()) 00

= exp(ikzcos()) L imJm(krsin())eim¢

m=-(X)

where K, = Jk2 - (2. The total field in region (II) is

pd(r, rp, z) 00 00

(1)

(2)

(3)

= L eim¢L[b~sinkz (z+d)+c~coskz(z+d)]Jm(knr) (4) m=-oo n=l

where kz = Jk2 - k;" J:n(kna) = 0, and the prime denotes differentiation with respect to the argument. The total transmitted field in region (III) is

00 roo pt(r, rp, z) = L eim¢ io ;Ptm(()Jm((r) exp [-iK,(z + d)] (d( .

m=-oo 0

(5)

Applying the Hankel transform to


:z [pi(r,q),z) +pT(r,q),z) +pS(r,q),z)L=o

{ :z [pd(r, q), z)L=o' r < a

0, r> a

(6)

yields • 00

;psm(() = -1 L kz( -c~ sin kzd + b~ cos kzd)h (kn , () (7) Ii

n=l

w here for a =J. (3 a

h(a,(3) = a 2 _ (32 [aJmH (aa)Jm((3a) - (3Jm(aa)Jm+1 ((3a)] (8)

otherwise

a2

h (a, a) = 2 [J!(aa) - Jm- 1 (aa)JmH (aa)]

The boundary condition for r < a at z = ° requires

pi(r,q),O) +pT(r,q),O) +pS(r,q),O) = pd(r,q),O).

(9)

(10)

Substituting ;psm(() into (10), multiplying (10) by Jm(kpr)r, and integrating with respect to r from ° to a, we get

00

2im h (k sin B, kp) - L ikz (-c~ sin kzd + b~ cos kzd) h n=l

(11)

where

12 = 100 Ii-I h (kn , ()h (kp , ()Cd( . (12)

Applying the Hankel transform to

a = {:z [pd(r, q), z)L=_d' az [pt(r, q), z)L=_d

0,

r<a (13)

r>a

gives

(14)

Similarly substituting ;ptm (() into another boundary condition

pt(r, q), -d) = pd(r, q), -d), r<a (15)

7.3 EM Scattering from a Circular Aperture in a Conducting Plane 179

and manipulating (15) yields

00

L ikzb~ 12 = c; h (kp, kp) . (16) n=l

The far-zone scattered and transmitted fields at distances Rs and R t are

cpS(R () A-.) = kcos(}sexp(ikRs -i7r/2) s, s, 'I' Rs

00

. L exp[im(¢ -1f/2)]~Sm(ksin(}s) (17) m=-oo

J..t(R 0 A-.) _ k cos (}t exp( -ikRt + i1f /2) (_ 'kd 0 ) '£' t, t, 'I' - Rt exp 1 cos t

00

. L exp[im(¢+1f/2)]~tm(ksin(}t). (18) m=-oo

The reflection coefficient {! (or transmission coefficient T) is a ratio of the power reflected from (or transmitted through) an aperture to the power impinging on an aperture. They are

(! = - 2k 2 () f: f: 1m{k;(b~ coskzd - c~ sinkzd)* a cos

m=-oon=l

. [(b~ sin kzd + c~ cos kzd)h (kn' kn ) - 2im h (k sin (), kn )] } (19)

T = 2k 2 () f: f: 1m (k;c~b~*) h (kn, kn ) . (20) a cos

m=-oo n=l

7.3 EM Scattering from a Circular Aperture in a Conducting Plane

Electromagnetic wave scattering from a circular aperture in a plane is an important canonical problem in diffraction theory [10-12]. A study of electromagnetic field penetration into a circular aperture is important for practical applications in electromagnetic compatibility and antenna radiation [13]. In this section we will analyze electromagnetic wave scattering from a circular aperture in a thick perfectly-conducting plane. For simplicity we normalize the fields E(t) and H(t) in a medium characterized by permittivity E and permeability J.t. Let's introduce the normalized fields E(t') and H(t') that are related by E(t) = foE(t'), H(t) = -i..,fiH(t'), and t = v'J.tOEot'. Maxwell's equations are then given in time domain by


Region (I)

/. Region (III) \ E' Et3

- -0· . -- - 6_ ~-- ~ ~-~ ~r~-~~-~ --- ~ ~~----~~---o r-~-~

\l X E(t') = in aH(t') at'

\l x H(t') = in aE(t') at'

(1)

(2)

where n is the refractive index of a medium. An electromagnetic wave Ei impinges from below on a circular aperture with radius a and thickness din a thick perfectly-conducting plane. The wavenumbers in regions (I), (II), and (III) are kj = wnj, (j = 1,2,3), respectively, where nj is the corresponding medium refractive index. It is convenient to represent the transverse E t and H t fields in terms of two eigenvectors ejm and eje (j = 1,2, 3) that are transverse to the z-direction in circular cylindrical coordinates (r, </> , z) . The eigenvectors ejm and eje are associated with the TM (transverse magnetic to the z-direction) and TE (transverse electric to the z-direction) waves [14) . In region (I) (z > d/2) the transverse transmitted field is

00 roo Etl = L io [lltm()e lm + llte(()e le] exp[iJo\;l(Z - d/2))d(

v=-oo 0

(3)

Htl = f 100 [lltm(()hlm + llte(()hle ] exp[iJo\;l (z - d/2))d(

v=-oo 0

(4)

where Jo\;l = J w2nr - (2 and

elm = hIe = [fiJo\;l(J~((r) - J Jo\;~v Jv((r)] eiv¢ (5)

[

A ivwn 1 A ,] • ¢ ele = hIm = r-r-Jv((r) - </> wnl(Jv((r) elV (6)


and the prime denotes differentiation with respect to the argument. Note that l' and J denote the unit vectors in cylindrical coordinates. In region (II) (-d/2 < Z < d/2 and r < a) the transverse field within the aperture is

00

Et2 = .L: .L:{[Avjexp(il\;vjz)+Bvjexp(-il\;vjz)]e2m v=-oo j

+ [Cvj exp(il\;~jz) + DVj exp( -il\;~jz)] e2e } (7)

00

Ht2 = .L: .L: {[(Avj exp(il\;vjz) - BVj exp( -il\;vjz)]h2m v=-oo j

+ [CVj exp(il\;~jz) - DVj exp( -il\;~jz)] h 2e } (8)

where I\;vj = Jw2n~ - (Xvj/a)2, I\;~j = Jw2n~ - (X~j/a)2, and

[1'il\;v~Xvj J~ (X2 r) - JI\;~V J v (X~j r)] eiv</> (9)

[ ,ivwn2J (X~j) J. X~jJ' (X~j )] iv</> r-- v -r -,!,wn2- -r e r a a v a

(10)

[ ,ivwn2 J (XVj) J. Xvj J' (XVj )] iv¢ r-- v -r -,!,wn2- -r e r a a v a

(11)

h 2e = [1' il\;~~X~j J~ (X~j r) _ J I\;~V Jv (X~j r) ] eiv¢ . (12)

Note that Xvj is the jth root of JvO = 0 and X~j is the jth root of J~(.) = o. In region (III) (z < -d/2) the total field consists of the incident, specularly reflected, and scattered components. For the TE wave incidence, the incident E-field takes the form of

Ei = fj exp(ik3x sin ()i + ik3z cos ()i) 00

= fjexp(ik3zcos()i) .L: i V Jv(k3rsin()i)eiV¢ (13) 1/=-00

For the TM wave incidence, the incident H-field Hi is represented by (13). The transverse scattered field in region (III) is

Et3 = vf;oo 100 [Wtm(()e3m + Wte(()e3e] exp[-iI\;3(Z + d/2)]d( (14)

Ht3 = - vf;oo 100 [Wtm(()h3m + Wte (()h3e] exp[ -iI\;3(Z + d/2)]d( (15)


(16)

(17)

We are now in a position to determine the modal coefficients Avj , Bvj , Gvj , and DVj by using the boundary conditions. The tangential E-field continuity at z = d/2 requires

E t1 (r,¢,d/2) = {~t2(r,¢,d/2), r<a r > a.

(18)

In matching the boundary conditions on the field continuities (18), it is convenient to utilize the orthogonality property associated with the eigenvectors. Consider

21 r21r ('Xl [En x him((')]. zrdrd¢ 7r 10 10 .

= 2~ 100 lltm(() 121r 100

[elm(() x him((')]· zrdrd¢d(

= -iwnllbl(lltm(()I(-4(f . (19)

A more detailed discussion on the eigenvector orthogonality property can be found in [14]. We define an operation < alb> as

Then

1 r 21r r < alb >= 27r 10 10 (a x b*) . zrdrd¢ = - < bla >* (20)

< E t2 lh1m ((') > = L [Avj exp(ilbvjd/2) + BVj exp( -ilbvjd/2)] < e2ml h lm((') >

j

+ L [Gvj exp(ilb~jd/2) + DVj exp( -ilb~jd/2)] < e2elhlm((') > .(21) j

In view of (19) and (21), (18) becomes

-iwnllbl(lltm(() = L [Avj exp(ilbvjd/2) + BVj exp( -ilbvjd/2)] j

[ . '() (2Jv((a) 1 . -lWnllbvjXvjJv Xvj 2 (2 - (Xvj/a)

+ L [Gvj exp(ilb~jd/2) + DVj exp( -ilb~jd/2)] j

. [-iw2nln2vJV(X~j)Jv((a)] .

Similarly applying the operation < E1Ih1e ((') > to (18) gives

(22)


-iwnIKI(fite (() = L [Cvj exp(iK~jd/2) + DVj exp( -iK~jd/2)] j

[. (X~j)2 J ( /) KI(J~((a) 1 . 1Wn2-- v Xvj 2 a (2 - (X~j/a)

The tangential H-field continuity at z = d/2 and r < a is

< H t2 le 2m >

(23)

= 100 [fitm (() < hIm hm > + fite (() < hIe hm >] d( (24)

< H t2 le2e >

= 100 [fitm(() < h Im le2e > +fite(() < h Ie le2e >] d( . (25)

Substituting fitm (() and fite (() into (24) and (25), and carrying out algebraic manipulation, we get

[Avq exp(iKvq d/2) - Bvq exp( -iKvq d/2)]

· { -iWn2KVqX~q~[J~(XVqW }

= L [Avp exp(iKvpd/2) + Bvp exp( -iKvpd/2)] p

· [-iwnIKVpKvqXVpXVqJ~(Xvp)J~(XVq)] h (p, q)

+ L [Cvp exp(iK~pd/2) + Dvp exp( -iK~pd/2)] p

· [-iw2nIn2VKVqXVqJv(X~p)J~(XVq)] h(q)

[Cvq exp(iK~qd/2) - Dvq exp( -iK~qd/2)]

[ . / 1( /2 2)J2( / )] · -lwn2KVq"2 X vq - V v Xvq

= L [Avp exp(iKvpd/2) + Bvp exp( -iKvpd/2)] p

· [-iw2nIn2vKvpXvpJv(X~q)J~(Xvp)J h(p)

+ L [Cvp exp(iK~pd/2) + Dvp exp( -iK~pd/2) ] p

.{ [-iwgnln~v2Jv(X~p)Jv(X~q)J Ig

+ [-iW~ (¥ r Jv(X~p)Jv(X~q)l I4(P,q)}

(26)

(27)


where

(28)

(29)

(30)

(31)

It is possible to obtain another set of simultaneous equations for the modal coefficients Avj , Bvj , Gvj , and DVj by using the boundary conditions at z = -d/2. The result is

[Avq exp( -iK,vqd/2) - Bvq exp(iK,vqd/2)]

· { -iwn2K,VqX~q~[J~(Xvq)f} = L [Avp exp( -iK,vpd/2) + Bvp exp(iK,vpd/2)]

p

· [iwn3K,vpK,vqXvpXvqJ~(Xvp)J~(xvq)lI1 (p, q)

+ L [Gvp exp( -iK,~pd/2) + Dvp exp(iK,~pd/2)] p

· [iw2n2n3vK,vqXvqJ~(XVq)Jv(X~p)] 12(q)

+ < (Hi + H T )le2m > (32)

7.4 Acoustic Radiation from a Flanged Circular Cylinder [15] 185

[Gvq exp( -i/),~qdj2) - Dvq exp(i/),~qdj2)]

[ . , 1(,2 2)J2(')] . -lwn2/),vq"2 X vq - V v Xvq

= L [Avp exp( -i/),vpdj2) + Bvp exp(i/),vpdj2)] p

. [iw2n2n3v/),vpXvpJ~(Xvp)Jv(X~q)] I2(p)

+ L [Gvp exp( -i/),~pdj2) + Dvp exp(i/),~pdj2)] p

.{ [iw3n~n3v2 Jv(X~p)Jv(X~q)] 13

+ [iW~: (X~PaX~q) 2 Jv(X~p)JV(X~q)lI4(p,q)} +«Hi+Hr)he> (33)

where II through 14 are obtained by replacing /),1 in (28) through (31) with /),3·

For the TE incidence, we note

< (Hi + H r)le 2m > = 0 (34)

< (Hi + H r)le2e > = -2iv cosBiwn2(X'~jja)Jv(X~j)

[ J~(k3asinBi) 1 . (k3 sin Bi)2 - (X~jja)2

(35)

and for the TM incidence

< (Hi + HT)hm > = 2iv/),vjXvj k3 sinBiJ~(Xvj)

[ Jv(k3asinBi) 1 (36) . (k3 sinBd 2 - (XVjja)2

< (Hi + H r)le2e > = 2iv kW~2VB Jv(X~j)Jv(k3asinBi) . (37) 3 sm i

The final formulation obtained in this section is shown to be somewhat analogous to the result in [10].

7.4 Acoustic Radiation from a Flanged Circular Cylinder [15]

A problem of acoustic wave radiation from a flanged circular cylinder was considered in [16]. In this section we will revisit the problem of an acoustic wave (velocity potential) pier, z) radiating from a flanged circular waveguide


z

Region (II) <Ilt

Region (I)

Fig. 7.4. A flanged circular cylinder

with radius a. The total wave in region (I) (r < a and z < 0) is a sum of the incident and reflected waves pi(r, z) and pr(r, z)

(1)

(2)

where Vo is the velocity amplitude of incident wave, k is the wavenumber, f3m is the roots of Jl(f3m) = 0, and CTm = Jk2 - (f3m/a)2. The transmitted field in region (II) (z > 0) is

pt(r, z) = 100 ~t(() exp (izJk2 - (2) Jo((r)(d( . (3)

The boundary condition at z = 0 requires

r<a (4)

r > a.

Applying the Hankel transform to (4) gives

Jk2 - (2~t(() = ~ l a [k - ~ AmCTmJo(f3mr/a)] Jo((r)rdr. (5)

From (5), we get

7.5 Acoustic Scattering from Circular Apertures in a Hard Plane [17] 187

~t(() = Vo [ka J ((a) _ ~ A aam(Jo(!3m)Jl((a)] (6) kVk2 _ (2 (1 ~ m (2 - (!3m/a)2

Another boundary condition at z = 0 and r < a

(7)

is rewritten as

~ [1 + ~ AmJo(!3mr/a)] = 100 ~t(()Jo((r)(d( . (8)

We substitute (6) into (8), multiply (8) by rJo(!3nr/a), and integrate with respect to rover (0, a) for n ~ 0 to get

(9)

where

[00 (3 Jr ((a) 1m = Jo [(2 _ (!3n/a)2][(2 _ (!3m/a)2]Vk2 _ (2 d( .

(10)

We note that (9) agrees with the result that is based on Morse's equation [16].

7.5 Acoustic Scattering from Circular Apertures in a Hard Plane [17]

Acoustic wave scattering from a single circular aperture in a thick plane has been studied in Sect. 7.2. In this section we will solve an acoustic scattering problem of two circular apertures in a thick hard plane. Consider an acoustic uniform plane wave impinging on two circular apertures in a thick hard plane. For simplicity, the wavenumbers in regions (I) through (IV) are assumed to be all identical with k. In region (I) (z > 0) an incident field pi (velocity potential) impinges on two circular apertures. Regions (II) and (III) (-h < z < 0, r < rl, and r' < r2) denote two circular apertures and region (IV) (z < -h) denotes a half-space. In region (I) the total field consists of the incident, reflected, and scattered waves. The incident (pi) and reflected (pr) components are

pi,r(x,y,z) = exp [iksinOi(xcos4>i + y sin 4>i) =f ikz cos Oi] . (1)

Since x = r cos 4> - l/2 = r' cos 4>' + l/2 and y = r sin 4> = r' sin 4>' ,


y ,

---®~: , ---L Regio (II) i :

. I

i I I

2

}---------------

z +

--l---~- I .' ~ ~ .

.... .' Regior( II) ) ~I L :' ' , ' / j",

:' I 'J'" :' Hard plane I! 9, " . I I

<1>t Transmitted

Rs

Fig. 1.5. Two circular apertures in a hard plane

. ( ikl sin Oi ) p"T(r,c!>,z)=exp =fikzcosOi - 2

00

. L imJm(krsinOi)exp[im(c!>-c!>i)] m=-oo

. , , ( ikl sin Oi) p"T(r , c!>,z)=exp =fikzcoSOi+ 2

00

, L imJm(kr'sinOi)exp[im(c!>'-c!>i)] m=-oo

(2)

(3)

where Jm (,) is the Bessel function of the first kind, In view of the superposition principle, the scattered wave in region (I) may be represented as a sum of two components


w.nere

(5)

(6)

and Ii, = Jk 2 - (2. In regions (II) and (III) the fields are, respectively, given as

00 00

pI! (1', rfy, z) = L eim¢ L m=-oo n=1

[a~ sin k;'(z + h) + b~ cos k;'(z + h)] Jm(k~1') (7) 00 00

pI!I(1",rfy',z) = L eim¢f L m=-oo n=1

[c~ sin k;'(z + h) + d~ cos k;'(z + h)] Jm(k~1") (8)

where k;' = Jk2 - (k;i')2, J:n(k~1'd = 0; k';' = Jk2 - (k;i')2 , J:n(k~1'2) = 0, and the prime' denotes differentiation with respect to the argument. In region (IV) the transmitted wave is a sum of two components

pt = ptl(1', rfy, z) + pt2(1", rfy', z) (9)

where

The boundary conditions at z = 0 require

Ops1 I = { 0:; I I ' oz z=O

z=O 0,

z=o

{ OpIII

= oz z=O

0,

(10)

(11)

(12)

(13)

pi(1', rfy, 0) + pT(1', rfy, 0) + pS(1', rfy, 0) = pI! (1', rfy, 0), l' < 1'1 (14)

pi(1", rfy', 0) + pT(1", rfy', 0) + pS(1", rfy', 0) = pI!I (1", rfy', 0), 1" < 1'2 .(15)

Applying the Hankel transforms to (12) and (13), respectively, yields


• 00

~Sl(() = -1 "'km(amcoskmh-bmsinkmh)Im(km () (16) m ",L...Jz n z n z 1 n'

n=l • 00

~~(() = ~l L k~ (c~ cosk~h - d~ sink~h) I:f'(k;:t, () (17) n=l

where for 0: -::f. (3

If:2(0:,(3) = 0:2r~2(32 [O:Jm+l(o:r1,2)Jm((3r1,2)

-(3Jm (o:rl,2)JmH ((3r1,2)] (18)

otherwise

If:2(0:,0:) = r~2 [J!(o:r1,2) - Jm- 1 (o:rl,2)Jm +1 (o:r1,2)]

Graf's addition theorem [18] gives 00

Jp((r')eiP¢1 = L Jm_p((l)Jm((r)eim¢ m=-oo

00

Jp((r)eip¢ = L Jp_m((l)Jm((r')eim¢/. m=-oo

(19)

(20)

(21)

Substituting (16), (17), and (20) into (14), multiplying (14) by Jm (kr;'r)r, and integrating over 0 < r < rl, we get

00

- L ik~(a~ cosk~h - b~ sink~h)I11 n=l

00 00

- L L ik~(~ cos k~h - d~ sin k~h)I12 p=-oon=l

= (a'; sink~h + b'; cosk~h) Ir'(k';, k';) (22)

where

I11 = 1000 ",-lIr'(k~,()Ir'(kr;',()(d( (23)

h2 = 1000 ",-lJm_p((l)Ir'(k,;,()If(k~,()(d(. (24)

Similarly from (16), (17), (21), and (15), we obtain


00 00

- L L ik~(a~ cos k~h - ~ sin k~h)I21 p=-oon=l

00

- Lik~(c~cosk~h-d~sink~h)I22 n=l

= (c';;' sin k~h + a;' cos k~h) I2'(k';;', k';;') (25)

where

122 = 100 ",-1 I2'(k~, ()I2' (k';;', ()Cd( (26)

121 = 100 ",-1 Jp-m((l)If(kh' ()I2' (k';;', ()Cd( . (27)

The additional boundary conditions at z = -h require

oiPt1 I = { oiPIl I ' oz oz z=-h z=-h 0,

(28)

OiPt21 = { O~:Il I ' oz z=-h

z=-h 0, (29)

iPt(r, ¢, -h) = iPIl (r, ¢, -h), r < r1 (30)

iPt(r', ¢', -h) = iPIlI (r', ¢', -h), r' < r2 . (31)

Applying the Hankel transforms to (28) and (29) gives respectively • 00

~!!() = ~ Lk~a~Ir'(k~,() (32) n=l

Substituting (20), (21), (32), and (33) into (30) and (31), we obtain

00 00 00

~)k~a~ 111 + L L ikfchh2 = b';;' Ir'(k'J\ k';;') (34) n=l . p=-oo n=l

00 00 00

L Likfa~I21 + Lik~c~I22 = d';;'I2'(k,;;"k';;'). (35) p=-oon=l n=l


The reflection coefficient ()// (or transmission coefficient T //) is a ratio of the time-averaged power reflected from (or transmitted through) aperture v to that impinging on aperture v where v is 1 or 2. Then the total reflection coefficient () and transmission coefficient T of two circular apertures are

() = (}l + (}2 (36)

with

(}l = - 2k 2 0 f f Im{k~*(a~ cosk~h - b~ sink~h)* r1 cos i m=-oo n=l

. [(a~ sin k~h + b~ cos k~h)I;n(k~, k~)

-2im exp ( - i~l sin Oi - im¢i) I;n(k sin Oi, k~)] } (37)

(}2 = - 2k 2 0 f flm{k~*(C~COSk~h-d~Sink~h)* r2 cos i m=-oo n=l

. [(C~ sin k~h + d~ cos k~h)I!{'(k~, k~)

-2im exp ( - i~l sin Oi - im¢i) I!{' (k sin Oi, k~)] } (38)

and

(39)

with

(40)

(41)

The far-zone scattered and transmitted fields at distances Rs and Rt are

and

pS(Rs, Os, ¢)

exp(ikRs - i1f/4) cos Os

Rsvsin Os

00 00

m=-oop=-oo

{ exp(im¢ - ip1f/2) [ - iei(I/2~;1(() + e-i(I/2~;2(()]

+(-1)mexp(im¢ + ipn/2) [e-i(I/2~;1(() - iei(I/2~;2(()]} . (42) (=ksm Os

7.6 Acoustic Radiation from Circular Cylinders in a Hard Plane [19] 193

CPt(Rt, ()t, ¢)

exp(ikRt - in /4) exp( -ikh cos ()t) cos ()t

RtV'sin()t

00 00

m=-oop=-oo

{(_l)m exp(im¢ - ipn/2) [iei<I/2~~1«() _ e-i<I/2~~2«()]

+ exp(im¢ + ip7r /2) [ - e-i<I/2~~1 «() + iei<I/2~~2«()] } I . (43) <=ksm(),

7.6 Acoustic Radiation from Circular Cylinders in a Hard Plane [19]

A study of electromagnetic wave radiation from flanged circular cylinders is important for practical applications in microwave antenna array problems. Acoustic wave radiation from a single flanged circular cylinder has been studied in Sect. 7.4. In this section we will study acoustic wave radiation from multiple circular cylinders in a hard plane. A theoretical analysis given in this section is similar to that in Sect. 7.5. Consider an N number of circular cylinders (v = 1,2,3, ... ,N) on an infinite hard plane where an acoustic wave emanates from the first flanged circular cylinder v = 1. In region (1) the total field (velocity potential) consists of the incident, reflected, and scattered waves, where the incident (cpi) and reflected (CPT) components are

cpi,T(X,y,Z) = e±ikz . (1)

Inside the vth cylinder the scattered component is represented as 00 00

cpBV = L eim,pv Lb:;'nexp(-ik:;'z)Jm(k;;!:.rv) (2) m=-oo n=1

where k::" = Jk2 - (k:;;)2, J:n (k::"nav) = 0, and the prime' denotes differentiation with respect to the argument. In region (N + 1) the transmitted wave, based on the superposition principle, is represented as

N

cpt = L cpt/-lh" ¢/-I' z) (3) /-1=1

00 roo cpt/-l(r/-l' ¢/-I' z) = L eim,p" io ~~«()Jm«(r/-l)eiI<Z(d(

m=-oo 0

(4)

where K, = Jk2 - (2 and cpt/-l(r/-l,¢/-I,z) is the wave transmitted through the JLth cylinder.

The boundary conditions at z = 0 for the vth circular cylinder for 1 :::; v:::; N require


8ptv

8z

Region (N+ 1)

z • I

!

TransmiHed

Fig. 7.6. Multiple circular cylinders in a hard plane

{ ~'" rv < av _ 8z z=o

z=o 0, rv > av

[pi(rl' cPl, 0) + pr(rl ' cPl, O)]<5Vl + pSV(rv, cPv, 0)

= pt(r, cP, 0), rv < av

x

(5)

(6)

where <5ij denotes the Kronecker delta. Applying the Hankel transform to (5) yields

(7)

7.6 Acoustic Radiation from Circular Cylinders in a Hard Plane [19] 195

where for 0: f:. (3

I;:' (0:, (3) = 2 a., (32 0: -

. [o:Jm+1 (o:a.,)Jm((3a.,) - (3Jm(o:a.,)Jm+1 ((3a.,)] (8)

otherwise 2

1;:'(0:,0:) = a; [J!(o:a.,) - Jm- 1 (o:a.,)Jm+1 (o:a.,)] (9)

Substituting (1), (2), and (3) into (6) yields 00 00

28.,1 + L eimcPv L b~n exp(-ik~z)Jm(k:!r.,) m=-oo n=l

N

= L tf!t,..(r,.., 41,.., z) . (10) ,..=1

Graf's addition theorem gives 00

m=-oo

We substitute (4), (7), and (11) into (10), multiply (10) by Jm(k'Z~r.,)r." and integrate over 0 < r., < a., to obtain

where

00 N 00 00

L b~nk~ h + L L L bf,nk~ exp[i(p - m)41.,,..]I2

n=l 1'=1 p=-oo n=l I"Fv

II = 100 11:-1 I;:'(k:!, ()I;:'(k~, ()Cd(

12 = 100 11:-1 Jm_p((l.,,..)I;:'(k~, ()I~(k~n' ()Cd( .

(12)

(13)

(14)

In case of single cylinder scattering (a1 f:. 0 and a., = 0 for v 2:: 2), (12) reduces to (9) in Sect. 7.4.

The reflection coefficient Gu (self-coupling) is a ratio of the reflected power to the incident power and the coupling coefficient G1., (cross-coupling) is a ratio of the power coupled into the vth cylinder to the incident power. They are given as


Cn - k!2 f f Re {ki"*lb~l If'(kr:" kr:,) } 1 m=-oon=l

+2Re(b~1) + 1

Clv = k!2 f f: Re {k~*lb~nI2 I~(k~n' k~J} 1 m=-oo n=l

The far-zone transmitted field at distance Rt is

q/(Rt,Ot,cp)

exp(ikRt - i7r/4) cosOt

RtvsinOt

. m'%;oo eim¢ {e-i7r/2cli~(() + ~pf:oo exp[i(p - m)CPlv]

. [ exp ( - i~7r _ i~l) _ i( _1)m exp C~7r + i~l) ]

.cli~(()}1 (=ksin(},


(15)

(16)

(17)

1. J. H. Lee and H. J. Eom, "Electrostatic potential through a circular aperture in a thick conducting plane," IEEE Trans. Microwave Theory Tech., vol. 81, no. 12, pp. 341-343, Feb. 1996.

2. R. L. Gluckstern, R. Li, and R. K. Cooper, "Electric polarizability and magnetic susceptibility of small holes in a thin screen," IEEE Trans. Microwave Theory Tech., vol. 38, no. 2, pp. 186-191, Feb. 1990.

3. R. L. Gluckstern and J. A. Diamond, "Penetration of fields through a circular hole in a wall of finite thickness," IEEE Trans. Microwave Theory Tech, vol. 39, no. 2, pp. 274-279, Feb. 1991.

4. J. G. Lee and H. J. Eom, "Magnetostatic potential distribution through a circular aperture in a thick conducting plane," IEEE Trans. Electromagn. Compat., vol. 40, no. 2, pp. 97-99, 1998.

5. K. F. Casey, "Low-frequency electromagnetic penetration of loaded apertures," IEEE Trans. Electromagn. Compat., vol. 23, no. 4, pp. 367-377, Nov. 1981.

6. K. H. Jun and H. J. Eom, "Acoustic scattering from a circular aperture in a thick hard screen," J. Acoust. Soc. Am., vol. 98, no. 4, pp. 2324-2327, Oct. 1995.

7. S. N. Karp and J. B. Keller, "Multiple diffraction by an aperture in a hard screen," Optica Acta, vol. 8, pp. 61-72, Jan. 1961.

8. C. J. Bouwkamp, "Theoretical and numerical treatment of diffraction through a circular hole," IEEE Trans. Antennas Propagat., vol. 18, no. 2, pp. 152-176, March 1970.

9. G. P. Wilson and W. W. Soroka, "Approximation to the diffraction of sound by a circular aperture in a rigid wall of finite thickness," J. Acoust. Soc. Am., vol. 37, no. 2, pp. 286-297, Feb. 1965.


10. A. Roberts, "Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen," J. Opt. Soc. Am. A., vol. 4, no. 10, pp. 1970-1983, Oct. 1987.

11. L. J. Palumbo and A. M. Platzck, "Diffraction by a circular aperture: a new approach," J. Opt. Soc. Am. A., vol. 4, no. 5, pp. 839-842, May 1987.

12. K. Hongo, "Diffraction of an electromagnetic plane wave by circular disk and circular hole," IEICE TI-ans. Electron., vol. E80-C, no. 11, pp. 1360-1366, Nov. 1997.

13. W. T. Cathey, Jr., "Approximate expressions for field penetration through circular apertures," IEEE TI-ans. Electromagn. Compat., vol. 25, no. 3, pp. 339-345, Aug. 1983.

14. R. E. Collin, Field Theory of Guided Waves, New York, IEEE Press, Second Edition, Chapter 5 and pp. 121-123, 1991.

15. H. J. Eom, T. J. Park, and S. Kozaki, "A series solution for acoustic radiation from a flanged circular pipe," Acustica, vol. 80, no. 3, pp. 315-316, May/June 1994.

16. W. E. Zorumski, "Generalized radiation impedances and reflection coefficients of circular and annular ducts," J. Acoust. Soc. Am., vol. 54, no. 6, pp. 1667-1673, 1973.

17. J. S. Seo, H. J. Eom, and H. S. Lee, "Acoustic scattering from two circular apertures in a thick hard plane," J. Acoust. Soc. Am., vol. 107, no. 5, Pt. 1, pp. 2338-2343, May 2000.

18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, New York, Dover Publications, p. 363, 1965.

19. J. S. Seo and H. J. Eom, "Acoustic scattering from flanged circular cylinders," Acustica, vol. 86, no. 5, pp. 780-783, Sept./Oct. 2000.

8. Annular Aperture in a Plane

8.1 Static Potential Through an Annular Aperture in a Plane

The polarizability of various aperture shapes in a conducting plane finds practical applications in low-frequency microwave scattering and penetration problems [1]. For instance, when an aperture size is small compared to an incident wavelength, the polarizability is a useful concept to estimate a field penetration into apertures. The behavior of polarizability for an annular aperture in a conducting plane has been well studied in [2-3] based on the variational method. In the present section we will revisit the problem of polarizability of an annular aperture in a thick conducting plane and analyze its static potential distribution through an annular aperture.

8.1.1 Electrostatic Distribution [4,5]

In this subsection we will investigate an electrostatic potential distribution through an annular aperture with a floating inner conductor. An electrostatic potential q,i(r, z) is applied to an annular aperture in a thick conducting plane. A conducting plane (r > band Izl < d/2) is at zero potential and the inner conductor (r < a and Izl < d/2) is electrically floated with potential V and charge Q. Regions (I), (II), and (III) are an upper half-space (z > d/2) with permittivity El, an annular aperture (radii: a and b, depth: d) with E2,

and a lower half-space (z < -d/2) with E3. In region (I) the total electrostatic potential is a sum of the incident and scattered components

q,i(r, z) = Eo(z - d/2)

q,S(r,z) = 100 ~S(()Jo((r)exp[-((z-d/2)](d(.

The electrostatic potential in region (II) is assumed to be

q,d(r, z) = V~: :~: + ~ (aneknz + bne-knz)R(knr)

where

(1)

(2)

(3)


200 8. Annular Aperture in a Plane

<1>' 1 Reg~n (I) 1 <1>'

~=d/2). b

~ - • 1 ~f%a~~~~~~ z = -dI2 Region (II)

<1>' 1

• r

Region (III)

(4)

and a constant kn is determined by R(knb) = 0. The transmitted electrostatic potential in region (III) is

q,t(r, z) = 100 ¥t(()Jo((r) exp[((z + d/2)](d( . (5)

The boundary condition on the field continuity at z = d/2 (Dirichlet boundary condition) requires

{V,

q,i(r, d/2) + q,S(r, d/2) = q,d(r, d/2), 0,

Taking the Hankel transform of (6) yields

O<r<a a<r<b otherwise.

(6)

8.1 Static Potential Through an Annular Aperture in a Plane 201

~S(() = ln~/a Jo((a) 0 Jo((b) + ~ (aneknd/2 + bne-knd/2) En(() (7)

where

~ 2 [ Jo((a) Jo((b)] 1 =n(() = -; No(kna) - No(knb) (2 - k; . (8)

The boundary condition on the field continuity (Neumann boundary condition) requires

101 (8tfJi I + 8tfJs I ) - 102 8tfJd I a < r < b . (9) 8z z=o 8z z=o - 8z z=o'

We substitute ~S(() into (9), multiply (9) by rR(kpr), and integrate with respect to r from a to b to obtain

E L 101 V J ~ ( knd/2 b -knd/2) I 101 0 P-Inb/a 2-E1~ ane + n e 1

= E2kp (apekpd/2 - bpe-kpd/2) [A(b) - A(a)]

where 2 1

A(r) = 7r2k2 N,2(k r) pop

h = 100 En(()Ep(()(2d(

12 = 100 [Jo((a) - Jo((b)]Ep(()d(

Ln = lab R(knr)rdr = 7r!; [No(~nb) - No(~na)] Similarly from the boundary conditions at z = -d/2, we obtain

EV 00

_3 -J + 10 "(a e-knd/2 + b eknd/2) I Inb/a2 3~ n n 1

= E2kp (ape-kpd/2 - bpekpd/2) [A(b) - A(a)] .

(10)

(11)

(12)

(13)

(14)

(15)

The total charge Q on the inner conductor must be conserved since the inner conductor is electrically floated. In order to determine a relationship between the charge Q and the potential V, we apply Gauss's law over the surface of the inner conductor. The relationship is

27rV Q = In b/ a [E2d + (101 + (3)WY]

00

-7rE1a2 Eo + 27ra L(anC1 + bnC2) (16) n=l


where

C1 = 13 (€le knd/2 + €3e-knd/2) - 2€2R'(kna) sinh(kndJ2) (17)

C2 = 13 (€le-knd/2 + €3eknd/2) - 2€2R'(k na) sinh(kndJ2) (18)

13 = 100 J1((a)En(()(d( (19)

'Y = 100 [Jo((a) - Jo((b)] J1 ~(a) d( . (20)

We assume that Q is linearly proportional to V, and Q = 0 when Eo = V = O. By setting Eo to zero and V to unity, we solve (10) and (15) for an and bn. Substituting an and bn into (16) yields the capacitance C

_ 8Q 27f 00

C = 8V = InbJa [€2d + (€1 + €3)a'Y] + 27fa ~(anC1 + bnC2) . (21)

The electric polarizability Xe (z) is

Xe(Z) == 47f lb q>d(r,z)rdr

7fV(b2 _ a2) 00

= + 47f "(a eknz + b e-knz ) L InbJa ~ n n n· n=l

(22)

8.1.2 Magnetostatic Distribution [4]

Consider a problem of magnetostatic distribution through an annular aperture in a thick conducting plane. Regions (I) (z > 0), (II) (-dJ2 < z < dJ2 and a < r < b), and (III) (z < -dJ2) denote an upper half-space, an annular aperture, and a lower half-space, respectively. In region (I) the total magnetostatic potential consists of the incident and scattered components. An incident magnetostatic potential is

q>i(r,¢,z) = x = rcos¢.

The scattered magnetostatic potential in region (I) takes the form of

In region (II) the magnetostatic potential is

q>d(r, ¢, z) 00

(23)

(24)

= L [amsinhkm(z + dJ2) + bmcoshkm(z + dJ2)] R(kmr)cos¢ (25) m=l

where

8.1 Static Potential Through an Annular Aperture in a Plane 203

(26)

Note that the prime denotes differentiation with respect to the argument

and a constant km is given by the condition 8lrd I = O. In region (III) the r==b

transmitted magnetostatic potential is

(27)

The boundary condition at z = d/2 requires

{ 8q,d I

8q,i + 8q,s _ 8z z==d/2'

8z IZ==d/2 8z IZ==d/2 -0,

a<r<b (28)

otherwise.


00

~S(() = - L (am coshkmd + bmsinhkmd) km5'm(() (29) m==l

where

(30)

An additional boundary condition at z = d/2 for a < r < b requires

q,i I + q,s I - q,d I z==d/2 z==d/2 - z==d/2 . (31)

It is expedient to apply to (31) the orthogonality property

(32)

where

A(r) = 71}k; [1- (kn1r)2] N{(Lr)2 . (33)

Multiplying (31) by R(knr)r and integrating with respect to rover (a, b), we get

00

Ln - L (am coshkmd + bm sinhkmd) kmI m==l

= (an sinh knd + bn cosh knd) [A(b) - A(a)] (34)


where

I = 100 Sn(()Sm(()(2d(

Ln = lb r2R(knr)dr = 7r!~ [NH~nb) - NH~na)] From the boundary condition at Z = -d/2, we obtain

00

L amkmI = bn [A(b) - A(a)] . m=1

The magnetic polarizability Xm(z) is shown to be

rb 8~d Xm(z) == 7r Ja r2 8z dr

00

(35)

(36)

(37)

= 7r L [an cosh kn (z + d/2) + bn sinh kn(z + d/2)] knLn. (38) n=1

8.2 EM Radiation from a Coaxial Line into a Parallel-Plate Waveguide [6]

Electromagnetic wave radiation from a coaxial line into a half-space or a parallel-plate waveguide was studied for material permittivity characterization and antenna feed application [7-9]. In this section we will analyze electromagnetic radiation from a coaxial line into a parallel-plate waveguide. Assume that an incident TEM wave emanates from a flanged coaxial line. Due to a symmetry of the problem geometry, the H-field has only a ¢-component. In region (I) (a < r < band Z < 0) the incident and reflected H-fields are

(1)

(2)

where {31 = WVJ.tf.1, kzn = {31 J1- (kn/{31)2, Rn(r) = J1(knr)No(knb) -N1 (knr)Jo(knb), Mo = l/Jln(b/a), and Mn = 7rkn/J2 - 2J6(knb)/J6(kna). Note that an eigenvalue kn is given by Jo(kna)No(knb) -No(kna)Jo(knb) = O. The transmitted H-field in region (II) (z > 0) is

H~(r, z) = 100 ii(() (eiKZ + e2iKhe-iKZ) J1((r)(d( (3)

where fi, = J {322 - (2 and the wavenumber is {32 = wy' J.tf.2 = 27r / >...

8.2 EM Radiation from a Coaxial Line into a Parallel-Plate Waveguide (6) 205

Fig. 8.2. A coaxial line radiating into a parallel-plate waveguide

The tangential E-field continuity at z = 0 requires

b<r<a otherwise.


- 102 1 [ 00 1 H(() = 10111:(1 _ e2i l<h) (1- eo)!3do(() - ~ cnkznfn(()

where

fo(() = _ Mo [Jo(b() - Jo(a()] (

fn(() = 2Mn( [Jo((b)Jo(kna) - Jo((a)Jo(knb)] 7rknJo (kna) (kn 2 - (2)

The tangential H-field continuity at z = 0 requires

H¢(r, O) = H~(r, 0) + H;(r, 0), a<r<b .

We multiply (8) by rMpRp(r)dr and integrate from a to b to obtain

Ie = (U - A)-1 r I

(4)

(5)

(6)

(7)

(8)

(9)

where C is a column vector of cp , U is a unit matrix, and the elements of matrices A and r are for p ~ 0


apo = 100 /31g(()fo(()fp(()(d(

apn = 100 kzng(()fn(()fp(()(d(, n ~ 1

'Yp = -apO - c5po

gee) = -i E2 cot(K,h) . El K,

(10)

(11)

(12)

(13)

It is convenient to transform the integral apn into fast convergent series by performing the residue calculus along a contour path in the (-plane. The results are

. Hi (.I /31 ~ 2 Aoo(c;m) aoo = -IV 4" cotfJ2h - 2" ~ 7rMoam hc;~ (14)

aOn = 7r~~n f a m4>n(C;m)AlO(C;m) m=O

(15)

7r/31 ~ apO = 2h ~ a m4>p(C;m)AOl (C;m)

m=O (16)

kznM; cot (v'/3,§ - k~h) , a = A11 (k )c5

pn 7rknJg(kna)v'/3'§ _ k~ n pn

- 7r~~n f am¢pn(c;m)An(c;m) (17) m=O

, dAn(()I where An (kn ) means d( (=k

n and

Ast(() = [Jo (sakn)Jo (takp)Jo (b()Hd1) (b()

-Jo(sakn)Jo (tbkp) Jo (a()Hd1) (b()

-Jo(sbkn)Jo(takp)Jo(a()H~l) (b()

+Jo (sbkn)Jo (tbkp)Jo (a()Hd1) (a()] E2, s, t = 1,0 (18) El

{ I, am = 2,

m=O m = 1,2,3, ...

C;m = J /3'§ - (m7r/h)2, m = 0,1,2, ...

2MoMn 4>n(() = 7rknJo(kna)(kn 2 _ (2)

( ) 4MpMn(2 ¢pn ( = 7r2kpknJo(kpa)Jo(kna)(kp 2 _ (2)(kn 2 _ (2) .

(19)

(20)

(21)

(22)

8.2 EM Radiation from a Coaxial Line into a Parallel-Plate Waveguide (6) 207

Performing the residue calculus, we represent the transmitted H-field in rapidly-convergent series

00

H~(r,z) = B1 L m=O

am m7r ( ) 1- (m7r) 2 COS h ZJ1 ~mr

(32 h

. [H~l)(~mb) - H~l)(~ma)]

_ ~ ~ B cos(m7r/h)z ~~ 2 (k; ~;.)

[JO(~mb)Jo(kna) - Jo(~ma)Jo(knb)lHi1) (~mr)

J1 (~mr)Jo(kna)H~l) (~mb) - Hi1) (~mr)JO(~ma)Jo(knb)

J1(~mr)[H~1)(~mb)Jo(kna) - H~l)(~ma)Jo(knb)l

10, r ~ b

No (knb)J1 (knr) - Jo(knb)N1(knr), a:::; r:::; b

0, r:::; a

(23)

h B - /¥o2 7rVo B - 2Cnf2kzn~mMnam TT _ (31 (1 - eo) d were 1 - - - , 2 - , yo - , an /-Lo 2hln(b/a) f1knJo(kna)h Wf1 MO

"'n = -";"k2"2 ---;k~;. Let '8 consider a coaxial line radiating into a half-space. When h -t 00,

the solution (9)-(12) is applicable with g(() = -f2/(f1"')' and it is expedient


to transform the integrals into fast convergent forms by performing a branchcut integration. We choose a similar deformed contour path as shown in Fig. 1.2 with a branch cut associated with a branch point ( = fh. The contour integration results give

f!;2 100 Mg . aOO = - - + f31 --;;-AOO(()ldV fl 0 ~~

(24)

(25)

(26)

(27)

where ( = f32 + iv.

8.3 EM Radiation from a Coaxial Line into a Dielectric Slab [10]

An open-ended coaxial line radiating into a half-space and a stratified dielectric medium was studied in [11-16]. In this section we will analyze the electromagnetic wave radiation from a flanged coaxial line into a dielectric slab that is displaced from an open-ended coaxial line aperture. A theoretical analysis given in this section is in continuation of Sect. 8.2. Assume that a TEM wave radiates from region (I) (a < r < band z < 0). The incident (TEM) and reflected (TMon waves) H-fields in region (I) are

. eif3z H¢(r,z) = Mo-

r (1)

(2)

.f 1 rr~ where f3 = w.fiiEl, f3n = V f32 - A;, Mo = ,Mn = ---;:======

Jln(b/a) 2 _ 2 JJ(Anb) JJ(Ana)

and Rn(r) = J1(Anr)No(Anb) - Nl (Anr)JO (Anb). The eigenvalue An is given by JO(Ana)No(Anb)-No(Ana)JO(Anb) = O. Regions (II), (III), and (IV) represent a background (wavenumber: k2 = w..,fiiE2), a dielectric slab of thickness

8.3 EM Radiation from a Coaxial Line into a Dielectric Slab [10] 209

z

Region (IV) k4

Region (II) k2 d1

8 ------ r ~

PEe

1 Pi p,l

~

Fig. 8.3. A flanged coaxial line radiating into a dielectric slab

h (k3 = w"ffI€3), and a half-space (k4 = wy'1i€4), respectively. The H-field in region (II) (0 < z < d1 ) is

Hf(r,z) = 100 [Ht«()eiI<2 Z + Hu«()e-iI<2 Z ] (J1«(r)d( (3)

where "'2 = Jk~ - (2. In region (III) (d1 < z < d2 ) the H-field is

H:II (r, z) = 100 [HtI«()eiI<3 Z + HUI «()e- iI<3 Z ] (J1 «(r)d( (4)

where "'3 = Jk~ - (2. In region (IV) (z > d2 ) the H-field is

HJv(r,z) = 100 Hlv «()exp[i"'4(Z - d2 )](J1«(r)d( (5)

where "'4 = Jk~ - (2. The tangential E- and H-field continuities at z = d2 give

H- «() = ('T}3 -'T}4) ei2 1<3 d2H+ «() (6) III 'T}3 + 'T}4 III

HIV«() = ( 2'T}3 ) ei l<3 d2 H+ «() (7) 'T}3 + 'T}4 III


where 'fJ3 = K,3/f3 and 'fJ4 = K,4/f4' The tangential E- and H-field continuities at z = dl similarly give

Hii(() = rei2K2dl Ht(() (8)

H+ = [ (1 + r)('fJ3 + 'fJ4)eiK2dl-iKSd2 ] H+ lII(() ('fJ3 + 'fJ4)e-iKSh + ('fJ3 - 'fJ4)eiKSh lI(()

(9)

where

r = ('fJ2 - 'fJ3)('fJ3 + 'fJ4)e-~KSh + ('fJ2 + 'fJ3)('fJ3 - 'fJ4)e~KSh . (10) ('fJ2 + 'fJ3)('fJ3 + 'fJ4)e-1KSh + ('fJ2 - 'fJ3)('fJ3 - 'fJ4)e1K3h

The tangential E-field continuity at z = 0 is

E;I (r, 0) = {Eo,:(r, 0) + E;(r, 0), a < r < b (11) otherwise.


-+ - f2 [ ~ 1 HlI (() - flK,2 (1- rei2K2dl) (1- eo)f3fo(() - ~ cnf3nfn(() (12)

where Mo

fo(() = -T [Jo((b) - Jo((a)]

fn(() = 2Mn( [Jo ((b)Jo (Ana) - Jo((a)Jo(An b)] . 7rAn(A; - (2)JO(Ana)

(13)

(14)

The tangential H-field continuity at z = 0 requires

H~I (r, 0) = H~(r, 0) + H;(r, 0), a < r < b. (15)

Substituting (8) and (12) into (15), multiplying (15) by rMmRm(r)dr, and integrating from a to b, we get a matrix equation

(16)

where C is a column vector of Cm , U is a unit matrix, and the elements of matrices A and Q for m ~ 0 are

amo = - roo f2f3 gl(() fm(()fo(()(d( (17) 10 flK,2 g2(()

amn = - roo f2f3n gl(() fm(()fn(()(d( (18) 10 (1K,2 g2(()

(19)

with

gl (() = 'fJ2'T/3 COS(K,3 h) COS(K,2dl) - 'fJ~ sin(K,3 h) Sin(K,2dl)

-i ['fJ2'fJ4 Sin(K,3h) COS(K,2dl) + 'T/3'fJ4 COS(K,3h) sin(K,2dl)] (20)

g2(() = 'fJ3'fJ4 COS(K,3 h) COS(K,2dl) - 'fJ2'T/4 sin(K,3 h) sin(K,2dt}

-i ['fJ~ sin(K,3 h) COS(K,2dl) + 'f/2'fJ3 COS(K,3h) sin(K,2dt}] . (21)

8.3 EM Radiation from a Coaxial Line into a Dielectric Slab [10] 211

It is convenient to transform amn into numerically-efficient forms by per-

1m (I,:)

Branch cut

Re(l,:)

Branch cut


forming a contour integral in the complex (-plane. The results are

aoo = !"§gl(() <>00 - L>7f~g~(() M(f Aoo(() I V E1 g2(() (p "'2 g2(() ( (=(p

+~ J3-fAoo(() G(v)dv '1 00 M2 I 2 0'" (=k4+ iv

(22)

(23)

(24)


_ i (3n 91(() MmMnA~1(O I 8 amn - -; "-2 92(() AmJO(Ama)JO(Ana) t;=An mn

--C ~)1r(3n 9~((~)) tPmn(()Al1 (()1 t;p "-2 92 .. t;=t;p

+~ 100

(3ntPmn(()Al1(()!t;=k4+iV G(v)dv (25)

where the prime I denotes differentiation with respect to ( and

Ats(() = £2 [JO (saAn)JO (taAm)JO (bOH61) (b() £1

-Jo(saAn)JO (tbAm)JO (a()H61) (b()

-JO(SbAn)Jo(taAm)Jo(a()H61) (b()

+JO(SbAn)Jo(tbAm)Jo(a()H61) (a()] , s, t = 1,0 (26)

2MoMn( ¢n(O = 1rAnJO(Ana)(A; _ (2) (27)

4MmMn(3 tPmn(() = 1r2AmAnJO(Ama)JO(Ana)(A~ _ (2)(A; _ (2) (28)

G(v) = 91(() I - 91(() I . (29) "-292 (() t;=k4+iv "-292 (() i<r+- K 4,t;=k4 +iv

The first term in amn is a residue contribution at ( = ±Am and the second term represents residue contributions at ( = (p, which are zeros of 92((). The third term in amn is a branch-cut integration along F3 and F4 associated with a branch-point at ( = k4 • In low-frequency limit, a single-mode solution (amn = 0 for m, n ~ 1) is given by

1 + aOO

I-aoo' Cn ~ 0,

The far-zone field in region (IV) is

HIV(R (J) '" _ 'TJ3k4COS(J <P' £192(()

. [(1- eo)f3!o(() - ~ cn(3n!n(()]

where R = vr2 + Z2.

(30)

R (31)

Next we consider two special cases corresponding to the half-space and attached-slab scattering problems. When k2 = k3 = k4 and d2 = 0 (half-space case), we obtain

8.4 EM Radiation from a Monopole into a Parallel-Plate Waveguide [17]

HJV(R,O) ~ :: (FO+ ~Fn) ei~R where

F. = (1- )RM, [Jo(k4bsinO) - Jo(k4aSinO)] o Col-'O k.O

4 sm Fn = 2cnf3nMnk4 sin 0

. [Jo(k4bSinO)JO(Ana) - Jo(k4asinO)JO(Anb)] 7rAn(A; - k~ sin2 O)JO(Ana)

When k2 = k3, d1 = 0, and k3 > k4 (attached-slab case), we get

roo ( €3f3 "'3 cos /'i,3d2 - i"'4 sin /'i,3d2 ) amo = - 10 €1/'i,3 "'4 cos /'i,3d2 - i"'3 sin /'i,3d2

·1 m «() 10 «()( J1 «(r )d(

and

213

(32)

(33)

(34)

(35)

(36)

. (37)

(38)

(39)

8.4 EM Radiation from a Monopole into a Parallel-Plate Waveguide [17]

A study of electromagnetic wave radiation from a coaxially-fed monopole antenna into a parallel-plate waveguide is important for practical applications in antenna feeder [18,19]. In this section we will present a scattering analysis for a coaxially-fed monopole antenna radiating into a parallel-plate waveguide. When an incident TEM wave radiates from a coaxial line, the total field in region (III) (a1 < r < a2 and permittivity: €3) consists of the incident, reflected, and scattered components


r

Region (II) Il, to2

z Incident wave

Region (II)

r

Fig. 8.5. A monopole antenna radiating into a parallel-plate waveguide

. e-ikaz H¢=---

TJr eikaz HT - __ _

¢ - TJr

E;II (r, z) = -. - R(l\,r) cos((z)d( . 2 100

1WE37r 0

(1)

(2)

(3)

In regions (I) (0 < r < a1 and permittivity: Ed and (II) (r > a2 and permittivity: E2) the scattered fields are respectively

(4)

• 00

E;I(r,z) = _1_ L Qm6mcos(h2mZ)H61)(6mr) WE2 m=O

(5)

where hpm = m7r/hp, ~pm = Jk~ - h~m' I\, = Vk~ - (2, kp = wJ/1Ep, TJ =

V/1/E3, R(l\,r) = JO(l\,r)E+(()-No(l\,r)E-((), H61)0 = JoO+iNo(')' JoO is the Oth order Bessel function, and NoO is the Oth order Neumann function.

Applying the Fourier cosine transform to the Ez continuity at r = aI, E!(al' z) + E;(al' z) + E;II (aI, z) = E;(al' z), yields

00

R(l\,al) = - L Pm E3 6mJo(6mal)E~(() . m=O EI

(6)

Similarly from the Ez continuity at r = a2, we obtain

8.4 EM Radiation from a Monopole into a Parallel-Plate Waveguide [17] 215

00

R(IW2) = - L qm 1':3 6mH~I)(6ma2)S!(() (7) m=O 1':2

where

(8)

We multiply the HI/> continuity at r = al for 0 < z < hI by cos(mr/ht}z and integrate to get

n = 0,1, ... (9)

where

21001,( )~1() It = - - - R 1Wl':: (d(

1f 0 /'i, n (10)

co = 2, en = 1 (n = 1,2, ... ), and R'O = dR(·)/d(·). Utilizing a residue calculus, it is possible to transform 11 into a rapidly-convergent series

00

It = L (_I)m+n [Pm 1':3 6mJO(6mal)11 m=O 1':1

1':3 (1)( )-] +qm -6mHo 6ma2 112 . 1':2

Similarly using the HI/> continuity at r = a2 for 0 < z < h2, we get

where

2100 1,( ) ~2 ( ) 12 = -;: 0 ;,R /'i,a2 '::n ( d(

00 .

= L(-I)m+n[qm 1':36mH~I)(6ma2)12 m=O 1':2

1 _ hI Ll18mnem ik3 1 - exp(i2k3ht) 1 - 2" /'i,lm Ll (/'i,lm) + 2alln(a2/al) (k~ - h~m)(k~ - h~n)

i 00 (v

- al ~ [1 - J6(/'i,val)/ J6(/'i,va2)]

1 - exp(i2(vhl)

(11)

(12)

(13)

(14)


12 = h2 Lb5mncm ik3 1 - exp(i2k3h2)

2 1£2m.:1(1£2m) 2a2In(a2/al) (k~ - h~m)(k~ - h~n)

i 00 (v

- a2?; [1- JJ(l£va2)/JJ(l£vadl

1 - exp(i2(vh2)

1 - X _ ik3 exp(ik3lh2 - hI!) - exp[ik3(h2 + hdl pq - pq 2ap In(adal) (k~ - h~n)(k~ - h~m)

X l2 =

i 00 (v

- ap ?; [Jo(l£v ad/Jo(l£v a2) - Jo(l£v a2)/Jo(l£v adl

exp(i(vlh2 - hI!) - exp[i(v(h2 + hdl

((; - h~n)((; - h~m)

2(-I)m h2msin(~m1f) 1fapl£~m.:1(1£2m) h~m - hin

(15)

(16)

(17)

Omn is the Kronecker delta, .:1(1£) = Jo(l£adNo(l£a2) - Jo(l£a2)No(l£ad, .:11 = JO(l£lma2)N1(l£lmad - J1(l£lmadNo(l£lma2), .:12 = J1(1£2ma2)NO(1£2mad -

JO(1£2madN1(1£2ma2), I£pm = Jk~ - h~m' I£v is determined by .:1(l£v) = 0,

and (v = Jk~ - I£~. The reflected plus scattered TEM wave at z = 00 is shown to be

H r HIlI - exp(ik3Z ) (1 L M ) ¢+ ¢ -- + 0- 0 TJr

(18)

where

(19)

(20)


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15. L. Anderson, G. Gajda, and S. Stuchly, "Analysis of an open-ended coaxial line sensor in layered dielectrics," IEEE Trans. Instrum. Meas., vol. 35, pp. 13-18, Mar. 1986.

16. S. Bakhtiari, S. I. Ganchev, and R. Zoughi, "Analysis of radiation from an openended coaxial line into stratified dielectrics," IEEE Trans. Microwave Theory Tech., vol. 42, pp. 1261-1267, July 1994.

17. H. J. Eom, Y. H. Cho, and M. S. Kwon, "Monopole antenna radiation into a parallel-plate waveguide," IEEE Trans. Antennas Propagat., vol. 48, no. 7, pp. 1142-1144, July 2000.

18. A. G. Williamson, "Radial-line/coaxial-line junctions: analysis and equivalent circuits," Int. J. Electron., vol. 58, no. 1, pp. 91-104, 1985.

19. Z. Shen and R. H. MacPhie, "Modal expansion analysis of monopole antennas driven from a coaxial line," Radio Sci., vol. 31, no. 5, pp. 1037-1046, Sept.-Oct. 1996.

9. Circumferential Apertures on a Circular Cylinder

9.1 EM Radiation from an Aperture on a Shorted Coaxial Line [1]

z

1<...-_ ........ _-'--_ _ r

(a) (b)

Fig. 9.1. A circumferential aperture on a shorted coaxial line

9.1.1 Field Analysis

Electromagnetic scattering and radiation from a circumferential aperture on a shorted coaxial line (dielectric-filled edge-slot antenna, DFES antenna) was considered in [2]. In this section we will revisit the problem of DFES antenna


220 9. Circumferential Apertures on a Circular Cylinder

radiation. Consider radiation from a shorted coaxial line whose outer conductor is circumferentially removed and replaced by a dielectric medium with permittivity E = EOEr. An incident TEM wave propagates from below inside a coaxial line that is shorted at z = O. The field in region (I) (a < r < b) consists of the incident and scattered waves

(1)

(2)

(3)

where r;, = Jk2 - (2, k = wVJii = 27r/>", R(r;,r) = Jo(r;,r)No(r;,a) -No (r;,r)Jo (r;,a) , and R'(r;,r) = dR(r;,r)/d(r;,r). The total field in region (II) (b < r < c) of permittivity E is

00

Ez II(r, z) = L Ro(r;,mr ) cos(amz) m=O

where e = Jk5 - (2 and ko = wJWo. The tangential E-field continuity condition at r = b requires

E (b z) _ {EZIl(b, z), zI, - 0,

-d<z<O otherwise.

Applying the Fourier cosine transform to (6) yields

EI(() = ~ ~~:;)b) Em(()

where

:;' (r) = ((-I)msin((d) ~m ., ((2 _ a~) .

The HI/> field continuity at r = b for -d < z < 0 is

HI/>I(b, z) + H~I(b, z) = HI/>Il(b, z) .

(4)

(6)

(7)

(8)

(9)

We multiply (9) by cos(asz) (8 = 0,1,2, ... ) and integrate with respect to z from -d to 0 to obtain

9.1 EM Radiation from an Aperture on a Shorted Coaxial Line [1]

where 2 ('Xl R'(K-b) ~ ~

It = -:; 10 K-R(K-b) '::s (()'::m (()d( .

Using the residue calculus, we get

dR'(K-b) I II = 2 R( b) €m 8ms

K- K- (=a",

00 i(( _1)m+s(1 _ e2i(d) I - ~ b[l- JJ(K-b)/JJ(K-a)]((2 - a;')((2 - a~) (=(n

ik(-I)m+s(l- e2ikd )

2bln (b/a)(k2 - a;')(k2 - a~)

where (n is given by R(K-b)k=(n = O. The E z field continuity at r = C is written as

E ( ) - {EZll(C, z), zIll c,z - 0 ,

-d<z<O otherwise.


- ~ Ro(K-mc) EllI (() = L...J (1) Gm (()

m=O Ho (~c)

where -i([1 - (_I)me- i (dj

Gm (() = (2 2 . -am

The H</> field continuity at r = C is given by

H</>Il(C, z) = H</>IlI(C, z) .

221

(10)

(11)

(12)

(13)

(14)

(15)

(16)

We multiply (16) by cos(asz) and integrate with respect to z from -d to 0 to get

(17)


where

00 H(l)'( ) h = ~ 1 - 0 ~c Gm(()Gs( -()d( .

271" -00 ~H~l)(~C) (18)

It is convenient to transform h into a numerically-efficient form by using the residue calculus. The evaluation of 12 is summarized in Subsect. 9.1.2 Appendix.

The scattered field at z = - 00 in region (I) is

H,p!(r,-oo) = (I+Lo)e-~z + fLn(()R'(K,r)ei(ZI_ (19) 'f} n=l (--(n

where 'f} is the intrinsic impedance ~, the eigenvalue (n is given by R(K,b)k=(n = 0, and

Lo = ksin(kd) f (-I) mRo(K,m b) (20) In(b/a) m=O (k2 - a~)

-2w€sin((d) ~ (-1)mRo(K,mb) Ln(() = bR'(K,b)[l- JJ(K,b)/JJ(K,a)] ~o ((2 _ a~)· (21)

The reflected power at z = -00 is

Pr = ~Re {127r lb ErI(r, z)H¢I(r, z)rdrd¢ }

= ~ In (b/a)ll + Lol2 . 'f}

(22)

(23)

The far-zone radiation field at distance R (r = R cos Os and z = R sin 0 s) is

H¢III(R, Os)

_ eikoR f w€ORo(K,m c) tanOs [( -1)m exp(ikodsinOs) - 1] (24)

- R m=O 71"H~1)(koccosOs)(k5sin20s-a~) .

9.1.2 Appendix

Consider

_ 1 JOO Hil)(~C)(2[1- (_I)me- i(d][l_ (_I)Sei(d] 12 - - ( ) d( .

271" -00 ~HOI (~C)((2 - a~)((2 - a;) (25)

When m + s is odd, 12 = O. When m + s is even, h is rewritten as

_ 1 JOO 2Hil)(~C)(2[1- (_I)mei (d] h - - ( ) d(.

271" -00 ~HOI (~C)((2 - a~)((2 - a;) (26)

Let's evaluate h along the contour path in the complex (-plane as shown in Fig. 1.2 of Sect. 1.1. The integrand has a pair of branch points corresponding

9.2 EM Radiation from Apertures on a Shorted Coaxial Line [3] 223

to e = 0 and two simple poles at ( = ±am when m = s. Integrating along the deformed contour path F1 , n, F3 , and F4 gives

_ d Hi 1) (ec) I 12 - '2 (1) E:m8ms + (13 + 14)

eHo (ec) (=a", (27)

where

koc[g(v)J2[(1 + iv)2 - (am /kO)2][(1 + iv)2 - (a s /ko)2]

JJ[g(v)c] ~NJ[9(v)c]dV (28)

(1 + iv)2 koc[g(v)J2[(1 + iv)2 - (am /kO)2][(1 + iv)2 - (a s /ko)2]

JJ[g(v)c] ~ NJ[g(v)c]dv (29)

g(v) = kov'v(-2i+v). (30)

d H(l)(ec) I Note that E:m'2 \1) '" O(1/ko) and (13 + 14) '" O(1/k5); hence

eHo (ec) (=a",

J. ~ d Hi 1) (ec) I . h· h f 1· . (k )

2 ~ E:mUms'2 (1) m 19 - requency 1m1t 0 -+ 00 . eHo (ec) (=a",

9.2 EM Radiation from Apertures on a Shorted Coaxial Line [3]

In Sect. 9.1 the radiation characteristics of a dielectric-filled edge-slot (DFES) antenna were studied. In the present section we will investigate a multiple dielectric-filled edge-slot antenna that is made by circumferentially and multiply removing the outer conductor of a shorted coaxial line. Consider a TEM wave exciting a multiple dielectric-filled edge-slot antenna consisting of an N number of periodic circumferential slots on the outer conductor of a coaxial line. The field representations in regions (I) and (III) are the same as those in Sect. 9.1. The field in region (II) is represented as

N-1 00 •

H¢Il(r, z) = L L 1WE ~(~mr) cosam(z + nT) n=O m=O ~m

·[u(z + d + nT) - u(z + nT)] (1)


Fig. 9.2 Multiple circumferential apertures on a shorted coaxial line

H (I)( ) H(2)( ) h Ro( ) - n 0 Ibmr n 0 Ibmr _ . I 2 2

were Ibmr - Pm (1) + qm (2) , Ibm - V k - am' k = HI (Ibm b) HI (lbmC)

WJJ-tfOfr, am = m1fjd, and u(·) is a unit step funCtion. A procedure of applying the boundary conditions is somewhat similar to

the dielectric-filled edge-slot antenna problem considered in Sect. 9.1, thus leading to

(2)

(3)

where 6ms is the Kronecker delta, em == 2(m= 0), 1 (m = 1,2, ... ), and

9.3 EM Radiation from Apertures on a Coaxial Line [4] 225

(4)

(5)

(6)

(7)

Note that ~ = Jk3 - (2 and ko = w.,fii€O is the wavenumber in region (III). When N = 1, (2) and (3) reduce to the single-aperture case considered in Sect. 9.1.

The reflection coefficient Fin from a feeding point at z = - (d + NT) is related to

Erl(r, z) I . Fin = Ei (r z) = -(1 + L~) exp[21k(d + NT)]

rl' z=-(d+NT)

(8)

n ~ ~ kRo(ll,mb) Lo = ~ ~ In(b/a)(k2 - a;")

.[(-I)m sink(d+nT) -sin(knT)]. (9)

The far-zone radiation field at distance R is

H (R () ) = ~ ~ iRo(ll,me)G~( -ko sin()s) eikoR (10) </JIll , s L...J L...J () H(l)(k () 'I1oR

n=O m=O 7rCOS s 0 oecos s "

where 'TIo = J 1-'/ EO is the intrinsic impedance of free-space and the observation point is at z = Rsin()s and r = R cos ()s'

9.3 EM Radiation from Apertures on a Coaxial Line [4]

Electromagnetic wave radiation from slots on a coaxial line is of practical interest in antenna and electromagnetic interference problems [5], In this section we will analyze electromagnetic wave radiation from a finite N number of circumferential periodic slots (apertures) on a coaxial line, A scattering analysis in this section is an extension of the single-aperture problem discussed in Sect, 9.1. An incident TEM wave is assumed to propagate along a slotted coaxial line. In region (I) (a < r < b and wavenumber: k) the incident and scattered H-fields are

(1)

(2)


Region (III) Region (II)

Fig. 9.3. Multiple apertures on a coaxial line

In region (III) (e < r and wavenumber: ko) the scattered H-field is

-iwfo 100 1 - (1) -i(z Hq,JlI(r, z) = 2;- -00 ~EJlI(()HI (~r)e d( (3)

where", = Jk2 - (2, 'f7 = VJi7€, R("'r) = Jo(",r)No(",a)-No(",r)Jo(",a), k =

wy1if = W.../J-tfrfO, R'(",r) = dR(",r)/d(",r), ~ = Jk3 - (2, and ko = wv'JifO. In region (II) (b < r < e and wavenumber: k) the H-field is

N-I 00 •

Hq,Jl(r,z) = L L IWfR~("'mr)cosam(z-nT) n=O m=O "'m ·[u(z - nT) - u(z - d - nT)] (4)

H (I)( ) H(2)( ) ( ) _ n 0 "'mr n 0 "'mr _ . /k2 2

where Ro "'mr -Pm (1) +qm (2) , "'m - V -am, am = HI ("'mb) HI ("'me)

m7r/d, and u(.) is a unit step function. The procedure for matching the boundary conditions is somewhat similar

to that in Sect. 9.1. By enforcing the boundary conditions on the tangential E and H-field continuities at r = band e, we obtain

9.3 EM Radiation from Apertures on a Coaxial Line [4] 227

(6)

where

(7)

(8)

(9)

It is convenient to transform hand [2 into numerically-efficient forms by utilizing the residue calculus.

The scattered field at z = ±oo in region (I) is

e±ikz 00 I Hq,I(r, ±oo) = L~-r- + L LT(()R'(lir)e±i(z

T} j=1 (=(j

(10)

where (j is determined by R(lib) = 0 and

L± = ~ f Ro(limb)G~(=t=k) o n=O m=O 2 In (b j a )

(11)

± _ ~1 ~ iWERo(limb)[l- (_I)me'fi(d] 'fi(nT Lj (() - =t= ~;;:o bR'(lib)[I- JJ(lib)jJJ(lia)]((2 _ a~) e . (12)

The reflection (PrJ Pi) and transmission (Pt/ Pi) coefficients are ILo 12 and

11 + Lt 12 , respectively. The far-zone radiation field at distance R (z = R cos B and r = RsinB) is


H¢II feR, 0) N-l 00 ik R

=LL~ n=O m=O 'floR

koRo(K,mc) [1- (-l)mexp(-ikodcosO)] exp(-ikonTcosO)

1TtanO H61) (kocsin 0) [(ko cos 0)2 - a~]

where 'flo = J /1/ EO is the intrinsic impedance of free-space.

9.4 EM Radiation from Apertures on a Coaxial Line with a Cover [6]

Fig. 9.4. Multiple apertures on a coaxial line with a dielectric cover

(13)

In Sect. 9.3 we have analyzed electromagnetic wave radiation from apertures on a coaxial line. In this section we will present an analysis of radiation from multiple circumferential apertures (slots) on a coaxial line covered with a dielectric layer. A similar scattering analysis for apertures on a dielectriccovered coaxial line was performed in [7-9]. A TEM wave is incident from left along a slotted coaxial line with a dielectric cover. Regions (I), (II), (III), and (IV) denote a coaxial line interior (a < r < b), apertures in the outer conductor (b < r < c), a dielectric layer (c < r < h), and a free space, respec-

9.4 EM Radiation from Apertures on a Coaxial Line with a Cover [6] 229

. eik1Z tively. The incident field components are assumed to be E~J(r, z) = -- and

r . eik1Z

H¢J(r,z) = -- where kl = wy'P,f.Of.rl and'T}l = ..jp,/(f.Of.rI). The scattered 'T}l r

fields in regions (I), (II), (III), and (IV) are

(1)

N-l 00 •

'"' '"' lWf.2 I 0 m7r H</>II(r,z) = L...J L...J ~RO(Kmr)cosd(z-nT) n=O m=O m

. [u(z - nT) - u(z - d - nT)] (2)

-iWf.4jOO 1 - 1 . H</>Jv(r,z) = -2- -EJv«()Hi \K4r )e-1(zd(

7r -00 K4 (4)

where R(Klr) = Jo(Klr)No(Kla) - No(Klr)Jo(Kla), Ki = ..jkl- (2, ki = H (I)( 0) H(2)( ° )

£ . N ( 0) n 0 Km r n 0 Kmr w.,fP,f.Of.ri or z = 1,2,3,4. ote Ro Km r = Pm (1) + qm (2) ,

HI (K~b) HI (K~C) K?n = ..jk~ - (m7r/d)2, and u(·) is a unit step function. The prime denotes differentiation with respect to an argument. Note that d denotes an aperture width and T is a distance between two adjacent apertures.

Following the Fourier transform procedure of matching the boundary conditions as was done in Sect. 9.3, we obtain

N-l 00 [H(I)( ° b) d 1 '"' '"' 0 Km iPnp + f.r2 _ em 8ml8n pn L...J L...J H(I)( ° b) ml f. 2 KO P m n=O m=O 1 Km rl m

N-l 00 [H(2)( ° b) d H(2)( ° b) 1 + '"' '"' 0 Km iPnp + f.r2 _ em 1 Km 8 18 n L...J L...J H(2)( ° ) ml f.rl 2 KO H(2)( ° ) m np qm n=O m=O 1 KmC m 1 KmC

= ~Gf(kl) (5)

N-l 00 [H(I)( ° ) d H(I)( ° ) 1 '"' '"' 0 Km C tJinp _ f.r2 _ em 1 Km C 8 18 pn L...J L...J H(I)( ° b) ml f. 3 2 KO H(I)( ° b) m np m n=O m=O 1 Km r m 1 Km

(6) N-l 00 [H(2)( ° ) d 1 '"' '"' 0 KmC tJinp _ f.r2 _ em 8 8 n = 0 + L...J L...J H(2) ( 0 ) ml f.r 3 2 KO ml np qm n=O m=O 1 Km C m


where Cm = 2 (m = 0), 1 (m = 1,2, ... ), 8ml is the Kronecker delta, and

~np = ~ ('" R'(/I",lb) Gn «(,)GP(-(,)d(, ml 211" i-oo /1",1 R (/I", 1 b) m l

00 () (1)( ) (1)'( ) tPnp _ ~ r A (, R1 /l",3 h + R1 /l",3 h

ml - 211" i-oo /1",3 [A«(,)R~O)(/I",3h) +R~O)'(/I",3h)]

·G~ «(,)Gf ( -(,)d(,

Rii) (/I",3r) = Hcl1) (/I",3r)HI2) (/1",3 c) - Hcl2) (/I",3r)Hl1) (/1",3 c) ,

H (l)( h) A«(,) = Er4/1",3 1 /1",4

Er3/1",4Hcl1) (/1",4 h)

Gn «(,) = -i(,[( _1)mei(d - 1] i(nT

m (,2 _ (m1l"/d)2 e .

The scattered fields at z = ±oo in regions (I) and (III) are

H ( ± ) = ~ ~ RO(/I",~b)G~(=Fk1) e±iklZ </>1 r, 00 L...J L...J 21n (b/a) r

n=O m=O 'TIl

+ ~ Lj«(,)R' (/I",l r )e±i(z I(=(i N-1 00 00 R ( 0 )

H</>IlI(r, ±oo) = ±WE3 L L L °a~c n=O m=Os=l -

a(,

·T(/I",3r)G~«(,)e-i(z k='F(s

(7)

(8)

i = 0,1 (9)

(10)

(11)

(12)

(13)

where (,j and (,S are determined by R(/I",lb) = 0 and ~ = 0, respectively, and

± ~ ~ WE1Ro(/I",~b)G~(=F(,) Lj «(,) = ~ ~ b(,R'(/I",l b)[l- JJ(/I",l b)/JJ(/I",la)]

~ = /1",3 [A«(,)R~O)(/I",3h) +R~O)'(/I",3h)]

T(/I",3r) = [Hi1) (/1",3 h) - A«(,)Hcl1) (/I",3h)]Hi2) (/I",3 r )

-[Hi2) (/1",3 h) - A«(,)Hcl2) (/I",3h)]Hi1) (/I",3r ) •

(14)

(15)

(16)

The solution at (, = ±('s represents a contribution from the surface wave propagating in the ±z directions. The reflection coefficient at z = 0 is

n = ~ ~ RO(/I",~b)G~(k1) m ~ ~o 2In(b/a) .

(17)

The far-zone radiation field at distance R (z = R cos 0 and r = R sin 0) is

9.5 EM Radiation from Apertures on a Circular Cylinder [10] 231

N-l 00 2 0 -( 0 )on (k ()) ik4R H</>IV(R'())=LL HOKm C ml-4COS e

n=O m=O 7r2g(())tan()H6 )(k4 hsin()) 114R

. [A(-k4COS())R~O)(g(())) +R~O)/(g(()))rl (18)

where g(()) = hy'k5 - ki cos2 () and 114 = y' /1,f(fOfr 4) is the intrinsic impedance of free space.

9.5 EM Radiation from Apertures on a Circular Cylinder [10]

z

..................... ....... ..... ........... ~7!f;~"" ......... . ........ . ...... .

77 ··.n:n~Tff~0ff.··· · ······ ~~z=~·i ::~:W6~d?J.LLi~iP8

Region (III) Region (II)

Fig. 9.5. Multiple apertures on a conducting circular cylinder

Electromagnetic wave radiation from circumferential apertures (slots) on a conducting cylinder was studied in [11-16] and applied to antenna and interference problems. A study of radiation from multiple apertures on a cylinder is useful for the design of microwave and millimeter wave array antenna of cylindrical structure. In this section we will analyze radiation from a finite (N) number of circumferential apertures on a conducting cylinder.


9.5.1 TE Radiation

Consider electromagnetic wave radiation from periodic circumferential apertures on a thick circular conducting cylinder. The incident TEO! wave is assumed to propagate along the cylinder. The scattered E-field has no variation in the 4>-direction due to a circular symmetry of the problem geometry. In region (I) (r < a) the total E-field is a sum of the incident and scattered components

(1)

(2)

where kc = 3.832/a, j3z = Jj32 - k~, 13 = w..jJ.Lf.rf.O = 2rr/>", and Ii =

J 132 - (2. In region (II) (a < r < b) the E-field is

N-1 00 •

E¢II(r,z) = L L -lWJ.LQ'(likr)sinak(z-nT) n=O k=1 lik ·[u(z - nT) - u(z - d - nT)]

where ~ = J 13'5 - (2 and 130 = w..[ii€O. The E¢ field continuity at r = a is given by

E ( ) _ { E¢II(a, z), ¢J a,z - 0 ,

nT < z < d+nT otherwise.

Taking the Fourier transform of (5) yields

EJ(() = - "E f: Q'(lika) liSr(() n=O k=1 lik J1 (lia)

where k iC;;d 1 =n(I")_ak[(-l)e -1 iC;;nT

~k ." - 1"2 2 e ." -ak

(3)

(5)

(6)

(7)

We multiply the Hz field continuity, H;J(a, z) + HzJ(a, z) = Hz II (a, z), by sina,(z - pT) and integrate with respect to z from pT to d + pT to get


(8)

where

h = 2- roo K,;o/,"~) S;:(()Sj( -()d( . (9) 2n i -00 1 K,a

Using the residue calculus, we transform h into a fast convergent series

d K,Jo(K,a) I h = -2 J ( ) 8kl8np

1 K,a (=ak

~. 2 A1 I (10) -akal ~ 1K, a(((2 _ a2)((2 _ a2) 3=1 k 1 (=(j

A1 = [( -1)k+1 + 1] exp(i(ln - pIT)

-( _l)k exp [i(ld + (n - p)Tll -( _1)1 exp [i(l(n - p)T - dll (11)

where (j are the roots of J1(K,a)k=(j = O. In addition, the tangential field continuities at r = b result in

(12)

where

12 = 2- roo ~H~1)(~b) sn(()SP(-()d( . 2n i-oo Hi1)(~b) k 1

(13)

Using the contour integral technique shown in Fig. 1.2 of Sect. 1.1, we transform h into a numerically-efficient integral

d~H(l)(~b) { 12 ="2 (~) 8kl 8np - akal [1 + (_l)k+l] h(gT)

H1 (~b) (=ak

-(-1)kI3 (d+gT)-(-1)II3 (-d+gT)} (14)

where 9 = n - p and


2 rOO ei.8olxle-.8ovlxl

13(x) = - 11"2 io ,83b[(1 + iv)2 - (ak/,80)2][(1 + iv)2 - (al/,8o)2] 1

dv. (15) Jl[,8obylv( -2i + v)] + Nf[,8obylv( -2i + v)]


(16)

where

N-1 00. Q'( ) 2[( 1)k 'fi(d 1] L=!=(() = L L lak K,k a K, - e - e'fi(nT

3 n=O k=l K,ka (Jo(K,a)((2 - aD (17)

and (j is determined by J1(Ka)k=(j = O. The reflection (ll), transmission (T), and scattering ((J') coefficients are

Pt T=-

Pi

= 11 + LT((-)12 +" k~Re(()J6(K,a) ILT(() 12 1 (18) 3 3 (j=.8. ~ K,2 Re(,8z)J6 (kca) 3

(j#,B. (=(j

Ps (J'=-Pi

II = Pr = f k~Re(()J~(K,a) IL-:-(() 12 1

Pi '-1 K,2 Re(,8z)Jo (kca) 3 3- (=~

(19)

(20)

where Pi, Ph Ps , and Pr are the incident, transmitted, scattered (radiated through apertures), and reflected powers, respectively, with

1I"a2wJ.t 2 Pi = 2:k2Jo (kca)Re(,8z)

c (21)

Ps = 1I"~d Re {f. ~ -!:J.t Q'(K,kb)Q*(K,kb)} (22)

The power conservation stipulates T + (J' + II = 1. The far-zone radiation field at distance R (z = R cos fJ and r = R sin fJ) is

E¢III(R, fJ)

ei.8oR N-1 00 Q'(K,k b) =~ LL 1I"K,k

n=O k=l

. {iwJ.tak[(-1)keXP(-i,80dCOSfJ) -1]eXP(-i,8onTCOSfJ)} (23) Hi1) (,8ob sin fJ)[(,8o cosfJ)2 - a%]


9.5.2 TM Radiation

Consider an incident TMOl wave traveling along a circular cylinder. The Hfield in region (I) (r < a) consists of

(24)

(25)

where kc = 2.405ja, j3z = Jj32 - k~, j3 = w..//l-fr€O, and ", = Jj32 - (2. In region (II) (a < r < b) the H-field is

N-l 00 •

H¢IJ(r, z) = L L IW€ ~("'mr) cosam(z - nT) n=O m=o"'m ·[u(z - nT) - u(z - d - nT)] (26)

H (I)( ) H(2)( ) n 0 "'mr n 0 "'mr . / 2

where RO("'mr) = Pm (1) + qm (2) , "'m = V j3 - a;', and HI ("'ma) HI ("'m b)

am = m 7r j d. In region (III) (r > b) the scattered field takes the form of

-iw€o roo 1 - (1) . H¢IJI(r,z) = ~ 1-

00 ~EIJI(()HI (~r)e-l(zd( (27)

where ~ = Jj33 - (2 and j30 = w../J.l€o. The E z and H¢ field continuities at r = a between regions (I) and (II)

yield

where

(29)

(30)

Using the residue calculus, we convert II into a fast convergent series


- d J1 (l\;a) I It = -2 J ( ) cmbm$bnp

I\; 0 I\;a (=a",

00 A I +.( 1 'L> a«(2 _ a2 )«(2 _ a2) 1=1 m $ (=(;

A1 = [(_l)m+$ + 1] exp(i(ln - pIT)

-( _1)m exp [i(ld + (n - p)TIl

-( _1)$ exp [i(l(n - p)T - dll

(31)

(32)

where (j are the roots of JO(l\;a)k=(j = O. In addition, the Ez and H¢ field continuities at r = b between regions (II) and (III) yield

(33)

where

12 = J..- /00 HP)(~b) Gn «()GP(-()d( . (34) 27f -00 ~H~1)(~b) m $

We transform 12 into a numerically-efficient integral

- _ d H~1)(~b) I { m+s -12 - 2" (1) cmbmsbnp - [1 + (-1) ]13 (gT)

~HO (~b) (=a",

-( _l)m 13(d + gT) - (_l)S 13 ( -d + gT)} (35)

where 9 = n - p and

2100 13(X) = 2"

7f 0

,85v( -2i + v)b[(l + iv)2 - (ami ,80)2][(1 + iv)2 - (asl,8o)2] 1

dv. (36) JJ[,8oby'v( -2i + v)] + NJ[,8oby'v( -2i + v)]


H¢J(r, ±oo) = i;LT«() -~f J1 (I\;r)e±i(z 1(=(; (37)

where

L±«() = =f ~1 f: Ro(l\;ma) t;;[( _l)me'fi(d - l]e'fi(nT

1 n=O m=O a h(t;;a)«(2 - a;') (38)


and (j is determined by JO(K;a)k=<i = O. The reflection (e), transmission (r), and scattering (u) coefficients are

Pt r=-

Pi

= 11 + Lt((-)12 + '"' k~Re(()Jr(K;a) ILt(()121 (39) J J <i=!3. ~ K;2 Re((3z)Jr (kca) J

<d!3z <=<i

Ps u=-

Pi

e = Pr = ~ k~Re(()Jr(K;a) IL-:-(() 12 1 p. ~ K;2Re((3 )J2(k a) J

• j=l z 1 c <=<i

where

7ra2w€ 2 Pi = --w-J1 (kca)Re((3z)

c

Ps = 7r~d Re {}; ~ iw:~m Ro(K;mb)R~(K;mb)* }

(40)

(41)

(42)

(43)

The far-zone radiation field at distance R (z = R cos 0 and r = R sin 0) is

H",III(R,O)

ei!3oR N-l 00 Ro(K;mb)

= ~ L L 7rtanO n=O m=O

. {W€O[I- (-I)m exp(-i(3odcosO)] exp(-i(3onT cos 0) } (44) H~l) ((3ob sin 0) [((30 cos 0)2 - a~]


1. J. K. Park and H. J. Eom, "Fourier transform analysis of dielectric-filled edgeslot antenna," Radio Sci., vol. 32, no. 6, pp. 2149-2154, Nov./Dec. 1997. Correction to "Fourier transform analysis of dielectric-filled edge-slot antenna," vol. 33, no. 3, May-June 1998.

2. D. L. Sengupta and L. F. Martins-Camelo, "Theory of dielectric-filled edge-slot antenna," IEEE Trans. Antennas Propagat., vol. 28, no. 4, pp. 481-490, July 1980.

3. J. K. Park and H. J. Eom, "Multiple dielectric-filled edge-slot antenna," Microwave Opt. Technol. Lett., vol. 21, no. 5, pp. 366-367, June 1999.

4. J. K. Park and H. J. Eom, "Radiation from multiple circumferential slots on coaxial cable," Microwave Opt. Technol. Lett., vol. 26, no. 3, pp. 160-162, Aug. 2000.

5. J. F. Kiang, "Radiation properties of circumferential slots on a coaxial cable," IEEE Trans. Microwave Theory Tech., vol. 45, no. 1, pp. 102-107, Jan. 1997.


6. J. K. Park and H. J. Earn, "Radiation from multiple circumferential slots on coaxial cable with a dielectric or plasma layer," J. Electromagn. Waves Appl., vol. 14, no. 3, pp. 359-368, 2000.

7. W. J. Dewar and J. C. Beal, "Coaxial-slot surface wave launchers," IEEE Trans. Microwave Theory Tech., vol. 18, no. 8, pp. 449-455, Aug. 1970.

8. J. R. Wait and D. A. Hill, "On the electromagnetic field of a dielectric coated coaxial cable with an interrupted shield," IEEE Trans. Antennas Propagat., vol. 23, no. 4, pp. 470-479, July 1975.

9. J. R. Wait and D. A. Hill, "Electromagnetic fields of a dielectric coated coaxial cable with an interrupted shield - Quasi-static approach," IEEE Trans. Antennas Propagat., vol. 23, no. 4, pp. 679-682, Sept. 1975.

10. J. K. Park and H. J. Eom, "Radiation from multiple circumferential slots on a conducting circular cylinder," IEEE Trans. Antennas Propagat., vol. 47, no. 2, pp. 287-292, Feb. 1999.

11. C. H. Papas, "Radiation from a transverse slot in an infinite cylinder," J. Math. Phys., vol. 28, pp. 227-236, Jan. 1950.

12. J. R. Wait, Electromagnetic Radiation from Cylindrical Structures, Elmsford, NY: Pergamon, 1959.

13. D. C. Chang, "Equivalent-circuit representation and characteristics of a radiating cylinder driven through a circumferential slot," IEEE Trans. Antennas Propagat., vol. 21, no. 6, pp. 792-796, Nov. 1973.

14. S. Papatheodorou, J. R. Mautz, and R. F. Harrington, "The aperture admittance of a circumferential slot in a circular cylinder," IEEE Trans. Antennas Propagat., vol. 40, no. 2, pp. 240-244, Feb. 1992.

15. C. M. Knop and L. F. Libelo, "On the leakage radiation from a circumferentially-slotted cylinder and its application to the EMI produced by TEM-coaxial rotary joints," IEEE Trans. Electromagn. Compat., vol. 37, no. 4, pp. 583-589, Nov. 1995.

16. S. Xu and X. Wu, "Millimeter-wave omnidirectional dielectric rod metallic grating antenna," IEEE Trans. Antennas Propagat., vol. 44, no. 1, pp. 74-79, Jan. 1996.

A. Appendix

A.1 Vector Potentials and Field Representations

Maxwell's equations in a time-harmonic case are

V x E =iwB-M

V x H = -iwD+J

V·D = Pe

V·B=Pm.

Based on the superposition principle, E and H are decomposed into

E=Ee+Em

H=He+Hm

(1)

(2)

(3) (4)

(5) (6)

where lEe and He are due to the electric current density J, and Em and H m

are due to the magnetic current density M, respectively. Maxwell's equations for E e and He are therefore

V x Ee = iWjlHe

V x He = -iwEEe +J

V ·De = Pe

V·Be =0.

Introducing the magnetic vector potential A such as

Be = V x A

and the Lorentz condition V . A = iWjlE<pe, we get

(V2 + k2 )A = -jlJ .

(7)

(8)

(9)

(10)

(11)

(12)

Note that <Pe is the electric scalar potential and k=wffi is the wavenumber. Similarly to determine Em and H m, we introduce the electric vector potential F where Dm = -V x F. We then obtain

(V2 + k2)F = -EM. (13)

Hence the total E and H are

240 A. Appendix

E = Ee + Em = iwA + _i_V(V. A) - ~V X F W/-l€ t:

(14)

H = He + Hm = .!.V X A + iwF + _i_V(V. F) . /-l w/-lE

(15)

Consider an electromagnetic wave propagation in the z-direction along an infinitely long waveguide with a uniform cross section. The vector potentials

z .......

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

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

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

y

Fig. A.I. Orthogonal coordinate system

in a source-free region of cylindrical waveguide satisfy the wave equations

(V2 + k2 )A = 0

(V2 + k2)F = 0 .

(16)

(17)

If we choose F = zFz(x, y, z) and A = 0, then Ez(x, y, z) = 0 and this type is called the TE wave (transverse electric to the wave propagation direction z). Similarly when A = zAz(x,y,z) and F = 0, we obtain the TM wave (transverse magnetic to the z-direction) with Hz(x,y,z) = O. The explicit field expressions in rectangular and cylindrical coordinates are shown in Tables A.l and A.2, respectively. If the wavenumber in the z-direction is k, then Ez = Hz = O. This type is referred to as the TEM wave.

A.1 Vector Potentials and Field Representations 241

Table A.I. Field representations in rectangular coordinates

Rectangular II TM wave TE wave

Ex i {)2 Az 1 {)Fz

WlJ,f {)x{)z f {)'11

Ey i {)2Az 1 {)Fz

WlJ,f {)'IJ{)z f {)x

. ({)2 2) Ez _1_ "jj2 + k 2 Az 0

WJl.f Z

Hx 1 {)Az i {)2 Fz

---IJ, {)'11 W/-tf {)x{)z

Hy 1 {)Az i {)2 Fz

---- ---IJ, {)x WlJ,f {)'II{)z

. ({)2 2) Hz 0 _1_ "jj2 + k 2 Fz

WJl.f Z

Table A.2. Field representations in circular cylindrical coordinates

Cylindrical II TM wave TE wave

Er i {)2 Az 1 {)Fz

WUf {)r{)z - fr iii

E</J i {)2 Az 1 {)Fz

WIJ,Er iii{)z f {)r

. ({)2 2) Ez _1_ "jj2 + k2 Az 0

WJl.f Z

Hr 1 {)Az i {)2 Fz

IJ,r iii ---WlJ,f {)r{)z

H</J 1 {)Az i {)2Fz

----WlJ,fr tiJ;{)z IJ, {)r

Hz 0 . ({)2 2) W~f {)Z2 + k2 Fz

Index

acoustic hybrid junction, 140 acoustic wave - double junction in rectangular

waveguide, 136 - hybrid junction, 140 - radiation from a circular aperture,

186 - radiation from circular apertures, 193 - scattering from a circular aperture,

176 - scattering from a rectangular

aperture, 153 - scattering from two circular

apertures, 187 annular aperture - electromagnetic radiation into a slab,

208 - electromagnetic scattering, 204 - electrostatic distribution, 199 - magnetostatic distribution, 202 - monopole antenna radiation, 213 antenna - dielectric-filled edge-slot, 220 - leaky wave, 11 - monopole, 213 - slotted coaxial line, 225 - slotted coaxial line with a dielectric

cover, 228 - slotted cylinder, 231

Bessel functions, XI

circular aperture - acoustic scattering, 176, 187 - electromagnetic scattering, 179 - electrostatic distribution, 173 - flanged acoustic, 186, 193 circular cylinder - corrugated, 106 circumferential aperture - on a coaxial line with a dielectric

cover, 228

- on a circular cylinder, 231 - on a coaxial line, 225 - on a shorted coaxial line, 220 coaxial line

corrugated, 97 radiation into a dielectric slab, 208

- radiation into a parallel-plate waveguide, 204 shorted with a cavity, 104

- with a cavity, 102 - with a gap, 101 corrugated circular cylinder, 105 corrugated waveguide, 17, 21, 105 cosine integral, 59 coupling coefficient, 195

dielectric-filled edge-slot antenna, 220 directional coupler - parallel-plate double slit, 111 - parallel-plate multiple slit, 115 dispersion relation - for groove guide, 93 - for inset dielectric guide, 89 - for multiple groove guide, 97 double bend, 133 double junction - acoustic rectangular waveguide, 136 - parallel-plate H-plane, 129

eigenvector, 180 - orthogonality, 182 electromagnetic penetration - into a double slit for directional

coupler, 111 - into a slitted parallel-plate waveguide,

74 - into a slitted rectangular cavity, 78 - into a circular aperture, 179 - into a slit, 63 - into multiple apertures in a cavity,

164

244 Index

- into multiple rectangular apertures, 160

- into multiple slits, 70 - into multiple slits for directional

coupler, 111 - into parallel slits, 82 electromagnetic radiation - from a parallel-plate waveguide into

a slab, 37 - from a coaxial line, 204 - from a coaxial line into a slab, 208 - from a DFES antenna, 220 - from a flanged parallel-plate

waveguide, 35 - from a monopole antenna, 213 - from apertures on a circular cylinder,

231 - from apertures on a coaxial line, 225 - from apertures on a coaxial line with

a dielectric cover, 228 - from grooves in a ground plane, 11 EM, XI

Fourier cosine transform, XII Fourier sine transform, XII Fourier transform, XII

Craf's addition theorem, 190, 195 Creen's formula, 49 groove - double, 21 - multiple, 6, 11, 17, 21, 29 - rectangular, 1 groove guide, 90 - multiple, 95

Hankel functions, XI Hankel transform, XIII hard surface, 140 Hertz vector potentials, 106 hybrid junction, 140 hybrid wave, 106

inset dielectric guide, 87 inverse transform, XII

Kirchhoff approximation, 3, 37 Kronecker delta, XI

leaky wave antenna, 11 Lorentz condition, 239

magic T, 140 magnetic polarizability, 153, 159

Maxwell's equations, 106, 239

parallel-plate waveguide - array, 47 - flanged, 35, 37 - flanged with window, 51 - obliquely-flanged, 48 - slitted, 74 PEe, XI polarizability, 157, 175, 176 - electric, 151, 202 - magnetic, 70, 204 potentials, 239 - electric vector potential, 239 - magnetic vector potential, 239

rectangular aperture - acoustic scattering, 153 - electromagnetic scattering, 160, 164 - electrostatic distribution, 150, 155 - magnetostatic distribution, 151, 157 reflection coefficient, 20, 41, 45, 51, 65,

100, 102, 104, 115, 117, 123, 124, 128, 132, 135, 139, 155, 179, 192, 195, 227, 230, 234, 237

residue calculus, 5, 10, 23, 62, 109, 206, 212,222

sine integral, 59 slit - electromagnetic scattering, 63, 70,

74, 78, 82 - electrostatic distribution, 57, 59 - magnetostatic distribution, 68 soft surface, 144 superposition principle, 60, 188, 239

T-junction - parallel-plate E-plane, 123 - parallel-plate H-plane, 122 - rectangular waveguide E-plane, 125 TE wave, 1, 240 TEM wave, 240 TM wave, 1, 240 transmission coefficient, 20, 41, 45, 48,

51,65,67, 72, 74, 100, 104, 115, 117, 123, 124, 128, 132, 135, 139, 155, 164, 179, 192, 227, 234, 237

unit vectors, XI

water wave scattering, 29 wave equation, 240

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