Dyadic Green’s function of an ideal hard surface circular waveguide …yakovlev/... · 2013. 2....

Dyadic Green’s function of an ideal hard surface

circular waveguide with application

to excitation and scattering problems

Victor A. Klymko, Alexander B. Yakovlev, Islam A. Eshrah,

Ahmed A. Kishk, and Allen W. Glisson

Center for Applied Electromagnetic Systems Research, Department of Electrical Engineering, University of Mississippi,University, Mississippi, USA

Received 1 September 2004; revised 28 February 2005; accepted 11 March 2005; published 22 June 2005.

[1] Green’s function analysis of ideal hard surface circular waveguides is proposed withapplication to excitation and scattering problems. A decomposition of the hard surfacewaveguide into perfect electric conductor and perfect magnetic conductor waveguidesallows the representation of dyadic Green’s function in terms of transverse electric (TE)and transverse magnetic (TM) waveguide modes, respectively. In addition, a termcorresponding to a transverse electromagnetic (TEM) mode is included in therepresentation of the Green’s dyadic. The TEM term is extracted in closed form from theeigenmode expansion of TM and TE modes in the zero-cutoff limit. The electric fielddistribution due to an arbitrarily oriented electric dipole source is illustrated forrepresentative TM, TE, and TEM modes propagating in the ideal hard surface circularwaveguide. The derived Green’s function is used in the method of moments analysis of anideal hard surface waveguide excited by a half-wavelength strip dipole antenna. Inaddition, the scattering of the TEM mode by a thin strip is studied in the ideal hard surfacecircular waveguide.

Citation: Klymko, V. A., A. B. Yakovlev, I. A. Eshrah, A. A. Kishk, and A. W. Glisson (2005), Dyadic Green’s function of an

ideal hard surface circular waveguide with application to excitation and scattering problems, Radio Sci., 40, RS3014,

doi:10.1029/2004RS003167.

1. Introduction

[2] Artificial surfaces with anisotropic impedance,characterized as hard and soft surfaces, have beenintroduced in electromagnetics in the past two decades.A planar hard surface is realized by loading a conductingsurface with dielectric-filled corrugations in the directionof wave propagation, whereas a soft surface is imple-mented by using transverse corrugations with respect tothe propagating direction [Kildal, 1988a, 1990]. A planarhard surface can also be realized by using longitudinalperiodic metal strips placed on the dielectric coating of ametal surface [Aas, 1991]. A hard surface cylindricalwaveguide is implemented by introducing longitudinaldielectric-filled corrugations [Kildal, 1990] or by liningthe perfect electric conductor (PEC) waveguide with adielectric coating loaded with narrow longitudinal PECstrips [Kishk and Morgan, 2001].

[3] Different approaches have been used for the anal-ysis of scattering from soft and hard surfaces. Lindell[1995] applied the image theory for a planar anisotropicsurface by decomposing the original source above thesurface in two components to generate the transverseelectric (TE) and transverse magnetic (TM) polarizedfields with respect to the direction of corrugations. Anopen hard surface is studied by Sipus et al. [1996] byconsidering a grounded dielectric slab illuminated by aspectrum of plane waves with no component of theelectric field in the direction of propagation alongthe surface. The three-dimensional scattering from theinfinite and semi-infinite plain surfaces with anisotropicimpedance is analyzed by Nepa et al. [2001], andscattering from the edges of anisotropic impedancesurfaces is studied by Manara et al. [2000]. The asymp-totic boundary conditions (ASBC) are applied by Kishket al. [1998] instead of using the Floquet mode expan-sions for the problem of scattering from a cylinder withdielectric-filled corrugations. The scattering from dielec-tric cylinders loaded with longitudinal or circumferential

RADIO SCIENCE, VOL. 40, RS3014, doi:10.1029/2004RS003167, 2005

Copyright 2005 by the American Geophysical Union.

0048-6604/05/2004RS003167$11.00

RS3014 1 of 15

metal strips illuminated at oblique incidence is studied byKishk and Kildal [1997] by using the ASBC approach.Also, the ASBC has been used by Kishk [2003] in theanalysis of cylindrical hard surfaces of arbitrarily shapedcross section.[4] Artificial hard surfaces are primarily utilized in

hard horn antennas in order to create a transverseelectromagnetic (TEM) wave propagation regime witha uniform field distribution over the horn aperture andzero cross-polarization [Kildal, 1988b]. The use of a hardhorn antenna increases directivity and decreases beamwidth for given aperture dimensions [Lier and Kildal,1988]. Also, an infinite planar array of open-endedrectangular waveguides with dielectric-loaded wallsresults in a better matching of the array to free spaceand increase of the aperture and gain to remove the scanblindness [Skobolev and Kildal, 1998]. The performanceof an open-ended circular waveguide with strip-loadeddielectric hard walls used as the element of antennaarrays is also studied by Skobolev and Kildal [2000].[5] Recent developments in the application of hard

surfaces are reported by Yang et al. [1999], wherein aphotonic band gap (PBG) structure implemented onsidewalls of rectangular waveguide has been used tocreate the TEM mode propagation. Also, a rectangularwaveguide with sidewalls covered by the printed dipole

frequency selective surface (FSS) has been designed toobtain the quasi-TEM mode [Cucini et al., 2004; Maci etal., 2005].[6] In this work, Green’s function analysis of an ideal

hard surface circular waveguide is proposed in order tocreate a TEM wave propagation with application toexcitation and scattering problems. The modal charac-terization of an ideal hard surface waveguide gives aphysical insight into the behavior of modes supported bypractical hard surface waveguides. The developedGreen’s function provides a powerful framework forthe efficient analysis of the ideal hard surface circularwaveguide, which serves as an initial step in the analysisand design of waveguides with physical hard surface. Inthe proposed approach, the ideal structure is decomposedinto PEC and perfect magnetic conductor (PMC) wave-guides, which independently support TM and TE modes,respectively. In addition, it is shown that the TEM modewith a uniform field propagates in the ideal hard surfacecircular waveguide. On the basis of the modal analysis ofthe structure, the electric dyadic Green’s function due toan arbitrarily oriented electric dipole source is obtainedin the eigenmode expansion form in terms of TM modesof the PEC waveguide, TE modes of the PMC wave-guide, and the TEM mode. The TEM term of the Green’sdyadic is extracted from the eigenmode expansion in thezero-cutoff limit. The electric field distribution due to anarbitrarily oriented electric dipole source is illustrated forrepresentative TM, TE, and TEM modes propagating inthe ideal hard surface circular waveguide. The Green’sfunction derived is used in the method of momentsanalysis of ideal hard surface waveguide excitation bya half-wavelength strip dipole antenna. In addition, thescattering of the TEM mode by a thin strip is studied inthe ideal hard surface circular waveguide.

2. Dyadic Green’s Function of an Ideal

Hard Surface Circular Waveguide

[7] The ideal hard surface circular waveguide is mod-eled by alternating PEC and PMC longitudinal strips (inthe propagating direction) with vanishing widths [Ruvioet al., 2003]. The PEC and PMC strips enforce theannulment of the longitudinal electric and magneticfields, respectively. This results in decomposition ofthe original hard surface waveguide into PEC andPMC waveguides, which individually support TM andTE waveguide modes, respectively. In addition, a hardsurface waveguide as a multiconductor transmission linealso supports a TEM mode. This mode is obtained as acontribution of TM and TE modes of PEC and PMCwaveguides in the zero-cutoff limit.[8] The concept of the ideal hard surface circular

waveguide as a composition of PEC and PMC wave-guides is illustrated in Figure 1.

Figure 1. PEC/PMC strip model for an ideal hardsurface circular waveguide. The hard surface behaveslike a PEC waveguide for TM modes and as a PMCwaveguide for TE modes. A TEM mode is obtained as acontribution of TM and TE modes in the zero-cutofflimit. The TEM mode ‘‘sees’’ the ideal hard surfacewaveguide as a combination of PEC and PMCwaveguides, and it satisfies the boundary conditions onthe ideal hard surface partially contributed by PEC andPMC waveguides, that is, @Er /@rjS = 0, @E� /@rjS = 0.See color version of this figure at back of this issue.

RS3014 KLYMKO ET AL.: HARD SURFACE WAVEGUIDE

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[9] In Figure 1, r is the normal direction to the wave-guide boundary and f is the tangential direction to thewaveguide boundary and transverse to the longitudinalstrips. When the hard surface behaves like a PEC conduc-tor, Ef and the normal derivative of Er are zero on thewaveguide surface S, that is,EfjS = 0 and @Er/@rjS = 0, andwhen the hard surface behaves like a PMC conductor, Erand the normal derivative of Ef are zero on the waveguidesurface S, that is, ErjS = 0 and @Ef/@rjS = 0, and vice versafor the magnetic field components.[10] Consider an ideal hard surface circular waveguide

with the impressed electric current source shown inFigure 2. The electric field in the waveguide (includingthe source region) is expressed in the integral form,

E rð Þ ¼ �jwm0ZVimp

G r; r0ð Þ � J r0ð ÞdV 0; ð1Þ

where G(r, r0) is the electric dyadic Green’s function ofthe ideal hard surface circular waveguide obtained as thesolution of a dyadic wave equation,

rrG r; r0ð Þ � k20G r; r0ð Þ ¼ Id r� r0ð Þ; ð2Þ

where k0 = wffiffiffiffiffiffiffiffiffie0m0

p.

[11] The dyadic Green’s function is obtained in theeigenmode expansion form as a superposition of threeparts associated with the TM modes of the PEC wave-guide, TE modes of the PMC waveguide, and the TEMmode of the ideal hard surface,

G r; r0ð Þ ¼ GTM r; r0ð Þ þGTE r; r0ð Þ þGTEM r; r0ð Þ: ð3Þ

[12] This corresponds to decomposition of the bound-ary value problem for the Green’s function into threeproblems. Thus GTM(r, r0) part of the Green’s functionsatisfies the first-kind boundary conditions on S (equiv-alent to those for the tangential electric field on PEC),

^̂RGTM r; r0ð Þ��S¼ 0; ð4Þ

where ^̂R is the normal in the radial direction (incylindrical coordinates) to S; GTE(r, r0) part satisfiesthe second-kind boundary conditions on S (equivalent tothe behavior of electric field on PMC),

^̂RrGTE r; r0ð ÞjS ¼ 0

^̂R �GTE r; r0ð Þ��S¼ 0: ð5Þ

[13] The TEM part of the Green’s function satisfiesmixed boundary conditions, which are equivalent tothose for normal and tangential components of theelectric field, @Er /@rjS = 0 and @Ef /@rjS = 0, (shownin Figure 1),

^̂R � r ^̂R �GTEM r; r0ð Þ� ��

S¼ 0

^̂R � r F̂ �GTEM r; r0ð Þ� ��

S¼ 0:

ð6Þ

[14] The electric Green’s dyadic of the ideal hardsurface waveguide is expressed in terms of solenoidaland irrotational parts [Collin, 1991; Johnson et al.,1979],

G r; r0ð Þ ¼ 12jwm0

PV ETEM rð ÞETEM r0ð Þ�

þX1s¼0

X1k¼1

ETMsk rð ÞETMsk r

0ð Þ�

þETEsk rð ÞETEsk r

0ð Þ

� L d r� r0ð Þ

k20: ð7Þ

[15] The solenoidal part is obtained in the form ofeigenmode expansion of TM, TE, and TEM modes andunderstood in the principal value (PV) sense [Yaghjian,1982; Chew, 1989; Eshrah et al., 2004]. The irrotationalpart includes a depolarizing dyadic L associated with aspecific principal exclusion volume [Yaghjian, 1980]. Aslice-pillbox principal volume with the normal in thepropagating z direction (Figure 2) [Eshrah et al., 2004;Yaghjian, 1980; Wang, 1982; Viola and Nyquist, 1988] isused in this formulation with L = ẑẑ. The main singularityof the Green’s function is contained in the solenoidal partand represents a source-plane singularity at z = z0. It isunderstood as an improper integral with the slice-pillboxprincipal volume.

Figure 2. Geometry of an ideal hard surface circularwaveguide with the impressed electric current source. Aslice-pillbox principal volume associated with thesource-plane singularity of the Green’s function at z =z0 is shown in the source region.


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[16] The electric field inside the waveguide given by(1) is uniquely defined as the sum of the improperintegral of the solenoidal part understood in the principalvalue sense and the integral of the irrotational part overthe entire volume of the impressed current source,

E rð Þ ¼ � 12limd!0

ZVimp�Vd

ETEM rð ÞETEM r0ð Þ�

þX1s¼0

X1k¼1

ETMsk rð ÞETMsk r

0ð Þ þ ETEsk rð ÞETEsk r

0ð Þ�

� J r0ð ÞdV 0 � ẑ Jz rð Þjwe0

: ð8Þ

[17] Below we derive the electric and magnetic fieldsof TM and TE eigenmodes of PEC and PMC wave-guides, respectively. The electric fields of TM and TEeigenmodes propagating in the positive, ‘‘+’’, and neg-ative, ‘‘�’’, z direction, EskTM±(r) and EskTE±(r), areexpressed in terms of electric vector wave functions[Collin, 1991],

ETM�sk rð Þ ¼ eTMsk r;fð Þ � ẑez;sk r;fð Þ�

e�gsk z

ETE�sk rð Þ ¼ eTEsk r;fð Þe

�gsk z: ð9Þ

[18] The corresponding magnetic fields of TM and TEeigenmodes are obtained in terms of magnetic vectorwave functions,

HTM�sk rð Þ ¼ �hTMsk r;fð Þe�gsk z

HTE�sk rð Þ ¼ �hTEsk r;fð Þ þ ẑhz;sk r;fð Þ

� e�gsk z: ð10Þ

[19] The longitudinal components of electric and mag-netic fields, ez,sk (r, f) and hz,sk (r, f), satisfy Helmholtz’sequation in cylindrical coordinates,

@2

@r2ez;sk r;fð Þ þ

1

r@

@rez;sk r;fð Þ

þ 1r2

@2

@f2ez;sk r;fð Þ þ k2c;skez;sk r;fð Þ ¼ 0

@2

@r2hz;sk r;fð Þ þ

1

r@

@rhz;sk r;fð Þ

þ 1r2

@2

@f2hz;sk r;fð Þ þ k2c;skhz;sk r;fð Þ ¼ 0: ð11Þ

[20] The solution of (11) is obtained by separating thevariables,

ez;sk r;fð Þ ¼ aTMsk Js kc;skr�

cos sfð Þþ bTMsk Js kc;skr

� sin sfð Þ

hz;sk r;fð Þ ¼ aTEsk Js kc;skr�

sin sfð Þþ bTEsk Js kc;skr

� cos sfð Þ ð12Þ

where Js is the Bessel function of the first kind of order sand kc,sk are the cutoff wave numbers of TM and TEmodes determined as roots of the characteristic equation,

Js kc;ska�

¼ 0: ð13Þ

This implies that the ideal hard surface circularwaveguide supports degenerate TM and TE modes withthe same cutoff wave numbers and the same propagation

constants, gsk =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2c;sk � k20

q.

[21] The solution in (12) represents the superpositionof two orthogonal polarizations (associated with the sineand cosine terms). Since one polarization can be obtainedfrom the other by 90 rotation of coordinates, only onepolarization will be considered in the derivations tofollow. Thus the transverse electric vector wave func-tions are expressed in terms of ez,sk and hz,sk,

eTMsk ¼ �gskk2c;sk

rtez;sk

¼ gskkc;sk

��^̂RaTMsk J 0s kc;skr

� cos sfð Þ

þ F̂ skc;skr

aTMsk Js kc;skr�

sin sfð Þ

ð14Þ

eTEsk ¼jwm0k2c;sk

ẑrthz;sk

¼ jwm0kc;sk

� ^̂R skc;skr

aTEsk

�Js kc;skr�

cos sfð Þ

þ F̂aTEsk J 0s kc;skr�

sin sfð Þ

ð15Þ

and the transverse magnetic vector wave functions areobtained as follows,

hTMsk ¼ �jwe0k2c;sk

ẑrtez;sk

¼ jwe0kc;sk

�� ^̂R s

kc;skraTMsk Js kc;skr

� sin sfð Þ

� F̂aTMsk J 0s kc;skr�

cos sfð Þ

ð16Þ

hTEsk ¼ �gskk2c;sk

rthz;sk

¼ gskkc;sk

�� ^̂RaTEsk J 0s kc;skr

� sin sfð Þ

� F̂ skc;skr

aTEsk Js kc;skr�

cos sfð Þ

; ð17Þ

where J 0s(kc,skr) = @@ kc;skrð ÞJs(kc,skr).


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[22] The transverse electric and magnetic vector wavefunctions (14)–(17) are normalized by power,

ZSw

eTM ;TEsk h

TM ;TEsk

� � ẑdS ¼ 1; ð18Þ

where Sw is the waveguide cross section.[23] The TE and TM cases are normalized separately

resulting in the following expressions for the unknowncoefficients. Thus for TM modes,

aTMsk ¼

ffiffiffiffiffiffiffiffiffiZTM0k

pkc;0k=g0k

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip J2 kc;0ka

� � 2� 2J2 kc;0ka� J1 kc;0ka� kc;0ka

þ J1 kc;0ka� � 2� �s ; s ¼ 0

ffiffiffiffiffiffiffiffiffiZTMsk

pkc;sk=gsk

a

ffiffiffip2

rJ 0s kc;ska� ; s 6¼ 0

8>>>>>>>>><>>>>>>>>>:

ð19Þ

and for TE modes the coefficients are

aTEsk ¼

kc;0k=g0k

affiffiffiffiffiffiffiffiZTE0k

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip J2 kc;0ka

� � 2� 2J2 kc;0ka� J1 kc;0ka� kc;0ka

þ J1 kc;0ka� � 2� �s ; s ¼ 0

kc;sk=gsk

affiffiffiffiffiffiffiffiZTEsk

p ffiffiffip2

rJ 0s kc;ska� ; s 6¼ 0;

8>>>>>>>><>>>>>>>>:

ð20Þ

where ZskTM =

gskjwe0

and ZskTE =

jwm0gsk

are the wave impedances

of TM and TE modes, respectively.[24] The coefficients given by (19) and (20) can be

written in the form

aTMsk ¼kc;sk

ffiffiffiffiffiffiffiffiffiZTMsk

pgsk

csk ;aTEsk ¼kc;sk

gskffiffiffiffiffiffiffiffiZTEsk

p csk ; ð21Þwhere

csk ¼1.a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip J2 kc;0ka

� � 2� 2J2 kc;0ka� J1 kc;0ka� kc;0ka

þ J1 kc;0ka� � 2� �s

; s ¼ 0

1.a

ffiffiffip2

rJ 0s kc;ska�

; s 6¼ 0:

8>>><>>>:

ð22Þ

[25] Following the same procedure for the other polar-ization, it can be easily shown that the coefficients bsk areequal to ask for both TE and TM modes. Substituting (9)together with (12), (14), (15), and the correspondingexpressions of the other polarization into the eigenmodeexpansion (7), we obtain the representation for the nine

components of the electric dyadic Green’s function of theideal hard surface circular waveguide,

Grr r; r0ð Þ ¼ 1

2jwm0PV GTEMrr þ

X1s¼0

X1k¼1

c2sk

(

��ZTMsk J

0s kc;skr�

J 0s kc;skr0� þ s2

k2c;skZTEsk

�Js kc;skr�

Js kc;skr0�

rr0

� e�gsk jz�z0j cos s f� f0ð Þð Þ)

ð23Þ

Grf r; r0ð Þ ¼ 1

2jwm0PV GTEMrf þ

X1s¼0

X1k¼1

c2sks

(

��ZTMsk J

0s kc;skr� Js kc;skr0�

kc;skr0þ ZTEsk

Js kc;skr� kc;skr

� J 0s kc;skr0� e�gsk jz�z0 j sin s f� f0ð Þð Þ

)ð24Þ

Grz r; r0ð Þ ¼ 1

2jwm0PV sign z� z0ð Þ

X1s¼0

X1k¼1

c2skZTMsk

kc;sk

gsk

(

� J 0s kc;skr�

Js kc;skr0�

� e�gsk jz�z0 j cos s f� f0ð Þð Þ)

ð25Þ

Gfr r; r0ð Þ ¼ 1

2jwm0PV GTEMfr �

X1s¼0

X1k¼1

c2sks

(

� ZTMsk J 0s kc;skr0� Js kc;skr�

kc;skrþ ZTEsk

Js kc;skr0� kc;skr0

�

� J 0s kc;skr�

e�gsk jz�z0 j sin s f� f0ð Þð Þ

)ð26Þ

Gff r; r0ð Þ ¼ 1

2jwm0PV GTEMff þ

X1s¼0

X1k¼1

c2sk

(

��

s2

k2c;skZTMsk

Js kc;skr�

Js kc;skr0�

rr0

þ ZTEsk J 0s kc;skr�

J 0s kc;skr0�

� e�gsk jz�z0j cos s f� f0ð Þð Þ)

ð27Þ


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Gfz r; r0ð Þ ¼ � 1


X1s¼0

X1k¼1

c2skZTMsk

s

gsk

(

�Js kc;skr� r

Js kc;skr0�

e�gsk jz�z0j

� sin s f� f0ð Þð Þ)

ð28Þ

Gzr r; r0ð Þ ¼ � 1


X1s¼0

X1k¼1

c2skZTMsk

(

� kc;skgsk

Js kc;skr�

J 0s kc;skr0� e�gsk jz�z0 j

� cos s f� f0ð Þð Þ)

ð29Þ

Gzf r; r0ð Þ ¼ � 1


X1s¼0

X1k¼1

c2skZTMsk

s

gsk

(

�Js kc;skr0� r0

Js kc;skr�

e�gsk jz�z0j

� sin s f� f0ð Þð Þ)

ð30Þ

Gzz r; r0ð Þ ¼ � 1

2jwm0PV

X1s¼0

X1k¼1

c2skZTMsk

k2c;sk

g2skJs kc;skr� (

� Js kc;skr0�

e�gsk jz�z0 j cos s f� f0ð Þð Þ

)

� d r� r0ð Þ

k20: ð31Þ

[26] The TEM term included in expansion (7) of thedyadic Green’s function is obtained in closed form as acontribution of TM and TE waveguide modes in thezero-cutoff limit, kc,sk ! 0. It is shown below that theTEM part of the Green’s function has only transversecomponents, Grr

TEM, GrfTEM, Gfr

TEM, and GffTEM.

[27] The zero-cutoff limit corresponds to the zero rootof the characteristic equation (13) (except for s = 0, thatis, limkc,sk!0 J0(kc,ska) = 1), which represents an addi-tional solution in the series expansion of TM and TEmodes associated with the TEM mode with the propa-gation constant g0 = limkc,sk!0 gsk = jk0.[28] The transverse electric and magnetic vector wave

functions of the TEM mode are obtained in the zero-

cutoff limit of the corresponding expressions of TM andTE modes, (14)–(17),

eTM0 ¼ jk0ATM lim

kc;sk!0

�� ^̂RJ 0s kc;skr

� cos sfð Þ

þ F̂sJs kc;skr� kc;skr

sin sfð Þ

eTE0 ¼ jwm0ATE lim

kc;sk!0� ^̂Rs

Js kc;skr� kc;skr

cos sfð Þ�

þ F̂J 0s kc;skr�

sin sfð Þ

ð32Þ

hTM0 ¼ jwe0ATM limkc;sk!0

� ^̂RsJs kc;skr� kc;skr

sin sfð Þ�

� F̂J 0s kc;skr�

cos sfð Þ

hTE0 ¼ jk0ATE lim

kc;sk!0� ^̂RJ 0s kc;skr

� sin sfð Þ

�� F̂s

Js kc;skr� kc;skr

cos sfð Þ

; ð33Þ

where the coefficients ATM and ATE are defined asfollows,

ATM ¼ limkc;sk!0

aTMskkc;sk

¼ limkc;sk!0

cskffiffiffiffiffiffiffiffiffiZTMsk

pgsk

ð34Þ

ATE ¼ limkc;sk!0

aTEskkc;sk

¼ limkc;sk!0

csk

gskffiffiffiffiffiffiffiffiZTEsk

p¼ lim

kc;sk!0

ATMffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZTMsk Z

TEsk

p ¼ ATMh0

; ð35Þ

with h0 =ffiffiffiffim0e0

qbeing the intrinsic impedance of free

space.[29] Bessel functions and their derivatives in (32) and

(33) are expanded in Taylor series,

Js kc;skr�

¼kc;skr� s2ss!

�kc;skr� sþ22sþ2 sþ 1ð Þ!þ :: ð36Þ

J 0s kc;skr�

¼kc;skr� s�12s s� 1ð Þ! �

sþ 2ð Þ kc;skr� sþ1

2sþ2 sþ 1ð Þ! þ ::; ð37Þ

which are used to find the limit of the followingexpressions,

limkc;sk!0

Js kc;skr� kc;skr

¼1=2; s ¼ 1

0; s > 1

8<: ð38Þ


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limkc;sk!0

J 0s kc;skr�

¼1=2; s ¼ 1

0; s > 1:

8<: ð39Þ

[30] Taking into account (38) and (39) in expressions(32) and (33) we obtain the representation of electric andmagnetic fields of the TEM mode (given in terms of TMand TE parts),

eTM0 ¼1

2jk0A

TM � ^̂R cos fð Þ þ F̂ sin fð Þn o

eTE0 ¼1

2jwm0

ATM

h0� ^̂R cos fð Þ þ F̂ sin fð Þ

n o ð40Þ

hTM0 ¼1

2jwe0ATM � ^̂R sin fð Þ � F̂ cos fð Þ

n o

hTE0 ¼1

2jk0

ATM

h0� ^̂R sin fð Þ � F̂ cos fð Þ

n o;

ð41Þ

which are normalized by power,ZSw

eTM0 þ eTE0�

hTM0 þ hTE0� �

� ẑdS ¼ 1; ð42Þ

resulting in the expression for ATM,

ATM ¼ 1jk0a

ffiffiffiffiffih0p

r: ð43Þ

[31] On the basis of the above derivations, the TEMterm of the Green’s function is obtained as the superpo-sition of dyadic products of two orthogonal polarizations(sine and cosine angular solutions) of TM and TE modesin the zero-cutoff limit,

GTEM

f; z;f0; z0ð Þ ¼ PV eTM ;cos0 fð ÞeTM ;cos0 f

0ð Þ�

þ eTM ;sin0 fð ÞeTM ;sin0 f

0ð Þþ eTE;cos0 fð Þe

TE;cos0 f

0ð Þþ eTE;sin0 fð Þe

TE;sin0 f

0ð Þg e�jk0 z�z0j j

ð44Þ

or in terms of Green’s function components,

GTEMrr f; z;f0; z0ð Þ ¼ GTEMff f; z;f

0; z0ð Þ

¼ PV h02pa2

cos f� f0ð Þe�jk0 z�z0j j ð45Þ

GTEMrf f; z;f0; z0ð Þ ¼ �GTEMfr f; z;f

0; z0ð Þ

¼ PV h02pa2

sin f� f0ð Þe�jk0jz�z0j; ð46Þ

which are included in the representations (23), (24), (26),and (27).

3. Excitation of Ideal Hard Surface

Circular Waveguide Modes by an Electric

Dipole Source

[32] The electric dyadic Green’s function derived in theprevious section has been used to obtain the electric fieldof representative TM, TE, and TEM modes excited in theideal hard surface circular waveguide by an electricdipole source. Figures 3, 4, and 5 demonstrate thetransverse electric field of TM01, TE01, and TEM modesat the distance of z = lg/2 (lg of the correspondingmode) from the source plane at z0 = 0. The modespropagate in the ideal hard surface circular waveguideof radius a = 11 mm.[33] Figure 3 shows the transverse electric field of the

TM01 mode at 12 GHz due to the r-directed electricdipole source, J(r) = R̂ > 1rd(r �

a2)d(f � p

2)d(z).

Obviously, the electric field distribution of this and, ingeneral, all TMsk modes is the same as in the PECwaveguide.

Figure 3. Transverse electric field of the TM01 modeplotted at z = lg

TM01/2 = 25.342 mm due to the r-directedelectric dipole at z0 = 0 mm.


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[34] The transverse electric field of the TE01 mode atthe same frequency due to the f-directed electric dipole,J(r) = F̂ 1rd(r �

a2)d(f)d(z), is shown in Figure 4. It should

be noted that the electric field of TEsk modes propagatingin the ideal hard surface waveguide (as in the PMCwaveguide) is similar to the distribution of magnetic fieldof the corresponding TMsk modes (propagating in thePEC waveguide).[35] Figure 5 demonstrates the transverse electric field

of the TEM mode at 10 GHz due to the f-directedelectric dipole, J(r) = F̂ 1rd(r �

a2)d(f � p

4)d(z). It can be

seen that the field is uniform over the waveguide crosssection, and polarization of the field depends on polar-ization of the source. Clearly, the ideal hard surfacecircular waveguide supports a TEM mode polarizedaccording to the orientation of the source.

4. Excitation of an Ideal Hard Surface

Circular Waveguide by a Strip Dipole

Antenna

[36] The Green’s function derived in Section II is usedin the method of moments analysis of hard surfacewaveguide excitation by a half-wavelength strip dipoleantenna (with geometry shown in Figure 6). The dipolelength is 2L = l0/2 and dipole width is 2b = 0.25 mm.The voltage delta gap source of 1 V is at the center of thedipole.

[37] The electric field integral equation for the currentinduced along the dipole (assuming only Jy componentof the current for a thin strip dipole) is obtained byenforcing a boundary condition for the total tangentialelectric field,

� jwm0ŷ �Z dþbd�b

Z L�L

^̂RGrr ^̂R0 þ ^̂RGrfF̂

0 þ F̂Gfr ^̂R0n

þ F̂GffF̂0o � ŷ0Jy y0ð Þdy0dz0 ¼ �Egap yð Þ; ð47Þ

where the integral in the left-hand side of (47) isthe electric field due to induced electric current andthe right-hand side represents the excitation field. Theelectric dyadic Green’s function used in this formula-tion has been derived for an ideal hard surface circularwaveguide terminated with the PEC ground plane atz = 0 (with the Green’s function components presentedin Appendix A). The integral equation (47) isdiscretized via the method of moments with thepiecewise-sinusoidal basis functions used for the currentexpansion (given in Appendix B). Galerkin’s projectiontechnique is used in the y direction along the strip andthe delta function testing in the z direction across thestrip. The method of moments impedance matrixelements are summarized in Appendix B.[38] Convergence of the method of moments is studied

for a different number of terms N used in the currentexpansion. The results shown in Figure 7 are obtained at

Figure 5. Transverse electric field of the TEM modeplotted at z = lg

TEM/2 = 15 mm due to the �-directedelectric dipole at z0 = 0 mm.

Figure 4. Transverse electric field of the TE01 modeplotted at z = lg

TE01/2 = 25.342 mm due to the �-directedelectric dipole at z0 = 0 mm.


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10 GHz (l0 = 30 mm) for the half-wavelength stripdipole shifted at x0 = 8 mm from the waveguide axis.[39] The current behavior is studied with respect to

the dipole position on the x axis as shown in Figure 8,where the current decreases as the dipole is shiftedfrom the waveguide axis because of the interactionwith the waveguide wall. The results presented in

Figure 8 for real and imaginary part of the current areobtained at 10 GHz for a half-wavelength strip dipole,with 5 basis functions used in the current expansion.[40] The electric field of the TEM mode excited by the

strip dipole at 10 GHz is obtained by integrating the fourtransverse components of the TEM term of the Green’sfunction with the current obtained from the method of

Figure 6. Half-wavelength strip dipole positioned at lg/4 from the PEC termination in the idealhard surface circular waveguide. The dipole is shifted by an arbitrary distance x0 from thewaveguide axis.

Figure 7. Behavior of the strip dipole current (real and imaginary parts) for a different number ofpiecewise-sinusoidal basis functions N used in the current expansion.


9 of 15

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moments solution (since all the other modes are evanes-cent in the hard surface waveguide at this frequency),

E rð Þ ¼ � jwm0XNn¼1

In

Z ynþDyyn�Dy

^̂RGTEMrr ^̂R0 þ ^̂RGTEMrf F̂

0n

þ F̂GTEMfr ^̂R0 þ F̂GTEMff F̂

0o � ŷ0Sn y0ð Þdy0: ð48Þ[41] The results of the TEM mode excitation by a half-

wavelength strip dipole in the ideal hard surface circularwaveguide of radius a = 11 mm at the frequency f =10 GHz are shown in Figure 9.

5. Scattering of the TEM Mode by Thin

Strip in the Ideal Hard Surface Waveguide

[42] The Green’s function derived in Section II is usedin the method of moments analysis of scattering of theTEM mode by a thin metal strip arbitrarily positioned inthe ideal hard surface circular waveguide as shown inFigure 10. The strip width is 2w = 0.25 mm, strip lengthis 2h = 15 mm, and the radius of waveguide is a =11 mm. The l component of the incident electric fieldof the TEM mode is obtained as follows,

Eincl f; zð Þ ¼1

2a

ffiffiffiffiffih0p

rl̂ � ^̂R cos fð Þ þ l̂ � F̂ sin fð Þ

� �e�jk0z;

ð49Þ

where l̂ = x̂cos(a) + ŷsin(a).

[43] The scattered electric field tangential to the strip iscalculated using the four components of the ideal hardsurface Green’s function, (23), (24), (26), and (27),

Escl rð Þ ¼ � jwm0̂l �Z w�w

Z h�h

^̂RGrr ^̂R0 þ ^̂RGrfF̂

0n þ F̂Gfr ^̂R0þ F̂GffF̂

0o � l̂0J l0ð Þdl0dz0: ð50Þ[44] The electric field integral equation is obtained by

enforcing the boundary condition for the total tangentialelectric field on the surface of the strip, which isdiscretized via the method of moments. The current isexpanded in terms of piecewise-sinusoidal basis func-tions in the l direction given in Appendix B. Galerkin’sprojection technique is applied for testing in the l coor-dinate and the delta function is used for testing in the zcoordinate. The method of moments impedance matrix issummarized in Appendix C.[45] The scattered electric field at the distance d = l0

from the strip is calculated as the integral of the TEMterm of the Green’s function with the current obtainedfrom the method of moments solution (assuming all theother modes are evanescent in the ideal hard surfacewaveguide at the operating frequency),

Esc rð Þ ¼ � jwm0XNn¼1

In

Z lnþDlln�Dl

^̂RGTEMrr ^̂R0 þ ^̂RGTEMrf F̂

0n

þ F̂GTEMfr ^̂R0 þ F̂GTEMff F̂

0o � l̂0Sn l0ð Þdl0: ð51Þ

Figure 8. Behavior of the strip dipole current (real and imaginary parts) with respect to the dipoleposition on the x axis.


10 of 15

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[46] The reflection coefficient at port 1 with coordi-nates (a/2, 0, �d ) is defined as the ratio of the scatteredfield over the incident field,

S11 ¼Escf a=2; 0; �dð ÞEincf a=2; 0; �dð Þ

; ð52Þ

and the transmission coefficient is defined as the ratio ofthe total field at port 2 with coordinates (a/2, 0, d) overthe incident field at port 1,

S21 ¼Escf a=2; 0; dð Þ þ Eincf a=2; 0; dð Þ

Eincf a=2; 0; �dð Þ: ð53Þ

[47] The realistic hard surface was modeled using thefinite difference time domain commercial software QuickWave [QWED, 1998] by placing 180 longitudinal metalstrips of azimuthal width w = 1 inside the dielectriccoating with the inner radius a = 11 mm, dielectricpermittivity er = 2.2, and thickness t = 6.8 mm calculatedas a quarter of the effective wavelength,

t ¼ l04

ffiffiffiffiffiffiffiffiffiffiffiffier � 1

p : ð54Þ

[48] The reflection and transmission coefficients forthe strip in the ideal hard surface waveguide calculatedusing (52) and (53) compared to the scattering parame-ters obtained using Quick Wave [QWED, 1998] for therealistic hard surface are shown in Figure 11. In themethod of moments code, the number of basis functionsused is 5 and the number of terms in the Green’s functionexpansion is 40 40.

6. Conclusion

[49] The Green’s function analysis of ideal hardsurface circular waveguide is based on the decompo-sition of the structure into PEC and PMC waveguides.The electric Green’s dyadic due to an arbitrarilyoriented electric dipole source is obtained in terms ofsolenoidal and irrotational parts. The solenoidal part isexpressed by the eigenmode expansion of TM and TEmodes of the PEC and PMC circular waveguides,respectively, with an additional term associated withthe TEM mode. The TEM term of the Green’s dyadicis obtained in closed form as a contribution of TM andTE modes in the zero-cutoff limit. It is shown that theTEM mode of the ideal hard surface circular wave-guide has a uniform field distribution over the wave-guide cross section, and polarization of the modenecessarily depends on polarization of the electricdipole source. The Green’s function of an ideal hard

Figure 10. Geometry of the ideal hard surface circular waveguide with an arbitrarily positionedl-directed thin metal strip. The strip has a fixed point at (xs, ys).

Figure 9. Field distribution of the TEM mode excitedby a half-wavelength strip dipole shifted at x0 = 8 mmfrom the waveguide axis in the ideal hard surface circularwaveguide terminated with the PEC ground plane.


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surface circular waveguide terminated with the PECground plane has been used in the method of momentsanalysis of the waveguide excitation by a half-wave-length strip dipole antenna. In addition, scattering ofthe TEM mode in the ideal hard surface waveguidehas been analyzed on the basis of the method ofmoments and the Green’s function derived in thiswork.

Appendix A: Dyadic Green’s Function for

Semi-Infinite Waveguide

[50] The Green’s function components of an ideal hardsurface circular waveguide terminated with the PECground plane at z = 0 are summarized as follows,

Grr r; r0ð Þ ¼ 1

jwm0PV

(h0

2pa2f0 z; z

0ð Þ cos f� f0ð Þ

þX1s¼0

X1k¼1

c2sk

(ZTMsk J

0s kc;skr�

J 0s kc;skr0�

þ s2

k2c;skZTEsk

Js kc;skr�

Js kc;skr0�

rr0

)

� fsk z; z0ð Þ cos s f� f0ð Þð Þ)

ðA1Þ

Grf r; r0ð Þ ¼ 1

jwm0PV

�h0

2pa2f0 z; z

0ð Þ sin f� f0ð Þ

þX1s¼0

X1k¼1

c2sks

�ZTMsk J

0s kc;skr� Js kc;skr0�

kc;skr0

þ ZTEskJs kc;skr� kc;skr

J 0s kc;skr0�

� fsk z; z0ð Þ sin s f� f0ð Þð Þ

ðA2Þ

Grz r; r0ð Þ ¼ 1

jwm0PV

�sign z� z0ð Þ

X1s¼0

X1k¼1

c2skZTMsk

kc;sk

gsk

� J 0s kc;skr�

Js kc;skr0�

gsk z; z0ð Þ

� cos s f� f0ð Þð Þ

ðA3Þ

Gfr r; r0ð Þ ¼ � 1

jwm0PV

�h0

2pa2f0 z; z

0ð Þ sin f� f0ð Þ

þX1s¼0

X1k¼1

c2sks

�ZTMsk J

0s kc;skr

0� Js kc;skr� kc;skr

þ ZTEskJs kc;skr0� kc;skr0

J 0s kc;skr�

� fsk z; z0ð Þ sin s f� f0ð Þð Þ

ðA4Þ

Figure 11. Scattering parameters for the strip in the ideal hard surface waveguide obtained fromthe method of moments solution and compared with the finite difference time domain simulationfor the same strip in the realistic hard surface.


12 of 15

RS3014

Gff r; r0ð Þ ¼ 1

jwm0PV

�h0

2pa2f0 z; z

0ð Þ cos f� f0ð Þ

þX1s¼0

X1k¼1

c2sk

�s2

k2c;skZTMsk

Js kc;skr�

Js kc;skr0�

rr0

þ ZTEsk J0s kc;skr�

J 0s kc;skr0�

� fsk z; z0ð Þ cos s f� f0ð Þð Þ

ðA5Þ

Gfz r; r0ð Þ ¼ � 1

jwm0PV


X1s¼0

X1k¼1

c2skZTMsk

s

gsk

�Js kc;skr� r

Js kc;skr0�

gsk z; z0ð Þ

� sin s f� f0ð Þð Þ

ðA6Þ

Gzr r; r0ð Þ ¼ � 1

jwm0PV


X1s¼0

X1k¼1

c2skZTMsk

kc;sk

gsk

� Js kc;skr�

J 0s kc;skr0� gsk z; z0ð Þ

� cos s f� f0ð Þð Þ

ðA7Þ

Gzf r; r0ð Þ ¼ � 1

jwm0PV


X1s¼0

X1k¼1

c2skZTMsk

s

gsk

�Js kc;skr0� r0

Js kc;skr�

gsk z; z0ð Þ

� sin s f� f0ð Þð Þ

ðA8Þ

Gzz r; r0ð Þ ¼ � 1

jwm0PV

�X1s¼0

X1k¼1

c2skZTMsk

k2c;sk

g2skJs kc;skr�

� Js kc;skr0�

gsk z; z0ð Þ cos s f� f0ð Þð Þ

� d r� r0ð Þ

k20; ðA9Þ

where

fsk z; z0ð Þ ¼

e�gsk z sinh gskz0ð Þ; z � z0

e�gsk z0sinh gskzð Þ; z < z0

8<: ðA10Þ

gsk z; z0ð Þ ¼

e�gsk z cosh gskz0ð Þ; z � z0

e�gsk z0cosh gskzð Þ; z < z0

8<: ðA11Þ

f0 z; z0ð Þ ¼

je�jk0z sin k0z0ð Þ; z � z0

je�jk0z0sin k0zð Þ; z < z0:

8<: ðA12Þ

Appendix B: Method of Moments (MOM)

Impedance Matrix for Strip Dipole Antenna

[51] Piecewise-sinusoidal basis functions used in thecurrent expansion are

Sn y0ð Þ ¼

sin k0 y0 � yn þ Dyð Þð Þ

sin k0Dyð Þ; yn�1 � y0 � yn

sin k0 yn þ Dy� y0ð Þð Þsin k0Dyð Þ

; yn � y0 � ynþ1

0; otherwise:

8>>>>><>>>>>:

ðB1Þ

[52] Themethod ofmoments impedancematrix elementsobtained in the discretization of integral equation (47) are

Zmn ¼1

4b

h0jk0pa2

z0

Z ymþDyym�Dy

Z ynþDyyn�Dy

Sn y0ð ÞSm yð Þ

� ðxcss1 y; y0ð Þ þ xssc1 y; y

0ð Þ þ xscs1 y; y0ð Þ

þ xccc1 y; y0ð ÞÞdy0dyþ 1

2b

X1s¼0

X1k¼1

c2skzsk

�ZTMsk

�Z ymþDyym�Dy

Z ynþDyyn�Dy

�J 0s kc;skr0�

J 0s kc;skr00

� Sn y

0ð Þ

� Sm yð Þxcsss y; y0ð Þdy0dyþ s

kc;sk

Z ymþDyym�Dy

Z ynþDyyn�Dy

� J 0s kc;skr0� Js kc;skr00�

r00Sn y

0ð ÞSm yð Þxsscs y; y0ð Þdy0dy

� skc;sk

Z ymþDyym�Dy

Z ynþDyyn�Dy

J 0s kc;skr00

� Js kc;skr0� r0

� Sn y0ð ÞSm yð Þxscss y; y0ð Þdy0dyþs2

k2c;sk

Z ymþDyym�Dy

�Z ynþDyyn�Dy

Js kc;skr0� r0

Js kc;skr00� r00

Sn y0ð ÞSm yð Þxcccs y; y0ð Þ

� dy0dy�þ ZTEsk

Z ymþDyym�Dy

Z ynþDyyn�Dy

�J 0s kc;skr0�

� J 0s kc;skr00�

Sn y0ð ÞSm yð Þxcccs y; y0ð Þdy0dy�

s

kc;sk

�Z ymþDyym�Dy

Z ynþDyyn�Dy

J 0s kc;skr0� Js kc;skr00�

r00Sn y

0ð Þ

� Sm yð Þxscss y; y0ð Þdy0dyþs

kc;sk

Z ymþDyym�Dy

Z ynþDyyn�Dy

� J 0s kc;skr00� Js kc;skr0�

r0Sn y

0ð ÞSm yð Þxsscs y; y0ð Þdy0dy

þ s2

k2c;sk

Z ymþDyym�Dy

Z ynþDyyn�Dy

Js kc;skr0� r0

Js kc;skr00� r00

� Sn y0ð ÞSm yð Þxcsss y; y0ð Þdy0dy�

; ðB2Þ


13 of 15

RS3014

where

r0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix20 þ y2

q; r00 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix20 þ y02

qxcss1 y; y

0ð Þ ¼ cos tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ�

� sin tan�1 y=x0ð Þ�

sin tan�1 y0=x0ð Þ�

xssc1 y; y0ð Þ ¼ sin tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ

� � sin tan�1 y=x0ð Þ

� cos tan�1 y0=x0ð Þ

� xscs1 y; y

0ð Þ ¼ sin tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ�

� cos tan�1 y=x0ð Þ�


xccc1 y; y0ð Þ ¼ cos tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ

� � cos tan�1 y=x0ð Þ


� xcsss y; y

0ð Þ ¼ cos s tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ� � �

� sin tan�1 y=x0ð Þ�


xsscs y; y0ð Þ ¼ sin s tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ

� � �� sin tan�1 y=x0ð Þ


� xscss y; y

0ð Þ ¼ sin s tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ� � �

� cos tan�1 y=x0ð Þ�


xcccs y; y0ð Þ ¼ cos s tan�1 y=x0ð Þ � tan�1 y0=x0ð Þ

� � �� cos tan�1 y=x0ð Þ


� ðB3Þ

and

z0 ¼1

jk01� e�jk0b þ e

�j2k0d

2e�jk0b � ejk0b� � �

zsk ¼1

gsk1� e�gsk b þ e

�2gsk d

2e�gsk b � egsk b� � �

:

ðB4Þ

Appendix C: MOM Impedance Matrix for

Strip Scatterer

[53] Method of moments impedance matrix elementsobtained in the analysis of scattering of the TEM modeby a thin strip arbitrarily positioned in the ideal hardsurface circular waveguide are summarized as follows:

Zmn ¼h0

4wpa21� e�jk0w

jk0

Z lmþDllm�Dl

Z lnþDlln�Dl

Sn l0ð ÞSm lð Þ

� xccc1 l; l0ð Þ þ xscs1 l; l

0ð Þ þ xssc1 l; l0ð Þ�

þ xcss1 l; l0ð Þ

� dl0dl þ 12w

X1s¼0

X1k¼1

c2sk1� e�gskw

gskZTMsk

Z lmþDllm�Dl

��

�Z lnþDlln�Dl

J 0s kc;skr�

J 0s kc;skr0� Sn l0ð ÞSm lð Þxcccs l; l0ð Þ

� dl0dl þ skc;sk

Z lmþDllm�Dl

Z lnþDlln�Dl

J 0s kc;skr� Js kc;skr0�

r0

� Sn l0ð ÞSm lð Þxscss l; l0ð Þdl0dl �s

kc;sk

Z lmþDllm�Dl

Z lnþDlln�Dl

� J 0s kc;skr0� Js kc;skr�

rSn l

0ð ÞSm lð Þxsscs l; l0ð Þdl0dl

þ s2

k2c;sk

Z lmþDllm�Dl

Z lnþDlln�Dl

Js kc;skr� r

Js kc;skr0� r0

Sn l0ð Þ

� Sm lð Þxcsss l; l0ð Þdl0dl�þ ZTEsk

Z lmþDllm�Dl

Z lnþDlln�Dl

�� J 0s kc;skr

� J 0s kc;skr

0� Sn l0ð ÞSm lð Þxcsss l; l0ð Þdl0dl� skc;sk

Z lmþDllm�Dl

Z lnþDlln�Dl

J 0s kc;skr� Js kc;skr0�

r0Sn l

0ð Þ

� Sm lð Þxsscs l; l0ð Þdl0dl þ s

kc;sk

Z lmþDllm�Dl

Z lnþDlln�Dl

� J 0s kc;skr0� Js kc;skr�

rSn l

0ð ÞSm lð Þxscss l; l0ð Þdl0dl

þ s2

k2c;sk

Z lmþDllm�Dl

Z lnþDlln�Dl

Js kc;skr� r

Js kc;skr0� r0

Sn l0ð Þ

� Sm lð Þxcccs l; l0ð Þdl0dl�

; ðC1Þ

where

xccc1 l; l0ð Þ ¼ cos tan�1 y=xð Þ � tan�1 y0=x0ð Þ

� � cos a� tan�1 y=xð Þ

� cos a� tan�1 y0=x0ð Þ

� xscs1 l; l

0ð Þ ¼ sin tan�1 y=xð Þ � tan�1 y0=x0ð Þ�

� cos a� tan�1 y=xð Þ�

sin a� tan�1 y0=x0ð Þ�

xssc1 l; l0ð Þ ¼ sin tan�1 y=xð Þ � tan�1 y0=x0ð Þ

� � sin a� tan�1 y=xð Þ


� xcss1 l; l

0ð Þ ¼ cos tan�1 y=xð Þ � tan�1 y0=x0ð Þ�

� sin a� tan�1 y=xð Þ�


xcccs l; l0ð Þ ¼ cos s tan�1 y=xð Þ � tan�1 y0=x0ð Þ

� � �� cos a� tan�1 y=xð Þ


� xscss l; l

0ð Þ ¼ sin s tan�1 y=xð Þ � tan�1 y0=x0ð Þ� � �

� cos a� tan�1 y=xð Þ�


xsscs l; l0ð Þ ¼ sin s tan�1 y=xð Þ � tan�1 y0=x0ð Þ

� � �� sin a� tan�1 y=xð Þ


� xcsss l; l

0ð Þ ¼ cos s tan�1 y=xð Þ � tan�1 y0=x0ð Þ� � �

� sin a� tan�1 y=xð Þ�


;

ðC2Þ


14 of 15

RS3014

with

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p; r0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix02 þ y02

px ¼ xs þ l cos að Þ; y ¼ ys þ l sin að Þ

x0 ¼ xs þ l0 cos að Þ; y0 ¼ ys þ l0 sin að Þ:

ðC3Þ

[54] Acknowledgments. This work was partially supportedby the National Science Foundation under grant ECS-0220218and NASA EPSCoR Cooperative Agreement NCC5-574. Theauthors thank the reviewers for their valuable comments.

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��I. A. Eshrah, A. W. Glisson, A. A. Kishk, V. A. Klymko, and

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[email protected]; [email protected])


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RS3014

Figure 1. PEC/PMC strip model for an ideal hard surface circular waveguide. The hard surfacebehaves like a PEC waveguide for TM modes and as a PMC waveguide for TE modes. A TEMmode is obtained as a contribution of TM and TE modes in the zero-cutoff limit. The TEM mode‘‘sees’’ the ideal hard surface waveguide as a combination of PEC and PMC waveguides, and itsatisfies the boundary conditions on the ideal hard surface partially contributed by PEC and PMCwaveguides, that is, @Er/@rjS = 0, @Ef/@rjS = 0.

RS3014 KLYMKO ET AL.: HARD SURFACE WAVEGUIDE RS3014

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