RECTANGULAR CONDUCTING WAVEGUIDE … In Electromagnetics Research, PIER 25, 111–129, 2000 RECTANGULAR CONDUCTING WAVEGUIDE FILLED WITH UNIAXIAL ANISOTROPIC MEDIA: A MODAL ANALYSIS

Progress In Electromagnetics Research, PIER 25, 111–129, 2000

RECTANGULAR CONDUCTING WAVEGUIDE FILLED

WITH UNIAXIAL ANISOTROPIC MEDIA: A MODAL

ANALYSIS AND DYADIC GREEN’S FUNCTION

S. Liu, L. W. Li, M. S. Leong, and T. S. Yeo

Communications and Microwave DivisionDepartment of Electrical EngineeringThe National University of SingaporeSingapore 119260

1. Introduction2. Basic Formulation of the Problem3. Fields in Source-free Rectangular Waveguides4. Dyadic Green’s Functions5. Applications of Dyadic Green’s Functions6. ConclusionReferences

1. INTRODUCTION

A general anisotropic medium is charaterized by permittivity tensorε and permeability tensor µ [1–3], whose form depends on the kindof anisotropy. At present, different anisotropic materials are widelyused in integrated optics and microwave engineering [4–8]. The tech-nology advances are making the production of substrates, dielectricanisotropic films and anisotropic material filling more and more con-venient. It shows the necessity of better charaterizing the anisotropicmedia and producing more realistic models for the components thatuse them.

Due to the complexity caused by the parameter tensors, the planewave expansion is often used in the analysis of anisotropic media. Andconsequently, the Fourier transform is widely applied [1–3, 6, 9, 10] andwill also be employed in this study. Using these methods, Uzunoglu et

112 Liu et al.

al. [9] found the solution of the vector wave equation in cylindricalcoordinates for a gyroelectric medium. Ren [10] furthered the workunder spherical coordinates in a similar procedure and obtained spher-ical wave functions and dyadic Green’s functions in gyroelectric media.Recently, the theory of TE- and TM-decomposition attracts the inter-ests of some researchers [11–13]. The TE- and TM-field decompositionis a method for solving electromagnetic problems involving certain classof boundaries and media that separate the field into reversely-handedcircularly polarized waves. It is now extended to the problems of gen-eral uniaxially anisotropic media [11] and bi-anisotropic media [12, 13]from that of isotropic media.

In this paper, the fields in a rectangular conducting waveguide filledwith a uniaxially anisotropic material are studied. The characteristicfeature of the uniaxial media is the existence of a distinguished axis.If one of the coordinate axes is chosen to be parallel to this distin-guished direction, it turns out that the parameter tensor is diagonalbut the element referring to the distinguished axis is different fromthe remaining two diagonal ones. Some features of the waveguidesof uniaxial anisotropic material have been analyzed and their appli-cations made by some authors [4, 7, 8]. It’s found in our study thatthe fields in a rectangular conducting waveguide filled with uniaxiallyanisotropic material are split into TE and TM modes after solvingthe eigenvalue equation. The calculated dispersion curves are thendepicted. There exist considerable effects of the material parameterson the cutoff frequencies of the propagating waves, especially the TMmodes. The dyadic Green’s functions for various kinds of anisotropicmedia with different structures have been studied by many authors[14–20]. The problems, however, are mostly analyzed in spectral do-main in terms of Fourier transform, due to the difficulty of findingthe expansion of the dyadic Green’s functions in terms of vector wavefunctions for anisotropic media. So far as we know, the dyadic Green’sfunctions for anisotropic media-filled waveguides with perfectly con-ducting walls have not been shown in any literature. In our study,the suitable vector wave functions can be selected separately for theTE modes and TM modes by applying the boundary conditions. Thedyadic Green’s functions are then derived using Ohm-Rayleigh method[21]. It’s shown that the expression obtained here is reducible to theisotropic case which has been obtained by Tai [21]. The magnetic typedyadic Green’s function due to electric source can be obtained from the

Rectangular conducting waveguide 113

ox

y

z

perfectly conducting walls

a

b

Figure 1. Geometry of a rectangular waveguide.

electric type one. As the application of the dyadic Green’s function,numerical results of the the modes excited by an infinitesimal electricdipole are also given.

Throughout this paper, the harmonic e−iωt time dependence is as-sumed and suppressed. Bold prints indicate vectors and an overheadbold face indicates a dyad(ic). Caps with bold face represent unitvectors.

2. BASIC FORMULATION OF THE PROBLEM

Consider a rectangular waveguide (Fig. 1) of which coordinates axissystem is represented by (x, y, z) . z is the direction of propagation.

The waveguide is filled with homogeneous electrically uniaxial aniso-tropic medium that is characterized by the following set of constitutiverelations:

D = ε0εr · E, (1a)B = µ0H, (1b)

where ε0 and µ0 are the free space permittivity and permeability,respectively.

We assume that the optics axis of the uniaxial media is orientedalong the z-axis, and the other two principal axes are oriented alongthe two remaining coordinate axes. So the permitivity tensors εr isgiven by

εr =

εt 0 00 εt 00 0 εz

. (2)

114 Liu et al.

It can be proved that εr takes the same form in both Cartesian andcylindrical coordinates systems. For this kind of media, the source-incorporated Maxwell’s equations are written as

∇ × E = iωµ0H, (3b)∇ × H = −iωε0εr · E + J. (3b)

Substituting (3a) into (3b) yields

∇ × ∇ × E − k20εr · E = iωµ0 · J, (4)

where k0 = ω√

ε0µ0 , which is the free space wavenumber.

3. FIELDS IN SOURCE-FREE RECTANGULARWAVEGUIDE

Now we study the eigenvalues and eigenvectors of the source-free vectorwave equation in an infinite uniaxial anisotropic medium. Let J = 0 ,(4) reduces to

∇ × ∇ × E − k20εr · E = 0. (5)

The characteristic waves corresponding to (5) can be examined in thespectral domain using Fourier transform

E(r) =∫∫∫ +∞

−∞E(k)eik·rdk (6)

where k = kxx + kyy + kzz . Substituting (6) to (5), we have∫∫∫ +∞

−∞(k2I − kk − k2

0εr) · E(k)eik·rdk = 0 (7)

where I = xx + yy + zz is the unit dyadic. For nontrivial solutionsof (7), it is required that the determinant of matrix (k2I− kk− k2

0εr)must be equal to zero. Performing the algebraic operations, we obtainthe characteristic equation finally as

εtk4c + (εt + εz)

(k2

z − k20εt

)k2

c + εz

(k2

z − k20εt

)2 = 0 (8)

where k2c = k2

x + k2y . Solving the characteristic equation, we have the

eigenvalues given by

k2c1 = k2

0εt − k2z , (9a)

k2c2 = k2

0εz − k2z

εz

εt. (9b)


It can be seen that kc1 is independent upon εz while kc2 is a functionof εz , which lead to the ordinary and extraordinary waves [1], respec-tively. To find the corresponding eigenvectors, we substitute theseeigenvalues back to (7). Finally we have

E1z = 0,

E1x cos(φk) + E1y sin(φk) = 0,

E1(φk, kz) = E1xx + E1yy, (10a)

for kc1 ; and

E2x = A(kz) cos(φk)E2z,

E2y = A(kz) sin(φk)E2z,

E2(φk, kz) = E2xx + E2yy + E2zz, (10b)

for kc2 ; where

A(kz) =εzkz

εt

√εz(k2

0 − k2z/εt)

(10c)

andφk = tan−1(ky/kx). (10d)

Obviously E1 takes TE modes, which can be expressed using thevector wave function M with z as the piloting vector [21]. In theCartesian coordinate system, E2z takes the form of ej(kx2x+ky2y+kzz) .From (3a) we have

H2z =1

iωµ0

[∂(E2 · y)

∂x− ∂(E2 · x)

∂y

]

= 0, (11)

which means E2 takes TM modes. So E2 can be expanded usingvector wave function N , and L with z as the piloting vector as well.For rectangular waveguides bounded at x = 0 and a and at y = 0and b by conducting walls, these vector wave functions can be writtenas [21]

M1eo(h) = ∇ ×

(cossin

kx1xcossin

ky1yejkzzz)

, (12a)

N2eo(h) =

1k2

∇ × ∇ ×(cos

sinkx2x

cossin

ky2yejkzzz)

, (12b)

L2eo(h) = ∇

(cossin

kx2xcossin

ky2yejkzz)

. (12c)

116 Liu et al.

0 2 4 6 8 10 120

5

10

15

εt = 2, εz = 5 TE10 TE01 TE20 TE11 TM

11 TE21 TM21 TE30 TE31 TM31

k za

k0a

Figure 2. Dispersion curves for the lowest 10 modes.

It has been shown that fields can be decomposed into TE and TMmodes. The observation is in agreement with Lindell’s analysis [11]that decomposition method is applicable to such problems with perfectelectric or magnetic conducting surfaces parallel to the z-axis.

To match the boundary condition,

z · E2(r)|boundary = 0 (13)

must be satisfied, because E1z is zero everywhere. (13) straightfor-wardly leads to that we should choose N2o and L2o as eigenfunctionsof E2(r) , and that

kx2 = m2π/a, m2 = 1, 2, · · · ; (14a)ky2 = n2π/b, n2 = 0, 1, 2, · · · . (14b)

In this case, x · E2(r)|boundary = 0 is automatically satisfied. So onthe boundary, E1(r) must satisfy

x · E1(r)|boundary = 0, (15)

which leads to that only M1e can be chosen to represent E1(r) , and

kx1 = m1π/a, m1 = 1, 2, · · · ; (16a)ky1 = n1π/b, n1 = 0, 1, 2, · · · . (16b)


Equations (14) and (16), together with (9), make the dispersion re-lations for a rectangular conducting waveguide filled with a uniaxialanisotropic material. The calculated dispersion curves for εz/εt = 2.5are depicted in Fig. 2 where a/b = 2 . Only the modes correspondingto the lowest 10 modes (which are listed in sequence in the legend ofFig. 2) of an isotropic rectangular waveguide are shown. It is found thatdue to the existence of the optics axis, the order of these modes changesfor the uniaxial material parameters. For example, TM11 is the fifthlowest mode in isotropic rectangular waveguide with the dimension ofa/b = 2 , but it becomes the 2nd lowest one in uniaxial rectangularwaveguide with the same dimension. This phenomenon suggests thatwe can select the propagating mode we want by changing the materialparameters. Since the optics axis of the uniaxial media only affects theTM modes in the present case, in Fig. 3 the cutoff frequencies of someTM modes are shown as εz increases from 1.0 to 10.0 (εt = 2.0) . Itcan be seen that the cutoff frequencies monotonically decrease as εz

increases.In Fig. 4–6, some lowest TM modes are plotted for a uniaxial (εz/εt

= 2.5) rectangular waveguide and an isotropic (εz/εt = 1.0) rectan-gular waveguide. The dimensions of the waveguide are selected so thata/b = 2 . It can be seen that due to the existence of the optics axis, thedistributions of the field intensity in the uniaxial waveguide are differ-ent from the isotropic ones. A greater εz produces more significantvariation of the intensity distribution.

4. DYADIC GREEN’S FUNCTIONS

In this section, we perform the eigenfunction expansion of dyadicGreen’s functions for a rectangular waveguide filled with uniaxial ma-terial making use of the Ohm-Rayleigh method.

Consider an electric source J expressed in terms of the Dirac-deltafunction δ(r − r′) and the unit dyadic I as follow:

J(r) =∫

V ′δ(r − r′)I · J(r′)dV ′. (17)

Due to the linearity of (4), the electric field can be related directly tothe source [21] through

E(r) = iωµ0

∫

V ′GEJ(r, r′) · J(r′)dV ′, (18)

118 Liu et al.

1 2 3 4 5 6 7 8 9 10

2

3

4

5

6

7

8

9

10

11

12

13

14

15

εt = 2 TM11 TM21 TM31 TM12 TM22 TM41

Cut

off F

requ

ency

(k ca

)

εz

Figure 3. The cutoff frequencies for the lowest 6 TM-modes versusεz .

a. TM11 uniaxial

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0500

10001500

00.02

0.040.06

0.08

b. TM11 isotropic

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0500

1000

00.02

0.040.06

0.08

Figure 4. Electrical field intensities of TM11 mode.

a. TM21 uniaxial

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0200400600

00.02

0.040.06

0.08

b. TM21 isotropic

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0100200300400

00.02

0.040.06

0.08

Figure 5. Electrical field intensities of TM 21 mode.


a. TM31 uniaxial

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0100200300400

00.02

0.040.06

0.08

b. TM31 isotropic

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0100200

00.02

0.040.06

0.08

Figure 6. Electrical field intensities of TM31 mode.

where GEJ denotes electric dyadic Green function due to electricsource, which is required to satisfy the dyadic Dirichlet condition n×GEJ = 0 on the conducting boundaries, and V ′ stands for the vol-ume occupied by the exciting current source. Substituting (17) and(18) into (4), we have

∇ × ∇ × GEJ(r, r′) − k20εr · GEJ(r, r′) = Iδ(r − r′). (19)

By inspection from the results of the previous section, GEJ can alsobe expanded using M1emn , N2omn and L2omn as

GEJ(r, r′) =∫ +∞

−∞dkz

∑

n,m

[M1emn(kz)a1emn(kz)

+N2omn(kz)b2omn(kz) + L2omn(kz)c2omn(kz)] (20)

where a, b and c are unknown coefficient matrices to be determined.(16) and (14) must be met to satisfy the boundary conditions. Forconvenience, we have used the notation mn to stand for m1, n1 andm2, n2 as well. Subsequently, we adopt the simplified notations M1e ,N2o and L2o instead of M1emn, N2omn and L2omn , respectively.

According to the Ohm-Rayleigh method we also expand the sourcefunction Iδ(r − r′) , similarly, into

Iδ(r − r′) =∫ +∞

−∞dkz

∑

n,m

[M1e(kz)P1e(kz)

+N2o(kz)Q2o(kz) + L2o(kz)V2o(kz)] . (21)

Taking the anterior scalar product of (21) with M1e′(−k′z),

N2o′(−k′z) , and L2o′(−k′

z) in turn and integrating the resultant equa-tions through the entire volume of the rectangular waveguide, we can

120 Liu et al.

determine the coefficient matrices P, Q and V given by

P1e(kz) =2 − δ0

πabk2c1

M ′1e(−kz), (22a)

Q2o(kz) =2 − δ0

πabk2c2

N ′2o(−kz), (22b)

V2o(kz) =2 − δ0

πabk22

L′2o(−kz). (22c)

To obtain (22), the following orthogonality relations among the vectorwave functions have been used, i.e.,

< M1e(kz), M1e′(−k′z) > =

πabk2c1

2(1 + δ0)δmm′δnn′δ(kz − k′

z), (23a)

< N2o(kz),N2o′(−k′z) > =

πabk2c2

2(1 − δ0)δmm′δnn′δ(kz − k′

z), (23b)

< L2o(kz),L2o′(−k′z) > =

πabk22

2(1 − δ0)δmm′δnn′δ(kz − k′

z), (23c)

< M1e(kz),N2o′(−k′z) > = 0, (23d)

< M1e(kz),L2o′(−k′z) > = 0, (23e)

< N2o(kz),L2o′(−k′z) > = 0, (23f)

where

δ0 =

1, m = 0 or n = 00, otherwise ,

δmm′ =

1, m = m′

0, m = m′ ,

δnn′ =

1, n = n′

0, n = n′ .

In (23), we’ve taken the operator < •, • > [22] which defines

< a, b >=∫∫∫

a · bdV. (24)

Substituting (20) and (21) into (19), and taking the anterior scalarproduct of (20) with M1e′(−k′

z), N2o′(−k′z) , and L2o′(−k′

z) in thesame way, we finally arrive at the following equation in matrix form:

[Ω][X] = [Θ], (25)


where [Ω] is a 3 × 3 matrix given by

[Ω] =

Ω11 0 00 Ω22 Ω23

0 Ω32 Ω33

, (26a)

with

Ω11 =πab

2(1 + δ0)k2

c1

(k2

1 − k20εt

),

Ω22 =πab

2(1 − δ0)k2

c2

[

k22 −

k20

k22

(εtk

2z + εzk

2c2

)]

,

Ω23 = −Ω32 = iπab

2(1 − δ0)

kz

k2k2

c2k20(εt − εz),

Ω33 = −πab

2(1 − δ0)k2

0

(εtk

2c2 + εzk

2z

).

In (26a), [X] and [Θ] are two column vectors given by

[X] =

a1e(kz)b2o(kz)c2o(kz)

, and [Θ] =

M ′1e(−kz)

N ′2o(−kz)

L′2o(−kz)

.

Solving (25), we have the solutions for a1e(kz), b2o(kz) and c2o(kz)as follows

a1ekz)=2 − δ0

πabk2c1

1k2

1−k210

M1e(−kz), (27a)

b2o(kz)=2 − δ0

πabk2c2

1εzk2

2

(k2

2−k220

) [βNNN2o(−kz) + βNLL2o(−kz)], (27b)

c2o(kz)=2 − δ0

πabk2c2

1εzk2

2

(k2

2−k220

) [βLNN2o(−kz) + βLLL2o(−kz)], (27c)

where the ordinary and extraordinary wave numbers are defined as

k210 = k2

0εt, (28a)

k220 = k2

0εt +(

1 − εt

εz

)

k2c2; (28b)

122 Liu et al.

and the coupling coefficients are given by

βNN = εtk2c2 + εzk

2z , (28c)

βNL =ikz

k2k2

c2(εt − εz), (28d)

βLN = − ikz

k2k2

c2(εt − εz), (28e)

βLL = − k2c2

k20k

22

[k4

2 − k20(εtk

2z + εzk

2c2)

]. (28f)

Hence, (20) can be rewritten as

GEJ(r, r′) =∫ +∞

−∞dkz

∑

m,n

2 − δ0

πabk2c1

1k2

1 − k210

M1e(kz)M ′1e(−kz)

+2 − δ0

πabk2c2

1εzk2

2

(k2

2 − k220

)

[

βNNN2o(kz)N ′2o(−kz)

+ βNLN2o(kz)L′2o(−kz) + βLNL2o(−kz)N ′

2o(−kz)

+ βLLL2o(kz)L′2o(−kz)

]

. (29)

In this way, the dyadic Green’s functions for rectangular waveguidesfilled with uniaxial anisotropic media are explicitly represented in theform of the eigenfunction expansion in terms of the rectangular vec-tor wave functions, as given in (29). However, for ease of practicalapplications and physical interpretation of possible novel phenomena,mathematical simplification to (29) is necessary. In order to apply theresidue theorem to (20), we must first extract the irrotational term in(20) which does not satisfy the Jordan lemma as pointed out in [21].To do so, we write

N2o(kz) = N2ot(kz) + N2oz(kz), (30a)L2o(kz) = L2ot(kz) + L2oz(kz), (30b)

and so are N ′ and L′ . The subscripts t and z denote their trans-verse vector components and their z-vector components respectively.In terms of these functions, (29) can be rewritten in the form

GEJ(r, r′) =∫ +∞

−∞dkz

∑

m,n

2 − δ0

πabk2c1

1k2

1 − k210



+2 − δ0

πabk2c2

1εzk2

2

(k2

2 − k220

)

[k2

0εz − k2c2

k2z

N2ot(kz)N ′2ot(−kz)

+ N2ot(kz)N ′2oz(kz) + N2oz(kz)N ′

2ot(−kz)

+k2

0εt − k2z

k2c2

N2oz(kz)N ′2oz(kz)

]

, (31)

where we have expressed L2ot(kz) and L2oz(kz) in terms of N2ot(kz)and N2oz(n, kz) , namely,

L2ot(kz) = − ik2

kzN2ot(kz), (32a)

L2oz(kz) =ikzk2

k2c2

N2oz(kz), (32b)

and similarly for the primed functions.The singular term in (31) is contained in the component N2oz(kz)

N ′2oz(−kz) [21]. From (21), we note that

zzδ(r − r′) =∫ ∞

−∞dkz

∑

m,n

2 − δ0

πab

[1

k2c2

N2oz(kz)N ′2oz(−kz)

+1k2

2

L2oz(kz)L′2oz(−kz)

]

=∫ ∞

−∞dkz

∑

m,n

2 − δ0

πabk2c2

k22

k2c2

N2oz(kz)N ′2oz(−kz). (33)

Making use of the identity

k22

k20εz

(k2

2 − k220

)k2

0εt − k2z

k2c2

= − k22

k20εzk2

c2

+εtk

22

k20ε

2z

(k2

2 − k220

) , (34)

we can split (31) into

GEJ(r, r′) = −∫ ∞

−∞dkz

∑

m,n

2 − δ0

πabk2c2

k22

k20εzk2

c2

N2oz(kz)N ′2oz(−kz)

+∫ +∞

−∞dkz

∑

m,n

2 − δ0

πabk2c1

1k2

1 − k210


+2 − δ0

πabk2c2

k22

k20εz

(k2

2 − k220

)

[k2

0εz − k2c2

k2z

N2ot(kz)N ′2ot(−kz)

+ N2ot(kz)N ′2oz(−kz) + N2oz(kz)N ′

2ot(−kz)

+εt

εzN2oz(kz)N ′

2oz(−kz)]

. (35)

124 Liu et al.

In view of (33), the first integral in (35) is equal to

− 1k2

0εzzzδ(r − r′), (36)

and the second integral can be evaluated by making use of the residuetheorem in the kz-plane. The final result is given after some mathe-matical manipulations by

GEJ(r, r′) = − 1k2

0εzzzδ(r − r′) +

i

ab

∑

m,n

1

k2c1kz1

M1e(±kz1)M ′1e(∓kz1)

+k2

20

k20εtk2

c2kz2

[

N2ot(±kz2) +εt

εzN2oz(∓kz2)

]

×[

N ′2ot(∓kz2) +

εt

εzN ′

2oz(∓kz2)]

, z >< z′ (37)

where z and z′ are the positions of the observation point and thesource point, respectively, measured along the z-direction, and

k2z1 = k2

10 − k2c1, (38a)

k2z2 = k2

20 − k2c2, (38b)

with k10 and k20 having been defined in (28).It can be observed that (37) is reducible to the isotropic case by

assuming that εt = εz = ε . In this case, k210 = k2

20 = k20ε and (37)

shall be simplified as:

GEJ(r, r′) = − 1k2

0εzzδ(r − r′) +

i

ab

∑

m,n

2 − δ0

k2ckz

[

Me(±kz)M ′e(∓kz)

+ No(±kz)N ′o(∓kz)

]

, z >< z′ (39)

which is exactly the same as that obtained by Tai [21].So far, we have obtained the electric dyadic Green’s function as given

in (37). With this electric DGF, the electric field can be obtained di-rectly from (18). To obtain the magnetic field due to an electric currentdistribution, we need to utilize the following Maxwell’s equation

∇ × E = iωµ0H, (40a)


ox

y

z

a

b

Figure 7. Excitation of an electric dipole.

or that in dyadics form

∇ × GEJ = GHJ . (40b)

If we express the magnetic field in terms of the magnetic dyadic Green’sfunction, we have the following integral:

H =∫∫∫

V ′GHJ(r, r′) · J(r′)dV ′. (41)

5. APPLICATION OF DYADIC GREEN’S FUNCTIONS

In this section, an infinitesimal dipole in y-direction is assumed asthe electric current source in the rectangular waveguide filled withuniaxial material. In a three-dimensional expression form, this sourcedistribution is expressed by

J(r′) = I0yδ(x′ − x0)δ(y′ − y0)δ(z′ − z0) (42)

where the point (x′, y′, z′) is located at the center of the rectangularwaveguide cross section as shown in Fig. 7, i.e.,

x0 =a

2, y0 =

b

2, z0 = 0. (43)

Substituting the above current distribution and (37) into (18), we have

126 Liu et al.

E(r) = − I0ωµ0

ab

∑

m,n

kx1

k2c1kz1

[

sin(

kx1a

2

)

cos(

ky1b

2

)]

M1e(±kz1)

± ik220ky2

k20εtk2k2

c2

[

sin(

kx2a

2

)

cos(

ky2b

2

)][

N2ot(±kz2)

+εt

εzN2oz(∓kz2)

]

, z >< z′. (44)

From (44), it is easy to find out that only the modes with odd m andeven n are excited. Since for TE waves, a uniaxial medium with theoptics axis in z-direction looks like an isotropic medium, the fields ofTE modes in the uniaxial anisotropic medium should have the samerepresentations as the well-known ones in the isotropic medium. There-fore, we should discuss the features of the TM-mode fields only. In oursubsequent numerical computation, the dimensions of the waveguideand the parameters of the material are selected as the same as thosein the previous section (i.e., a/b = 2, εt = 2.0 , and εz = 2.5) , forease and meaningfulness of comparison. Fig. 8 and Fig. 9 show theelectric field intensity and Ez distributions of TM12 mode and TM32

mode in the transverse cross-section. It is seen that the distribution ofthe guided TM modes in a uniaxial anisotropic medium has changedas compared with those guided ones in an isotropic medium.

6. CONCLUSION

The electromagnetic fields in rectangular conducting waveguides filledwith uniaxial anisotropic media are characterized in this paper. Withthe optics axis of the uniaxial media oriented along the z-axis of thewaveguide, the fields can be decomposed into TE and TM modes withdifferent propagation wave numbers. The ordinary wave takes theform of TE modes while the extraordinary wave takes the form of TMmodes. The calculated dispersion curves are depicted and the effects ofthe material parameters on the cutoff frequencies are shown. The cut-off frequencies change considerably, especially for TM modes. The fielddistributions are drawn and compared with those in the isotropic rect-angular waveguide. The electric type dyadic Green’s function due toan electric source is derived by using eigenfunctions expansion and theOhm-Rayleigh method. The irrotational term is properly extracted.


a. Intensity Distribution

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0500

10001500

00.02

0.040.06

0.080 0.02 0.04 0.06 0.08 0.1

0

0.01

0.02

0.03

0.04

0.05b. Ez Distribution

Figure 8. The distribution of TM12 mode in the transverse cross-section: (a) Electric field intensity and (b) Ez distribution.

a. Intensity Distribution

00.02

0.040.06

0.080.10

0.010.020.030.040.05

0200400600

00.02

0.040.06

0.080 0.02 0.04 0.06 0.08 0.1

0

0.01

0.02

0.03

0.04

0.05b. Ez Distribution

Figure 9. The distribution of TM32 mode in the transverse cross-section: (a) Electric field intensity and (b) Ez distribution.

It shows that the expression obtained can be readily reduced to thatin the case of filled isotropic media. The magnetic type dyadic Green’sfunction due to an electric source is also obtained from the electric onethrough Maxwell equations. As an application of the dyadic Green’sfunction, numerical results of the the modes excited by an infinitesimalelectric dipole are presented and discussed.

ACKNOWLEDGMENT

This project is supported in part by the Ministry of Education, Re-public of Singapore by a research grant RP3981676.

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Documents

RECTANGULAR CONDUCTING WAVEGUIDE … In Electromagnetics Research, PIER 25, 111–129, 2000 RECTANGULAR CONDUCTING WAVEGUIDE FILLED WITH UNIAXIAL ANISOTROPIC MEDIA: A MODAL ANALYSIS