RECTANGULAR CONDUCTING WAVEGUIDE FILLED WITH UNIAXIAL ANISOTROPIC MEDIA: A MODAL ANALYSIS AND DYADIC GREEN S FUNCTION

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1 Progress In Electromagnetics Research, PIER 5, , 000 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 Division Department of Electrical Engineering The National University of Singapore Singapore Introduction. Basic Formulation of the Problem 3. Fields in Source-free Rectangular Waveguides 4. Dyadic Green s Functions 5. Applications of Dyadic Green s Functions 6. Conclusion References 1. INTRODUCTION A general anisotropic medium is charaterized by permittivity tensor ɛ and permeability tensor µ [1 3], whose form depends on the ind of anisotropy. At present, different anisotropic materials are widely used in integrated optics and microwave engineering [4 8]. The technology advances are maing the production of substrates, dielectric anisotropic films and anisotropic material filling more and more convenient. It shows the necessity of better charaterizing the anisotropic media and producing more realistic models for the components that use them. Due to the complexity caused by the parameter tensors, the plane wave expansion is often used in the analysis of anisotropic media. And consequently, the Fourier transform is widely applied [1 3, 6, 9, 10] and will also be employed in this study. Using these methods, Uzunoglu et

2 11 Liu et al. al. [9] found the solution of the vector wave equation in cylindrical coordinates for a gyroelectric medium. Ren [10] furthered the wor under spherical coordinates in a similar procedure and obtained spherical wave functions and dyadic Green s functions in gyroelectric media. Recently, the theory of TE- and TM-decomposition attracts the interests of some researchers [11 13]. The TE- and TM-field decomposition is a method for solving electromagnetic problems involving certain class of boundaries and media that separate the field into reversely-handed circularly polarized waves. It is now extended to the problems of general uniaxially anisotropic media [11] and bi-anisotropic media [1, 13] from that of isotropic media. In this paper, the fields in a rectangular conducting waveguide filled with a uniaxially anisotropic material are studied. The characteristic feature of the uniaxial media is the existence of a distinguished axis. If one of the coordinate axes is chosen to be parallel to this distinguished direction, it turns out that the parameter tensor is diagonal but the element referring to the distinguished axis is different from the remaining two diagonal ones. Some features of the waveguides of uniaxial anisotropic material have been analyzed and their applications made by some authors [4, 7, 8]. It s found in our study that the fields in a rectangular conducting waveguide filled with uniaxially anisotropic material are split into TE and TM modes after solving the eigenvalue equation. The calculated dispersion curves are then depicted. There exist considerable effects of the material parameters on the cutoff frequencies of the propagating waves, especially the TM modes. The dyadic Green s functions for various inds of anisotropic media with different structures have been studied by many authors [14 0]. The problems, however, are mostly analyzed in spectral domain in terms of Fourier transform, due to the difficulty of finding the expansion of the dyadic Green s functions in terms of vector wave functions for anisotropic media. So far as we now, the dyadic Green s functions for anisotropic media-filled waveguides with perfectly conducting walls have not been shown in any literature. In our study, the suitable vector wave functions can be selected separately for the TE modes and TM modes by applying the boundary conditions. The dyadic Green s functions are then derived using Ohm-Rayleigh method [1]. It s shown that the expression obtained here is reducible to the isotropic case which has been obtained by Tai [1]. The magnetic type dyadic Green s function due to electric source can be obtained from the

3 Rectangular conducting waveguide 113 y perfectly conducting walls b o a x z 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 electric dipole are also given. Throughout this paper, the harmonic e iωt time dependence is assumed and suppressed. Bold prints indicate vectors and an overhead bold face indicates a dyad(ic). Caps with bold face represent unit vectors.. BASIC FORMULATION OF THE PROBLEM Consider a rectangular waveguide (Fig. 1) of which coordinates axis system is represented by (x, y, z). ẑ is the direction of propagation. The waveguide is filled with homogeneous electrically uniaxial anisotropic medium that is characterized by the following set of constitutive relations: D = ɛ 0 ɛ r E, B = µ 0 H, (1a) (1b) where ɛ 0 and µ 0 are the free space permittivity and permeability, respectively. We assume that the optics axis of the uniaxial media is oriented along the z-axis, and the other two principal axes are oriented along the two remaining coordinate axes. So the permitivity tensors ɛ r is given by ɛ r = ɛ t ɛ t 0. () 0 0 ɛ z

4 114 Liu et al. It can be proved that ɛ r taes the same form in both Cartesian and cylindrical coordinates systems. For this ind of media, the sourceincorporated Maxwell s equations are written as Substituting (3a) into (3b) yields E = iωµ 0 H, H = iωɛ 0 ɛ r E + J. (3b) (3b) E 0ɛ r E = iωµ 0 J, (4) where 0 = ω ɛ 0 µ 0, which is the free space wavenumber. 3. FIELDS IN SOURCE-FREE RECTANGULAR WAVEGUIDE Now we study the eigenvalues and eigenvectors of the source-free vector wave equation in an infinite uniaxial anisotropic medium. Let J = 0, (4) reduces to E 0ɛ r E =0. (5) The characteristic waves corresponding to (5) can be examined in the spectral domain using Fourier transform + E(r) = E()e i r d (6) where = x x + y ŷ + z ẑ. Substituting (6) to (5), we have + ( I 0ɛ r ) E()e i r d =0 (7) where I = x x + ŷŷ + ẑẑ is the unit dyadic. For nontrivial solutions of (7), it is required that the determinant of matrix ( I 0 ɛ r) must be equal to zero. Performing the algebraic operations, we obtain the characteristic equation finally as ɛ t 4 c +(ɛ t + ɛ z ) ( z 0ɛ t ) c + ɛ z ( z 0ɛ t ) =0 (8) where c = x + y. Solving the characteristic equation, we have the eigenvalues given by c1 = 0ɛ t z, c = 0ɛ z z ɛ z. ɛ t (9a) (9b)

5 Rectangular conducting waveguide 115 It can be seen that c1 is independent upon ɛ z while c is a function of ɛ z, which lead to the ordinary and extraordinary waves [1], respectively. To find the corresponding eigenvectors, we substitute these eigenvalues bac to (7). Finally we have E 1z =0, E 1x cos(φ )+E 1y sin(φ )=0, E 1 (φ, z )=E 1x x + E 1y ŷ, (10a) for c1 ; and E x = A( z ) cos(φ )E z, E y = A( z ) sin(φ )E z, E (φ, z )=E x x + E y ŷ + E z ẑ, (10b) for c ; where ɛ z z A( z )= ɛ t ɛz (0 (10c) z/ɛ t ) and φ = tan 1 ( y / x ). (10d) Obviously E 1 taes TE modes, which can be expressed using the vector wave function M with ẑ as the piloting vector [1]. In the Cartesian coordinate system, E z taes the form of e j(xx+yy+zz). From (3a) we have H z = 1 [ (E ŷ) (E ] x) =0, (11) iωµ 0 x y which means E taes TM modes. So E can be expanded using vector wave function N, and L with ẑ as the piloting vector as well. For rectangular waveguides bounded at x = 0 and a and at y = 0 and b by conducting walls, these vector wave functions can be written as [1] ( cos M e 1 (h) = o sin x1x cos N e (h) = 1 o L e o (h) = ) sin y1ye jzz ẑ ( cos sin xx cos sin yye jzz ẑ ( cos sin xx cos sin yye jzz)., (1a) ), (1b) (1c)

6 116 Liu et al. z a ε t =, ε z = 5 TE 10 TE 01 TE 0 TE 11 TM 11 TE 1 TM 1 TE 30 TE 31 TM Figure. Dispersion curves for the lowest 10 modes. 0 a It has been shown that fields can be decomposed into TE and TM modes. The observation is in agreement with Lindell s analysis [11] that decomposition method is applicable to such problems with perfect electric or magnetic conducting surfaces parallel to the z-axis. To match the boundary condition, ẑ E (r) boundary =0 (13) must be satisfied, because E 1z is zero everywhere. (13) straightforwardly leads to that we should choose N o and L o as eigenfunctions of E (r), and that x = m π/a, m =1,, ; (14a) y = n π/b, n =0, 1,,. (14b) In this case, x E (r) boundary =0 is automatically satisfied. So on the boundary, E 1 (r) must satisfy which leads to that only M 1e x E 1 (r) boundary =0, (15) can be chosen to represent E 1 (r), and x1 = m 1 π/a, m 1 =1,, ; (16a) y1 = n 1 π/b, n 1 =0, 1,,. (16b)

7 Rectangular conducting waveguide 117 Equations (14) and (16), together with (9), mae the dispersion relations for a rectangular conducting waveguide filled with a uniaxial anisotropic material. The calculated dispersion curves for ɛ z /ɛ t =.5 are depicted in Fig. where a/b =. Only the modes corresponding to the lowest 10 modes (which are listed in sequence in the legend of Fig. ) of an isotropic rectangular waveguide are shown. It is found that due to the existence of the optics axis, the order of these modes changes for the uniaxial material parameters. For example, TM 11 is the fifth lowest mode in isotropic rectangular waveguide with the dimension of a/b =,but it becomes the nd lowest one in uniaxial rectangular waveguide with the same dimension. This phenomenon suggests that we can select the propagating mode we want by changing the material parameters. Since the optics axis of the uniaxial media only affects the TM modes in the present case, in Fig. 3 the cutoff frequencies of some TM modes are shown as ɛ z increases from 1.0 to 10.0 (ɛ t =.0). It can be seen that the cutoff frequencies monotonically decrease as ɛ z increases. In Fig. 4 6, some lowest TM modes are plotted for a uniaxial (ɛ z /ɛ t =.5) rectangular waveguide and an isotropic (ɛ z /ɛ t =1.0) rectangular waveguide. The dimensions of the waveguide are selected so that a/b =.Itcan be seen that due to the existence of the optics axis, the distributions of the field intensity in the uniaxial waveguide are different from the isotropic ones. A greater ɛ z produces more significant variation of the intensity distribution. 4. DYADIC GREEN S FUNCTIONS In this section, we perform the eigenfunction expansion of dyadic Green s functions for a rectangular waveguide filled with uniaxial material maing use of the Ohm-Rayleigh method. Consider an electric source J expressed in terms of the Dirac-delta function δ(r r ) and the unit dyadic I as follow: J(r) = δ(r r )I J(r )dv. (17) V Due to the linearity of (4), the electric field can be related directly to the source [1] through E(r) =iωµ 0 V G EJ (r, r ) J(r )dv, (18)

8 118 Liu et al. 15 Cutoff Frequency ( c a) ε t = TM 11 TM 1 TM 31 TM 1 TM TM Figure 3. The cutoff frequencies for the lowest 6 TM-modes versus ɛ z. ε z a. TM 11 uniaxial b. TM 11 isotropic Figure 4. Electrical field intensities of TM 11 mode. a. TM 1 uniaxial b. TM 1 isotropic Figure 5. Electrical field intensities of TM 1 mode.

9 Rectangular conducting waveguide 119 a. TM 31 uniaxial b. TM 31 isotropic Figure 6. Electrical field intensities of TM 31 mode. where G EJ denotes electric dyadic Green function due to electric source, which is required to satisfy the dyadic Dirichlet condition n G EJ =0 on the conducting boundaries, and V stands for the volume occupied by the exciting current source. Substituting (17) and (18) into (4), we have G EJ (r, r ) 0ɛ r G EJ (r, r )=Iδ(r r ). (19) By inspection from the results of the previous section, G EJ be expanded using M 1emn, N omn and L omn as + G EJ (r, r )= d z [M 1emn ( z )a 1emn ( z ) n,m can also +N omn ( z )b omn ( z )+L omn ( z )c omn ( z )] (0) where a, b and c are unnown coefficient matrices to be determined. (16) and (14) must be met to satisfy the boundary conditions. For convenience, we have used the notation mn to stand for m 1,n 1 and m,n as well. Subsequently, we adopt the simplified notations M 1e, N o and L o instead of M 1emn, N omn and L omn, respectively. According to the Ohm-Rayleigh method we also expand the source function Iδ(r r ), similarly, into + Iδ(r r )= d z [M 1e ( z )P 1e ( z ) n,m +N o ( z )Q o ( z )+L o ( z )V o ( z )]. (1) Taing the anterior scalar product of (1) with M 1e ( z), N o ( z),and L o ( z)inturn and integrating the resultant equations through the entire volume of the rectangular waveguide, we can

10 10 Liu et al. determine the coefficient matrices P, Q and V given by P 1e ( z )= δ 0 πabc1 M 1e( z ), Q o ( z )= δ 0 πabc N o( z ), V o ( z )= δ 0 πab L o( z ). (a) (b) (c) To obtain (), the following orthogonality relations among the vector wave functions have been used, i.e., < M 1e ( z ), M 1e ( z) > = πab c1 (1 + δ 0 )δ mm δ nn δ( z z), (3a) < N o ( z ), N o ( z) > = πab c (1 δ 0 )δ mm δ nn δ( z z), (3b) < L o ( z ), L o ( z) > = πab (1 δ 0 )δ mm δ nn δ( z z), (3c) < M 1e ( z ), N o ( z) > =0, (3d) < M 1e ( z ), L o ( z) > =0, (3e) < N o ( z ), L o ( z) > =0, (3f) where { 1, m =0orn =0 δ 0 =, 0, otherwise { δ mm 1, m = m = 0, m m, { δ nn 1, n = n = 0, n n. In (3), we ve taen the operator <, > [] which defines < a, b >= a bdv. (4) Substituting (0) and (1) into (19), and taing the anterior scalar product of (0) with M 1e ( z), N o ( z), and L o ( z) in the same way, we finally arrive at the following equation in matrix form: [Ω][X] =[Θ], (5)

11 Rectangular conducting waveguide 11 where [Ω] is a 3 3 matrix given by [Ω] = Ω Ω Ω 3, 0 Ω 3 Ω 33 (6a) with Ω 11 = πab (1 + δ 0)c1 ( 1 0ɛ ) t, Ω = πab (1 δ 0)c [ ( 0 ɛt z + ɛ z c ) ], Ω 3 = Ω 3 = i πab (1 δ 0) z Ω 33 = πab (1 δ 0)0 ( ɛt c + ɛ z z c 0(ɛ t ɛ z ), ). In (6a), [X] and [Θ] are two column vectors given by [X] = a 1e( z ) b o ( z ), and [Θ] = M 1e ( z) N o ( z). c o ( z ) L o( z ) Solving (5), we have the solutions for a 1e ( z ), b o ( z ) and c o ( z ) as follows a 1e z )= δ 0 1 πabc1 1 M 1e ( z ), (7a) 10 b o ( z )= δ 0 1 πabc ( ɛ z 0)[β NN N o ( z )+β NL L o ( z )], (7b) c o ( z )= δ 0 1 πabc ( ɛ z 0)[β LN N o ( z )+β LL L o ( z )], (7c) where the ordinary and extraordinary wave numbers are defined as 10 = 0ɛ t, 0 = 0ɛ t + ( 1 ɛ ) t ɛ c; z (8a) (8b)

12 1 Liu et al. and the coupling coefficients are given by β NN = ɛ t c + ɛ z z, β NL = i z c(ɛ t ɛ z ), β LN = i z c(ɛ t ɛ z ), (8c) (8d) (8e) β LL = [ c 4 0 0(ɛ t z + ɛ z c) ]. (8f) Hence, (0) can be rewritten as + { G EJ (r, r δ0 )= d z 1 M 1e ( z )M 1e( z ) m,n πabc [ β NN N o ( z )N o( z ) + δ 0 πabc ( ɛ z 0) + β NL N o ( z )L o( z )+β LN L o ( z )N o( z ) ]} + β LL L o ( z )L o( z ). (9) In this way, the dyadic Green s functions for rectangular waveguides filled with uniaxial anisotropic media are explicitly represented in the form of the eigenfunction expansion in terms of the rectangular vector wave functions, as given in (9). However, for ease of practical applications and physical interpretation of possible novel phenomena, mathematical simplification to (9) is necessary. In order to apply the residue theorem to (0), we must first extract the irrotational term in (0) which does not satisfy the Jordan lemma as pointed out in [1]. To do so, we write N o ( z )=N ot ( z )+N oz ( z ), L o ( z )=L ot ( z )+L oz ( z ), (30a) (30b) and so are N and L. The subscripts t and z denote their transverse vector components and their z-vector components respectively. In terms of these functions, (9) can be rewritten in the form + { G EJ (r, r δ0 1 )= d z πabc1 1 M 1e ( z )M 1e( z ) 10 m,n

13 Rectangular conducting waveguide 13 + δ 0 1 πabc ( 0 ɛ z c ɛ z 0)[ z N ot ( z )N ot( z ) + N ot ( z )N oz( z )+N oz ( z )N ot( z ) + 0 ɛ t z ]} c N oz ( z )N oz( z ), (31) where we have expressed L ot ( z ) and L oz ( z )interms of N ot ( z ) and N oz (n, z ), namely, L ot ( z )= i N ot ( z ), (3a) z L oz ( z )= i z c N oz ( z ), (3b) and similarly for the primed functions. The singular term in (31) is contained in the component N oz ( z ) N oz ( z) [1]. From (1), we note that ẑẑδ(r r )= d z m,n δ 0 πab [ L oz ( z )L oz( z ) δ 0 = d z πab m,n c Maing use of the identity c N oz ( z )N oz( z ) ] c N oz ( z )N oz( z ). (33) 0 ɛ ( z 0) 0 ɛ t z c = ɛ t 0 ɛ zc + ( 0 ɛ z 0), (34) we can split (31) into G EJ (r, r δ 0 )= d z πab m,n c 0 ɛ zc N oz ( z )N oz( z ) + { δ0 1 + d z πabc1 1 M 1e ( z )M 1e( z ) 10 m,n + δ 0 πabc 0 ɛ ( 0 ɛ z c z 0)[ z N ot ( z )N ot( z ) + N ot ( z )N oz( z )+N oz ( z )N ot( z ) + ɛ t ɛ z N oz ( z )N oz( z ) ]}. (35)

14 14 Liu et al. In view of (33), the first integral in (35) is equal to 1 0 ɛ ẑẑδ(r r ), (36) z and the second integral can be evaluated by maing use of the residue theorem in the z -plane. The final result is given after some mathematical manipulations by G EJ (r, r )= 1 0 ɛ ẑẑδ(r r )+ i z ab 0 { 1 m,n [ N ot (± z )+ ɛ t + 0 ɛ tc z [ N ot( z )+ ɛ t N ɛ oz( z ) z c1 M 1e (± z1 )M 1e( z1 ) z1 ] N oz ( z ) ɛ z ]},z > < z (37) where z and z are the positions of the observation point and the source point, respectively, measured along the ẑ-direction, and z1 = 10 c1, z = 0 c, (38a) (38b) with 10 and 0 having been defined in (8). It can be observed that (37) is reducible to the isotropic case by assuming that ɛ t = ɛ z = ɛ. In this case, 10 = 0 = 0ɛ and (37) shall be simplified as: G EJ (r, r )= 1 0 ɛẑẑδ(r r )+ i [ δ 0 ab M e (± z )M m,n c e( z ) z ] + N o (± z )N o( z ), z > < z (39) which is exactly the same as that obtained by Tai [1]. 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 directly from (18). To obtain the magnetic field due to an electric current distribution, we need to utilize the following Maxwell s equation E = iωµ 0 H, (40a)

15 Rectangular conducting waveguide 15 y b z o a x Figure 7. Excitation of an electric dipole. or that in dyadics form G EJ = G HJ. (40b) If we express the magnetic field in terms of the magnetic dyadic Green s function, we have the following integral: H = G HJ (r, r ) J(r )dv. (41) V 5. APPLICATION OF DYADIC GREEN S FUNCTIONS In this section, an infinitesimal dipole in ŷ-direction is assumed as the electric current source in the rectangular waveguide filled with uniaxial material. In a three-dimensional expression form, this source distribution is expressed by J(r )=I 0 ŷδ(x x 0 )δ(y y 0 )δ(z z 0 ) (4) where the point (x,y,z ) islocated at the center of the rectangular waveguide cross section as shown in Fig. 7, i.e., x 0 = a, y 0 = b, z 0 =0. (43) Substituting the above current distribution and (37) into (18), we have

16 16 Liu et al. E(r) = I 0ωµ 0 ab ± i 0 y 0 ɛ t c + ɛ t ɛ z N oz ( z ) { [ x1 ( a ) ( b ) ] m,n c1 sin x1 cos y1 M 1e (± z1 ) z1 [ ( a ) ( b ) ][ sin x cos y N ot (± z ) ]}, z > < z. (44) From (44), it is easy to find out that only the modes with odd m and even n are excited. Since for TE waves, a uniaxial medium with the optics axis in z-direction loos lie an isotropic medium, the fields of TE modes in the uniaxial anisotropic medium should have the same representations as the well-nown ones in the isotropic medium. Therefore, we should discuss the features of the TM-mode fields only. In our subsequent numerical computation, the dimensions of the waveguide and the parameters of the material are selected as the same as those in the previous section (i.e., a/b =, ɛ t =.0, and ɛ z =.5), for ease and meaningfulness of comparison. Fig. 8 and Fig. 9 show the electric field intensity and E z distributions of TM 1 mode and TM 3 mode in the transverse cross-section. It is seen that the distribution of the guided TM modes in a uniaxial anisotropic medium has changed as compared with those guided ones in an isotropic medium. 6. CONCLUSION The electromagnetic fields in rectangular conducting waveguides filled with uniaxial anisotropic media are characterized in this paper. With the optics axis of the uniaxial media oriented along the z-axis of the waveguide, the fields can be decomposed into TE and TM modes with different propagation wave numbers. The ordinary wave taes the form of TE modes while the extraordinary wave taes the form of TM modes. The calculated dispersion curves are depicted and the effects of the material parameters on the cutoff frequencies are shown. The cutoff frequencies change considerably, especially for TM modes. The field distributions are drawn and compared with those in the isotropic rectangular waveguide. The electric type dyadic Green s function due to an electric source is derived by using eigenfunctions expansion and the Ohm-Rayleigh method. The irrotational term is properly extracted.

17 Rectangular conducting waveguide 17 a. Intensity Distribution b. E z Distribution Figure 8. The distribution of TM 1 mode in the transverse crosssection: (a) Electric field intensity and (b) E z distribution. a. Intensity Distribution b. E z Distribution Figure 9. The distribution of TM 3 mode in the transverse crosssection: (a) Electric field intensity and (b) E z distribution. It shows that the expression obtained can be readily reduced to that in the case of filled isotropic media. The magnetic type dyadic Green s function due to an electric source is also obtained from the electric one through Maxwell equations. As an application of the dyadic Green s function, numerical results of the the modes excited by an infinitesimal electric dipole are presented and discussed. ACKNOWLEDGMENT This project is supported in part by the Ministry of Education, Republic of Singapore by a research grant RP REFERENCES 1. Chen, H. C., Theory of Electromagnetic Waves, McGraw-Hill, New Yor, Kong, J. A., Electromagnetic Wave Theory, The nd ed., John Wiley & Sons, New Yor, 1986.

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