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Spheroidal Wave Functions in Electromagnetic Theory Le-Wei Li, Xiao-Kang Kang, Mook-Seng Leong Copyright  2002 John Wiley & Sons, Inc ISBNs: 0-471-03170-4 (Hardback); 0-471-22157-0 (Electronic) Dyadic Green’s Functions in Spheroidal Systems 3.1 DYADIC GREEN’S FUNCTIONS To analyze the electromagnetic radiation from an arbitrary current distribution located in a layered inhomogeneous medium, the dyadic Green’s function (DGF) technique is usually adopted If the geometry involved in the radiation problem is spheroidal, the representation of dyadic Green’s functions under the spheroidal coordinates system should be most convenient If the source current distribution is known, the electromagnetic fields can be integrated directly from where the DGF plays an important role as the response function of multilayered dielectric media If the source is of an unknown current distribution, the method of moments [87], which expands the current distribution into a series of basis functions with unknown coefficients, can be employed In this case, the DGF is considered as a kernel of the integral; and the unknown coefficients of the basis functions can be obtained in matrix form by enforcing the boundary conditions to be satisfied Dyadic Green’s functions in various geometries, such as single stratified planar, cylindrical, and spherical structures, have been formulated [ 14,88-901 In multilayered geometries, the DGFs have also been constructed and their coefficients derived Usually, two types of dyadic Green’s functions, electromagnetic (field) DGFs and Hertzian vector potential DGFs, were expressed Three methods that are common and available in the literature: the Fourier transform technique (normally, in planar structures only), the wave matrix operator and/or transmission line (frequently, in planar structures) method, and the vector wave eigenfunction expansion method (in regular structures 61 62 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS where vector wave functions are orthogonal) In general, two domains are assumedin formulations of the DGFs: i.e the time domain and the spectral (or frequency) domain, where the spatial variables T and T’ are still in use However, it must be noted that the spectral domain has a different meaning in the derivation of the DGFs for planar stratified media This is because the Fourier transform is frequently utilized there to transform part of the spatial components from the conventional spectral domain to a Fourier transform domain The conventional spectral (or frequency) domain in this case is referred to a~ the spatial domain and the Fourier transform domain as the partial spectral domain, where spectral components such as k, associated with the discontinuity along the direction are considered In a planar stratified geometry [ 141, Lee and Kong [91] in 1983 employed the Fourier transform to deduce the DGFs in an anisotropic medium; Sphicopoulos et al [92] in 1985 used an operator approach to derive the DGFs in isotropic and achiral media; Das and Pozer [93] in 1987 utilized the Fourier transform technique; Vegni et al [94] and Nyquist and Kzadri [95] in 1991 made use of wave matrices in the electric Hertz potential to obtain DGFs and their scattering coefficients in isotropic and achiral media; Pan and Wolff [96] employed scalarized formulas, and Dreher [97] used the Fourier transform and method of lines to rederive the DGFs and their coefficients in the same medium; Mesa et al [98] applied the equivalent boundary method to obtain the DGFs and their coefficients in two-dimensional inhomogeneous bianisotropic media; Ali et al [99] in 1992 used the Fourier transform, and Li et al [loo] in 1994 employed vector wave eigenfunction expansion to formulate the DGFs and formulated their coefficients in isotropic and chiral media; Bernardi and Cicchetti [loll again employed Fourier transform and operator technique to the same medium but with backed conducting ground plane; Barkeshli utilized the Fourier transform technique in 1992 and 1993 to express the DGFs and uniaxial [102] media, dielectric/magnetic media their coefficients in anisotropic media [104]; Habashy et al [105] in 1991 applied the [103], and gyroelectric Fourier transform technique to work out the DGFs in arbitrarily magnetized linear plasma For the casesof a free space (or unbounded space), a singlelayered medium, or a multilayered structure, many references exist, such as various representations by Pathak [lOS], C avalcante et al [107], Engheta and Bassiri [108], Chew [go], Gl isson and Junker [1091, Krowne [l lo], Lakhtakia [ill-1131, Lindell [114], T oscano and Vegni [1151, and Weiglhofer [116-1201 Since a large number of publications are available, it is impractical to list all of them here In a multilayered cylindrical geometry [14], the DGFs in the chiral media and the specific coefficients were given in 1993 by Yin and Wang [121] Later in 1995, the unified DGFs in chiral media and their scattering coefficients in general form were formulated by Li et al [122] the DGFs in achiral In a multilayered spherical geometry [14,123,124], media and their scattering coefficients were generalized in 1994 by Li et al FUNDAMENTAL FORMULATION 63 11251 This work was then extended in 1995 to the DGFs in chiral media, and their scattering coefficients were formulated again by Li et al [126] In a spheroidal geometry, the dyadic Green’s functions in an unbounded medium were constructed in 1995 by Giarola [127] and by Li et al [128], respectively Also, scattering DGFs in the presence of (1) a perfectly conducting prolate spheroid [1271 and (2) a dielectric spheroid that can reduce to a conducting spheroid by letting the permittivity approach infinity [128] were represented It is shown in [128] that formulation of the DGFs in spheroidal structures is difficult and that the difficulty is due to the following two facts: (1) no recursive relations of the spheroidal angular and radial functions can be obtained by the methods generally used for the more common special functions of mathematical physics (the existing recurrence relations of Whittaker type are, as stated by Flammer [I], actually identities, not recursion formulas); and (2) the coupling series coefficients of the scattered fields must be calculated numerically by the inversion of coefficients of matrices However, the formulation in [127] is valid only when its spherical limit is approached, since the orthogonality of Eqs (7) and (8) in [1271 is valid only in the limit when the spheroid approaches a sphere Later in 2001, Li et al [13] formulated not only the DGFs in a two-layered spheroidal structure, but also the corresponding matrix equations for their scattering coefficients due to the spheroidal interface The DGFs in a multilayered spheroidal structure in general form were recently formulated by Li et al [15,129] as an extension 3.2 FUNDAMENTAL FORMULATION To analyze the EM fields in spheroidal structures, we consider a prolate spheroidal geometry of multilayers as shown in Fig 3.1 Here all the spheroidal interfaces are assumed to have the same interfocal distance d Oblate spheroidal problems can be analyzed by a procedure similar to that presented here or by the symbolic transformations, < + *it and c $ tic, where c = $d (Ic is the wave propagation constant, as indicated in Chapter 2) Assume that the space is divided by N - spheroidal interfaces into N regions, as shown in Fig 3.1 The spheroidally stratified regions are labeled, respectively, f = 1,2,3, N The EM radiated fields Ef and Hf in the fth (field) region (f = 1,2,3, ) N) due to the electric and magnetic current distributions J, and MS located in the sth (source) region (s = 1,2,3, , N), as shown in Fig 3.1, can be expressed by VxVx VxVxHf Ef - kfEf= -kfHf= [aqJr-@XWf] [iw&fMf+@xJ)f] b, (3.la) Sfs, (3.lb) where 6f, denotes the Kronecker delta (= for f = s and for f # s), kf= w J PfEf (1 + iaf/wEf) is the wave propagation constant in the fth 64 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS f2 Y Fig 3.1 Geometry of a multilayered prolate spheroid under coordinates (c, q, q5), FUNDAMENTAL 65 FORMULATlON layer of the multilayered medium, and &f, pf, and af identify the permittivity, permeability, and conductivity of the medium, respectively A time dependence exp(-iwt) is assumed to describe the EM fields throughout the book Moreover, the media reduce to free space if pf = ~0 The EM fields excited by an electric current source J, and a magnetic current distribution A& can be expressed in terms of integrals containing dyadic Green’s functions as follows [14,100,122,125]: Ef (r) = iops @$r, sss c,;(r, Hf (r) = iw~~ dV’ M,(T’) dV’, l (f > sss J&‘) T’) V T’) l (3.2a) V lJ/ @Z(T, r’) l MS@‘) dV’ V +;‘(T, T’) l J,(d) dV’, (3.2b) where the prime denotes the coordinates ( -(f and GHJ (Y, Y’), and C$$ (T, T’) and C$$ (T, T’), as follows: Tai [141 defined $$J’ (T, rl) and GCf ‘) (T, r’) as the electric and magnetic HJ dyadic Green’s functions of the first kind [i.e., ~L{‘)(T, T’) and C:~)(T, r’>]; and @&$(Y, Y’) and cCfs’ (T, rl> as the electric and magnetic dyadic Green’s HIM functions of the second kind [i.e., GLi”(r, rl) and E~;‘(T, r’)] Substituting Eqs (3.2a) and (3.2b) into Eqs (3.la) and (3.lb), respectively, we obtain (3.4a) (3.4b) 66 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS where r stands for the unit/identity dyad and 6(r - #) identifies the Dirac delta funct& Since c,J ( T, T’) and -(f (r, rl> are related by the upper elements of G,, Eqs (3.3a) and (3.3b) and @$,, rl> and @$T, rl) by the lower elements of Eqs (3.3a) and (3.3b), we not need to derive all of them Instead, only (f ) the formulations of -(f (T, rl) and Z&&T, T’) are considered here The G,, following boundary conditions at the spheroidal interface, < = cf, are satisfied by various types of dyadic Green’s functions after Eqs (3.2a) and (3.2b) are substituted into the EM field boundary conditions: +Gl EJ+a4 @f -[ I - (x - @f ’ (3.5a) HM (3.5b) where i Wf Gf +1 +1 J stands for the ruling that either the upper or lower elements of the matrices should be taken at the same time In fact, Eqs (3.5a) and (3.5b) represent four equations if all the upper and lower elements are considered, respectively from the -(fs) GE, Furthermore, the DGF $$, rl> can be obtained (T, r’) by making the simple duality replacements E ) H, H+ E,J-+M,M-+-J,p u,and~+p 3.3 3.3.1 UNBOUNDED DYADIC GREEN’S FUNCTIONS Method of Separation of Variables According to Collin [88], the scalar Green’s function g( T, r’) satisfies the following differential equation: (V2 + k2) g(r, T’) = -6(r - r’> (3.6) In a source-free region, the solution of the EM fields, Emn and Hmn, for the wave modes mn can be found by using the well-known method of separation of variables, and is given by the radial function 7f(k, c) and the angular functions O(k,q) and @(k, 4) as follows: * IFl(k, - A/R(‘) (c,