# Cấu trúc sóng chức năng trong điện lý thuyết P3

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## Cấu trúc sóng chức năng trong điện lý thuyết P3

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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....

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1. 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) 3 Dyadic Green’s Functions in Spheroidal Systems 3.1 DYADIC GREEN’S FUNCTIONS To analyze the electromagnetic radiation from an arbitrary current distribu- tion 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 di- rectly from where the DGF plays an important role as the response function of multilayered dielectric media. If the source is of an unknown current distri- bution, 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, electro- magnetic (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
2. 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 trans- form 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 scat- tering 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 Cic- chetti [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 their coefficients in anisotropic uniaxial [102] media, dielectric/magnetic media [103], and gyroelectric media [104]; Habashy et al. [105] in 1991 applied the Fourier transform technique to work out the DGFs in arbitrarily magnetized linear plasma. For the casesof a free space (or unbounded space), a single- layered 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]. In a multilayered spherical geometry [14,123,124], the DGFs in achiral media and their scattering coefficients were generalized in 1994 by Li et al.
3. 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 con- ducting 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 spher- oidal 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 Whit- taker 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 spher- oidal geometry of multilayers as shown in Fig. 3.1. Here all the spheroidal interfaces are assumed to have the same interfocal distance d. Oblate spher- oidal 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 - 1 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) re- gion (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 2 VxVx Ef - kfEf= [aqJr-@XWf] b, (3.la) 2 VxVxHf -kfHf= [iw&fMf+@xJ)f] Sfs, (3.lb) where 6f, denotes the Kronecker delta (= 1 for f = s and 0 for f # s), kf= w J PfEf (1 + iaf/wEf) is the wave propagation constant in the fth
4. 64 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS f2 Y Fig. 3.1 Geometry of a multilayered prolate spheroid under coordinates (c, q, q5),
5. FUNDAMENTAL FORMULATlON 65 layer of the multilayered medium, and &f, pf, and af identify the permit- tivity, 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, T’) l J&‘) dV’ sss V - (f > c,;(r, T’) l M,(T’) dV’, (3.2a) sss V Hf (r) = iw~~ lJ/ V @Z(T, r’) l MS@‘) dV’ +;‘(T, T’) l J,(d) dV’, (3.2b) where the prime denotes the coordinates ( -(f 4 and GHJ (Y, Y’), and C$$(T, T’) and C$$ (T, T’), as follows: Tai [141 defined $$J’ (T, rl) and GCf ‘) (T, r’) as the electric and magnetic HJ dyadic Green’s functions of the first kind [i.e., ~L{‘)(T, T’) and C:~)(T, r’>]; and @&(Y, Y’) and cCfs’ (T, rl> as the electric and magnetic dyadic Green’s HIM functions of the second kind [i.e., GLi”(r, rl) and E~;‘(T, r’)]. Substituting Eqs. (3.2a) and (3.2b) into Eqs. (3.la) and (3.lb), respec- tively, we obtain (3.4a) (3.4b) 6. 66 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS where r stands for the unit/identity dyad and 6(r - #) identifies the Dirac delta funct& Since c,J ( T, T’) and -(f 4 (r, rl> are related by the upper elements of G,, Eqs. (3.3a) and (3.3b) and @,, rl> and @T, rl) by the lower elements of Eqs. (3.3a) and (3.3b), we do not need to derive all of them. Instead, only the formulations of -(f 4 (T, rl) and Z&&T, T’) are considered here. The G,, (f ) following boundary conditions at the spheroidal interface, < = cf, are satisfied by various types of dyadic Green’s functions after Eqs. (3.2a) and (3.2b) are substituted into the EM field boundary conditions: +Gl -[ I @f - (x - EJ+a4 (3.5a) @f ’ HM (3.5b) where iWf Gf +1 +1 Jstands for the ruling that either the upper or lower elements of the matrices should be taken at the same time. In fact, Eqs. (3.5a) and (3.5b) represent four equations if all the upper and lower elements are con- sidered, respectively. Furthermore, the DGF$$, rl> can be obtained from the -(fs) GE, (T, r’) by making the simple duality replacements E --) H, H+--E,J-+M,M-+-J,p--u,and~+p. 3.3 UNBOUNDED DYADIC GREEN’S FUNCTIONS 3.3.1 Method of Separation of Variables According to Collin [88], the scalar Green’s function g( T, r’) satisfies the fol- lowing 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, 7. UNBOUNDED DYADIC GREEN’S FUNCTIONS 67 - C’S$&, 7) + D’Sg;(c, q), (3.7b) @(k, qb)= E cos(mqb)+ F sin(m$), (3.7c) where m and n identify the eigenvalue parameters; A, B, A’, B’, C, D, C’, D’, E, and F are constants; and P,“(aJ) and .Cr(cq p) denote the generalized Legendre functions in general [9]. However, Py (c, c) and Lr (c, 8. DYADIC GREEN’S FUNCTIONS IN SPHEROiDAL SYSTEMS (3.9) where r> and T< denote the coordinate vector T, where c is taken as max( 9. UNBOUNDED DYADIC GREEN’S FUNCTlONS 69 given in Appendix A. Unfortunately, those sets of vector wave functions are neither orthogonal among themselves, nor, in general, orthogonal to the other sets. Thus, it is inconvenient to employ them to construct dyadic Green’s functions by the conventional method described by Tai [14]. Therefore, a combined method, developed from the two methods above, is presented for the layered spheroids. 3.3.4 Unbounded Green’s Dyadics One way to formulate the dyadic Green’s functions is to solve Eqs. (3.4a) and (3.4b) for them; a second is to employ the following relations between the Green’s dyadics and the scalar Green’s function in unbounded space, accord- ing to Tai [14] and Collin [SS]: cEJO(T, T’) = [l+$vv.] Fg(V?] , (3.12a) GHJO(T, T’) = v x [%l(T, T’J] = vg(T, T’) X f, (3.12b) where the subscript 0 next to EJ and HJ stands for the unbounded space. In terms of the above-defined spheroidal vector wave functions in an explicit bivector form, the electric dyadic Green’s functions given in Eqs. (3.12a) and (3.12b) can be obtained after substitution of Eq. (3.9) for < 2
10. 70 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS (c’,~‘, 4’). The first term of Eq. (3.13a) stands for the nonsolenoidal con- tribution and can be obtained by using the method given by Tai [14, pp. 128-129, 1541. It is worth mentioning that the singularity of the Green’s functions was a controversial issue in the late 1970’s. Now, the issue of irrotational DGF’s has been well resolved and is no longer the problem to the electromagnetics com- munity. In this chapter, the irrotational part of the Green’s dyadic is found from a combination of two contributions: one of them taken directly from the unit delta dyadic, and the other obtained from the first order derivative of the Green’s function at the discontinuity point at 5 = = GEJO(T, T’)Sfs + G:;;(c r’), CH; 1(T, T’) = (f cH.J&, T’)bfs + @$;‘,(T, r’), (3.14b) where the scattering DGF ctfs) EJs(y, T’) and &;, 1 (T, P’) describe the addi- (f tional contribution of the multiple reflection and transmission waves in the presence of the boundaries produced by the dielectric media, while the un- bounded dyadic Green’s functions GE JO (VJrl) and GH JO (T, rl) , given respec- tively by Eqs. (3.12a) and (3.12b), represent the contribution of the direct waves from radiation sources in an unbounded medium. The superscript (fs) denotes the layers where the field point and the source point are located, respectively, and the subscript s identifies the scattering dyadic Green’s func- tions. When the antenna is located in the sth region, the scattering dyadic Green’s function in the fth region must be of a form similar to that of the unbounded Green’s dyadic. To satisfy the boundary conditions, however, the additional spheroidal vector wave functions M, ‘$)Jc; q, 5,4) and Ni$f,Jc; q, C, (6) shdd be included to account for the effects of multiple transmissions and reflections. For easeof determination of the scattering coefficients, the sets of vector wave functions M*(l) Emfl ,M) and Ng;l JGE) are used in construction of the scattering DGFs. ‘MA@) ,(c, c) and ‘A$‘L1 ,(c, t) are defined as follows: Emfl 7 9 (3.15a)
11. SCATTERING GREEN’S DYADICS 71 (3.15b) where X denotes either A4 or Iv. For a two-layered spheroidal geometry, the dyadic Green’s functions have been given by Li et al. [13,128]. Therefore, the scattering dyadic Green’s func- tions in each region of a multilayered spheroidal structure can be formulated in a similar fashion. In this section, the following three casesare discussed. 3.4.1 Scattering Green’s Dyadics in the Inner Region (f = 1) (3.16a) (3.16b) 3.4.2 Scattering Green’s Dyadics in the Intermediate Regions (2
12. 72 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS (3.17a) (3.17b) 3.4.3 Scattering Green’s Dyadics in the Outer Region (f = N) . 00 00 -(f 4 %Jsh~‘~ = $c c n=m m=O 13. DETERMINATION OF SCATTERING COEFFICIENTS 73 + (3.18a) [ @;‘,(r, r’) =$25 n=m m=O +V -xM fEmn + + (3.18b) [ HWe brn09 SflV, and sfl are Kronecker delta functions. cS = &d and cf = $d, where k, and kf are, respectively, the wave propagation constants in the media where the source and field points are located. d$f~p'Z"M'N)' BC’x,*P~“)(M~N) C(*“I’PY~)(~Y~), and ,D(~“Y’?JY’)(~‘N) are u&no&n scattering fEmn ’ fzrnn fZmn coefficients to be determined from the EM field boundary conditions. 3.5 DETERMINATION OF SCATTERING COEFFICIENTS 3.5.1 Nonorthogonality and Functional Expansion In Eqs. (3.16a) to (3.18b), the DGFs are expressed in terms of appropriate spheroidal vector wave functions using the principle of scattering superposi- tion. Because of the lack of general orthogonality of the spheroidal vector wave functions, the Green’s dyadics are expressed in a different way, where the coordinate unit vectors are also combined in the construction, as shown in Eqs. (3.16), (3.17), and (3.18). The unknown scattering coefficients of the DGFs above can be determined from the EM field boundary conditions at the multilayered spheroidal interfaces (at c = cf, where the subscript f denotes the fth region of the spheroidal multilayers), as Eqs. (3.5a) and (3.5b). Af- ter substitution of Eqs. (3.14a) and (3.14b) into Eqs. (3.15a) and (3.15b),
14. 74 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS respectively, the following relations of vector wave functions are used in the vector operations: (3.19b) (3.19c) (3.19d) These relations are the same as those of vector wave functions in the orthog- onal coordinate systems [ 1,141. Because of the orthogonality of the trigonometric functions, the coefficients of the same $-dependent trigonometric function in Eqs. (3.5a) and (3.5b) must be equal, component by component; the equalities must hold for each corresponding term in the summation over m. For the summation over n, however, the individual terms in the series cannot be decomposed term by term because of the nonorthogonality of the spheroidal radial functions. This causes difficulty in determining the unknown scattering coefficients. To solve for the unknown coefficients, the following expanded intermediate forms [10,241 are introduced: for m > 1, (3.20a) (3.20b) 2 3/2 m grzn(C) ’ p,“_i:t(rl) = (l - 73 > sn Cc7rl), (3.20~) t=o (3.20d) (3.20e) (3.20f) (3.20h) 15. DETERMINATION OF SCATTERING COEFFICIENTS 75 (3.2Oi) (3.2Oj) (3.20k) (3.201) and for m = 0, 2 Ii-$W l Pt(rl) = (1 - ?7-2)1’2s;(c,7j), (3.20m) t=o 5 I$(4 l pl!+to = (1 - ?jJ2)3’2s;(c,r& (3.20n) t=o (3.200) (3*2OP) ww (3.20r) (3.20s) (3.20t) (3.20~) (3.20~) where Sr (c, 77)and St@, 77)are spheroidal angular functions, PgIt+t (7) and Pt+Jq) are associated Legendre functions, and the intermediates Ityln (t = 0, 1,2,. . . and e = 1,2,. . . , 12) have been provided in closed form in Appendix B. The individual terms in the summation over t must be matched term by 16. 76 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS term by considering the orthogonality of the associated Legendre functions P,“-t+&). - By substitution of the equations above, all factors that are func- tions of 7 are replaced by a seriesof associated Legendre functions, which are orthogonal functions in the interval -1 5 q 5 1. Thus, the equations used to determine the unknown coefficients constitute an infinite system of coupled linear equations. 3.5.2 Matrix Equation Systems Finally, the equations used to determine the unknown coefficients constitute an infinite system of coupled linear equations and the unknown coefficients can be solved for from the following matrix equation systems: (&?) = (4hJ’ l (rh) l (rh) 7 (3.21) where h = +x, -x, +y, -y, and x, respectively. Also in Eq. (3.21), (A&) is the matrix of the unknown coefficients to be determined, (Ahv) and (rh) are the matrices of constant elements obtained from the functional expansions, and (rh) is the constant matrix in which the integrals of source currents are involved. For an (N - 1)-layered spheroidal structure, if the truncation number of the summation over n is chosen as NT, which means that n is takenasm,m+l,...,m+A+- 1 as an approximation (within the controlled accuracy) to the infinite summation for a given m, the matrices in Eq. (3.21) can be expressed subsequently. Only the solution of the scattering coefficients of the electric DGFs is presented here, to avoid unnecessary repetition. 3.5.2.1 The matrix The matrix on the left-hand side of (3.21) is found in its explicit form to be (3.22) where T denotes the transpose of a matrix and the element matrices are defined as T Ah dhM f$%m' dhM f zm,m+l ’ dhM fzm,m+2’ ----- ” l ’ dhM fgm,m-Wh-1 1 (3.23) f= T dhN dhN f zm,m+l’ dhp f,m,m+2’ ” dh,N fom,m+NT-l fgm,m’ l l 1 ) for f = 1,2,3 ,..., IV- 1, and T Bh t3hM fZm,m’ t?hM f zm,m+l ’ BhM fEm,m+2’ ----- ’ l ’ ’ BhM f\$wn+NT-l 1 (3.24) f= T l?hN f zm,m’ 13hN fgm,m+l’ ghN fzm,m+2’ ’ ’ l ’ 13hN f;m,m+NT-1 1
17. DETERMINATION OF SCATTERING COEFFICIENTS 77 for f =2,3,4 ,..., N. In the special case of the single-layered spheroidal structure (N = 2), AhM1:m,m Ah* 1°m,m+l Ah% lzm,m+2 AhM lzm,m+NT-1 AhN fm,m AhnP lem,m+l AhXr lEm,m+2 AhN lzm,m+Nr-1 (few) = ----- (3.25) f3hM 2gm,m+NT-1 ahN 2em,m 13hfl 2Em,m+l t?hN 2Em,m+2 t3hN 2Em,m+NT-1 3.5.2.2 The matrix Ah In a similar fashion, the first matrix on the right- ( > hand side of Eq. (3.21) is expressed explicitly as ah0 nh 1 - - oh 2 (3.26a) (Ah >
18. 78 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS where the element matrices are given by ‘i w wp 0; 0; l l l o;-2 op o[ 0; w; w; 0; l l l o;-2 oy 0; . . . . . . . . . . . . . . . . . . . . . . of-2 wi-l ‘Wf nh = f-1 f-1 f-1 f-1 * f-1 -1 f-1 O1 O2 O4 O3 t l of-2 f -1 wf f O1 O2 O3 O4 Of f f f f l ** f . . . . . . . . . . . . . . . . . . O1 O2 O3 O4 . Oflz y-2 r-2 N-2 N-2 l * #-; O3 O4 oN-l ON iON- ON-1 N-1 N-1 l *’ ONI1 N-l N-l . . . ON-3 1 ON-2 1 ON-l 1 ON \ 1 N-2 . . . ON-3 2 02 oN-l 2 ON2 of+1 of+2 . ON-3 ON-2 oN-l ON f-l f-l * f-l f-l f-l f-l wf+l of+2 l ... q-3 q-2 q-1 0; . f f . . . ON-3 wN-2 wN-1 ON N-2 N-2 N-2 N-2 . ON-3 ON-2 ” N-l N-l WE-; - w; - 1I (3.26b) In Eq. (3.26b) the submatrices Of are defined: for 1 < I < N - 1, 2 5 f 5 IV- 1, as Of 00 00 0 0 0 0 1= 0000’ ( 1 0 0 0 0 andforlQ
19. DETERMINATION OF SCATTERING COEFFICIENTS 79 with the number of 0 elements being NT; the submatrices Wf f are given for f=lby and for 2 - f < N - 1 by < - while the submatrices W:-l are derived for f = N as - u~~3)‘t(C*‘~*-1) -v,h(3)7t(C*‘&Ll) - u;‘3”t(cN,
20. 80 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS V;xo’t ( CP) tfq) = (cp, Sq) v;;;;tm+,)(c,, cq) v+(i) 4 v+@)‘t 3m(m+2) ( cp ~+m(m-WT--l) (cp,