3 Dyadic Green’s Functions in Spheroidal Systems To analyze the electromagnetic radiation from an arbitrary current distribu- tion located in a layered inhomogeneous medium, the dyadic
Trang 13
Dyadic Green’s Functions
in Spheroidal Systems
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
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)
Trang 262 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
where vector wave functions are orthogonal) In general, two domains are assumed in 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 cases of 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
Trang 3to 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
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 = $d (Ic is the wave propagation constant, as indicated in Chapter 2) Assume that the space is divided by N - 1 spheroidal interfaces into N regions, as shown in Fig 3.1 The spheroidally stratified regions are labeled, respectively,
f = 1,2,3, N The EM radiated fields Ef and Hf in the fth (field) re- gion (f = 1,2,3, ) N) due to the electric and magnetic current distributions
J, and MS located in the sth (source) region (s = 1,2,3, , N), as shown
in Fig 3.1, can be expressed by
Trang 464 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
Trang 5FUNDAMENTAL FORMULATlON 65
layer of the multilayered medium, and &f, pf, and af identify the permit- tivity, permeability, and conductivity of the medium, respectively A time dependence exp(-iwt) is assumed to describe the EM fields
book Moreover, the media reduce to free space if pf = ~0
throughout the The EM fields excited by an electric current source J, and a magnetic current distribution A& can be expressed in terms of integrals containing dyadic Green’s functions as follows [14,100,122,125]:
Ef (r) = iops
sss
@$r, T’) l J&‘) dV’
V (f >
- sss
and M, , and V identifies the volume occupied by the sources in the second region
Substituting Eqs (3.2a) and (3.2b) back into (3.la) and (3.lb), we have
-(f 4 the relationships between the various types of Green’s dyadics, GEJ (T, rl> -(f 4
and GHJ (Y, Y’), and C$$ (T, T’) and C$$ (T, T’), as follows:
Tai [ 141 defined $$J’ (T, rl) and GCf ‘) HJ (T, r’) as the electric and magnetic dyadic Green’s functions of the first kind [i.e., ~L{‘)(T, T’) and C:~)(T, r’>]; and @&$(Y, Y’) and cCfs’ HIM (T, rl> as the electric and magnetic dyadic Green’s functions of the second kind [i.e., GLi”(r, rl) and E~;‘(T, r’)]
Substituting Eqs (3.2a) and (3.2b) into Eqs (3.la) and (3.lb), respec- tively, we obtain
(3.4a)
(3.4b)
Trang 666 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
where r stands for the unit/identity dyad and 6(r - #) identifies the Dirac delta funct&
Since c,J ( T, T’) and G,, -(f 4 ( r, rl> are related by the upper elements of Eqs (3.3a) and (3.3b) and @$,, rl> and @$T, rl) by the lower elements
of Eqs (3.3a) and (3.3b), we do not need to derive all of them Instead, only
stands for the ruling that either the upper or lower elements
of the matrices should be taken at the same time In fact, Eqs (3.5a) and (3.5b) represent four equations if all the upper and lower elements are con- sidered, respectively Furthermore, the DGF $$, rl> can be obtained
-(fs)
from the GE, ( T, r’) by making the simple duality replacements E ) H,
H+ E,J-+M,M-+-J,p u,and~+p
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:
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,<) = Apn”(c,t) + Bc:(c,t>
- A/R(‘) (c,<) + B’R$,$(c,c),
- O(k, 77) = CP;i, r]) + DLr(c, 7)
(3.7a)
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-
- C’S$&, 7) + D’Sg;(c, q),
@(k, qb) = E cos(mqb) + F sin(m$),
(3.7b) (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, <) are specifically referred to as the first and second kinds of radial functions R$$(c, <) and &k (c, c) [I], respectively They can also be considered as the generalized spherical Bessel functions of the first and second kinds since they have properties similar to those of jn(kr) and gn(kr) in spherical coordinates Therefore, the third and fourth kinds of radial functions &’ (c <) and Ln(c, (4) [) can also be constructed in terms of the first and second”kihds, similar to those of the third and fourth kinds of spherical Bessel functions [i.e., the Hankel functions hi’) (kr) and hp)(kr) of the first and second kinds] To simplify the representation of radial functions
of different kinds, the radial function of the ith kind, R$&(c, <) (i = 1,2,3, or 4), takes the usual form; therefore, &L(c, c) and R$,$(c, c) represent outgoing and incoming waves, respectively In a similar form of associated Legendre function Pz (7-j) in the spherical case, the angular function for a spheroidal case is chosen as SF (c, q) [ 1,7]
Thus, the scalar wave eigenfunctions are given by [ 1,7,9]
where for the fields inside the spheroid, the first kind of radial function (i = 1)
is taken, and for the fields outside the spheroid, the third kind (i = 3) is used because of the present time dependence chosen For the intermediate region between the two spheroidal interfaces, both the first and third kinds of radial functions are used in the formulation of DGFs
3.3.2 Unbounded Scalar Green’s Function
In terms of the scalar spheroidal wave functions above, the scalar Green’s function has been formulated [l] and is given by
00
c
2-Jm0 m ysn Cc9 rlls?Cc7 rl’) zy! [m(4 -
N
Trang 8(3 9)
where r> and T< denote the coordinate vector T, where c is taken as max(<, t’) and min(<, c’), respectively, while the coordinates 7;1,4 and v’, 4’ should be adopted correspondingly Also in Eq (3.9), &me is the Kronecker delta and Nmn is the normalization factor of the angular function of the first kind [l]
3.3.3 Appropriate Spheroidal Vector Wave Functions for Construction
of DGFs
To construct the electromagnetic dyadic Green’s functions in the layered me- dia, two methods are usually employed for mathematical expressions One of them is to apply coordinate tensors to represent the dyadic Green’s functions [89], the other is to use the vector wave functions to construct dyadic Green’s functions [ 1 41 Mat hemat ically, the expressions of Green’s dyadics in the lat- ter method are more compact than those in the former method However, to express
bivector
other
dyadic Green’s functions in terms of vector wave functions in their form requires that these vector wave functions be orthogonal to each
To develop the dyadic Green’s functions for spheroidally layered geometry,
a very convenient and compact method is to employ the spheroidal vector wave functions for construction Several kinds of vector wave functions have been proposed, and almost all of them have been used in the study of plane wave scattering by a single spheroid [lo], a system of two spheroids [11,12,29], and
a layered spheroid [33] H owever, not all of these vector wave functions can be used conveniently to construct dyadic Green’s functions for electromagnetic radiation problems Although it is possible to use these vector wave functions with the piloting vectors Fj, & and & they are still inconvenient spheroidal vector wave functions Since V x (6, & +) # 0 in spheroidal coordinates,
Thus, the most appropriate spheroidal vector wave functions for the construc- tion of Green’s dyadics are defined in terms of the scalar eigenfunctions above
as follows:
(3.11a) (3.11b) These vector eigenfunctions were obtained by using the scalar eigenfunctions I/J shown above as the generating function and the coordinate vector Z&Z
or ? as the piloting (or constant) vector The explicit forms of the spheroidal vector wave functions under the alternative spheroidal coordinate system are
Trang 9UNBOUNDED 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+ $vv.] Fg(V?] ,
GHJO(T, T’) = v x [%l(T, T’J] = vg(T, T’) X f,
(3.12a) (3.12b) where the subscript 0 next to EJ and HJ stands for the unbounded space
In terms of the above-defined spheroidal vector wave functions in an explicit bivector form, the electric dyadic Green’s functions given in Eqs (3.12a) and (3.12b) can be obtained after substitution of Eq (3.9) for < 2 <’ as
Trang 10(c’,~‘, 4’) The first term of Eq (3.13a) stands for the nonsolenoidal con- tribution and can be obtained by using the method given by Tai [14, pp 128-129, 1541
It is worth mentioning that the singularity of the Green’s functions was a controversial issue in the late 1970’s Now, the issue of irrotational DGF’s has been well resolved and is no longer the problem to the electromagnetics com- munity In this chapter, the irrotational part of the Green’s dyadic is found from a combination of two contributions: one of them taken directly from the unit delta dyadic, and the other obtained from the first order derivative of the Green’s function at the discontinuity point at 5 = <‘ The total effects of the two parts make the present form of the irrotational contribution to the Green’s dyadic
By using the principle of scattering superposition, the dyadic Green’s function can be considered as the sum of the unbounded Green’s dyadic in Eqs (3.13a) and (3.13b) and a scattering Green’s dyadic to be determined The Green’s dyadic is therefore given by [14]
-(f 4
(f 1 CH; (T, T’) = cH.J&, T’)bfs + @$;‘,(T, r’), (3.14b) where the scattering DGF ctfs) EJs(y, T’) and &;, (f 1 (T, P’) describe the addi- 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 ( VJ rl) 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 ease of 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@) Emfl ,(c, c) and ‘A$‘L1 ,(c, t) are defined as follows:
(3.15a)
Trang 1171
(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 cases are 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<fgv-1)
Trang 1272 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
(3.17a)
(3.17b)
00 00 -(f 4
Trang 13DETERMINATION OF SCATTERING COEFFICIENTS 73
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
coefficients to be determined from the EM field boundary conditions
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),