Xem mẫu
- 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
- 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.
- 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 = $d
(Ic is the wave propagation constant, as indicated in Chapter 2). Assume
that the space is divided by N - 1 spheroidal interfaces into N regions, as
shown in Fig. 3.1. The spheroidally stratified regions are labeled, respectively,
f = 1,2,3, . . .. N. The EM radiated fields Ef and Hf in the fth (field) re-
gion (f = 1,2,3, . . .) N) due to the electric and magnetic current distributions
J, and MS located in the sth (source) region (s = 1,2,3, . . ., N), as shown
in Fig. 3.1, can be expressed by
2
VxVx Ef - kfEf= [aqJr-@XWf] b, (3.la)
2
VxVxHf -kfHf= [iw&fMf+@xJ)f] Sfs, (3.lb)
where 6f, denotes the Kronecker delta (= 1 for f = s and 0 for f # s),
kf= w J PfEf (1 + iaf/wEf) is the wave propagation constant in the fth
- 64 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
f2
Y
Fig. 3.1 Geometry of a multilayered prolate spheroid under coordinates (c, q, q5),
- FUNDAMENTAL FORMULATlON 65
layer of the multilayered medium, and &f, pf, and af identify the permit-
tivity, permeability, and conductivity of the medium, respectively. A time
dependence exp(-iwt) is assumed to describe the EM fields throughout the
book. Moreover, the media reduce to free space if pf = ~0.
The EM fields excited by an electric current source J, and a magnetic
current distribution A& can be expressed in terms of integrals containing
dyadic Green’s functions as follows [14,100,122,125]:
Ef (r) = iops @$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 $$J’ (T, rl) and GCf ‘) (T, r’) as the electric and magnetic
HJ
dyadic Green’s functions of the first kind [i.e., ~L{‘)(T, T’) and C:~)(T, r’>];
and @&$(Y, Y’) and cCfs’ (T, rl> as the electric and magnetic dyadic Green’s
HIM
functions of the second kind [i.e., GLi”(r, rl) and E~;‘(T, r’)].
Substituting Eqs. (3.2a) and (3.2b) into Eqs. (3.la) and (3.lb), respec-
tively, we obtain
(3.4a)
(3.4b)
- 66 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
where r stands for the unit/identity dyad and 6(r - #) identifies the Dirac
delta funct&
Since c,J ( T, T’) and -(f 4 (r, rl> are related by the upper elements of
G,,
Eqs. (3.3a) and (3.3b) and @$,, rl> and @$T, rl) by the lower elements
of Eqs. (3.3a) and (3.3b), we do not need to derive all of them. Instead, only
the formulations of -(f 4 (T, rl) and Z&&T, T’) are considered here. The
G,, (f )
following boundary conditions at the spheroidal interface, < = cf, are satisfied
by various types of dyadic Green’s functions after Eqs. (3.2a) and (3.2b) are
substituted into the EM field boundary conditions:
+Gl
-[ I
@f
- (x
- EJ+a4 (3.5a)
@f ’
HM
(3.5b)
where
iWf
Gf
+1
+1 Jstands for the ruling that either the upper or lower elements
of the matrices should be taken at the same time. In fact, Eqs. (3.5a) and
(3.5b) represent four equations if all the upper and lower elements are con-
sidered, respectively. Furthermore, the DGF $$, 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,
- 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,
- DYADIC GREEN’S FUNCTIONS IN SPHEROiDAL SYSTEMS
(3.9)
where r> and T< denote the coordinate vector T, where c is taken as max(
- 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+ $vv.] Fg(V?] , (3.12a)
GHJO(T, T’) = v x [%l(T, T’J] = vg(T, T’) X f, (3.12b)
where the subscript 0 next to EJ and HJ stands for the unbounded space.
In terms of the above-defined spheroidal vector wave functions in an explicit
bivector form, the electric dyadic Green’s functions given in Eqs. (3.12a) and
(3.12b) can be obtained after substitution of Eq. (3.9) for < 2
- 70 DYADIC GREEN’S FUNCTIONS IN SPHEROIDAL SYSTEMS
(c’,~‘, 4’). The first term of Eq. (3.13a) stands for the nonsolenoidal con-
tribution and can be obtained by using the method given by Tai [14, pp.
128-129, 1541.
It is worth mentioning that the singularity of the Green’s functions was a
controversial issue in the late 1970’s. Now, the issue of irrotational DGF’s has
been well resolved and is no longer the problem to the electromagnetics com-
munity. In this chapter, the irrotational part of the Green’s dyadic is found
from a combination of two contributions: one of them taken directly from the
unit delta dyadic, and the other obtained from the first order derivative of
the Green’s function at the discontinuity point at 5 = = GEJO(T, T’)Sfs + G:;;(c r’),
CH; 1(T, T’) =
(f
cH.J&, T’)bfs + @$;‘,(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)
- 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
- 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
- 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),
- 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 $-dependent trigonometric function in Eqs. (3.5a) and (3.5b)
must be equal, component by component; the equalities must hold for each
corresponding term in the summation over m. For the summation over n,
however, the individual terms in the series cannot be decomposed term by
term because of the nonorthogonality of the spheroidal radial functions. This
causes difficulty in determining the unknown scattering coefficients.
To solve for the unknown coefficients, the following expanded intermediate
forms [10,241 are introduced: for m > 1,
(3.20a)
(3.20b)
2 3/2 m
grzn(C) ’ p,“_i:t(rl) = (l - 73 > sn Cc7rl), (3.20~)
t=o
(3.20d)
(3.20e)
(3.20f)
(3.20h)
- DETERMINATION OF SCATTERING COEFFICIENTS 75
(3.2Oi)
(3.2Oj)
(3.20k)
(3.201)
and for m = 0,
2 Ii-$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
- 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
- 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 >
- 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
- 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,
- 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,
nguon tai.lieu . vn
