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Using the T-matrix, the scattering amplitude dyad F(Os, cPs; Oi, cPi) can be calculated. For a plane wave incident in the direction ki = (Oi' cPi),
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-='LE'ne = ei E 0 e ik r ~ '
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the incident field coefficients are, from (1.4.67),
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+ 1)) Z C -mn (0 i, 'f/i . ei E 0 'n "') ~ 'Yrnn n n + 1 1 aE(N) = (_1)m_ _(2n+1) inILmn(Oi,cPi) eiEo mn 'Ymn n(n + 1) i
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= (1)m -1 (2n - (
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(2.7.13) (2.7.14)
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The asymptotic far-field solutions of M mn and N mn are given in (1.4.69a) and (1.4.69b). From (2.7.4), (2.7.5), (2.7.13), and (2.7.14), we have
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lim a~r;:) M mn (kr, 0, cP)
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1 (2n' + 1) .n'~ - - '( '+ 1) . Z C-rn'nl(Oi, cPi) . eiEo '1rn'n' n n
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7.1 T-Matrix and Relation to Scattering Amplitudes
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(_1)m,_1_ (2n' + 1) . .n,ILm'n,(Oi,cPi) . . et "(m'n' n '( n , + 1) 't 't
. "(mnCmn(O, cP)~et
lim a~~) N mn(kr, 0, cP)
- L.J Tmnm'n' (-1)
_ " [ (21)
1 (2n' + 1) 'n'- - '(' 1)' 't C-m'n,(Oi, cPi) . eiEo "(m'n' n n +
1 (2n'+1). 'n,B-m'n,(Oi,cPi) .A.E] . et 0 "(m'n' n '(' + 1) 't n 't
-1..) '-n 1 ikr . "(mn B mn (0 , 'I' 't kr e
Using (2.7.15) and (2.7.16) in (2.7.3) and the definition ~f the scattering amplitude dyad F gives the following relation expressing F in terms of the T-matrix elements
F 0, cP,; 0, cP ) =
Using the optical theorem, the extinction cross section for incident direction (Oi, cPi) can be calculated from (2.4.24). To find the scattering cross section for incident direction (Oi, cPi) and incident polarization (3, w,e note that (2.7.18) Substituting (2.7.17) in (2.7.18) and making use of orthogonality relations for vector spherical harmonics, we have
167[2 ~{I L..J n' (7s/3(e i, i) = k2 L..J ~ z (-1) m ' "(-m'n'
n,m T(ll) C ,,(e. ~, [ mnm'n' -m n m'n'
.) . ~ + T(12)
B-m'nl(ei, i)
.~] 1
+ I Lin' ( -1 )m' "(-m'n'
[T,\;~~'n'C-m'n' (9i , <Pi) . h T,\;~~,}Lm'n;(9i , <Pi) . jJ]
with (3 = V o~h. Hence, given the T-matrix elements, the scattering matrix elements for F can be computed according to (2.7.17).
7.2 Unitarity and Symmetry
In this section, we shall derive the unitarity and symmetry properties of the T-matrix. The unitarity property is a result of energy conservation for a nonabsorptive scatterer, and symmetry is a result of reciprocity.
A. Unitarity
The total field is a summation of incident and scattered fields. We use (2.7.2) and (2.7.3) and also the fact that regular wave solution is a combination of outgoing waves (Hankel functions of first kind) and incoming waves (Hankel functions of second kind), that is, in = (h~l) + h~2 )/2. We use superscript (2) to denote that the spherical Hankel function of the seco~d kind is used in the vector wave function. Defining the scattering matrix S as
_ _ [T(ll) S = 1 + 2T = 1 + 2 T(21)
and using the expression for
T(12)] T(22)
=i L{ (Sgl)a~(M) +SI\~2)a~(N))
E and E
in Section 7.1 gives the total field
Mz(kr,e, )
(Sz\:l)a~(M) +Sz\:2)a~(N ) Nz(kr,O, )}
+ ~ L {af'M) M~2\kr, 0, ) + af<N) N~2)(kr, 0, )}
7.2 Unitarity and Symmetry
In the far field, we make asymptotic approximations of the spherical vector
wave functions of (1.4.69a) and (1.4.69b). Thus,
_ eikr _ e-ikr _ lim E(r) = -k W 1 (O, ) + -k-W2(O, ) kr---+oo 2 r 2 r
Wl(O, ) =
{(S~~~'n,a~~, + S~~~'n,a~;n') Cmn(O, )
(2.7.23) (2.7.24)
+ S(22) (N ) 'B (lJ A..)} -n-l + ( S(21) mnm'n,am'n' mnm'n,am'n' Z mn u, 'I' 'Ymn Z
W2(O, ) =
L {a~~Cmn(O, ) + a'::~ Bmn;O, )} 'Ymn i n+
and are perpendicular to f. The complex Poynting vector is then
2 S = E x H* = 417(:2 r 2) {IW1(t'1, )1 -IW2(O, )12
+e- 2ikr [W 2(O, ). W~(O, )]
_e 2ikr [Wl(O, ).w;(O, )]}
The sum of the latter two terms in the curly brackets is purely imaginary. Hence, the time-average Poynting's vector is
(8) = ~Re(E x H*) = 817(:r)2 {IW 1(O, )1 -IW2(O, )1 }
If the scattering object is lossless, i.e., the imaginary part of permittivity ts is zero, the scatterer is nonabsorptive and integration of (2.7.26) over a spherical surface at infinity should give zero. Hence,
dO {IW 1 (O, )1 -IW 2 (O, )1
-* _ m(n+m)!Cmn(O, ) - (-1) (n _ m)! C-mn(O, ) -* m(n+m)!Bmn(O, ) = (-1) (n _ m)! B-mn(O, )
Using (2.7.24), (2.7.28), and (2.7.29), and the orthogonality relation of vector spherical harmonics, we have