Spherical Waves in .NET

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1.4 Spherical Waves
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The spherical waves and outer spherical waves are as described in 1. We first express the unit dyadic delta function in terms of vector spherical waves. We multiply (1.4.66) by e- ip .r ' j(21r and then integrate over dp. The left-hand-side becomes
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For the right-hand-side, a typical term is
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(::)3 e -ip.r' P -mn (()p, >p) Rg L mn (pr, (),
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p2 41rin - 1 , " dp -()3' Rg L- mn (-pr ,() , > ) Rg Lmn(pr, 0, (2.1.30) o 21r I-mn The equality above is a result of using (1.4.63). We also note that Rg L- mn (-pr', 0', ') = (_1)n+l Rg L-mn(pr', e', '); Rg M -mn( -pr', e', ') = (-1tRgM- mn (pr',O', >') and RgN-mn(-pr',O', >') = (-1t+lRgN- mn (pr', e', '). Thus using (1.4.63)-(1.4.65) in the integration, we have .
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+ Rg Mmn(pr, 0, Rg M -mn(pr', ()', >') + Rg N mn(pr, e, ) Rg N -mn(pr', e', ') }
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We let Go be expanded into linear combinations of vector spherical wave functions and substitute into the equation \7 x \7 x Go - k 2 G o = 10(1' - 1"), and we use (2.1.31) on the right-hand-side. Balancing coefficients then gives
+ Rg Mmn(pr, 0, Rg M -mn(pr', 0', >')
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Note that the vector spherical waves Rg L mn accounts for the field in the source region. Performing integration over dp gives [Tsang and Kong, 1980J
G or, -') - - 0(1' k 2 1"L (- r rr
+ 'k "'( l)m L...J'I,
It is useful to note the symmetry relation
n,m M -mn(kr, 0, RgMmn(kr', 0', >')
~N_mn(kr,O, RgNmn(kr',O', >')
for r > r' for r'
RgM-mn(kr, 0, Mmn(kr', 0', >') +RgN-mn(kr,O, Nmn(kr',O', >')
where superscript t denotes the transpose of the dyad. The symmetry relation is a result of reciprocity.
Huygens' Principle and Extinction Theorem
Huygens' principle is an exact relation that expresses the field in a region of space to the fields on a surface that encloses the region. Thus if the surface fields are determined, then the fields at any point in space can be calculated readily. Extinction theorem is useful in formulating integral equations for electromagnetic scattering problems. Both Huygens' principle and extinction theorem can be derived using the wave equation and the vector Green's theorem.
2 Huygens' Principle and Extinction Theorem
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Figure 2.2.1 Electromagnetic scattering with source] in volume Va.
Vector Green's theorem states that for any two vectors P and Q in a region of space V,
dV { P . \7 x \7 x Q - Q . \7 x \7 x P}
{Q x
\7 x P - P x \7 x
where 5 is a surface that encloses volume V and n is the outward normal of 5 that points away from volume V. Consider electromagnetic wave scattering with source] in region 0 that has permittivity E and permeability J.L while region 1 has permittivity EI and permeability J.L1 (Fig. 2.2.1). Let 51 denote the surface that encloses V1 with outward normal n, and let 5 00 be the surface at infinity with outward normal noo that encloses the all of space. From the Maxwell equations, we have \7 x \7 x E - k 2E = iwJ.L] (2.2.2)
\7 x \7 X E 1 - k 1E 1 = 0
with the boundary conditions n 2. E = n x El and n x H The free space Green's function Go obeys the equation
= nx
Hi on 51.
However, 1" as given in Go(1',1") can be in either region 0 or region 1. Let p = E in (2.2.1) and Q = Go(1',1") . a and V = Va in the vector Green's theorem of (2.2.1), with a being an arbitrary constant vector. We have
dV { E(1') . V x V x Go(1',1") . a - Go(1', 1") . a . V x V x E(1') }
dSit { Go(1', 1") . a x V x E - E x V x Go(1',1") . a}
dSit oo ' { Go (1', 7<') . a x V x E - E x V x Go(1',7<') .
The surface integral over Soo vanishes because of the radiation condition. Using (2.2.4a) and (2.2.2) in (2.2.5), we have
dV { E(1') . ac5(1' - 1") - (Go(1', 7<') . a) . iWJL1(1') }
dSit. {(Go(1', 1") . a) x V x E - E x V x (Go (1', r') . a)} (2.2.6)
The Dirac delta function in (2.2.6) only contributes if 1" is in region O. The other terms can be simplified by using the symmetry relation of Green's function,
where superscript t denotes the transpose of the dyad and the properties of triple scalar product of vectors. Thus