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1 1.1 1.2 1.3 1.4
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2 Dyadic Green's Function Green's Functions Plane Wave Representation Cylindrical Waves Spherical Waves Huygens' Principle and Extinction Theorem Active Remote Sensing and Bistatic Scattering Coefficients Optical Theorem Reciprocity and Symmetry Reciprocity Reciprocal Relations for Bistatic Scattering Coefficients and Scattering Amplitudes Symmetry Relations for Dyadic Green's Function Eulerian Angles of Rotation T-Matrix T-Matrix and Relation to Scattering Amplitudes Unitarity and Symmetry Extended Boundary Condition Extended Boundary Condition Technique Spheres 8.2.1 Scattering and Absorption for Arbitrary Excitation 8.2.2 Mie Scattering of Coated Sphere Spheroids References and Additional Readings
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Dyadic Green's Function
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1.1 Green's Functions
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The Green's function is the solution of the field equation for a point source. Using the principle of linear superposition, the solution of the field due to a general source is just the convolution of the Green's function with the source. The equation for the Green's function for the scalar wave equation is
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+ k~)g(r, 1") =
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-0(1' - 1")
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where 0(1' - 1") is the three-dimensional Dirac delta function with the source located at 1". The solution of (2.1.1) is [Kong, 1990; Ishimaru, 1991]
g(r,r') = g(r - 1") =
47r r - r
The dyadic Green's function relates the vector electromagnetic fields to vector current sources. From the Maxwell equations in frequency domain with exp( -iwt) time convention
\7 x E
= iWJ-LH
(2.1.3) (2.1.4) (2.1.5) (2.1.6)
\7 x H = -iWE E \7 . J-LH = 0 \7. EE = p
it follows that the electric field obeys the vector wave equation
\7 x \7 x E - k;E
= iWJ-l]
k; = w2J-LE.
\7 x \7 x E - k~E = 0
In source-free region, ] = 0, and we have
The free space dyadic Green's function satisfies the equation
\7 x \7 x G(r,r') - k~G(r, 1") = 10(1' - 1")
where 1 is the unit dyad. Using the dyadic Green's function [Tai, 1971; Tsang et al. 1985], the electric field is equal to the convolution of the dyadic Green's function with the current source.
E(r) = iWJ-L
dr' G(r, 7"') . J(r')
1.2 Plane Wave Representation
The solution to (2.1.8) is
G(r, 1") =
(1 + :~
\7\7) g(r, 1")
and can be veriEed by directly substi~ting (2.1.10) into (2.1.8) and noting that \7 x \7 x (1g) = \7\7g - \7. (\7g)1 and that 9 obeys (2.1.1). For the usage of Green's function in scattering problem, it is often useful to express the dyadic Green's function of unbounded medium in different coordinate systems. In the following sections, we shall treat the cases of plane wave representation, cylindrical wave representation, and spherical wave representation.
1.2 Plane Wave Representation
The plane wave representation is particularly useful when one considers scattering in the presence of layered media. The plane wave representation can be derived by solving (2.1.1), with Fourier transforms. The Fourier transform of (2.1.1) gives
g(r - 1") = (2:)3
The Dirac delta function in plane wave representation is
c5(r - 1") = (2:)3
i: i:
dk ik{F-r')
where k = kxx + kyY + kzz and dk = dkxdkydk z . Substituting (2.1.11) and (2.1.12) into (2.1.1) gives
1 g(k) = k 2 _ k 2
where k 2 = k; + k~ + k;. Integration can be performed for one of the threefold integral of dkxdkydk z . In remote sensing applications, the z axis is usually chosen to be the vertical direction. To perform the contour integration over dk z , we note that 1m k o > 0, so that a pole exists at k oz = (k; - k~ - k~)1/2 in the upper-half k z plane and a pole exists at -koz in the lower half k; plane with k z = Re k z + ilm k z ' Thus for z > z' we deform upward and pick up the contribution of the pole at k oz , and for z < z' we deform downward and pick up the contribution of the pole at -kozo Thus
g(r - r') =
(2~)21: dkl- ik.L.(r.L -~~::iko.IZ-Z'1