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2 BASIC THEORY OF ELECTROMAGNETIC SCATTERING
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For the case of spheroids, the surface is governed by the equation x 2 + y 2 z2 (2.8.95) --2-=--- + 2" = 1 a c For a > c the spheroid is oblate, and for a < c the spheroid is prolate. In spherical coordinates, (2.8.95) assumes the form sin 2 0 cos2 OJ -1/2 r=r(O) = [ - - + -2a2 c and from (2.8.36) we obtain
(2.8.96)
r + 0r 2 sin 0 cos 0
(:2 -:2)
(2.8.97)
The T-matrix elements can be evaluated numerically in a straightforward manner. The integration over J071" dO sin 0 can be broken into a sum of {71"/2 {71"/2 }0 dO sin 0 + } 0 dO' sin 0' with 0' = 7f - O. Since r(7f - 0) use of the property
= r(O) for the case of spheroids and we make
(2.8.98)
(2.8.99)
P;:"(cos(7f - 0)) = (-l)n+mp;:,,(cosO) it follows that Jmnm'n' - Jmnm'n' - 0
if n
(11) _ (22) _
+ n' =
even, and
Jmnm'n' - Jmnmln' - 0
if n
(12)
(21)
(2.8.100) (2.8.101)
(2.8.102)
+ n' =
odd. Hence,
Pmnm'n'
for n
= Umnm'n' = 0
+ n' = odd,
Rmnm'n' = Smnm'n' = 0
8.3 Spheroids
for n +n' = even. By making use of the relation between the negative degree associated Legendre polynomial and that of the positive degree in (1.4.38), it follows that
..) J (~J
( -m )n( -m' )n'
J(i j ) mnm'n' J( ij) mnm'n'
for ij = 11 or 22 for ij = 12 or 21
(2.8.103)
Small Spheroids
For small spheroids, we shall only keep the dipole term n = n' = 1. The leading term of the real part and the leading term of the imaginary part of J~~m'n' terms will be kept. The results can be compared to that of Section 2.3 of 1, which is based on the separation of variable approach for the Laplace equation. However, we will include, as in the spheres case, an additional term that ensures the optical theorem be satisfied. The only nonzero T-matrix elements are T~;~'l with
T~;~'l = 6mm ,Tm
where
To = ito - t~
(2.8.104)
(2.8.105)
TI = itl - ti 2 k 3 a 2 c (Es/E - 1) to = 9 (1 + VdAc) 2 k 3 a 2 c (Es/E - 1) tl = - -_"':""':-'----'9 (1 + vdAa)
(2.8.106)
(2.8.107)
(2.8.108) is given in (1.2.21).
and A a and A c are given in (1.2.26)-(1.2.29), and
2 BASIC THEORY OF ELECTROMAGNETIC SCATTERING
REFERENCES AND ADDITIONAL READINGS
Artken, G. B. (1995), Mathematical Methods for Physicists, 4th edition, Academic Press, San Diego. Barber, P. W. and S. C. Hill (1989), Light Scattering by Particles: Computational Methods, World Scientific Publishing, Singapore. Bohren, C. F. and D. R. Huffman (1983), Absorption and Scattering of Light by Small Particles, Wiley-Interscience, New York. Born, M. and E. Wolf (1975), Principles of Optics, 5th edition, Pergamon Press, New York. Chew, W. C. (1990), Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York. Ishimaru, A. (1978), Wave Propagation and Scattering in Random Media, 1 and 2, Academic Press, New York. Ishimaru, A. (1991), Electromagnetic Wave Propagation, Radiation, and Scattering, PrenticeHall, Englewood Cliffs, New Jersey. Jackson, J. D. (1999), Classical Electrodynamics, 3rd edition, John Wiley & Sons, New York. Kerker, M. (1969), The Scattering of Light, and Other Electromagnetic Radiation, Academic Press, New York. Kong, J. A. (1990), Electromagnetic Wave Theory, 2nd edition, John Wiley & Sons, New York. Mischenko, M. I., J. W. Hovenier, and L. D. Travis, Eds. (2000), Light Scattering by Nonspherical Particles, Theory, Measurement and Applications, Academic Press, New York. Morse, P. M. and H. Feshback (1953), Methods of Theoretical Physics, McGraw-Hill, New York. Peterson, B. and S. Strom (1973), T matrix for electromagnetic scattering from an arbitrary number of scatterers and representation of E(3), Phys. Rev. D, 8, 3661-3678. Stratton, J. A. (1941), Electromagnetic Theory, McGraw-Hill, New York. Tai, C. T. (1971), Dyadic Green's Function in Electromagnetic Theory, International Textbook, Scranton, PA. Tsang, L. and J. A. Kong (1980), Multiple scattering of electromagnetic waves by random distribution of discrete scatterers with coherent potential and quantum mechanical formulism, J. Appl. Phys., 51, 3465-3485. Tsang, L., J. A. Kong, and R. T. Shin (1985), Theory of Microwave Remote Sensing, WileyInterscience, New York. Waterman, P. C. (1965), Matrix formulation of electromagnetic scattering, Proc. IEEE, 53, 805-811.
Scattering of Electromagnetic Waves: Theories and Applications Leung Tsang, Jin Au Kong, Kung-Hau Ding Copyright 2000 John Wiley & Sons, Inc. ISBNs: 0-471-38799-1 (Hardback); 0-471-22428-6 (Electronic)