Appendices, References, and Index

10 Scattering and absorption properties of nonspherical particles 359 Further reading The book edited by Mishchenko et al. (2000a) is a major systematic source of information on calculations, measurements, and applications of electromagnetic scattering by nonspherical and heterogeneous particles. Further information can be found in the special journal issues edited by Shafai (1991), Barber et al. (1994), Hovenier (1996), Lumme (1998), Mishchenko et al. (1999a,b), and Videen et al. (2001) as well as in the conference proceedings edited by Schuerman (1980), Wriedt et al. (1996), Wriedt and Eremin (1998), Obelleiro et al. (1999), and Videen et al. (2000b). The book by Colton and Kress (1998) treats mathematical and numerical aspects of the inverse scattering problem for electromagnetic and acoustic waves. Scattering by randomly and preferentially oriented spheroids and finite circular cylinders in the geometrical optics limit is discussed by Yang and Cai (1991), Arnott and Marston (1991), Macke and Mishchenko (1996), Kokhanovsky and Nakajima (1998), Langley and Marston (1998), and Marston (1999). Listed in Sassen and Ar- nott (1998) are several feature journal issues discussing optical phenomena associated with natural ice crystals. Appendix A Spherical wave expansion of a plane wave in the far-field zone In this appendix we derive Eq. (2.57) following the approach described by Saxon (1955b). We begin with the well-known expansion of a plane wave in scalar spherical harmonics (Jackson 1998, page 471): ∞ l ⋅ ′ = π l ′ ∗ ′ exp(ir r ) 4 i jl (rr ) Ylm(rˆ)Ylm(rˆ ), (A.1) l=0 m=−l ′ .l . = r ′ = r = l − 1 d / sin y / where rˆ , rˆ , the jl (y) y / / are spherical Bessel func- r r′ y dy 0 y 0 tions of the first kind, and the Ylm(rˆ) are scalar spherical harmonics. The latter are defined as (2l +1)(l − m)! Y (rˆ) = Pm(cosϑ)eimϕ, (A.2) lm 4π (l + m)! l ϑ ϕ m where and are spherical angular coordinates of the unit vector rˆ and Pl are associated Legendre functions defined in terms of Legendre polynomials Pl as fol- lows: d m 1 dl P m(x) = (−1)m (1− x2 )m 2 P (x), P (x) = (x2 −1)l (A.3) l dxm l l 2ll! dxl with x ∈[−1, 1]. Using the asymptotic form (Arfken and Weber 1995, p. 682) π . = 1 − l jl (y) sin y /, (A.4) y→∞ y 2 0 360 Appendix A 361 we have ! . .1 ′ = 1 i − ′ + 1 π − ′ − 1 π jl (rr ) "exp irr i l/ expirr i l/2. (A.5) rr′→∞ rr′ 2 # 2 0 2 03 Substituting this expression in Eq. (A.1) and making use of the completeness relation for spherical harmonics (Jackson 1998, p. 108) ∞ l ∗ ′ = ϑ − ϑ′ ϕ −ϕ′ = − ′ Ylm(rˆ)Ylm(rˆ ) δ(cos cos )δ( ) δ(rˆ rˆ ) (A.6) l=0 m=−l and the symmetry relation − ′ = − l ′ Ylm( rˆ ) ( 1) Ylm(rˆ ), (A.7) we derive, after simple algebra, i2π ! e−irr′ eirr′ 1 exp(irr′rˆ ⋅rˆ′) = "δ(rˆ + rˆ′) − δ(rˆ − rˆ′) 2. (A.8) rr′→∞ r′ # r r 3 Appendix B Wigner functions, Jacobi polynomials, and generalized spherical functions Jacobi polynomials, Wigner functions, and generalized spherical functions are closely related special functions which were introduced in classical analysis, the quantum theory of angular momentum, and the theory of representations of the rotation group, respectively (Szegő 1959; Wigner 1959; Gelfand et al. 1963). Because differences in notational conventions in various publications may lead to confusion, we present in this appendix a short consistent summary of the main properties of these functions. Wigner d-functions are defined as s ϑ = + − + − dmn( ) (s m)!(s m)!(s n)!(s n)! (cos 1 ϑ)2s−2k +m−n (sin 1 ϑ)2k −m+n × (−1)k 2 2 , (B.1) k!(s + m − k)!(s − n − k)!(n − m + k)! k where s, m, and n are integers, 0 ≤ ϑ ≤ π , and the sum is taken over all integer values = of k that lead to non-negative factorials. Thus the summation index runs from kmin − = + − s ϑ = ≥ max(0, m n) to kmax min(s m, s n). Therefore, dmn( ) 0 unless kmax kmin, which is equivalent to requiring that s ≥ 0 and −s ≤ m, n ≤ s. Making the substitu- tions k → s − n − k, k → s + m − k, and k → m − n + k, respectively, we derive the following alternative expressions: s ϑ = − s−n + − + − dmn( ) ( 1) (s m)!(s m)!(s n)!(s n)! (cos 1 ϑ)m+n+2k (sin 1 ϑ)2s−m−n−2k × (−1)k 2 2 , (B.2) k!(s − m − k)!(s − n − k)!(m + n + k)! k 362 Appendix B 363 s ϑ = − s+m + − + − dmn( ) ( 1) (s m)!(s m)!(s n)!(s n)! (cos 1 ϑ)2k −m−n (sin 1 ϑ)2s+m+n−2k × (−1)k 2 2 , (B.3) k!(s + m − k)!(s + n − k)!(k − m − n)! k s ϑ = − m−n + − + − dmn( ) ( 1) (s m)!(s m)!(s n)!(s n)! (cos 1 ϑ)2s−2k −m+n (sin 1 ϑ)2k +m−n × (−1)k 2 2 . (B.4) k!(s − m − k)!(s + n − k)!(m − n + k)! k Equations (B.1)–(B.4) imply that the d-functions are real and have the following symmetry properties: s ϑ = − m−n s ϑ = − m−n s ϑ = s ϑ dmn( ) ( 1) d−m,−n( ) ( 1) dnm( ) d−n,−m( ), (B.5) s −ϑ = − m−n s ϑ = s ϑ s = δ dmn( ) ( 1) dmn( ) dnm( ), dmn(0) mn, (B.6) s π −ϑ = − s−n s ϑ = − s+m s ϑ s π = − s−n δ dmn( ) ( 1) d−mn( ) ( 1) dm,−n( ), dmn( ) ( 1) −mn, (B.7) δ where mn is the Kronecker delta: $1 if m = n, δ = mn % &0 if m ≠ n. Denoting x = cosϑ and requiring that ϑ ∈[0, π ] yield 1 ϑ = 1 + 1 ϑ = 1 − cos 2 2 (1 x), sin 2 2 (1 x). (B.8) Substituting Eq. (B.8) in Eq. (B.1) and modifying the resulting formula, we obtain s ϑ = s − (m−n) 2 + −(m+n) 2 − s−n−k dmn( ) Amn (1 x) (1 x) ( 1) k (s + m)!(s − m)!(s − n)! × (1+ x)s+m−k (1− x)k −m+n, (B.9) k!(s + m − k)!(s − n − k)!(k − m + n)! where − s−n ! + 11 2 s = ( 1) (s n)! Amn " 2 . (B.10) 2s #(s + m)!(s − m)!(s − n)!3 Finally, recalling the Leibniz rule, N N k N −k d . N! ! d . 1 ! d . 1 / [ f (x)g(x)] = " / f (x)2 " / g(x)2, (B.11) dx 0 k!(N − k)! " dx 0 2 " dx 0 2 k =0 # 3 # 3 and applying it to f (x) = (1+ x)s+m, g(x) = (1− x)s−m, and N = s − n, we can rewrite Eq. (B.9) in the form s−n d . d s (ϑ) = As (1− x)(m−n) 2 (1+ x)−(m+n) 2 / [(1+ x)s+m (1− x)s−m ]. (B.12) mn mn dx 0 364 Scattering, Absorption, and Emission of Light by Small Particles The Jacobi polynomial of degree q is given by Eq. (4.3.1) of Szegő (1959): q (−1)q d . P(a,b)(x) = (1− x)−a (1+ x)−b / [(1+ x)q+b (1− x)q+a], (B.13) q 2q q! dx 0 where q is a non-negative integer and a > −1 and b > −1 are real. Comparing Eq. (B.12) with Eq. (B.13), we obtain the following expression of the Wigner d-functions in terms of the Jacobi polynomials for n ≥ |m|: ξ ! + + 11 2 s ϑ = mn q!(q a b)! − a 2 + b 2 (a,b) dmn( ) " 2 (1 x) (1 x) Pq (x), (B.14) 2(a+b) 2 #(q + a)!(q + b)!3 = − = + = − ξ = ≥ where a n m, b n m, q s n, and mn 1. The condition n |m| ensures that a ≥ 0 and b ≥ 0, thereby preventing singularities for x = ±1. Using the symmetry relations of Eq. (B.5), it is straightforward to show that Eq. (B.14) can be used for arbitrary m and n, provided that = − = + = − 1 + a |m n|, b |m n|, q s 2 (a b), (B.15) $1 for n ≥ m, ξ = mn % (B.16) &(−1)m−n for n < m. The orthogonality property of the Jacobi polynomials (Eq. (4.3.3) of Szegő 1959) and Eqs. (B.14)–(B.16) lead to the following orthogonality property of the d- functions: π s s′ 2 dϑ sinϑ d (ϑ)d (ϑ) = δ ′. (B.17) mn mn + ss 0 2s 1 The completeness property of the Jacobi polynomials (Szegő 1959) and Eqs. (B.14) + 1 s ϑ = + and (B.17) imply that functions s 2 dmn( ) with s smin, smin 1, form a com- plete orthonormal system of functions on [0, π ], where = smin max(|m|, |n|). This means that if a real-valued function f (ϑ) defined on the closed interval [0, π ] is square integrable on this interval, i.e., if π dϑ sinϑ [ f (ϑ)]2 < ∞, (B.18) 0 η ≥ then there exists a unique set of coefficients s (s smin) such that the series expansion ∞ ϑ = η s ϑ ϑ ∈ π f ( ) s dmn( ), [0, ] (B.19) = s smin holds in the following sense: 2 π S ϑ ϑ ϑ − η s ϑ = lim d sin f ( ) sdmn( ) 0. (B.20) S →∞ 0 = s smin Appendix B 365 Conversely, if a real-valued function f (ϑ) on [0, π ] admits the expansion of Eq.

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