Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

Using the integral representation for the translation coe cients, making the transformation ϕ0 → ϕ0 + π, changing the variable of integration from β to π − β and using the identities (A.24) and (A.25), yields

Amn,m n (−kr0) = (−1)n+n Amn,m n (kr0) ,

Bmn,m n (−kr0) = (−1)n+n +1Bmn,m n (kr0) .

Further, since

Amn,m n (kr0) = (−1)n+n A−m n ,−mn (kr0) ,

Bmn,m n (kr0) = (−1)n+n +1B−m n ,−mn (kr0) ,

we obtain

Recurrence relations for the scalar and vector addition theorem has also been given by Chew [32, 33], Chew and Wang [35] and Kim [117]. The relationship between the coe cients of the vector addition theorem and those of the scalar addition theorem has been discussed by Bruning and Lo [29], and Chew [32].

290 C Computational Aspects in E ective Medium Theory

z

Fig. C.1. Integration surfaces SR , S∞ and Sz

The integral can be decomposed into three integrals

Imm1 n = Imm1,R n + Imm1,∞ n + Imm1,z n ,

where Imm1,R n is the integral over the spherical surface SR of radius 2R cen-

1,∞

tered at r0l, Imm n is the integral over the surface of a half-sphere S∞ with radius R∞ in the limit R∞ → ∞, and Imm1,z n is the integral over the xy- plane Sz (the plane z = 0). The choice of the integration surfaces is shown in Fig. C.1.

Using the identity r0p = r0l + rlp and replacing, for convenience, the variables rlp, θlp and ϕlp by r, θ and ϕ, respectively, we obtain

for Ke · r = Ksr cos θ, the orthogonality of the associated Legendre functions and the relation Ke · r0l = Ksz0l, we end up with

− (KsR) h(1)n (2ksR) (jn (2KsR)) .

1,∞

To compute Imm n we use the stationary point method. Using the inequality Im{Ks} > 0 and the asymptotic expressions of the spherical Hankel functions of the first kind

1,∞

we see that Imm n vanishes as R∞ → ∞.

To evaluate Imm1,z n we pass to cylindrical coordinates, integrate over ϕ, and obtain