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

280 B Wave Functions

which yields

C001 ,0n (kz0) = (−1)n √2n + 1jn (kz0) ,

C003 ,0n (kz0) = (−1)n √2n + 1h(1)n (kz0) .

Using these starting values, the Cmm,mn coe cients can be computed from the recurrence relation (B.64) with m = n + 1,

where Cmm,mn can be obtained from C00,0n for all values of m ≥ 1 and n ≥ m, while (B.65) can then be used to compute Cmn,mn for n ≥ m + 1.

In general, the translation addition theorem for vector spherical wave functions can be written as [43, 213]

n =1 m =−n

+Amn,m n (kr0)N m n (kr1).

As in the scalar case, integral and series representations for the translation coe cients can be obtained by using the integral representations for the vector spherical wave functions. First we consider the case of regular vector spherical wave functions. Using the integral representation (B.26), the relation r = r0 + r1, and the vector spherical wave expansion

282 B Wave Functions

Inserting the spherical wave expansion of the plane wave exp(jk ·r0) and integrating over α, we derive the following series representation for the translation coe cients:

×a1 (m, m | n , n, n ) u1m−m n (kr0) ,

×b1 (m, m | n , n, n ) u1m−m n (kr0) ,

where

We note that the coe cients a1(·) and b1(·) can be expressed in terms of the coe cients a(·) by making use of the recurrence relations for the associated Legendre functions. As in the scalar case, the integration with respect to the azimuthal angle α gives

+ τn|m|(β)τn|m |(β) ejkz0 cos β sin β dβ

and

If the translation is along the z-axis the double summation in (B.66) and (B.67) reduces to a single summation over the index n , and we have

Passing to the radiating vector spherical wave functions we consider the integral representation (B.28) and the relation r = r0 + r1. For r1 > r0, this representation can written as

In (B.70) and (B.71) the integration with respect to the azimuthal angle α can be analytically performed and in the case of axial translation, the integral representations for the translation coe cients A3mn,m n and Bmn,m3 n read as

The vector addition coe cients Amn,m n and Bmn,m n can be related to the scalar addition coe cients Cmn,m n . For axial translations and positive values of m, these relations can be obtained by partial integration and by using the integral representations for the addition coe cients, the associated Legendre equation (A.13) and the recurrence relation (A.17). We obtain [150]

and

m

Bmn,mn (kz0) = jkz0 Cmn,mn (kz0) .

nn (n + 1) (n + 1)

286 B Wave Functions

For negative values of the index m, the following symmetry relations can be used for practical calculations

A−mn,−mn (kz0) = Amn,mn (kz0),

B−mn,−mn (kz0) = −Bmn,mn (kz0).

The translation coe cients for axial translation can be obtained from the above recurrence relations. In general, through rotation of coordinates, the numerical advantages to a common axis can be exploited and a transformation from Oxyz to Ox1y1z1 can be accomplished through three steps [150]:

1.The coordinate system Oxyz is rotated with the Euler angles α = ϕ0, β = θ0 and γ = 0, where (r0, θ0, ϕ0) are the spherical coordinates of the position vector r0.

2.The rotated coordinate system is axially translated with r0.

3.The translated coordinate system is rotated back to the original orientation with the Euler angles α = 0, β = −θ0 and γ = −ϕ0.

The rotation-axial translation-rotation scheme gives

n

Xmn,m n (kr0) = Dmmn (ϕ0, θ0, 0)Xm n,m n (kr0) m =−n

×Dmn m (0, −θ0, −ϕ0),

where X stands for A or B. The translation addition theorem can be written in matrix form as

where the pair (p, q) takes the values (1, 1), (3, 3), and (3, 1), and

We conclude this section by noticing some symmetry properties of the translation coe cients for the inverse transformation:

C

Computational Aspects

in E ective Medium Theory

In this appendix we compute the basic integrals appearing in the analysis of electromagnetic scattering from a half-space of randomly distributed particles. Our derivation follows the procedures described by Varadan et al. [236], Tsang and Kong [223, 226], and Tsang et al. [228].

C.1 Computation of the Integral Imm1 n

The integral Imm1 n is

Imm1 n = ejKe·r0p u3m −mn (ksrlp) dV (r0p) ,

Dp

where Ke = Ksez , and the integration domain Dp is the half-space z0p ≥ 0, excluding a spherical volume of radius 2R centered at r0l. The volume integral can be transformed into a surface integral by making use of the following result. Let u and v be two scalar fields satisfying the Helmholtz equation in the bounded domain D, with the wave numbers Ks and ks, i.e.,

∆u + Ks2u = 0, ∆v + ks2v = 0.

Then, from Green’s theorem we have

where S is the boundary surface of the domain D, and n is the outward unit normal vector to S. Using this result we transform the integral Imm1 n as follows: