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

278 B Wave Functions

If the translation is along the z-axis, the addition theorem involves a single summation

For radiating spherical wave functions we consider the integral representation (B.2) and the relation r = r0 + r1. For r1 > r0, we express u3mn as (cf. (B.5))

and use the spherical wave expansion of the quasi-plane wave Q(k, β, α, r1) to derive

and use the spherical wave expansion of the plane wave exp(jk · r1) to obtain

n =0 m =−n

Inserting the spherical wave expansion of the quasi-plane wave Q(k, β, α, r0) into (B.61) yields

Cmn,m3 n (kr0) = 2jn −n jn a (m, m | n , n, n ) u3m−m n (kr0) . (B.62)

n

Finally, we note the integral representation for the translation coe cients in the specific case of axial translation:

Recurrence relations have been derived for the Cmn,mn coe cients [150]. The derivation is simple for the case of axial translation and positive m. Using the integral representation (B.60) (or (B.63)) and the recurrence relations for

We note that the convention Cmn,mn = 0 for m > n or m > n , is assumed in the above equations. For the recurrence relationships to be of practical use, initial values are needed. This is accomplished by using the integral representations

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