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

260 A Spherical Functions

The following orthogonality relations are important for solving scattering problems:

π

Pnm(cos θ)Pnm(cos θ) sin θ dθ = δnn ,

0

π

τnm(θ)τnm(θ) + m2πnm(θ)πnm (θ) sin θ dθ = n(n + 1)δnn ,

0

π

[πnm(θ)τnm(θ) + τnm(θ)πnm (θ)] sin θ dθ = 0.

0

For negative arguments we have the symmetry relation

Pnm(− cos θ) = (−1)n−mPnm(cos θ)

and consequently,

B

Wave Functions

In this appendix, we summarize the basic properties of the characteristic solutions to the scalar and vector wave equations. We recall the integral and orthogonality relations, and the addition theorems under coordinate rotations and translations.

B.1 Scalar Wave Functions

The spherical wave functions form a set of characteristic solutions to the Helmholtz equation and are given by [215]

u1mn,3 (kr) = zn1,3 (kr) Pn|m| (cos θ) ejmϕ,

where n = 0, 1, . . . ,, m = −n, . . . , n, (r, θ, ϕ) are the spherical coordinates of the position vector r, zn1 designates the spherical Bessel functions jn, zn3 stands for the spherical Hankel functions of the first-order h(1)n , and Pn|m| denotes the normalized associated Legendre functions. u1mn is an entire solution to the Helmholtz equation (solution which are defined in all of R3) and u3mn is a radiating solution to the Helmholtz equation in R3 − {0}.

The Green function or the fundamental solution is defined as

and straightforward di erentiation shows that for fixed r R3, the fundamental solution satisfies the Helmholtz equation in R3 − {r }. The expansion of the Green function in terms of spherical wave functions is given by

262 B Wave Functions

The spherical harmonics are defined as

Ymn (θ, ϕ) = Pn|m| (cos θ) ejmϕ

and they satisfy the orthogonality relation

2π π

Ymn (θ, ϕ) Ym n (θ, ϕ) sin θ dθ dϕ = 2πδm,−m δnn .

00

The spherical wave functions can be expressed as integrals over plane waves [26, 70, 215]

where (k, β, α) are the spherical coordinates of the wave vector k. We note that (B.2) is valid for z > 0 since only then the integral converges.

The spherical wave expansion of the plane wave exp(jk ·r) is of significant importance in electromagnetic scattering and is given by [215]

The quasi-plane waves have been introduced by Devaney [46], and for z > 0, they are defined as

2π π −j∞

Q(k, β, α, r) = 2 ∆(β, α, β , α )ejk (β ,α )·r sin β dβ dα ,

00

where

If β and β belong to [0, π], the expression of ∆ simplifies to

∆(β, α, β , α ) = δ(cos β − cos β)δ(α − α).

As in (B.3), the expansion of quasi-plane waves in terms of radiating spherical wave functions is given by

The function ∆ allows to formally analytically continue a function f (β, α), defined for real β, onto the complex values of β. In particular we see that

2π π

where β can assume all values lying on the contour [0, π2 − j∞), and therefore

2π π2 −j∞

f (β, α)ejk(β,α)·r sin β dβ dα

00

2π π

00

where we have used the definition of Q. In view of (B.2) and (B.5), we deduce that the integral representation for the radiating spherical wave functions is given by

In the far-field region, a plane wave can be expressed as a superposition of incoming and outgoing spherical waves. To derive this expression, we start with the series representation (B.3) written as

use the asymptotic behavior of the spherical Bessel functions for large values of the argument

take into account the completeness relations for spherical harmonics,

δ (er − ek ) = δ(ϕ − α)δ(cos θ − cos β)

266 B Wave Functions

The superscript ‘1’ stands for the regular vector spherical wave functions while the superscript ‘3’ stands for the radiating vector spherical wave functions. It

is useful to note that for n = m = 0, we have M 100,3 = N 100,3 = 0. M 1mn, N 1mn is an entire solution to the Maxwell equations and M 3mn, N 3mn is a radiating

solution to the Maxwell equations in R3 − {0}.

The vector spherical wave functions can be expressed in terms of vector spherical harmonics as follows:

In the far-field region, the asymptotic behavior of the spherical Hankel functions for large value of the argument yields the following representations for the radiating vector spherical wave functions [40]:

The orthogonality relations on a sphere Sc, of radius R, are

[er × M mn(kr)] · [er × M m n (kr)] dS(r) = πR2zn2 (kR)δm,−m δnn ,

Sc

×δm,−m δnn ,