Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2 Theory
can be separated into three independent differential equations using the ansatz Ψ(r, θ, φ) =
R(r) · Θ(θ) · Φ(φ):
1
R
(
d
dr
sin θ
Θ
r 2 dR )
+ k 2 r 2 = α
dr
(
sin θ dΘ )
= β − α sin 2 θ
dθ
d
dθ
1 d 2 Φ
Φ dφ 2 = −β
Setting β = m 2 and α = n(n + 1) where m = 0, 1, 2, . . . and n = m, m + 1, . . . we obtain the
complete solution Ψ for the wave equation, and from this the vector harmonics ⃗ M = ⃗∇ × (⃗rΨ)
and ⃗ N =
⃗∇× ⃗ M
k
.
The solution of the radial part of the wave equation R(r) is given by a linear combination of
the spherical Bessel functions j n (ρ) = √ π
2ρ J n+1/2(ρ) and y n = √ π
2ρ Y n+1/2(ρ) where J ν (ρ) and
Y ν (ρ) are Bessel functions of first and second kind, and ρ = kr. For the outgoing scattered
wave the appropriate linear combination is given by one of the spherical Bessel functions of
the third kind or spherical Hankel functions, h (1)
n (ρ) = j n (ρ) + iy n (ρ).
The zenith part of the wave equation Θ(θ) is solved by associated Legendre functions Pn m (cos θ).
For the following calculations it is convenient to define the angle-dependent functions π n =
Pn
1
sin θ and τ n = dP1 n
dθ .
It can be shown that the solution for the scattered electric field is given by
⃗ Escat =
∞
∑ i n 2n + 1
)
E 0
(ia nNn ⃗ − b nMn ⃗
n(n + 1)
n=1
where the applicable vector harmonics are given by
⃗M n = cos φ π n (cos θ) h (1)
n (ρ) ê θ − sin φ τ n (cos θ) h (1)
n (ρ) ê φ
⃗N n = cos φ n(n + 1) sin θ π n (cos θ) h(1) n (ρ)
ρ
ê r + cos φ τ n (cos θ) [ρh(1) n (ρ)] ′
ρ
ê θ −
− sin φ π n (cos θ) [ρh(1) n (ρ)] ′
ρ
ê φ
and the coefficients a n and b n are
a n = m ψ n(mx) ψ ′ n(x) − ψ n (x) ψ ′ n(mx)
m ψ n (mx) ξ ′ n(x) − ξ n (x) ψ ′ n(mx)
; b n = ψ n(mx) ψ ′ n(x) − m ψ n (x) ψ ′ n(mx)
ψ n (mx) ξ ′ n(x) − m ξ n (x) ψ ′ n(mx)
6