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