Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2.2 Single scattering – Mie theory
Figure 2.1: Coordinate system of single scattering. The coordinate system (θ, φ) of
Mie scattering at a spherical particle with radius a is defined by the wave vector ⃗ kin
and the electric field vector ⃗ Ein of the incoming light wave. The wave vectors of the
incoming and outgoing waves⃗ kin and⃗ kout span the scattering plane.
The scattered light intensity at a certain point⃗r must then be
∫∫
I(⃗r) ∝
G(⃗r,⃗r 1 )G ∗ (⃗r,⃗r 2 )Ψ in (⃗r 1 )Ψ ∗ in(⃗r 2 ) d⃗r 1 d⃗r 2
For a dilute system with pointlike scatterers the perturbation series of eqn. 2.3 immediately
justifies treating multiple scattering of electromagnetic waves as random walks of photons
with different numbers of scattering events: The photons travel in free space (described by
Green’s functions G 0 , which have the form of spherical waves) until they hit a particle and are
scattered into the surrounding space, where they again propagate freely, hit another particle,
and so on.
The random walk picture can still be upheld if the size of the particles is not negligible.
However, in this case one needs to consider how the particles distribute the incoming intensity
into their surrounding to describe the random walk properly. If a spherical particle can be
considered as an acceptable approximation for the actual particle shape, one can use the
solution given by G. Mie [42] and others. In the next section, we will follow the approach of
[16] to derive the distribution of the scattered light around a single particle.
2.2 Single scattering – Mie theory
The problem of an electromagnetic wave scattered by a spherical particle clearly has a spherical
symmetry. Therefore it is convenient to treat the problem in a polar coordinate system
with the scattering particle of radius a located in the origin, and wave vector and polarization
of the incident light defining the angular coordinates θ = 0 and φ = 0. The wave equation in
polar coordinates
1
r 2
(
∂
r 2 ∂Ψ )
+ 1
∂r ∂r r 2 sin θ
(
∂
sin θ ∂Ψ )
+
∂θ ∂θ
1
r 2 sin 2 θ
∂ 2 Ψ
∂φ 2 + k2 Ψ = 0
5