Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2 Theory
In scattering theory, the mathematical and physical framework for studying and understanding
scattering events, the interaction of scattering particles with electromagnetic waves is described
as the solution of a particular partial differential equation, the so-called wave equation.
For many systems, like for example scattering of light at a single spherical particle, one can
solve this equation exactly. Multiple scattering samples however, with millions and billions of
randomly distributed scattering particles, have to be described by an approximate, collective
solution, as the exact solution can be obtained neither analytically nor numerically.
2.1 The wave equation
The scattering system we will consider in the following consists of randomly distributed scatterers
with dielectric permittivity ɛ scat in a surrounding medium with ɛ surr . We assume both
media to be non-magnetic (i.e. permeabilities µ scat = µ surr = 1) which corresponds to the usual
experimental conditions and does not bring any structural changes into the calculations.
The vector wave equations for a multiply scattered electromagnetic wave can be derived from
Maxwell’s equations as
∇ 2 ⃗ E(⃗r) +
ω 2
c 2 0
ɛ(⃗r)⃗ E(⃗r) = 0 and ∇
2 H(⃗r) ⃗ +
ω 2
ɛ(⃗r)⃗ H(⃗r) = 0
c 2 0
where ⃗ E(⃗r) and ⃗ H(⃗r) are the electric and magnetic field components, ɛ(⃗r) is either ɛscat or
ɛ surr , ω is the light frequency, and c 0 is the vacuum speed of light.
Instead of directly solving the above vector equations it is however convenient to find a solution
for the scalar wave equation
∇ 2 Ψ(⃗r) + ω2 ɛ(⃗r)Ψ(⃗r) = 0 (2.1)
c 2 0
As it will be demonstrated in sec 2.2, one can construct two vector harmonics M ⃗ = ⃗∇ × (⃗c Ψ)
and N ⃗ =
⃗∇× M ⃗
k
from this solution using a suitable pilot vector ⃗c. M ⃗ and N ⃗ are orthogonal
solutions of the vector wave equation, so the complete solution for the electric field is the
linear combination ⃗ E = AM ⃗ + BN, ⃗ and the magnetic field H ⃗ can be calculated from ⃗ E using
Maxwell’s equations.