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