Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2 Theory
The scalar wave equation 2.1 can be written in form of the inhomogenous differential equation
∇ 2 Ψ(⃗r) + k 2 Ψ(⃗r) = −V(⃗r)Ψ(⃗r)
where k 2 [ ]
= ω2 ɛ
c 2 surr , and V(⃗r) = ω2
0
c ɛ(⃗r) − ɛsurr is the scattering potential. a Its solution at a
2
0
certain point⃗r is given by
∫
Ψ(⃗r) = Ψ in (⃗r) +
G 0 (⃗r,⃗r 1 )V(⃗r 1 )Ψ(⃗r 1 ) d⃗r 1 (2.2)
where Ψ in is the part of the incoming wave that has not been scattered before. G 0 is termed the
bare Green’s function and describes the propagation of the electromagnetic field in a medium
without scatterers. It is defined by
∇ 2 G 0 (⃗r,⃗r 1 ) + k 2 G 0 (⃗r,⃗r 1 ) = −δ(⃗r,⃗r 1 )
and is given by
G 0 (⃗r,⃗r 1 ) = e−ik|⃗r−⃗r 1|
4π|⃗r −⃗r 1 |
By applying eqn. 2.2 recursively, the wave function can be expanded into a perturbation series
∫
Ψ(⃗r) = Ψ in (⃗r) +
G 0 (⃗r,⃗r 1 )V(⃗r 1 )Ψ in (⃗r 1 ) d⃗r 1 +
∫∫
+ G 0 (⃗r,⃗r 1 )V(⃗r 1 )G 0 (⃗r 1 ,⃗r 2 )V(⃗r 2 )Ψ in (⃗r 2 ) d⃗r 1 d⃗r 2 + · · · (2.3)
It would be convenient to split off the incoming wave Ψ in like Ψ(⃗r) = ∫ G(⃗r,⃗r ′ )Ψ in (⃗r ′ ) d⃗r ′ .
This introduces the total Green’s function G, which describes the electromagnetic field at a
certain position⃗r due to a disturbance at another point⃗r ′ . It has a perturbation series
∫
G(⃗r,⃗r ′ ) = G 0 (⃗r,⃗r ′ ) + G 0 (⃗r,⃗r a )V(⃗r a )G 0 (⃗r a ,⃗r ′ ) d⃗r a +
∫∫
+ G 0 (⃗r,⃗r a )V(⃗r a )G 0 (⃗r a ,⃗r b )V(⃗r b )G 0 (⃗r b ,⃗r ′ ) d⃗r a d⃗r b + · · ·
and the formal definition
∇ 2 G(⃗r,⃗r ′ ) + ω2 ɛ(⃗r)G(⃗r,⃗r ′ ) = −δ(⃗r,⃗r ′ )
c 2 0
[a] Elsewhere [16, 56], the scattering potential is defined as V(⃗r) = ω2 [ɛ
c 2 surr − ɛ(⃗r)] (or similar), so that the wave
0
equation becomes ∇ 2 Ψ(⃗r) + k 2 Ψ(⃗r) = V(⃗r)Ψ(⃗r). This definition makes the perturbation series eqn. 2.3 look less
intuitive as half of the integrals seem to be subtracted.
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