Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2.5 Photon flux from a surface
The main contribution of the flux comes from the immediate neighborhood of dS, so that
the photon density can be replaced by its first-order Taylor expansion around the origin.
Evaluation of the integral then yields
j ∓ = ρ 0v
4 ± vl∗
6
( ) ∂ρ
∂z
0
where the photon density and its derivative have to be taken at the origin.
If dS is located at a boundary, there will be no flux from outside the sample, but internal
reflections will create an apparent flux from this direction, which is related to the outgoing
flux by the reflectivity R of the surface: j in = R · j out . This gives the boundary conditions
ρ − 2l∗
3
ρ + 2l∗
3
1 + R
1 − R · ∂ρ
∂z = 0 for an upper boundary, i.e. j + = R · j − and
1 + R
1 − R · ∂ρ
∂z = 0 for a lower boundary, i.e. j − = R · j +
The point where the photon density drops to zero is not located at the boundary but at a
distance 2l∗ 1+R
3 1−R · ∂ρ
∂z
in front of it. We can identify this distance with the average penetration
depth z 0 which we have used above for the image point method, as this also is the point where
the photon density vanishes. Assuming a constant gradient of the photon density distribution
close to the surface, one obtains |z 0 | = 2l∗ 1+R 1+R
3 1−R
≈ 0.67
1−R l∗ . Other solutions are mostly of the
order of |z 0 | ≈ 0.7 for the limit of non-reflective surfaces (see e.g. [53], [54]).
2.5 Photon flux from a surface
The experimentally accessible quantity in multiple scattering experiments is not the photon
density distribution inside the sample, but the light intensity emitted from a sample surface.
It is obvious that this intensity (or photon flux density) is proportional to the number of
photons, i.e. the photon density, at the surface. One can therefore easily calculate the expected
intensities for the various experimental situations from eqns. 2.9 and 2.10.
2.5.1 Backscattering geometry
Backscattering experiments are usually performed with thick samples, where the influence of
the rear and side surfaces on the photon density distribution can be neglected, and the sample
is well described as an infinite half-space.
In backscattering geometry, photon paths of all lengths contribute to the intensity at the surface,
so that the latter is not fully described by the solution of the diffusion equation, which is
valid only for long photon paths. It has been shown however [44] that correct results can be
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