Coherent Backscattering from Multiple Scattering Systems - KOPS ...

2.2 Single scattering – Mie theory
Figure 2.2: Vibration ellipse. The tip of the electric field vector of a light wave traces
out an ellipse with semimajor axis r 1 , semiminor axis r 2 , azimuth γ, and ellipticity
η = arctan(r 2 /r 1 ).
with the size parameter x = ka, the relative refractive index m = k scat
k
, and the Riccati-Bessel
functions ψ n (z) = z j n (z) and ξ n (z) = z h (1)
n (z). The prime indicates differentiation with
respect to the argument in parentheses.
⃗M has no radial component, and in the far-field limit with large ρ = kr the radial component
of N ⃗ is negligible compared to the transverse component, as the spherical Hankel function is
asymptotically given by h (1)
n (kr) ≈ − (−i)n e ikr
ikr
. The scattered electromagnetic field is therefore
basically transverse, and
E θ ≈ E 0 ·
E φ ≈ −E 0 ·
e ikr
−ikr · cos φ · S 2(cos θ) with S 2 =
e ikr
−ikr · sin φ · S 1(cos θ) with S 1 =
n c
∑
n=1
n c
∑
n=1
2n + 1
n(n + 1) (a nτ n + b n π n )
2n + 1
n(n + 1) (a nπ n + b n τ n )
Both sums converge, so that the series can be terminated after n c steps without causing major
inaccuracies. A good assumption is n c ≈ x.
We now focus on a certain scattering plane, which is defined by the wave vectors of the
incoming and emerging waves. Intensity and polarization of the scattered light are then
merely a function of the scattering angle θ and the polarization of the incoming light with
respect to the scattering plane.
In a non-absorbing medium, both incoming and scattered electromagnetic waves can be characterized
by a Stokes vector [30]
⎛ ⎞ ⎛
⎞
I
1
⃗p = ⎜Q
⎟
⎝U⎠ = ⎜cos(2η) cos(2γ)
⎟
⎝cos(2η) sin(2γ) ⎠
V
sin(2η)
7