Coherent Backscattering from Multiple Scattering Systems - KOPS ...

3 Setups
measuring the backscattering of a sample with known scattering properties as a function of
the incoming laser power P, which is determined with a calibrated powermeter (FieldMaxII
from Coherent) from a reflection on a glass plate in the laser beam at the entrance of the
setup. A fit of the powers P(d θ ) as a function of the diode signals d θ with a polynomial then
yields a calibration function for each photodiode (see fig. 5.1).
As reference sample we use a block of teflon, the backscattering cone of which has a FWHM
of the order of 0.03 ◦ for visible light (see sec. 5.2.2). This is much narrower than the angular
resolution of the wide-angle setup, so that teflon can be considered to give a purely incoherent
signal proportional to P α d (θ).
3.3 Small Angle Setup
In sec. 2.7 the coherent backscattering cone was presented as a superposition of Gaussian
distributions j(s) · e − sin2 θ
3 k 2 sl ∗ , whose width is a function of the path length s of the timeinverted
photon paths. Absorption and localization both affect mainly long paths, which
contribute essentially at the very tip of the backscattering cone. Their reduced contribution
results in a rounding of the tip of the backscattering cone. With a high-resolving setup it
should therefore be possible to observe both phenomena in coherent backscattering. The
same setup could also be used to measure the transport mean free paths of samples with
extremely narrow backscattering cones and thus complement the wide angle setup.
3.3.1 Precision requirements
For an estimate of the required setup precision we assume for a moment that absorption
(or localization) results in an abrupt cutoff at path length s = L. The narrowest Gaussian
that contributes to the backscattering cone is therefore the one with standard deviation σ =
√
3/(2k 2 Ll ∗ ) = √ 1/(2k 2 Dτ). The angular width of the conetip rounding must be of the same
order of magnitude.
The absorption of a sample like the titania powder R700 (see sec. 4.2) with absorption time
τ = 2 ns and diffusion coefficient D = 15 m2 /s at wavelength λ = 590 nm will therefore require
to properly resolve an angular range of θ round ≈ ±0.02 ◦ . To observe localization, which sets
in after a localization length l a = 340 mm [48], a similar resolution is necessary.
Test calculations show that the intensity difference between a localizing and a non-localizing
sample is less than 0.1% of the maximum of the cooperon (fig. 3.4). As the detection must
be able to capture the maximal backscattered intensity at θ = 0, plus some external radiation
and electronic noise that can never be avoided completely, while still providing the necessary
intensity resolution, the digital range of the detection must be at least 2 14 , better 2 15 − 2 16 .
3.3.2 Optical setup
In the small angle setup a 4-megapixel 16-bit monochrome CCD camera (Alta U4000 from
Apogee) is placed opposite the sample (fig. 3.5). The camera can be cooled thermoelectrically
28