Depolarized light scattering due to double scattering

We have investigated theoretically and experimentally the contribution of double scattering to the depolarized scattering intensity. This work was motivated by a recent molecular scattering theory calculation of Oxtoby and Gelbart, who found that the intensity of the depolarized light scattered by a fluid near the critical point arises primarily from successive independent scattering events; however, the dependence of the depolarization ratio on sample size predicted by Oxtoby and Gelbart cannot be directly tested, since they considered an impractical geometry, a small illuminated sphere concentric with a large spherical scattering volume. We have calculated for a practical scattering geometry the depolarized intensity due to double scattering by simply integrating over all possible double scattering events. Although our measurements were performed on a fluid near the critical point, our calculations apply to any system which has a differential scattering cross section with an angular dependence given by the dipole intensity distribution, $\ensuremath{\sigma}={\ensuremath{\sigma}}_{0}{sin}^{2}\ensuremath{\varphi}$, where ${\ensuremath{\sigma}}_{0}$ depends on the properties of the scattering system and $\ensuremath{\varphi}$ is the angle between the polarization vector of the incident light and the direction of propagation of the scattered light. (The scattering integrals that we have derived can also be straightforwardly evaluated for a system whose cross section has a different angular dependence.) In the scattering geometry considered here a vertically (V) polarized focused laser beam illuminates a parallelepiped sample volume, which is viewed at a 90\ifmmode^\circ\else\textdegree\fi{} scattering angle by a detector which is in the horizontal plane. Our principal theoretical result is that the ratio of the intensity of the horizontally (H) polarized component of the doubly scattered light, ${I}_{\mathrm{VH}}^{d}$, to the (total) intensity of the vertically polarized component of the scattered light, ${I}_{\mathrm{VV}}$, should be given by $\frac{{I}_{\mathrm{VH}}^{d}}{{I}_{\mathrm{VV}}}=\frac{1}{4} \ensuremath{\pi}{\ensuremath{\sigma}}_{0}S$, where $S$ is the height of the sample seen by the detector; $S$ is assumed to be small compared to the horizontal dimensions of the sample. There is an additional "collision-induced" contribution to the depolarized intensity which we have not calculated, but if the sample height $S$ is greater than the laser-beam diameter, this contribution will be dependent of $S$. Thus the prediction is that plots of the measured depolarization ratio $\frac{{I}_{\mathrm{VH}}^{d}}{{I}_{\mathrm{VV}}}$ (which includes both the double scattering and collision-induced contributions) as a function of $S$ will be straight lines with slopes given by $\frac{1}{4}\ensuremath{\pi}{\ensuremath{\sigma}}_{0}$. We have measured the depolarization ratio for xenon as a function of sample height, wavelength, and temperature, and the results are in agreement with the predictions of the double-scattering theory [except very close to the critical point where higher-order scattering is important, i.e., the theory is applicable for $\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}\ensuremath{\gtrsim}{10}^{\ensuremath{-}3}$]. We conclude that depolarization ratio measurements may be generally useful in determining absolute scattering cross sections for a variety of scattering systems. Refinements of the technique are suggested which would make possible absolute cross-section measurements of high accuracy even for systems which are not strong scatterers.