Optical transport lengths quantifying depolarization in experiments on random media

When light diffuses in random media, the intensity decays and at the same time the polarization state is scrambled. As explanation for this apparent inseparability, theoretical work has identified a profound relation between length scales of optical transport and depolarization. Here, we experimentally confirm and quantify their proportionality with thickness-dependent depolarization measurements in colloidal suspensions of microscopic constituents. The observed equivalences accurately predict the nonlinear evolution rate of depolarization over a large range of penetration depths. Our results provide a simple relation, that connects light diffusion in strongly scattering media with measurable polarization signatures over wide spectral ranges and scatterer concentrations.

Figure 1

(a) Sketch of the randomization of $\mathbf{k}$ in real space for two different ratios of ${\ell }^{*}/\ell$. A shorter transport mean free path ${\ell }^{*}$ implies faster randomization. (b) A random walk of the optical property $\mathrm{\Delta }$ in the birefringence or dichroism plane represents the randomization of polarization [see Eq. (3)]. (c) Schematic of the experiment. Polarized and collimated LED light is sent through a colloidal suspension of fixed size spheres over varying thicknesses. The outgoing light is analyzed with polarization optics and a photodiode and the resulting Mueller matrix is obtained

Figure 2

(a) Light transport quantities (solid lines) $\ell$ in blue (dark gray) and ${\ell }^{*}$ in green (light gray) show distinct spectra. Measured depolarization length scales $z_b^{\mathrm{T}}$ in blue (dark gray) symbols and $B_b^{-1}$ in green (light gray) symbols follow $\ell$ and ${\ell }^{*}$ for linear (square) and circular (circle) polarization [Eq. (7)]. (b) Measured depolarization (symbols) with corresponding fits to Eq. (5) (solid lines) as a function of sample thickness $z$ for a wavelength $\lambda =530\mathrm{nm}$. Each data set belongs to one of the bases, linear $L$ (square), linear $L45$ (triangle), and circular $C$ (circle). The transient regime is characterized by $z_b^{\mathrm{T}}$. The diffusion coefficient $B_b$ gives rise to the linear growth of depolarization in the stationary regime. Its inverse $B_b^{-1}$ is the corresponding depolarization length. All graphs are drawn at a volume fraction of $\varphi =1.1\%$ and sphere size $2r=1.5\mathrm{\mu m}$.

Figure 3

Depolarization length $B_b^{-1}$ (a, b) and transient length $z_b^{\mathrm{T}}$ (c, d) plotted as functions of ${\ell }^{*}$ and $\ell$, respectively. Data points were obtained from parameters ${\mathbf{A}}_b$ and ${\mathrm{\Sigma }}_b$ using fits to Eq. (5), while $\ell$ and ${\ell }^{*}$ were computed from Mie theory using the known values of ($\varphi , x$). Each color and symbol corresponds to a different volume fraction of scatterers measured at different wavelengths. The left and right columns correspond to linear ($L$) and circular ($C$) polarization, respectively.