Figure 1

Schematic representation of the Mach-Zehnder interferometer used in the setup. The first beamsplitter (BS1) divides the light between a reference arm and a signal arm. The light in the signal arm is focused on the sample by lens L1. The transmitted speckle pattern is collimated by lens L2 and recombined with the reference beam at beamsplitter BS2. Since the reference arm is a few millimeters shorter than the sample arm, the signal pulse does not overlap the reference pulse temporally. Finally, aperture $D1$ selects an area that is smaller than the typical speckle size and polarizer P1 blocks light with a polarization perpendicular to that of the reference beam in order to increase the signal to noise ratio. The beam containing the signal pulse and the reference pulse is propagated into a scanning interferometer (FTIR). When a sample is placed in the signal arm, only a fraction of the incident light reaches the FTIR. In order to balance the interferometer, we use beamsplitters (BS1 and BS2) that reflect approximately 4%.Reuse & Permissions

Figure 2

Time-resolved intensity transmission for a $10.1\mu \mathrm{m}$ thick sample consisting of $\mathrm{Ti}{\mathrm{O}}_2$ grains. The observed nonexponential decay (solid line) agrees with the finite-area correction, Eq. (26), over four decades. Theoretical curves are obtained from Eq. (7) and the shape of the filter. The dotted line was corrected for the time-dependent detection efficiency due to focusing, using Eq. (26) with $w_s=15\mu \mathrm{m}$ and $w_d=10\mu \mathrm{m}$ as estimated from the experimental configuration and a fitted diffusion constant of $D=27.0{\mathrm{m}}^2{\mathrm{s}}^{-1}$. The dashed line is the uncorrected curve for the same diffusion constant. The inset shows an example of the interference signal at the detector as a function of the delay length in the scanning interferometer. The average intensity transmission is obtained from 50 such measurements performed on different areas of the sample.Reuse & Permissions

Figure 3

Diffuse traversal time ${\tau }_t$ (triangles pointing upward), and scaled decay time ${\tau }_d{\pi }^2∕6$ (triangles pointing downward), as a function of sample thickness. ${\tau }_d$ is found by fitting the exponential decay of $J\left(t\right)$ (Method II); the error bars indicate the values found from fitting the first part of the decay (lowest value) and the last part of the decay (highest value). The dashed line is the theoretical value of ${\tau }_d$. The diffuse traversal time is found from the time-resolved intensity transmission obtained by numerically evaluating Eq. (15) (Method I). Without taking into account detection efficiency, theory (dotted line) predicts that the two time scales converge for thick samples. When the theory is compensated for the effect of a finite detection area (solid line), this convergence is lost, in agreement with the experimental data.Reuse & Permissions

Figure 4

Probability distributions for the delay time ${\varphi }^{\prime }$ (top) and the weighted delay time $W$ (bottom) as measured in a $10.1\mu \mathrm{m}$ thick $\mathrm{Ti}{\mathrm{O}}_2$ sample. The dimensionless parameter $Q$ characterizes the width of these distributions. We find $Q=0.47$ from a fit of the theoretical curves given by Eqs. (17, 19) (solid lines). The average values of ${\varphi }^{\prime }$ and $W$ are used in Methods IV and V, respectively, to find the diffusion constant.Reuse & Permissions

Figure 5

Diffuse traversal time ${\tau }_t$ measured using three different techniques. Method I (squares) calculates ${\tau }_t$ from the time-resolved intensity. Method IV (triangles) obtains ${\tau }_t$ from the measured optical phase alone, whereas Method V (circles) uses the intensity-weighted phase information. The excellent agreement indicates that the transmitted field is described by Gaussian distribution. The solid line are the theoretical values for a diffusion constant of $D=25.5{\mathrm{m}}^2{\mathrm{s}}^{-1}$.Reuse & Permissions

Figure 6

Measured field correlation function $C_E$ (real part: circles, imaginary part: squares) for a $\mathrm{Ti}{\mathrm{O}}_2$ sample of thickness $10.1\mu \mathrm{m}$ as a function of the frequency difference $\Omega$ between two optical frequencies. The horizontal axis is scaled by ${\tau }_t^{-1}$ as found by measuring $⟨{\varphi }^{\prime }⟩$. Excellent agreement with theory (solid lines) confirms the diffusion model with boundary corrections.Reuse & Permissions

Figure 7

Measured correlation functions for the intensity $C_I\left(\Omega \right)$ (diamonds), and the weighted delay time $C_W\left(\Omega \right)$ (circles), in a $10.1\mu \mathrm{m}$ thick $\mathrm{Ti}{\mathrm{O}}_2$ sample. The solid lines are the $C_1$ approximations for both correlation functions. Except for some spurious oscillations, agreement with theory is evident.Reuse & Permissions

Figure 8

Measured diffusion constant for samples of different thicknesses obtained in five different ways. The top figure shows the diffusion constant obtained from the diffuse traversal time ${\tau }_t$. We measured the traversal time directly in the time domain (Method I, squares), by extracting the average of the delay time $⟨{\varphi }^{\prime }⟩$ (Method IV, circles), and the weighted delay time $⟨W⟩$ (Method V, triangles). Error bars are only presented for $⟨{\varphi }^{\prime }⟩$, but the errors in the two other values are comparable. The diffusion constants in the middle figure are calculated from the decay time ${\tau }_d$ which is obtained by fitting the decay of the transmitted flux (Method II). The bottom plot displays the diffusion constants obtained from fitting the intensity correlation function (Method III). Except for the two thinnest samples, the consistency of the five different measurement methods is evident.Reuse & Permissions