Varying the Effective Refractive Index to Measure Optical Transport in Random Media

We introduce a new approach for measuring both the effective medium and the transport properties of light propagation in heterogeneous media. Our method utilizes the conceptual equivalence of frequency variation with a change in the effective index of refraction. Experimentally, we measure intensity correlations via spectrally resolved refractive index tuning, controlling the latter via changes in the ambient pressure. Our experimental results perfectly match a generalized transport theory that incorporates the effective medium and predicts a precise value for the diffusion constant. Thus, we directly confirm the applicability of the effective medium concept in strongly scattering materials.

Figure 1

The experimental setup. The sample is placed in a pressurized chamber and illuminated by a HeNe laser. A single polarization of the transmitted speckle is recorded by a CCD camera.

Figure 2

RIT autocorrelation coefficient as a function of $\Delta n_h$. Data points show measurement results for 3 slabs of polyethylene filters ($L=1$, 1.5, and 2 mm for black squares, red circles, and blue triangles, respectively). The lines show the fit to theory [Eq. (5)].

Figure 3

Measured relative change of $n_e$ as a function of $\Delta n_h$. The value of $\Delta n_e/n_e$ is extracted by measuring this spectral shift. The refractive index of air is calculated based on the Edlen formula [23]. Inset: The shifted cross-correlation coefficient $C$ versus the spectral shift $\Delta \nu$ for certain pressures (from right to left: $\Delta P=0.0$, 0.075, 0.18, 0.28, 0.37, and 0.46 bar). The peak position of each curve denotes the spectral shift, which is proportional to the relative change in the effective refractive index. The spiky feature at zero position is an artifact of the slight inhomogeneity over the detection efficiency of CCD pixels.

Figure 4

The measurement parameters $a\equiv \frac{4\pi \nu \delta }{{\tau }_{\delta }}$ and $b\equiv \frac{1}{{\pi }^2{\tau }_d}$ are plotted versus the inverse of the thickness squared. Black squares and gray (red) dots correspond to the index tuning and time-resolved measurements, respectively. The slope of each data set is equal to the diffusion constant $D$ measured by each specific method. The fact that the two slopes are equal proves the consistency of our index tuning method with the time-resolved measurements.