Non-Gaussian Correlations between Reflected and Transmitted Intensity Patterns Emerging from Opaque Disordered Media

The propagation of monochromatic light through a scattering medium produces speckle patterns in reflection and transmission, and the apparent randomness of these patterns prevents direct imaging through thick turbid media. Yet, since elastic multiple scattering is fundamentally a linear and deterministic process, information is not lost but distributed among many degrees of freedom that can be resolved and manipulated. Here, we demonstrate experimentally that the reflected and transmitted speckle patterns are robustly correlated, and we unravel all the complex and unexpected features of this fundamentally non-Gaussian and long-range correlation. In particular, we show that it is preserved even for opaque media with thickness much larger than the scattering mean free path, proving that information survives the multiple scattering process and can be recovered. The existence of correlations between the two sides of a scattering medium opens up new possibilities for the control of transmitted light without any feedback from the target side, but using only information gathered from the reflected speckle.

Popular Summary

When coherent light (like a laser) passes through or reflects off most materials, it scatters and creates what is known as a speckle pattern, a seemingly random intensity pattern. These speckles limit the precision of many types of imaging applications, including microscopic images of biological processes and ground-based telescopic views of space. Common sense suggests that reflected and transmitted light are completely uncorrelated—no information can be obtained regarding the transmitted light by measuring the reflected one and vice versa. However, some of our team recently predicted that interference effects should lead to such correlations. We experimentally investigate this prediction, and we find that this correlation not only exists but is much richer and more complicated than expected.

We shine a helium-neon laser at a 45-degree angle onto a slab of glycerol with suspended particles of titanium dioxide and look for correlations between the speckle patterns of the transmitted and reflected light. By exploring a large range of sample thicknesses and scattering mean-free paths, we show that for large optical densities, the reflected and transmitted intensity profiles exhibit a long-range anticorrelation. For thinner systems, this is accompanied by a previously unforeseen long-range positive correlation. We develop a perturbative theory that describes both contributions and accurately represents the experimental results.

The presence of correlations between the reflected and transmitted light proves that information of what is happening on the far side of an opaque medium can, in principle, be obtained by only measuring reflected light, thus opening the way to novel noninvasive imaging techniques.

Figure 1

(a) Experimental setup. A scattering slab, formed by a suspension of ${\mathrm{TiO}}_2$ particles in glycerol, is illuminated by a laser beam incident at an angle of approximately 45°. The speckle patterns on the two surfaces, $T(x,y)$ and $R(x,y)$ are recorded with two identical imaging systems. (b) Examples of samples with thickness $L=20\mu \mathrm{m}$ but different ${\mathrm{TiO}}_2$ concentrations: from left to right, $5\mathrm{g}/{\mathrm{d}\mathrm{m}}^3$, $10\mathrm{g}/{\mathrm{d}\mathrm{m}}^3$, and $40\mathrm{g}/{\mathrm{d}\mathrm{m}}^3$, which correspond to a mean free path of (60, 20.4, and 9.8) $\pm 2.5\mu \mathrm{m}$, respectively.

Figure 2

Typical measured speckle patterns in transmission (a) and reflection (b), for a sample with $L=20\mu \mathrm{m}$ and $\ell \simeq 20\mu \mathrm{m}$. (c) Cross-correlation product between the speckle patterns in (a) and (b). (d) Correlation function $C^{RT}(\mathrm{\Delta }x,\mathrm{\Delta }y)$ obtained after additional ensemble averaging from ${10}^4$ realizations of the disorder. The long-range character of the correlation function, which extends far beyond the size of a speckle spot, is clearly visible.

Figure 3

Average reflection-transmission correlation function $C^{RT}$ for different values of $L$ and $\ell$ and the optical thickness $b=L/\ell$. Left column: 3D numerical simulations of 2D maps of $C^{RT}(\mathrm{\Delta }x,\mathrm{\Delta }y)$. Center and right columns: Experimental results. For clarity, both 2D maps of $C^{RT}(\mathrm{\Delta }x,\mathrm{\Delta }y)$ and cross sections along the line $\mathrm{\Delta }y=0$ (indicated as a dotted line in the 2D maps) are displayed. Two regimes are identified. For moderate optical thickness ($b\lesssim 1$), the correlation function is dominated by a narrow peak with a negative side lobe. For large optical thicknesses ($b>1$), the correlation function is dominated by a wide negative dip.

Figure 4

Diagrams contributing to the $C^{RT}(\mathrm{\Delta }\mathbf{r})$ correlation. An intensity correlation depends on two intensities (four fields) that propagate through the sample, therefore involving four inputs and four outputs. Shaded tubes represent diffusive paths and open circles stand for scatterers; single solid lines stand for averaged fields and single dashed lines for their complex conjugates. The diagram in panel (a) is representative of the class of $C_2$ diagrams describing the negative contribution of the correlation function at large optical thickness ($L\gg \ell$). Panel (b) represents the class of $C_0$-type diagrams that contribute to the positive peak dominant in the regime $\ell \sim L\gg \lambda$.

Figure 5

Experimental data to determine the mean free path of the three different sample concentrations we used. In red are the data corresponding to the concentration of 50 mg of ${\mathrm{TiO}}_2$ in 10 mL of glycerol, in blue the concentration of 150 mg of ${\mathrm{TiO}}_2$ in 10 mL of glycerol, and in yellow the data corresponding to the concentration of 400 mg of ${\mathrm{TiO}}_2$ in 10 ml of glycerol. The black dashed lines represent the Lambert-Beer law fitting to the corresponding experimental data.

Figure 6

Typical raw average intensity measurement in reflection (a) and transmission (b), with the averaged intensity distribution over 2000 realizations of disorder in reflection (c) and transmission (d). The inhomogeneous background is clearly visible in both $⟨R⟩$ and $⟨T⟩$. (e) Direct cross-correlation of $R$ and $T$ (without subtracting the background). (f) Cross-correlation between $\delta R$ and $\delta T$ as described in the main text.

Figure 7

Comparison between the two correlation functions $⟨\overline{C^{RT}}(\mathrm{\Delta }\mathbf{r})⟩$ and $C^{RT}(\mathrm{\Delta }\mathbf{r})$. The transverse distance $\mathrm{\Delta }r=\mathrm{\Delta }x$ is varied along the direction of the illumination plane. The parameters of the 3D numerical simulation are $L/\ell =1.5$, $k\ell =10$, ${\theta }_a\simeq 45°$.

Figure 8

Correlation function $C^{RT}$ calculated from numerical simulation of the wave equation in a 3D slab. (a) Full correlation. (b) Connected part of the correlation [second term of the rhs of Eq. (c2)]. (c) $C_1$ contribution of the correlation [first term of the rhs of Eq. (c2)]. The parameters are $k\ell =15$, $L/\ell =1$, ${\theta }_a\simeq 45°$.

Figure 9

Comparison of two R-T correlation functions calculated from numerical simulation of the wave equation in a 3D slab. (a) Correlation function $C^{RT}(\mathrm{\Delta }\mathbf{r})=⟨\delta R(\mathbf{r})\delta T(\mathbf{r}+\mathrm{\Delta }\mathbf{r})⟩/⟨R(\mathbf{r})⟩⟨T(\mathbf{r}+\mathrm{\Delta }\mathbf{r})⟩$ built from fluctuating parts of the fields. (b) Correlation function ${\tilde{C}}^{RT}=⟨\delta \tilde{R}(\mathbf{r})\delta \tilde{T}(\mathbf{r}+\mathrm{\Delta }\mathbf{r})⟩/⟨\tilde{R}(\mathbf{r})⟩⟨\tilde{T}(\mathbf{r}+\mathrm{\Delta }\mathbf{r})⟩$ built from the full fields (see text for details). The parameters are $k\ell =15$, $L/\ell =1$, ${\theta }_a=45°$.

Figure 10

Comparison of two R-T correlation functions calculated from numerical simulation of the wave equation in a 2D slab. (a) Moderate optical thickness $L=\ell$ and shifted incidence ${\theta }_a\simeq -35°$. (b) Large optical thickness $L/\ell =7$ at normal incidence ${\theta }_a=0$. The other parameter is $kl=10$.

Figure 11

Typical diagrams contributing to the connected four-field correlations in R-T. Panel (a) corresponds to the case where the fields exchange at the entrance, propagate, and exchange inside the medium. After this exchange, they can travel diffusively through long distances to eventually be measured at the desired points. Panel (b) corresponds to the case where the two pairs of fields propagate first together, then exchange inside the medium, and, at the end, have to exchange again to eventually be measured at two different points. The output vertex of this diagram is identical to the output vertex of the $C_1$ correlation.

Figure 12

Analytical predictions for the $C_2^{RT}$ correlation (black solid line), $C_2^{\mathrm{out}}$ correlation (red dotted line), and $C_2^{\mathrm{in}}$ correlation (green dashed-dotted line) compared with the simulation of 2D wave propagation in a disordered medium (blue dashed line). The parameters of the simulations are $k\ell =10$, $L/\ell =10$, ${\theta }_a=0$.

Figure 13

$C^{RT}(\mathrm{\Delta }\mathbf{r})$ calculated from 3D numerical simulations of the wave propagation in a disordered slab of moderate optical depth. The direction $\mathrm{\Delta }y=0$ is defined as the intersection of the incidence plane with the sample surface. The parameters are $L/\ell =1$, $k\ell =15$, ${\theta }_a\simeq 75°$.

Figure 14

Dependence of $C^{RT}(\mathrm{\Delta }\mathbf{r})$ on the illumination angle ${\theta }_a$, along the illumination direction $\mathrm{\Delta }y=0$ (horizontal axis of Fig. 13). The parameters of the 3D simulation are $L/\ell =1$, $k\ell =15$.

Figure 15

Leading diagrams contributing to $C_0^{RT}$. Shaded tubes represent diffusive paths (ladders); single solid lines stand for averaged fields, and single dashed lines for their complex conjugates. The extra scatterer located near the surface boundary can connect the ladders in four different ways.

Figure 16

Analytical prediction for the $C_0^{RT}$ correlation (normalized to unity) represented in Fig. 15 for two different illumination angles, (a) ${\theta }_a=0$ and (b) ${\theta }_a=45°$, and their cut (top figure) along the direction $\mathrm{\Delta }y=0$, which is defined as the intersection of the incidence plane with the sample surface. The parameters are $\lambda =632\mathrm{nm}$, $L=50\mu \mathrm{m}$, $\ell =15\mu \mathrm{m}$. The diffusive approximation was used for the ladder diagrams.