Largest eigenvalue statistics of wavefront shaping in complex media

TL;DR

This work develops a complete statistical description of the enhancement factor in wavefront shaping through complex media by combining experiments, simulations, and exact random-matrix results. It shows that, for finite input/output channels, the Laguerre–Wishart ensemble accurately predicts the mean and full distribution of the largest eigenvalue $\lambda_{\max}$ beyond the Marčenko–Pastur law, while long-range mesoscopic correlations induce giant fluctuations not captured by the uncorrelated model. In the limit $M_1 \gg M_2$, the LW predictions converge to a GUE-based description, linking practical focusing performance to universal random-matrix statistics; however, thick, strongly scattering samples exhibit enhanced fluctuations due to correlations, revealing fundamental limits and a practical route to quantifying long-range effects from modest datasets. The findings have broad implications for biomedical imaging, optical metrology, optical trapping, and communications through scattering media, where accurate prediction of the enhancement and its variability is essential.

Abstract

In wavefront shaping, waves are focused through complex media onto one or more target points, and the resulting intensity enhancement is quantified by the enhancement factor. While reproducible enhancement is crucial in experiments, the fluctuations of the enhancement factor remain largely unexplored. Here, we combine experiments, simulations, and exact analytical results using random matrix theory to determine its full distribution in multi-point focusing. Our theoretical framework goes beyond the Marčenko-Pastur law-valid only in the limit of a large number of channels-by accurately predicting the mean enhancement in finite-channel experiments and its fluctuations whenever long-range mesoscopic correlations are negligible (e.g., in weakly scattering media or under limited wavefront control). Notably, in the strongly scattering regime, experiments and simulations reveal giant fluctuations in the enhancement factor, which we attribute directly to long-range mesoscopic correlations. From a fundamental perspective, our results provide a direct method for quantifying long-range correlations in wavefront-shaping-based focusing, even with a limited number of input control channels. On the applied side, they enable the accurate prediction of the enhancement factor and a lower bound on its fluctuations, which are crucial for biomedical imaging, optical metrology, optical trapping, and communication through scattering media.

Figures (10)

• Figure 1: The wavefront-shaping concept to focus light through a scattering medium onto multiple target points is shown. (a) A sketch of the speckle pattern in transmission when an unshaped wavefront is incident onto a scattering medium is demonstrated. (b) The incident wavefront is shaped by a spatial light modulator with $M_1$ degrees of freedom at the input, focusing light onto two selected target points (output channels) $M_2 = 2$.
• Figure 2: The mean and probability density of the enhancement factor $\lambda_{\max}$ are shown. The mean enhancement $\langle\lambda_{\max}\rangle$ is plotted versus the number of input channels $M_{1}$ at fixed $M_{2}=2$ (a) and versus the number of output channels $M_{2}$ at fixed $M_{1}=100$ (b). The predictions from our LW ensemble theory (solid blue) obtained from the exact analytical result in (\ref{['Qmax']}) agree quantitatively with the experiments (red open circles), whereas clear deviations from the Marčenko–Pastur law (dashed black) are clearly visible in panels (a,b). With no fitting parameter, the predicted probability densities $P(\lambda_{\max}) = Q'_{\max}(\lambda_{\max})$ from Eq. (\ref{['Qmax']}) likewise match quite well the experimental distributions for small $M_1$ in (c) and for all values of $M_2$ for $M_1 = 100$ in (d).
• Figure 3: Panels (a) and (b) show respectively the mean and the standard deviation of the largest eigenvalue, $\lambda_{\max}$ (i.e., the enhancement factor), as a function of the number of input channels $M_1$, for different numbers of target channels $M_2 = 1, 2, 3$ (indicated by color). Open circles denote numerical results obtained from Laguerre–Wishart (LW) matrices, while solid circles correspond to the exact analytical predictions derived from the cumulative distribution function of $\lambda_{\max}$ [Eq. (\ref{['Qmax']})]. Solid lines show the asymptotic large-$M_1$ results [Eqs. (\ref{['large_nu']}) and (\ref{['std_dev']})], demonstrating excellent agreement between numerical, exact, and asymptotic results.
• Figure 4: Fluctuations of the enhancement factor, quantified by the standard deviation $\Delta\lambda_{\max}\equiv\sqrt{\langle\lambda_{\max}^2\rangle-\langle\lambda_{\max}\rangle^2}$, are shown versus the number of input channels $M_{1}$ at fixed $M_2 = 2$ and $M_2 = 3$ (a) and versus the number of output channels $M_{2}$ at fixed $M_{1}=100$ and $M_{1}=1000$ (b). Our LW-ensemble theory (solid lines) agrees well with the experiments (open circles) up to $M_{1}\approx 100$ in (a) and for all $M_{2}$ at fixed $M_1 = 100$ in (b). For $M_{1}>100$, the experiments exhibit larger fluctuations than predicted (a,b), which are attributable to long-range mesoscopic correlations within the sample.
• Figure 5: The mean enhancement $\langle\lambda_{\max}\rangle$ versus the number of input channels $M_{1}$ at fixed $M_{2}=2$ and $M_{2}=3$ (a) and versus the number of output channels $M_{2}$ at fixed $M_{1}=400$ and $M_{1}=1000$ are shown for our analytic LW-ensemble theory (dashed black line), experiments (black open circles), and numerical simulations (red open squares for $k_0L = 1$, blue open diamonds for $k_0L = 50$, and magenta asterisks for $k_0L = 200$, all with normalized transport mean free path $k_0 l_t = 20.6$). All results agree across $M_{1}$ and $M_{2}$, demonstrating that long-range mesoscopic correlations do not affect the mean enhancement. Fluctuations, quantified by the standard deviation $\Delta\lambda_{\max}\equiv\sqrt{\langle\lambda_{\max}^{2}\rangle-\langle\lambda_{\max}\rangle^{2}}$ versus the number of input channels $M_{1}$ at fixed $M_{2}=2$ (c), and versus the number of output channels $M_{2}$ at fixed $M_{1}=400$ (d), exhibit a slight experimental excess and a pronounced excess in simulations, which grows with slab thickness. These giant fluctuations in thick slabs reveal long-range mesoscopic correlations beyond the correlation-free LW model.