## Abstract

Focusing scattered light using wavefront shaping provides interesting perspectives from which to image deep in opaque samples, as e.g., in nonlinear fluorescence microscopy. Applying these techniques to *in vivo* imaging remains challenging due to the short decorrelation time of the speckle in depth, as focusing and imaging has to be achieved within the decorrelation time. In this paper, we experimentally study the focus lifetime after focusing through dynamical scattering media when iterative wavefront optimization and speckle decorrelation occur over the same timescale. We show experimental situations corresponding to a broad distribution of decay rates of the scattering paths, where the focus presents significantly higher stability than the surrounding speckle.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

In recent years, several wavefront-shaping techniques have been developed to partially compensate for the scattering induced by a disordered medium and form a diffraction-limited focus using the scattered light [1], possibly at depth and non-invasively [2]. However, a major limitation to the application of these techniques to the imaging of real biological systems is the temporal decorrelation induced by minute changes of the optical index inhomogeneity: the decorrelation time of biological tissues can be in the millisecond range [3,4]. As a consequence, fast wavefront-shaping systems are required for focusing light in these systems [4,5], and the lifetime of the focus formed is also limited by this decorrelation time [6]. Two main approaches have been developed to focus within this decorrelation time. On the one hand, digital optical phase conjugation (DOPC), relying on the phase conjugation of a measured wavefront, is a fast non-iterative technique capable of focusing in the millisecond range [7,8]. On the other hand, iterative optimizations can focus almost as fast through a scattering medium [9–12]. While DOPC methods are very appealing for tackling fast decorrelating media, optimization methods remain inescapable in many bio-imaging scenarios, particularly those based on two-photon fluorescence [13–16].

So far, the lifetime of the focus obtained after wavefront shaping has only been studied for DOPC [6,7]. In particular, it has been shown experimentally and theoretically that the temporal correlation function in intensity ${{g}_2}({ t})$, used to quantify the decorrelation dynamics of the speckle, also causes temporal decay of the focus intensity after phase conjugation. An analogous experiment has also been conducted with iterative optimization in the frequency domain [17], but in a stationary medium where the decorrelation is induced by tuning the incident wavelength rather than by a physical displacement of the scatterers. In this paper, the authors demonstrate experimentally and theoretically that the focus intensity degradation is proportional to the spectral correlation function in intensity of speckle after shifting the laser frequency. In [6,7,17], the medium can be considered to be static during the wavefront-shaping procedure, resulting in a proportionality between the timescales of speckle decorrelation and focus degradation. On the other hand, if the wavefront-correction procedure is much slower than the decorrelation time, focusing of the scattered photons cannot be achieved [9].

However, an interesting question remains: What happens if we perform an iterative optimization through a dynamical scattering media, where wavefront correction and decorrelation occur over similar timescales? In this paper, we investigate which scattering medium properties (mean decorrelation time, width of decay rate distribution) impact the characteristic lifetime of the focus obtained in such a scenario. In particular, we show that there are experimental situations in which the focus can be significantly more stable than the surrounding speckle pattern when, for instance, the medium is heterogeneous. Finally, we experimentally demonstrate that this phenomenon can be observed in acute brain slices.

## 2. PRINCIPLE

#### A. Analytical Model

The output speckle pattern formed after a scattering media can be seen as a coherent sum of fields resulting from different scattering sequences (i.e., the diffusion path of a photon in the medium) with random phases and amplitudes [18]. Inside a dynamical scattering medium, each scattering sequence dephases, which induces a decorrelation of the output speckle pattern. Individual scattering sequence decorrelates with a given decay rate $\Gamma $ (see SI). Considering all the scattering sequences provides the distribution of decay rates ${G}(\Gamma )$ of the scattering medium [19]. This distribution can’t be directly measured unlike the temporal intensity correlation function:

In the specific case of a monodisperse colloidal solution, the decay rate of a scattering path is proportional to the number of scattering events encountered, so the distribution of decay rates is directly proportional to the path length distribution [18]. For more complex scattering media—for example, where some parts are static and others are moving or when the distribution of scatterer size isn’t known (i.e., biological tissue)—there is no simple relation between these two quantities.

Performing wavefront shaping with a spatial light modulator (SLM) means adding the appropriate phase to the incident wavefront to sum constructively some sequences at a desired target to form a sharp focus. Inside a dynamic scattering media, the phases of the scattering sequences shift, which induces a degradation of the focus after ending wavefront shaping. The normalized degradation of the focus can be expressed in a similar way as Eq. (2) (see SI):

with ${{G}_m}(\Gamma )$ being the distribution of decay rates of the focus. If the sequences selected to focus through a dynamical scattering medium present the same decay rate distribution as the full set of sequences forming the speckle, the focus and speckle should have the same decorrelation dynamics, as was observed in DOPC. Hypothetically, if more stable scattering sequences could be favored during wavefront shaping (by selecting either more stable scatterers or shorter sequences), the focus should present a higher stability than the initial speckle before optimization.

For a continuous iterative optimization, if decorrelation and optimization occur over similar timescales, it is not clear which scattering sequences will be used to form a focus. We experimentally investigate this question with a custom fast wavefront-shaping system [9]. We use this setup to focus through synthetic homogeneous scattering media of various stabilities and scattering strengths, but also stratified media with a “static” and “dynamic” part, as can be the case for biological samples. In this latter case, the width of the decay rate distribution of the different scattering sequences in the medium can be broad. We study the degradation of the focus in these various scenarios and investigate under which conditions the focus could present an enhanced stability.

#### B. Experimental Setup

Figure 1 describes the experimental wavefront-shaping setup. A phase-only spatial light modulator (Kilo-DM segmented, Boston Micromachines) shapes the incident wavefront of a CW laser $\lambda = {532}\,\,{\rm nm}$ (Coherent Sapphire). The SLM is conjugated to the back focal plane of a microscope objective (${10 \times}$, 0.25), which illuminates a scattering sample. The polarized output speckle is simultaneously imaged onto a CCD camera (Allied Vision Technologies Manta G-046B) and on a mono-detector (PMT, Hamamatsu H10721-20). A continuous iterative wavefront optimization algorithm is implemented to maximize the intensity of one speckle grain collected by the PMT. In short, the optimization is obtained using the Hadamard input basis. At each iteration, half the pixels are modulated in phase, while the PMT signal is monitored, and the optimal phase is added to the correction mask. We combined a fast acquisition card (NI PXIe-6361) and a fast FPGA board (NI PXIe-7962R) to reach a speed of 4.1 kHz per mode [9]. For all experiments, the full Hadamard basis is successively optimized five times in 1.25 s.

Our experimental system is capable of measuring successively the temporal correlation function in intensity ${{ g}_2}({ t})$ and the focus degradation after ending the optimization ${{ I}_{\rm focus}}({t})$. Indeed, the PMT signal provides ${{I}_{\rm focus}}({ t})$, and ${{g}_2}({t})$ is measured with the CCD camera [20,21].

#### C. Fitting Model

Establishing a complete theory describing which dynamic scattering sequences are selected during an iterative optimization to focus remains difficult and outside the scope of this paper. Instead, we use a simple analysis of our experimental results. As was done previously to quantify the ${{g}_2}$ function [19], we express both decorrelation functions (of the speckle ${{ g}_2}$ and of the focus ${{ I}_{\rm focus}}$) as a function of the moments of their decay rate distribution:

with $\tau $ being the mean decorrelation time, $\alpha $ a correlation term at large times that result from the presence of static sequences, and ${\sigma _\Gamma }$ the standard deviation of the decay rate distribution. Each of these three quantities are defined for the decorrelation of the speckle and of the focus. Additional moments could be added to this expression but they are, experimentally, harder to extract due to noise.

Therefore, we can compare for all experiments the decay rate distribution (through the estimate of its mean value and standard deviation) for the speckle and the focus. If the focus uses statistically the same scattering sequences as the speckle, these distributions (and therefore their first moments) should be identical. On the other hand, if more stable scattering sequences are favored during wavefront shaping, the mean value of the decay rate distribution of the focus should be higher and the width of its distribution should be narrower. Moreover, if the medium contains static scattering sequences, and if they are favored by the wavefront-shaping optimization, the correlation term at large times $\alpha $ will be larger for the focus.

## 3. RESULTS

We have used different samples in order to understand in which situation a stable focus can be formed. We studied first the case of a colloidal solution in a multiple scattering regime, where the difference of sequence decay rate results from difference in their length. The width of the path length distribution of such a medium (and therefore its decay rate distribution) is not tunable. To study the impact of the width of the decay rate distribution, we then designed a second category of samples, composed of a thin dynamical layer above a static layer in multiple scattering regime. In this medium, part of the light travels ballistically through the dynamical layer. Therefore, static scattering sequences exist through the sample. By varying the size of the scatterers inside the colloidal solution, the decay rate distribution of the dynamical scattering sequences could be tuned. For small polystyrene beads, scattered light was highly unstable. In this situation, our optimization scheme was only capable of compensating for the static sequences. On the other hand, for large polystyrene beads, the decorrelation was slower. In this last situation, our system was both capable of compensating static and dynamical scattering. Finally, we achieved qualitatively the same result through acute brain slices from the brainstem.

#### A. Monodisperse Colloidal Solution

The first sample used was a 500 µm thick solution of ${{\rm TiO}_2}$ (Sigma Aldrich 224227) in glycerol with a mass concentration of 20 g/l (${\ell_s} = {70}\,\, \unicode{x00B5}{\rm m}$ and ${ l}^* = {200}\,\, \unicode{x00B5}{\rm m})$ [9]. Light propagation through the sample is therefore in a regime of multiple scattering. A schematic of a few dynamical sequences is illustrated in Fig. 2(a). The decorrelation time of a scattering sequence in a monodisperse solution is directly related to the number of scattering events [18]. Furthermore, tuning the temperature modifies the viscosity of the sample, thus allowing us to tune its mean decorrelation time.

The temperature of the sample was first adjusted at 16°C to obtain an average decorrelation time of the speckle of ${\tau _{\rm speckle}} = {70}\,\,{\rm ms}$. The resulting mean focus degradation and its standard deviation are shown on Fig. 2(b). For this dynamical sample, a mean focus decorrelation time of ${\tau _{\rm focus}} = {70}\,\,{\rm ms}$, with a standard deviation of 21 ms, was measured over 500 realizations.

We then measured the average value of the focus decorrelation time and its standard deviation for different media, obtained by changing the viscosity of the solution via the sample temperature, such that the decorrelation time of the speckle was ranging from 50 to 150 ms. In Fig. 2(c), the ratio ${\tau _{\rm focus}} / {\tau _{\rm speckle}}$ is shown. This ratio is constant and close to unity for all tested stabilities. Interestingly, an individual realization of the focus may have a characteristic decorrelation time different from that of the speckle, but on average they are identical. We also did not measure any significant difference between the standard deviation of the decay rate distribution of the focus and of the speckle for these experiments. For these samples, this value ranged from ${{1 \cdot 10}^{ - 3}}$ to ${{3 \cdot 10}^{ - 3}}\,\,{{\rm ms}^{ - 1}}$. Considering all experiments, the mean $R_2$ obtained by fitting ${{g}_2}$ was 0.99 $ + / - $ 0.01. The mean $R_2$ obtained by fitting each individual focus decorrelation was ${0.96}\,\,{ + / - }0.01$. This lower value originates from the dynamic speckle background that overlaps with the focus. A residual offset $\alpha $ of the order of 1% can be measured in these experiments; it results from imperfection in the measurements: finite FOV on the camera to measure ${{g}_2}$ and insufficient averaging to remove the speckle background for the focus. This residual offset is of the order of the fit accuracy, and was therefore neglected.

So far, using monodisperse colloidal solutions, we had not found any optimization procedures that may favor stable scattering sequences. The decay rate distribution might be too narrow to observe different stabilities between the focus and speckle. To investigate further, we synthetized dynamical scattering media with a wider decay rate distribution of the different scattering sequences.

#### B. Combination of Layers of Static and Dynamic Scatterers

In a second experiment, we synthetized dynamical scattering media that exhibits a larger decay rate distribution [see Fig. 3(a) for a diagram of the scattering sequences existing in our media]. By superimposing two scattering media, a thick static layer and a thin dynamical layer, we were able to control the percentage of dynamical scattering sequences exiting the sample. In this experiment, we wanted to investigate which sequences (fixed or dynamic, and which ones among the dynamic) would form a focus after optimization. In the first experimental situation, we designed a sample where the dynamical sequences were decorrelating too fast to be corrected by our wavefront-shaping system. In the second situation, we designed a sample where the dynamical sequences were slower and may be compensated by our system [9].

The first medium was a solution of polystyrene beads (Polybead Carboxylate Microspheres 0.35 µm) in water positioned over a thick static scattering medium. The thickness of the layer of the dynamical scattering solution was 1 mm.

The percentage of ballistic photons through the scattering solution was controlled by adjusting the polystyrene bead concentration. Using Mie theory, the concentration required to obtain a given mean free diffusion path in the dynamical medium can be computed. This percentage ranges from 46% (${l_s} = {1.3}\,\,{\rm mm}$) to 100% (no scattering solution). The addition of a strongly scattering static layer ensures that we are overall in the multiply scattering regime.

For each solution, the measurement of ${{g}_2}$ confirms that the speckle resulting from dynamically scattered photons decorrelates in less than a millisecond [see Fig. 3(b), dashed lines]. We also observe that the ${{ g}_2}$ function reaches a plateau (${\alpha _{\rm speckle}}$) for large decorrelation times, indicating that static sequences contribute to the speckle pattern. For each sample, the decorrelation of the focus averaged over 100 realizations is plotted in Fig. 3(b) (solid line). Yet, for all tested concentrations of colloidal solutions [Fig. 3(b)], the wavefront correction system was able to form a focus which decorrelated slower than the speckle and eventually reached a plateau. As our system is not fast enough to compensate for the dynamically scattered photons, the value of the plateau (${\alpha _{\rm focus}}$) was larger than that measured for the speckle (${\alpha _{\rm speckle}}$), showing that the focus contains a larger amount of static sequences [Fig. 3(b)]. As the scattering mean free path decreases, more and more photons were scattered by the dynamical layer [Fig. 3(c)]. Nevertheless, in all cases, the focus obtained by wavefront shaping was mostly formed by static scattering sequences. Interestingly, some slowly decorrelating (on the order of seconds) scattering sequences also contributed to the focus. These more stable sequences are probably snake-like sequences that encounter only very few forward scattering events. Through these samples, we see that the mode distribution used to form a focus is very different from that of the speckle. Moreover, Fig. 3(d) shows that the enhancement of the focus intensity by optimization (as defined in Vellekoop and Mosk [23]) was larger for smaller bead concentrations in the dynamical samples. Indeed, the dynamical scattering speckle can be seen as an extra measurement noise that reduces the enhancement [24].

We then synthetized a similar dynamical medium but with larger colloidal beads (Polybead Carboxylate Microspheres, 1 µm). The larger beads sustained higher viscosity forces, which decreased the decay rates of the dynamical scattering sequences. By tuning the concentration (therefore ${l_s}$), we simultaneously controlled the percentage of fixed sequences that exited from the sample and the mean decorrelation time of the dynamical sequences. For all prepared samples, the mean speckle decorrelation times ranged from 50 to 250 ms, and the proportion of fixed scattering sequences ranged from almost 0 to 80%. Our wavefront-shaping system should therefore be capable of optimizing the phase of the wavefront travelling through any of these sequences [9]. The standard deviation of the decay rate distribution ranged from ${{2 \cdot 10}^{ - 3}}$ to ${{11 \cdot 10}^{ - 3}}$ ${{\rm ms}^{ - 1}}$.

The results for ${l_s}$ ranging from 0.6 to 2.3 mm are presented in Fig. 4. Examples of focus and speckle decorrelation are presented in Fig. 4(a) for ${l_s}=0.6$, with their fits and residuals. Considering all experiments, the mean $R^2$ obtained was ${0.97} { \pm } {0.01}$. The blue squares and red diamonds indicate the average proportion of static sequences [Fig. 4(b)], mean decorrelation time [Fig. 4(c)], and standard deviation of the decay rate distribution [Fig. 4(d)] for the speckle and focus, respectively. The data obtained for the focus were averaged over 100 realizations.

For large ${\ell_s}$, most of the scattering sequences inside the scattering medium are static. Therefore, there is not much possibility of increasing the number of static scattering sequences contributing to the focus. For small ${\ell_s}$, the proportion of static sequences increases at the focus. In all cases, the focus exhibits, on average, a twofold higher mean decorrelation time as compared to the speckle, which also results in a narrower width of the decay rates distribution for the focus. In a monodisperse colloidal scattering solution, these more stable sequences should correspond to snake-like sequences that encounter only few forward scattering events. The intensity enhancement, as defined in Vellekoop and Mosk [23], follows a linear trend similar to that previously reported [9], ranging from 20 for ${\tau _{\rm speckle}} = {25}\,\,{\rm ms}$ up to 120 for ${\tau _{\rm speckle}} = {225}\,\,{\rm ms}$.

To conclude, the key element to form a focus with stable sequences seems to be the width of the decay rate distribution of the different scattering sequences. The broader the distribution of the different scattering sequences, the more stable sequences the focus contains and the longer its lifetime.

#### C. Biological Samples

As a last experiment, we investigated whether our optimization algorithm allows the achievement, in biological tissues, of a focus more stable than the speckle [25]. This would of course be very beneficial to performing non-linear imaging after wavefront correction, since an increased focus stability would provide additional time for the formation of a fluorescence image.

Our sample was a 300 µm thick acute slice of mouse brain (${l_s} \sim {40}\,\, \unicode{x00B5}{\rm m}$ [26]). To keep the slices alive, a stream of a solution of 125 mM NaCl, 2.5 mM KCl, 2 mM ${{\rm CaCl}_2}$, 1 mM ${{\rm MgCl}_2}$, 1.25 mM ${{\rm NaH}_2}{{\rm PO}_4}$, 26 mM ${{\rm NaHCO}_3}$, and 25 mM glucose, bubbled with 95% ${{\rm O}_2}$ and 5% ${{\rm CO}_2}$, was imposed around the wafer [27]. Every effort was made to keep the brain slices alive for the duration of the experiment. The scheme of the system used to maintain the slice’s acuity is shown on Fig. 5(a), and a typical widefield image of a slice is shown on Fig. 5(b).

Figure 5(c) shows the temporal correlation function of the speckle (solid blue line). A first rapid decorrelation (slope at the origin of ${\sim}{100}\,\,{\rm ms}$) is followed at length by a slower decorrelation with a typical timescale on the order of 5 s. We did not study here the microscopic origin of these different decorrelation times. As in the previous experiment, we observed that the focus obtained by optimization (solid red line) is on average (over 500 realizations) more stable than the speckle and presents an average enhancement of ${33} \pm {10}$. The red dotted line and red crossed line respectivelyshow the least stable and most stable focus. These two optimizations led to identical enhancement on the order of 30. It seems that there are no benefits (or drawbacks) in term of enhancement to favor stable sequences. Interestingly, the most stable focus generated here seems to be almost perfectly stable (over tens of seconds).

## 4. DISCUSSION AND CONCLUSION

We have shown that, in contrast to the previous wavefront-shaping experiments, the focus does not always have the same decay rate distribution as the surrounding speckle. We have shown in particular that in specific conditions, an increase of the focus mean decorrelation time by a factor of 2 (Fig. 4) up to several orders of magnitude (Fig. 3) is obtained, as compared to the speckle stability.

Our interpretation of these results is as follows. At each iteration of the optimization, some stable and dynamic sequences are corrected to interfere constructively with the focus. Rapidly, the dynamic sequences decorrelate and no longer contribute to the focus, while the stable sequences still do. Therefore, iteration after iteration, more and more stable sequences accumulate, leading to an enhanced stability of the focus. Ultimately, all the SLM modes available could compensate only for the stable sequences [as observed in Fig. 5(c)]. The key to achieving a more stable focus seems to be the width of the decay rate distribution of the scattering sequences. The wider this distribution, the easier it is to promote stable diffusion sequences by optimization. On the contrary, a narrow distribution (as through a monodisperse solution) did not allow for an iterative optimization of the wavefront or the selection of more stable sequences, at least for the range of parameters investigated.

Another key element to obtaining a more stable focus is the speed at which the optimization is done. If the optimization is too fast compared to the decorrelation time of the medium, this effect does not appear (as in DOPC). In our case, we observed that if the optimization time is of the order of the mean decorrelation time of the medium, a focus more stable than the speckle can be formed. The influence of these two parameters (width of decay rate distribution of the scattering sequences and speed at which the optimization is done) remains difficult to analyze experimentally, and further numerical studies beyond the scope of this paper may be required to fully describe their respective roles. Additional studies of the impact of the optimization algorithm on the stability of the focus could highlight optimization strategies that could further promote the emergence of a more stable focus.

We believe these results are of great interest, particularly for biomedical imaging. For instance, during *in vivo* imaging of a mouse brain (the skull having been removed and replaced by a glass coverslip), part of the light propagates through or around blood vessels, thus imposing a very rapid decorrelation of the speckle. Despite this, a wavefront-correction system should be able to focus the photons scattered by static structures or having a slow dynamic (cells, myelinated axons, etc.) if the fraction of dynamically scattered light remains low.

Another important scenario is imaging through the skull. The skull would then act as a nearly static scatterer, and the brain tissue would be the dynamical scatterer. In this case, we expect the wavefront-correction system to preferentially correct the scattering by the skull. An interesting case is the correction of the wavefront in the presence of a ballistic but aberrant wavefront inside a tissue. The scattered light rapidly decorrelates, whereas the aberrated light will be relatively stable. A correction of the wavefront would then preferentially correct aberrations.

One last perspective could be to extend this study to the broadband regime. Mounaix *et al*. demonstrated a selection of short scattering sequences by exploiting the short coherence length of a pulsed laser through a homogeneous scattering media [28]. By exploiting this effect, one could obtain a further increase in the stability of the focus.

## Funding

H2020 European Research Council (278025, 724473); Agence Nationale de la Recherche (ANR-10-IDEX-0001-02 PSL* Research University, ANR-10-INSB-04-01 France-BioImaging infrastructure, ANR-10-LABX-54 MEMOLIFE, ANR-15-CE19-0011-01, ANR-15-CE19-0011-03 ALPINS); Université Pierre et Marie Curie (UPMC).

## Acknowledgment

We thank Claudio Moretti, Dimitri Dumontier, Stephane Dieudonné, and Mariano Casado for valuable help with the acute brain slice experiment, and Cathie Ventalon for advice and comments on the project. S. G. is a member of the Institut Universaire de France. B. B. was funded by a Ph.D. fellowship from UPMC under the program “Interface pour le Vivant”.

See Supplement 1 for supporting content.

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