## Abstract

Optical imaging deep inside scattering media remains a fundamental problem in bioimaging. While wavefront shaping has been shown to allow focusing of coherent light at depth, achieving it non-invasively remains a challenge. Various feedback mechanisms, in particular acoustic or nonlinear fluorescence-based, have been put forward for this purpose. Noninvasive focusing in depth on fluorescent objects with linear excitation is, however, still unresolved. Here we report a simple method for focusing inside a scattering medium in an epidetection geometry with a linear signal: optimizing the spatial variance of low-contrast speckle patterns emitted by a set of fluorescent sources. Experimentally, we demonstrate robust and efficient focusing of scattered light on a single source and show that this variance optimization method is formally equivalent to previous optimization strategies based on two-photon fluorescence. Our technique should generalize to a large variety of incoherent contrast mechanisms and holds interesting prospects for deep bioimaging.

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

## 1. INTRODUCTION

Disordered media, such as biological tissues, are a major hindrance to
retrieving information in depth with light, in particular for imaging.
Propagation of light is strongly perturbed due to refractive index
inhomogeneities. In the multiple scattering regime, ballistic light is
exponentially attenuated with depth, and coherent light gives rise to an
extended speckle pattern [1]. As a
consequence, all point scanning or wide-field imaging techniques rapidly
fail beyond a few hundred microns in tissues. Wavefront shaping techniques
have emerged as an extremely effective way to inverse the effect of
scattering and focus light to a diffraction-limited spot [2]. This is achieved by first
measuring a feedback signal from the targeted focal point and then
correcting the incident wavefront with a spatial light modulator (SLM).
Initially, the feedback was measured with a detector placed behind the
scattering medium [3–6], or recovered from
a single implanted guide star [7,8]. Using two-photon
fluorescence as feedback, it was shown in Ref. [9] that a focused spot could be
retrieved, even from an extended object. This was later demonstrated
noninvasively and used to scan the focused spot, thanks to the memory
effect, in order to image in depth [10,11]. A wide set of
feedback techniques nowadays allows one to focus in depth inside complex
media [12]. Linear fluorescence
microscopy remains an inescapable tool in life science [13–16], allowing superficial layers of a biological
sample to be imaged with high resolution and a variety of contrasts.
However, the general problem of focusing in depth, noninvasively, using
linear fluorescence feedback, on extended or multiple targets, was tackled
only very recently. Because one-photon fluorescence is linear with respect
to the excitation, maximizing it as in Refs. [9–11] does not converge to a single
diffraction-limited focus. Alternatives have been proposed to focus light
using a linear signal [17,18]. In Ref. [17], speckle correlations are used to
produce a sharp focused beam at the object plane, whereas Daniel
*et al.* [18] exploit the variations of the emitted fluorescence when
the speckle illumination is translated across the object. However, both
these approaches require the presence of at least some memory effect, thus
they only work for relatively thin samples.

Here, we demonstrate that by choosing as a metric for the wavefront optimization, not the total linear fluorescence backscattered by the medium, but the spatial variance of the fluorescence speckle pattern it reflects, we can successfully generate a single diffraction-limited focus inside the scattering medium. This approach is simple and does not require any memory effect, which opens interesting perspectives for imaging in depth. As the fluorescent light is spatially incoherent, the speckle patterns originating from different fluorescent sources will sum incoherently. If the sources are not within the spatial memory effect range of the scattering medium, all the generated speckles are uncorrelated. Their incoherent sum thus produces a speckle pattern whose contrast is more or less lowered depending on the number of sources [1]. Hence, maximizing the variance tends to concentrate the excitation on a single source. We finally show that once a single bead has been isolated, the centroid of the diffuse spot allows, within a few microns, to localize the target.

## 2. PRINCIPLE

Contrary to a common misconception, a fluorescent object, despite being spatially and temporally incoherent, can generate speckle through a complex medium. However, the speckle pattern that one recovers–that we will refer to as fluorescent speckle by convenience–is low contrast. First, each fluorescent source, emitting broadband and unpolarized light, generates a speckle whose contrast is decreased: each polarization and spectral band form independent speckles that are summed incoherently [1]. Second, speckles from all the fluorescent sources, excited inside the medium, are also added incoherently. The overall contrast of the final speckle is directly linked to the number of independent speckles [1], but only decreases with a mild square root dependence. On the other hand, its intensity scales linearly with the excitation intensity in the case of linear fluorescence.

Here, we take advantage of the product of these two by optimizing the linear fluorescence spatial standard deviation $\sigma $ to focus the illumination on a single fluorescent target. We point out here that since spatial variance (Var) is just the square of the standard deviation ($\mathrm{Var}={\sigma}^{2}$), we refer for simplicity to variance optimization, but the experimentally fitted parameter throughout the process is the standard deviation.

The principle of the method is depicted in Fig. 1. We excite $N$ fluorescent targets, hidden behind a scattering medium, with a speckle illumination Fig. 1(B). Linear and isotropic emitted fluorescence forms a low-contrast speckle, ${I}_{\mathrm{fluo}}$ in Fig. 1(C). The latter is the incoherent sum of the $N$ uncorrelated speckles generated by all the targets, weighted by the excitation intensity each target receives. We must recall here that all of these $N$ uncorrelated speckle patterns are intrinsically lowered in contrast due to the broad fluorescence spectrum. However, the overall contrast of the fluorescent speckle, $C({I}_{\mathrm{fluo}})$, is related to the number of excited fluorescent targets and scales as $1/\sqrt{N}$ [1]. Consequently, if one shapes the illumination such that fewer targets are excited, the contrast should increase. In other words, the number of excited targets is closely linked to the contrast of the fluorescent pattern. In particular, $C({I}_{\mathrm{fluo}})$ is maximum when only a single target is excited, while the other $N-1$ receive almost zero intensity. To enhance the intensity on this target and form a focus with a significant signal-to-background ratio (SBR), we also need to maximize the total fluorescence signal. Therefore, the metric reads $C({I}_{\mathrm{fluo}})\times {I}_{\mathrm{fluo}}$, which corresponds to the standard deviation of the fluorescent speckle, $\sigma ({I}_{\mathrm{fluo}})$. Using spatial standard deviation as a metric allows us to create a nonlinear feedback signal.

Our variance-based wavefront shaping technique consists in optimizing the phase $\mathrm{\Phi}$ of each input mode (modulated with the SLM) to maximize $\sigma ({I}_{\mathrm{fluo}})$, and consequently, the variance $\mathrm{Var}({I}_{\mathrm{fluo}})=\sigma {({I}_{\mathrm{fluo}})}^{2}$.

To have a better understanding of how one mode is optimized, we derive the relation that links the spatial variance of the fluorescent speckle $\mathrm{Var}({I}_{\mathrm{fluo}})$ to the laser excitation ${I}_{\mathrm{exc}}$. ${I}_{\mathrm{fluo}}$ is the incoherent sum of the fluorescence emitted by each target, ${I}_{\mathrm{fluo}}=\sum _{k}{I}_{\mathrm{fluo}}^{(k)}$. Since we consider linear fluorescence, we have for all the targets ${I}_{\mathrm{fluo}}^{(k)}\propto {I}_{\mathrm{exc}}^{(k)}$, where ${I}_{\mathrm{exc}}^{(k)}$ corresponds to the excitation intensity on target $k$. We thus obtain ${I}_{\mathrm{fluo}}\propto \sum _{k}{I}_{\mathrm{exc}}^{(k)}$.

Its spatial variance, which is the second central moment of ${I}_{\mathrm{fluo}}$, reads $\mathrm{Var}({I}_{\mathrm{fluo}}(x,y,\mathrm{\Phi}))\propto \sum _{k}{I}_{\mathrm{exc}}^{(k)}{(\mathrm{\Phi})}^{2}$. In addition, ${I}_{\mathrm{exc}}^{(k)}(\mathrm{\Phi})$ evolves as a sine wave with respect to $\mathrm{\Phi}$, which gives

Equation (1) explicitly highlights that spatial variance introduces a nonlinearity of order 2 with respect to the excitation intensity. This justifies the fact that even though the response of the fluorescent targets is a linear signal, its variance generates a nonlinear feedback signal. This new optimization scheme enables light focusing on a single fluorescent target, similar to [11] and other works using total two-photon fluorescence as feedback. In our Supplement 1, we validate our model and show that standard deviation experimental data, $\sigma ({I}_{\mathrm{fluo}})$, fit well with $\sqrt{\mathrm{eq}\xb7(1)}$.

## 3. EXPERIMENTAL DEMONSTRATION

In the following section, we report on the experimental implementation of our variance-based optimization with linear fluorescence. We first describe the optical setup, then detail the optimization algorithm and present experimental results.

#### A. Experimental Setup

The experimental setup is shown in Fig. 2. A cw laser ($\lambda =532\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, Coherent Sapphire) is expanded on a phase-only microelectromechanical systems (MEMS) SLM (Kilo-DM segmented, Boston Micromachines), so that all the 1024 SLM segments are illuminated. The SLM is conjugated to the back focal plane of a microscope objective (Zeiss W “Plan-Apochromat” $20\times $, NA 1.0) exciting orange fluorescent beads (Invitrogen FluoSpheres, 1.0 μm) placed behind three layers of parafilm. Complementary measurements allow us to claim that such a scattering medium has virtually no memory effect in the plane of the beads. The excitation beam (diameter $<6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$) underfills the illumination objective back aperture (diameter $\simeq 20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$), which reduces the actual illumination NA. The scattered one-photon fluorescence emission is collected by the same microscope objective (epidetection configuration) and detected by a first camera: CAM1 (sCMOS, Hamamatsu ORCA Flash). These recorded fluorescence images are then used to optimize the variance. We use a dichroic mirror (DM) shortpass 550 nm (Thorlabs) and filters (Fs): a 532 nm longpass (Semrock) and a 533 nm notch (Thorlabs). A second microscope objective (Olympus “MPlan N” $50\times $, NA 0.75) images the plane of the beads in transmission onto a CCD camera (Allied Vision, Manta G-046B) as a passive control. This part of the setup allows us to: (1) get an image of the beads in bright field using white light (MORITEX, MHAB 150W), and (2) monitor the speckle illumination and verify our ability to focus it on a single fluorophore.

#### B. Variance Optimization Algorithm

The optimization algorithm we use works as follows. For each iteration, we modulate the phase of half of the pixels according to one Hadamard mode (binary basis whose entries are either +1 or −1) on the SLM. One-half of the pixels (corresponding to entries “$+1$”, for example) of the selected mode are discretely (with ${N}_{\mathrm{step}}$) modulated in phase from 0 to $2\pi $. The other half of the pixels (corresponding to entries “$-1$”) are not modulated and act as a reference. For each one of the ${N}_{\mathrm{step}}$, we acquire a fluorescent speckle pattern on CAM1 and calculate its standard deviation. In our experiment we use ${N}_{\mathrm{step}}=8$. At each iteration, the algorithm finds the phase that maximizes the spatial standard deviation, and thus the spatial variance. A new optimal phase mask is then calculated by adding to the previous mask the Hadamard mode with this optimal phase. This mask is applied to the SLM before starting the next iteration. When the full Hadamard basis has been optimized, the algorithm restarts with the first mode of the basis. We can monitor using the control camera how a single focus with high SBR is formed. Note that we expect that this method would also be compatible with other optimization strategies proposed to focus on complex media, such as genetic algorithms [19].

#### C. Focusing Light on a Single Target

With the optimization algorithm we have detailed in the previous subsection, we report on the results obtained through five different optimization procedures. Each time we ran 1500 iterations and used 1024 input Hadamard modes. Our scattering medium is made of three layers of parafilm (thickness $\simeq 400\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$), which corresponds approximately to $\simeq 2{l}_{s}$ and $\simeq 0.5{l}_{t}$. Scattering properties of parafilm (mean free paths, spectral bandwidth, and polarization) are characterized in Supplement 1. The resulting speckle pattern illuminates 12 beads. The latter are dried on a glass coverslip (thickness $\simeq 160\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$) and placed on top of the scattering layers of parafilm. These experimental conditions (scattering medium and fluorescent sample) are kept identical for the five optimizations.

The spatial standard deviation, $\sigma ({I}_{\mathrm{fluo}})$, is estimated throughout the entire optimization process from the full image recorded on CAM1. Figure 3(A) represents the evolution of the fluorescent speckle throughout the entire optimization (for Opt. #1). It shows that maximizing the variance not only enhances the contrast of the speckle pattern but also shrinks its envelope when a single bead is selected. Both of these two effects contribute to focus light.

The corresponding excitation in the plane of the beads [see Fig. 3(B)] is monitored with the control camera placed in transmission (CAM2). As can be seen, the variance enhancement of the fluorescent speckle is achieved by exciting only one target, i.e., focusing the illumination on the bead. However, the position of the focus cannot be determined in advance. Additionally, if one changes the initial speckle illumination and/or simply rearranges the input modes sorting order, another focus may potentially be generated. Over five realizations, two other spatial foci on different beads are obtained [Fig. 3(B)]. We observe clearly a single focus spot for each illumination. As one can notice, each focus is distorted due to the presence of the bead that absorbs and diffracts light because we image in transmission. Therefore, the maximum intensity is not correctly estimated and SBR cannot be precisely quantified from CAM2 images, but is definitely underevaluated. A proper measurement would require removing the beads from the field of view, which was not possible in our experiment.

In Fig. 3(C), we plot
$\sigma ({I}_{\mathrm{fluo}})$ throughout the whole process. In all
cases, it increases continuously and reaches a plateau after
$\simeq 1000$ iterations, which is consistent with
the number of input modes (1024 SLM pixels). We also note that
$C({I}_{\mathrm{fluo}})$ does not converge to a unique value.
The latter is indeed really sensitive to the position of the envelope
across the field of view. Furthermore, we estimate, *a
posteriori*, the contrast of a subarea of the full images
recorded on CAM1, denoted $C({I}_{\text{center}})$, in order to investigate only the
contribution of the speckle without its envelope shape. We crop
fluorescent images [Fig. 3(A)] and set height and width such that it captures all
the intensity pixels above 80% of the maximum intensity of
${I}_{\mathrm{fluo}}$. Since the envelope shrinks
throughout the optimization, the cropped area is increasingly small.
Contrast of the fluorescent speckle pattern, $C({I}_{\text{center}})$, significantly increases during the
first iterations [Fig. 3(F)] and rapidly converges to $C({I}_{\text{center}})\simeq 0.13$ much below 1, because collected
fluorescence is broadband and unpolarized. At the beginning, the
optimization tends to shape the speckle to excite a single bead, which
seems to be the best scenario to substantially increase the variance.
Once a single focus is obtained, the variance enhancement mainly comes
from the total fluorescence enhancement. This leads to a higher SBR of
the generated focus [Fig. 3(D)]. We also report on the performance of our
variance-based optimization in more demanding regimes in
Supplement 1.

#### D. Fluorescent Targets Localization

At high depth, the memory effect range is too small to form an image by raster scanning the focus across the sample. Nevertheless, fluorescent speckles after optimization contain spatial information on the position of their emitters. We take advantage of using a two-dimensional (2D) detector (as CAM1), required for the optimization, to extract information on the position of the excited emitter. Indeed, when light is focused on a single target, the envelope of the fluorescent speckle is shrunk compared to the initial one; see Fig. 3(A). The observed diffuse spot is simply the propagation of a localized emitted fluorescence through the scattering medium.

We can estimate the position of the emitter by computing a 2D centroid localization of the fluorescence envelope obtained at the end of the optimization. We actually perform two separate one-dimensional (1D) fits, for both directions, on the projected data, as can be seen in Fig. 4(A). We can notice that the three successive optimizations focusing on the same bead (Opt. #2, #3, and #4), give rise to similar diffuse spots located in an area whose typical size $\simeq 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. This means that, in our case, this technique would not distinguish two different targets that are not at least 10 μm far apart. A better accuracy should be achieved by repeating the measurement several times. Here, our beads sample is sufficiently sparse (beads are at least $>20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ apart) so that our localization technique distinguishes the three beads. Our localization technique is essentially limited by the signal-to-noise ratio (SNR) of the envelope. The latter depends not only on fluorescence generated outside of the beads (for instance autofluorescence of parafilm) and detector noise, but also on the way the sample is imaged on the camera: if out-of-focus, the diffuse spots emitted by the beads get larger and less intense, due to diffraction. However, if one manages to form a focus on one bead, it means (as detailed in Supplement 1) that we have a limited amount of noise, and localization should also be possible. Note also that localization accuracy is impacted by the inhomogeneity of the scattering medium. If the medium has an uneven thickness, or inhomogeneous scattering properties, the centroid of the diffuse spot would be shifted relative to the position of the bead.

To show the consistency of the estimated positions based on centroid localization (in epidetection, CAM1), we superimpose the real position of the beads (in bright field, CAM2) or directly refer to the positions of the foci [Fig. 4(B)]. Images are rescaled in microns by taking into account the magnification of our system (objective + tube lens) and the pixel size on both cameras. In Fig. 4(C), we show that the estimated relative positions of the beads (based on fluorescent envelope localization, CAM1) are in good agreement with the ones retrieved with the control camera (CAM2). The time needed to focus on different beads is the main limitation of this technique.

## 4. DISCUSSION

To summarize, we developed a new all-optical mechanism that focuses light inside scattering media. This technique relies on the use of a nonlinear feedback signal, the spatial variance of the fluorescent speckle. It ensures the generation of a single diffraction-limited focus. An important advantage of our approach is that it works with a linear signal, such as one-photon fluorescence, and can potentially be interesting for Raman imaging. Unlike recent works, no memory effect is required, which makes our technique even applicable with very thick scattering media as long as we detect fluorescence with sufficient signal-to-noise to perform the optimization. As an example, it might be used to optically extract mouse brain information through the skull ($L\simeq {l}_{t}$), as the fluorescent speckle contrast can be measured if the number of sources (i.e., neurons) is reasonable, as described in Ref. [20]. More precisely, light focusing is achievable only if the noise level is low compared to the initial contrast of the fluorescent speckle pattern. The latter depends essentially on the number of excited targets and the scattering properties of the medium. For instance, if we increase the number of targets or make the scattering medium thicker, the overall contrast is reduced. Below a certain value set by the experimental noise, the spatial variance is no longer correctly estimated and no focus is formed at all. We provide an elaborated study about the experimental limitations of our method in Supplement 1.

As in any optimization process, the main drawback is the relatively long time scale required to perform all the iterations. Our optimization scheme is mainly limited by the acquisition time of the fluorescent images on CAM1. Even with an sCMOS camera, we need an exposure time of few tens of microseconds. If we run 1500 iterations and register eight fluorescent images to optimize each mode as done in Fig. 3, measurement time is on the order of 10 min. Also, increasing the laser excitation power is also a possibility, but bleaching of the fluorescent targets becomes an issue, since the optimization process forms a focus on the target. Controlling the laser power throughout the optimization would limit this effect.

Additionally, the target bead cannot be chosen in advance and depends on
the optimization parameters. However, its position can be estimated
*a posteriori*, and focusing on different targets can
be achieved by performing multiple optimizations with different initial
conditions. We finally exploit the optimized fluorescent speckles to
retrieve information about the relative position of the beads based on the
centroid localization of their envelopes.

Our approach is in principle applicable to three-dimensional (3D) objects, although deeply buried targets are less likely to be selected by the optimization process, because they are less excited at the beginning of the optimization, due to light scattering.

## Funding

H2020 European Research Council (724473).

## Acknowledgment

SG is a member of the Institut Universitaire de France.

See Supplement 1 for supporting content.

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