suppReferences

# Single-shot Digital Optical Fluorescence Phase Conjugation Through Forward
Multiply Scattering Samples

Tengfei Wu  Saints-Pères Paris Institute for the Neurosciences, CNRS UMR 8003,
Université de Paris, 45 rue des Saints-Pères, Paris 75006, France  Yixuan
Zhang  Saints-Pères Paris Institute for the Neurosciences, CNRS UMR 8003,
Université de Paris, 45 rue des Saints-Pères, Paris 75006, France  Baptiste
Blochet  Saints-Pères Paris Institute for the Neurosciences, CNRS UMR 8003,
Université de Paris, 45 rue des Saints-Pères, Paris 75006, France  Payvand
Arjmand  Saints-Pères Paris Institute for the Neurosciences, CNRS UMR 8003,
Université de Paris, 45 rue des Saints-Pères, Paris 75006, France  Pascal
Berto  Saints-Pères Paris Institute for the Neurosciences, CNRS UMR 8003,
Université de Paris, 45 rue des Saints-Pères, Paris 75006, France  Sorbonne
Université, CNRS, INSERM, Institut de la Vision, 17 Rue Moreau, 75012 Paris,
France  Institut Universitaire de France, Paris, France  Marc Guillon  Saints-
Pères Paris Institute for the Neurosciences, CNRS UMR 8003, Université de
Paris, 45 rue des Saints-Pères, Paris 75006, France

Aberrations and multiple scattering in biological tissues critically distort
light beams into highly complex speckle patterns. In this regard, digital
optical phase conjugation (DOPC) is a promising technique enabling in-depth
focusing. However, DOPC becomes challenging when using fluorescent guide-stars
for four main reasons: The low photon budget available, the large spectral
bandwidth of the fluorescent signal, the Stokes shift between the emission and
the excitation wavelength, and the absence of reference beam preventing
holographic measurement. Here, we demonstrate the possibility to focus a laser
beam through multiple-scattering samples by measuring speckle fields in a
single acquisition step with a reference-free, high-resolution wavefront
sensor. By taking advantage of the large spectral bandwidth of forward
multiply scattering samples, Digital Fluorescence Phase Conjugation (DFPC) is
achieved to focus a laser beam at the excitation wavelength while measuring
the broadband speckle field arising from a micron-sized fluorescent bead.

## Introduction

#### In-depth laser focusing.

Laser light focusing inside or through aberrant and scattering sample is a
major challenge which is of interest for in-depth imaging [1], photo-
stimulation [2] or phototherapy [3]. At depth larger than the scattering mean
free path $\ell_{s}$ (typically $\simeq 100~{}{\rm\mu m}$ in brain tissues
[4]) light is scrambled into a complex random-like speckle pattern. Imaging
ability is thus critically altered by reduced signal to noise ratios, even
under non-linear photo-excitation [5].

#### Adaptive optics and wavefront shaping.

In-depth light control has then been addressed both by adaptive optics [6] and
wavefront shaping [7] approaches. Adaptive optics (AO) efficiently compensates
for optical aberrations. It relies on the weak turbulence assumption and
considers that a spatially coherent light beam (e.g. originating from a guide-
star or a laser) only experiences smooth wavefront distortions. This
assumption remains valid shallow in the sample ($z\lesssim\ell_{s}$), in
transverse planes conjugated with smooth and large-scaled refractive index
mismatches. For discontinuous, small-scaled or typical volumetric refractive
index mismatches, multiple light path trajectories (_i.e._ scattering) are
involved and interfere, resulting not only in wavefront distortions but also
in highly contrasted intensity fluctuations. At depths larger than $\ell_{s}$,
the fraction of scattered light exceeds ballistic energy, potentially by
several orders of magnitude. More complex optical control is then demanded,
referred as “wavefront shaping” techniques in the literature [7]. A system
compensating tissue distortion at various depths must be able to measure and
correct both low varying wavefront distortions – attributed to aberrations –
and highly scrambled wavefronts induced by scattering.

#### Iterative vs. single-shot phase conjugation.

Aberrations and scattering corrections rely on the principle of phase
conjugation. The latter is based on the use of a spatial light modulator (SLM)
placed in the excitation and/or the detection path of an imaging system. Phase
conjugation can then be achieved either iteratively or by direct wavefront
measurement. In the iterative approaches, the correcting pattern is either
obtained by fully characterizing the transmission matrix from several
measurements [8, 9] or by sequentially updating the pattern at the SLM in
order to maximize a proper metric that ensures focusing. This metric can be
the intensity at a given position [10], a non-linear signal [11, 12, 13, 14],
an image quality metric [15, 16] or a variance contrast [17]. Those approaches
have proven successful to correct both scattering and aberration in a wide
range of scenarios [7]: coherent [10], incoherent [18], with [18] or without
guide-stars [11, 19]. However, these techniques demand multiple sequential
measurements at the expense of speed, which ultimately becomes an issue,
especially in the case of rapidly evolving media such as living animals [20].

Alternatively, direct wavefront conjugation is also possible and potentially
faster. It consists in first measuring the wavefront arising from a guide-star
(and emerging from the scattering medium) and then to correct wavefront
distortions in a second step. This solution is effective when using coherent
guide-stars [21, 22] since interferometric measurements is then possible.
Conversely, for incoherent guide-stars such as fluorescent, the outgoing
wavefield can only be measured with reference-less wavefront sensing
techniques. However, in this case, only multi-acquisition schemes combined
with phase-retrieval optimization algorithms have allowed measuring complex
fluorescence wavefronts through scattering media so far [23, 24], and single-
shot fluorescence wavefield measurement has been limited to low-order
aberration compensation in AO schemes [6, 25, 26]. Noteworthy, direct
scattering compensation by single-shot amplitude-only acquisition was also
demonstrated based on a common-path speckle-interferometric measurement [27]
but then ignoring ballistic low-order aberrated light. Furthermore, in ref.
[27], DOPC by binary amplitude-only beam modulation was achieved, at the
expense of a drastic loss in focusing efficiency [28, 29]. Therefore, a
single-shot measurement of both aberrant and scattered fluorescence wavefronts
originating from an incoherent guide-star hidden by a complex medium has never
been achieved. The solution we propose relies on a typical AO scheme.

#### Wavefront sensors.

Fluorescence has established itself as the gold standard optical contrast for
bio-microscopy since being minimally invasive, bio-specific, compatible with
living samples, and even having the possibility to be genetically encoded
through an increasing number of optogenetic tools, including a large choice of
fluorescent proteins with a wide range of photo-sensitive properties. For this
reason, fluorescence has also been primarily exploited to perform AO and
compensate low-order aberrations [6]. In bio-imaging, AO is typically
implemented by measuring the wavefronts with a Shack-Hartman wavefront sensor
(WFS) [30, 31]. WFS are compact single-shot reference-free instruments,
compatible with broadband light sources and providing the complex amplitude of
a light-beam (phase and intensity). They are thus ideally suited to measure
fluorescence signals. WFS actually measures the wavefront gradient and thus
requires a numerical integration step. So far, the use of WFS has mostly been
limited to smooth and low-order aberrations measurements [6, 25, 32] although
they have proved to be capable of providing high spatial resolution of smooth
objects [33, 34]. In the context of astronomy, it was shown that in the case
of strong turbulences, singular points of zero intensity associated with
screw-phase dislocations (or optical vortices) appear and degrade AO
performances [35], especially because deformable mirrors cannot compensate
phase dislocations. AO, in its traditional implementation, has thus only been
considered for compensating the first few low-order Zernike polynomials
contribution to aberrations [25] and has long ignored the contribution of
optical vortices to the phase structures, so preventing its use for
compensating high-order multiple-scattering processes. Measuring random
speckle wavefields has long remained a challenge because of the high spatial
density of intrinsic optical vortices [36, 37, 38], that are responsible for a
so-called “hidden phase” [39] canceled by the integration step [40]. Optical
vortices are associated with a vanishing intensity at the singularity location
and a non-conservative and diverging grid-distortion vector-field. Numerical
[41] and experimental [42] efforts have been made to detect optical vortices
using Shack-Hartmann WFS, but despite Fried’s suggestion to reconstruct the
corresponding “hidden-phase” [39], only individual isolated vortices could be
successfully restored in practical experiments [43, 44, 45]. Recently, we
proposed a solution that experimentally demonstrated the possibility to
quantitatively rebuild complex wavefields containing high densities of optical
vortices [46]. By performing image processing of the Helmholz decomposition,
and relying on the quantization of optical vortex charges, we demonstrated
accurate experimental speckle measurement with a reference-less WFS (see Fig.
S3).

#### Spectral sensitivity of scattering samples.

Although WFS are compatible with broadband guide-stars, performing high-
resolution digital optical phase conjugation (DOPC) using broadband light
beams like fluorescent signals and/or ultrashort laser pulses through multiple
scattering samples is complicated. Multiple scattering usually involves large
distributions of path length trajectories, and thus results in narrow
spectral-correlation bandwidths, and blurred broadband speckle patterns of low
contrast whose phase and amplitude are not defined. Nevertheless, biological
tissues are multiple scattering samples having the specific property to
exhibit large anisotropy factors [4]. They can be typically modeled
numerically by stacking thin low-angle-scattering phase plates [47, 48, 49].
We have demonstrated that despite the multiple scattering process, the large
anisotropy factor $g$ of biological tissues is responsible for large spectral
bandwidths [50, 51, 49] scaling as
$\Delta\lambda=\lambda^{2}\ell^{\ast}/L^{2}$, where $L$ is the slab thickness
($\ell_{s}\ll L\ll\ell^{\ast}$) and $\ell^{\ast}=\ell_{s}/(1-g)$ is the
transport mean-free-path. An “achromatic” speckle plane lying in a virtual
image plane located at $2/3$ of the slab thickness was identified [51, 49]
(See Supp. Fig. S12). The intermediate regime of biological tissues thus
bridges the gap between the community of AO and wavefront shaping [52, 53, 12,
54, 55] by involving both the contribution of ballistic light (altered by
geometrical aberrations) on the one hand, and the contribution of multiply
scattered light on the other hand.

Here, we demonstrate the possibility to compensate multiple scattering in a
single-shot DOPC experiment in the absence of a reference beam. The system we
use, based on a high-resolution WFS, is shown to be compatible with broadband
light sources like fluorescent guide-stars. This demonstration is based on a
rigorous quantitative analysis of spectral correlations of multiply forward
scattering samples and the resulting highly multimodal speckle beams. First,
we demonstrate the key contribution of optical vortices to the wavefield
structure for phase conjugation applications. Second, we discuss the
robustness of speckle phase conjugation to spectral shift thanks to the large
spectral correlation width of forward multiply scattering samples [50, 51,
49]. Third, DOPC performances are experimentally investigated as a function of
the required number of photons per spatial modes. Finally, based on the former
quantitative analysis, we demonstrate digital ($30~{}{\rm nm}$-wide)
fluorescence phase conjugation (DFPC) through $500~{}{\rm\mu m}$ of paraffin
samples.

## DOPC with a wavefront sensor

Fig. 1: Principle of DOPC with a WFS. Scheme of the DOPC experiment (A). A
guide-star laser beam is focused by a objective lens (Obj.1) onto the input
face of a scattering sample S. The scattered light, collected by a second
objective (Obj.2), is sent onto a phase-only spatial light modulator (SLM)
conjugated to a WFS (WFS) through a polarizer P. The phase (C) and the
intensity (D) of the beam are measured while the SLM is exhibiting a flat
phase. The smooth contribution of phase (C) is also rebuilt (B) for
comparison. In a second step, a playback laser beam is injected backward
thanks to a pellicle beam-splitter. With no phase modulation at the SLM, a
speckle pattern (E) is observed on the camera (Cam.). When displaying the
measured phase pattern (C) onto the SLM, a sharp focus is observed behind the
$720~{}{\rm\mu m}$-thick spinal chord slice from a mouse (F), exhibiting a
$\simeq 200$ enhancement factor (G).

The scheme of the experiment is shown in Fig. 1. A DOPC microscope was built
up based on a liquid-crystal SLM (LCoS X10468-01, Hamamatsu, Japan) conjugated
with a high-resolution WFS in an “in-line” configuration to simplify the
alignment of the WFS and SLM [56, 57] (See Supp. Fig. S7). In contrast with
usual “pupil” AO systems, in “conjugated AO”, both the SLM and the WFS are
conjugated to the scattering sample plane [56]. As a result, the maximum
number of spatial modes arriving at the WFS does not depend on the sample but
is solely limited by the pupil size of the imaging objective lens (Obj.1 in
Fig. 1). For usual “pupil” AO designs, the speckle grain size at the SLM/WFS
decreases with an increasing size of the illuminated sample region, so
resulting in correspondingly large number of modes. This chosen “conjugate” AO
configuration further allows us to conjugate both the WFS and the SLM to the
achromatic plane of the forward scattering samples, which also turns out to be
the best conjugating plane for optimizing the isoplanatic patch [58].

The WFS we use here is a home-made quadri-wave lateral shearing interferometer
[33] made of a checkerboard phase grating conjugated to a plane located at a
few millimeters from a 4Mpx sCMOS camera (Prime BSI Express, Teledyne-
Photometrics). The possibility to accurately measure the full speckle phase
pattern from a WFS based on the algorithm of Ref. [46] is shown in Supp. Fig.
S3 by comparing with a digital holographic measurement. In the DOPC setup, the
maximum number of spatial modes arriving at the WFS was tuned to $\simeq 3000$
by optical design. This value was chosen based on a prior quantitative
experimental characterization of the WFS using the setup shown in Fig. S1.
Also based on the experimental results of this first scaling step, the
expected evolution of the DOPC performances is plotted as a function of the
number of modes in Fig. S4. The process for optimizing the distance between
the phase mask grating and the camera is described in Supp. Fig. S6.

First, we demonstrate DOPC with a WFS at a single monochromatic wavelength. In
Fig. 1, the principle is illustrated using a fixed $720~{}{\rm\mu m}$-thick
spinal chord slice from a mouse as a sample. In a first step, a guide-star
point source is created at the rear of the tissue by focusing a $635~{}{\rm
nm}$ diode laser beam, after prior spatial cleaning through a monomode fiber
(see Supp. Fig. S5). A flat phase is displayed onto the SLM to record the
output wavefield with the WFS, intensity (Fig. 1D) and phase (Fig. 1C). In a
second step, the phase measured at the WFS (Fig. 1C) is conjugated and
addressed to the SLM, which is then illuminated by a back-propagating playback
laser at the same wavelength. A sharp focus is then observed at the readout
camera (Fig. 1F) exhibiting an enhancement factor [10, 13] as large as
$I_{focus}/\left<I_{speckle}\right>\simeq 200$ (Fig. 1G). This value can be
compared to the maximum theoretical expectation which is qualitatively given
by the number of measured and controlled modes [10, 59]. In our system, this
number was set to $\simeq 3000$ by optical design (a factor $\simeq 15$ above
the experimental result and a factor $\simeq 10$ above one might have been
expecting from prior calibration experiment (Fig. S4)). The observed
difference between the experimental enhancement factor and its theoretical
maximum value may involve partial depolarization of the playback laser by the
scattering medium, slight misalignment between the SLM and the WFS (Fig. S7),
and imperfect wavefront reconstruction. Interestingly, when ignoring the
contribution of optical vortices in the phase reconstruction process (Fig.
1B), such as currently done by all Shack-Hartmann-based AO systems, no focus
is obtained (Fig. 1E). The reason is that the fraction of ballistic light
through this sample is negligible. This result demonstrates that high-
resolution WFS enables reference-less single-shot DOPC through multiply
scattering media. The ability to compensate multiple scattering is
substantiated by the size of the isoplanatic patch, measured to be as small as
a single speckle grain size (Supp. Fig. S9). DOPC through a multimode fiber,
of interest for micro-endoscopy techniques, is also demonstrated in Supp. Fig.
S10. The dimensions of the focal spot in Fig. 1F were measured to be
$1.2~{}{\rm\mu m}$-FWHM and $1.5~{}{\rm\mu m}$-FWHM along two orthogonal
directions, respectively. These values are very close to the diffraction limit
achievable with our objective lens (Obj. 2 in Fig. 1A): $0.51\lambda/{\rm
NA}=1.31~{}{\rm\mu m}$, where NA is the numerical aperture of Obj. 2. The
possibility to get focal spot dimensions smaller than the NA of the focusing
lens is made possible thanks to the scattering medium increasing the effective
focusing aperture [10]. In Supp. Fig. S8, we further demonstrate that using
computer generated holography, it is possible to exploit the angular memory
effect of a diffuser to project more complex intensity patterns through the
scattering sample.

## Contribution of optical vortices to DOPC

Fig. 2: Importance of the optical vortices for efficient DOPC through an out-
of-focus $1^{\circ}$ surface diffuser. The SLM and WFS are conjugated to the
virtual image of the guide-star focus. Displaying a flat phase on the SLM (A)
results in a speckle at the playback camera (B) while full phase compensation
(C) yields a focus (D) containing $25\%$ of light energy (amplitude set to one
unit). The smooth contribution to the phase (E), typically rebuilt in AO
experiments does not provide a focus (F) while the vortex contribution to the
phase (G) does (H).

With a Shack-Hartmann-like WFS, the wavefront is obtained by integrating the
measured gradient vector field. Usual reconstruction algorithms only
considered the curl-free contribution of the vector field, so resulting in a
conservative potential [60, 40]. The non-conservative solenoidal contribution
of optical vortices, responsible for the so-called “hidden phase” [39] has so
far been ignored in typical AO schemes. Here, we thus specifically study the
contribution of both the conservative phase and the hidden-phase to DOPC
performances. Results are shown in Fig. 2. A speckle is created by
illuminating a holographic diffuser ($1^{\circ}$ diffuser, Edmund Optic) with
a collimated laser beam. The diffuser was defocused by $2~{}{\rm mm}$ from the
conjugate plane of the SLM/WFS so yielding a speckle pattern containing many
optical vortices. When a flat phase is displayed onto the SLM, the playback
laser beam yields a fully-developed speckle pattern on the camera (Fig. 2A and
2B). When conjugating the full phase measured at the WFS including both phase
contributions, a focus is obtained containing $25\%$ or light energy (Fig. 2C
and 2D). Its amplitude was normalized for comparison with other cases.
Interestingly, when considering only the smooth (_i.e._ conservative)
contribution of the phase, which is solely rebuilt in AO systems, the laser
energy is slightly concentrated to the central region but no focus is obtained
in this scattering condition which cannot be considered as low-order optical
abberations (Fig. 2E and 2F). In contrast, we observed that the vortex phase
allows focusing $\simeq 2/3$ of the maximum energy, corroborating the idea
that the structure of phase in speckle patterns is driven by vortex phase
singularities [61]. Experimentally, the fraction of focused energy was
calculated by computing the ratio between the energy in the focused spot and
the integration of the whole transmitted speckle signal obtained without phase
conjugation, which fully falls in the field of view of our camera for forward
scattering media (see Supp. Fig. S15). Hereafter, we characterize DOPC
performances by measuring the fraction of refocused energy rather than the
enhancement factor. The enhancement factor is more suitable to cases where the
number of controlled modes is much smaller than the number of modes of the
scattering medium, which is the case of the sample used in Fig. 1. Conversely,
for more forward scattering samples, the total transmitted energy is
measurable and the fraction of refocused energy becomes comparable to one.
This result demonstrates that rebuilding the non-conservative contribution of
the gradient vector field measured by WFS is of critical importance to
efficiently compensate scattering in DOPC experiments.

## DOPC from a spectrally-detuned guide-star

Fig. 3: Fraction of refocused energy as a function of spectral detuning
between the refocusing laser $\lambda_{rf}$ and the guide-star laser
($\lambda_{GS}=690/5nm$). Samples consist in $250~{}{\rm\mu m}$-thick (red
curves) and $500~{}{\rm\mu m}$-thick (blue curves) samples of paraffin. In
both cases, refocusing efficiency is plotted in two conditions: with the
SLM/WFS conjugated to the “achromatic” plane (solid arrows in (A) and solid
fitting lines in (B)) and with the guide-star plane (dashed arrows in (A) and
dashed fitting lines in (B)).

In constrast to monochromatic DOPC that we have just demonstrated, the
implementation of DOPC with a fluorescent guide-star faces the specific
challenge of robustness to spectral width and detuning. It is indeed necessary
to be able to both measure broadband speckles and refocus a laser beam at a
wavelength shorter than the average fluorescence wavelength to take into
account the $\simeq 25~{}{\rm nm}$ Stokes’ shift of typical organic
fluorophores. Then, we considered studying the robustness of DOPC to spectral
detuning by shifting the wavelength of the playback laser as compared to the
guide-star’s. In a former article, we showed that a fixed $1~{}{\rm mm}$-thick
brain slice can exhibit a spectral correlation width as large as $\simeq
200~{}{\rm nm}$ [50]. Although experiencing multiple scattering events over
optical trajectories much larger than the wavelength, we identified that, in
virtue of the large anisotropy factor of such media ($g$ close to $1$), all
trajectories are snake-like [62] and the dispersion in trajectories path
lengths is much smaller than their average length [51]. Consequently, the
spectral bandwidth, scaling as the inverse of the dispersion in optical path
delays, was derived to be
$\Delta\lambda=\frac{\lambda^{2}\ell^{\ast}}{L^{2}}$. Both analytical
derivation [51] and numerical simulations [49], supported by experimental
results demonstrated that the speckles generated through the scattering sample
had a spectral-correlation maximum in a virtual plane located at $2/3$ of the
slab thickness. Noteworthy, not only speckle intensities are achromatic in
this “achromatic” plane but also their spatial phases [51] and even their
spectral phases [49].

We thus conjugated our SLM/WFS to this “achromatic” plane in order to both
maximize coherence of the broadband guide-star speckle and to make DOPC robust
to spectral detuning. To demonstrate the relevance of this configuration, we
compare its performances to the case where the SLM/WFS are conjugated to the
guide-star plane. Data shown in Fig. 3 are obtained for two sample
thicknesses, $L=250~{}{\rm\mu m}$ and $L=500~{}{\rm\mu m}$, obtained by
stacking two and four layers of parafilm, respectively. The achromatic plane
location, that we evidenced experimentally (see Supp. Fig. S12), is identified
by solid-lined arrows in Fig. 3A and the guide-star planes by dashed-lined
arrows. A focused laser spot having a fixed wavelength
$\lambda_{GS}=690~{}{\rm nm}$ was used as a guide-star and DOPC performances
measured as the playback laser was spectrally scanned ($\lambda_{rf}$).
Experimentally measured fraction of refocused energy are plotted in Fig. 3B
and fitted by Lorentzian profiles (in agreement with theory [51]):

$F(\lambda_{rf})=\frac{F_{0}}{1+\left(\frac{\lambda_{rf}-\lambda_{0}}{\Delta\lambda}\right)^{2}}$
(1)

Fitting parameters are given in Table 1.

|  $F_{0}$ ($\%$) |  $\lambda_{0}$ (nm) |  $\Delta\lambda$ (nm)
---|---|---|---
SLM/WFS conjugated to the achromatic plane
$L=250~{}{\rm\mu m}$ | $11.4$ | $665$ | $126$
$L=500~{}{\rm\mu m}$ | $6.5$ | $671$ | $53.7$
SLM/WFS conjugated to the guide-star plane
$L=250~{}{\rm\mu m}$ | $5.34$ | $680$ | $26.1$
$L=500~{}{\rm\mu m}$ | $2.23$ | $689$ | $36.0$
Table 1:  Lorentzian fitting parameters for DOPC efficiency as a function of
spectral detuning.

As expected, a larger spectral width is observed when conjugating the SLM/WFS
to the achromatic plane for both samples, and the largest spectral width and
peak energy are obtained for the thinnest sample. The maximum fraction of
refocused energy is $\simeq 12\%$ for the $L=250~{}{\rm\mu m}$-thick sample
and roughly half for $L=500~{}{\rm\mu m}$. Less expected is the fact that a
larger focused energy is obtained at the achromatic plane than at the guide-
star plane, even at the guide-star wavelength. We partially attribute the
improvement of DOPC at the achromatic plane to the smaller density of optical
vortices in this plane – roughly half the one in the guide-star plane – so
potentially resulting in less phase reconstruction errors. Surprisingly also,
the peak of the focused-energy distribution is slightly blue-shifted with
respect to the guide-star wavelength. This shift is in favor of one photon
fluorescence excitation but its explanation is not obvious (we ruled out a
possible SLM-calibration error), all the more because it differs between the
cases where the SLM/WFS are conjugated to the achromatic plane
($\delta\lambda\simeq 22~{}{\rm nm}$) or the guide-star plane
($\delta\lambda\simeq 5~{}{\rm nm}$). Importantly, we experimentally checked
that the DOPC spectral widths measured by detuning the playback laser line are
consistent with both the speckle correlation width of the medium (Supp. Fig.
S11) and with the DOPC spectral widths measured by broadening the spectrum of
the playback laser (Supp. Fig. S13). Our results are thus relevant to non-
linear optical imaging modalities requiring multi-line or broadband ultra-
short laser beams.

## Required photon budget

Fig. 4: Efficiency of DOPC as a function of the guide-star photon budget. The
fraction of refocused energy is measured as a function of the average number
of collected photon per camera pixels at the WFS and fitted. Here, the speckle
patterns were generated using a $1^{\circ}$ holographic diffuser located 3mm
away from the focused laser guide-star at $635~{}{\rm nm}$. The SLM and WFS
are conjugated to the virtual image of the guide-star (like in Fig. 2).
Vertical bars represent twice the experimental standard deviations
($\pm\sigma$) computed over 5 measurements.

The photon budget available from guide-stars for performing DOPC is usually
low either because of speed requirement or just because of the limited photon
budget of the guide-star itself. Furthermore, through a scattering medium,
this energy is split among all outgoing spatial modes. Under these conditions,
increasing the numerical aperture of the collecting objective lens
($.25~{}{\rm NA}$ in our case) does not help since it does not increase the
spatial-mode density of photon at the diffuser output, which solely matters.
We thus investigated the photon budget required per spatial mode to perform
WFS-based speckle phase conjugation. In Fig. 4, the fraction of focused energy
$F$ is plotted as a function of the mean speckle intensity at the WFS
$\left<I\right>$, expressed in photo-electron per WFS-pixel. The measurements
are fitted with a theoretical model from Ref. [59]:
$F=F_{0}\left(1+\frac{I_{0}}{\left<I\right>}\right)^{-1}$, with $I_{0}$, the
required average intensity to get half the optimal performances.
Experimentally, we obtain that this model allows a good fitting of
experimental data for a parameter $I_{0}\simeq 20$ photo-electrons per pixel.
Considering that our DOPC system was designed to measure and control $\simeq
3000$ spatial modes with our $4$ Mpx WFS-camera, our measurements yield a
required photon flux of $\simeq 3\times 10^{4}$ photo-electons per spatial
mode. We could also measure that the ratio between the playback speckle extent
(without DOPC) and the playback focal spot size (with DOPC) is $\simeq 62$, so
giving a number of spatial modes for the diffuser equal to
$\pi/4*62^{2}=3019$, in excellent agreement with our optical design estimate.

## Single-shot DFPC from a fluorescent guide-star

Fig. 5: DFPC using $1~{}{\rm\mu m}$ fluorescent beads as guide-stars. The
absorption (Abs.) and fluorescence emission (Fluo.) spectra of the beads are
shown in (A) together with the excitation laser line at $532~{}{\rm nm}$
(Exc.) and the band-pass emission filter $565/30$ (Em.). Beads are hidden
behind a thickness $L=500~{}{\rm\mu m}$ of a paraffin samples (B). The SLM/WFS
are conjugated to the achromatic plane of the scattering samples and the
measured fluorescence speckle is shown in (C) together with its intensity
histogram exhibiting a mean fluorescence intensity of $33$ photo-electrons per
pixel. The playback laser beams is then sent through the sample after phase
conjugation both at the average emission wavelength ($\lambda=565~{}{\rm nm}$)
and at the excitation wavelength ($\lambda=532~{}{\rm nm}$) when considering
the smooth phase contribution only (D) or the full reconstructed phase (E).
The focii quality in d and e are degraded by the refractive index mismatch
between the fluorescent micro-bead and the embedding PVA matrix.

Finally, we demonstrate the ability of our system to perform single-shot DFPC
using fluorescent micron-size spheres. Fluorescence excitation was achieved
using a $532~{}{\rm nm}$ laser line and fluorescence was collected by stacking
a pair of two emission filters to efficiently block the excitation light (Fig.
5A). The resulting filter bandwidth of $30~{}{\rm nm}$ has been chosen so as
to match as much as possible the spectral width of fluorescence, within the
limit of the spectral correlation bandwidth of the scattering sample (Fig. 3B
and Supp. Fig. S11). The beads were spin-coated in a PVA solution on a
coverslip and then covered with 4 layers of parafilm (resulting in a
$500~{}{\rm\mu m}$-thick scattering sample) and another coverslip on top (Fig.
5B). The excitation of the fluorescent beads was achieved by focusing the
$532~{}{\rm nm}$ laser from the rear of the sample with excitation intensity
$<10~{}{\rm kW.cm^{-2}}$. Noteworthy, neither the power of our laser source
($<5~{}{\rm mW}$) or the single-photon fluorescence-absorption mechanism used,
allowed us to efficiently excite single isolated micro-beads through the
scattering medium. However, several solutions have been investigated to
generate guide-stars in-depth [7], especially using two-photon excitation [11,
6]. _In fine,_ the total fluorescence photon budget at the WFS was $1.4\times
10^{8}$ photo-electrons, corresponding to a mean photo-electron budget per
pixel equal to $33$ (see Fig. 4). The SLM/WFS was conjugated to the achromatic
plane of the scattering sample and the fluorescence speckle signal was imaged
and measured at the WFS (Fig. 5C). Finally, the conjugate phase was addressed
to the SLM without (Fig. 5D) and with (Fig. 5E) including optical vortices,
and the playback laser beam was sequentially tuned to both the average
fluorescence wavelength ($565~{}{\rm nm}$) and the excitation wavelength
($532~{}{\rm nm}$). In both cases, complete phase reconstruction outperforms
the regular AO scheme only compensating for ballistic light, further
demonstrating the ability to robustly mistune the playback laser wavelength to
take into account the fluorophores Stokes’ shift at the expense of a slight
loss in refocused energy ($14.6\%$ _vs._ $17.9\%$). When conjugating the
SLM/WFS to the guide-star plane, no focus could be obtained. Noteworthy, the
focus quality obtained with fluorescent microbeads (Fig. 5) are noticeably
degraded as compared to all other focii we could obtain in all other
conditions, including for spectrally broad or spectrally shifted readout
lasers (See Supp. Fig. S14). The focii degradation observed in Fig. 5D and 5E
is due to the presence of the fluorescent micro-bead on the beam path because
its refractive index is not matched with the one of the PVA embedding matrix
[17]. Moreover, it was not possible to shift the playback focus away from the
microbead without inducing a important focus-energy degradation because the
isoplanatic patch of our sample is typically comparable to bead size itself.
To characterize the DFPC performances, we thus measured the fraction of
playback light energy in the main focus as described in Supp. Fig. S15.
Performances are quantified in Table 2 for three samples thicknesses.

| $L=250~{}{\rm\mu m}$ | $L=375~{}{\rm\mu m}$ | $L=500~{}{\rm\mu m}$
---|---|---|---
DFPC at the average fluorescence wavelength ($\lambda_{rf}=565{\rm nm}$)
with vortices | $11.3\%\,(\pm 5.8)$ | $9.3\%\,(\pm 4.1)$ | $7.1\%\,(\pm 6.1)$
without vortices | $6.8\%\,(\pm 2.3)$ | $3.9\%\,(\pm 0.9)$ | $3.7\%\,(\pm 2.1)$
DFPC at the excitation wavelength ($\lambda_{rf}=532{\rm nm}$)
with vortices | $8.7\%\,(\pm 3.3)$ | $9.5\%\,(\pm 4.7)$ | $6.5\%\,(\pm 4.8)$
without vortices | $4.7\%\,(\pm 1.7)$ | $2.8\%\,(\pm 1.5)$ | $2.3\%\,(\pm 1.2)$
Table 2:  DFPC average efficiency ($\pm$ experimental standard deviations) for
a playback laser wavelength $\lambda_{rf}$ at the average emission wavelength
($\lambda_{rf}=565\pm 2.5~{}{\rm nm}$) and at the excitation wavelength
($\lambda_{rf}=532~{}{\rm nm}$) from fluorescent light emitted by micro-beads
($\lambda_{GS}=565\pm 15~{}{\rm nm}$) as a function of paraffin thickness.

## Discussion

In this work, we have demonstrated the ability to perform single-shot DOPC
from incoherent guide-stars hidden behind forward scattering samples. This
progress has been made possible thanks to two conceptual improvements: the
ability to rebuild complex speckle patterns with a high-resolution WFS on the
one hand and the large spectral bandwidth revealed by scattering media with
large anisotropy factors on the other hand. Our results not only underline the
importance of rebuilding optical vortices which have typically been ignored by
Shack-Hartmann WFS users but they also prove the key role of associated spiral
phase structures for DOPC performances. We also demonstrate the ability to use
fluorescent micro-beads as guide-stars which are associated with three issues:
the weak fluorescence photon budget challenging the maximum number of modes
one may rebuild, the $\simeq 30~{}{\rm nm}$ spectral bandwidth of the
fluorescence signal, and the Stokes’ shift between the fluorescence and the
excitation spectra. The spectral width of DOPC efficiency through
$500~{}{\rm\mu m}$-thick paraffin samples is characterized, and qualitatively
matches the spectral correlation width of the medium. The spectral width of
the scattering medium is driven by the standard deviation of optical path
length trajectories and impose an upper limit for the spectral width of the
emission filter. Beyond this limit, the speckle cannot be considered as
achromatic and only an averaged wavefront can be defined, wherein optical
vortices are scrambled, so reducing the focusing ability of the AO system. In
this regard, we also demonstrate the importance of conjugating the WFS and the
SLM with the achromatic plane of the scattering slab by demonstrating the
degraded performances when conjugation is achieved with the guide-star plane.
In the end, the emission filter width is a compromise between the contrast of
the fluorescence speckle at the WFS and the photon budget. We estimated that
our solution requires an average photon budget per spatial mode of the
scattering medium of the order of $3\times 10^{4}$.

In AO systems, the SLM/WFS are often placed in the Fourier plane. Our system
did not allow us to easily switch from the sample plane to the Fourier plane
and a fair comparison might deserve further experimental characterization. As
an argument in favor of our configuration, in the Fourier plane, the beam
scattered by a forward scattering medium experiences a transverse homothetic
dilation for a wavelength shift, so that the maximum number of
measurable/controllable modes is limited by
$\left(\frac{\lambda}{\delta\lambda}\right)^{2}$. In contrast, the achromatic
plane is the origin of the longitudinal axial dilation responsible for the
transverse dilation at infinity [51, 49]. No such transverse dilation occurs
and the speckle correlation is solely limited by the intrinsic spectral
correlation width of the scattering medium. Consequently, for broadband light
manipulation, the number of modes we can manipulate is not limited in the
achromatic plane, in contrast to the Fourier plane where fluorescence speckles
are blurred. This Fourier-conjugated configuration was in particular chosen by
the authors of Ref. [23] where a $10~{}{\rm nm}$ filter was used to increase
the coherence of the fluorescence signal and improve the number of measurable
modes. Further quantitative comparison with the technique Ref. [27] would be
deserved, especially regarding the DOPC performances as a function of the
nature of phase distortions (including low order aberrations or not) as well
as the photon budget per mode.

Noteworthy, provided that our technique enables efficient multiple scattering
compensation through samples, in particular those leaving no ballistic light,
the guide-star size should not exceed the autocorrelation width of the
original speckle pattern. Besides the technical challenge to generate isolated
guide-stars, especially in densely labeled samples, the isoplanatic patch size
associated with the obtained readout focus appears as critically small for
imaging applications. This intrinsic constraint imposed by such samples makes
our achievement more relevant to applications such as in-depth
photostimulation, thermal phototherapy, in-depth caged-drug photo-delivery,
light-induced guided neuronal growth, non-imaging cell activity monitoring or
micro-endoscopy through fiber bundles. Brain cells monitoring through the
skull also appears as especially interesting thanks to the distance separating
the scattering medium (the skull) from the object of interest (the brain),
yielding an isoplanatic patch size of $\simeq 15~{}{\rm\mu m}$ [14, 63].

Beside opening a solution for performing DOPC, this work motivates further
fundamental investigations about forward scattering media. First, we could not
attribute the spectral blue-shift observed in Fig. 3 to a mis-calibration of
our DOPC system. The difference in the amount of blue-shift between the two
sample thicknesses ($L=250~{}{\rm\mu m}$ and $L=500~{}{\rm\mu m}$) and between
the two imaging conditions (conjugation with the guide-star plane or the
achromatic plane) suggests a physical reason for blue-shifted light to yield
larger DOPC efficiency than at the guide-star wavelength. Blue-shifted light
may benefit more from phase conjugation because of the multiple scattering
process scales as $\lambda^{2}$ in forward scattering media [49] (in contrast
to the $\lambda^{4}$-scaling in the Rayleigh scattering case [4]). Second, by
contributing to bridging the gap between AO and wavefront shaping, this work
about broadband forward scattering media interrogates about the relevance and
the possibility to define a theoretical discrimination criterion between
optical aberrations and light scattering. The distortion matrix concept has
supported this distinction [53]. Usage has typically used the term aberrations
for low-order wavefront distortions only although high-order smooth
distortions exhibit no fundamental difference. However, the presence of
optical vortices in the wavefront suggests making out from wording aberrations
since optical vortices necessarily originate from at least three interfering
beams [64] and thus suppose several optical trajectories.

## Materials and Methods

#### Samples preparation.

The $720~{}{\rm\mu m}$ acute slice of spinal cord of a 6 days mouse was fixed
in paraformaldeyde and mounted in fluoromount between two coverslips separated
by a $800~{}{\rm spacer}$. Nail varnish was used to minimize evaporation. The
sample was left vertically in place for 24 hours for gravity drifting to stop.

Parafilm samples were prepared by stacking 2, 3 or 4 layers of parafilms
between two coverslips and gently pressed to minimize air spacing and thus
surface scattering at interfaces. The thicknesses of the stacks were measured
thanks to a calibrated translation stage by observing the back reflexion of
the guide-star on the internal coverslip surfaces right aside from the sample.
To ensure isotropic scattering, the parafilm layers were oriented at
$90^{\circ}$ for the two-layer samples and at $0^{\circ}$, $45^{\circ}$,
$90^{\circ}$ and $-45^{\circ}$ for the 4-layer samples.

The beads samples were prepared by stacking parafilm diffusers onto a spin-
coated layer of beads embeded in poly-vinyl alcool (PVA). The layer was
prepared by spin-coating at $10000~{}{\rm rpm}$ a $10^{-4}$-weight solution of
$1~{}{\rm\mu m}$ fluorescent microbeads (Fluospheres F8820, Thermofisher
Scientific) in a $2\%$-weight solution of PVA. For the 2-parafilm-layer
sample, two $\\#1$-coverslips were used to add a $\simeq 500~{}{\rm\mu m}$
spacing between the layer of beads and the parafilm.

#### Optical setups.

A detailed description is provided in the supplementaries. In brief, two
optical systems were used, a first one for comparing speckle field
reconstruction by wavefront sensing and digital holography (see Supp. Fig. S1)
and a second one to perfom DOPC through scattering samples using a WFS (Supp.
Fig. S5). To obtain Fig. 2, the first objective lens (Obj. 1) was removed to
illuminate the sample with a collimated laser beam. In the DOPC experiments,
the SLM and WFS were conjugated to the so-called “achromatic plane”, located
in a virtual plane lying at a depth equal to $2/3$ of the sample thickness (or
equivalently, the fluorescent guide-start depth). The location of the output
plane of the scattering sample was first identified thanks to white light
illumination and the objective was then moved forward of the right amount
towards the scattering sample.

In the DOPC experiment, the guide-star was generated by using a focused laser
beam originating from fiber-coupled diode laser ($532~{}{\rm nm}$, $635~{}{\rm
nm}$) or a super-continuum laser (ElectroVis-470, LEUKOS, France) combined to
a computer-tunable filter box (Bebop, LEUKOS, France). The laser was focused
into the sample with a $\times 20$, $0.4~{}{\rm NA}$ objective lens and the
scattered light was collected using a $\times 10$, $0.25~{}{\rm NA}$ objective
lens.

In the fluorescent guide-stars experiment in Fig. 5, the microbeads were
excited from the rear of the scattering medium using a focused $532~{}\rm nm$
laser. The excitation intensity was $<10~{}{\rm kW.cm^{-2}}$ and the exposure
time at the WFS was $1~{}{\rm s}$.

The WFS design is similar to the one described in Ref. [32]. It is made of a
two-dimensional $20~{}{\rm\mu m}$-step checkerboard $0-\pi$-phase grating
conjugated to a CMOS camera (BSI Prime Express, Photometrics) thanks to a
relay telescope. An additional $-75~{}{\rm mm}$ diverging lens was fixed at
the C-mount thread of the camera to minimize field curvature introduced by the
relay telescope. The final global magnification of the imaging-relay system
was $\simeq 3$.

In the first optical system comparing digital holography to wavefront sensing
(Supp. Fig. S1), a $635~{}{\rm nm}$ diode laser was spatially filtered by a
monomode fiber and collimated to illuminate a $1^{\circ}$ holographic diffuser
(Edmund Optic). This diffuser was tuned out-of-focus the WFS and an iris was
placed in an intermediate Fourier plane to tune the speckle grain size. For DH
experiments, the grating of the WFS was removed and a reference arm generated
from a beam splitter placed before the scattering sample. The reference beam
and the “signal” speckle beam were combined together with an angle larger than
the opening angle of the speckle, thanks to a pellicle beam-splitter located
in the Fourier plane of the relay telescope of the WFS.

Finally, a polarizer was added in the optical path of the WFS because
scattering media, even forward scattering, slightly depolarize light beams
[65]. In our case, we measured that, for a thickness $L=500~{}{\rm\mu m}$ of
paraffin, the polarization rate was $0.83$. This slight depolarization thus
accounts for $\simeq 17\%$ of degradation in DOPC performances since our SLM
can only modulate a single polarization component.

## Acknowledgments

The authors thank Boris Lamotte D’Incamps and Brigitte Delhomme for the
preparation of biological specimens. MG acknowledges Sylvain Gigan for
encouraging and stimulating discussions.

## Funding

This work was partially funded by the technology transfer company SATT ERGANEO
(Project 600) and the French Research National Agency (projects SpeckleSTED
ANR-18-CE42-0008-01 and MaxPhase ANR-20-CE42-0006). MG acknowledges support
for Institut Universitaire de France.

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Supplementary Materials
Single-shot Digital Optical Fluorescence Phase Conjugation
Through Forward Multiply Scattering Samples

##  S1 Quantitative measurement of complex field with a wavefront sensor

###  S1.1 Experimental setup

In this section, we experimentally demonstrate that our wavefront sensor (WFS)
can quantitatively measure a scattered speckle field from a single-shot
acquisition by comparing the retreived phase and amplitudes to the one
obtained by off-axis digital holography (DH), which we consider as a ground
truth. Fig. S1 shows the full experimental setup, in which the collimated beam
with a wavelength of $635~{}{\rm nm}$ passes through a polarized beam
splitter, separated into the reference beam path and the signal beam path. In
the signal beam path, the beam interacts with a $5^{\circ}$ holographic
diffuser (Edmund Optic), and the complex field is imaged on the WFS by a
telescope with a lens set of f=$100~{}{\rm mm}$ and f=$200~{}{\rm mm}$,
respectively. In order to tune the speckle density, an adjustable diaphragm is
placed in the intermediate Fourier plane of the telescope. The WFS is composed
by a two-dimensional $20~{}{\rm\mu m}$-step checkerboard phase grating, placed
$3~{}{\rm mm}$ away from the camera sensor, a telescope with a lens set of
f=$100~{}{\rm mm}$ and f=$200~{}{\rm mm}$, and a divergence lens with a focal
length f=$-50~{}{\rm mm}$, which is used to correct the field curvature. A
beam block (ZBB) is placed closed to the intermediate Fourier plane of the
telescope in the WFS, to only select the first diffracted order of the grating
pattern (blocking the zero order and higher orders). In the reference beam
path, a set of mirrors are used to adjust the length of the optical path and
the tilt angle between two beams to optimize the contrast of interference
fringes in DH measurement. A lens with a focal length f=$50~{}{\rm mm}$ and a
pellicle beam splitter is used to image the reference beam on the camera. Two
half waveplates are placed respectively in the two paths to adjust the
relative beam intensity. A beam blocker BB is used to collect the useless beam
in the experiments. In the data acquisition step, for both methods we take two
measurements with and without scattering sample and compare the relative
complex field between the two measurements for each method. In order to be
sure that the two methods measure the same scattering signal, we first measure
the field without scattering sample in DH (i.e. open both the two paths and
remove the grating G and beam blocker ZBB). Then we insert the scattering
sample and measure the scattering field in DH. After finishing the measurement
in DH, the reference beam path is blocked and we place the grating G and beam
blocker ZBB to do the measurement with scattering sample in WFS. Finally, we
remove the scattering sample to do another measurement in WFS. 50 frames are
recorded and averaged for each measurement for better signal to noise ratio.

Fig. S1: Full experimental setup of the demonstration of the quantitative
measurement of complex field by a lateral shearing interferometer WFS. WP:
Half wave plater; PBS: Polarized beam splitter; S: Scattering sample; L: Lens;
D: Diaphragm; G: Grating; ZBB: zero-order beam blocker; BB: Beam blocker;
Pellicle BS: Pellicle beam splitter; DL: Divergence lens; Cam: Camera

###  S1.2 Phase gradient measurement from a lateral shearing interferometer
WFS

Off-axis DH can directly measure a complex field (phase and amplitude) by
extracting the first order of the fringe pattern in the Fourier space. In
contrast, the WFS does not measure directly the phase but the phase gradient.
Here we describe in detail the procedures to extract the phase gradient
information from the raw camera image of a quadriwave lateral shearing
interferometer (QWLSI) [33, 34]. A detailed description of the full gradient
recovery process from QWLSI measurements is given in Ref. [66]. Fig. S2
briefly summarizes and illustrates the main steps.

Fig. S2: Phase gradient reconstruction procedure from a quadri-wave lateral
shearing interferometer WFS.

###  S1.3 Measurement of complex scattering field WFS vs DH

In Fig. S3, we compare the complex field reconstruction obtained from both the
WFS and off-axis DH, thanks to the optical system shown in Fig. S1. In these
images, the number of vortices in the beam at the WFS (_i.e._ the number of
spatial modes) was $1581$. The intensity image in the DH part is obtained by
blocking the reference beam for direct intensity recording. Both the phase
difference and intensity difference are shown, demonstrating the good
agreement between WFS and DH measurements. A remaining low-spatial frequency
content appears in the phase difference pattern, probably due to cumulative
integration errors of phase gradients estimates over the large number of
modes.

Fig. S3: Comparison of speckle field reconstruction using a lateral shearing
interferometer WFS and off-axis digital holography (DH).

Fig. S4: Experimental estimation of the maximum number of modes that our WFS
can rebuild to achieve DOPC. The number of vortices (_i.e._ the number of
modes) is varied by tuning an iris diaphragm in a Fourier plane (D in Fig.
S1). The phase is measured with the WFS and compared to the phase measured by
DH, the latter phase pattern being considered as the ground truth. Then,
digital optical phase conjugation (DOPC) is numerically simulated based on the
phase difference, both to compute the enhancement factor (A) and the fraction
of refocused energy (B). Experimental data points are fitted using empirical
curves to underline and quantify the global trend. As expected, the slope of
the DOPC for small number of modes is close to one [10, 59].

##  S2 Experimental setup and calibration for digital optical phase
conjugation with a wavefront sensor

###  S2.1 Experimental setup

In this section, we mainly show the detailed experimental setup for performing
the digital optical phase conjugation (DOPC) with a WFS. A collimated guide-
star (GS) laser beam passes through a pellicle beam splitter (PBS) and an
objective Obj.1 (x20, 0.4NA) focuses the beam that generates a guide-star (or
excites a fluorescent micro-bead guide-star) at the rear of the scattering
sample S. The scattered field is imaged onto the SLM by an afocal telescope,
which is composed by an objective Obj.2 (x10 0.25NA), a diverging lens DL with
a focal length f=$-50~{}{\rm mm}$ and a wavefront correcting lens with a focal
length f=$1000~{}{\rm mm}$ placed in front of the SLM. As compared to a
regular Galileo telescope made of two converging lenses in a 4-f
configuration, our system has the advantage to allow large magnification
($\times 110$) between the sample plane and the SLM plane over a shortened
physical distance. Then, a telescope with a lens set of f=$300~{}{\rm mm}$ and
f=$150~{}{\rm mm}$ is used to conjugate the SLM plane to the WFS, which has
the same components as the previous experimental setup described in Sec.S1.1.
A removable mirror (not shown) can be placed between the pellicle beam
splitter and the WFS grating G to enable the recording of the reference
pattern. After measuring the scattered field, the conjugated phase is loaded
on the SLM. A collimated ”playback” laser beam is reflected by a polarized
beam splitter and propagates back to the SLM. The linear polarizer P, used to
polarize the beam originating from the scattering sample is also used to clean
the polarization of the ”playback” laser beam and match the modulating
polarization axis of the liquid-crystal SLM. The wavefront modulated by the
SLM then passes through the scattering sample. The focus of the ”playback”
laser beam at the guide-star position is monitored on a camera through Obj.1
thanks to a tube lens TL (f=$150~{}{\rm mm}$).

Fig. S5: Full and equivalent experimental scheme of the optical phase
conjugating microscope. A) Full experimental system. PBS: Pellicle beam
splitter; BB: Beam block; S: Scattering sample; DL: Diverging lens; L:
converging lens; SLM: Spatial light modulator; P: Polarizer; G: grating; ZBB:
Zero-order beam block; F: spectral filter used for fluorescence experiment but
removed otherwise; Cam: Camera; TL: Tube lens. B) Equivalent representation of
the speckle beam control arm. The configuration in A) allows having a large
magnification imaging system with reduced size on the optical table.

###  S2.2 Optimization of the distance between the grating and the camera

For a lateral shearing interferometer WFS, the grating-camera distance d
drives a trade-off between the phase sensitivity, phase gradient dynamic and
spatial resolution. In our case, spatial resolution is of critical importance
to measure complex wavefields containing as many spatial modes as possible.
The number of modes roughly equals the number of optical vortices of a fully
developed speckle pattern. Fig. S6A shows the optical vortex distribution in a
measured complex scattering field. The yellow and blue peaks denote the random
locations of vortices of charges $+1$ and $-1$, respectively. $\Delta A$
stands for the Laplacian of the vector potential A, which account for the
solenoidal contribution in the Helmholtz decomposition of the phase gradient
vector field measured at the WFS, as detailed in Ref. [46]. $\Delta A$ yields
peaks (in theory Dirac functions) at vortices locations. Segmentation splits
the $\Delta A$ image into multiple small regions according to the locations of
the peaks. A following integration within each small region gives the specific
vortex charge in each location, according to Stokes’ theorem [46]. The
histogram of the distribution of the estimated charges of the optical vortices
in the complex scattering field is shown in Fig. S6C, in which three peaks
occurs at charge +1/-1 and 0. Here we use the histogram contrast, which is
defined by the ratio between the width of the central peak w and the distance
L between the central peak and the side peak, to evaluate the vortex-
discrimination ability of the WFS. The ratio $\frac{w}{L}$ is then plotted as
a function of the distance d. The optimal distance d is finally obtained by
computing the minimum coordinate of a parabolic fitting curve.

Fig. S6: Optimization of the distance grating-camera in the WFS for speckle
patterns measurement. (A) Laplacian of the vector potential shows the location
of the optical vortices in a speckle field. (B) Image segmentation identifies
the optical vortices map for topological charge measurement. (C) Histogram of
the measured charges of optical vortices. The optimized grating-camera
distance can be determined by the contrast of the histogram, defined as the
ratio $w/L$ where $w$ is the peak width and $L$ the distance between peaks.
(D) Plot of the measured contrast as a function of grating-camera distance.

###  S2.3 Alignement and calibation of SLM and WFS

In order to compensate the scattering field in the DOPC experiments, a proper
alignment and calibration of SLM and WFS is highly required. Optical elements
conjugating the SLM to the WFS may induce some geometric transformation such
as translation, rotation, and dilation between the SLM and WFS plane. In our
experiments, we calibrate alignment issues thanks to a custom-designed vortex
array. Specifically, we design a 20x20 vortex array with alternating $+1$ and
$-1$ charges. Four additional symmetry-breaking vortices were added to the
array to eliminate symmetry ambiguity problems. The grid thus contain $404$
optical vortices in total. A phase map corresponding to this optical vortices
array is shown in Fig. S7A. This pattern is loaded on the sub-region of the
SLM lying in the field of view of the WFS. A collimated beam then illuminates
the SLM and the reflected beam is then measured by the WFS. The $\Delta A$ map
is then computed and compared to the generated vortex pattern. The histogram
of the measured optical vortices is shown as an inset in Fig. S7A. The
calibration of the SLM is achieved by centering the peaks of the vortex charge
histogram on $\pm 1$ values. A typical comparison between a still non-aligned
measured vortex location and the loaded true vortices array is shown in Fig.
S7B. To correct this mismatch, we evaluate the geometric transformation of the
two vortices arrays by using the built-in function in Matlab (”imregtform”,
Matlab Image Processing Toolbox). With the evaluated transformation
coefficients, we introduce a numerical correcting transform to the pattern
measured by the WFS to align the two patterns as shown in Fig. S7C. We then
compute the phase difference between the loaded phase map (in Fig. S7A) and
the phase map measured by the WFS in Fig. S7D, where all phase vortices
perfectly cancel out. This flat phase pattern ensures the possibility to
achieve phase conjugation.

Fig. S7: Alignement and calibation of the SLM and WFS. (A) Input phase map
generated by a specific vortices array, used to align the mismatch between the
SLM and the WFS plane. Inset is the histogram of the measured optical
vortices. (B) Merged vortices maps of the input and the direct measurement
from the WFS before alignment. (C) Merged vortices maps of the input and the
measurement after alignment. (D) Phase difference between the input phase map
and the measured phase map with alignment.

##  S3 Combination of DOPC and computer generated holography

In this section, we demonstrate two more DOPC related experiments with the
measurement of the complex scattering field by WFS. We show that as long as
the scattering is properly compensated, one can deliver a specifically
designed hologram through the scattering sample. The experimental setup
presented in Sec.S2.1 is used. The scattering sample here is a 1∘ holographic
diffuser (Edmund Optics), ensuring a sufficient memory effect for the hologram
delivery. The collimated beam with a wavelength of $635~{}{\rm nm}$ passes
through the object Obj. 1 and forms a virtual guide-star, locating 3mm away
from the diffuser surface. The SLM and WFS are conjugated to the guide-star
plane. After measuring the complex scattering field, DOPC is performed through
the scattering sample, as is shown in Fig. S8A the phase map and the refocus.
In the next step, by adding another $-1$-charged optical vortex phase map
(Fig. S8B) into the measured DOPC phase pattern, instead of only having a
focus, we obtain a donut-shaped optical vortex beam on the camera, as is shown
in (Fig. S8C). Similarly, other hologram patterns can also be delivered
through the scattering sample with the measured complex scattering field.
Several computer generated holograms could then be generated using a
Gerchberg–Saxton algorithm [67]. The delivery results are respectively shown
in Fig. S8D. Such hologram pattern delivery can be interesting for e.g. the
application of photostimulation.

Fig. S8: Computer generated holography combined with DOPC for patterned
excitation. (A) The phase map used to perform DOPC through a 1∘ holographic
diffuser. Inset is the focus after DOPC. (B) Hologram pattern for generating
+1 charge optical vortex. (C) The resulted phase map for delivering a donut
(+1 charge information) through the scattering sample. Inset is the result
with DOPC. (D) Hologram patterns and corresponding focused patterns through
the diffuser.

Fig. S9: Size of the isoplanatic patch (IP) through the $720~{}{\rm\mu
m}$-thick spinal cord slice in Fig. 1. Phase conjugation yields a sharp focus
that can be translated by shifting the pattern aside at the SLM (A) and its
intensity decreases away from the guide-star location (A). The evolution of
the peak intensity of the focus with the amount of translation is plotted in
(B) (hollow orange circles) and compared to the autocorrelation plot profile
of the speckle pattern measured in the absence of phase conjugation (green
curve). The ability to translate the focused spot defines the dimensions of
the isoplanatic patch (IP). The matching between the two curves demonstrates
that phase conjugation effectively compensates multiple scattering events
through the slice.

##  S4 DOPC through a multimode fiber

This section aims at demonstrating the potential of our DOPC approach to
refocus a laser beam through a multimode fiber. Specifically, we use a
$1~{}{\rm cm}$ long multimode fiber (FG050UGA, thorlabs), and the virtual
guide-star is generated at S = $300~{}{\rm\mu m}$ away from the input fiber
facet, as is shown in Fig. S10A. The output facet is conjugated to the SLM and
WFS and the complex field is measured by WFS. We show in Fig. S10B the
playback camera image with flat phase displayed at the SLM, showing a speckle
intensity. The image after DOPC is compared in Fig. S10C, which gives a more
intense focus on the camera (same colorbar). The complex field measured by the
WFS is shown in Fig. S10C as insets. By translating the guide-star to
different axial positions from S = $200~{}{\rm\mu m}$ to S = $500~{}{\rm\mu
m}$, we implement DOPC on all the cases, as we show in Fig. S10D. We can
notice that the peak intensity of the focus decreases with increase of S,
while the size of the focus (defined as the full-width- half-maximum) gets
larger, and it follows the relation of $\sim\frac{\lambda{S}}{D}$, which is
shown as a fitting plot in Fig. S10D.

Fig. S10: DOPC through a multimode fiber. (A) Schematic of the experiments.
(B) The recorded image on the monitoring camera without DOPC. (C) DOPC results
at S = $300~{}{\rm\mu m}$ with the complex field measured by WFS. Insets are
respectively the phase and intensity. (D) DOPC results for various distances
of S and the plot of the size of the focus as a function of S.

Fig. S11: Spectral cross-correlation product of speckles transmitted through
paraffin samples of thicknesses $L=250~{}{\rm\mu m}$ (red) and
$L=500~{}{\rm\mu m}$ (blue) when illuminated by a collimated laser beam. A
reference speckle is considered at the arbitrary wavelength
$\lambda_{GS}=690~{}{\rm nm}$. Experimental data are fitted using Lorentizan
curves
($C(\lambda_{rf})=1/[1+(\lambda_{GS}-\lambda_{rf})^{2}/w_{\lambda}^{2}]$) both
in the case when the camera records speckles while conjugated to the virtual
entrance plane image (dashed lines) and the virtual achromatic plane image
(continuous lines). For $L=250~{}{\rm\mu m}$ (red plots), the measured FWHM
are $2w_{\lambda}({\rm GS})=37~{}{\rm nm}$ and $2w_{\lambda}({\rm
achr})=126~{}{\rm nm}$ in the entrance plane and the achromatic plane,
respectively. For $L=500~{}{\rm\mu m}$ (blue plots), the measured FWHM are
$2w_{\lambda}({\rm GS})=29~{}{\rm nm}$ and $2w_{\lambda}({\rm achr})=90~{}{\rm
nm}$ in the entrance plane and the achromatic plane, respectively.

Fig. S12: Achromatic plane location evidenced by axial cross-correlation
product of speckles at two nearby wavelengths ($\lambda_{1}=525~{}{\rm nm}$
and $\lambda_{2}=550~{}{\rm nm}$). The beams are transmitted through paraffin
samples of thicknesses $L=250~{}{\rm\mu m}$ (red) and $L=500~{}{\rm\mu m}$
(blue) when illuminated by a collimated laser beam. Experimental data are
fitted using Lorentizan curves ($C=1/[1+(z-z_{achr})^{2}/w_{z}^{2}]$), both
revealing a correlation maximum at $z_{achr}\simeq\frac{L}{3n}$, with $L$ the
slab thickness and $n$ the average refractive index of paraffin, in agreement
with theory. Noteworthy, the $L/3$ position of the achromatic plane must be
corrected by the average refractive index mismatch between the slab (paraffin)
and the embedding medium (air). Measured achromatic planes:
$z_{achr}(250~{}{\rm\mu m})=61.4~{}{\rm\mu m}$, $z_{achr}(500~{}{\rm\mu
m})=109~{}{\rm\mu m}$ _vs._ theoretical achromatic planes:
$z_{achr}^{th}(250~{}{\rm\mu m})=59.5~{}{\rm\mu m}$,
$z_{achr}^{th}(500~{}{\rm\mu m})=119~{}{\rm\mu m}$ for an estimated value
$n=1.4$.

Fig. S13: Effect of the spectral width of the playback laser on the fraction
of refocused light energy. Both for the cases of paraffin samples of
thicknesses $L=250~{}{\rm\mu m}$ (red) and $L=500~{}{\rm\mu m}$ (blue), the
guide-star ($\lambda_{GS}=690~{}{\rm nm}$) was focused at a physical distance
of $500~{}{\rm\mu m}$ from the output surface of the sample as shown in (A).
DOPC was then achieved using a broadband playback light beam of central
wavelength $\lambda_{rf}=\lambda_{GS}$ and of bandwidth $\delta\lambda$ FWHM.
Fraction of refocused energy as a function of $\delta\lambda$ is shown in (B)
for the SLM/WFS conjugated to the achromatic plane (solid line fits) and the
guide-star plane (dashed line fits). Experimental data are fitted using
Lorentizan functions
$f(\delta\lambda)=\alpha/[1+(\delta\lambda/\Delta\lambda)^{2}]$, with
$\Delta\lambda_{achr}(L=250)=70.4~{}{\rm nm}$,
$\Delta\lambda_{GS}(L=250)=44~{}{\rm nm}$,
$\Delta\lambda_{achr}(L=500)=22~{}{\rm nm}$,
$\Delta\lambda_{GS}(L=500)=19.7~{}{\rm nm}$. For the measurements in the
achromatic plane, a set of three realizations was achieved and the
experimental variance is shown as bars.

Fig. S14:  Images of the focii of the readout laser. Two paraffin samples of
thicknesses $L=250~{}{\rm\mu m}$ and $L=500~{}{\rm\mu m}$ are considered. The
playback laser has wavelength detuned from the guide-star laser (A) and
spectral width increased (B). In B, the spectrum is centered on the guide-star
wavelength. Images in A and B correspond to data-points of curves shown in
Fig. 3 and Supp. Fig. S13, respectively.

Fig. S15: Experimental estimation of the enhancement factor and the fraction
of refocused energy, depending on the scattering sample at play. For the
experiment with the spinal chord slice in Fig. 1, the energy is not fully
collected by the objective lens. The enhancement factor is thus considered,
defined as the ratio between the peak intensity of the focus (B) and the
average speckle intensity (computed in the yellow rectangle where the speckle
is uniform in (A)). Conversely, for the DFPC experiment shown in Fig. 5, light
is scattered at smaller angles allowing full light-energy collection by the
objective lenses and not allowing measuring the peak intensity of the focus
because of the focus degradation by the microbead. The fraction of refocused
energy can then be measured, and is then computed as the ratio between the
energy in a small circle surrounding the bead (D) and the total light energy
on the camera (integrating the energy in the yellow rectangle in (C)). In both
conditions, the background noise was subtracted.