## Abstract

PSF engineering is of utmost interest, in particular for microscopy, but remains mostly restricted to weakly scattering or transparent samples. We report a method to design at will the spatial profile of transmitted coherent light after propagation through a strongly scattering sample. We compute an operator based on the experimentally measured transmission matrix, obtained by numerically adding an arbitrary mask in the Fourier domain prior to focusing. We demonstrate the strength of the technique through several examples: propagating Bessel beams, thus generating foci smaller than the diffraction-limited speckle grain; donut beams; and helical beams. We characterize the three-dimensional profile of the achieved foci and analyze the fundamental limitations of the technique. Our approach generalizes Fourier optics concepts for random media and opens in particular interesting perspectives for super-resolution imaging through turbid media.

© 2017 Optical Society of America

## 1. INTRODUCTION

Generating a specific optical point-spread function (PSF) has been one of the cornerstones of modern
microscopy. This is conventionally done by inserting a phase or amplitude mask in the Fourier plane of
the imaging system. For instance, Durnin *et al.* generated Bessel beams using a spatial
filter for beam shaping [1]. Nowadays, holographic methods using a
spatial light modulator (SLM) are the most versatile [2,3]. These techniques allow flexible beam shaping in a wide range of
applications such as super-resolution microscopy [4,5], 3D microscopy [6,7], optical tweezers [8–10], and particle trapping [11]. However, all these studies typically require high-quality
optics and demand little or no sample aberrations.

Light propagation in materials with optical index heterogeneities is affected by scattering. In scattering materials such as white paint or biological tissue, multiple scattering is at the origin of an intricate interference light field at the output of the medium, also known as speckle pattern [12]. Although the size of a speckle grain is diffraction-limited, this complex interference figure is detrimental for all conventional imaging systems. Nonetheless, this scattering process is linear and deterministic and thus even strongly scattering materials can be described by a transmission matrix (TM) [13].

Wavefront-shaping techniques have recently emerged as a powerful technique for controlling the output field using SLMs or phase-conjugate mirrors [14]. SLMs provide up to several million degrees of freedom to design the input field at will, in order to control the propagation of light after the medium. Over the last decade, pioneering works have proven the capability to drastically increase light intensity at one or several output positions of a disordered system such as white paint [13,15], multimode fibers [16], or biological samples [17]. However, the resulting focal spots have sizes comparable to a speckle grain, and thus are diffraction-limited [18]. This limit can be improved by increasing the numerical aperture of the imaging system [19] or controlling the optical near field [20]. Nonetheless, the achieved focus still has the same size as a speckle grain.

Recently, Di Battista *et al.* [21] overcame this
size limit by mechanically inserting an annular mask just after the scattering medium, prior to
iteratively optimizing the focus intensity at a distance via wavefront shaping. Due to the filtering of
the low spatial frequencies, the resulting speckle and optimized spot was narrower than the initial
speckle. After removing the filter, the narrow spot—effectively a Bessel-like beam—remained intense,
over a background speckle pattern wider than the focus.

Herein we report the first formulation, to our knowledge, of a TM-based operator with a controllable PSF that enables deterministic focusing after propagation through a multiply scattering medium. We build this new operator by numerically applying a well-chosen mask (that can be arbitrarily designed in amplitude and/or phase) in a virtual Fourier plane of the output modes of the experimentally measured transmission matrix. We then demonstrate experimentally that a focus with the corresponding PSF can be obtained after the medium by performing digital phase conjugation on this operator. To demonstrate the robustness and wide applicability of our technique, we generate and characterize a variety of useful PSFs. First, we generate a Bessel beam focus using an amplitude annulus mask and show that its central FWHM is narrower than the size of a speckle grain, therefore demonstrating deterministic sub-speckle focusing without mechanical masking as in [21]. We also demonstrate donut-mode generation with various topological charges and helical foci. Characterization of the axial properties of Bessel and helical PSFs shows the potential of the technique for 3D wave control and for subspeckle imaging, extending previous studies to 3D control of the focus with only a transverse measurement.

## 2. PRINCIPLE OF THE EXPERIMENT

A particular class of wavefront-shaping methods relies on the measurement of the optical transmission matrix (TM), denoted $H$ in this paper, which contains the relationship between the input field and output field [22]. Its complex coefficients ${h}_{{\mathbf{X}}^{\prime}\mathbf{X}}$ connect the optical complex field at the output [${\mathbf{X}}^{\prime}=({x}^{\prime},{y}^{\prime})$ camera pixel coordinates] to the input field [$\mathbf{X}=(x,y)$ SLM pixel coordinates] by

Experimentally, the TM is measured by displaying a set of input fields on the SLM and recording the corresponding output fields on the camera, and requires the medium to be stable during the whole measurement process (a few minutes in our case). The digital phase conjugation of the TM, ${H}^{\u2020}$, where $\u2020$ stands for the conjugate transpose, enables focusing at any output position [13] or scanning of the focus [23]. The resulting spot has a size given approximately by the spatial correlation of the output speckle, i.e., diffraction-limited [12,18]. It effectively sets to a common phase all contributions arriving at this position. However, it is possible to completely tune the phase and amplitude distribution of the $k$-vectors forming this focus, and therefore control at will the PSF.

In Fig. 1, we detail how we build a new operator based on the experimentally measured TM to generate an arbitrary mask and generate the corresponding PSF at the focus. We first numerically perform a two-dimensional spatial Fourier transform on the TM, noted ${\mathfrak{F}}_{2D}$, of every output field. We define ${\widehat{h}}_{{\mathbf{K}}^{\prime}\mathbf{X}}={\mathfrak{F}}_{2D}({h}_{{\mathbf{X}}^{\prime}\mathbf{X}})$, where ${\mathbf{K}}^{\prime}=({k}_{{x}^{\prime}},{k}_{{y}^{\prime}})$ is the wave vector associated with ${\mathbf{X}}^{\prime}$. This numerical operation is equivalent to computing the TM in a Fourier plane of the output imaging plane. In order to generate a given PSF, we then multiply in the $k$-space the field in the pupil plane by mask $M$ (amplitude and/or phase), corresponding to this PSF. We thus obtain a new numerically filtered coefficient in the Fourier domain:

We then return to the spatial domain ${\mathbf{X}}^{\prime}\mathbf{X}$ by taking the inverse Fourier transform of ${\widehat{h}}_{{\mathbf{K}}^{\prime}\mathbf{X}}^{\mathrm{filt}}$:

The resulting operator ${H}^{\mathrm{filt}}$ can now be used instead of $H$ to perform focusing and scanning with the chosen PSF. We compute the complex input field using the phase conjugation of ${H}^{\mathrm{filt}}$ [22], and we display its phase on the phase-only SLM to generate the corresponding PSF at the chosen position at the output of the scattering sample. As discussed in [22], the penalty of displaying only the phase information of the phase-conjugation solution is just a mild factor 2 in signal to noise, and does not compromise the shape of the PSF. Nonetheless, encoding phase and amplitude on the SLM is also possible [24].

## 3. EXPERIMENTAL DEMONSTRATION

Figure 2 sketches the experimental setup to measure the transmission matrix $H$, compute ${H}^{\mathrm{filt}}$, and generate and characterize the formed PSF in three dimensions. A cw laser ($\lambda =800\text{\hspace{0.17em}}\mathrm{nm}$, MaiTai, Spectra Physics) is split between a reference and sample path. In the sample path, a phase-only SLM (LCOS-SLM, Hamamatsu X10468) subdivided in $64\times 64$ macropixels is conjugated with the back focal plane of a microscope objective, which illuminates a scattering medium made of ZnO nanoparticles (thickness $\simeq 100\text{\hspace{0.17em}}\mathrm{\mu m}$). Another microscope objective is used to image the transmitted speckle. The objective is placed onto a motorized stage (Thorlabs, Z825B) to scan the imaging plane axially. For the TM measurement, the output beam is recombined with the reference on a beam splitter and the hologram is recorded on a charge-coupled device (CCD) camera (Allied Vision, Manta $G$-046). The reference beam is blocked during focusing and PSF characterization.

We first demonstrate the generation of a Bessel-like beam using an annular mask. The irradiance profile of the ideal beam is described by a zero-order Bessel function of the first kind, which propagates with an associated complex field $\propto {J}_{0}({k}_{r}r)\mathrm{exp}(-i{k}_{z}z)$. The experimental result is represented in Fig. 3. The applied mask during the numerical filtering step is an annular amplitude mask with an inner ring size 58% of the pupil size (see Supplement 1 for details). This value has been found to be a good compromise between the loss in intensity at the output, the ability to detect the foci over the background speckle, and a significant narrowing of the FWHM of the central spot, as with a mechanical mask [21]. Using ${({H}^{\mathrm{filt}})}^{\u2020}$, we obtain in the imaging plane a Bessel-like focus standing over a background speckle. We compare its FWHM to a standard focus obtained with the standard TM ${H}^{\u2020}$. In the same experimental conditions, the Bessel-like central lobe FWHM is 23% narrower than a standard speckle focus grain. Both intensity profiles are fitted to an ideal Bessel and Gaussian profile, respectively. The central lobe of a Bessel beam is surrounded by a decaying set of side-lobe rings. Each lobe carries approximately the same amount of energy as the central spot, whereas Gaussian beams contain 50% of their total energy within their FWHM [25]. Achieving a smaller FWHM than a diffraction-limited focus entails penalties such as loss of energy in the central peak (40% lower than a standard beam). Inherent to wavefront shaping in complex media techniques, the PSF is not perfect, but rather stands over a speckle pattern that remains in the background, with the peak to background ratio proportional to the number of input modes [15].

In the generation of Bessel-like PSFs, an extra penalty is added due to the low transmission of the virtual annular mask, which steeply decreases the energy in the focus while the background remains the same (see Supplement 1 for a quantitative characterization of this effect). In order to detect the focus with sufficient SNR, a high number of degrees of freedom, i.e., SLM pixels, is required; here we used $N=4096$ input pixels. We therefore demonstrate sub-speckle focusing after propagation through a scattering medium, using a Bessel-like beam, without ever physically inserting a mask in the Fourier plane.

Since any arbitrary phase and/or amplitude mask can be computed and placed onto the virtual pupil field, we also demonstrate the generation of donut-shaped beams, which are closely related to single-ringed Laguerre–Gaussian beams (${\mathrm{LG}}_{0}^{m}$). The corresponding mask to be applied in the numerical filtering step is a spiral phase plate distribution presenting a continuous and gradual phase change from 0 to $2\text{\hspace{0.17em}}m\pi $ around the optical axis, where $m$ is to the integer number of $2\pi $ cycles in the pupil plane [26] and corresponds to a topological charge. The ring pattern dimensions increase with $m$. Experimental results of focusing (${\mathrm{LG}}_{0}^{m}$) beams, with $m$ from 1 to 4, are presented in Fig. 4 using $N=1024$ SLM pixels. The transmittance $M$ of the numerical phase mask in the Fourier domain applied during the numerical filtering step, as defined in Eq. (2) and illustrated in Fig. 4, reads

Several beam profiles such as Bessel beams also have very interesting propagation properties along the $z$ axis. Our setup enables the scan of the $z$ axis thanks to a motorized stage under the collecting microscope objective. Experimentally, we can scan over a few Rayleigh ranges (${z}_{R}=10\text{\hspace{0.17em}}\mathrm{\mu m}$) on both sides of the focal plane. In Fig. 5(a), $yOz$ cross sections of Bessel-like beam obtained in the same conditions as in Fig. 3(a) are reported. As expected, we prove without ambiguity that the generated Bessel-like beam has a longer depth of focus during propagation along the $z$ axis relative to standard Gaussian beams, for which the depth of focus is the Rayleigh length. Here, we observe experimentally that our Bessel-like beam has a depth of focus 1.7 times longer than the Gaussian focus.

As a last example, we demonstrate the generation of double-helix point spread functions (DH-PSFs). This 3D design has two dominant lobes in the imaging plane, whose angular orientation rotates with the axial ($z$) position [5,7]. This profile is realized by applying a particular phase mask during the numerical filtering step leading to ${H}^{\mathrm{filt}}$, illustrated in Fig. 5(c), which reads [5]

## 4. DISCUSSION

We have demonstrated focusing with various PSFs, defined by arbitrary control of a mask in phase and amplitude in the virtual Fourier domain. It is to be noted that, as in all methods relying on focusing through a complex medium, the focus stands over a background speckle resulting from the incomplete phase conjugation [15], the signal to background of the focus increasing linearly with the number of segments controlled on the SLM. However, our technique goes well beyond spatial-shaping techniques, relying on intensity optimization in the imaging plane; namely, our approach allows for fine control of the focus shape, in amplitude and phase, that cannot simply be achieved using optimization approaches on the intensity. In the latter, the maximal intensity scales inversely with the number of target points [15] or the size of the focusing area, as in acousto-optic and photoacoustic techniques [28,29]. Similarly, in our approach, the maximal intensity in the PSF decreases with its complexity but the amount of energy in the PSF remains the same.

This PSF engineering method through complex media is also not limited to amplitude and phase modulation
in the Fourier domain and could be, for instance, extended to an arbitrary polarization mask if
measuring a polarization-resolved transmission matrix [30], or
even to more complex spectrally dependent PSFs thanks to the spectral dependence of the speckle [31,32]. Moreover, the
Fourier domain of the plane of interest may not be accessible in practice as in [21], where the amplitude mask was placed after the medium and the speckle observed
at a distance. In contrast, our technique would work in any plane, even at the output surface of the
medium. Furthermore, a single transmission matrix measurement can be used to focus at different
positions and allows rapid switching between various PSFs, as the numerical filtering step and phase
conjugation are realized *a posteriori*. Using a simple quadratic phase mask, the
focusing plane can also be translated longitudinally at will.

On the downside, in order to perform an accurate spatial Fourier transform, the reference beam during the transmission matrix measurement needs to be a well-defined plane wave, which requires a reference arm with interferometric stability during the measurement process. Additionally, the resolution of the generated mask is related to the sampling in the Fourier domain and therefore depends mostly on the sampling in the imaging plane (on the CCD): generating an accurate mask requires sampling at least a few CCD pixels per speckle grain over an extended output spatial region of at least a few times the PSF size. Note that, however, the resolution of the mask is advantageously independent of the number and resolution of the SLM segments, which only affect the signal to background ratio of the focus after phase conjugation. Increasing the number of SLM segments involves a longer measurement time of the experimental transmission matrix, thus requiring increased stability of the scattering sample. In our implementation, a transmission matrix containing $N=4096$ input patterns can be measured in approximately 30 min, mainly limited by the refresh rate of the liquid crystal SLM, while the scattering sample is stable along a few hours. It is also worth pointing out that the signal to background ratio is crucially affected by the total transmission of the virtual mask (as pointed out by [21] for physical masks).

Finally, these presented results are strictly valid only in the regime of multiple scattered light. In this regime, the mixing between input degrees of freedom (pixels of the SLM) and output degrees of freedom (spatial [13], polarization [33], and spectrum [34]) is maximal, as illustrated by the fact that the transmission matrix can be shown to be well described by random matrix theory [13]. In a weaker scattering regime—for instance, for surface scattering or when the thickness of the medium is less than a few transport mean free paths (with a significant ballistic or forward scattering component)—the transmission matrix formalism is still valid so the PSF-shaping methodology is still applicable. However, due to imperfect mixing, the signal to noise ratio, resolution, fidelity, and achievable PSF may be degraded. For example, a thin scatterer might not allow for efficient amplitude modulation in the pupil plane.

## 5. CONCLUSION

In conclusion, we have reported the first formulation to our knowledge of an operator, built upon the experimental transmission matrix, that enables deterministic focusing of any arbitrary PSF after propagation through a multiply scattering sample. We have illustrated the strength of this technique by generating Bessel, “donut,” and double helix beams through a scattering sample with a simple use of this new operator, and characterized their transverse and longitudinal properties. The method can readily be extended to other complex media, from biological tissues to multimode fibers. The possibility of arbitrarily generating complex PSF through multiply scattering media opens up new opportunities in several fields, in particular for 3D and super-resolution microscopy as well as optical manipulation and trapping [35]. A particularly interesting extension would be to exploit more directly the high spectral diversity offered by multiply scattering media [31] to generate highly complex, spectrally varying PSFs that could have applications in coherent control or nanophotonics.

## Funding

European Research Council (ERC) (278025); National Science Foundation (NSF) (1310487, 1611513).

## Acknowledgment

The authors thank Andréane Bourges, Hugo Defienne, and Gaetan Gauthier for discussions. S.G. is a member of the Institut Universitaire de France. R.P. acknowledges NSF support.

See Supplement 1 for supporting content.

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