## Abstract

The rarely considered case when the optical radiation passes through the weakly scattering medium, e.g. mid-density atmospheric fog with the number of scattering events up to 10 was investigated in this paper. We demonstrated an improvement of focusing of a laser beam (λ=0.65 µm) passed through the 5 mm-thick layer of scattering suspension of 1 µm polystyrene microbeads diluted in a distilled water. For the first time the low-order aberration corrector – wide aperture bimorph deformable mirror with 48 electrodes configured in 6 rings was used to optimize a far-field focal spot. We compared efficiencies of the algorithm that optimized the positions of the focal spots on Shack-Hartmann type sensor and the algorithm that optimized the peak brightness and the diameter of the far-field focal spot registered with a CCD. We experimentally demonstrated the increase of the peak brightness of the far-field focal spot by up to 60% due to the use of the bimorph deformable mirror for beam focusing through the scattering medium with concentration values of scatterers ranged from 10^{5} to 10^{6} mm^{−3}.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A medium is considered to be turbid or scattering if it has a pronounced optical inhomogeneity caused by impurities of particles with a refractive index that differs from a refractive index of the medium. Striking examples are atmospheric aerosol, haze, fog, sea water, clouds, ground glass, biological tissues, etc. [1,2]. In a turbid medium a part of a laser beam energy is absorbed, while the rest is redistributed in space, forming a halo of a scattered light, which makes contours of objects look blurred and prevents efficient beam focusing. The solution of this problem is of particular significance for such applications as pattern recognition, wireless data transmission, medical noninvasive diagnosis, and some others [3,4].

In 2007 the technique that used wavefront shaping approach to focus the light through strongly scattering media was developed [5]. The authors demonstrated that a liquid-crystal phase modulator can be used to shape the wavefront in order to focus the light through even strongly scattering objects. Since 2007 a number of techniques were developed to overcome the problem of imaging and focusing through a scattering medium [6,7]. Clear images of an object could be obtained using holographic techniques by reversing a scattering process [8]. The technique called multispectral multiple-scattering low-coherence interferometry [9] uses coherence and spatial gating to produce images of optical properties of a tissue up to 9 mm deep with millimeter-scale resolution. When there is no access to the space behind a scattering layer, a non-invasive approach to image fluorescent objects can be used [10]. In order to focus light or to reconstruct images of an object placed within or behind a strongly scattering medium spatial light modulators are often used [11–14]. In [15] authors demonstrated steady-state focusing of coherent light through dynamic scattering media. In [16] authors demonstrated numerically and experimentally that by using a transmission matrix inversion method peak-to-background intensity ratio of the focus can be higher than that achieved by conventional methods. The authors of [17] developed the new optical time-reversal focusing technique called Time Reversal by Analysis of Changing wavefronts from Kinetic targets that allowed to obtain a focal peak-to-background strength of 204. In [18] authors developed the first full-polarization digital optical phase conjugation system that, on average, doubles the focal peak-to-background ratio achieved by single-polarization digital optical phase conjugation systems. The authors of [19] introduced a phase control technique using programmable acoustic optic deflectors to image the objects in dynamically changing living biological tissue.

Basically, when we are talking about light propagation through an optically inhomogeneous medium, methods of adaptive optics for active control of light might be applied. If the refractive index of an inhomogeneous medium changes smoothly and continuously in space, then the wavefront of the radiation can be correctly defined and optimized using the principles of conventional adaptive optics. On the other hand, if the medium has high concentration of random inhomogeneities, i.e. biological tissues, the image of a target object hidden behind this medium could be completely lost. In that case the wavefront shaping technique can be applied [5]. The regimes described above just represent different ends of a continuum. Our paper is devoted to the so-called cross-over regime [20] — when the light passed through the scattering medium is partially coherent, the so-called averaged wavefront is not completely scrambled, and methods of classical adaptive optics still could be applied in order to improve beam focusing. In this paper we use the term “averaged wavefront” — this is the time-averaged phase surface reconstructed from the Shack-Hartmann sensor spot field that is produced by a mixture of ballistic (unscattered), quasi-ballistic (“snake”) and diffusive components [21] of the scattered radiation. The idea of our work was to control and modify the direction of the quasi-ballistic and diffusive photons in order to increase the peak intensity and decrease the diameter of the far-field focal spot formed by the beam that passed through the scattering medium.

In the following sections of the paper numerical and experimental results will be discussed. The developed model based on Shack-Hartmann technique to estimate the distortions of the averaged wavefront of the laser beam propagated through the scattering medium as well as the experimental confirmation of the simulation results will be described in part 2. The results of the numerical estimations of focusing enhancement that could be achieved with the use of bimorph deformable mirror would be demonstrated in part 3. An experimental setup will be described and the results of an improvement of focusing of a laser beam passed through the scattering medium using the bimorph deformable mirror will be demonstrated and discussed in part 4.

## 2. Numerical estimations and experimental measurements of scattered laser beam distortions

#### 2.1. Simulation of laser beam propagation

The propagation of the light through the inhomogeneous medium is generally described by the radiative transfer equation [22]:

*t*is time, $\vec{\rm r}$ is the vector position, $\vec{\rm s}$ is the incident direction of propagation, $f(\vec{s}^{\prime},\vec{s})$ is the scattering phase function derived from the appropriate scattering theory (e.g. Mie or Rayleigh theory), ${\rm d}\Omega ^{\prime}$ is the solid angle spanning $\vec{s}^{\prime}$ and

*c*is the speed of the light in the surrounding medium.

The radiative transfer equation can be summarized as follows: the change of radiance along a line of sight (left-side term of Eq. (1)) corresponds to the loss of radiance due to the extinction of incident light (first right-side term of Eq. (1)) plus the amount of radiance that is scattered from all other directions $\vec{s}^{\prime}$ into the incident direction $\vec{s}$ (second right-side term of Eq. (1)). The total extinction represented by the first right-side term of Eq. (1) equals to the radiance lost due to scattering of the incident light in all other directions, minus the radiance that is absorbed during each light-particle interaction. Radiative transfer equation is applicable for the wide range of turbid media: however, the analytical solutions are only available in rather simple circumstances where assumptions and simplifications are introduced to reduce the equation to a more tractable form. Since there are no analytical solutions available to the transport equation in realistic cases, a number of numerical techniques have been developed and utilized, such as method of path integrals, methods of the diffusion approximation, method of the light scattering by Brownian particles, and method of the small-angle approximation [23].

However, these approaches are based on the theoretical results obtained under different assumptions and thus are not universal. Such methods as diffusion approximation and light scattering by Brownian particles are suitable only for the scattering media with the small anisotropy factor (not our case). The method of small-angle approximation actually can be used for the highly anisotropic medium (our case), but this method does not take into account the diffuse component of scattered light. It was not suitable for our research because one of our goals was to investigate the influence of both quasi-ballistic and diffuse components of the scattered light on the quality of the beam focusing. There are two variations of the method of path integrals: analytical (modified method of Brownian particles) and stochastic (Monte Carlo method). The most versatile and widely used numerical solution is based on a statistical Monte Carlo technique [24–26] which gives an approximate solution of the radiation transport equation for almost all conditions of the problem, i.e. the arbitrary configurations of the medium and the boundary conditions. Monte Carlo method involves modelling of the behavior of the individual elementary parts of a physical system. In particular, for the problem of the light propagation this method takes into account the quantum nature of the light and simulates the behavior of the photon flux [27].

We used Monte Carlo simulation to model the laser beam propagation through the scattering medium. As a scattering medium we used a suspension of 1 µm polystyrene microbeads (produced by Magsphere, Inc., Pasadena, USA) with the refractive index of 1.582 [28] diluted in distilled water (refractive index equals to 1.33) within the 5 mm-thick glass cuvette. In our simulations and experiments the concentration value, or a number of microbeads per cubic millimeter, of the scattering suspension was varied approximately from 10^{5} to 10^{6} mm^{−3}. Scattering coefficients [25] of such media were calculated using Mie theory [25] and found to be in the range of 0.2–2 mm^{−1}, which means that the corresponding mean free paths were varied from 5 mm to 0.5 mm. Altogether, the scattering volume with the parameters described above (the diameter, the refractive index, and the concentrations of scattering particles as well as the thickness of the scattering volume) could be considered as an equivalent to the mid-density atmospheric fog layer of 0.3–5 km length, based on the principle of similarity [24]. According to this principle, if the phase functions and albedo of the single scattering event are similar for different media, then the intensity values in different points with the same non-dimensional coordinates also should be equal.

We simulated the laser beam as the photon flux (approximately 2.5×10^{11} photons) with uniform distribution of photons within the aperture. The distance between two scattering events (photon-particle interaction), called free path length, was calculated as defined in [25]: $l ={-} {{\ln ({\xi _l})} / {{\mu _s}}}$, where ${\mu _s}$— scattering coefficient, described above, ${\xi _l}$— random number uniformly distributed within the interval [0, 1). To calculate the propagation directions of photons we used Henyey–Greenstein phase function because it accurately approximates the angular scattering dependence of single scattering event for the range of concentrations used in modelling [26] and provides high calculation speed. In order to define the new direction of propagation of each photon after the scattering event, we calculated scattering and azimuthal angles. Scattering angle θ (Eq. (2)) is the angle between the current and new direction of photon propagation (Henyey-Greenstein phase function allows to calculate the cosine of this angle). Azimuthal angle φ (Eq. (3) is the angle between the projection of the new direction on the plane perpendicular to the initial direction of the photon propagation (X-Y plane at Fig. 1) and X-axis.

Here $g$— anisotropy factor [25], ${\xi _\theta },{\xi _\varphi }$— random numbers uniformly distributed within the interval [0, 1). The new photon position after scattering event was calculated according to the known free path length and the direction of propagation. If the photon stayed within the scattering medium, the parameters *l*, $\theta$ and $\varphi$ were calculated again. For each simulated photon we obtained the information about the total travelled optical path length, final position, final direction of propagation, and total number of the encountered scattering events. In such a way, we estimated the intensity distribution of the scattered beam at the specified plane located after the scattering volume, when all of the photons travelled through the medium.

#### 2.2. Estimation of scattered beam distortions

In order to numerically estimate laser beam distortions caused by light scattering we developed the model based on Shack-Hartmann technique [29] and implemented it in the simulation of the laser beam propagation. When the beam propagated through the scattering layer, it fell on the microlens array, placed at the distance of 450 mm after the boundary of the scattering medium. The microlens array consisted of small microlenses with the diameter of 150 µm and focal length of 6 mm, that split the beam into a number of smaller beams and focused them at the plane of the imaging sensor. The principle scheme of the simulation of the laser beam (photons flux) propagation through the scattering medium using Monte Carlo method as well as measurements of the beam distortions using Shack-Hartmann technique are presented on Fig. 1.

Our approach of numerical estimations of the scattered beam distortions was rather simple and relied on the principles of geometrical optics. The simulated photon that left the scattering medium had the trajectory of propagation defined by direction cosines $c {X_1}$, $c {Y_1}$, $c {Z_1}$:

The photon was traced to the plane of the microlens array and then retraced to the plane of the imaging sensor, where its coordinates ${x_f}$, ${y_f}$, ${z_f}$ were calculated using the formulas:

*f*— focal length of the microlens.

In each sub-aperture of the imaging sensor (that corresponds to the particular microlens) we obtained a focal spot with the intensity distribution that corresponded to the number and positions of the photons registered inside this particular sub-aperture. The resultant simulated field of focal spots is called hartmannogram (Fig. 2(a)). The inner blue circle corresponded to the initial diameter (4 mm) of the collimated beam before entering the scattering volume. The outer blue circle corresponded to the area (4.8 mm diameter) of the Shack-Hartmann sensor where averaged wavefront was analyzed.

There are three components of a scattered photon flux [30]: ballistic, on-axis (or “snake”) and off-axis (or diffusive) photons. Ballistic photons travel through a turbid medium without interaction with the scatterers and do not change the initial trajectory. This coherent component of the scattered light is the most important for imaging and focusing applications. On-axis photons undergo few scattering events and travel in near-forward paths along a trajectory that is close to the initial direction of the beam propagation. These photons play an important role for imaging and focusing when the thickness of the scattering medium layer increases, because the number of ballistic photons decreases exponentially in this case [3]. Off-axis photons scatter in all directions and form a noncoherent component of the scattered light. As noted above, ballistic photons are most valuable for focusing, but their number decreases exponentially when the layer thickness or scatterers concentration increases. When the concentration value of the scattering medium increases, the number of “snake” and diffusive photons grows up as well [3]. Our previous numerical and experimental investigations conducted with Shack-Hartmann sensor demonstrated [31], that the increase of the concentration value from 10^{5} to 10^{6} mm^{−3} led to the displacements of the peripheral focal spots located between two blue circles in Fig. 2(a) from its reference positions outwards (direction of displacements are displayed as arrows in Fig. 2(b)). According to Shack-Hartmann technique, these displacements are proportional to the local slopes of the wavefront of the incident radiation [32–34]. If the center of the particular focal spot is located in the center of the sub-aperture, then the wavefront within this sub-aperture will be considered flat. On the other hand, if the focal spot is shifted away from the center of the sub-aperture, then the wavefront inside this sub-aperture will be tilted. The local wavefront tilts $\partial W(x,y)/\partial x$, $\partial W(x,y)/\partial y$ are derived from the displacements (${S_x}$, ${S_y}$) of the spot centroid and the focal length *f* of a microlens [35]. Solving least-square problem, the expansion coefficients could be found, and wavefront could be obtained by approximation, for example, using Zernike polynomials ${Z_i}(x,y)$ [36,37]:

- ${a_i}$— Zernike coefficient,
- ${Z_i}(x,y)$—
*i*^{th}Zernike polynomial value at the point $(x,y)$, *N*— number of Zernike polynomials,- $f$— focal length of the microlens array,
- ${S_x}$, ${S_y}$— focal spot centroid displacements.

During the numerical estimations we varied the concentration value of the suspension of polystyrene microbeads from 1.3×10^{5} mm^{−3} to 10^{6} mm^{−3}. We simulated the propagation of the photon flux through the scattering medium with 8 particular concentration values from this range and estimated the resultant averaged wavefront distortions in terms of Zernike polynomials using Shack-Hartmann technique described above. Due to the axial symmetry nature of the Mie scattering process, only symmetrical distortions were obtained in the simulation. And, what was interesting and important, there was not only defocus (Zernike coefficient #3), like it was supposed previously [38], but also low- and high-order spherical distortions: Zernike coefficients #8 and #15 (Table 1).

Another interesting result obtained from the simulation was that the amplitude of the distortions (peak-to-valley) increased with the increase of the scatterers concentration. It was due to the increase of the number of “snake” and diffusive photons and simultaneous decrease of the number of ballistic photons, that led to the displacements of the centers of the focal spots on the hartmannogram from the centers of the sub-apertures.

To verify the numerical results, we assembled the experimental setup (Fig. 3). Collimated laser beam with the diameter of 4 mm passed through the small (18 mm × 18 mm × 5 mm) transparent glass cuvette (thickness was 5 mm), filled with the suspension of 1 µm-diameter polystyrene microbeads diluted in distilled water, and registered on the Shack–Hartmann sensor produced by Active Optics NightN Ltd. [39]. The sensor consisted of a digital CCD camera Basler A302fs (1/2-inch sensor with the size of the receiving area equaled to 6.4 × 4.8 mm) and a microlens array (focal length — 6 mm, diameter of a single microlens — 150 µm, total number of microlenses — greater than 1300). The CCD camera frame rate was 30 Hz. Shack-Hartmann sensor was moved 450 mm away from the cuvette (as it was in the model) in order to decrease the impact of the totally diffused photons on the quality of the measurements.

Scattered beam distortions (averaged wavefront) were analyzed in the circular aperture of 4.8 mm, the center of which coincided with the center of the camera sensor [31]. Increase of the aperture up to 4.8 mm for beam analysis above the initial beam size of 4 mm was necessary in order to analyze the impact of “snake” and diffusive photons on the distortions of scattered light. The averaged wavefront measured by Shack-Hartmann sensor was approximated by Zernike polynomials. The resultant symmetric Zernike polynomial #3, #8 and #15 [34] values as well as PV and RMS values are presented in Table 2.

As we can see from Table 2, the impact of low-order Zernike terms is predominated in the averaged wavefront distortions: the increase of the concentration of scattering particles in the turbid medium resulted in the increase of Zernike term #3 value (from 0.04 to 0.91 µm) and Zernike term #8 value (from 0.06 to 0.59 µm).

Figure 4 shows the comparison of the dependence of PV values on the concentration values of scatterers. Both the model and the experimental curves demonstrated the similar trends of increase of the amplitude of distortions with the increase of the concentration values of the scatterers.

Differences between model and experimental curves could be due to the fact that in the model we simulated a flat-top intensity distribution (Fig. 4(b)) using random number generator but did not emulate the intensity distribution of the real laser beam (Fig. 4(c)).

## 3. Numerical simulation of the correction of the scattered laser beam distortions

In order to estimate the potential efficiency of focusing of the scattered laser beam, we performed numerical simulation of the correction of the experimentally measured distortions of the averaged wavefront by means of the bimorph deformable mirror.

#### 3.1 Bimorph deformable mirror

In previous section of this paper we demonstrated that low-order axial-symmetric distortions predominated in the averaged wavefront of scattered light. To compensate for such distortions, bimorph deformable mirrors are commonly used [40–42]. A conventional bimorph mirror consists of a passive substrate made of glass, silica or cooper and two active piezoceramic disks firmly glued to the substrate (Fig. 5(a)). The inner piezo disk has one “ground” electrode and one control electrode to change the overall curvature of the mirror surface, the outer piezo disk has a grid of control electrodes to compensate for various aberrations (Fig. 5(b)). Number of control electrodes depends on the type of the distortions to be corrected for. Generally, bimorph mirror contains additional piezoceramic disk with one round control electrode to change the overall curvature of the surface.

Bimorph deformable mirrors are able to effectively compensate for low-order wavefront distortions. In our case, it was very important because, as we mentioned above, low-order Zernike terms #3 and #8 had the most significant impact in the distortions of scattered light. The bimorph mirror made by Active Optics NightN Ltd. that we used to conduct the experiments described below had 48 control electrodes configured in 6 rings (Fig. 3(b)). Initial flatness of the reflective surface of the mirror was 0.3 µm, total stroke was ±20 µm, control voltages range was −200…+300 V. Such a configuration of the mirror allowed us to compensate for the distortions of the beam caused by scattering as well as to correct for misalignments of optical elements and initial deformation of the surface of the mirror. This mirror was able to reproduce such distortions as defocus (Zernike term #3), 4th order (Zernike #8) and 6th order (Zernike #15) spherical distortions with PV=25 µm (RMS=0.03 µm), 4 µm (0.005 µm), and 2 µm (0.04 µm), correspondingly.

#### 3.2 Algorithm and results of numerical simulation of correction

In order to numerically estimate how effective the compensation of scattered beam distortions could be, we applied standard least-square fit-error minimization algorithm [43]. We used experimentally measured response functions of the bimorph mirror electrodes and calculated displacements of the focal spots of the experimentally obtained hartmannogram (section 2 of this paper).

Response function of the single electrode of the bimorph mirror represents deformation of the surface of the mirror influenced by the voltage applied to this particular electrode. In our case the response function of each electrode of the mirror was measured by Shack-Hartmann sensor and expressed as the vector of the displacements of the focal spots on the sensor. The main steps of the algorithm of numerical correction were as follows:

- 1. Calculate the displacements of the focal spots of the experimentally obtained hartmannogram.
- 2. Calculate the control voltages to be applied to the electrodes of the mirror to minimize the displacements measured at step 1.
- 3. Calculate new displacements of the focal spots that we would obtain if we apply the voltages calculated at step 2.
- 4. Retrieve the resultant averaged wavefront from the residual displacements of the focal spots, approximate it with Zernike polynomials, and calculate its RMS.

The results of numerical correction of the distortions of the laser beam passed through the scattering medium with concentration value equaled to 6.06×10^{5}mm^{−3} by means of the response functions of the 48-electrodes bimorph mirror are presented on Fig. 6.

As it was outlined in the previous sections of this paper, the averaged wavefront of the scattered laser beam contained not only defocus but spherical distortions also. Zernike approximation of the averaged wavefront depicted in Fig. 6(b) and Fig. 6(e) gave us the following values: defocus (Zernike #3) — 0.1 µm, spherical distortions (Zernike #8 and #15) — 0.1 µm and 0.06 µm, correspondingly. The total PV of the averaged wavefront was equal to 0.64 µm prior to adaptive correction, calculated Strehl ratio was equal to 0.5. During the optimization process Zernike coefficient values were decreased 10 times, the calculated Strehl ratio after numerical correction was increased up to 0.98. The results of numerical correction of the scattered beam distortions (Fig. 6(d) – Fig. 6(f)) confirmed that bimorph deformable mirror can efficiently flatten the averaged wavefront of scattered light.

## 4. Experimental laser beam focusing

In order to investigate the focusing of the scattered laser beam we assembled the experimental setup depicted in Fig. 7. Radiation from the fiber-coupled diode laser source with the wavelength of 0.65 µm [44] fell on the lens *L1*, and collimated beam of 6 mm diameter then passed through the 5 mm-thick glass cuvette filled with the scattering medium. The scattered beam passed through the 3x matching telescope (lenses *L2* and *L3*), placed at approximately 20 mm away from the cuvette, and reflected from the bimorph deformable mirror. The telescope imaged the plane where we wanted to measure laser beam distortions to the plane of the reflective surface of the bimorph mirror. After the reflection from the mirror, the beam fell on the 10x telescope (lenses *L4* and *L5*) that matched the aperture of the mirror and the aperture of the microlens array of the Shack-Hartmann sensor. Part of the beam was directed to Shack-Hartmann sensor to estimate the averaged wavefront while another part was reflected by the beam splitter (*BS*) and focused on the CCD camera to analyze the intensity distribution of the focal spot. Control software processed the data from both Shack-Hartmann sensor and CCD camera and calculated the set of correcting voltages to be applied to the electrodes of the mirror.

We compared efficiencies of two optimization algorithms for scattered laser beam focusing. The goal of the algorithm №1 [45] was to minimize displacements of the focal spots on Shack-Hartmann sensor from their reference positions by means of calculating the conjugated averaged wavefront.

- $\Delta {x_k}$—
*k*^{th}focal spot displacement along X axis, - $\Delta {y_k}$—
*k*^{th}focal spot displacement along Y axis, *M*— number of electrodes,- ${u_j}$— voltage value at the
*j*^{th}electrode, - ${b_j}({x_k},{y_k})$— response function of the
*j*^{th}electrode.

When the conjugated averaged wavefront was reconstructed by the mirror, we expected to see the improved quality of the focal spot in the far-field.

The goal of the algorithm №2 [46] was to optimize the intensity distribution and diameter of the far-field focal spot directly. We used the following merit function for the optimization process:

where ${D_x}$, ${D_y}$— diameters of the focal spot at*X*and

*Y*cross-sections, ${I_{\max }}$— peak brightness value within the focal spot. Such criteria allowed to increase the peak brightness and to decrease the diameter of the focal spot simultaneously. As an optimization procedure we used iterative “hill-climbing” algorithm [46] — it cycles through all of the mirror electrodes, setting the probing voltage value to each electrode sequentially and controlling the merit function value.

Before conducting experiments with the scattering medium, we calibrated the setup. The glass cuvette was filled with distilled water (without scattering microbeads). Then we ran the optimization procedure (algorithm №2) that intended to achieve the diffraction limited focal spot in far-field (on the CCD camera). When near diffraction limited focal spot was achieved, we stored the vector ${V_{init}}$ of correcting voltages of the mirror electrodes, and set the resultant spot field on Shack-Hartmann sensor as a reference one. In such a way, we took into account all of the misalignments and imperfections of the optical elements of the experimental setup so that we could measure only the distortions of the laser beam induced by the scattering particles.

After the calibration was done, we injected the first drop of the scattering suspension into the cuvette — concentration value of the medium became 1.3×10^{5} mm^{−3}. Shack-Hartmann sensor registered displacements of focal spots from its reference positions, while CCD camera registered increase of the diameter and decrease of the peak brightness of the far-field focal spot. Then we ran the optimization algorithm, found optimal correcting voltages, stored the results and set the initial voltages ${V_{init}}$ to the electrodes of the mirror. After that we increased the concentration value of the scattering medium by injecting the next drop of the suspension into the cuvette and repeated the procedure. For each concentration value considered in the experiment we registered intensity distributions of the focal spot before and after optimization procedure and calculated focusing efficiency *E* as follows:

*I*— peak brightness value within the focal spot on the CCD prior to optimization and

_{before}*I*— after optimization.

_{after}We conducted a set of measurements and found that the algorithm №1 allowed to increase the focusing efficiency *E* by 13% on the average in under 200 ms, while the algorithm №2 — by 60% on the average in under 5 minutes for the concentration value 3.3×10^{5} mm^{−3} (Fig. 8).

Comparing the cross-sections of the focal spots before optimization (solid blue curve) and after optimization with algorithm #1 (dashed orange curve) it can be seen from Fig. 8 that there are a lot of energy concentrated under the dashed curve from 9th to 20th pixel and from −8th to −20th pixel (averaged intensity value is 13 units in shades of gray on vertical axis). At the same time there is no such a pedestal for the solid blue curve but the FWHM for this curve is wider. That is where the energy came to. The algorithm №2 was more efficient than the algorithm №1 comparing the increase of peak intensity of the focal spot due to a few reasons discussed at the end of this section. Because the algorithm №2 was not limited by the possible approximation errors of the method of least squares, this algorithm was able to collect quasi-ballistic and some of diffuse light from the peripheral area of the scattered beam (as discussed in section 2.2 and depicted in Fig. 2) and retrace it to the focal spot more efficiently.

The results for a higher concentration value (6.2×10^{5} mm^{−3}) were a bit worse — 10% increase of focusing efficiency for the algorithm №1 and 45% — for the algorithm №2. Such an efficiency drop could be explained by the fact that the number of “snake” photons decreased while the number of diffusive photons increased with the increase of the concentration value of the scatterers [3]. Large concentration values caused multiple scattering regime when the photons became diffusive and simply did not reach the aperture of the mirror. Thus, the smaller the number of “snake” photons passed through the scattering medium — the smaller was the influence of the deformable mirror on the focusing efficiency of the laser beam.

The first reason why the algorithm №1 was less effective compared to the algorithm №2 was that the algorithm №1 was based on the minimization of the displacements of focal spots on Shack-Hartmann sensor that typically led to the increase of the quality of the far-field focal spot. But it has been shown in [47] that the minimization of the functional of displacements of Shack-Hartmann focal spots does not always lead to the optimal wavefront (i.e., flat wavefront) because different combinations of the focal spot positions could correspond to the same value of root-mean-square error but to completely different amplitudes of the wavefront distortions (Fig. 9). This being said, the root-mean-square error that is close to the minimal value does not always correspond to the minimal amplitude of the wavefront. The second reason is that the method of least squares used in the algorithm №1 can provide larger approximation error in case of multiple light scattering compared to the case when there is no scattering and the wavefront is defined correctly. In our case we were dealing with partially coherent beam and the information of the beam phase was also partly lost. The direct improvement of the focus by optimization procedure showed better focal spot brightness — we just somehow were correcting with the mirror actuators the path difference between different parts of the beam. And we were not actually correcting any phase.

## 5. Summary

We made numerical and experimental investigations of a laser beam propagation through the turbid medium and demonstrated that symmetric defocus and spherical distortions predominated in the averaged wavefront of the scattered light. We experimentally performed laser beam focusing through the scattering medium using two optimization techniques. The algorithm based on the optimization of the far-field focal spot demonstrated a 60% increase of efficiency of focusing of a scattered laser beam and could be applied in case of quasi static scattering medium. The algorithm based on the minimization of displacements of focal spots on Shack-Hartmann sensor from its reference positions demonstrated lower focusing efficiency (up to 13%), but it was significantly faster and thus could be used in case of dynamic scattering medium. Both of the optimization techniques could be applied in atmospheric adaptive optical systems.

## Funding

Russian Science Foundation (20-19-00597, 20-69-46064); Ministry of Science and Higher Education of the Russian Federation (0146-2016-0001).

## Acknowledgments

The study was funded by Russian Science Foundation (section 2.1 - by project # 20-19-00597, section 2.2 - by project # 20-69-46064) and Ministry of Science and Higher Education of the Russian Federation # 0146-2016-0001 (section 3.1).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **H. C. van de Hulst, “* Light scattering by small particles*,” John Wiley & Sons, Inc.: New York (1957).

**2. **J. Berne Bruce and R. Pecora, “Dynamic light scattering: with applications to chemistry, biology, and physics”, Courier Corporation (2000).

**3. **A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics **6**(5), 283–292 (2012). [CrossRef]

**4. **A. N. Bashkatov, A. V. Priezzhev, and V. V. Tuchin, “Laser technologies in biophotonics,” Quantum Electron. **42**(5), 379 (2012). [CrossRef]

**5. **I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. **32**(16), 2309 (2007). [CrossRef]

**6. **Y. Zhang, Y. Chen, Y. Yu, X. Xue, V. V. Tuchin, and D. Zhu, “Visible and near-infrared spectroscopy for distinguishing malignant tumor tissue from benign tumor and normal breast tissues in vitro,” J. Biomed. Opt. **18**(7), 077003 (2013). [CrossRef]

**7. **J. W. Goodman, W. H. Huntley Jr., D. W. Jackson, and M. Lehmann, “Wavefront reconstruction imaging through random media,” Appl. Phys. Lett. **8**(12), 311–313 (1966). [CrossRef]

**8. **H. Kogelnik and K. S. Pennington, “Holographic imaging through a random medium,” J. Opt. Soc. Am. **58**(2), 273–274 (1968). [CrossRef]

**9. **T. Matthews, M. Medina, J. Maher, H. Levinson, W. Brown, and A. Wax, “Deep tissue imaging using spectroscopic analysis of multiply scattered light,” Optica **1**(2), 105 (2014). [CrossRef]

**10. **J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature **491**(7423), 232–234 (2012). [CrossRef]

**11. **O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics **6**(8), 549–553 (2012). [CrossRef]

**12. **S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. **104**(10), 100601 (2010). [CrossRef]

**13. **D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express **20**(2), 1733–1740 (2012). [CrossRef]

**14. **C. Hsieh, Y. Pu, R. Grange, G. Laporte, and D. Psaltis, “Imaging through turbid layers by scanning the phase conjugated second harmonic radiation from a nanoparticle,” Opt. Express **18**(20), 20723–20731 (2010). [CrossRef]

**15. **C. Stockbridge, Y. Lu, J. Moore, S. Hoffman, R. Paxman, K. Toussaint, and T. Bifano, “Focusing through dynamic scattering media,” Opt. Express **20**(14), 15086–15092 (2012). [CrossRef]

**16. **J. Xu, H. Ruan, Y. Liu, H. Zhou, and C. Yang, “Focusing light through scattering media by transmission matrix inversion,” Opt. Express **25**(22), 27234–27246 (2017). [CrossRef]

**17. **E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica **1**(4), 227–232 (2014). [CrossRef]

**18. **Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through scattering media by full-polarization digital optical phase conjugation,” Opt. Lett. **41**(6), 1130–1133 (2016). [CrossRef]

**19. **D. Feldkhun, O. Tzang, K. H. Wagner, and R. Piestun, “Focusing and scanning through scattering media in microseconds,” Optica **6**(1), 72–75 (2019). [CrossRef]

**20. **I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express **23**(9), 12189–12206 (2015). [CrossRef]

**21. **I. Galaktionov, A. Kudryashov, J. Sheldakova, A. Byalko, and G. Borsoni, “Laser beam propagation and wavefront correction in turbid media,” Proc. SPIE **9617**, 96170D (2015). [CrossRef]

**22. **E. Berrocal, D. Sedarsky, M. Paciaroni, I. Meglinski, and M. Linne, “Laser light scattering in turbid media Part I: Experimental and simulated results for the spatial intensity distribution,” Opt. Express **15**(17), 10649–10665 (2007). [CrossRef]

**23. **E. A. Vorob’eva and I. P. Gurov, “Models of Propagation and Scattering of Optical Radiation in Randomly Inhomogeneous Media,” Moscow Meditsina (2006).

**24. **E. P. Zege, A. P. Ivanov, and I. L. Katsev, “* Image transfer through a scattering medium*,” Springer-Verlag, Berlin: New York, 131 (1991).

**25. **L. Wang and S. Jacques, “MCML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Prog. Bio. **47**(2), 131–146 (1995). [CrossRef]

**26. **S. L. Jacques and S. A. Prahl, “Henyey-Greenstein scattering function,” web-address: http://omlc.org/classroom/ece532/class3/hg.html, last accessed – February 2 (2018).

**27. **I. V. Meglinski, “Quantitative assessment of skin layers absorption and skin reflectance spectra simulation in the visible and near-infrared spectral regions,” Physiol. Meas. **23**(4), 741–753 (2002). [CrossRef]

**28. **X. Ma, J. Lu, S. Brocks, K. Jacob, P. Yang, and X.-H. Xin, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. **48**(24), 4165–4172 (2003). [CrossRef]

**29. **W. H. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. **70**(8), 998–1005 (1980). [CrossRef]

**30. **H. Ramachandran, “Imaging through turbid media,” Curr. Sci. **76**, 1334 (1999).

**31. **I. V. Galaktionov, J. V. Sheldakova, and A. V. Kudryashov, “Wavefront analysis of the laser beam passed through turbid medium,” Quantum Electron. **45**(2), 143–144 (2015). [CrossRef]

**32. **B. Platt and R. J. Shack, “History and principles of Shack-Hartmann wavefront sensing,” J. Refract. Surg. **17**(5), S573–S577 (2001). [CrossRef]

**33. **J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of the wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A **11**(7), 1949 (1994). [CrossRef]

**34. **R. G. Lane, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. **31**(32), 6902 (1992). [CrossRef]

**35. **J. Sheldakova, T. Cherezova, A. Alexandrov, A. L. Rukosuev, and A. V. Kudryashov, “The use of M2 meter to correct for high-power laser aberrations,” Proc. SPIE Int. Soc. Opt. Eng. **5708**, 352 (2005). [CrossRef]

**36. **J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Proc. of Applied optics and optical engineering 27–39 (1992).

**37. **A. Nikitin, J. Sheldakova, A. Kudryashov, G. Borsoni, D. Denisov, V. Karasik, and A. Sakharov, “A device based on the Shack-Hartmann wave front sensor for testing wide aperture optics,” Proc. SPIE **9754**, 97540K (2016). [CrossRef]

**38. **S. Restaino, “Adaptive Optics for Horizontal Propagation Applications,” Proc. of AOIM 27 (2013).

**39. **A. G. Aleksandrov, V. E. Zavalova, A. V. Kudryashov, A. L. Rukosuev, P. N. Romanov, V. V. Samarkin, and J. Sheldakova, “Shack-Hartmann wavefront sensor for measuring the parameters of high-power pulsed solid-state lasers,” Quantum Electron. **40**(4), 321–326 (2010). [CrossRef]

**40. **J. Sheldakova, V. Samarkin, A. Kudryashov, and A. Rukosuev, “Laser beam formation by adaptive optics,” Proc. SPIE **7913**, 79130I (2011). [CrossRef]

**41. **A. Kudryashov, A. Alexandrov, A. Rukosuev, V. Samarkin, P. Galarneau, S. Turbide, and F. Châteauneuf, “Extremely high-power CO2 laser beam correction,” Appl. Opt. **54**(14), 4352–4358 (2015). [CrossRef]

**42. **A. Kudryashov, V. Samarkin, A. Alexandrov, J. Sheldakova, and P. Romanov, “Wide aperture (more than 500 mm) deformable mirrors for high power laser beam correction,” Proc. SPIE **8960**, 89601G (2014). [CrossRef]

**43. **I. V. Galaktionov, A. V. Kudryashov, J. Sheldakova, A. A. Byalko, and G. Borsoni, “Measurement and correction of the wavefront of the laser light in a turbid medium,” Quantum Electron. **47**(1), 32–37 (2017). [CrossRef]

**44. **V. E. Karasik and V. M. Orlov, “Laser vision systems,” Moscow, MVTU named after Bauman, 478 (2013).

**45. **A. A. Aleksandrov, A. V. Kudryashov, A. L. Rukosuev, T. Y. Cherezova, and Y. V. Sheldakova, “An adaptive optical system for controlling laser radiation,” J. Opt. Technol. **74**(8), 550–554 (2007). [CrossRef]

**46. **J. Sheldakova, A. L. Rukosuev, and A. V. Kudryashov, “Genetic and hill-climbing algorithms for laser beam correction,” Proc. SPIE **5333**, 106–111 (2004). [CrossRef]

**47. **A. V. Kudryashov, V. V. Samarkin, Y. V. Sheldakova, and A. G. Aleksandrov, “Wavefront compensation method using a Shack-Hartmann sensor as an adaptive optical element system,” Optoelectron.Instrument.Proc. **48**(2), 153–158 (2012). [CrossRef]