## 2 Materials and Methods
### Shack-Hartmann Technique  
Shack-Hartmann sensor [61; 62; 63; 64; 65; 66] is well-known device that is widely used in large and diverse sets of applications, primarily to measure distortions of the wavefront of the radiation passed through different media--turbulent and/or scattering atmosphere, biological tissues, etc. Since, in our work, we consider the case of moderately scattering medium where average number of scattering events is from 0 to 10--the so-called cross-over scattering regime [67; 68]--we assumed (and proved it experimentally) that the Shack-Hartmann technique still could be applied to measure the scattered laser beam distortions. We will cover this point later in Section 2.4. In a conventional Shack-Hartmann technique, the wavefront of the incident radiation is divided into a number of sub-apertures by means of lenslet array. Lenslet array is a thin, flat base with a grid of micro lenses (lenslets) etched on the base. Each lenslet commonly has diameter from 100 to 300 \(\upmu\)m and focal length \(f\) from 3 mm to 8 mm. Radiation passes through these lenslets and forms a set of focal spots at the measurement (sensor) plane (Figure 2a). Since the diameter of each lenslet is rather small, it is assumed that the wavefront \(W\) is flat within a single lenslet, and the only aberration it has is tip-tilt. In case a tip-tilt equals zero (i.e., a wavefront is flat and parallel to the plane of the lenslet), radiation is focused to the center of the corresponding sub-aperture of the sensor. In case a wavefront has non-zero tip-tilt within a lenslet, then the focal spot will be displaced (\(S_{x}\), \(S_{y}\)) from the center of the sub-aperture proportional to the tip-tilt value (Figure 2b). In other words, if we measure these displacements \(S_{x}\) and \(S_{y}\) of the focal spot per \(X\) and \(Y\) axis, correspondingly, we will \begin{table}
\begin{tabular}{c c c} \hline \hline
**Criteria** & **MCML** & **Developed Model** \\ \hline Transmittance, \% & 94.380 & 94.377 \\ Reflectance, \% & 5.620 & 5.623 \\ \% of single scattered photons & 25.080 & 25.060 \\ \% of double scattered photons & 23.910 & 23.910 \\ \% of photons with scattering order = 3 & 16.280 & 16.288 \\ \% of photons with scattering order = 4 & 9.110 & 9.120 \\ \% of photons with scattering order = 5 & 4.660 & 4.660 \\ \% of photons with scattering order = 6 & 2.410 & 2.410 \\ \% of photons with scattering order = 7 & 1.370 & 1.370 \\ \% of photons in circle (radius = 100 \(\upmu\)m) & 16.540 & 16.550 \\ \% of photons in ring (r1 = 100 \(\upmu\)m, r2 = 200 \(\upmu\)m) & 3.350 & 3.330 \\ \% of photons in ring (r1 = 200 \(\upmu\)m, r2 = 300 \(\upmu\)m) & 3.410 & 3.420 \\ \% of photons in ring (r1 = 300 \(\upmu\)m, r2 = 400 \(\upmu\)m) & 3.390 & 3.370 \\ \% of photons in ring (r1 = 400 \(\upmu\)m, r2 = 500 \(\upmu\)m) & 3.300 & 3.290 \\ \% of photons in ring (r1 = 500 \(\upmu\)m, r2 = 600 \(\upmu\)m) & 3.160 & 3.160 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model verification: comparison of results obtained from the developed model and MCML. obtain the values of partial derivatives \(\partial W/\partial x\) and \(\partial W/\partial y\) of the wavefront \(W\) within each sub-aperture (Equation (5)):  
\[ \begin{cases}\frac{\partial W(x,y)}{\partial x}\\ \frac{\partial W(x,y)}{\partial y}\end{cases} =\frac{1}{f} \begin{cases}S_{x}\\ S_{y}\end{cases} , \tag{5}\]  
where \(\frac{\partial W(x,y)}{\partial x}\), \(\frac{\partial W(x,y)}{\partial y}\)--partial derivatives of the wavefront \(W\),  
\(f\)--local spot of each lenslet,  
\(S_{x}\), \(S_{y}\)--displacements of a focal spot within a sub-aperture per \(X\) and \(Y\) axis. On the other hand, in order to analytically describe and visualize the wavefront surface, one can use the polynomial approximation; for example, B-Splines [69] or Zernike polynomials [70; 71; 72; 73], which are commonly used in optics. These polynomials are orthogonal within the unity circle and are analytically expressed in polar coordinates as follows:  
\[Z_{n}^{l}(\rho,\theta)=