## 2 Materials and Methods
### Hybrid Model Implementation
When the concentration value of the scattering medium increases, the number of on-axis and diffusive photons increases as well. In addition, all of these photons together form the extra focal spots located between yellow and red circle on Figure 3.  
At this point, we could apply the Shack-Hartmann technique to estimate the distortions of scattered laser radiation. We calculated the displacements of focal spots \(S_{x}\) and \(S_{y}\) (Equation (5)) and solved the system of linear equations (Equation (9)) to find the unknown coefficients \(a_{i}\) of Zernike polynomials. During the numerical estimations, we varied the concentration value of the scattering medium from 1.3 \(\times\) 10\({}^{5}\) to 10\({}^{6}\) mm\({}^{-3}\). We simulated the propagation of the photon flux through the scattering medium with eight particular concentration values from this range. Figure 4 shows some of hartmannogram images obtained on Shack-Hartmann during simulation of laser beam propagation through a scattering medium with different concentrations. It can be observed that the spots within the initial aperture of the beam that equaled 4 mm are presented as delta functions centered in the center of the corresponding sub-apertures--this is because the number of ballistic photons is much bigger than the others. The peripheral focal spots outside of the initial diameter are much wider in diameter, but they also have the clear-cut center. This is due to the impact of rather big portion of on-axis photons (Figure 4a-c). However, with the increase of the concentration of scatterers, the intensity distribution within each peripheral focal spot became uneven without the strongly marked center (Figure 4d-f). This is due to the impact of diffusive photons. As is known, conventional Shack-Hartmann sensor allows for the measurement of the wavefront of radiation. However, when a laser beam passes through a scattering medium, it does not have the wavefront in strict physical sense since part of the radiation is scattered. Figure 5 explains what we actually measure with a Shack-Hartmann sensor in this work. Figure 3: Shack–Hartmann focal spot field (hartmannogram) obtained after the beam propagation through a scattering layer. The inner yellow circle corresponded to the initial diameter (4 mm) of the collimated beam before entering the scattering volume. The outer red circle corresponded to the area (4.8 mm diameter) of the Shack–Hartmann sensor where the distortions were analyzed. Initially, before the scattering medium, the beam has a flat-incident wavefront. The scatterers with which radiation interacts become point sources of secondary spherical waves with their own wave vectors \(\overset{\rightarrow}{k}\). At the same time, part of the radiation passes through the medium without scattering \(\overset{\rightarrow}{k}_{0}\) (so-called ballistic component), retaining the original flat wavefront. As a result, a focal spot is formed at the focal plane of the lenslet. In addition, the displacement of this focal spot from the optical axis of the lenslet is proportional to the local slope of a so-called \"average wavefront\", which is composed of a set of, generally speaking, independent wavefronts corresponding to different point sources (i.e., scatterers in the medium). Moreover, we measure and analyze this average wavefront. Due to the axial symmetry nature of the Mie scattering process, only symmetrical distortions of the average wavefront were obtained in the simulation. The obtained coefficients of symmetrical Zernike polynomials (#3, #8, #15, #24) are presented on Figure 6.  
Figure 4: Hartmann images obtained from the simulation of laser beam propagation through a scattering medium with different concentration values: (**a**) 1.3 \(\times\) 10\({}^{5}\) mm\({}^{-3}\), (**b**) 4.4 \(\times\) 10\({}^{5}\) mm\({}^{-3}\), (**c**) 7.4 \(\times\) 10\({}^{5}\) mm\({}^{-3}\), (**d**) 9.4 \(\times\) 10\({}^{5}\) mm\({}^{-3}\), (**e**) 10\({}^{6}\) mm\({}^{-3}\), and (**f**) 1.2 \(\times\) 10\({}^{6}\) mm\({}^{-3}\).