# Non-Classic Atmospheric Optical Turbulence: Review
## 5 Theoretical Predictions for Non-Classic Turbulence-Light Interactions  
In this section, we will overview the extended Huygens-Fresnel (EHF) method for characterizing the interactions between an optical beam and atmospheric turbulence described with non-Kolmogorov and anisotropic power spectra on the order of the second-order statistical moments of the field. Several important second-order statistics can be generally deduced from the knowledge of the cross-spectral density (CSD) matrix [86] of the beam. EHF is currently the most popular method used for predictions of CSD propagation of beams radiated by sources with any spectral composition, degree of coherence and polarization state. For a beam originating in the source plane \(z=0\) and propagating close to the positive \(z\) direction,  
Figure 12: The two-SLM setup for generation of any pre-modulated beam (\(SLM_{1}\)) and simulating its interaction with non-Kolmogorov turbulence (\(SLM_{2}\)). \(LD\)—laser diode, \(NDF\)—neutral density filter, \(L\)—lens, \(CMOS\)—complementary metal oxide semiconductor camera, \(PC_{1}\), \(PC_{2}\) and \(PC_{3}\) are personal computers. (**a–e**) Generated frame-like beams passing through simulated turbulence with different \(\alpha\); (**f**) comparison of cross-sections. Printed with permission from Reference [85]. the EHF integral relates the components of the CSD matrix in the source \(\overleftrightarrow{W}^{(0)}\) and in the field \(\overleftrightarrow{W}\) by the following law:  
\[W_{ij}(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=\iint W_{ij}(\mathbf{r}_{1}^{ \prime},\mathbf{r}_{2}^{\prime};\omega)K(\mathbf{r}_{1},\mathbf{r}_{2}; \mathbf{r}_{1}^{\prime},\mathbf{r}_{2}^{\prime};\omega)d^{2}\mathbf{r}_{1}^{ \prime}d^{2}\mathbf{r}_{2}^{\prime}, \tag{39}\]  
where \(\mathbf{r}_{a}^{\prime}\) (\(\alpha=1,2\)) are the 2D source points (\(z=0\)), \(\mathbf{r}_{a}\) (\(\alpha=1,2\)) are the 3D field points, \(\omega\) is the angular frequency, and the integration is taken twice over the source plane. Additionally, \(i\) and \(j\) refer to the mutually orthogonal components \(x\) and \(y\) of the electric field, transverse to optical axis \(z\). Propagator \(K\), being the correlation of two spherical wave Green's functions, takes for the 3D power spectrum \(\Phi_{n}(\mathbf{\kappa})\) the following form [87]:  
\[K(\mathbf{r}_{1},\mathbf{r}_{2};\mathbf{r}_{1}^{\prime},\mathbf{r }_{2}^{\prime};\omega) =\left(\frac{k}{2\pi z}\right)^{2}\exp\left[-ik\frac{(\mathbf{r}_ {1}^{\prime}-\mathbf{r}_{1}^{\perp})^{2}-(\mathbf{r}_{2}^{\prime}-\mathbf{r} _{2}^{\perp})^{2}}{2z}\right]\] \[\times\exp\left[-2\pi k^{2}z\int_{0}^{1}d\xi\int_{0}^{\infty}d^{2 }x\Phi_{n}(\mathbf{\kappa})\right. \tag{40}\] \[\times\left.[1-\exp[\xi\mathbf{r}_{d}^{\perp}\cdot\mathbf{\kappa}+(1 -\xi)\mathbf{r}_{d}^{\prime}\cdot\mathbf{\kappa}^{\perp}]]\right].\]  
Here, \(\mathbf{r}_{d}^{\perp}=\mathbf{r}_{2}^{\perp}-\mathbf{r}_{1}^{\perp}\), \(\mathbf{r}_{d}^{\prime}=\mathbf{r}_{2}^{\prime}-\mathbf{r}_{1}^{\prime}\) and \(\mathbf{\kappa}^{\perp}=(\kappa_{x},\kappa_{y},0)\). The first line of propagator \(K\) in Equation (40) corresponds to free-space diffraction, while the rest attributes to the effects of turbulence. To arrive at propagation law (39), the Markov approximation was employed, under which the correlation in the forward directions is modeled as the delta-function. Under the assumption of isotropic turbulence, the turbulence-related part of \(K\) reduces further to the following expression:  
\[\exp\left[-4\pi^{2}k^{2}z\int_{0}^{1}d\xi\int_{0}^{\infty}d\kappa\Phi_{n}( \kappa)[1-J_{0}[|(1-\xi)\mathbf{r}_{d}^{\perp})+\xi(\mathbf{r}_{d}^{\prime})| \kappa^{\perp}]\right], \tag{41}\]  
where \(\kappa^{\perp}=|\mathbf{\kappa}^{\perp}|\) and \(J_{0}\) are the zero-order Bessel functions of the first kind. The two-fold integral can be either solved numerically or approximated for small values of the Bessel function argument. The anisotropic spectrum can also be first converted to an isotropic spectrum, and then the procedures above can be applied.