# Non-Classic Atmospheric Optical Turbulence: Review
## 5 Theoretical Predictions for Non-Classic Turbulence-Light Interactions
For example, in the case of a horizontal link, the transformation can be done by implementing the change of variables \(\mu_{\alpha}\kappa_{\alpha}\rightarrow\kappa_{a}\), \(\alpha=x,y\) and considering new power spectrum depending only on function \(|\kappa|^{\perp}\) in the new variables. From the knowledge of the field CSD matrix components \(W_{ij}\), the evolution of the spectral density \(S\) and the degree of coherence \(\mu\) can be found as follows:  
\[S(\mathbf{r};\omega)=Tr\overleftrightarrow{W}(\mathbf{r},\mathbf{r};\omega), \tag{42}\]  
and  
\[\eta(\mathbf{r}_{1},\mathbf{r}_{2};\omega)=\frac{Tr\overleftrightarrow{W}( \mathbf{r}_{1},\mathbf{r}_{2};\omega)}{\sqrt{S(\mathbf{r}_{1};\omega)}\sqrt{ S(\mathbf{r}_{2};\omega)}}. \tag{43}\]  
A variety of polarization properties can also be determined the most important of which is the degree of polarization [86]:  
\[P(\mathbf{r};\omega)=\sqrt{1-4\frac{Det[\overleftrightarrow{W}(\mathbf{r}, \mathbf{r};\omega)]}{Tr^{2}[\overleftrightarrow{W}(\mathbf{r},\mathbf{r}; \omega)]}}. \tag{44}\]  
Here, \(Tr\) and \(Det\) stand for matrix trace and determinant. Second-order statistics of stationary optical beams radiated by a large number of sources with various spectral compositions, spectral density distributions as well as coherence and polarization states were theoretically analyzed in interactions with non-Kolmogorov turbulence (e.g., [88; 89; 90; 91; 92; 93]). The beams discussed the first three of these references, referring to the broad class of Gaussian Schell-Model (GSM) beams (either vectorial or treated under scalar approximation). These beams are the extensions of the Gaussian laser beams to any coherence state (from coherent to incoherent) and any polarization state (from polarized to unpolarized). They can either be considered at a fixed frequency \(\omega\)[88; 91], or a model spectral composition can be employed [90]. Figure 13 illustrates the average spreading of a GSM beam with growing propagation distance from the source, defined as follows:  
\[w(z;\omega)=\frac{\iint r^{\perp 2}S(\mathbf{r}^{\perp};\omega)d^{2}r^{\perp}}{ \iint S(\mathbf{r}^{\perp};\omega)d^{2}r^{\perp}}, \tag{45}\]  
for several values of \(\alpha\). Figure 14 illustrates the changes in the degree of coherence \(\eta\) and the degree of polarization \(P\) as the GSM propagates in non-Kolmogorov turbulence [90]. Interaction of the electromagnetic GSM (EM GSM) beams with anisotropic turbulence along a vertical path (up-link) was examined in Reference [94]. The refractive-index structure constant profile was included in accordance with the well known Hafnagel-Valley model [95]. Figure 15 presents the behavior of the spectral density of a coherent (A) and partially coherent (B) EM GSM beam in the up-link configuration in the presence of the anisotropic, non-Kolmogorov turbulence with \(\alpha=3.5\) and several values of the effective anisotropy parameter \(\zeta\sim\mu_{z}^{-1}\). For \(\zeta=1\), the turbulence reduces to isotropic, while for \(\zeta>1\), it accounts for anisotropic structure. In the vertical propagation case, the anisotropy only acts as a strength modifier of turbulence. We also note that for a coherent beam, its impact is larger than in the case of a partially coherent beam: in the latter case, most of the diffraction is caused by source correlation at any rate. The evolution of the EM GSM along a horizontal path in an anisotropic turbulence was considered in Reference [96]. Figure 16 presents the transverse cross-sections of three EM GSM beams with different source coherence states (top row corresponds to more coherent beam) on passing in such turbulence at different ranges. While a more coherent beam acquires an elliptical profile (vertically stretched) on passing at a sufficient distance from the source, the less coherent beam retains its initial circular cross-section, proving to be more resilient to turbulence fluctuations.