## 1 Introduction
\[\alpha_{2}=\frac{n^{\prime}}{w_{0}^{2}}+\frac{1}{\sigma_{0}^{2}}+\frac{1}{\delta_{ jj}^{2}}-\frac{ik}{2z}, \tag{10}\]  
\[\alpha_{3}=\frac{1}{\sigma_{0}^{2}}+\frac{1}{\delta_{jj}^{2}}. \tag{11}\]  
To obtain the analytical outcome of Equation (8), \(I_{l-m}(\cdot)\) is first expanded as:  
\[I_{l-m}(2\alpha_{3}\rho_{01}\rho_{02})=\sum\limits_{s=0}^{\infty}\frac{( \alpha_{3}\rho_{01}\rho_{02})^{2s+l-m}}{s!\Gamma(s+l-m)}, \tag{12}\]  
Then, we recall the following integral formula [46]:  
\[\exp\left[\frac{ik\rho\rho_{01}}{z}\cos(\theta-\theta_{01})\right]=\sum_{l=- \infty}^{\infty}i^{l}I_{l}\left(\frac{k\rho\rho_{01}}{z}\right)\exp[il(\theta- \theta_{01})], \tag{5}\]0  
where \(\Gamma(. )\) and \({}_{1}F_{1}(.;.;. )\) are Gamma and Kummer functions, respectively. The cross-spectral density of the partially coherent circular flattened Gaussian vortex beam propagating in turbulent biological tissues can be analytically expressed as:  
\[ \begin{split}& W_{jj}(\rho,\theta,\rho,\theta,z)=\frac{k^{2}I_{ l0}}{4z^{2}}\sum\limits_{n=1}^{N}\sum\limits_{n^{\prime}=1}^{N}\sum\limits_{l=- \infty}^{\infty}\sum\limits_{s=0}^{\infty}\frac{b_{n}b_{n^{\prime}}\Gamma^{2} (p_{1})(\alpha_{1}\alpha_{2})^{-p_{1}}\alpha_{3}^{2s+l-m}}{(l!)^{2s}\Gamma(s+ l-m)}\Big{(}\frac{k^{2}\rho^{2}}{4z^{2}}\Big{)}^{l}\\ &\quad\times\exp\Bigl{(}-\frac{k^{2}\rho^{2}}{4\alpha_{1}z^{2}}- \frac{k^{2}\rho^{2}}{4\alpha_{2}z^{2}}\Bigr{)}{}_{1}F_{1}\Big{(}p_{2};l+1; \frac{k^{2}\rho^{2}}{4\alpha_{1}z^{2}}\Bigr{)}{}_{1}F_{1}\Big{(}p_{2};l+1; \frac{k^{2}\rho^{2}}{4\alpha_{2}z^{2}}\Bigr{)},\end{split}  \tag{14}\]  
where the auxiliary parameters \(p_{1}\) and \(p_{2}\) are given by:  
\[p_{1}=l+s-0.5m+1, \tag{15}\]  
\[p_{2}=-s+0.5m. \tag{16}\]  
Apparently, \(W_{xy}(\rho,\rho,\theta,\theta,z)\) and \(W_{yx}(\rho,\rho,\theta,\theta,z)\) are equal to zero. The matrix of the cross-spectral density for the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues \(\overset{\leftrightarrow}{W}(\rho,\rho,\theta,\theta,z)\) is given by:  
\[\overset{\leftrightarrow}{W}(\rho,\rho,\theta,\theta,z)=\left[ \begin{array}[] {cc}W_{xx}(\rho,\rho,\theta,\theta,z)&W_{xy}(\rho,\rho,\theta,\theta,z)\\ W_{yx}(\rho,\rho,\theta,\theta,z)&W_{yy}(\rho,\rho,\theta,\theta,z)\end{array}  \right], \tag{17}\]  
The average intensity and the degree of polarization for the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues turn out to be [44]:  
\[I(\rho,\theta,z)=\mathrm{Tr}\overset{\leftrightarrow}{W}(\rho,\rho,\theta, \theta,z)=W_{xx}(\rho,\rho,\theta,\theta,z)+W_{yy}(\rho,\rho,\theta,\theta,z), \tag{18}\]  
\[P(\rho,\theta,z)=\left\{1-\frac{4\mathrm{det}\overset{\leftrightarrow}{W}( \rho,\rho,\theta,\theta,z)}{\left[\mathrm{Tr}\overset{\leftrightarrow}{W}( \rho,\rho,\theta,\theta,z)\right]^{2}}\right\}^{1/2}=\frac{\left|W_{xx}(\rho, \rho,\theta,\theta,z)-W_{yy}(\rho,\rho,\theta,\theta,z)\right|}{W_{xx}(\rho, \rho,\theta,\theta,z)+W_{yy}(\rho,\rho,\theta,\theta,z)}, \tag{19}\]  
where \(\mathrm{Tr}\) and \(\mathrm{det}\) mean the trace and the determinant, respectively. The degree of polarization of the partially coherent circular flattened Gaussian vortex beam in the initial plane yields:  
\[P^{(0)}=\frac{\left|I_{x0}-I_{y0}\right|}{I_{x0}+I_{y0}}. \tag{20}\]