## 1 Introduction  
Since the intensity distribution of a flat-topped beam is almost uniform, the flat-topped beam has the potential for application in some special fields. So far, research on the propagation factor, kurtosis parameter, focal shift, fractional Fourier transform, spectral change, and laser coupling of the flat-topped beam has been reported [1; 2; 3; 4; 5; 6; 7]. The properties of a flat-topped beam passing through a misaligned optical system, an apertured optical system, and an unapertured optical system have been extensively studied [8; 9; 10; 11; 12; 13; 14]. The average intensity, scintillation index, and Stokes parameters of the flat-topped beam have also been demonstrated in the turbulent atmosphere and ocean [15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. It was found that an efficient optical system could be proposed to generate flat-topped beams [25], while complex amplitude modulation only requires a phase spatial light modulator [26]. Many different mathematical models have been proposed to describe the properties of the flat-topped beam [27; 28; 29; 30], with one of the successful models being the circular flattened Gaussian beam [31; 32]. The advantage of this beam is that its intensity distribution varies continuously from the maximum value to zero. The characteristics of circular flattened Gaussian beams passing through an apertured and unapertured optical system in the turbulent atmosphere have been investigated [33; 34]. Under the non-approximate condition, the vector structure properties of a circular flattened Gaussian beam have been revealed [35]. Partially Coherent Circular Flattened Gaussian Vortex Beams Propagating in a Turbulent Biological Tissue  
Because of the symmetry of the studied beam, a cylindrical coordinates system is an optimal choice. The longitudinal axis namely \(z\)-axis is the propagating axis; \(z=0\) is the initial plane, and \((\rho_{0},\,\theta_{0})\) are the transverse coordinates in the initial plane. The cross-spectral density matrix is used to describe the second-order coherence and the polarized characteristics of partially coherent optical beams. The 2 \(\times\) 2 matrix of cross-spectral density for a partially coherent circular flattened Gaussian vortex beam in the initial plane \(z=0\) is described by [44]  
\[\overset{\leftrightarrow}{W}(\rho_{01},\rho_{02},\theta_{01},\theta_{02},0)= \left[ \begin{array}{ll}W_{xx}(\rho_{01},\rho_{02},\theta_{01},\theta_{02},0) &W_{xy}(\rho_{01},\rho_{02},\theta_{01},\theta_{02},0)\\ W_{yx}(\rho_{01},\rho_{02},\theta_{01},\theta_{02},0)&W_{yy}(\rho_{01},\rho_{02 },\theta_{01},\theta_{02},0)\end{array} \right], \tag{1}\]  
with \(W_{xx}(\rho_{01},\rho_{02},\theta_{01},\theta_{02},0)\) and \(W_{yy}(\rho_{01},\rho_{02},\theta_{01},\theta_{02},0)\) being given by  
\[ \begin{split} W_{jj}(\rho_{01},\rho_{02},\theta_{01},\theta_{02}, 0)=&\,I_{0}\sum_{n=1}^{N}b_{n}\exp\left(-\frac{n\rho_{01}^{2}}{w_{0}^{2 }}\right)\sum_{n^{\prime}=1}^{N}b_{n^{\prime}}\exp\left(-\frac{n^{\prime}\rho _{02}^{2}}{w_{0}^{2}}\right)\exp[im(\theta_{01}-\theta_{02})]\\ &\times\exp\left[-\frac{\rho_{01}^{2}+\rho_{02}^{2}-2\rho_{01}\rho_{02 }\cos(\theta_{01}-\theta_{02})}{\delta_{jj}^{2}}\right]\end{split}  \tag{2}\]  
where \(j=x\) or \(y\). \((x,y)\) are the transverse coordinates of the Cartesian coordinates system, \(I_{j0}=\left|E_{j0}\right|^{2}\) and \(E_{j0}\) are the characteristic amplitude, \(b_{n}=(-1)^{n-1}N!/(Nn!(N-n)! )\) is the weight coefficient, \(N\) is an integer and is assumed to be greater than or equal to 2, \(w_{0}/\sqrt{n}\) is the Gaussian waist, \(m\) is the topological charge, \(\delta_{jj}\) is the transverse coherent length in the \(j\)-direction. A Schell model source is employed in Equation (2). Here, the simplest case is considered, namely, \(W_{xy}(\rho_{01},\,\rho_{02},\,\theta_{01},\,\theta_{02},\,0)=W_{yx}(\rho_{01}, \,\rho_{02},\,\theta_{01},\,\theta_{02},\,0)=0\), which means that the two orthogonal components are irrelevant in the initial plane.