## 1 Introduction
The intensity in the initial plane is given by \(W_{xx}(\rho_{0},\,\rho_{0},\,\theta_{0},\,\theta_{0},\,0)+W_{yy}(\rho_{0},\, \rho_{0},\,\theta_{0},\,\theta_{0},\,0)\). Therefore, the intensity in the initial plane is determined by the parameters \(N\) and \(w_{0}\), as shown in Figure 1. \(w_{0}=2\)\(\upmu\)m in Figure 1a and \(N=2\) in Figure 1b. Though the flatness of the partially coherent circular flattened Gaussian vortex beam depends on the two parameters \(N\) and \(w_{0}\), the role of the parameter \(N\) is more obvious than that of the parameter \(w_{0}\). By using the extended Huygens-Fresnel integral formula, the cross-spectral density of the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues can be expressed as [41; 45]:  
\[ \begin{array}{c}W_{jj}(\rho,\theta,\rho,\theta,z)=\frac{k^{2}}{4\pi^{2}z^{2}} \int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{2\pi}W_{jj}(\rho_{01 },\rho_{02},\theta_{01},\theta_{02},0)<\exp[\psi(\rho_{01},\theta_{01},\rho, \theta)+\psi^{*}(\rho_{02},\theta_{02},\rho,\theta)]>\\ \times\exp\left\{\frac{k}{2}[\rho_{02}^{2}-\rho_{01}^{2}-2\rho_{02}\rho\cos( \theta_{02}-\theta)+2\rho_{01}\rho\cos(\theta_{01}-\theta)]\right\}\rho_{01} \rho_{02}d\rho_{01}d\theta_{02}d\theta_{01}d\theta_{02},\end{array}  \tag{3}\]  
with \(<\exp[\psi(\rho_{01},\theta_{01},\rho,\theta)+\psi^{*}(\rho_{02},\theta_{02}, \rho,\theta)]>\) being given by  
\[<\exp[\psi(\rho_{01},\theta_{01},\rho,\theta)+\psi^{*}(\rho_{02},\theta_{02}, \rho,\theta)]>=\exp\left[-\frac{\rho_{01}^{2}+\rho_{02}^{2}-2\rho_{01}\rho_{0 2}\cos(\theta_{01}-\theta_{02})}{\sigma_{0}^{2}}\right], \tag{4}\]  
where \(k=2\pi/\lambda\). \(\lambda\) is the wavelength of the incident beam. The asterisk denotes the complex conjugation, \(|\sigma_{0}|=0.22(C_{n}^{2}k^{2}z)^{-1/2}\), the parameter \(C_{n}^{2}\) is the structural constant of the refraction index in the biological tissue. The spherical wave lateral coherence length in turbulent atmosphere is characterized by \(\sigma_{0}=(0.545C_{n}^{2}k^{2}z)^{-3/5}\). Moreover, the order of magnitude of \(C_{n}^{2}\) in turbulent biological tissues is generally 1-10\({}^{2}\) m\({}^{-1}\), while the order of magnitude of \(C_{n}^{2}\) in turbulent atmosphere is generally 10\({}^{-15}\)-10\({}^{-13}\) m\({}^{-2/3}\). In order to perform the integral in Equation (3), the following mathematical formulae are used [46]  
\[\exp\left[\frac{ik\rho\rho_{01}}{z}\cos(\theta-\theta_{01})\right]=\sum_{l=- \infty}^{\infty}i^{l}I_{l}\left(\frac{k\rho\rho_{01}}{z}\right)\exp[il(\theta- \theta_{01})], \tag{5}\]  
\[\int_{0}^{2\pi}\exp[-im\theta_{01}+2c\rho_{01}\rho_{02}\cos(\theta_{01}- \theta_{02})]d\theta_{01}=2\pi\exp(-im\theta_{02})I_{m}(2c\rho_{01}\rho_{02}), \tag{6}\]  
\[J_{l}(x)=\frac{i^{l}}{2\pi}\int_{0}^{2\pi}\exp(-ix\cos\varphi-il\varphi)d\varphi, \tag{7}\]  
where \(J_{l}(\cdot)\) and \(I_{m}(\cdot)\) are \(l\)th-order Bessel and \(m\)th-order modified Bessel functions of the first kind, respectively. The cross-spectral density of the partially coherent circular flattened Gaussian vortex beam propagating in turbulent biological tissues can be rewritten as:  
\[W_{jj}(\rho,\theta,\rho,\theta,z)=\frac{k^{2}I_{l0}}{z^{2}}\int_{0}^{\infty} \int_{0}^{\infty}\sum_{n=1}^{N}\sum_{\
u=1}^{N}\sum_{-\infty}^{\infty}b_{n}b_{ \
u}J_{l}\left(\frac{k\rho\rho_{01}}{z}\right)_{l}\left(\frac{k\rho\rho_{02}}{ z}\right)\exp(-\alpha_{1}\rho_{01}^{2}-\alpha_{2}\rho_{02}^{2})I_{l-m}(2\alpha_{ 3}\rho_{01}\rho_{02})\rho_{01}\rho_{02}d\rho_{01}d\rho_{02}, \tag{8}\]  
with the auxiliary parameters \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\) being defined by:  
\[\alpha_{1}=\frac{n}{w_{0}^{2}}+\frac{1}{\sigma_{0}^{2}}+\frac{1}{\delta_{jj}^ {2}}+\frac{ik}{2z}, \tag{9}\]  
Figure 1: Distribution of normalized intensity in the initial plane \(z=0\). **(a)**\(w_{0}=2\) μm, **(b)**\(N=2\).