## 4 Conclusions  
The propagating feature of a partially coherent circular flattened Gaussian vortex beam was investigated in a turbulent biological tissue. The analytical expression for the cross-spectral density of the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues was presented. As a result, the average intensity and the degree of polarization were obtained by employing  
Figure 11: Effect of the structural constant on the degree of polarization distribution in the selected investigation plane of the turbulent biological tissue. All the parameters are the same as in Figure 6.  
Figure 12: Effect of \(P^{(0)}\) on the normalized average intensity distribution (**a**) and the degree of polarization distribution (**b**) in the investigation plane \(z=3\) μm of the turbulent biological tissue. \(w_{0}=2\) μm, and the other parameters are the same as in Figure 1.  
the cross-spectral density matrix. By numerical simulation, the distributions of normalized average intensity and degree of polarization for a partially coherent circular flattened Gaussian vortex beam were demonstrated in a selected study plane of a turbulent biological tissue.  
The effects of the two beam parameters, the topological charge, the two transverse coherent lengths, and the structural constant on the distribution of normalized average intensity in the turbulent biological tissue were examined. When the waist size of the fundamental Gaussian part \(w_{0}\) or the parameter \(N\) increase, the divergence of the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues decreases. When the topological charge \(m\) or the structural constant \(C_{n}{}^{2}\) decrease, the divergence of the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues also decreases. When the selected investigation plane is sufficiently far away, the distribution of normalized average intensity is almost independent of the two transversal coherent lengths. Otherwise, the beam size increases with the increase of the two transverse coherent lengths.  
When the two transverse coherent lengths are not the same, the degree of polarization for a partially coherent circular flattened Gaussian vortex beam in a turbulent biological tissue is variable. Upon propagation in a turbulent biological tissue, the range of change for the degree of polarization narrows. With the increase of the waist size of the fundamental Gaussian part \(w_{0}\), the range of change for the degree of polarization also increases. The curve of the degree of polarization hardly varies with the parameter \(N\). When the topological charge \(m\) increases, the degree of polarization increases. As the structural constant \(C_{n}{}^{2}\) decreases, the variation range for the degree of polarization becomes wider.  
When propagating in a turbulent biological tissue, a partially coherent circular flattened Gaussian vortex beam has the advantage of rapidly reproducing the flat-topped average intensity distribution. The axial propagation distance required to reproduce the flat-topped average intensity distribution is very small (it was only 0.1 mm in our calculation). When the flat-topped average intensity is reproduced, the degree of polarization distribution is stable, and the variation range of the degree of polarization is also small. The results obtained here are of great significance for the potential application of a partially coherent circular flattened Gaussian vortex beam in medical imaging and medical diagnosis.  
[PERSON] and [PERSON] conceived of and wrote the paper; [PERSON] and [PERSON] performed the numerical simulations; [PERSON] and [PERSON] analyzed and evaluated the results.  
This research was supported by the National Natural Science Foundation of China under Grant No. 11574272 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY16A040014.  
The authors declare no conflict of interest.