## 3 Numerical Simulation and Results  
On the basis of the analytical expressions obtained in Section 2, the characteristics of the partially coherent circular flattened Gaussian vortex beam in turbulent biological tissues are simulated numerically. In this paper, numerical simulation was carried out using Mathematica 10.0 software; \(\lambda\) was set to be 0.6328 \(\upmu\)m in the following calculations. Since the average intensity of a partially coherent circular flattened Gaussian vortex beam propagating in turbulent biological tissues is determined by \(w_{0}\), \(N\), \(m\), \(\delta_{xx}\), \(\delta_{yy}\), and \(C_{n}{}^{2}\), it is necessary to understand the effects of these parameters. Figure 2 shows the normalized average intensity distribution of partially coherent circular flattened Gaussian vortex beam with different \(w_{0}\) values on the selected study plane in the turbulent biological tissue. The investigation planes were selected as \(z=0\), 3 \(\upmu\)m, 5 \(\upmu\)m, and 10 \(\upmu\)m, respectively. Other parameters in Figure 2 were set to \(N=2\), \(m=3\), \(\delta_{xx}=2\)\(\upmu\)m, \(\delta_{yy}=4\)\(\upmu\)m, \(P^{(0)}=0.5\), and \(C_{n}{}^{2}=10^{-4}\)\(\upmu\)m\({}^{-1}\). In the initial plane \(z=0\), the partially coherent circular flattened Gaussian vortex beam is a solid beam with the maximum normalized average intensity on the longitudinal axis. In the investigation plane of the turbulent biological tissue near the initial plane, the partially coherent circular flattened Gaussian vortex beam is a dark-hollow beam. When the axial propagating distance in the turbulent biological tissue augments, the valley value in the distribution of on-axis normalized average intensity gradually increases and eventually vanishes. When \(z=10\)\(\upmu\)m, the flat-topped average intensity distribution is reproduced. When the waist size of the fundamental Gaussian part \(w_{0}\) increases, the divergence of the partially coherent circular flattened Gaussian vortex beam in the turbulent biological tissue decreases. The effect of the parameter \(N\) on the distribution of normalized average intensity in the selected investigation plane of the turbulent biological tissue is demonstrated in Figure 3, where \(w_{0}=2\)\(\upmu\)m, and the other parameters are the same as those in Figure 2. With the increase of the parameter \(N\), the beam size of the partially coherent circular flattened Gaussian vortex beam in the initial plane becomes bigger, and the flat-topped length in the initial plane becomes longer. With the increase of the parameter \(N\), the valley value in the on-axis normalized average intensity also accelerates upward, while the divergence of the partially coherent circular flattened Gaussian vortex beam in the turbulent biological tissue decreases. This results in a beam size, in the case of \(N=3\), being smaller than in the case of \(N=2\) in the investigation plane \(z=5\)\(\upmu\)m. The flat-topped average intensity distribution is also reproduced in the investigation plane \(z=10\)\(\upmu\)m. In the investigation plane of the turbulent biological tissue, the effect of the topological charge \(m\) on the distribution of normalized average intensity is shown in Figure 4, where \(w_{0}=2\)\(\upmu\)m. The other parameters in Figure 4 are the same as in Figure 2. In the research plane of the turbulent biological tissue, the beam size, in the case of \(m=4\), is larger than that in the case of \(m=3\). When the topological charge \(m\) increases, the divergence of the partially coherent circular flattenedFigure 3: Effect of the parameter \(N\) on the normalized average intensity distribution in the selected investigation plane of the turbulent biological tissue. \(w_{0}=2\) μm, and the other parameters are the same as in Figure 2.  
Figure 2: Effect of the waist size of the fundamental Gaussian part on the normalized average intensity distribution in the selected investigation plane of the turbulent biological tissue.