# Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence
## 3 Analysis of Numerical Simulation Results
### Beam Wandering Effect  
The laser transmission through an oceanic turbulent atmosphere also induces a spot wander effect. In general, we use the variation of beam centroid position to describe the beam wandering effect, and the beam centroid can be expressed as follows [29]:  
\[\boldsymbol{\rho}_{c}=\frac{\iint\boldsymbol{\rho}I(\boldsymbol{\rho})\;\mathrm{ d}\boldsymbol{\rho}}{\iint I(\boldsymbol{\rho})\;\mathrm{d}\boldsymbol{\rho}} \tag{23}\]  
where \(\boldsymbol{\rho}\) is the transverse vector, and \(I\) is the light intensity. The beam centroid in the horizontal and vertical directions is as follows:  
\[x_{c}=\frac{\iint xI(x,y)\mathrm{d}x\mathrm{d}y}{\iint I(x,y)\mathrm{d}x \mathrm{d}y} \tag{24}\]  
\[y_{c}=\frac{\iint yI(x,y)\mathrm{d}x\mathrm{d}y}{\iint I(x,y)\mathrm{d}x \mathrm{d}y} \tag{25}\]  
Figure 4: The variation of the scintillation index \(\beta_{1}^{2}(r)\) with \(C_{n}^{2}\) and \(r\) for no aberration and four typical aberrations (\(Z_{4}\), \(Z_{7}\), \(Z_{11}\), \(Z_{25}\)) at different \(a_{i}\), respectively. (**a**) \(Z_{4}\); (**b**) \(Z_{7}\); (**c**) \(Z_{11}\); (**d**) \(Z_{25}\). Then, we can obtain the root-mean-square (RMS) value of the beam centroid wandering as follows:  
\[\sigma_{p}=\sqrt{\langle\mathbf{\rho}_{\varepsilon}^{2}\rangle}=\left[\iint\iint \left(\mathbf{\rho}_{1}\cdot\mathbf{\rho}_{2}\right)I(\mathbf{\rho}_{1})I(\mathbf{\rho}_{2}) \mathrm{d}\mathbf{\rho}_{1}\mathrm{d}\mathbf{\rho}_{2}\right]^{1/2}/\iint I(\mathbf{\rho}) \mathrm{d}\mathbf{\rho} \tag{26}\]  
Here, we set the weight coefficient to be \(a_{i}=0.00\), \(0.06\), \(0.26\), and \(0.45\), respectively, Figure 5 illustrates the variation of the RMS of beam wandering for four typical single aberration conditions as the turbulence strength changes from \(2.5\times 10^{-16}m^{-2/3}\) to \(7.5\times 10^{-14}m^{-2/3}\). From Figure 5, it can be observed that as the turbulence intensity increases, the centroid drift phenomenon of the beam spot becomes more pronounced, and different types of aberrations have varying effects on beam wandering. (1) In weak turbulence, the RMS of beam wandering of \(Z_{4}\) aberration decreases as the weight coefficient increases. When the turbulence strength grows to a certain value, the RMS of beam wandering begins to increase with the higher weight coefficient of \(Z_{4}\) aberration. Compared to the no aberration state, the RMS of beam wandering of \(Z_{4}\) aberration is always larger, particularly under conditions of high weight coefficients and strong turbulence, with the maximum difference reaching \(0.15\) cm. (2) In weak turbulence, the variation of the weight coefficient of \(Z_{7}\) aberration has  
Figure 5: The variation of the RMS of beam wandering with \(C_{n}^{2}\) for no aberration and four typical aberrations (\(Z_{4}\), \(Z_{7}\), \(Z_{11}\), \(Z_{25}\)) at different \(a_{i}\), respectively. (**a**) \(Z_{4}\); (**b**) \(Z_{7}\); (**c**) \(Z_{11}\); (**d**) \(Z_{25}\). a negligible impact on the beam wandering effect. As the turbulence strength increases, the fluctuation range of the RMS value of the beam wandering becomes more significantly affected by the weight coefficient. When the turbulence strength \(C_{n}^{2}=7.45\times 10^{-16}m^{-2/3}\), the fluctuation range of the RMS of the beam wandering for \(Z_{7}\) aberration is 0.01 cm, while when the turbulence strength increases to \(7.5\times 10^{-14}m^{-2/3}\), the RMS fluctuation range of the beam wandering for \(Z_{7}\) aberration grows up to 0.17 cm. Additionally, compared to the no-aberration state, the RMS of beam wandering for \(Z_{7}\) aberration is always larger at any turbulence strength, with the maximum difference being 0.18 cm. (3) Compared to the no-aberration state, the RMS of the centroid drift for \(Z_{11}\) aberration is consistently larger, and the difference increases as the turbulence intensity and the aberration weight coefficient become larger, with the maximum difference reaching 0.17 cm.