# Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence
## 3 Analysis of Numerical Simulation Results
### Intensity Scintillation Effect  
In laser atmospheric transmission, the normalized intensity fluctuation variance \(\beta_{I}^{2}\), also known as the intensity scintillation index, is commonly used to quantitatively describe the magnitude of intensity fluctuations. The on-axis scintillation index can be expressed as follows [37]:  
\[\beta_{I}^{2}=\frac{\left\langle I^{2}\right\rangle-\left\langle I\right\rangle ^{2}}{\left\langle I\right\rangle^{2}} \tag{21}\]  
In this formula, \(I\) is the light intensity on the axis at the receiver, and \(<\cdot>\) denotes the ensemble average. In order to more clearly illustrate the differences between the on-axis scintillation index of each aberration and the aberration-free state, we introduce the relative variation rate \(R_{a_{i}}\), where \(R_{a_{i}}=\left(\beta_{I,a_{i}}^{2}-\beta_{I,a_{0}}^{2}\right)/\beta_{I,a_{0}}^{2}\) and \(\beta_{I,a_{i}}^{2}\) is the scintillation index for the weight coefficient \(a_{i}\). Here, we set the weight coefficient to be \(a_{i}=0.00,0.06\) and \(0.26\), respectively. Figure 3 illustrates the variation of \(\beta_{I,m}^{2}\) and \(R_{a_{i}}\) for four typical single aberration conditions, as the turbulence strength changes from \(2.5\times 10^{-16}m^{-2/3}\) to \(10^{-13}m^{-2/3}\). From Figure 3, it can be observed that (1) in weak turbulence, compared to the no-aberration state, \(Z_{4}\) aberration shows a suppressing effect on the on-axis scintillation index, especially at higher weight coefficients. When the turbulence strength grows up to a certain value, \(Z_{4}\) aberration begins to have a promoting effect on the on-axis scintillation index, and the higher the weight coefficient, the more significant the promoting effect. Additionally, the maximum absolute value of the relative change rate of the scintillation index due to \(Z_{4}\) aberration \(|R_{a_{4}}|\) is 6%. (2) Compared to the no-aberration state, \(Z_{7}\) aberration always has a promoting effect on the on-axis scintillation index, especially at stronger turbulence or higher weight coefficients. The maximum absolute value of the relative change rate of the scintillation index due to \(Z_{7}\) aberration \(|R_{a_{7}}|\) is 9.13%. (3) In weak turbulence, \(Z_{11}\) aberration always has a promoting effect on the on-axis scintillation index, and the higher the aberration weight coefficient, the larger the relative change rate of the on-axis scintillation index. After the turbulence strength increases to a certain value, \(Z_{11}\) aberration begins to have a suppressing effect on the on-axis scintillation index, and the larger the  
Figure 3: The variation of \(\beta_{L,m}^{2}\) and \(R_{a}\) with \(a_{i}\) for no aberration and four typical aberrations (\(Z_{4}\), \(Z_{7}\), \(Z_{11}\), \(Z_{25}\)) under different \(C_{n}^{2}\). (**a**) \(Z_{4}\); (**b**) \(Z_{7}\); (**c**) \(Z_{11}\); (**d**) \(Z_{25}\). aberration weight coefficient, the smaller the relative change rate of the on-axis scintillation index. The maximum value of \(|R_{a_{11}}|\) in this case is 5.8%. (4) \(Z_{25}\) aberration always has a suppressing effect on the on-axis scintillation index, especially at higher weight coefficients. The maximum value of \(|R_{a_{25}}|\) in this case is 5%. Thus, we can infer that compared to the no-aberration state, the range of fluctuations in the relative change rate of the scintillation index for each aberration satisfies \(Z_{7}>Z_{4}>Z_{11}>Z_{25}\). In practical applications, it is necessary to consider situations with a certain receiving area, as laser beams typically have a defined area. The aperture scintillation index for a circular region with a specific receiving radius is given by [38]:  
\[\beta_{I}^{2}(r)=\frac{\left\langle I(r)^{2}\right\rangle-\left\langle I(r) \right\rangle^{2}}{\left\langle I(r)\right\rangle^{2}} \tag{22}\]  
where \(I(r)\) represents the total received light intensity over a circular region with a radius of \(r\).