# Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence
## 3 Analysis of Numerical Simulation Results
### Probability Density Distribution of Light Intensity  
The probability of distribution of light intensity fluctuations provides a fundamental statistical description of this random process. Oceanic atmospheric turbulence is typically weak to moderate [39], under which condition, the probability density distribution of log-intensity fluctuations for the received laser spot generally follows a normal distribution. Skewness \(\gamma_{1}\) and kurtosis \(\gamma_{2}\) are used to assess how the sample distribution deviates from a Gaussian distribution with the same mean and variance [29], calculated as follows:  
\[\gamma_{1}=\mu_{3}/\mu_{2}^{3/2},\gamma_{2}=\mu_{4}/\mu_{2}^{2}-3 \tag{29}\]  
where \(\mu_{3}\) is the third moment of the logarithmic intensity probability density distribution, and \(\mu_{4}\) is the fourth moment. Skewness reflects the asymmetry of the probability density distribution relative to the mean. If the probability distribution is symmetric about the mean (such as in a normal distribution), the skewness is zero; otherwise, the skewness is non-zero. Kurtosis reflects the concentration of the probability distribution relative to a normal distribution. If kurtosis is positive, the probability distribution is more concentrated than the normal distribution; otherwise, it is more dispersed. Here, we set the weight coefficient to be \(a_{i}\) = 0.00, 0.06, 0.16, 0.26, 0.36, 0.40, and 0.45, \(C_{n}^{2}=7.45\times 10^{-16}m^{-2/3}\), and \(r\) = 0 mm, and all the other parameters involved were the same as those in Figure 6. Figure 7a-d show the probability density distribution of the log-intensity of the spot on the target after the Gaussian beam has transmitted over 1.1 km under various aberration conditions. To provide a deeper analysis of the target intensity distribution, Figure 6e-h show the corresponding mean (Mean), standard deviation of log-intensity fluctuation (Std), skewness (S), and kurtosis (K). Figure 7: The variations in the mean, standard deviation, skewness, and kurtosis of the intensity probability density distribution with respect to \(a_{i}\) for no aberration and four typical aberrations (Z\({}_{4}\), \(Z_{7}\), \(Z_{11}\), \(Z_{25}\)) at \(C_{n}^{2}=7.45\times 10^{-16}m^{-2/3}\), respectively. (**a–d**) Intensity probability density distributions; (**e**) mean; (**f**) standard deviation; (**g**) skewness; (**h**) kurtosis. As shown in Figure 7, the probability density distribution curves for each aberration exhibit different trends as the weight coefficient varies: (1) As the weight coefficient increases, the peak of the log-intensity distribution for \(Z_{4}\) aberration gradually rises. Moreover, at lower weight coefficients, the absolute values of skewness and kurtosis are relatively small, and the probability density distribution curves approach a normal distribution. (2) As the weight coefficient of the aberration increases, the log-intensity probability density curve for \(Z_{7}\) becomes more concentrated around the mean, exhibits better symmetry, and increasingly follows a normal distribution. (3) As the weight coefficient increases, the log-intensity probability density distribution curve for \(Z_{11}\) aberration shifts to the right overall. Furthermore, as the weight coefficient grows higher, the deviation of the log-intensity probability density distribution curve from a normal distribution gradually increases as well. (4) For \(Z_{25}\) aberration, when the weight exceeds 0.26, the logarithmic intensity probability density distribution curve gradually deviates from a normal distribution and shifts to the right overall. The peak logarithmic intensity gradually increases, and the concentration of energy continues to rise. Additionally, from Figure 7, we can also conclude that when \(C_{n}^{2}=7.45\times 10^{-16}m^{-2/3}\), the probability density distribution of the on-axis log-intensity for laser beams with different aberrations approximately follows a log-normal distribution.