# Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence
## 2 Numerical Simulation Method
### Construction of Turbulent Phase Screens  
To obtain reasonable simulation results from the aforementioned methods, a core issue that needs to be addressed is how to introduce a random phase \(\varphi\) that satisfies the characteristics of atmospheric turbulence in each iteration process. To tackle this problem, a commonly used method for constructing phase screens is based on Fourier transformation. The fundamental principle involves generating a complex spatial random field using the refractive index spectrum of turbulence and a complex Gaussian random number matrix. Then, an inverse Fourier transformation is performed to obtain the two-dimensional spatial distribution of phases. This process can be represented as follows [31]:  
\[\varphi_{H}(x,y)=\sum_{mm}\sum_{m}C_{mm,m}\exp\left[i2\pi\left(f_{x_{m}}x+f_{ y_{mm}}y\right)\right] \tag{11}\]  
where \(C_{mm,m}=h\left(f_{x_{mm}},f_{y_{mm}}\right)\sqrt{2\pi k^{2}\Delta z\Phi_{n}( \kappa)}\Delta f_{x}\Delta f_{y}\), \(f_{x_{mm}}\) and \(f_{y_{mm}}\) represent the discrete spatial frequencies in the x-direction and y-direction, respectively; \(\Delta f_{x}=1/L_{x}\), and \(\Delta f_{y}=1/L_{y}\), where \(L_{x}\) and \(L_{y}\) represent the grid sizes in the x-direction and y-direction; \(\Delta z\) is the spacing between phase screens along the \(z\)-axis; and \(\Phi_{n}(\kappa)\) is the power spectral density function of the atmospheric refractive index fluctuation.  
In the near-sea surface environment, the maritime atmospheric turbulence power spectrum can be represented as follows [32]:  
\[\Phi_{n}(\kappa)=0.033C_{n}^{2}\Big{[}1-0.061\kappa/\kappa_{H}+2.836(\kappa/\kappa _{H})^{7/6}\Big{]}\frac{\exp\big{(}-\kappa^{2}/\kappa_{H}^{2}\big{)}}{\big{(} \kappa^{2}+\kappa_{0}^{2}\big{)}^{11/6}} \tag{12}\]  
where \(\kappa\) indicates the spatial frequency magnitude, and \(l_{0}\) and \(L_{0}\) are the outer and inner scales of turbulence, respectively. \(\kappa_{0}=2\pi/L_{0}\) is the outer scale wave number parameter, while \(\kappa_{H}=3.41/l_{0}\) is the inner scale wave number parameter. \(C_{n}^{2}\) is the atmospheric refractive index structure constant, also known as optical turbulence strengths.  
It is worth noting that the phase screens generated by the Fourier transform method clearly exhibit a deficiency in low-frequency components. In this study, a higher-order harmonic compensation technique is employed to enhance the low-frequency components. In this study, a low-frequency sub-harmonic compensation technique has been employed to enhance the low-frequency components of the phase screens. Setting the specific harmonic order \(N_{p}=3\), the new phase screen \(\varphi(x,y)\) can be expressed as follows [33]:  
\[\begin{array}{ll}\varphi(x,y)&=\varphi_{H}(x,y)+\varphi_{L}(x,y)\\ &=\sum\limits_{mm=-\infty}^{\infty}\sum\limits_{m=-\infty}^{\infty}C_{mm,m} \exp\big{[}i2\pi\big{(}f_{x_{mm}}x+f_{y_{mm}}y\big{)}\big{]}+\sum\limits_{p=1} ^{N_{p}}\sum\limits_{mm=-1}^{1}\sum\limits_{nm=-1}^{1}C_{mm,nn}\exp\big{[}i2 \pi\big{(}f_{x_{mm}}x+f_{y_{mm}}y\big{)}\big{]}\end{array} \tag{13}\]  
where \(\varphi_{L}(x,y)\) represents the low-frequency compensation component of the phase screen, \(P\) denotes the sub-harmonic order, and \(\Delta f_{p}=1/(3^{p}L)\) is the frequency interval between the sub-harmonics.