# Effect of Optical Aberrations on Laser Transmission Performance in Maritime Atmosphere Turbulence
## 2 Numerical Simulation Method
### Numerical Simulation Method for Laser Atmospheric Transmission  
The numerical simulation of light propagation in random media begins with the propagation equation of light, which is described by [PERSON]'s wave equation [29]. The optical field \(E\) of a laser transmitting through atmospheric turbulence satisfies the following [PERSON]'s wave equation:  
\[\
abla^{2}E+k^{2}n^{2}E+2\
abla(E\cdot\
abla\ln(1+n_{1}))=0 \tag{1}\]  
where \(\
abla^{2}\) is the Laplacian operator, \(n\) is the atmospheric refractive index, \(k=2\pi/\lambda\) is the wave number, and \(\lambda\) is the wavelength of the incident light. For light propagation in the atmosphere, \(\lambda\ll l_{0}\) the polarization term on the left side of the equation is disregarded. It is assumed that only small-angle forward scattering occurs, and no backward scattering is present. Under the paraxial approximation along the \(z\)-axis, the optical field can be  
Figure 1: Algorithm simulation flowchart. expressed as \(E=ue^{ikz}\), and the propagation equation of the light wave in a medium with refractive index \(n=1+n_{1}\) follows the standard parabolic equation, as shown below:  
\[\frac{\partial u}{\partial z}=\frac{i}{2k}\
abla_{\perp}^{2}u+ikn_{1}u \tag{2}\]  
where \(n_{1}\) is the refractive index fluctuation, a small quantity. The standard parabolic equation forms the foundational basis for the numerical simulation and theoretical investigation of light propagation in random media. As shown in Figure 2, The multi-layer phase screen numerical simulation method models a turbulent atmosphere as a series of very thin phase screens. When the phase variations induced by fluctuations in atmospheric refractive index are sufficiently small, the processes of vacuum transmission and atmospheric phase modulation can be treated as independent. This method divides the atmospheric medium into a sequence of parallel plates, each with a thickness of \(\Delta z\). The optical field incident on each plate propagates to its rear surface according to the vacuum solution of the generalized parabolic equation, where it experiences phase modulation induced by the plate. The field then propagates through a vacuum and undergoes phase modulation again at the subsequent plate, thereby sequentially constructing the final optical field. During the propagation process, we analyze the optical field traveling from the plane at \(z=z_{i-1}\) through a parallel plate with thickness \(\Delta z\) to the plane at \(z_{i}=z_{i-1}+\Delta z\). This process can be expressed as a combination of the results from vacuum propagation over a distance \(\Delta z\) and the phase modulation induced by the phase screen. The phase modulation is as follows [29]:  
\[S(r,z_{i})=k\int_{Z_{i-1}}^{Z_{i}}n_{1}(r,z)dz \tag{3}\]  
i.e.,  
\[u(r,z_{i})\approx\exp\left[\frac{i}{2k}\int_{Z_{i-1}}^{Z_{i}}\
abla_{\perp}^{2 }dz\right]\exp\left[iS(r,z_{i})\right]u(r,z_{i-1}) \tag{4}\]  
Due to the randomness of the magnitude of the phase modulation \(S\), it is impossible to obtain an analytical solution for the above equation; numerical methods must be used. Common methods include the finite difference method, and the Fourier transform method, with this study employing the latter. Taking the Fourier transform of the above equation yields,  
\[F[u(r,z_{i})]=\exp\left[-\frac{i\Delta z}{2k}\left(K_{x}^{2}+K_{y}^{2}\right) \right]\times F\left[e^{is(r,z_{i})}u(r,z_{i-1})\right] \tag{5}\]  
Figure 2: Simulation model for laser transmission accounting for optical aberrations in marine atmospheric turbulence. where \(K_{x}^{2}\) and \(K_{y}^{2}\) is the spatial wave numbers. Performing the inverse Fourier transform yields the final transmission equation.