## 2 Atmospheric Laser Communication Link Model  
Due to the rapid motion of airborne platforms, the compression of the air near the optical turret results in a non-uniform density distribution, causing the significant distortion of the light beam as it passes through the flow field. The assessment of this distortion typically employs optical path difference as a criterion [19]. In the context of short distances, the Navier-Stokes equations indicate that the motion of an airborne optical turret in the atmosphere induces changes in the air density [20]. The velocity influences the gradient distribution of the air density, while the flight altitude dictates the environmental temperature, static pressure, and static air density. Upon selecting the flight altitude and velocity, steady-state solutions can be computed for the flow field to determine the gradient distribution of the air density. The aircraft is assumed to be in constant motion with velocity \(v\) at altitude \(h\). We build a fluid computing space around the gondolas, with \((x,y,z)\) representing the spatial position. The relationship between air density and refractive index can be established through the Gladstone-Dale relationship [19; 21], as follows:  
\[n(h,v,x,y,z)=1+K_{\rm GD}\rho(h,v,x,y,z), \tag{1}\]  
where \(\rho(h,v,x,y,z)\) represents the air density function at the position \((x,y,z)\) within the computational domain under the current altitude \(h\) and flight velocity \(v\) conditions; \(n(h,v,x,y,z)\) denotes the refractive index at that point; and \(K_{\rm GD}\) represents the wavelength-dependent Gladstone-Dale constant, expressed as follows:  
\[K_{\rm GD}(\lambda)=2.23\times 10^{-4}(1+7.52\times\frac{10^{-3}}{\lambda^{2}}). \tag{2}\]  
In geometrical optics theory, the optical path length (OPL), \(L_{\rm OPL}\), is defined as the integral of the refractive index along the propagation path of a light ray. Given the relatively uniform density distribution in the upstream direction within the flow field, optical path lengths are computed separately for different emission directions, \(\theta\), to distinguish the beam distortion caused by different regions of the flow field. Equation (1) provides the variation in the optical path length for the emitted beam at a propagation distance \(L\), as follows:  
\[OPL(h,v,x,y)=\int\limits_{0}^{L}n(h,v,x,y,z)dz, \tag{3}\]  
The optical path difference (OPD) is defined as the difference in the optical path length experienced by different rays as they propagate through an inhomogeneous and dynamic flow field, as follows:  
\[OPD(h,v,x,y)=OPL(h,v,x,y)-\left\langle OPL(h,v,x,y)\right\rangle, \tag{4}\]  
In Equation (4), \(\left\langle OPL(h,v,x,y)\right\rangle\) represents the spatial average of \(OPL(h,v,x,y)\) within a specific range. The phase difference can be characterized using the optical path difference, expressed as follows:  
\[\varphi(h,v,x,y)=\sum\frac{2\pi}{\lambda}(OPD(h,v,x,y)), \tag{5}\]Thus, the light field distribution after the optical beam with an amplitude of \(A(x,y)\) is affected by aerodynamic optical effects, which can be represented as follows:  
\[U(h,v,x,y)=A(x,y)\exp[-\mathrm{i}\varphi(h,v,x,y)], \tag{6}\]  
Assuming the receiving aperture area is \(D_{\mathrm{R}}\), the basic model of the communication system under an intensity modulation-direct detection (IM-DD) scheme can be represented as follows:  
\[i_{\mathrm{RX}}=f(h)RP_{\mathrm{RX}}+N=f(h)RUI(h,v,x,y)^{2}D_{\mathrm{R}}+N, \tag{7}\]  
Equation (7) provides the following parameters: \(i_{\mathrm{RX}}\) represents the received photocurrent, \(R\) denotes the detector sensitivity, \(P_{\mathrm{RX}}\) signifies the received optical power, and \(N\) describes the zero-mean additive Gaussian white noise with variance \(\sigma_{N}^{2}\).