## 2 Airborne Air-to-Air Channel Laser Communication Link Analysis and Modeling
\[a_{n}=\frac{\left[A_{n}(mx)/m+n/x\right]\text{Re}[\zeta_{n}(x)]-\text{Re}[ \zeta_{n-1}(x)]}{\left[A_{n}(mx)/m+n/x\right]\zeta_{n}(x)-\zeta_{n-1}(x)} \tag{9}\]  
\[b_{n}=\frac{\left[mA_{n}(mx)/m+n/x\right]\text{Re}[\zeta_{n}(x)]-\text{Re}[ \zeta_{n-1}(x)]}{\left[mA_{n}(mx)/m+n/x\right]\zeta_{n}(x)-\zeta_{n-1}(x)} \tag{10}\]where \(m\) is the complex refractive index of the scattering element; \(x\) is the scattering element scale factor, \(x=2\pi r/\lambda\); \(r\) is the radius of the scattering element; and \(n\) is the order of calculation of the Mie scattering coefficient, which is generally taken as  
\[n=\max\Bigl{(}x+4x^{1/3}+2,|mx|\Bigr{)}+15 \tag{11}\]  
The scattering element absorption coefficient calculation model is given below as follows [16]:  
\[Q_{ext}=\frac{2}{x^{2}}\sum_{n=1}^{\infty}(2n+1)\text{Re}(a_{n}+b_{n}) \tag{12}\]  
The modeling analysis of the pulse-broadening phenomenon resulting from the interaction between the photons transmitted via the airborne air-to-air channel and the turbulence is shown in Figure 3.  
In Figure 3, the mean free path between turbulent bubbles, denoted as \(l_{p}\), follows a Gaussian distribution with a mean of 1 m and a standard deviation of 0.9 m. Let \(\varepsilon\) be a uniformly distributed random number in the range [0, 1]. The angle of incidence of a photon is \(\theta_{1}\) in rad and the angle of refraction of the photon is \(\theta_{2}\) in rad. The direction vector of the outgoing photon is \(\mathbf{S_{i2}}\), the refractive index of the turbulent bulb is \(n_{2}\), the reflectivity is \(R_{T}\), and the refractive index is \(T_{R}\) (where \(r_{s}\) and \(r_{p}\) are the reflection coefficients of the \(s\) component and the \(p\) component). 1. When \(\varepsilon\leq R_{T}\), reflection (including total reflection) occurs between the transmitted photon and the turbulent bubble From [PERSON]'s equation, it follows that [17] \[T_{R}=1-R_{T},\ R_{T}=(r_{s}^{2}+r_{p}^{2})/2\] (13) Combined with [PERSON]'s theorem, this gives [18] \[r_{s}=-\frac{\sin(\theta_{1}-\theta_{2})}{\sin(\theta_{1}+\theta_{2})};\ r_{p}=\frac{\tan(\theta_{1}-\theta_{2})}{ \tan(\theta_{1}+\theta_{2})}\] (14)  
When \(n_{1}>n_{2}\) and \(\sin\theta_{1}>n_{2}/n_{1}\), the transmitted photon undergoes total reflection and the unit vector in the direction of the outgoing photon can be described by Equation (15). In this expression, \(\mathbf{S_{r1}}\) represents the unit vector in the photon incidence direction and \(\mathbf{S_{n}}\) represents the unit vector in the direction normal to the photon incidence, satisfying \(\mathbf{S_{r1}}\cdot\mathbf{S_{n}}\leq 0\). \[\frac{dN}{dr}=\sum_{i=1}^{3}\frac{N_{i}}{\sqrt{2\pi}\ln\sigma_{g,i}}\exp\left[ -\frac{\left(\ln r-\ln r_{g,i}\right)^{2}}{2\ln^{2}\sigma_{g,i}}\right] \tag{7}\]0
2. When \(\varepsilon>R_{T}\), refraction occurs between the transmitted photon and the turbulent bubble. Let \(\mathbf{S_{k}}\) be the unit vector of the outgoing normal photon, \(\mathbf{S_{k}}=(\mathbf{C}-\mathbf{B})/R\); let \(\mathbf{A}\) be the coordinates of the photon incident turbulence bubble point; let \(\mathbf{B}\) be the coordinates  
\[\frac{dN}{dr}=\sum_{i=1}^{3}\frac{N_{i}}{\sqrt{2\pi}\ln\sigma_{g,i}}\exp\left[ -\frac{\left(\ln r-\ln r_{g,i}\right)^{2}}{2\ln^{2}\sigma_{g,i}}\right] \tag{7}\]1 Schematic diagram of photons’ interaction with turbulent bubbles. Let  
\[M_{coef}= \begin{bmatrix}\cos\theta_{1}&-1\\ 1&-\cos\theta_{1}\end{bmatrix} ^{-1} \begin{bmatrix}\cos\theta_{2}\\ \cos(\theta_{1}-\theta_{2})\end{bmatrix}  \tag{16}\]  
The unit vector of the photon exit direction and the coordinates of the exit point under refractive conditions are calculated as follows:  
\[\mathbf{S_{r2}}=M_{coef}(1)\cdot\mathbf{S_{rt}}+M_{coef}(2)\cdot\mathbf{S_ {k}} \tag{17}\]  
\[\mathbf{B}=\mathbf{S_{rt}}\cdot 2\cos\theta_{2}\cdot R+\mathbf{A} \tag{18}\]  
To establish a free-space optical channel simulation model, this paper conducted laser transmission simulation research to achieve air-to-air airborne high-capacity communication.