## 2 Light Field Transmission Model
Under this assumption, the stable light field distribution \(U_{F,TX}(x,y)\) inside the single-mode fiber can be expressed as follows [12]:  
\[U_{F,TX}(x,y)=A_{0}\frac{J_{0}(\frac{U_{01}\sqrt{x^{2}+y^{2}}}{a})}{J_{0}(U_{0 1})}, \tag{4}\]  
In the equation, \(x,y\) denotes the coordinate at the F-face, \(A_{0}\) is the light field intensity coefficient, \(J_{0}\) represents the zeroth-order Bessel function of the first kind, \(a\) is the fiber core radius, and the real parameter \(U_{01}\) can be determined by solving the eigenvalue equation as follows:  
\[U_{01}\frac{J_{1}(U_{01})}{J_{0}(U_{01})}=W\frac{K_{1}(W_{01})}{K_{0}(W_{01})}, \tag{5}\]  
In the equation, \(U_{01}=a\sqrt{n_{1}^{2}k_{0}^{2}-{\beta_{01}}^{2}}\) and \(W_{01}=a\sqrt{{\beta_{01}}^{2}-n_{2}^{2}k_{0}^{2}}\) are real parameters, \(J_{1}\) is the first-order Bessel function of the first kind, \(K_{0}\) is the zeroth-order modified Bessel function of the second kind, \(K_{1}\) is the first-order modified Bessel function of the second kind, \(n_{1}\) is the refractive index of the fiber core, \(n_{2}\) is the refractive index of the fiber cladding, \(k_{0}=2\pi/\lambda_{0}\) is the wave number in vacuum, \(\lambda_{0}\) is the wavelength in vacuum, and \(\beta_{01}\), which is the longitudinal propagation constant of the LP01 mode, can be determined by solving the LP01 eigenvalue function. By substituting into Equation (4), the numerical result for \(U_{01}\) is obtained. Finally, the two-dimensional distribution of the stable light field inside the single-mode fiber at a wavelength of \(\lambda_{0}\) can be computed. The free-space transmission process from the F-face to the T-face can be simulated using Diffraction-limited Coherent Imaging Theory [13]. Assuming the lens is a circular thin lens with a radius \(R\) and a focal length \(f\), the exit pupil position of the laser transceiver system corresponds to the location of the thin lens. The pupil function \(P(\xi,\eta)\) is expressed as follows:  
\[P(\xi,\eta)= \begin{cases}1,\sqrt{\xi^{2}+\eta^{2}}\leq R\\ 0,\sqrt{\xi^{2}+\eta^{2}}>R^{\prime},\end{cases}  \tag{6}\]  
In the equation, \(\xi,\eta\) denotes the coordinate at the exit pupil. The process of light field propagation from the fiber emission end face to the front surface of the exit pupil can be \begin{table}
\begin{tabular}{c c c c} \hline
**Process/Status** & **Light Field Distribution** & **Theory** & **Approximation Conditions** \\ \hline F-face & \(U_{F,TX}(x,y)\) & \begin{tabular}{c} Step-index Fiber Light \\ Field Distribution Theory \\ \end{tabular} & \begin{tabular}{c} Single-mode Weakly \\ Guiding Fiber \\ \end{tabular} \\ \hline F-face\(\rightarrow\)T-face & \(U_{T,TX}(x_{i},y_{i})\) & \begin{tabular}{c} Diffraction-limited \\ Coherent Imaging Theory \\ \end{tabular} & \begin{tabular}{c} Paraxial Approximation Ideal \\ Thin Lens Ideal Circular Stop \\ \end{tabular} \\ \hline T-face & \(U_{T,RX}(x_{i},y_{i})\) & \begin{tabular}{c} Johnson Transformation \\ System Theory \\ \end{tabular} & \begin{tabular}{c} Wide-Sense Stationary Random \\ Process for Rough Surfaces \\ \end{tabular} \\ \hline T-face\(\rightarrow\)F-face & \(U_{F,RX}(x,y)\) & \
\begin{tabular}{c} Diffraction-limited \\ Coherent Imaging Theory \\ \end{tabular} &
\begin{tabular}{c} Paraxial Approximation Ideal \\ Thin Lens Ideal Circular Stop \\ \end{tabular} \\ \hline Fiber coupling & - & Fiber Coupling Theory & - \\ \hline \end{tabular}
\end{table} 
Table 1: Explanation table of laser emission and echo reception process. described by the Fresnel diffraction integral theory. The incident light field distribution at the exit pupil \(U_{l}(\xi,\eta)\) can be expressed as follows:  
\[ \begin{array}{l}U_{l}(\xi,\eta)=\frac{\exp(jk_{0}d_{0})}{j\lambda d_{0}}\exp[ \frac{jk_{0}}{2d_{0}}(\xi^{2}+\eta^{2})]\cdot\iint_{\xi}U_{F,TX}(x,y)\times\\ \exp[\frac{jk_{0}}{2d_{0}}(x^{2}+y^{2})]\cdot\exp[-\frac{jk_{0}}{d_{0}}(x_{\xi }^{2}+y\eta)]dxdy\end{array}  \tag{7}\]  
In the equation, \(d_{0}\) represents the distance from the fiber emission end face to the exit pupil.