## 2 Numerical Methods
### Heterodyne Detection Processes with ISAL  
Heterodyne detection is an important technique for implementing high-resolution ISAL images. As a kind of coherent detection, highly steady coherence between the local oscillator signal and echo signal in the heterodyne detection process should be maintained. However, the spatial decoherence of the echo induced by turbulence could reduce the SNR and heterodyne efficiency, while the temporal decoherence of the echo induced by turbulence could introduce an extra random phase, which deteriorates the phase stability of the echo and makes the coherent synthesis impossible. Therefore, it is necessary to analyze the effect of turbulence on ISAL imaging based on the heterodyne detection principle. Here, the signal transmission and coherent detection processes under the atmospheric turbulence are given. It is assumed that both the local oscillation \(E_{L}(x,y,t)\) and the return signal \(E_{S}(x,y,t)\) can be expressed as  
\[E_{L}(x,y,t)=E_{L}(x,y)\exp(i\omega_{L}t) \tag{3}\]  
\[E_{S}(x,y,t)=E_{S}(x,y)\exp(i\omega_{S}t) \tag{4}\]  
where \(\omega_{L}\) and \(\omega_{S}\) are the frequency of the local and return signals, respectively. After \(E_{L}\) and \(E_{S}\) are coherently detected on the beam-combining mirror, the heterodyne signal is as follows:  
\[ \begin{array}{rcl}i(t)&=&\iint|E_{S}(x,y,t)+E_{L}(x,y,t)|^{2}dxdy\\ &=&\iint|E_{S}(x,y,t)|^{2}+|E_{L}(x,y,t)|^{2}\\ &&+E_{S}^{*}(x,y,t)E_{L}(x,y,t)+E_{S}(x,y,t)E_{L}^{*}(x,y,t)dxdy\end{array}  \tag{5}\]  
where the last two items are beat frequency items, and by adding the phase items \(\phi_{S}(x,y)\), \(\phi_{L}(x,y)\), the above equation can be written as  
\[i(t)=\iint|E_{S}(x,y)|^{2}+|E_{L}(x,y)|^{2}+2|E_{L}(x,y)E_{S}(x,y)|\cos[( \omega_{S}-\omega_{L})t+\phi_{S}(x,y)-\phi_{L}(x,y)]dxdy \tag{6}\]  
where \(\phi_{L}(x,y)\) is the initial phase of the local signal and \(\phi_{S}(x,y)\) is the phase fluctuation caused by atmospheric turbulence. It should be noted that detection can only be performed if the reference and return signals are in the same modes. In order to achieve good coherence with the reference light, the mode-matching requirements must be satisfied. The transverse mode of the laser can be described by the Hermite-Gaussian polynomial \(H_{m,n}(x,y)\) with orthogonal normalization and completeness:  
\[\iint H_{m,n}(x,y)H_{k,l}^{*}(x,y)dxdy=\delta_{m,k}\delta_{n,l} \tag{7}\]  
The light field in the atmosphere can be decomposed according to the Laguerre-Gaussian (L-G) mode or Hermite-Gaussian (H-G) mode. First, the echo signal electric field \(E_{S}(x,y)\) is expanded according to the H-G polynomial:  
\[E_{S}(x,y)=\sum C_{m,n}^{S}H_{m,n}(x,y) \tag{8}\]  
where  
\[C_{m,n}^{S}=\iint E_{S}(x,y)H_{m,n}^{*}(x,y)dxdy \tag{9}\]In Equation (9), \(C_{m,n}^{S}=\left|C_{m,n}^{S}\right|e^{i\phi_{m,n}}\left|C_{m,n}^{S}\right|^{2}\) represents the proportion of \(m\) and \(n\) mode compositions in the echo signal field, and \(\phi_{m,n}\) represents the additional phase of \(m\) and \(n\). The local reference light field \(E_{L}(x,y)\) is generally a \(TEM_{00}\) mode:  
\[E_{L}(x,y)=C_{0,0}^{L}H_{0,0}(x,y) \tag{10}\]  
Based on the detection conditions, we can see that when the reference light is in \(TEM_{00}\) mode, only the component of the \(TEM_{00}\) mode in the echo signal field participates in the beat frequency. Therefore, the mode decomposition is performed to obtain the coefficients and phases of the \(TEM_{00}\) mode of the echo. Then, Equation (6) can be written as  
\[i(t)=\left|C_{0,0}^{L}\right|^{2}+\left|C_{0,0}^{S}\right|^{2}+2\left|C_{0,0}^ {L}\right|\left|\left|C_{0,0}^{S}\right|\cos\left[(\omega_{S}-\omega_{L})t+ \phi_{0,0}^{S}-\phi_{0,0}^{L}\right] \tag{11}\]  
where \(C_{0,0}^{S}=\left|C_{0,0}^{S}\right|e^{i\phi_{0,0}^{S}}\) is the coefficient of the echo signal's \(TEM_{00}\) mode.