## 2 Theory and Model  
### Fast Calculation of Far-Field Intensity Distribution with PSF Convolution  
In our model assumptions, the propagation process of the laser from emission system to target can also be regarded as an imaging process of infinite point light source, and the schematic diagram is shown in Figure 1. Far-field intensity can be obtained by calculating the _PSF_ of the point light source imaging system, in which a series of effects, such as limited aperture, platform jitter, atmospheric turbulence, and thermal blooming are considered. Considering the finite aperture and aberration of the laser emission system, the _OTF_ of the imaging system is the normalized autocorrelation function of the generalized pupil function and can be expressed as [24]:  
\[ \begin{array}{c}OTF(\lambda zf_{x},\lambda zf_{y})=\\ \frac{\int\int_{-\infty}^{\infty}p(x,y)p(x-\lambda zf_{x},y-\lambda zf_{y}) exp(i\vartheta(x,y))exp(-i\vartheta(x-\lambda zf_{x},y-\lambda zf_{y}))dxdy}{\int \int_{-\infty}^{\infty}p^{2}(x,y)dxdy}\end{array}  \tag{1}\]  
Figure 1: Schematic diagram for (**a**) the propagation process of laser from emission system to target and (**b**) the imaging process of infinite point light source. where \(p(\lambda zf_{x},\lambda zf_{y})\) is aperture function, \(\lambda\) is laser wavelength, \(z\) is focal length, \(f_{x}\),\(f_{y}\) are the coordinates of spatial frequency, and \(\psi(\lambda zf_{x},\lambda zf_{y})\) is the aberration function of the aperture. When turbulence, jitter, and other random phase disturbances are considered, \(OTF_{L}\) with long exposure, can be expressed as the ensemble average of the random phase,  
\[OTF_{L}(\lambda zf_{x},\lambda zf_{y})= \tag{2}\] \[\frac{\int\int_{-\infty}^{\infty}p(x,y)p(x-\lambda zf_{x},y- \lambda zf_{y})\left(exp(i\psi(x,y))exp\left(-i\psi(x-\lambda zf_{x},y-\lambda z f _{y})\right)\right)dxdy}{\int\int_{-\infty}^{\infty}p^{2}(x,y)dxdy}\]  
\(\psi\) is expressed as the sum of turbulence effect, \(\psi_{turbulence}\); jitter, \(\psi_{jitter}\); thermal blooming effect, \(\psi_{TB}(x,y)\); aperture of launch system; and initial aberration of system, \(\psi_{p}(x,y)\), as follows:  
\[\psi(x,y)=\psi_{turbulence}(x,y)+\psi_{jitter}(x,y)+\psi_{TB}(x,y)+\psi_{p}(x,y) \tag{3}\]  
It is assumed that the phase of turbulence effect, thermal blooming effect, and jitter are independent of each other, and that the aberration caused by the thermal blooming effect will not change when the thermal blooming effect of the CW laser reaches steady state. In this case, the thermal blooming effect is considered as the initial aberration of the aperture \(\psi_{p+\text{TB}}(x,y)=\psi_{TB}(x,y)+\psi_{p}(x,y)\). Considering the phase disturbances above, the aberration function can be expanded to:  
\[ \begin{array}{c}\left\langle exp(i\psi(x,y))exp\left(-i\psi(x-\lambda zf_{x}, y-\lambda zf_{y})\right)\right\rangle\\ =\left\langle\exp(i\psi_{jitter}(x,y))\right\rangle\exp\left(-i\psi_{jitter}(x -\lambda zf_{x},y-\lambda zf_{y}))\right\rangle\\ \times\left\langle\exp(i\psi_{turbulence}(x,y))\exp\left(-i\psi_{turbulence }(x-\lambda zf_{x},y-\lambda zf_{y})\right)\right\rangle\\ \times exp\left(i\psi_{p+TB}(x,y)\right)exp\left(-i\psi_{p+TB}(x-\lambda zf_{x },y-\lambda zf_{y})\right)\end{array}  \tag{4}\]  
Furthermore, if the turbulence and jitter phase are generalized stationary, then the ensemble mean value of the jitter effect and turbulence effect is independent of the \(x\), \(y\) coordinates, and moves it outside the integral, then Equation (2) can be expressed as:  
\[