## 4 Conclusions  
A fast calculation method of far-field intensity distribution with _PSF_ convolution for thermal blooming and turbulence effects on HEL propagation is proposed in this paper. We calculate the far-field intensity of the laser propagation system from the point of view of the point source imaging system to calculate the PSF. A series of effects, such as infinite aperture, platform jitter, atmospheric turbulence, and thermal blooming, are all regarded as optical transfer functions of imaging systems. When turbulence effect, thermal blooming effect, and jitter phase are considered to be independent of each other, and when the thermal blooming effect of continuous laser reaches a steady state, the aberration caused by the thermal blooming effect will not change; the fast calculation results of \(PSF_{\text{FSI}}\) can be obtained by convoluting the _PSF_ of each effect. In the study, the results calculated by this method under different aberration, turbulence, and thermal conditions for the horizontal transmission scene and uplink transmission scene are verified by those calculated by the 4D wave optics simulation program. The results show that the method is in good agreement with the calculation results of the 4D wave optics simulation program under the linear effect, and there are some differences between the two under the condition of the turbulent thermal blooming effect, but as a rapid evaluation method, it has a high accuracy. This difference mainly comes from the assumption that the interaction between turbulence and thermal blooming is independent. The fast calculation model and method mentioned in this paper can provide a reference for the engineering application of the rapid evaluation  
Figure 14: The change curve comparison between the fast calculation results and 4D wave optics simulation results of (**a**) \(Enc_{\text{axis}}\) and (**b**) \(Enc_{\text{peak}}\) with facula expansion multiple \(\beta\) under S8, S9, and S10 scenarios.  
Figure 15: The relative error between the fast calculation results and 4D wave optics simulation results for \(Enc_{\text{axis}}\) and \(Enc_{\text{peak}}\) under S8, S9, and S10 scenarios.  
for HEL propagation. In particular, it has great practical value for HEL evaluations, which require large amounts of far-field intensity information quickly. However, it is important to point out that the results calculated by the proposed method are relatively accurate when the thermal blooming effect is weak, but when the thermal blooming effect is strong and the interaction between turbulence and thermal blooming is obvious, the deviation of the results will increase significantly.  
For further work, we will expand the assessment capability of the fast calculation method of far-field intensity distribution with _PSF_ convolution, such as considering other shapes of aperture and the beam passing through other random media, such as rain, fog aerosols, and ocean turbulence.  
Conceptualization, P.Z. and H.M.; methodology, P.Z.; software, P.Z. and J.Z.; validation, C.F., C.Q., and H.M.; formal analysis, P.Z. and C.F.; investigation, C.Q. and H.L.; resources, P.Z.; data curation, H.M., W.Z., and X.L.; writing--original draft preparation, H.M.; writing--review and editing, H.M.; visualization, H.M. and H.L.; supervision, C.F.; project administration, J.Z.; funding acquisition, H.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China under Grant 61905002 and Natural Science Research Projects in Anhui Universities under Grant KJ2019A0210.  
Not applicable.  
Data Availability Statement: Data sharing is not applicable to this article.  
The authors declare no conflict of interest.