## 1 Introduction  
A series of propagation effects, such as turbulence effect and thermal blooming effect, will be produced in the process of high energy laser (HEL) propagation in the atmosphere [1]. Many works have verified that these propagation effects, laser beam aberration, and platform jitter will cause serious degradation of laser beam quality, which will bring adverse effects to the actual application of laser propagation engineering [2; 3; 4; 5; 6]. In the engineering application of high-energy lasers, the beam quality is an important metric. The rapid evaluation of the beam quality of laser far-field long exposure can provide an important reference for the practice and optimization of laser systems. At present, researchers generally use the method of scaling law to analyze the beam quality of laser far-field long exposure facula; that is, through a 4D wave optics program to simulate the far-field peak intensity and the beam expansion radius under a certain ring energy under different turbulence, thermal blooming, initial beam quality, and other parameters, and use the characteristic parameters describing these effects to fit the results to achieve the characteristics parameter scaling of far-field beam quality. In 1976, [PERSON] [7] obtained the expression of far-field facula peak power of a Gaussian beam and a truncated Gaussian beam by using the generalized thermal distortion parameter, and obtained the scaling formula of laser far-field facula expansion based on the RMS approximation theory, assuming that the far-field distribution is Gaussian distribution. In 1977, [PERSON] et al. [8] presented the expression of the peak power of the far-field facula of the platform beam laser, but lacked the scaling results of the spread of the platform beam facula. Subsequently, [PERSON] [9] obtained the scaling law of the far-field facula expansion of the platform beam. Based on the Bradley Herrmann thermal distortion parameter [10], [PERSON] [11] obtained the scaling relationship of the facula spread under the combined effect of the turbulence thermal of the platform beam. In 2006, [PERSON] [12] found that, under the same thermal distortion parameters, the peak intensity of laser up-line and down-line propagation was significantly different, and the influence of the thermal blooming phase screen at different positions in the propagation path on the peak value of far-field facula was also different. By introducing propagation scaling and path weight function to improve the thermal distortion parameters, [PERSON] could reduce the dependence of the Fresnel number and extinction path change in the scaling relationship. In 2008, [PERSON] [13] modified the thermal distortion parameter to make it suitable for the evaluation of the defocused laser propagation effect. In 2010, [PERSON] [14; 15] obtained the scaling rule of laser atmospheric horizontal propagation at the 1 \(\upmu\)m and 4 \(\upmu\)m band by using this parameter, and analyzed the improved method of thermal distortion parameter \(N_{d}\) under different turbulence effects [14; 15; 16], pointing out that, when the turbulence effect is weak, it is assumed that the turbulence and thermal blooming effect are independent; when the turbulence effect is strong, the thermal blooming effect acts on the basis of turbulence expansion. Based on the above study of the scaling law, we can summarize that there are some limitations in the evaluation of laser far-field facula with the scaling law. First, due to the difference of the diffraction effect of the facula shape and the difference of the propagation scene, different scaling relations are needed for different turbulence conditions, the initial light field distribution, and the propagation scene. Second, the scaling law is only for the expansion of facula under the specific normalized annular energy in the scaling; if the normalized energy of the ring is changed, the fitting form of the scaling law will also be changed.