Laser beam propagating in atmosphere is affected by many effects such as turbulence, diffraction, self-focusing, and thermal blooming, which is an extremely complex physical process. Although the numerical simulation method can be used to predict the effects of different atmospheric conditions and beam parameters on laser propagation, its computational efficiency is insufficient for practical applications. Therefore, the study of scaling law becomes essential. However, on the one hand, most studies on the scaling law of laser atmospheric propagation focus on beam spreading. On the other hand, the thermal blooming effect causes the focus shift of a focused Gaussian beam, and the actual focus point is shifted to the position in front of the target plane, which severely degrades the beam quality at the target plane. In this paper, to achieve the maximum peak intensity at the target plane, the scaling law for the optimal focal length of a focused Gaussian beam under thermal blooming effect is investigated.

Thermal blooming effect of a laser beam propagating through atmosphere can be described by the paraxial wave equation and the hydrodynamic equation. A time-dependent four-dimensional code to simulate the focused Gaussian beam propagating in atmosphere is developed by using the multi-phase screen method, fast Fourier transform method, and difference method. In this paper, the method of pre-defocusing is employed to obtain the maximum peak intensity at the target plane. To predict the optimal focal length f_opt under varying atmospheric conditions and beam parameters, this paper employs the controlled variable method to numerically analyze the effects of six scaling factors (initial laser power P, atmospheric wind speed v, atmospheric absorption coefficient α, initial beam width w₀, target plane distance z₀, and wavelength λ)on f_opt. Through physical analysis and extensive numerical calculations, a formula of the optimal focal length f_opt relating with the six scaling factors is established.

In this paper, the method of pre-defocusing is employed to determine the optimal focal length of the lens, thereby achieving the maximum peak intensity at the target plane (Figs. 1 and 2). First, three scaling factors (P, v, and α) are considered , which significantly affect the intensity of thermal blooming effect. The generalized thermal distortion parameter N is a physical quantity used to describe the intensity of thermal blooming effect. A scaling law between f_opt and N is investigated [Fig. 3 and Eq. (10)]. In contrast, when the other three scaling factors (z₀,w₀,and_{λ) are changed, both the intensity of the thermal blooming effect and the intensity of the diffraction effect are altered (Figs. 4?6). Therefore, it is necessary to modify Eq. (10). The scaling law of fopt relating with N, z0 and w0 is modified on the basis of Eq. (10) [Fig. 7 and Eq. (13)]. This scaling law can predict the optimal focal length under various atmospheric conditions and beam parameters. When N<11 and Fresnel number NF>4, this scaling law is applicable, and the mean relative error is 5.2% [Fig. 8 and Eq. (13)]. Additionally, it is shown that the increase of peak intensity at the target plane after focusing is greater when N is larger (Fig. 9).}

In this paper, a scaling law for the optimal focal length of a focused Gaussian beam considering atmospheric thermal blooming effect is studied. The method of pre-defocusing is employed to determine the optimal focal length of the lens, thereby achieving the maximum peak intensity at the target plane. Using the controlled variable method, the influence of six scaling factors (P, v, α, z₀, w₀, and_{λ) on the optimal focal length fopt is obtained by varying them, respectively. Then, the scaling law of fopt relating with the generalized thermal distortion parameter N, the target plane distance z0 and the initial beam width w0 is established. This scaling law is applicable when N<11 and Fresnel number NF>4. This scaling law can predict the optimal focal length under various atmospheric conditions and beam parameters, and the mean relative error is 5.2%. Additionally, it is shown that the increase of peak intensity at the target plane after focusing is greater when N is larger. It is worth noting that laser atmospheric propagation is a very complex physical process. In practical application, the beam quality at the target plane is also related to the beam quality at the source plane, beam truncation ratio, tracking accuracy, and atmospheric turbulence. The results obtained in this paper are useful for the application of laser atmospheric propagation.}