Laser beams propagating in the atmosphere suffer from adverse effects due to the atmospheric optical characteristics and laser system features, which broaden the beam radius and weaken the encircled mean intensity. The wave-optics-based four-dimensional codes work with redundant inputs and slow speed, failing to meet the requirements of rapid assessment for practical applications. Researchers have made efforts to develop new methods, holding reasonable accuracy, calculating quickly and easily, without consideration of the mesh size and computational stability as wave optics programs. Integrated with characteristic parameters of laser system and atmosphere, the scale law has received much attention and is widely used in system design and applications with lots of computation.

Current laser beam propagation scale law is based on radius-square-sum (RSS) assumption, meaning that the resulting far-field radius is the root of the sum of radii squared of the individual effect contributions. The RSS assumption lacks scientific foundation and may bring some errors in use. Besides, though the accuracy of scale law is crucial for reliable analysis, few reports on the accuracy of scale models have been released. Furthermore, previous attention was focused mainly on flat-top source, thus the effect of new features of Gaussian source, such as truncating extent, on far-field spot has not been well studied.

Theoretical analysis and numerical simulations are used to build the scale model. Analytical expression of 63.2% encircled power radius in the far-field of infinite Gaussian source is deduced on the basis of Huygens-Fresnel principle, showing that the radius is a function of wavelength, distance and aperture. When the Gaussian source is truncated, split-step wave optics simulations are used to obtain the far-field radii corresponding to 63.2% and 86.5% encircled power. Referring to the analytical expression of infinite Gaussian source, a radius scale function for truncated Gaussian source is built, and the scale exponents are given for different truncating factors. On the basis of established turbulent spread radius expression of infinite Gaussian beam, a radius scale model is given for truncated Gaussian source propagating through turbulence, showing that the scale exponent varies with the value of truncating factor.

When the mutual interaction among diffraction, beam quality, jitter of platform and optical turbulence is considered, the generally used RSS assumption is improved to a modified version which is named MRSS method. This new method introduces three scale exponents and an exponent term which consists of the ratio of two different characteristic radii in order to promote the model's applicability. For Gaussian source with truncating factor of 22 propagating in vacuum, the split-step wave optics simulations are operated in a wide range of parameter space shown in Table 2, with Fresnel number changing from 1.0 to 6003.4. The far-field radius scale models based on RSS assumption and MRSS method are built respectively, and the exponents are fixed with the help of genetic algorithm. Comparison with numerical simulations shows that the mean relative errors of the results from the model based on MRSS method are smaller than those based on RSS assumption.

A similar process is conducted to build the scale model of far-field radius and encircled mean intensity for the Gaussian source with truncating factor of 22 propagating in turbulent atmosphere. The numerical simulations are conducted with the Hufnagel-Valley optical turbulence profile, and with the propagating distance and other parameters varying in a wide range shown in Table 3. Comparison with numerical simulations shows that the accuracy of the model based on MRSS method is higher than that based on RSS assumption.

When the Gaussian source is truncated, the far-field radius of free diffraction in vacuum and turbulent spread in atmosphere is affected by the truncating factor, as the scale exponents vary with Fa, as shown in Fig. 1 and Fig. 2(b), respectively. For the scale models based on RSS assumption, aVR gives a mean relative error of 3.12%, as shown in Fig. 4(c), while aLR gives a mean relative error of 4.15%, as shown in Fig. 6(c). For the scale models based on MRSS method, aV gives a mean relative error of 1.55%, as shown in Fig. 4(c), while aL gives a mean relative error of 1.92%, as shown in Fig. 7(c). The mean relative error of mean intensity is 8.33% based on RSS assumption, and 3.80% based on MRSS method. In summary, the accuracy of the models based on MRSS method is higher than those based on RSS assumption.

The expression of aL based on MRSS method is equivalent to ad for ideal Gaussian beam propagating in vacuum, and to aV when the interaction among diffraction, beam quality and jitter of platform is considered. When only turbulence spread is considered, the optical quality of aL works well with the optical quality of turbulence spread radius, as shown in Fig. 8.

The scale models of far-field radius and encircled mean intensity for truncated Gaussian source are built in vacuum and turbulent atmosphere. Comparison with split-step wave optics simulations shows that the proposed MRSS method is able to improve the accuracy and applicability of scale models. The results are discussed for Gaussian source with truncating factor of 22 and far-field radius of 63.2% encircled power ratio. However, scale exponents and accuracy for other conditions need more research.