Ground-based laser systems can remove centimeter-scale space debris in the low-Earth orbit region. However, as the high-power laser beam propagates through the atmosphere, it encounters significant challenges. When the beam’s power exceeds the atmosphere’s critical power for the self-focusing effect, the beam quality at the target diminishes due to this nonlinear effect.

Interestingly, Airy beams exhibit self-accelerating characteristics, making them potentially advantageous for bypassing obstacles. However, in homogeneous self-focusing media, an Airy beam can lose its self-accelerating traits if its power is exceedingly high. This leads to pressing inquiries: Does the nonlinear self-focusing effect in an inhomogeneous atmosphere disrupt the self-accelerating nature of Airy beams? Is an Airy beam better suited than a Gaussian beam for ground-based laser space debris removal? How to enhance the target quality of Airy beams? Hence, analyzing the influence of nonlinear self-focusing on the attributes and quality of upward-propagating Airy beams in the atmosphere becomes crucial.

Under the paraxial approximation, the beam characteristics of diffraction and self-focusing nonlinearity were described via a nonlinear Schr?dinger equation. However, solving the nonlinear Schr?dinger equation analytically for an Airy beam propagating in the atmosphere is challenging. In this study, the nonlinear Schr?dinger equation was solved numerically using the multiphase screen method. As the altitude increased, the nonlinear refractive index decreased, and the nonlinear self-focusing effect became negligible at sufficiently high altitudes. Consequently, an Airy beam that propagated upward in the atmosphere experienced two stages: inhomogeneous atmospheric propagation (comprising both diffraction and self-focusing effects) and free space propagation (with only the diffraction effect).

As the exponential truncation factor of the Airy beams increases, the value of the B integral also increases (Fig.1), indicating a strengthening of the nonlinear self-focusing effect. The real focus of Airy beams shifts to the target due to self-focusing in an inhomogeneous atmosphere, a behavior distinct from that of Gaussian beams (Fig.2). By employing the preliminary defocusing method, an Airy beam maintains its Airy profile at the target even when the beam power significantly exceeds the critical power of the self-focusing effect in the atmosphere, and the intensity at the target notably increases (Fig.5). Specifically, a formula for the focal length of the preliminary defocusing of the Airy beams is obtained, and this is also confirmed (Fig.4). With the preliminary defocusing method, the self-accelerating characteristics of the Airy beams remain unaffected by the nonlinear self-focusing effect in an inhomogeneous atmosphere, even when the beam power significantly surpasses the critical power (Figs.6 and 7). This differs from the behavior of Airy beams in a homogeneous atmosphere. Given the same beam power, the intensity of the Airy beam at the target surpasses that of the Gaussian beam (Fig.8). Additionally, the Airy beam’s resistance to the nonlinear self-focusing effect in an inhomogeneous atmosphere exceeds that of the Gaussian beam (Fig.8).

In this study, the influence of nonlinear self-focusing on the characteristics and quality of Airy beams, as they are propagated from the ground through the atmosphere to space orbit, is numerically investigated. The strengthening of the self-focusing effect with the increasing exponential truncation factor of the Airy beams is observed. It is found that the Airy profile can be maintained at the target, even when the beam power significantly exceeds the critical power of the self-focusing effect, when the preliminary defocusing method is used, leading to a significant increase in target intensity. Furthermore, a formula for the focal length of the preliminary defocusing of Airy beams is derived. The self-accelerating characteristic of Airy beams is shown to be preserved with the preliminary defocusing method, proving beneficial for avoiding obstacles in the path. Under the same beam power, the target intensity of the Airy beam is found to be significantly higher than that of the Gaussian beam, suggesting that Airy beams are deemed more suitable than Gaussian beams for ground-based laser space debris removal.