The self-focusing effect on the beam quality of Hermite–Gaussian beams propagating upwards through the inhomogeneous atmosphere is studied. The analytical formula of the beam width is derived, and its validity is confirmed. Furthermore, the analytical formulas of the actual focal length and ${M^2}$-factor are also derived. It is found that the self-focusing effect in the inhomogeneous atmosphere results in beam quality degradation. Under the same initial beam width and the same beam power, as the beam order $m$ increases, the actual focal length is farther away from the target, and the spot size on the target and the ${M^2}$-factor increase; namely, the beam quality degrades further. In addition, it is shown that the beam quality can be improved by the phase compensation.

中文翻译：

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自聚焦对在非均匀大气中向上传播的厄米-高斯光束质量的影响

研究了在非均匀大气中向上传播的厄米-高斯光束的自聚焦效应。推导了光束宽度的解析公式，并验证了其有效性。此外，还推导了实际焦距和$ {M ^ 2} $-因子的解析公式。发现在不均匀大气中的自聚焦效应导致光束质量下降。在相同的初始光束宽度和相同的光束功率下，随着光束阶数$ m $的增加，实际焦距距离目标越来越远，并且目标和$ {M ^ 2} $上的光斑大小因素增加；即，光束质量进一步降低。另外，示出了可以通过相位补偿来改善光束质量。