Gao Hongfei, Chen Jiahao, Zhou Yimin, Wang Fei, Cai Yangjian, Zhou Guoquan
Opt Express. 2025 Jun 30;33(13):27130-27147. doi: 10.1364/OE.564810.
The Airy transform, endowed with its unique properties, is capable of converting a conventional Gaussian beam into an Airy beam. Could this transform be extended to unlock the potential for generating a broader series of specific beams? In this paper, a more comprehensive generalized Airy derivative transform is proposed, within which the conventional Airy transform is encapsulated as the zeroth-order case, while further extensions are incorporated. To further clarify the transmission characteristics, analytical expressions are derived for the center of mass, the dimension of the light spot, the divergent angle, and the M-square factor determined by the first- and second-order moments of the light intensity at the output plane of a transformed Gaussian beam processed through an Airy derivative transform optical system of arbitrary order. In addition, the propagation properties of the transformed Gaussian beam in free space after leaving the output plane are also investigated. The light field of the transformed Gaussian beam during free-space propagation remains expressed as a summation of the Airy function's derivatives from zeroth-order to -order, with modifications occurring in the corresponding weighting factors. During propagation in free space, the transformed Gaussian beam follows a ballistic trajectory within the - or - plane. As an experimental example, the first-order Airy derivative transform of the Gaussian beam is demonstrated. The experimental realization of the first-order Airy derivative transform is relatively straightforward and can be generalized to the generalized Airy derivative transform. Consequently, the generalized Airy derivative transform offers an alternative method for the generation of specific beams, thus broadening the transform's functionality.
艾里变换具有其独特的性质，能够将传统的高斯光束转换为艾里光束。这种变换能否扩展以挖掘生成更广泛系列特定光束的潜力呢？在本文中，提出了一种更全面的广义艾里导数变换，其中传统的艾里变换被封装为零阶情况，同时纳入了进一步的扩展。为了进一步阐明传输特性，针对通过任意阶艾里导数变换光学系统处理后的变换高斯光束在输出平面处光强的一阶和二阶矩所确定的质心、光斑尺寸、发散角和M²因子，推导了其解析表达式。此外，还研究了变换高斯光束在离开输出平面后在自由空间中的传播特性。变换高斯光束在自由空间传播期间的光场仍表示为从零阶到负阶的艾里函数导数的总和，只是相应的加权因子有所修改。在自由空间中传播时，变换高斯光束在x或y平面内遵循弹道轨迹。作为一个实验示例，展示了高斯光束的一阶艾里导数变换。一阶艾里导数变换的实验实现相对简单，并且可以推广到广义艾里导数变换。因此，广义艾里导数变换为生成特定光束提供了一种替代方法，从而拓宽了该变换的功能。
