Generalized Airy derivative transform of a Gaussian beam.

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.