A two-dimensional field that is a product of three Airy beams is proposed and investigated. It is shown that the Fourier image of this field has a cubic phase and a radially symmetric intensity with a super-Gaussian decrease. Propagation of the product of three Airy beams in a Fresnel zone is investigated numerically.
The general astigmatic transform, or two-dimensional non-separable linear canonical transform of a Hermite-Laguerre-Gaussian beam, is investigated by theoretical means. Some corollaries that apply to Hermite-Gaussian and Laguerre-Gaussian beam propagation are presented and discussed.
The problem of evaluation of higher derivatives of Airy functions in a closed form is investigated. General expressions for the polynomials which have arisen in explicit formulae for these derivatives are given in terms of particular values of Gegenbauer polynomials. Similar problem for products of Airy functions is solved in terms of terminating hypergeometric series.
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