Properties of Airy transform of elegant Hermite-Gaussian beams

•The Airy transform of an elegant higher-order mode is first investigated.•The analytically general formulae of centroid and beam spot size are derived.•The effect of the Airy control parameters on the Airy transform is investigated.•The experimental results are in good agreement with the theoretical results.•The Airy transform of elegant and standard Hermite-Gaussian beams is compared. Airy transform of elegant Hermite-Gaussian beams is theoretically and experimentally investigated. The analytically general formula of the Airy transform of elegant Hermite-Gaussian beams is presented. The analytically general expressions of the centroid and the beam spot size of an elegant Hermite-Gaussian beam passing through an Airy transform optical system are derived. The effects of the Airy control parameters and the transverse mode numbers on the intensity distribution, the centroid, and the beam spot size of elegant Hermite-Gaussian beams passing through Airy transform optical systems are investigated, respectively. The experimental results coincide with the theoretical results. Also, the Airy transform of the elegant and the standard Hermite-Gaussian beams is compared. The analytical expression of the Airy transform of the elegant Hermite-Gaussian beam is more concise than that of the standard Hermite-Gaussian beam. Although the optical fields in the x-direction of the Airy transform of the elegant and the standard Hermite-Gaussian beams with transverse mode number m > 1 are both the sum of the zero-order derivative up to mth-order derivative of the Airy function, the weight coefficient of the same order derivative of the Airy function is different for the elegant and the standard transformed Hermite-Gaussian beams. The centroids of the elegant and the standard Hermite-Gaussian beams passing through the same Airy transform optical system are the same. However, the beam spot size of the elegant transformed Hermite-Gaussian beam is less than or equal to that of the standard transformed Hermite-Gaussian beam. The meaning of elegance is also embodied in the Airy transform of the elegant optical beams. This research reveals the difference between the Airy transform of the elegant and the standard Hermite-Gaussian beams.