Ince–Gaussian series representation of the two-dimensional fractional Fourier transform

Abstract

We introduce the Ince–Gaussian series representation of the two-dimensional fractional Fourier transform in elliptical coordinates. A physical interpretation is provided in terms of field propagation in quadratic graded-index media whose eigenmodes in elliptical coordinates are derived for the first time to our knowledge. The kernel of the new series representation is expressed in terms of Ince–Gaussian functions. The equivalence among the Hermite–Gaussian, Laguerre–Gaussian, and Ince–Gaussian series representations is verified by establishing the relation among the three definitions.

© 2005 Optical Society of America

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