Ince–Gaussian beams in a quadratic-index medium

The propagation of Ince-Gaussian beams in media where the refractive index varies quadratically with the distance from the optical axis is studied. Explicit expressions for the complex beam parameter and the longitudinal phase shift are derived and discussed. Ince-Gaussian eigenmodes with constant width can be obtained by satisfying a relation between the beam width and the quadratic-medium coefficient. The derivation has included the possibility of propagation of Ince-Gaussian beams in complex lenslike media having quadratic transverse variations of the index of refraction and the gain or loss.