THE GENERALIZED FRESNEL TRANSFORM AND ITS APPLICATION TO OPTICS

We consider the mathematical properties of a class of linear transform s, which we call the generalized Fresnel transforms, and which have wi de applications to several areas of optics. Well-known transforms, suc h as the fractional Fourier transform and the Fresnel transform, can b e seen to be special cases of this general transform. By use of an ana logy with a quantum harmonic oscillator, we derive mathematical expres sions for functions which remain invariant under such transforms, a re sult which has important applications in the theory of laser resonator modes.