The generalized Fresnel transform and its application to optics

We consider the mathematical properties of a class of linear transforms, which we call the generalized Fresnel transforms, and which have wide applications to several areas of optics. Well-known transforms, such as the fractional Fourier transform and the Fresnel transform, can be seen to be special cases of this general transform. By use of an analogy with a quantum harmonic oscillator, we derive mathematical expressions for functions which remain invariant under such transforms, a result which has important applications in the theory of laser resonator modes.

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