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Effect of Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory. I -- Stable Resonators and Gaussian Beams
preprintposted on 2023-11-30, 19:53 authored by Pierre Pellat-Finet, Éric Fogret
The first part of the paper is devoted to diffraction phenomena that can be expressed by fractional Fourier transforms whose orders are real numbers. According to a scalar theory, diffraction acts on the amplitude of the electric field as well as on its spherical angular spectrum, and Wigner distributions can be defined on a space-frequency phase space. The phase space is equipped with an Euclidean structure, so that the effects of diffraction are rotations of Wigner distributions associated with optical fields. Such a rotation is shown to split into two specific elliptical rotations. Wigner distributions associated with transverse modes of a resonator are invariant in these rotations, and a complete theory of stable optical resonators and Gaussian beams is developed on the basis of this property, including waist existence and related formulae, and naturally introducing the Gouy phase.