Effect of Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory. I -- Stable Resonators and Gaussian Beams

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.