  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.
