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

Abstract

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 spacefrequency 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 theses 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.

A Light-Ray Approach to Fractional Fourier Optics

“…The proof of Equation ( 65) is given in Appendix C. Equation (65) can also be obtained in the framework of fractional Fourier optics [12]. It has been deduced here from the previous analysis, based on ray transfers and homogeneous matrices.…”

“…So far A, A , B and B are defined up to a multiplicative factor. We may choose additional conditions to define them more accurately: to make the link with diffraction, according to what has been done in a previous article [12], we shall impose…”

“…where ρ = (ρ x , ρ y ) and φ = (φ x , φ y ). Equation (42) shows that in the reduced phase-space the transfer from ray (r, Φ) to ray (r , Φ ) is represented by a four-dimensional rotation that splits into two rotations of angle −α, each rotation operating in a two-dimensional subspace of the reduced phase space [12]. From a physical point of view, matrices in Equation (42) are dimensionless.…”

“…Nevertheless, the composition of the associated matrices physically makes sense only if reduced variables on C are the same for both transfers. The problem has already been analyzed [12] and the result is as follows: the composition makes sense if, and only if, the radius of C is R C such that…”

“…7.1.1. Matrix form of Snell's Law (Refraction) [12] A light ray (r, Φ) is incident on D at point M (coordinates r), see Figure 10. The refracted ray is written (r , Φ ) and, since it passes through M, we have r = r. If e n denotes the unit vector normal to D at M, the incident angle θ is the angle taken from e n to e u , and the refracted angle θ is the angle from e n to e u , where e u is along the incident ray and e u along the refracted ray (see Section 2).…”