The Phase-Space Approach to Optical System Theory

Abstract

Phase-space optics is introduced as an alternative to conventional Fourier optics. The phase-space approach to optical system theory is briefly reviewed. The phase-space interpretation of Gaussian beam propagation serves as an example to illustrate the convenience and utility of phase-space optics.

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

“…As far as we know, no work has been dealing with establishing such a link, which constitutes the main subject of the present paper, and clearly, the result will be that the effect of diffraction is a rotation of the Wigner distribution. We do not obtain a shearing, as proposed by several authors [1,8,10], because these authors consider diffraction between two planes, and not between spherical caps as we do. Our approach will make the above mentioned breaking between Fresnel and Fraunhofer phenomena disappear: generally, Fraunhofer diffraction is physically obtained from Fresnel diffraction by continuously increasing the distance at which the diffracted irradiance is observed; its effect on the Wigner representation will be deduced from the effect of Fresnel diffraction by continuously varying the rotation angle up to −π/2.…”

“…The link has been made with real-order fractional Fourier transformations, whose effects are also rotations in an appropriate phase space [1,[6][7][8][9]. Nevertheless, the effect of Fresnel diffraction or of propagation in free space (through Fresnel transforms), considered between two transverse planes, is generally seen like a "horizontal" shear of the Wigner distribution [1,8,10], not a rotation. We notice that Lohmann expresses Fraunhofer diffraction as a π/2-rotation, but does not generalize to Fresnel diffraction [8], so that the previous descriptions introduce a breaking between the effects of Fresnel or Fraunhofer phenomena, a shearing or a rotation.…”

“…The optical field is described by a function of two real spatial variables, that is a function of a 2-dimensional vector variable, so that the corresponding phase space is 4-dimensional and the Wigner distribution is a function of four real variables. Almost every paper on the Wigner function deals with functions of time (1-dimensional variable); even when dealing with optical fields, most authors (excepting Lohmann [8]) restrict themselves to functions of one spatial variable, whose associated Wigner functions are easier to produce [15] and draw [1,10]. In the present paper we will manage with Wigner distributions in four variables, so that the effect of diffraction will be a 4-rotation.…”