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Effect of Fresnel Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory

Pierre Pellat-Finet, Éric Fogret, Pierre-Emmanuel Durand

Published Dec 1, 2014• 0 citations
OpenAlex

Abstract

According to a scalar theory, diffraction affects the electric field amplitude as well as its angular spectrum, providing a natural space–frequency phase space on which the Wigner distribution can be defined. The phase space is equipped with an Euclidean structure, so that the effect of diffraction is a rotation of the Wigner distribution associated with the optical field. The rotation is shown to split into two specific rotations. Wigner distributions associated with transverse modes of a resonator are invariant in such rotations, and a complete theory of optical resonators is developed on the basis of this property, including waist existence and related formulae, and naturally introducing the Gouÿ phase.

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According to a scalar theory, diffraction affects the electric field amplitude as well as its angular spectrum, providing a natural space–frequency phase space on which the Wigner distribution can be defined. The phase space is equipped with an Euclidean structure, so that the effect of diffraction is a rotation of the Wigner distribution associated with the optical field. The rotation is shown to split into two specific rotations. Wigner distributions associated with transverse modes of a resonator are invariant in such rotations, and a complete theory of optical resonators is developed on the basis of this property, including waist existence and related formulae, and naturally introducing the Gouÿ phase.
The Fresnel diffraction of periodic objects at rational fractions of the Talbot distance is described in terms of the Wigner distribution function (WDF). The analysis provides a heuristic model for understanding the formation of the diffraction patterns as well as for evaluating the complex amplitude at any fractional Talbot plane. Furthermore, certain symmetry properties of the Fresnel-diffracted wave field can be derived directly from the WDF. Additionally, a discussion is given on how periodic signals and information about the phase are encoded in the WDF.
Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
The propagation of a weak optical pulse through a resonant two-level atomic medium is discussed in the paraxial approximation. Transverse propagation effects within that approximation are treated exactly by development of the method of Fresnel transforms. In particular, for a temporally short pulse with an amplitude of arbitrary transverse spatial dependence it is shown that longitudinal propagation is not affected by diffractive spreading. Thus diffraction does not attenuate the characteristic ringing of one-dimensional pulse propagation in resonant media.
Various quasiprobability distributions have been developed in the past using the Hilbert space of the single-mode light field. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique Hermitian phase operator defined on the Hilbert space. Recently, however, Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and Barnett and Pegg [J. Mod. Optics 36, 7 (1989)] introduced a new formalism that does allow the construction of a Hermitian phase operator and associated phase eigenstates. In this paper we develop a quasiprobability distribution associated with the number and phase operators of the single-mode light field in the new formalism. The new distribution, which we call the number-phase Wigner function, has properties analogous to the Wigner function. We also derive the number-phase Wigner representation of number states, phase states, general physical states, coherent states, and the squeezed vacuum. We find this new representation has features that are related to the number and phase properties of states. For example, the number-phase Wigner representation of a number state is nonzero only on a circle, while the representation of a phase state is only nonzero along a radial line.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.