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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Role of semiclassical description in the quantumlike theory of light rays

Renato Fedele, Vladimir I. Man’ko

Published Nov 1, 1999• 26 citations
Publisher

Abstract

An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.

DOI
10.1103/physreve.60.6042

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An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
It is shown that the standard classical picture of charged-particle beam transport in paraxial approximation may be conveniently replaced by a Wigner-like picture in a semiclassical approximation. In this effective description, the classical phase-space equation for electronic rays is replaced by a von Neumann--like equation, where the transverse emittance plays the role of $\ensuremath{\Elzxh}$. Relevant remarks concerning the quantumlike corrections for an arbitrary potential in comparison with the standard classical description of the beam transport are given.
The premise of this submission is to identify the role of the Wigner function [1] in paraxial wave optics, where it can be naturally introduced on the basis of the analogy between the paraxial wave equation and the Schrodinger equation of nonrelativistic quantum mechanics. Also known as the Wolf function, it describes the cross section of a paraxial beam using a real-valued function of displacement from axis and transverse components of the wave vector. This can be perceived as a quasi-probability distribution over the system's phase space, following the dynamics of a state space density of an ensemble of free noninteracting particles. In a (2+1)–dimensional picture this corresponds to the propagation of straight linear rays where the Wolf function describes the local phase space intensity. There is no notion of relative phase but negative values can be present, providing an alternative basis for description of interference phenomena. This corresponds exactly to the concept of pencils of positive and negative rays, developed by E. C. G. Sudarshan in the late 70’s and 80’s [2, 3].
It is shown that the relation between ray optics and wave optics is analogous to the relation between the classical mechanics of a point particle and quantum mechanics. The time coordinate of quantum mechanics has to be replaced with a length coordinate and the light wavelength λ takes the place of Planck’s constant h. A well-known result of quantum mechanics is that it coincides with classical mechanics in the limit h → 0. Correspondingly, wave and ray optics must coincide in the limit λ → 0.
In quantum mechanics, systems can be described in phase space in terms of the Wigner function and star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to initial canonical coordinates and momenta. The reduction to a system of ordinary differential equations pertains rigorously at any fixed order in hbar\\hbarhbar. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into semiclassical transport models.