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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Wigner distribution function for finite signals

Kurt Bernardo Wolf, Natig M. Atakishiyev, S. M. Chumakov

Published Jul 1, 1997• 3 citations
Publisher

Abstract

We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.

DOI
10.1117/12.277653

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We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.
We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N→∞) reproduces Fourier paraxial optics.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called oscillator model in dimension 2 j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j + 1, which is known to tend to the canonical oscillator when j tends to infinity. In particular, we compare plots of our discrete Wigner functions for the su(2) oscillator with the well known plots of Wigner functions for the canonical quantum oscillator. This comparison supports our approach to discrete Wigner functions. 1
We define a quasiprobability distribution SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) which describes the quantum statistics of the photon number and phase observables of a single-mode field (or, equivalently, a harmonic oscillator). The properties of SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) are the photon number and phase analogies of the properties of Wigner's original function, which describes the position and momentum observables. For example, the marginals of SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) are the continuous phase and the discrete photon-number probability distributions. We give examples of the SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) representation of various states and show, in particular, that SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) displays the quantum interference associated with Schr\"odinger cat states. We also describe how SNP{\mathit{S}}_{\mathrm{NP}}SNP​(n,\ensuremath{\theta}) can be determined from quantities that are, in principle, measurable.
The paper presents a review of some applications of the Wigner distribution function to optics. The Wigner distribution function F(x,u) of a signal describes the signal in space x and spatial frequency u, simultaneously, and can be considered as a local spatial frequency spectrum of the signal. Although derived in terms of Fourier optics, the description of a signal by means of its Wigner distribution function closely resembles the ray concept in geometrical optics. Some examples are given to show this resemblance. Properties of the Wigner distribution function are discussed, showing, for instance, its relation to Heisenberg’s uncertainty principle and to some well‐known radiometric quantities. The concept of the Wigner distribution function is not restricted to deterministic signals; it can easily be extended to stochastic signals. Some examples of stochastic signals are considered. The propagation of signals through linear systems can readily be expressed in terms of Wigner distribution functions. Again, the description of systems by Wigner distribution functions can be interpreted directly in terms of geometrical optics: for Luneburg’s first‐order system, for instance, such a description immediately yields the ray transformation matrix of the system, and the relation between the input and the output Wigner distributioin function reads Fo(x,u) = Fi(Ax+Bu,Cx+Du). Transport equations for the Wigner distribution function in a general linear medium are derived. In the Liouville approximation they take the form of a first‐order partial differential equation, which allows, once more, a geometric‐optical interpretation: along the path of a geometric‐optical light ray the Wigner distribution function has a constant value.