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SPIE milestone series

The Wigner distribution function of partially coherent light

MJ Martin Bastiaans

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The concept of the Wigner distribution function is applied to stochastic signals, in particular to partially coherent light. Relations between the Wigner distribution function, the power spectrum and related subjects are formulated. Some inequalities for the Wigner distribution function are derived; one of these inequalities leads to the definition of an overall degree of coherence. The overall degree of coherence of a stochastic signal remains invariant when the signal propagates through a lossless linear system.

Related papers

The Wigner Distribution Function of Partially Coherent Light

Martin J. Bastiaans

The concept of the Wigner distribution function is applied to stochastic signals, in particular to partially coherent light. Relations between the Wigner distribution function, the power spectrum and related subjects are formulated. Some inequalities for the Wigner distribution function are derived; one of these inequalities leads to the definition of an overall degree of coherence. The overall degree of coherence of a stochastic signal remains invariant when the signal propagates through a lossless linear system.

Optica Acta International Journal of Optics•Sep 1, 1981•74 citations

The Wigner distribution function and its applications to optics

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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.

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SPIE milestone series•Jan 1, 2006•0 citations

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New class of uncertainty relations for partially coherent light

Martin J. Bastiaans

A class of uncertainty relations for partially coherent light is derived; the uncertainty relations in this class express the fact that the product of the effective widths of the space-domain intensity and the spatial-frequency-domain intensity of the light has a lower bound and that this lower bound is inversely proportional to the overall degree of coherence. The different members of this class of uncertainty relations correspond to different choices of the quantity that measures the overall degree of coherence. One particular uncertainty relation has the property of being compatible with the ordinary uncertainty principle; the corresponding overall degree of coherence might therefore be considered the best possible choice of all the quantities that measure the overall degree of coherence of the light.

Journal of the Optical Society of America A•Jul 1, 1984•50 citations
The concept of the Wigner distribution function is applied to stochastic signals, in particular to partially coherent light. Relations between the Wigner distribution function, the power spectrum and related subjects are formulated. Some inequalities for the Wigner distribution function are derived; one of these inequalities leads to the definition of an overall degree of coherence. The overall degree of coherence of a stochastic signal remains invariant when the signal propagates through a lossless linear system.
The concept of the Wigner distribution function is applied to stochastic signals, in particular to partially coherent light. Relations between the Wigner distribution function, the power spectrum and related subjects are formulated. Some inequalities for the Wigner distribution function are derived; one of these inequalities leads to the definition of an overall degree of coherence. The overall degree of coherence of a stochastic signal remains invariant when the signal propagates through a lossless linear system.
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.
The Wigner Distribution Function (WDF), defined for a phase space representation in quantum mechanics, offers an alternate form for representing signals simultaneously in Fourier reciprocal variables such as time, and frequency. A purely nonnegative function that is closely related to the WDF, is the so‐called ’’spectrogram’’. In this paper we describe three types of optical setups for the production of the spectrogram. They use coherent, partially coherent or temporally incoherent light as illumination. As application example speech signals are processed for a speaker verification.
The Wigner distribution function of an optical signal, which can be considered as the momentary temporal and local spatial spectrum of the signal, is defined. Equations are derived which describe the transport of the Wigner distribution function in a medium, e.g. a plasma, that is weakly inhomogeneous in space and time, and exhibits weak dispersion for the temporal as well as the spatial frequency variable. The transport equations are compared with the eikonal equation in geometrical optics giving, as a geometrical-optical formulation of the transport equations, that along the path of a geometrical-optical light ray the Wigner distribution function has a constant value.
A class of uncertainty relations for partially coherent light is derived; the uncertainty relations in this class express the fact that the product of the effective widths of the space-domain intensity and the spatial-frequency-domain intensity of the light has a lower bound and that this lower bound is inversely proportional to the overall degree of coherence. The different members of this class of uncertainty relations correspond to different choices of the quantity that measures the overall degree of coherence. One particular uncertainty relation has the property of being compatible with the ordinary uncertainty principle; the corresponding overall degree of coherence might therefore be considered the best possible choice of all the quantities that measure the overall degree of coherence of the light.