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Journal of the Optical Society of America A

New class of uncertainty relations for partially coherent light

Martin J. Bastiaans

Published Jul 1, 1984• 50 citations
Publisher

Abstract

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.

DOI
10.1364/josaa.1.000711

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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.
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.
A general prescription is given for finding uncertainty relations that dictate the lower bounds on the measures of spread corresponding to two different representations of a partially coherent wave field or mixed quantum state, for a given measure of overall coherence or purity.In particular it is shown that the coherent modes of the fields/states that achieve the lower bounds are independent of the measure of purity being used, and that this measure determines only the amount in which these modes contribute.Our results are important in the design of optical systems with partially coherent light and in quantum mixed states, for which maximal joint localization is desired.These ideas are illustrated for the case of optical beams, pulses propagating in dispersive media and quantum phase.
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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 problem of bipartite entanglement in partially coherent paraxial vector light fields is addressed. A generalized uncertainty principle suited for the polarization-spatial degrees of freedom is introduced. Partial transpose is implemented through the obtained generalized uncertainty principle. Partial transpose is shown to be necessary and sufficient in detecting entanglement for a class of partially coherent vector light fields which have a spatial part to be Gaussian. An experimental realization of the studied entangled states using classical optical interferometry is outlined.