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Wigner Distributions and Phase Space in Optics

INTRODUCTION

Our call for papers on Wigner distributions and phasespace in optics received a hearty response. In this fea-ture issue we have gathered papers that show, from quitedifferent perspectives, the relevance and promise of phasespace for optics. It was satisfying to receive such a rangeof viewpoints on the principal object of attention: quasi-probability distributions on phase space, i.e., Wignerfunctions. The phase-space function defined by EugeneWigner on statistical grounds for quantum mechanics hasproved to be a remarkably pliable tool in optics. In quan-tum optics, in fact, phase space is the canvas depictingthe very nature of light. However, it is also clear thatother, closely related, ideas play key roles in a variety ofphase-space methods for optics.

Given the interdisciplinary character of this research,we somewhat arbitrarily have grouped papers from theabstract to the concrete, and we have further subdividedthe latter. Within each group papers appear in alpha-betical order. The reader will appreciate from the refer-ences of these papers how lines of disparate research haveconverged. In contrast to the melding of space and timein relativity theory, the fusing of coordinate and fre-quency domains to form optical phase space has been aslower, more incremental process. As becomes clearfrom this collection, this process has been spurred in partby studies of short pulses, of image and information pro-cessing, and of semiclassical issues related to the ray–wave connection in various domains of wave physics.

A number of fundamental issues arise in several of thepapers. Localization, for example, is a central notion inphase space, together with the well-known uncertaintyrelation, which sets a lower bound on the products of themoments of the Wigner distribution. Generalized uncer-tainty relations are also discussed here, as is their con-nection to various types of nonclassical states of light(such as squeezed and correlated states) and to beam pu-rity measures. Similarly, the fractional Fourier transfor-

mation has received considerable interest since the earlynineties, especially in the signal processing and opticscommunities, and it arises repeatedly in these papers.The Wigner distribution of the fractional Fourier trans-form is a rotated version of the Wigner distribution of theoriginal function. Furthermore, the integral projection ofthe Wigner distribution onto any oblique axis in phasespace yields the squared magnitude of the associated frac-tional Fourier transform of the function. It is not sur-prising, therefore, that the fractional Fourier transform isso widely used here. For the analysis of discrete signals,several of the papers propose discrete analogs of this andother transforms that act on models of discrete phasespaces. Another unifying theme relates to the Gaussianprofiles that couple coherent-state representations, Gabortransforms, and windowed Fourier transforms, amongother spectrograms.

We hope that, given the breadth of this research, mostof the readers of the Journal of the Optical Society ofAmerica A will find in these papers insights and gemsthat spark their own ideas and interests. The topic fea-tured in this special issue is so evidently a rich field, andit continues to be enriched by the efforts of many re-searchers, including the authors of these papers. We aregrateful for their contributions to this collection and tothe staff at the OSA manuscript office, who were also es-sential to its creation.

Gregory William ForbesV. I. Man’ko

Haldun M. OzaktasR. Simon

Kurt Bernardo WolfFeature Editors

Wigner Distributions andPhase Space in Optics