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Journal of Modern Optics

Higher-order Coherence Functions in the Space-frequency Domain

G. S. Agarwal, E. Wolf

Published Aug 1, 1993• 14 citations
Publisher

Abstract

Some years ago a new coherence theory in the space-frequency domain was formulated. Subsequent research has revealed some difficulties regarding the representation, in this theory, of the higher-order correlation functions. In the present paper the origin of these difficulties is discussed and a new representation of the higher-order correlation functions is introduced, in terms of the coherent modes of the second-order theory.

DOI
10.1080/09500349314551551

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Some years ago a new coherence theory in the space-frequency domain was formulated. Subsequent research has revealed some difficulties regarding the representation, in this theory, of the higher-order correlation functions. In the present paper the origin of these difficulties is discussed and a new representation of the higher-order correlation functions is introduced, in terms of the coherent modes of the second-order theory.
In Part I of this investigation [ WolfE., J. Opt. Soc. Am.72. 343 ( 1982)] new representations were introduced for the cross-spectral density of a steady-state source of any state of coherence. The central concept in that formulation was the notion of a coherent source mode (a natural mode of oscillation). In the present paper the theory is developed further and new representations are obtained for the cross-spectral densities of all orders, both of the source and of the field that the source generates. These representations involve only the previously introduced coherent source modes and the moments of certain random coefficients that characterize the statistical properties of the source. The results provide a new mathematical framework for analyzing coherence properties of all orders of stationary sources and of stationary fields. Some potential applications of the theory are mentioned.
A new representation is obtained for the cross‐spectral density function of stationary optical wavefields. We show that under very general conditions it is possible to construct an ensemble of monochromatic wave functions, all of the same frequency ω, with the property that the cross‐spectral density function W(r−1,r−2,ω) of the field is expressible as a correlation between the values of the wave functions at the points r−1 and r−2. The representation is intimately related to a new mode expansion of the field and has a number of potential applications to various problems that involve wavefields of any state of coherence.
An account is given of optical coherence theory as it existed in the 1950's when, essentially, only thermal sources and slow detectors were available and it sufficed to characterize the coherence and polarization properties of an electromagnetic field by second-order correlation functions. The necessity for a more sophisticated specification of the coherence properties of fields, involving the use of higher-order correlation functions, first hinted at by the Hanbury-Brown-Twiss experiments and emphasized by the advent of the laser, is then introduced. Both classical and quantum mechanical treatments are discussed, each leading, in almost all cases of practical interest, to similar results. It is seen that the development of fast detectors enabled the higher-order correlation functions to be measured using photon counting techniques and an account of experimental work verifying the complete theory using such techniques is given. The review concludes with a discussion of some developments in associated areas whose origin lies in the resurgence of interest in coherence theory during the last decade.