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Physical Review Letters

Relationship between Complete Coherence in the Space-Time and in the Space-Frequency Domains

Mayukh Lahiri, E. Wolf

Published Aug 2, 2010• 15 citations
Publisher

Abstract

We present some new results relating to properties of completely coherent optical fields. Our analysis elucidates the relationship between the theories of such fields in the space-time and in the space-frequency domains. We also show that the concept of cross-spectral purity, introduced by L. Mandel many years ago, plays an important role in clarifying this relationship.

DOI
10.1103/physrevlett.105.063901

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We present some new results relating to properties of completely coherent optical fields. Our analysis elucidates the relationship between the theories of such fields in the space-time and in the space-frequency domains. We also show that the concept of cross-spectral purity, introduced by L. Mandel many years ago, plays an important role in clarifying this relationship.
The concept of cross-spectral purity of light, established by Mandel for stationary optical fields, is extended to nonstationary fields. Separability conditions that ensure the cross-spectral purity of nonstationary light are derived in the space-frequency and space-time domains. It is also shown that the property of cross-spectral purity is not strictly preserved on propagation of fields with any appreciable spectral bandwidth. Further, we introduce a method for generating cross-spectrally pure (stationary and nonstationary) fields from spatially incoherent light sources by means of achromatic Fourier-transform systems.
Cross-spectral purity, a concept introduced by Mandel in 1961, refers to an optical field having identical spectra at two points whose superposition yields the same spectrum. Here, we generalize this concept after discretizing the modal structure of the field, so that cross-purity can then apply to any optical degree-of-freedom (DoF). Making use of two binary DoFs (polarization and two spatial modes), the separability of the associated 4×4 coherence matrix is the condition for cross-purity. We find that the rank of the coherence matrix (the number of non-zero eigenvalues) plays a surprising yet crucial role in determining optical cross-purity: whereas rank-1 and rank-2 fields are always cross-pure, rank-3 fields in contrast are never cross-pure, and only a subset of rank-4 fields are cross-pure.
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.
Cross-spectral purity was conceived in the context of coherence research and has remained largely unknown to the wider optics community. Over the past few decades, coherence researchers have made progress in uncovering the intricate properties of this fundamental concept. Cross-spectral purity has been shown to be intimately connected to the concept of spatiotemporal separability, generalizing the concept of separability from fields to correlation functions. It has an observable effect on a multitude of experiments, and purity is in fact assumed in many well-known experiments. In this tutorial, we discuss the basic definitions related to cross-spectral purity and investigate some special cases. Additionally, we show how to generate and detect this property, and consider its experimental significance in areas such as ultrashort pulses, quantum optics, scattering, and ghost imaging.