Elsevier

Progress in Optics

Volume 55, 2010, Pages 285-341
Progress in Optics

Chapter 5 - The Structure of Partially Coherent Fields

https://doi.org/10.1016/B978-0-444-53705-8.00005-9Get rights and content

Publisher Summary

The general framework of optical coherence theory is now well established and has been described in numerous publications. This chapter provides an overview of recent advances, both theoretical and experimental, that have been made in a number of areas of classical optical coherence. These advances have been spurred on by the introduction of the space-frequency representation of partially coherent fields, and an increased emphasis on the spatial coherence properties of wave fields. The fundamental experiment to measure spatial coherence is Young's double-slit experiment. A number of important optical processes are influenced by the coherence properties of the wave field. Results relating to the propagation of partially coherent wavefields highlight some of the significant results relating to optical beams. The influence of coherence on focusing is summarized and reviewed, along with the scattering of partially coherent wave fields and its relation to inverse scattering problems is discussed. It has been shown that spatial correlation functions have interesting topological properties associated with their phase singularities; these properties and the relevant literature are discussed. The coherent mode representation and its applications are described and several techniques for the numerical simulation of wave fields with a prescribed statistical behavior are explained.

Section snippets

PREFACE

The general framework of optical coherence theory is now well established and has been described in numerous publications (see Beran & Parrent, 1964; Born & Wolf, 1999; Goodman, 1985; Mandel & Wolf, 1995; Marathay, 1982; Perina, 1985; Schouten & Visser, 2008; Troup, 1967; Wolf, 2007b). In this article, we provide an overview of recent advances, both theoretical and experimental, that have been made in a number of areas of classical optical coherence. These advances have been spurred on by the

THE SPACE-FREQUENCY REPRESENTATION

Optical coherence theory is the study of the statistical properties of light and their influence on the observable characteristics of optical fields. The beginnings of coherence theory can be traced back to Verdet (1865), who estimated the spatial coherence of sunlight on the Earth's surface, and van Cittert (1934) and Zernike (1948), who calculated the evolution of the spatial coherence of light propagating from an incoherent source.1

PARTIALLY COHERENT FIELDS IN YOUNG'S EXPERIMENT

The state of coherence of a wave field is intimately related to its ability to form an interference pattern. The relation between the visibility of the fringes that are produced in Young's celebrated experiment (see Young, 1804 and Young, 1807) and the state of coherence of the field at the two pinholes was first studied by Zernike (1938).3 To see this relation in the

THE EVOLUTION OF PARTIALLY COHERENT BEAMS

In was noted in Section 2 that coherence functions obey certain propagation equations: the mutual coherence function Γ(r1,r2,τ) satisfies a pair of wave equations and the cross-spectral density function W(r1,r2,ω) satisfies a pair of Helmholtz equations. The coherence functions therefore have a well-defined behavior as they propagate; however, other properties derived from those coherence functions are not solutions of a differential equation and can evolve in nontrivial and unexpected ways on

FOCUSING OF PARTIALLY COHERENT WAVE FIELDS

The classical theory of focusing deals with monochromatic wave fields that can be scalar or vectorial in nature; an excellent overview is given by Stamnes (1986). In the present section, we examine the focusing of partially coherent scalar fields. In particular, the effect of the state of coherence of the field in the exit pupil on the distribution of the spectral density and the coherence properties of the field in the focal region will be discussed. In addition, the focal shift phenomenon

SCATTERING OF PARTIALLY COHERENT WAVE FIELDS BY RANDOM AND DETERMINISTIC MEDIA

The scattering of wave fields by a particulate medium such as, for example, a colloidal suspension, is a problem of fundamental importance. The seminal work by Mie (1908) laid the groundwork for an entire field of study. Here we review the scattering of partially coherent fields from both deterministic and random media. Also, the role of coherence in inverse problems is explained.

We consider first a fully coherent field, propagating in free space, that is incident on a deterministic scatterer.

PHASE SINGULARITIES OF COHERENCE FUNCTIONS

Researchers have long noticed that the phase of a wave field has an unusual behavior in the neighborhood of its zeros of amplitude; an early instance of this was described by Sommerfeld (1964) in his textbook on optics. Looking at the structure of a wave field consisting of plane waves of different frequency and direction, he noted that in most regions the field behaves locally like a plane wave, the exception being the behavior in the neighborhood of zeros. He concluded,

However, just because

THE COHERENT MODE REPRESENTATION

The coherent mode representation introduced by Wolf (1982)4 is the expansion of the cross-spectral density function of a partially coherent source or partially coherent field into a diagonal representation of orthogonal modes, of the form W(r1,r2,ω)=nλn(ω)ϕn(r1,ω)ϕn(r2,ω), where the eigenvalues λn(ω) are non-negative quantities. The modes ϕn(r,ω) are orthogonal with

NUMERICAL SIMULATION OF PARTIALLY COHERENT FIELDS

As already noted, determining the free-space propagation of a partially coherent field typically involves the evaluation of one or more four-fold integrals, a difficult prospect even with modern computing power. The application of the coherent mode representation (described in Section 8) allows one to reduce the problem to a finite sum of two-fold integrals, but even these integrals may be difficult to evaluate – and the coherent mode representation of the field may not be available. Because of

DIRECT APPLICATIONS OF COHERENCE THEORY

The theory of optical coherence plays an important indirect role in many optical applications in which the statistical properties of light must be understood in order to evaluate their effect on system performance. There are also a number of applications, however, based directly on the manipulation of the state of spatial and temporal coherence.

Several of these applications have been touched upon in previous sections. For instance, it has been suggested by Gbur and Visser (2003a), Pu et al.

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