Chapter 5 - The Structure of Partially Coherent Fields
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 satisfies a pair of wave equations and the cross-spectral density function 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 where the eigenvalues are non-negative quantities. The modes 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.
References (0)
Cited by (118)
Research progress on manipulating spatial coherence structure of light beam and its applications
2023, Progress in Quantum ElectronicsMeasurement sensitivity of DUV scatterfield microscopy parameterized with partial coherence for duty ratio-varied periodic nanofeatures
2022, Optics and Lasers in EngineeringDevelopment and characterization of a source having tunable partial spatial coherence and polarization features
2020, OptikCitation Excerpt :High spatial coherence leads to interference of beam with its scattered or reflected parts leading to a speckle pattern [3]. This speckle pattern sometimes works as a deterrent and reduces the applicability of laser in several important areas such as high resolution optical imaging, optical communication through atmosphere etc [4,5]. In addition, a single mode laser field (highly coherent and monochromatic) on propagation through atmosphere travels in single path, produces fluctuations in intensity due to speckle formation, making the communication noisy, error prone and unreliable [5–7].
SEM Nano: An Electron Wave Optical Simulation for the Scanning Electron Microscope
2022, Microscopy and MicroanalysisFlexible Construction of a Partially Coherent Optical Array
2024, Photonics