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Optics Express

Degree of coherence for electromagnetic fields

Jani Tervo, Tero Setälä, Ari T. Friberg

Published May 19, 2003• 376 citations
Publisher

Abstract

The relationship between the visibility of fringes and the degree of spatial coherence in electromagnetic two-pinhole interference is assessed. It is demonstrated that the customary definition of the degree of coherence of an electromagnetic field is flawed and a new quantity, free of the formal drawbacks, is introduced. The new definition, which is shown to be consistent with known results for Gaussian statistics, has some unusual properties characteristic only for electromagnetic fields. The degree of coherence is measurable by a sequence of interference experiments.

DOI
10.1364/oe.11.001137

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We emphasize that the visibility of interference fringes is not an appropriate characteristic for the electromagnetic degree of coherence. Our definition is consistent with the notions of complete coherence. All points put forward in the Comment were already noted in our previous publication [Opt. Express11, 1137 (2003)].

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The relationship between the visibility of fringes and the degree of spatial coherence in electromagnetic two-pinhole interference is assessed. It is demonstrated that the customary definition of the degree of coherence of an electromagnetic field is flawed and a new quantity, free of the formal drawbacks, is introduced. The new definition, which is shown to be consistent with known results for Gaussian statistics, has some unusual properties characteristic only for electromagnetic fields. The degree of coherence is measurable by a sequence of interference experiments.
The first-order coherence condition is derived in terms of correlations between different field modes. Fluctuations of the electromagnetic field are shown to play an important role in the concept of coherence. The coherent states are shown to be the only ones which reduce these fluctuations to their minimum value. For the case of a field in which the different field modes are independent, the degree of first-order coherence and the visibility of interference fringes are shown to be specified completely by these fluctuations. Measurements of this visibility yield information on the statistical properties of radiation fields. For squeezed states, this visibility is expressed as a function of squeezed parameters. Moreover, the difference between generalized coherent states becomes manifest in measurements of this visibility.
Abstract A well-known result for the interference of two single-mode fields is that the degree of coherence and the degree of indistinguishability are the same when we consider the detection of a single photon. In this article, we present the relation between the degree of coherence, path indistinguishability and the fringe visibility considering interference of multiple numbers of single-mode fields while being interested in the detection of a single photon only. We will also mention how Born’s rule of interference for multiple sources is reflected in these results.
This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.
An electromagnetic Young's interference experiment is considered with arbitrary, spectrally partially coherent and partially polarized electric fields at the pinholes. Any correlations that exist at the openings lead to modulations of the Stokes parameters on the observation screen. The electromagnetic degree of coherence can be viewed, in direct analogy to the usual scalar case, as a measure of the modulation contrasts of not just the optical intensity but also the polarization state. The electromagnetic degree of coherence can be obtained experimentally via four visibility measurements using appropriate wave plates. If the field at the pinholes is fully polarized, a periodic variation of the polarization state appears on the observation screen, leading to a Pancharatnam–Berry phase. It is shown that this geometric phase has a straightforward but elegant relationship to the Stokes parameters at the pinholes.
We analyze the recently introduced Polarization contrast parameters and their relation to degrees of coherence in electromagnetic Young's interference experiment in the case of unpolarized illumination. We show that in the general case only two of the parameters can be made non-vanishing if suitable unitary transformations are applied to the field at pinholes. The result implies elegant relations between the contrast parameters and the intrinsic degrees of coherence in the case of unpolarized light at pinholes.