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Journal of the Optical Society of America A

Coherence and the spatial distribution of intensity

F. Gori, Giorgio Guattari, Massimo Santarsiero

Published Apr 1, 1993• 47 citations
Publisher

Abstract

We address the following problem: Can two wave fields with different coherence properties produce the same optical intensity everywhere in the space? Limiting ourselves to paraxial propagation, we prove that in the one-dimensional case the answer is negative. On the other hand, in the two-dimensional case we show through examples that the answer is affirmative. Some consequences are discussed.

DOI
10.1364/josaa.10.000673

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We address the following problem: Can two wave fields with different coherence properties produce the same optical intensity everywhere in the space? Limiting ourselves to paraxial propagation, we prove that in the one-dimensional case the answer is negative. On the other hand, in the two-dimensional case we show through examples that the answer is affirmative. Some consequences are discussed.
For the nondispersive propagation of waves in one dimension, there is no essential difference between the concepts of longitudinal and temporal coherence, and these terms are often used interchangeably. For the dispersive propagation of waves, such as electrons or neutrons in vacuo, or light in a medium, this is no longer the case. We analyze the coherence properties of a dispersively propagating beam of radiation, and discuss the observation of its spatial and temporal coherence properties by means of a two-beam interferometer.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
Beam propagation beyond the paraxial approximation is studied in an optically written waveguide structure. The waveguide structure, which leads to diffractionless light propagation, is imprinted on a medium consisting of a five-level atomic vapor driven by an incoherent pump and two coherent spatially dependent control and plane-wave fields. We first study beam propagation in a single optically written waveguide and find that the paraxial approximation does not provide an accurate description of the probe propagation. We then employ coherent control fields such that two parallel and one tilted Gaussian beams produce a branched waveguide structure. The tilted beam allows selective steering of the probe beam into different branches of the waveguide structure. The transmission of the probe beam for a particular branch can be improved by changing the width of the titled Gaussian control beam as well as the intensity of the spatially dependent incoherent pump field.
The premise of this submission is to identify the role of the Wigner function [1] in paraxial wave optics, where it can be naturally introduced on the basis of the analogy between the paraxial wave equation and the Schrodinger equation of nonrelativistic quantum mechanics. Also known as the Wolf function, it describes the cross section of a paraxial beam using a real-valued function of displacement from axis and transverse components of the wave vector. This can be perceived as a quasi-probability distribution over the system's phase space, following the dynamics of a state space density of an ensemble of free noninteracting particles. In a (2+1)–dimensional picture this corresponds to the propagation of straight linear rays where the Wolf function describes the local phase space intensity. There is no notion of relative phase but negative values can be present, providing an alternative basis for description of interference phenomena. This corresponds exactly to the concept of pencils of positive and negative rays, developed by E. C. G. Sudarshan in the late 70’s and 80’s [2, 3].