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.
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.
Introduction In the preceding chapters we have been largely concerned with the simplest coherence effects of optical fields, namely those which depend on the correlation of the field variable at two space-time points ( r 1 , t 1 ) and ( r 2 , t 2 ). As we have seen, these effects include the most elementary coherence phenomena involving interference, diffraction and radiation from fluctuating sources. In this chapter we present an extension of the theory to cover more complicated situations, which have to be described by correlations of higher order, i.e. by correlations of the field variables at more than two space-time points or the expectation values involving various powers and products of the field variables. Situations of this kind have become of considerable importance since the development of the laser and of nonlinear optics. The basic difference between the statistical properties of thermal light and laser light can, in fact, only be understood by going beyond the elementary second-order correlation theory. Many of the higher-order coherence phenomena are most clearly manifest in the photoelectric detection process, which can only be adequately described by the quantum theory of detection or by taking into account the quantum features of the field, both of which will be studied in the succeeding chapters. However, because the field is still described classically in the semi-classical theory of light detection, and also because the classical field description provides a natural stepping stone to the quantum description of field correlations, we will now discuss the general description of field correlations of all orders on the basis of the classical theory of the fluctuating wavefield.
Photoelectric correlation measurements of optical fields are now becoming commonplace and, as is well known, they require fourth order correlation functions of the field for a description. Nevertheless, the fourth order theory of optical coherence has not received a great deal of attention. This is the subject of the present paper. When interference effects are combined with intensity correlations, we encounter correlation functions of the general form Γ(2,2) ≡〈V* (r−1,t1)V* (r−2,t2)V (r−3,t 3)V (r−4,t4)〉, which are functions of four space‐time arguments. Some properties of such fourth order correlations, as they relate to interference and intensity correlation effects, are derived. The space arguments are generally suppressed in the calculation. The correlation functions are expressible as multiple Fourier transforms of certain fourth order spectral densities, which are functions of three frequencies in the stationary state. The superposition law for fourth order spectral densities is derived for independent light beams, and it is shown that the fourth order spectral densities exhibit the reproducing property under superposition only when the light obeys thermal statistics. Next the superposition law is found for the spectral densities of the photoelectric signals generated by light beams falling on photodetectors. Investigation of the photoelectric signal spectrum yields information on the spectrum of the light intensity fluctuations, and removes the measurement from the domain of optics to the domain of electronics. It is shown that the photoelectric spectral densities reproduce under superposition when the two light beams in question obey a certain factorization condition, that is analogous to the cross‐spectral purity condition for second order correlations. Finally, it is shown that, under certain circumstances, all the relevant fourth order correlation properties of the optical field are expressible in terms of simpler two‐frequency spectral functions.
An account is given of optical coherence theory as it existed in the 1950's when, essentially, only thermal sources and slow detectors were available and it sufficed to characterize the coherence and polarization properties of an electromagnetic field by second-order correlation functions. The necessity for a more sophisticated specification of the coherence properties of fields, involving the use of higher-order correlation functions, first hinted at by the Hanbury-Brown-Twiss experiments and emphasized by the advent of the laser, is then introduced. Both classical and quantum mechanical treatments are discussed, each leading, in almost all cases of practical interest, to similar results. It is seen that the development of fast detectors enabled the higher-order correlation functions to be measured using photon counting techniques and an account of experimental work verifying the complete theory using such techniques is given. The review concludes with a discussion of some developments in associated areas whose origin lies in the resurgence of interest in coherence theory during the last decade.
The coherence properties of electromagnetic fields are reviewed, based on both the classical and quantum theories. The elementary concepts, employed frequently in the discussion of interference phenomena, are summarized. The well-known interference phenomena are described in terms of second-order coherences. The coherence theory in space-frequency domain is introduced and the coherent mode representation is presented. The generation and propagation of coherence of light are analysed and it is shown that the coherence of light is developed as light propagates. The quantum theory goes parallel with the classical theory, via the optical equivalence theorem. There are, however, certain nonclassical characteristics of light, which may not be easily understood in classical terms. These nonclassical phenomena are believed to originate from the particle aspects of light. The quantum effect on the interfernce phenomena is analysed and finally the outlook of the future research is briefly mentioned.
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.
Introduction We will now study the properties of fluctuating electromagnetic fields, paying attention mainly to the optical region of the electromagnetic spectrum. It seems hardly necessary to stress that every electromagnetic field found in nature has some fluctuations associated with it. Even though these fluctuations are, as a rule, much too rapid to be observed directly, one can deduce their existence from suitable experiments that provide information about correlations between the fluctuations at two or more space-time points. The simplest manifestations of correlations in optical fields are the well-known interference effects that arise when two light beams that originate from the same source are superposed. With the availability of modern light detectors and electronic circuitry of very short resolving time, other types of correlations in optical fields began to be studied in more recent times. These investigations, as well as the development of lasers and other novel types of light sources, led to a systematic classification of optical correlation phenomena and the complete statistical description of optical fields. The area of optics concerned with such questions is now generally known as optical coherence theory . The first investigations of coherence phenomena are due to Verdet (1865, 1869) and von Laue (1907a, b). Some early investigations of Stokes (1852) and Michelson (1890, 1891a, b, c, 1892, 1920) although not explicitly mentioning coherence – because this concept is of a much later origin – have also contributed to the clarification and development of this subject.