In view of the misunderstanding which surrounds the use of semiclassical methods in the treatment of optical coherence problems, a simple analysis is given which illustrates the generality of these methods. In particular, it is pointed out that, under commonly occurring circumstances, these methods apply at arbitrarily low light levels, and give correct results for the photoelectric counting of photons. The conclusions are illustrated by treating the interference between laser light and thermal light to demonstrate that this nontrivial case, recently treated by quantum field theory, is correctly described by semiclassical methods.
In view of the misunderstanding which surrounds the use of semiclassical methods in the treatment of optical coherence problems, a simple analysis is given which illustrates the generality of these methods. In particular, it is pointed out that, under commonly occurring circumstances, these methods apply at arbitrarily low light levels, and give correct results for the photoelectric counting of photons. The conclusions are illustrated by treating the interference between laser light and thermal light to demonstrate that this nontrivial case, recently treated by quantum field theory, is correctly described by semiclassical methods.
The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcell's explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The representation of the classical wave amplitude, of the light by Gabor's complex analytic signal appears naturally in this treatment. It is shown that the correlation of counts registered by N separate photodetectors at N points in space is determined by a 2 N th order correlation function of the complex classical field. The variance of the individual counts is shown to be expressible as the sum of terms representing the effects of classical particles and classical waves, in analogy to a well-known result of Einstein relating to black-body radiation. Since the theory applies to correlation effects obtained with any type of light it applies, in particular, to the output of an optical maser, although, for a maser operating on one mode, correlation effects are likely to be very small.
It is pointed out that it is impossible to distinguish between photons and classical wave packets by measurement of photoelectric counting statistics for any state of a light beam for which a classical description exists. Therefore, an experiment with a thermal light source recently proposed by Karp as a test for the existence of photons can yield no information on this question, even if it should be able to distinguish between two different statistical models of a light beam. The quantum nature of the optical field can, however, be inferred from other photoelectric measurements, some of which are briefly discussed.
Introduction Up to now the electromagnetic field has been treated as a classical field, describable by c -number functions. The great success of classical electromagnetic theory in accounting for a variety of optical phenomena, particularly those connected with wave propagation, interference and diffraction, amply justifies the classical approach. Moreover, as we have seen in the preceding chapters, in some cases the classical wave theory also gives a good account of itself in the treatment of the interaction of electromagnetic fields. For example, it is able to describe such seemingly non-classical effects as photoelectric bunching and the photo-electric counting statistics. It might almost seem that there is little justification for going beyond the domain of classical wave theory in optics. On the other hand, it can be argued that optics lies well and truly in the quantum domain, in the sense that we often encounter situations in which very few quanta or photons are present. In the microwave region of the electromagnetic spectrum, and at still longer wavelengths, the number of photons in each mode of the field is usually very large, and we are justified in treating the system classically. However, in the optical region the situation is usually just the opposite. As we show in Section 13.1, for light produced by practically all sources other than lasers, the average number of photons per mode is typically much less than unity.
Recently, Chen et al. [Phys. Rev. A 84, 033835 (2011)] reported observation of anticorrelated photon coincidences in a Mach-Zehnder interferometer whose input light came from a mode-locked Ti:sapphire laser that had been rendered spatially incoherent by passage through a rotating ground-glass diffuser. They provided a quantum-mechanical explanation of their results, which ascribes the anticorrelation to two-photon interference. They also developed a classical-light treatment of the experiment and showed that it was incapable of explaining the anticorrelation behavior. Here we show that semiclassical photodetection theory, i.e., classical electromagnetic fields plus photodetector shot noise, does indeed explain the anticorrelation found by Chen et al. The key to our analysis is properly accounting for the disparate time scales associated with the laser's pulse duration, the speckle-correlation time, the interferometer's differential delay, and the duration of the photon-coincidence gate. Our result is consistent with the long-accepted dictum that laser light which has undergone linear-optical transformations is classical-state light, so that the quantum and semiclassical theories of photodetection yield quantitatively identical results for its measurement statistics. The interpretation provided by Chen et al. for their observations implicitly contradicts that dictum.
In this paper we formulate the general problem of determining the photoelectron "counting" distribution resulting from an electromagnetic field impinging on a quantum detector. Although the detector model used was derived quantum mechanically, our treatment is wholly classical and includes all results known to date. This combination is commonly referred to as the semiclassical approach. The emphasis, however, lies in directing the problem towards optical communication. The electromagnetic field is assumed to be the sum of a deterministic signal and a zero-mean narrow-band Gaussian random process, and is expanded in a Karhunen-Loève series of orthogonal functions. Several examples are given. It is shown that all the results obtainable can be written explicitly in terms of the noise covariance function. Particular attention is given to the case of a signal plus white Gaussian noise, both of which are band-limited to\pm BHz. Since the result is a fundamental one, to add some physical insight, we show four methods by which it can be obtained. Various limiting forms of this distribution are derived, including the necessary conditions for those commonly accepted. The likelihood functional is established and is shown to be the product of Laguerre polynomials. For the problem of continuous estimation, the Fisher information kernel is derived and an important limiting form is obtained. The maximum a posteriori (MAP) and maximum-likelihood (ML) estimation equations are also derived. In the latter case the results are also functions of Laguerre polynomials.
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.