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Journal de physique

Harmonic analysis and P-representation of generalized Gaussian fields

M. Rousseau

Published Jan 1, 1974• 0 citations
Publisher

Abstract

We establish classically the probability distribution (P-distribution) for the complex amplitudes of multimodal and unimodal optical fields belonging to the class of generalized Gaussian fields recently introduced. From this result follows immediately the unique P-representation for this class of fields. However, all the generalized Gaussian fields, except the chaotic field, are not fully stationary. To each of them we can associate a stationary equivalent field which possesses the same intensity. We give the P-representation of this stationary equivalent field in the monomode case.

DOI
10.1051/jphys:01974003503019300

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We establish classically the probability distribution (P-distribution) for the complex amplitudes of multimodal and unimodal optical fields belonging to the class of generalized Gaussian fields recently introduced. From this result follows immediately the unique P-representation for this class of fields. However, all the generalized Gaussian fields, except the chaotic field, are not fully stationary. To each of them we can associate a stationary equivalent field which possesses the same intensity. We give the P-representation of this stationary equivalent field in the monomode case.
In the general theory of optical coherence, the following problem discussed in the present paper, arises: to determine the statistical properties of a field represented by an analytic signal from the knowledge of the statistical properties of the corresponding real field. It is shown by the use of the characteristic functionals that in order to determine the joint probability distributions of the complex field at N space-time points, the knowledge of the complete statistical description of the real field is required; on the other hand, the moments of the complex field up to that order can be determined from the knowledge of the moments of the real field up to the same order. The results are illustrated by explicit calculations relating to the Gaussian random process, which, as is well known, characterizes the fluctuations of thermal light. A converse of a well-known theorem of Kac and Siegert relating to a Gaussian random process is derived as an immediate consequence of our analysis.
Stationary Gaussian optical fields are also often called thermal fields because they are generated by natural or thermal sources. By using the amplitude of the field, we show that thermal fields are only a particular case of the more general class of Gaussian fields. Such nonthermal Gaussian fields can be obtained experimentally, and we calculate some properties, concerning particularly the effect of Hanbury Brown and Twiss, photocounting, and coincidence experiments. We show that the thermal fields are the less chaotic or incoherent Gaussian fields. Finally, we introduce pseudo-Gaussian fields which appear in some experiments of diffusion with laser light. They are non-Gaussian fields, but their intensity has the same properties as that of a Gaussian field.
We develop a general representation for ensembles of non-stationary random pulses in terms of statistically uncorrelated, time-delayed, frequency-shifted Gaussian pulses which are classical counterparts of coherent states of a quantum harmonic oscillator. We show that the two-time correlation function describing second-order statistics of the pulses can be expanded in terms of the complex Gaussian pulses. We also demonstrate how the novel formalism can be applied to describe recently introduced Gaussian Schell-model pulses and pulse trains generated by typical mode-locked lasers.
This paper is concerned with the determination of the quantum statistics of the photoelectric detection, and the E-field detection of multiple-mode superposed chaotic and coherent radiation. The Glauber P-representation is utilized extensively, and results are obtained by a fully quantum-mechanical treatment. The generating function, the mean, and the variance of the photocount statistics are derived for illumination times of arbitrary duration and for modes of arbitrary statistical interdependence. For short illumination times, the complete photocount distribution is obtained. The distribution has the same form as that for a single mode of superposed chaotic and coherent radiation. The multiple-mode distribution, however, is nonstationary. The difference between experimental determinations of the mean and variance which utilize periodic sampling and those which utilize random sampling is derived. A method for determining the P-representation for modulated fields is described, and the detection statistics are derived. The statistics of the E field are found to be Gaussian, and it is found that for multiple-mode classical statistics and quantum statistics differ.
The comparison between the classical and quantum theories of optical coherence is presented by using an idea used in the problems of modulation of light beams of classical fields. In such problems it is assumed that it is possible to vary the mean light intensity of a beam without changing its statistical properties defined by a set of coherence functions. We generalize this idea and introduce precisely the class of optical fields which are consistent for modulation. It is shown that all the quantum fields of this class have a positive PPP representation and are strictly equivalent to classical fields. Moreover, when the field is assumed to be stationary, an interpretation of this condition is given which in particular makes precise the relations between photon-counting and light-intensity measurements. Finally it is shown that all the quantum fields without PPP representation cannot be consistent for modulation, and the condition of consistency for modulation appears as a characteristic property of fields strictly equivalent to classical ones.