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Journal of the Optical Society of America A

Nonstationary elementary-field light randomly triggered by Poisson impulses

Carlos R. Fernández‐Pousa

Published Apr 18, 2013• 5 citations
Publisher

Abstract

A stochastic theory of nonstationary light describing the random emission of elementary pulses is presented. The emission is governed by a nonhomogeneous Poisson point process determined by a time-varying emission rate. The model describes, in the appropriate limits, stationary, cyclostationary, locally stationary, and pulsed radiation, and reduces to a Gaussian theory in the limit of dense emission rate. The first- and second-order coherence theories are solved after the computation of second- and fourth-order correlation functions by use of the characteristic function. The ergodicity of second-order correlations under various types of detectors is explored and a number of observables, including optical spectrum, amplitude, and intensity correlations, are analyzed.

DOI
10.1364/josaa.30.000932

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A stochastic theory of nonstationary light describing the random emission of elementary pulses is presented. The emission is governed by a nonhomogeneous Poisson point process determined by a time-varying emission rate. The model describes, in the appropriate limits, stationary, cyclostationary, locally stationary, and pulsed radiation, and reduces to a Gaussian theory in the limit of dense emission rate. The first- and second-order coherence theories are solved after the computation of second- and fourth-order correlation functions by use of the characteristic function. The ergodicity of second-order correlations under various types of detectors is explored and a number of observables, including optical spectrum, amplitude, and intensity correlations, are analyzed.
We examine the coherence properties and photon statistics of stationary light obtained by the superposition of nonstationary emissions occurring at random times, in accordance with a homogeneous Poisson point process. The individual emissions are assumed to be in a coherent, chaotic, or nnn state. The statistical nature of the emission times results in fluctuations of the relative contributions of different emissions at a given time. This is manifested by an additional positive term, exhibiting particlelike properties, in the normalized second-order correlation function. Thus, the photon-counting variance is increased. For coherent emissions, interference between the randomly delayed emissions produces additional wavelike noise. In the limit when the emissions overlap strongly, the field exhibits the correlation properties of chaotic light, regardless of the statistics of the individual emissions. In the opposite limit, when emissions seldom overlap, the light intensity is describable by a shot-noise stochastic process, and the detected photocounts show an enhanced particlelike noise, which has its largest value when the counting time is long. In that limit, the photocounts obey the Neyman type-AAA and generalized Polya-Aeppli distributions, when the individual emissions are coherent and chaotic, respectively. When the individual emissions correspond to the nnn state, the Poisson emission times result in bunching which reduces or eliminates the inherent antibunching associated with the nnn state.
We examine the coherence properties of stationary light obtained by the superposition of nonstationary independent emissions occurring at random space–time points. The positions of the points are independent and uniformly distributed over the volume of the source. The emission times fluctuate in accordance with a stationary renewal point process. This process admits of super-Poisson, sub-Poisson, or Poisson behavior. The individual emissions are assumed to be in a coherent, chaotic, or n state. The statistical nature of the random-emission space–time points plays an important role in determining the coherence properties and photon statistics of the total field. This is manifested in the normalized second-order correlation function, which turns out to have the usual form for chaotic light with two additional terms. The first of these is determined by the statistical nature of the individual emissions (it is positive for coherent or chaotic, but zero for single-photon, emissions). The second term is governed by the statistics of the primary excitations (it is positive for super-Poisson, zero for Poisson, and negative for sub-Poisson excitations). Both additional terms become small for light with a high degeneracy parameter (many total photons per emission lifetime). The behavior of the light is then asymptotically chaotic. In the opposite limit, when the degeneracy parameter is small (or the emissions are instantaneous), the correlation properties of the primary excitations are directly transferred to the correlation properties of the photons. The first-order spatial-coherence properties of the field turn out to be identical with those of chaotic light (i.e., the van Cittert–Zernike theorem is obeyed), although the second-order properties differ. The photon-counting distribution reflects the character of the correlation function. Thus sub-Poisson primary excitations, together with n-state emissions, can lead to sub-Poisson photon counts under certain conditions. Such nonclassical light may be made arbitrarily intense if interference effects are eliminated by detecting many spatial modes. As an example of our theory, we cite a Franck–Hertz experiment excited by a space-charge-limited electron beam. If the electron excitations are represented as a sub-Poisson renewal point process, and the photon emissions as one-photon states, the light generated should be antibunched and sub-Poisson.
A representation of the mutual coherence function (MCF) of a light pulse as an incoherent sum of partially-coherent elementary pulses is introduced. It is shown that this MCF can be decomposed into fully and partially-coherent constituents and three different pulse models of partially-coherent constituents are constructed: single elementary-pulse fluctuations, emission of elementary fields driven by white noise, and elementary pulses triggered by Poisson impulses. The fourth-order correlation function of this last model includes as limit cases those of the fluctuating-pulse and noise-driven-emission models. These results provide a means of extending elementary-field models to higher-order coherence theory.
A general theory of the intermediate state interaction in the second order optical process is developed by extending the stochastic theory of line shape. We introduce the stochastic models of the intermediate state interaction and discuss the general features of the secondary emission spectrum, focusing particular attention on how the correlation between the incident and the scattered photons are affected by the intermediate state interaction. It is clarified how the spectral features or the relative weights among the various components of the second order optical process are determined by a few physical parameters; the radiative damping rate and the modulation amplitude and rate. Further we discuss the relationships between the first and the second order optical processes.
Random fluctuations in the phase and the amplitude of a source field induce random fluctuations in the field intensity at the network output, which in turn give rise to electrical noise in the detector current. The characteristics of this source-induced noise depend on the properties of the source and on the network transfer function. In the present study we consider a stationary optical source with a finite coherence time coupled to an arbitrary, linear, time-independent network. General expressions for the power-spectral density of the source-induced noise are derived in terms of the network transfer function and the second- and the fourth-order correlation functions of the source field in the frequency domain. The incoherent limit and the case of a source that has two uncorrelated polarization modes are considered in detail, and a simple method for the estimation of the source-induced noise is suggested.
A general theory of transient light scattering is developed on the basis of our stochastic theory of intermediate state interaction. The transient behaviors of various components of the second order optical process under a pulsed irradiation are discussed and model calculations of the time-resolved emission spectra are performed. It is clarified that in the fast modulation limit the Raman component has a time constant determined by the phase memory time of the intermediate state while the time constant of the luminescence component is given by the radiative damping rate. The transient behaviors of the scattered light intensity are also discussed.