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Journal of Physics A Mathematical and General

Spectral analysis with short data batches (photon correlation spectroscopy)

Christopher J. Oliver

Published Apr 1, 1979• 22 citations
Publisher

Abstract

This paper studies the limiting accuracy with which power spectra and the values of spectral parameters for random signals can be achieved from short batches of data. Initially, general expressions for arbitrary data are derived; these are then restricted for simplicity to wide-band signals. In addition, comparison is made with computer simulation for two well defined models, namely, photodetection of light of constant intensity and heterodyne photodetection of narrow-band Gaussian-Lorentzian light. It is shown that there is no analytical difference between operation in time or frequency space for batch data, and also that long data sets can be analysed at least as well, in terms of the accuracy with which spectral parameters can be determined by fitting, by averaging over many short batches as by processing the set as a whole.

DOI
10.1088/0305-4470/12/4/018

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This paper studies the limiting accuracy with which power spectra and the values of spectral parameters for random signals can be achieved from short batches of data. Initially, general expressions for arbitrary data are derived; these are then restricted for simplicity to wide-band signals. In addition, comparison is made with computer simulation for two well defined models, namely, photodetection of light of constant intensity and heterodyne photodetection of narrow-band Gaussian-Lorentzian light. It is shown that there is no analytical difference between operation in time or frequency space for batch data, and also that long data sets can be analysed at least as well, in terms of the accuracy with which spectral parameters can be determined by fitting, by averaging over many short batches as by processing the set as a whole.
In recent years the utilization of "optical mixing" spectroscopic techniques in laser light scattering experiments has proved to be a successful method for the investigation of the dynamical properties of physical systems. By illuminating a photosensitive detector with the scattered light and measuring the spectrum or autocorrelation function of the resulting photocurrent, one can, in general, obtain the ensemble-average time dependence of specific collective excitations within the scattering medium. We present here a detailed quantitative analysis of the statistical errors inherent in such measurements due to the stochastic nature of both the scattering and photoemission processes. We determine the statistical errors on the optical-intensity correlation function as measured by two photocounting digital correlator models and on the intensity spectrum as measured by a "self-beat" optical mixing spectrometer. From these errors and a generalized least-mean-squares fitting procedure we calculate the uncertainty on the measured correlation time (linewidth) for the case of a Gaussian optical field with an exponential intensity correlation function (a Lorentzian spectrum). Scaling relationships are given which permit our numerical results to be applied to an arbitrary set of experimental parameters.
The authors investigate the statistical errors which arise in digital autocorrelation of photon counting fluctuations due to the finite duration of experiments. An expression for the accuracy with which individual points of the autocorrelation function of intensities can be measured is derived analytically and evaluated for Gaussian- Lorentzian light. The appropriate weighting for a 'least-squares' fitting procedure is determined and the error in spectral linewidth, obtained by this procedure from experiment, is calculated.
An analysis of the variance of the structure function coefficients of a train of digital photodetections, both unnormalized and with various normalizations, is made and compared with known corresponding results for the photon correlation function. A gaussian-lorentzian model light field is used in both direct and homodyne detection. Numerical simulations confirm and extend the analytical results, which show certain circumstances when the structure function may be a useful alternative measure of spectral content.
In the framework of the signal processing approach to single-atom resonance fluorescence with spectral resolution, we diagrammatically derive an analytical formula for arbitrary-order spectral correlation functions of the scattered fields that pass through Fabry-Perot interferometers. Our general expression is then applied to study correlation signals in the limit of well separated spectral lines of the resonance fluorescence spectrum. In particular, we study the normalized second-order temporal intensity correlation functions in the case of the interferometers tuned to the components of the spectrum and obtain interferential corrections to the approximate results derived in the secular limit. In addition, we explore purely spectral correlations and show that they can fully be understood in terms of the two-photon cascades down the dressed state ladder.
A method for computing the approximate photocount statistics for Gaussian light is presented. This method may be used for the superposition of coherent radiation with chaotic radiation of arbitrary spectral shape. Data is presented for Gaussian-, triangular-, and square-shaped spectra as well as for Lorentzian-shaped spectra. The results show that the photocount statistics are significantly dependent upon spectral shape for intermediate time-bandwidth products. The results also show that the Bedard, Chang, and Mandel approximation gives a better fit to a Gaussian-shaped spectrum then to a Lorentzian-shaped spectrum. Significant differences between the photocount statistics for a time-bandwidth product of 10 and Poisson statistics were also obtained; even for a signal-to-noise ratio of 40:1.
An expression is derived for the capacity of a noiseless photon channel as a function of bandwidth, carrier frequency, and average signal power. This expression is an extension of a formula derived by Gordon which is applicable in the limit of narrow bandwidth. By comparison of the two results, it is shown that Gordon's formula is highly accurate over a large range of bandwidth, and that significant deviations occur only when the average signal power is extremely small. The derivation is based on a model of transmitting photons at different frequencies, distinguishable to the limit of the Heisenberg principle, which allows distinguishable states to have the same energy. This degeneracy, which in the narrow-band limit is simply related to the time-bandwidth product, plays a role in the (maximum entropy) energy distributions. These distributions are shown to vary from noise-like (where the rms fluctuations exceed the mean value) to Poisson as the time-bandwidth product varies from unity (or nondegenerate) to infinity. For any degeneracy, the distribution is the discrete analog of its continuous (classical) counterpart. For infinite degeneracy, when photon frequencies may take a continuum of values, the channel capacity is achieved for a signal power spectral density which has the same form as the average energy of a quantized oscillator in thermal equilibrium. As a concluding result, it is shown that for a noiseless channel with fixed average power but without any frequency restrictions, there exists an upper bound to channel capacity which is proportional to the square root of the average signal power.