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Quantum Optics Journal of the European Optical Society Part B

Noise on photon correlation data. I. Autocorrelation functions

Klaus Schätzel

Published Aug 1, 1990• 118 citations
Publisher

Abstract

An adequate analysis of photon correlation data requires knowledge about the statistical accuracy of the measured data. For the model of gamma-distributed intensities, that is including the effect of a finite intercept, the full covariance matrix is calculated for all the channels of the photon autocorrelation functions. A thorough discussion of multiple sample time correlation illuminates the importance of temporal averaging effects at large lag times. A practical estimation scheme is given for the noise in photon correlation data from a multiple sample time measurement.

DOI
10.1088/0954-8998/2/4/002

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An adequate analysis of photon correlation data requires knowledge about the statistical accuracy of the measured data. For the model of gamma-distributed intensities, that is including the effect of a finite intercept, the full covariance matrix is calculated for all the channels of the photon autocorrelation functions. A thorough discussion of multiple sample time correlation illuminates the importance of temporal averaging effects at large lag times. A practical estimation scheme is given for the noise in photon correlation data from a multiple sample time measurement.
The results of a theory of noise on photon correlation data is presented for the rather general case of a detector signal which can be modeled as a Gamma process. This model includes finite intercept effects and may even be used for heterodyne measurements. Multiple sample time measurements are considered with particular care and significant decreases of noise with increasing sample time are typically found at small as well as at large lag (and sample) times, if the standard normalization procedure is replaced by the symmetric or compensated normalization schemes. Large estimator covariances are observed across lag time ranges up to one coherence time. This type of correlated noise is typical for correlation measurements and may easily lead to overfitting problems. Experimental verifications are included for the noise calculations.
The standard expressions for the noise that is due to photon fluctuations in thermal background radiation typically apply only for a single detector and are often strictly valid only for single-mode illumination. I describe a technique for rigorously calculating thermal photon noise, which allows for arbitrary numbers of optical inputs and detectors, multiple-mode illumination, and both internal and external noise sources. Several simple examples are given, and a general result is obtained for multimode detectors. The formalism uses scattering matrices, noise correlation matrices, and some fundamentals of quantum optics. The covariance matrix of the photon noise at the detector outputs is calculated and includes the Hanbury Brown and Twiss photon-bunching correlations. These correlations can be of crucial importance, and they explain why instruments such as autocorrelation spectrometers and pairwise-combined interferometers are competitive (and indeed common) at radio wavelengths but have a sensitivity disadvantage at optical wavelengths. The case of autocorrelation spectrometers is studied in detail.
Experimental results for the noise of photon-correlation and photon-structure functions are presented. The noise dependences on the mean count rate and on the sampling time are investigated by means of a specifically built linear fluctuation analyzer, and the results are compared with theoretical predictions. The predicted mean count rate dependence and sampling time dependence of the noise are confirmed by the experimental analysis. Some discrepancies between analytical expressions and experimental results are observed in the crossover predictions. The results show that the structure function has better noise performance for relatively high mean count rates and for long sampling times, which strongly suggests a wider use of this statistical estimator in measurements requiring good accuracy, as in the case of uniform sampling.
Digital autocorrelation of optical signals is discussed with particular reference to the technique of clipping, reported briefly in an earlier publication. A detailed analysis of the relationship between the autocorrelation functions of clipped and unclipped photon-counting fluctuations is given. By evaluating the generating function for the joint distribution of intensity fluctuations, corrections to the earlier results are calculated which arise if the time over which the signal is sampled is not negligible compared with any coherence time of the light. The effect of clipping in heterodyne experiments is also analysed.
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.
The theoretical treatment of clipped photocount autocorrelation of Gaussian light, introduced by Jakeman and Pike, has been extended to include the effects of finite-duration sampling intervals and incomplete spatial coherence. A system of ``interval partitioning'' is used to derive theoretical expressions for 〈Nq(0)N(τ)〉 and 〈N0(0) N0(τ)〉, the single- and double-clipped photocount autocorrelation functions. Here N(t) is the number of photons detected in a sampling interval centered at t and Nq(t) is the clipped count at level q. Each original interval is systematically partitioned in space and time. Multidimensional generating functions are applied to the sets of subintervals with the object of expressing g(1) (t), the first-order field correlation function, in terms of measurable quantities. It is shown that under certain well-defined conditions, 〈Nq(0) N(τ)〉−〈Nq〉〈N〉=γq|g(1)(τ)|2 and 〈1−N0〉−2−[1−2〈N0〉+〈N0(0)N0(τ)〉]−1=φ|g(1)(τ)|2, where, for a given set of experimental parameters, γq and φ can be taken as simple multiplicative factors independent of the delay time τ. For comparative purposes, a similar analysis is performed for the unclipped correlation function 〈N(0)N(τ)〉. Finally, computer calculated values of various quantities of interest are presented for several representative cases.