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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Noise on multiple-tau photon correlation data

Klaus Schätzel, Rainer Peters

Published Jun 1, 1991• 27 citations
Publisher

Abstract

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.

DOI
10.1117/12.44160

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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.
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.
Photon correlated beams, generated by parametric down-conversion in nonlinear crystals (PDC), have been proved successfully for the measurement of quantum efficiency, of single photon detectors [1, 2, 3, 4] and can be used for measuring transmittance, as well. The main feature relies on the realization of two correlated quantum channels yielding coincident events. To reduce uncertainty [4, 5, 6] and determine its ultimate limit, contributions to statistical noise in a real experimental set-up are here investigated, and a general model is proposed to calculate the maximum likelihood best estimators and evaluate the probability of coincidence. In typical measurement scheme a channel is set as trigger (idler, i) and attention is paid to catch at least the same amount of correlated photons in the other channel (signal, s) [4]. Unwanted uncorrelated photons due to PDC itself result in a decreased degree of correlation, noise fluctuations in coincidence measurements and asymmetry between the two channels, whose mean photon rate can be quite different. We indicate by with the mean rate of correlated photons in s and i channels We address the total mean photon rates in both channels as where terms take into account also additive background photon rates due to straylight and detector noise. Actually statistical noise is also due to losses of correlated photons by optical elements, non ideal detectors and electronic devices in both channels. We distinguish, therefore, between optical losses ( transmittance of the correlated photon path), detection losses detectorquantum efficiency), electronic losses associated to the dead times time during which the detection system is unable to detect photons). It is usual to identify dead times as non-extending and extending: in the first case, all photons following a revealed photon within the fixed time interval are ignored, while in the second case any incoming photon produces or prolongs Eventually the finite duration of the time coincidence window, modifies the distribution probability of coincident events. Let us consider the distribution probability of single channel
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.
Correlation between channels of the normalized photocount-rate correlation function g(2)(τ) becomes significant at high count rates and leads to a number of data-analysis problems. We derive an expression for channel correlation that is valid for a detector area of arbitrary extent and compare the theoretical predictions with measured values. A data-analysis procedure is demonstrated that employs the theoretical expression for channel correlation and provides a rigorous test of an assumed fitting function. The procedure facilitates the use of the cumulant method in determining the polydispersity of scatterers. An expression for the statistical bias of g(2)(τ) is also derived and compared with measured values.
Optical measurements in quantum communication and quantum radiometry are quite often based on two-photon correlated channels and the coincidence between them. To quantify the noise level in these measurements, the authors focus on the statistics of photon coincidence in a basic experimental setup, accounting for all experimental effects contributing to the noise increment. By the proposed theory the measurement performance is evaluated in terms of noise fluctuations. Particularly, an ultimate limit in noise reduction is established by the optimal estimation model herein presented.