What is the gain in performing a dual cross correlation and averaging the two cross correlation functions - What are the dangers ?

There seems to be a certain misunderstanding in the effective benefits of  computing both cross correlation functions, namely the one obtained from correlating input 0 with input 1 and simultaneously the one obtained by correlating input 1 by input 0. Averaging these two is believed to strongly reduce the noise on the resulting correlation function.

Clearly, such a claim needs a defined statistical basis to ensure it is valid, and in here, we will try to evaluate the advantages (if any) and disadvantages of using such a scheme.

At first, and to define the precise historical context, we have to refuse that this scheme is at all new - the ALV-5000 Correlator Series allowed the simultaneous computation of both cross correlation since it’s first versions back in the late 80’s of the last century and likewise the averaging of these two could have been performed as well - however, there is good reason not to do so - at least not blindly, since certain, rather strict, conditions have to be met not to obtain unwanted distortions.

Stationary, Ergodic Processes

The first class of statistical processes we want to focus on are called “Wide Sense Stationary” - these are, at least for the sake of this article, all such processes which have fluctuation times smaller than the actual measurement duration and do not change their statistical property different from changes which are zero on average, faster than the actual measurement duration and bounded in their higher moments for measurement time -> infinity. Typical members of this class are the statistical processes created by scattering experiments on freely diffusing particles (DLS, DWS). Well known non-members are number fluctuation experiments (such as FCS) or scattering experiments on non ergodic samples, such as gels.

For a stationary and ergodic process, one can separate two distinct noise contributions on the correlation function - those caused by the limited number of “information transmitters” (e.g. the photons for scattering experiments) and those caused by the limited time of observation of the statistical process itself. Since most correlation experiment deal with photons from a either a scattering experiment or fluorescence photons, the first contribution is usually refered to as “Photon Noise” and the second contribution is usually refered to as “Diffusion Noise” or “Intensity Noise”.

It is, without deep algebra involved, easy to see, that these two noise contributions are principally independent from each other - no matter how many photons we use as “information transmitter”, the noise will, from a certain point on, simply be limited by the diffusional noise. Vice versa, the diffusional process is totally independent from the illumination and thus the number of photons scattered (at least, in first order - this assumption will fail for extremely high light densities, such as 100 mW laser power focused to 10 µm spot size, for example, since the light pressure on the particle may start to hinder the free diffusion).

Naturally, computing the two cross correlation functions in the case of a “Wide Sense Stationary” process will process two correlation functions which are strongly dependent in the diffusional noise (since the same statistical process has been observed) and strongly independent in the photon noise. Strong dependency in the diffusional noise is strictly necessary, without, two more or less different correlation functions would be obtained and the average of both correlation functions would show severe distortions.

Quite obviously, this means, that the photon noise contribution is the only contribution which can be reduced by averaging both cross correlation functions, the diffusional noise will at best stay the same. Unfortunately, the photon noise contribution seldom is the major noise source - as a rule of thumb, if the experiment yields more than 10 photons / fluctuation time, the photon noise already becomes small compared to the diffusional noise. This limit is easily reached though - for the usual scattering experiments, fluctuation times of a few 10 µs are already fast and these require count rates of a few 10 kHz only to no longer being photon noise limited. However, what could be the maximum gain in the photon noise for a strongly photon noise limited situation ? The answer is easy to give - if we assume the correlation function to be strongly photon noise limited we gain exactly a factor of two in the variance, or a factor of SQRT(2) in the Standard Deviation. There is no way of performing any better. However, a Multiple Tau Digital Correlator quickly reduces the photon noise contribution anyway, namely by increasing the sampling with the lag time and besides “esoteric” experimental conditions (say 500 cps, 1 µs fluctuation time), a Multiple Tau Correlation function simply can not be photon noise limited over the entire lag time regime ! This means, the gain in practice must be much lower, and in fact it is - below you will find the effective gain in using the average of the two cross correlation functions over just using one cross correlation function. All data was measured using ALV-6010-160s and recomputed for the different ALV-correlators:

Example 1, DLS, Silica Sample, 500 kHz count rate, multi-component, fluctuation times 10 µs and  500 µs,

Starting Sampling Time Max. Gain in Std.Dev Averaged Gain in Std.Dev from t= 0 ... 10 x fluctuation time 3.125 ns (ALV-7004/FAST) 14 % 4 % 6.25 ns (ALV-6010-160) 10 % 3 % 25 ns (ALV-7004 or ALV-6000) 5 % below 1 % 125 ns (ALV-5000/EPP) 2.2 % much below 1 %

Clearly, for the above sample, the increase in the statistical accuracy is small, much smaller than the expected 41 % increase in Std.Dev. In fact, it is so small, that the natural change in the actual realization of the statistical process will typically show larger variation than the gain. It is hard to believe, that this scheme would indeed yield to any better result than simply using one of the cross correlation functions for this example. Still, the ALV-MultiCorr Series (ALV-7004 and ALV-7004/FAST) offers the parallel computation of both cross correlations (thus CROSS 0/1 and CROSS 1/0, for example) anyway and these can be averaged whenever desired. The same holds true for the ALV-5000/60X0 Series in DUAL mode.
 

Non Stationary, Non Ergodic Processes

For non ergodic and/or non stationary processes, the two cross correlation functions are simply no longer identical in their expectation values and averaging them leads to a “synthetic” correlation function which can show severe distortions in comparison to the expectation of either or cross correlation function. In these cases (FCS, for example), this scheme must not by applied !

”Pseudo Cross Correlation Detectors”

If a “pseudo cross correlation detector”, such as the ALV / SO-SIPD or the ALV-Dual APD is used, this scheme can usually safely be applied besides the sub 100 ns lag time range. Below this, “detectors see each other” effects come into play, in particular for APD based detectors at low light levels, because these detectors emit light in the avalanche generation process, which, with some probability will be detected by the “other” detector via the beam splitting mechanism (no matter if fiber or standard optics be used). Clearly, in this case the two cross correlation will not at all be identical in expectation for this lag time range. Notably, the gain in statistical accuracy will be identical for the pseudo-cross correlation scheme to the one outlined in the example, not better, not worse. Very good beam splitting devices and mechanics must be used in addition, because every “instationarity” in the detection process from one detector to the other will principally yield to the same problems as in the case of instationary processes. Since the gain in statistical accuracy is usually very little only, ALV advises it’s customers not to average the two cross correlation functions - simply to be on the safe side. Still, the ALV-Correlator Software for WINDOWS has full support for this scheme as well. Nevertheless, for Static Light Scattering experiments, the first order estimator (average count rate) always will be computed as the sum of both input channel contribution.
 

Use High Quantum Efficiency Detectors instead !

The best way to avoid photon noise limitation of the experiment is to use the highest Q.E. detectors. The more photons, the better. Since the increase in Q.E. for standard detectors (e.g. PMT) to high Q.E. detectors (e.g. APD) can be as much as a factor of 20 for 633 nm wavelength (and even dramatically more for the near IR), it should be clear, that not the at best 41% gain in Std.Dev. using the cross correlation averaging scheme makes a certain (extremely rare) experiment feasible, but, if at all, the gain of 20 in count rate using better detectors (which, by the way, would increase the statistical accuracy at best by a factor of 20 in Std.Dev or a factor 400 in variance). If an experiment would need a month run time to obtain reasonable results with a PMT, this could be reduced to half a month using the averaged cross correlation scheme - but it would take just below 2 hours when using an APD instead !