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It seems that a common statistical model for the count numbers of a photomultiplier is a Poisson distribution whose parameter $\lambda$ equals to the square-root of the number of counts.(e.g. http://www.sciencedirect.com/science/article/pii/S1350448711005750).

This in particular, implies that the variance of the resulting statistic

  1. increases with the number of photons to detect,
  2. is not directly dependent from the duration of the counting process.

I did not manage to find the basis of this modeling choice. If somebody has some intuitive idea or a good reference I will apreciate. (I am not physicist and maybe I make a bad interpretation of the modeling applied to PM)

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    $\begingroup$ Note that poissonian distributions only hold for some light sources (and integration times). Typically only laser or laser-like sources are poissonian, while thermal sources are super-poissonian (higher variance) and some quantum sources such as single-photon sources can be drastically sub-poissonian. $\endgroup$ – Emilio Pisanty Jan 21 '13 at 17:19
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    $\begingroup$ the absolute variance increases but the 'relative uncertainty' (square root of the variance divided by the number of detected photons) decreases with the square root of the number of detected photons $\endgroup$ – Andre Holzner Jan 21 '13 at 20:31
  • $\begingroup$ Thanks for the answers. It really allows me to have a more accurate understanding of the problem I am working on. To complement my question, I wonder if there is any source of error which is specific to the PM device (e.g. background signal) ? $\endgroup$ – peuhp Jan 22 '13 at 8:13
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I haven't read that paper, but here is a physical reason for why the arrival of photons at a detector is modelled as a Poisson process -

Assuming a source of photons (say, for example a tungsten lamp) and a photodetector, there is no predetermined way of predicting when a photon is going to reach the detector, or with what energy. The emission of photons in the direction of the detector is a random (stochastic) process, which is described perfectly by a Poisson process.

However, if you've ever seen the distribution of intensities from a photodetector (although the same will be true for almost any stochastic process) the resulting distribution is Gaussian. This is a result of the Central Limit Theorem. This is mostly because the photodetector doesn't show you the output of a single photon, but it averages over the arrival of several photons. Therefore for a certain period of averaging, the variance of the resultant distribution is deterministic.

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  • $\begingroup$ Thanks. I think I understood my mistake: the poisson statistic does not really model an error about the PM but more about the distribution of photon number actually reaching the PM. Am I correct? $\endgroup$ – peuhp Jan 21 '13 at 14:48
  • $\begingroup$ Yup! That's correct. :) $\endgroup$ – Kitchi Jan 21 '13 at 15:15
  • $\begingroup$ @kitchi, I believe peuhp's setup is for photon counting, where each pulse from the photomultiplier results from a single photon. So your second statement would not apply. Of course, it will apply to the resulting signal if the electronics downstream from the photomultiplier is averaging the pulses. $\endgroup$ – Bobbi Bennett Jan 21 '13 at 15:55
  • $\begingroup$ I'm with @BobbiBennett on this. Single photon counting is par for the course in particle physics. It is not uncommon to see DAQ thresholds set so "one quarter os the SPE central value" where SPE is "single photo electron". $\endgroup$ – dmckee --- ex-moderator kitten Jan 21 '13 at 17:07
  • $\begingroup$ @BobbiBennett - I wasn't aware that there were detectors sensitive enough to detect a single photon. That's pretty cool! $\endgroup$ – Kitchi Jan 21 '13 at 19:23
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Here are three examples where poisson statistics are wrong or slightly-wrong for a PMT (photomultiplier tube) in photon-counting mode. These are rather unusual cases -- poisson statistics occur almost always -- but maybe they will help you better understand how poisson statistics come about.

(1) A fancy apparatus deterministically emits exactly one photon per second, and the PMT captures every one of those photons.

Explanation: When there are a fixed number of photons, and almost all of those photons successfully reach and trigger the PMT, then the fact that one photon was measured makes it less likely that there are other photons out there to be captured. This is an extremely unusual case: By contrast, you can imagine, say, a PMT capturing light from a faraway star. The star is emitting grillions of photons; the fact that one flew into your PMT neither raises nor lowers the probability that any other photon from the star will fly into your PMT. In general, Poisson statistics appear when each event does not affect the probability of occurrence of other events.

(2) Every second, my laser fires, and then PMT receives a burst of photons arriving almost-simultaneously (within a picosecond).

Explanation: The PMT needs some recovery time between photons to register them as separate events. There is almost definitely a poisson distribution of how many photons reach the PMT, but there will NOT be poisson distribution in how many photon-counts are registered ... there can only be zero or one current pulse within a picosecond. (This fact is often ignored because under many circumstances, there is negligible chance that two photons arrive so close together. But in pulsed-laser experiments it's often important.) Another way of thinking about this is: The fact that one pulse is measured decreases (to zero) the probability that another pulse will be measured, because of the PMT's dead time. Again, Poisson statistics appear when each event does not affect the probability of occurrence of other events.

(3) The PMT photon-counting threshold is set too high, so even if a photon arrives, it only has a (say) 30% chance of triggering a count.

Explanation: Well, there WILL be a poisson-distribution of photon counts, but the mean number of photon-counts will be only 30% of the mean number of photons. This is sort of a stupid example, sorry.

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