statistics for photon counts in photomultiplier 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 


*

*increases with the number of photons to detect,

*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)
 A: 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.
A: 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.
