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→Modeling the HPD spectrum
[[Image:HPDspectra.png|thumb|300px]]
[[Image:HPDspectra.png|thumb|300px]]
The HPD spectra collected from large pulses should essentially be a convolution of the Poisson distribution, in the limit of high mean count, with the electronic noise function, which is known to be broad enough to make individual photon peaks impossible to resolve. Both of these functions are Gaussian and the result should be a Gaussian as well. The measured spectra show a peculiar deviation (shown on the right.)
The HPD spectra collected from large pulses should essentially be a convolution of the Poisson distribution, in the limit of high mean count, with the electronic noise function, which is known to be broad enough to make individual photon peaks impossible to resolve. Both of these functions are Gaussian and the result should be a Gaussian as well. The measured spectra, however, show a peculiar deviation (shown on the right.) Clearly, because individual photon peaks cannot be seen with this device and given the large number of photons incident on the HPD window, these cannot be discrete photon peaks. Note also the the shift of the spectrum with increased intensity. Additionally, measurements showed on scaling of the distance between peaks with the gain (adjusted by varying the high voltage bias on the device.)
Our proposed explanation involves ions ejected with some constant probability upon the incidence of accelerated electrons. This positive ion is accelerated in the opposite direction to the path of the electron, back to the bialkali layer. Collision with this layer generates more electrons, which are accelerated and multiplied just as the first. In this model, the lone peak of a single photo-electron (of width ''σ<sub>1</sub>'') is joined by another centered about some value ''p'' (of width ''σ<sub>2</sub>'') representing the mean number of electrons generated by the ion's collision with the bialkali layer. The ratio of integrals of these peaks (''α'') represents the probability of the ion's ejection for every primary photo-electron. Increasing input to ''m'' photo-electrons essentially means a convolution of this spectrum with itself ''m''-times.
A fit of this model to the data was attempted with all parameters remaining free except the gain and the position in the spectrum corresponding to zero photo-electrons.