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→Modeling the HPD spectrum
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.)
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.
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 primary peak of a single photo-electron (''h<sub>1</sub>'' of width ''σ<sub>1</sub>'') is joined by another centered about some value ''p'' (of the form ''h<sub>2</sub>'' with 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.
This implies a distribution in the total collected charge ''q'' from ''m'' primary photo-electrons, summed over all possible values of ''m'' weighted by its Poisson distribution:
<math>\sum\limits_{m=0}^{\infty} \frac{e^{-\lambda} \lambda^m}{m!} f_m(q)</math>
Where ''f<sub>m</sub>'' is the ''m''-times self-convolution of the original primary and secondary peaks of the form:
<math>f_m(q) = [h_1(q) + h_2(q)] \
A fit of this model to the data was attempted, assuming the initial peaks are Gaussians, with all parameters remaining free except the gain and the position in the spectrum corresponding to zero photo-electrons. It turned out, however, that the model could not account for the long right-hand tail of the distribution, seen in the above figure. Evidently, the naive assumption that the secondary electron emission spectrum generated by the ion is inaccurate. A crude introduction of asymmetry in the peak of secondaries was introduced by another Gaussian with its location (''r''), normalization (''β'') and width (''σ<sub>3</sub>'') as another set of free parameters.
A fit of this model to the data was attempted, assuming the initial peaks are Gaussians, with all parameters remaining free except the gain and the position in the spectrum corresponding to zero photo-electrons. It turned out, however, that the model could not account for the long right-hand tail of the distribution, seen in the above figure. Evidently, the naive assumption that the secondary electron emission spectrum generated by the ion is inaccurate. A crude introduction of asymmetry in the peak of secondaries was introduced by another Gaussian with its location (''r''), normalization (''β'') and width (''σ<sub>3</sub>'') as another set of free parameters.
</math>