Analytical model for the photomultiplier single photoelectron response including the electron back-scattering contribution

Abstract

Many models exist to describe the single photoelectron response of single photon counting photomultipliers. Generally to describe the spectral region between the fully amplified primary photoelectron peak and the electronics pedestal an \emph{ad hoc} function is used (often an exponentially modified gaussian) attributing this region to `noise'. In this paper, following the physical description of back-scattered primary photoelectrons at the first dynode described in the "The Photomultiplier Handbook" by A.~G. Wright published by Oxford University Press, we derive an analytical function describing these partially amplified primary photoelectron at the first dynode. This function depends only on intrinsic parameters of the photomultiplier such as the gain at the first dynode and the intrinsic resolution of the dynode chain following the first. Furthermore, analytical descriptions of the fully amplified peak and very low charge signals are derived. The model has been successfully validated with data from two different photomultipliers acquired with a low-noise amplifier.

Figures (16)

• Figure 1: Kinetic energy distribution of secondary electrons emitted by a dynode after having been hit with primary electrons with energy $E_p$. Figure adapted from Figure 5.8 of Ref. Wright.
• Figure 2: Black spots: Total discrete probability density for detectable secondary electrons emitted at the first dynode. Blue triangles: contribution from fully amplified electrons. Red squares: contribution from partially amplified electrons. Here $\eta = 0.3$ and $G_1=15$.
• Figure 3: Noise spectra of the different components of the read-out spectra. All spectra are referred to the input of the digitizer in power density per Hertz. The feature at 2--3 MHz is instrumental from the Rohde & Schwarz FPL1003 spectrum analyser. Also note the pick-up noise structure at frequencies peaking at 100 MHz. The attenuation introduced by the waveform integration for 30 ns is shown in black (referred to the right scale). This plot refers to the photomultiplier Hamamatsu R5912-100.
• Figure 4: Left: Typical spectral response of Hamamatsu R5912-100 obtained with a light source at 405 nm (red histogram) and with light-off (black histogram). The light-only spectrum is obtained by subtracting the dark histogram from the light-on one after applying to the first one a global shift and a scaling factor to equalize the pedestal peak contributions (blue histogram). Right: subtracted spectral response.
• Figure 5: Typical spectral response of Hamamatsu 6233. See Fig. \ref{['fig:spectrum-5912-100']} caption.