We present a description of the operation of a multi-pixel detector in the presence of non-negligible dark-count and cross-talk effects. We apply the model to devise self-consistent calibration strategies to be performed on the very light under investigation.Several concepts and technologies have been proposed that lead to the development of detectors such as visible-light photon counters (VLPC) [18], superconductive transition edge sensors (TES) [19], time-multiplexed detectors [20][21][22], hybrid photodetectors (HPD) [23,24] and Silicon photomultipliers (SiPM) [25]. Irrespective of the concept and design features, these detectors may in general be classified in terms of photon detection efficiency, spectral response, time development of the signal, dead time, and notable photon number resolving capability. As of today, the ideal detector has yet to appear and the optimal choice is application specific. This paper focuses on SiPMs, detectors featuring unique characteristics that are achieved by a rapidly evolving technology. Silicon photomultipliers consist of a high density (by now limited to ∼ 2000 cells/mm 2 ) matrix of diodes with a common output. Each diode (or cell) is operated in a limited Geiger-Mueller (GM) regime, in order to achieve gains at the level of 10 6 . Quenching mechanisms are implemented to avoid establishing self-sustaining discharges. These detectors are sensitive to single photons triggering GM avalanches and can be endowed with a dynamic range well above 100 photons/burst. The photon detection efficiency (PDE) depends on the sensor design and specification, but it may well exceed 60%. Moreover, SiPM are genuine photon-number resolving detectors in that they measure light intensity simply by the number of fired diodes. Compactness, robustness, low cost, low operating voltage, and power consumption are also added values against traditional photodetectors. On the other hand, SiPMs are affected by significant dark count rates (DCR), associated to cells fired by thermally generated charge carriers. Moreover, the GM avalanche development is known to be associated to the generation of photons [26], which may in turn trigger secondary avalanches and result in relevant cross-talk. Whether DCR and cross-talk may be directly measured, it is clear that they are folded in the detector response to any signal and need to be modelled to properly assess the statistical properties of the light field being investigated. This paper reports the experimental validation of two models, on the way to a self-consistent characterization of the SiPM response. Experimental set-upThe detector response to a weak light field is shown in Fig. 1(a), featuring the sensor output signal after a high-bandwidth amplifier with a gain of 50. The different bands in the image, obtained in persistency mode, correspond to samples with different numbers of triggered cells, i.e. different numbers of detected photons. The photon-number resolving properties are also clear in Fig. 1(b) that shows the corresponding spectrum as obtained ...

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The breakdown behaviour of SiPMs (Silicon PhotoMultiplier) with pixel sizes of 15×15, 25×25, 50×50, and 100×100 µm 2 , manufactured by KETEK, has been investigated. From the currentvoltage characteristics measured with and without illumination by LED light of 470 nm wavelength, the current-breakdown voltage, V I , and from linear fits of the voltage dependence of the SiPM gain, measured by recording pulse-area spectra, the gain-breakdown voltage, V G , have been obtained. The voltage dependence of the Geiger-breakdown probability was determined from the fraction of zero photoelectron events with LED illumination. By comparing the results to a model calculation, the photodetection-breakdown voltage, V P D , has been determined. Within experimental uncertainties, V I and V P D are equal and independent of pixel size. For V G , a dependence on pixel size is observed. The difference V I − V G is about 1 V for the SiPM with 15 µm pixels, decreases with pixel size and is compatible with zero for the SiPM with 100 µm pixels.

Methods are developed, which use the pulse-height spectra of SiPMs measured in the dark and illuminated by pulsed light, to determine the pulse shape, the dark-count rate, the gain, the average number of photons initiating a Geiger discharge, the probabilities for prompt cross-talk and after-pulses, as well as the electronics noise and the gain fluctuations between and in pixels. The entire pulse-height spectra, including the background regions in-between the peaks corresponding to different number of Geiger discharges, are described by single functions. As a demonstration, the model is used to characterise a KETEK SiPM with 4384 pixels of 15 µm×15 µm area for voltages between 2.5 and 8 V above the breakdown voltage. The results are compared to other methods of characterising SiPMs. Finally, examples are given, how the complete description of the pulse-eight spectra can be used to optimise the operating voltage of SiPMs, and a method for an in-situ calibration and monitoring of SiPMs, suited for large-scale applications, is proposed.