IntroductionSilicon Photomultipliers (SiPMs) are well recognized due to exceptional photon

number and timing resolution. However, due to specific SiPM drawbacks such as high dark and correlated noises (crosstalk, afterpulsing), and nonlinearity (limited number of pixels, relatively slow pixel recovery time) their applications are mostly associated with detection of weak light pulses of nanosecond time scale [1, 2].

In contrast with the typical scintillation and Cherenkov detection - photon number and time resolving applications - accelerator Beam Loss Monitoring (BLM) systems with Cherenkov fibre readout have to precisely reconstruct the temporal profile of the light pulse. This is a rather challenging task for any photon detector because of the high dynamic range (~106) from a few photons to a few percent of destructive losses and presumably an arbitrary signal waveforms i.e. localisation of losses (Fig. 1) [3, 4].

This study, in advance of an initial approach [5], is focused on the consideration of a transient SiPM response as a reward-renewal Markov process formed by non-homogeneous Poisson process of photon arrivals and exponential pixel recovery process conditional on previous firing of the pixel.

Challenges of intense light signal detection due to SiPM nonlinearity

Nonlinearity of a photodetector response inevitably degrades its resolution (Fig. 2 a). SiPM nonlinearities are associated with losses of photons due to limited number of pixels and non-instant pixel recovery (dead time) [2] (Fig. 2 b). Moreover, incomplete recovering of pixels during detection of long light pulses results in a distortion of the probability distribution of Gain to lower mean values of Gain and to higher excess noise (Fig. 2 c).

Reward-renewal model of transient SiPM response Renewal process is considered as a Poisson process of photon arrivals with

exponential distribution of inter-arrival times and Bernoulli detection process with exponential recovering of probability to detect the next photon (Fig. 3 a).

Reward process is considered as a process of multiplication with a random gain when the gain depends on time lapsed from the previous firing of the same pixel t with

respect to recovery time τrec (Fig. 3 a).

Renewal equation and mean reward rate are evaluated accordingly known reward-renewal theory and its applications, and the analytical results are shown on Fig. 3 b.:

Experimental studies of transient SiPM response Experiments have been carried out with rectangular pulses (8 ns rise & fall

times) of a 440 nm LED with variable intensity, repetition rate, and pulse width (Fig. 4). SiPM response has been read out without a preamplifier at a 50 Ohm input to avoid any possible saturation at high output signal level in analog bandwidth from DC to 1 GHz. Single electron response was measured with a 20 dB, 4 GHz external amplifier Mini-Circuits ZX60-4016E+. Hamamatsu MPPC S10362-33-050C of 50 μm cell size 3x3 mm2 area was used as a well known and very popular SiPM representative.

ConclusionThe transient SiPM response reveals a rather complex dynamic behavior with a

strong dependence on the mean number of detected photons per pixel per recovery time. Reward-renewal Markov process model has been applied to SiPM response analysis for the first time and promising analytical results have been obtained. In the case of a rectangular light pulse detection the model allows to get an analytical expression for the mean SiPM response as a result of renewal process with a mean detection rate and reward process with a mean gain affected by incomplete recovering of pixels, which are in agreement with analytical and experimental results [5].

AcknowledgmentsThis work has been supported in part by the EC under Grant Agreement 329100 “SiPM in-

depth” and the STFC Cockcroft Institute Core Grant No ST/G008248/1.

References[1] P. Buzhan et al., Silicon photomultiplier and its possible applications, NIMA 504, 48–52, 2003. [2] S. Vinogradov et al., “Efficiency of Solid State Photomultipliers in Photon Number Resolution,” IEEE TNS 58, 9–16, 2011.[3] D. Di Giovenale, L. Catani, and L. Frohlich, “A read-out system for online monitoring of intensity and position of beam losses in electron linacs,” NIMA 665, 33–39, 2011.[4] L.J. Devlin, C.P. Welsch, E.N. delBusto, 'Update on Beam Loss Monitoring at CTF3 for CLIC', IBIC 2013, Oxford, UK, 2013.[5] S. Vinogradov, A. Arodzero, R.C. Lanza, and C.P. Welsch., 'SiPM response to long and intense light pulses', NIMA 787,

148-152, 2015.[6] S. Vinogradov, “Probabilistic analysis of Solid State Photomultiplier performance”, Proc. SPIE 8375, 83750S, 2012.

Sergey.Vinogradov@liverpool.ac.uk Sergey.Vinogradov@cockcroft.ac.uk Vin@sci.lebedev.ru +44 746 305 63 30 +44 192 560 31 97 +7 916 677 5797

Challenges of arbitrary waveform signal detection by SiPM in beam loss monitoring systems with

Cherenkov fibre readout

Sergey Vinogradov1,2,3, Lee Devlin1,2, Eduardo Nebot del Busto1,4,Maria Kastriotou1,4, and Carsten P. Welsch1,2

1 - Department of Physics, University of Liverpool, UK2 - Cockcroft Institute of Accelerator Science and Technology, Daresbury, UK

3 - P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia4 - CERN, Geneva, Switzerland

Fig. 1. a) Beam Loss Monitoring (BLM): 150 MeV electrons in a beam line; an optical fibre for loss electrons to Cherenkov light conversion; SiPM (MPPC) as an upstream photon detector. b) SiPM (MPPC) readout of typical BLM signal: raw output (red, left scale) and a result of deconvolution (black, right scale) with single electron pulse response function (red, inset) [3].

Fig. 2. Degradation of SiPM response and performance due to nonlinearity [2, 6]a) General schematic outline: nonlinearity of calibration curve (output vs input) affects resolution; b) Photon number resolution models: binomial distribution, fixed dead time, exponential recovery.c) Probability density function of SiPM Gain: degradation from ~ delta-function due to incomplete recovering of pixels at high photon arrival rates.

Fig. 3. Reward-renewal Markov process model of SiPM response to high-intensity step-function light pulse:a) Response components & contributions: probability density function (PDF) of inter-arrival times of potential events, transient pixel recovery process recovering its PDE during “dead time”, PDF of actual detected event times; evolution function (mean detection rate) of events after the first detection (plots are given for intensity of 3 phe/pixel/recovery).b) Mean SiPM response as a result of renewal process with a mean detection rate and reward process with a mean gain affected by incomplete recovering of pixels.

Fig. 4. Comparative responses of Hamamatsu MPPC S10362-33-050C of 50 μm cell size 3x3 mm2 area to rectangular light pulse of uniform illumination over active area; green trace is a LED pulse profile; blue trace is a MPPC response. Horizontal scale is 20 μs/div, vertical scales are 2, 10, 50, and 200 mV/div correspondingly. Direct readout of MPPC output signal to 50 Ohm oscilloscope input: DC – 1 GHz analogue bandwidth.

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