Statistical Modeling of SiPM Noise Sergey Vinogradov Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia National Research Nuclear University «MEPhI», Moscow, Russia Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 1
Scope & outline Photon detection – a series of stochastic processes described by statistics ◙ ◙ SiPM response – a result of the stochastic processes – a random variable ◙ SiPM “noise” has a double meaning: as well as a “signal” ◙ Physics point of view – specific nuisance contributions to the response ◙ Dark counts - DCR ◙ Crosstalk - CT ◙ Afterpulsing – AP ◙ Statistics point of view – standard deviation of the response ◙ All above +… ◙ Multiplication – Gain ◙ Photo-conversion!!! – PDE ◙ Statistics of the response times ◙ Statistics of the response quantities ◙ Statistics of the response transients ◙ ◙ Excess noise factor – ENF – as a Figure of Merit for all noise contributions Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 2
Definitions of statistics ◙ Statistics – wide sense – math related to random variables ◙ Random variable is fully described by its probability distribution Pr(X) ◙ Statistic – narrow sense – any function on probability distribution Statistical moments n th order m n of random variable X: ∞ 𝑗 𝑜 Pr(𝑌 = 𝑗) ― 𝑛 𝑜 (𝑌) = σ 𝑗=0 X – discrete r.v. ∞ 𝑦 𝑜 Pr 𝑌 = 𝑦 𝑒𝑦 ― 𝑛 𝑜 (𝑌) = X – continuous r.v. −∞ ― first moment Mean (µ) = 𝑛 1 ; 2 ― second central moment – Variance ( σ 2 ) = 𝑛 2 − 𝑛 1 𝑸𝒔 𝒀 = 𝒚 σ μ Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 3
Statistics of random times ◙ How probability distribution of signal and noise event times are transformed by photon detection processes Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 4
Statistics of random times ~ Poisson affected by dead time Pure Poisson – exponential distribution of ∆t ~ Poisson affected by afterpulsing Poisson process: https://doi.org/10.1016/j.nima.2015.07.009 = − DCR t No events in t : Pr( t | 0) e = − − DCR t Interval between events : Pr( t | 0) 1 e Pr (∆t) or PDF– full characterization Mean and Variance – not so obvious − = DCR t Probability density function: PDF ( t ) DCR e because affected by recovery losses = = = Mean time t 1/ DCR Standard deviation ( t ) t 1/ DCR and correlated events Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 5
Analysis based on probability density function (PDF) = histogram of interarrival times ◙ Common-sense analysis of DCR and correlated events ◙ Contributions are fitted separately in specific time frames P. Eckert et al, NIMA 2010 F. Acerbi et al, TNS 2016 Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 6
Analysis based on cumulative distribution function (CDF) = list of interarrival times ◙ Complimentary CDF approach: probabilities of independent “zero” dark and “zero” correlated events in ∆t are multiplied: = = − CCDF CCDF CCDF CCDF 1 CDF compliment ary CDF total corr dark ( ) ( ) − = − − 1 F ( t , P , DCR ) 1 F ( t , P ) 1 F ( t , DCR ) = − − total corr corr corr dark F ( t ) 1 exp( DCR t ) dark So, now we' ve got all to extract pure distributi on of correlated events F ( t , P ) : corr corr ( ) = − − F ( t , P ) 1 1 F ( t , N , DCR ) exp( DCR t ) corr corr total ph And analyse F ( t , P ) or f ( t , P ) without any assumption s on its shape corr corr corr corr Moreover, we can further extend (C)CDF approach for separation of crosstalk and afterpulse s = CCDF CCDF CCDF ....... corr CT AP Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 7
CCDF time distribution: dark and correlated events Pcorr 1- Pcorr Correction on lost dark counts at Tmax should be applied: = − CCDF exp( DCR T ) dark _ correction max Experimental example – courtesy of E. Popova, D. Philippov … , NSS/MIC 2016 Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 8
CCDF time distribution: reconstruction of correlated event CDF More details – next talk by Eugen Engelmann @ ICASiPM ( ) = − − F ( t , P ) 1 1 F ( t , N , DCR ) exp( DCR t ) corr corr total ph Now we are ready to play with correlated event CDF or PDF applying any models and analysis. For example, typical assumption on exponential time distribution, but take care Pcorr about multiple exponents (see backup): − t = − F ( t , P , ) 1 P e ... and more corr corr corr corr corr − t P = corr f ( t , P , ) e ... and more corr corr corr corr corr Experimental example – courtesy of E. Popova, D. Philippov , NSS/MIC 2016 Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 9
Empirical CDF (list mode) vs PDF (histogram): highest precision due to full information from all data points ◙ [1] Y. Cao, L. Petzold , “Accuracy limitations and the measurement of errors in the stochastic simulation…”, J. Comput. Phys. 212, 2006 Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 10
Statistics of random quantities ◙ How probability distributions of signal and noise quantities are transformed from input to output Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 11
Statistics of random quantities ◙ Pr( Nout|Nin ) – full characterization ◙ Mean and Variance – partial characterization Photons (Poisson) photoconversion (Bernulli) photoelectrons (Poisson) Dark+photoelectrons (Poisson) triggering (Bernulli) primary avalanches (Poisson) Primary avalanches (Poisson) CT AP , (TBD) secondar y avalanches (TBD) All avalanches(TBD) multiplication (TBD) output electrons (TBD) Pr(N out ) Process j Pr(N in ) Process i ( N ) ( N ) out in N N out in Input photon distribution Photodetection processes Output charge distribution Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 12
Correlated stochastic processes of CT & AP Single primary event N≡1 Poisson number of primaries <N>= μ Process Models e.g. SiPM dark electron spectrum e.g. SiPM photoelectron spectrum Random Photo events Non-random (Dark) event Random primary (Photo) events … Primary … Ra Random AP events Random AP events Geometric 1 st event nd om Chain Process 2 nd event CT ev ent No event s Non-random (Dark) event Random primary (Photo) events … … µ Branching … Random CT events … Random CT events λ Poisson … … … Process … … … S. Vinogradov, IEEE NSS/MIC 2009, TNS 2011, NDIP 2011 Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 13
Analytical results for CT & AP statistics Models Geometric chain process Branching Poisson process 30% p=0 L=3 Poisson ( μ ) Poisson ( μ ) Primary event Non-random Non-random single Crosstalk p=0.2 L=3 single ( N ≡1) ( N ≡1) distribution Total event Compound p=0.5 L=3 Borel ( λ ) Generalized Poisson ( μ, λ ) Geometric ( p ) Probability of k events Poisson ( μ, p ) distribution 20% − s 1 ~ ( ) ( ) ( ) ( ) ( ) − − + − − − − − − k 1 1 p s k 1 p k 1 L e k exp k 1 p k exp k P(X=k) k ! k ! 10% 1 1 E[X] − − − − 1 p 1 p 1 1 + p ( 1 p ) Var[X] ( ) ( ) − − − − 2 2 3 3 0% ( 1 p ) ( 1 p ) 1 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 1 3 k, number of events + = + + + 2 3 1 p p ( p )... 1 p ENF − + − 1 1 ln( 1 p ) 2 1.0E+07 Experiment Pct=10% Borel 1.0E+06 Pct=40% Geometric Dark Count Rate [Hz] 1.0E+05 1.0E+04 1.0E+03 1.0E+02 1.0E+01 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Threshould [fired pixels] S. Vinogradov, NDIP 2011 (experiment – R. Mirzoyan, 2008 (MEPhI SiPM)) S. Vinogradov, NDIP 2011 (experiment – LPI, Hamamatsu MPPC) Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 14
Statistics of transient signals - stochastic functions ◙ How random quantities are developed in time Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 15
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