Statistics and models of SiPM nonlinearity and saturation Sergey Vinogradov Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia National Research Nuclear University «MEPhI», Moscow, Russia Sergey Vinogradov Statistics of SiPM nonlinearity and saturation ICASiPM 14-06-2018 Schwetzingen, Germany 1
Scope & outline Photon detection – a series of stochastic processes described by statistics ◙ ◙ It could be described as a filtered marked correlated history-dependent point process ◙ SiPM response – a result of the stochastic process – a random variable Nonlinearity and saturation – the most sophisticated topic in SiPM statistics ◙ ◙ Binomial nonlinearity – detection of short light pulses (Tpulse < Trecovery) ◙ Conventional model: Poisson Npe in N pixels ◙ Adjustments to CT to account for ◙ Crosstalk ◙ Recovery ◙ Recovery nonlinearity – detection of long light pulses (Tpulse > Trecovery) ◙ Conventional model: non-paralizible counting with dead time ◙ Advanced model of exponential recovery process ◙ Advanced+ model of Markov reward-renewal process Sergey Vinogradov SiPM nonlinearity and saturation ICASiPM 13-06-2018 Schwetzingen, Germany 2
Statistics of linear photon detection ◙ Full characterization of random variable Nout resulted from photon detection processes Probability distribution Pr( Nout ) conditional on Pr( Nin) ― Mean < Nout > ― Variance ( Nout ) or ( N ) out ◙ Partial characterization (the most demanded in practice): Mean and Var of Nout conditional on Mean and Var of Nin ― Supported by Burgess variance theorem ― ENF approach for independent process chains allows to analyse specific noise contributions ◙ Linear detection: responsivity R = <Nout>/<Nin> = const Resolution σ / µ is degraded by specific ENFs Res in < Res process_i < Res process_j < Res out = Res in ·√ENF 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 SiPM nonlinearity and saturation ICASiPM 13-06-2018 Schwetzingen, Germany 3
Statistics of nonlinear photon detection ◙ Nonlinear: R = R ( Nph ) 50 Nonlinearity = random losses Random losses SSPM signal, fired pixels Ns TBD!!!! Losses depend on load Nph σ out Output resolution is “improved” Photon signal Calibrated resolution is degraded distribution σ out < σ in < σ calibrated SSPM equivalent photon distribution even for ideal nonlinear detector σ in µ out Calibration Excess noise of nonlinearity SSPM signal distribution 2 2 2 Res 1 = = = calib calib out ENF σ calibrated 2 2 2 2 Res d µ in µ calibrated in in in out d in 0 0 50 100 ◙ Nonlinear statistics: Number of photons, Nph Linear processes + Nonlinear processes => Severe complications in statistics (quantity/history/mutually-dependent processes) ― New nonlinear distribution Pr( Nout ), < Nout >, σ (Nout) ― New nonlinear responsivity R = R ( Nph ) ― New ENF of nonlinearity Sergey Vinogradov SiPM nonlinearity and saturation ICASiPM 13-06-2018 Schwetzingen, Germany 4
Binomial distribution of SiPM response ◙ Binomial distribution – detection of short light pulses (Tpulse < Trecovery) Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 5
Binomial distribution is presumed B. Dolgoshein et al., 1998 - 2003 Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 6
Urn model with non-random Npe is approximated to normal distribution with binomial µ and σ A. Stoykov et al., On the limited amplitude resolution of multipixel Geiger-mode APDs, JINST, 2007 Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 7
SiPM binomial nonlinearity: losses of photons firing the same pixel ◙ Binomial distribution of fired pixels N det in SiPM ( Tpulse < Trecovery ) N − pe N ! ( ) − = − N N = − N pix k pix det Pr( N , N , )) p p 1 p p 1 e pix − det pix ( N N )! N ! pix det det − N = = − = − pe P ( 0 ) exp E [ N ] N 1 P ( 0 ) Var [ N ] N 1 P ( 0 ) P ( 0 ) det pix det pix N pix N − pe exp 1 N ENF N = = pix + + pe nonlin Excess noise factor - S. Vinogradov et al., IEEE NSS/MIC 2009 PNR ENF 1 ... nonlin N N 2 N pe pe pix N pix Losses of simultaneous photons in a pixel results in nonlinearity and excess noise E.B. Johnson et al., IEEE NSS/MIC 2008 J. Barral, Study of SiPMs, MPI, 2004 Sergey Vinogradov SiPM nonlinearity and saturation ICASiPM 13-06-2018 Schwetzingen, Germany 8
Binomial vs Poisson 0.15 Poisson, Npe=10 Binomial, Npe=10 Poisson, Npe=100 0.1 Binomial, Npe=100 Probability 0.05 0 0 0.5 1 Fraction of fired pixels Pixel load, Npe/Npix Pixel load, Npe/Npix Sergey Vinogradov SiPM nonlinearity and saturation ICASiPM 13-06-2018 Schwetzingen, Germany 9
Adjustments of binomial model ◙ Crosstalk Typical approach: extension of mean Npe to include mean CT events µ CT ― Npe → Npe (1+µ CT ) + N N (1 ) − pe − pe CT = − → − N N So, Mean N N 1 e pix N 1 e pix det pix pix Reasonable from common sense, looks nice and simple but… In general, incorrect because Poisson with CT is not Poisson What about Variance - N det ??? ― What about Resolution ??? – And finally - probability distribution ??? Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 10
Adjustments of binomial model ◙ Recovery Typical approach: extension of mean Npix to include mean retriggering events µ RT ― Npix → Neff = Npix (1+µ RT )=Npix(1+Tpulse/Trec) N pe − N T − pe + pulse T N (1 ) = − → + − pix N puls e T So, Mean N N 1 e p ix N (1 ) 1 e r ec det pix p ix T r ec Reasonable from common sense, looks nice and simple… In general, possible because Poisson Npe over fixed Neff is still binomial ― More reasons in support ??? What about Variance - N det ??? ― What about Resolution ??? – And finally - probability distribution ??? ◙ Main question: what about lower Gain due to incomplete recovery? Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 11
Adjustments of binomial model ◙ Crosstalk + Recovery Typical approach: both corrections are applied as fitting parameters N N pe pe − − = − → − N N N N 1 e N 1 e pix eff _ 2 So, Mean det pix eff _1 So, the same concerns and questions Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 12
Recovery nonlinearity of SiPM response ◙ Recovery nonlinearity – detection of long light pulses (Tpulse > Trecovery) Photoelectron mean arrival rate per pixel = λ Dead time Sergey Vinogradov Nuisance Parameters: ENF ICASiPM 13-06-2018 Schwetzingen, Germany 13
SiPM recovery nonlinearity T T pulse rec Nonparalizible dead time model Probability distribution (~ Gaussian) W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Ch. XI, John Willey & Sons, Inc., 1968 t = + Ns 1 dead t = 2 Ns + 3 ( 1 ) M.Grodzicka, NSS 2011 dead Dead-time non-linear Resolution 1 1 ( ) − R Nph 5 10 9 ( ) − R Nph 50 10 9 0.1 Recovery nonlinearity of SiPM ( ) − R Nph 500 10 9 → ENF S. Vinogradov et al., IEEE NSS/MIC 2009 0.01 0.01 1 10 3 1 10 4 1 10 5 = + 1 10 100 ENF ) ( ) 1 1 Nph 1 10 5 eq Nph ( dead Number of photons per pulse Tpulse=20Treset Tpulse=2Treset Tpulse<<Treset 2 nd Advanced SiPM Workshop, CTA, Geneva, 24 February 2014 Sergey Vinogradov 14
SiPM recovery nonlinearity: advanced model of exponential recovery Non-paralizible dead time model for SiPM ( ENF ) 500 Exponential RC recovery model ( Gain, PDE, ENF ) 2 p.e. 400 1 p.e. Losses of sequential photons due to incomplete pixel recovery 3 p.e. 300 Events 0 p.e. 4 p.e. 200 5 p.e. Photoelectron mean arrival rate per pixel = λ 100 6 p.e. 7 p.e. 0 0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 Charge, electrons = = 1 + PDE & Gain ( t ) 0 ENF dead Dead time ( ) t + 2 − − 1 PDE & Gain ( t ) ~ 1 exp ENF − rec RC + 1 2 Recovery S. Vinogradov, SPIE Adv. Photon Counting 2012 Sergey Vinogradov International Conference on New Photo-detectors, July 6-9, 2015, Moscow, Troitsk, Russia 15
Recommend
More recommend
