Statistical Modeling of SiPM Noise Sergey Vinogradov Lebedev - - PowerPoint PPT Presentation

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 nth order mn of random variable X: ― 𝑛𝑜(𝑌) = σ𝑗=0

∞ 𝑗𝑜 Pr(𝑌 = 𝑗)

X – discrete r.v. ― 𝑛𝑜(𝑌) = ׬

−∞ ∞ 𝑦𝑜 Pr 𝑌 = 𝑦 𝑒𝑦

X – continuous r.v. ― first moment Mean (µ) = 𝑛1 ; ― second central moment – Variance (σ2) = 𝑛2 − 𝑛1

2

μ σ

𝑸𝒔 𝒀 = 𝒚 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

https://doi.org/10.1016/j.nima.2015.07.009

Pr(∆t) or PDF– full characterization Mean and Variance – not so obvious because affected by recovery losses and correlated events

~ Poisson affected by dead time ~ Poisson affected by afterpulsing Pure Poisson – exponential distribution of ∆t

Poisson process: No events in : Pr( | 0) Interval between events : Pr( | 0) 1 Probability density function: ( ) Mean time 1/ Standard deviation ( ) 1/

DCR t DCR t DCR t

t t e t e PDF t DCR e t DCR t t DCR 

−  −  − 

  =   = −  =   =  =  =

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:

( ) ( )

( )

....... s afterpulse and crosstalk

• f

separation for approach (C)CDF extend further can we Moreover, shape its

• n

s assumption any without ) , (

• r

) , ( analyse And ) exp( ) , , ( 1 1 ) , ( : ) , ( events correlated

• f
• n

distributi pure extract to all got ve we' now So, ) exp( 1 ) ( ) , ( 1 ) , ( 1 ) , , ( 1 CDF ary compliment 1

AP CT corr corr corr corr corr ph total corr corr corr corr dark dark corr corr corr total dark corr total

CCDF CCDF CCDF P t f P t F t DCR DCR N t F P t F P t F t DCR t F DCR t F P t F DCR P t F CDF CCDF CCDF CCDF CCDF  =   − − =  − − = −  − = − − =  =

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

Experimental example – courtesy of E. Popova, D. Philippov… , NSS/MIC 2016

Correction on lost dark counts at Tmax should be applied:

) exp(

max _

T DCR CCDF

correction dark

 − =

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 8

CCDF time distribution: reconstruction of correlated event CDF

Pcorr

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 about multiple exponents (see backup):

more and e P P t f more and e P P t F

corr corr

t corr corr corr corr corr t corr corr corr corr

... ) , , ( ... 1 ) , , (

 

  

− −

= − =

( )

) exp( ) , , ( 1 1 ) , ( t DCR DCR N t F P t F

ph total corr corr

  − − =

Experimental example – courtesy of E. Popova, D. Philippov , NSS/MIC 2016

More details – next talk by Eugen Engelmann @ ICASiPM

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

◙ How probability distributions of signal and noise quantities are transformed from input to output Statistics of random quantities

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 11

◙ Pr(Nout|Nin) – full characterization ◙ Mean and Variance – partial characterization

Statistics of random quantities

Input photon distribution Photodetection processes Output charge distribution Pr(Nin) Pr(Nout) Process i Process j

) (

in

N 

in

N

• ut

N

) (

• ut

N 

Photons (Poisson) photoconversion (Bernulli) photoelectrons (Poisson) Dark+photoelectrons (Poisson) triggering (Bernulli) primary avalanches (Poisson) Primary avalanches (Poisson) , (TBD) secondar CT AP       y avalanches (TBD) All avalanches(TBD) multiplication (TBD)

• utput electrons (TBD)

 

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 12

Correlated stochastic processes of CT & AP

Branching Poisson Process Geometric Chain Process Poisson number of primaries <N>=μ e.g. SiPM photoelectron spectrum Single primary event N≡1 e.g. SiPM dark electron spectrum Process Models

Non-random (Dark) event Primary 1st event 2nd event No event Random AP events …

Ra nd

• m

CT ev ent s Random Photo events

Random primary (Photo) events Random AP events

…

… … Non-random (Dark) event Random CT events

… … …

… … … Random primary (Photo) events Random CT events

… …

λ

µ

• 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

Models

Geometric chain process Branching Poisson process Primary event distribution Non-random single (N≡1) Poisson (μ) Non-random single (N≡1) Poisson (μ) Total event distribution Geometric (p) Compound Poisson (μ, p) Borel (λ) Generalized Poisson (μ, λ)

P(X=k)

( )

p pk − 

−

1

1

         

 − − s p s

e L

1 1

~



( ) ( )

! exp

1

k k k

k

   −  

−

( ) ( )

! exp

1

k k k

k

      − −   + 

−

E[X] p − 1 1 p − 1   − 1 1   − 1 Var[X]

2

) 1 ( p p −

2

) 1 ( ) 1 ( p p − +  

( )

3

1   −

( )

3

1   − ENF p + 1 )... ( 2 3 1 ) 1 ln( 1 1 1 1

3 2

p p p p   + + +  − + = −

1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Threshould [fired pixels] Dark Count Rate [Hz] Experiment Borel Geometric

Analytical results for CT & AP statistics

Pct=40% Pct=10%

• S. Vinogradov, NDIP 2011 (experiment – R. Mirzoyan, 2008 (MEPhI SiPM))
• S. Vinogradov, NDIP 2011 (experiment – LPI, Hamamatsu MPPC)

0% 10% 20% 30% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

k, number of events Probability of k events p=0 L=3 p=0.2 L=3 p=0.5 L=3

Crosstalk

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

Statistics of filtered marked correlated point processes

Signal Intensity Dark noise, Background Dark event Correlated event Dark event Single Electron Response (IRF) Random Gain (amplitude) False signal detection Discriminator True signal detection Signal events Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 16

Mean and Variance of filtered marked Poisson point process

 

 

 

       

dt t h d N V V t h ENF N dt t Vout d t V Var V t h ENF V t V Var t PDE N t R Gain q V t h V t Vout E specific SIPM t X Var t t t X COV REMARK t h A A t t IRF COV X COV t Y Var t d t N t h A t IRF E t X E t Y E N i iidrv A t h A t IRF t t IRF t Y N i iidrv t Poissonian N t t t X

ser sptr ph pe ser noise ser sptr ph gain pe

• ut

t noise gain ser

• ut

sptr ph ph fall load ser ser t t in in A t t in

• ut

t in

• ut

i i N i i

• ut

i N i i in

) ( ) ( ) ( )] ( [ ) ( )] ( [ ) ( ) ( ) ( )] ( [ )] ( [ ) ( ) ( ] [ : ) ( 1 ) ]( [ ] [ )] ( [ ) ( ) ( )] ( [ )] ( [ )] ( [ ) ... 1 ( ) ( ) ( ) ( ) ( ) ... 1 ( ) ( ) (

2 2 2 2 2 2 2 2 2 2 1 1

   +     = = +    =    =      =  →  −  =           +  =  −  =   =   =  = = −  = − = = − − − =

→  =  = =

  

                 

 

/ / 2

Example: baseline fluctuation due to DCR ( ) ( ) Var =V 2

fall rise

t t fall rise ser

• ut

t DCR const h t e e V ENF DCR

 

  

− −

= = = − +   

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 17

Figure of Merit for all noise contributions ◙ How to compare noises and optimize SiPM operations: Excess Noise Factor – ENF – approach

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 18

ENF: two definitions ◙ ENF in multiplying photodetectors (APD, PMT): ◙ ENF in amplifying electronics:

Noise factor (F) measures degradation of the signal-to-noise ratio (SNR), caused by components in a signal chain.

2 2 2 2 1

( ) 1 ( )

M

m M ENF m M M  = = +

2 2 2 2 2 2 in

• ut
• ut

in in

• ut

ENF Res Res     = =

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 19

ENF: two definitions are equal for Poisson input

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 20

2 2 2 2

1 ( 1) 1 ( ) 1 ( ) Input quantity N Poisson ( ) ( 1)

M M in in in in

ENF X vs ENF X N Fano N M M ENF X N ENF X    = + = = +    = = 

Resolution at output is a product of Resolution at input and √ENF(Xin≡1) Total ENF for a sequence of specific processes is approximately

ENF as a measure of SNR (resolution) degradation

) 1 ( ) ( 1 ) ( ) ( 1 ) 1 ( ) 1 ( 1 ) ( ) (

2 2

  = =  +  =

in in in in

• ut
• ut

in

• ut

X ENF X RES X Fano X Fano Y X RES Y RES  ... ) ( ) ( ) (

2 1 process process in total in

• ut

ENF ENF X RES ENF X RES Y RES   =  = Input photon distribution Photodetection processes Output charge distribution RES(Xin) RES(Yout)= RES(Xin)√ENF ENFi ENFj

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 21

ENF related to Gain, PDE and DCR ◙ Multiplication (single electron) ◙ Photon detection (single photon) ◙ Dark counts (Poisson dark+pe)

pe dcr

N t DCR ENF  + = 1 PDE PDE PDE PDE ENFpde 1 ) 1 ( 1

2

= −  + = SiPM

• f

most for 05 . 1 ... 01 . 1 1

2 2

 + =

gain gain gain

ENF Gain ENF 

Xin=1 PDF(Yout) ? PDF(Xin)=δ(x) <Yout>=Gain Photoelectrons RES(Yout) Dark counts Xin=1 PDF(Yout) = Bernoulli PDF(Xin)=δ(x) <Yout>=PDE <Yout>=Npe+Nd

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 22

Pr(0)

Methods of ENF measurements

1 10 100 1000 10000 100000 1000000 10000000 1 2 3 4 5 6 7 8 9 10 Amplitude [Single Electron Response] Events

Crosstalk amplitude histogram N=1

N=1 non-random primary

_________________________________

N ϵ Poisson random primaries

Nph could also be measured by reference PD

2 2 2 2 2 2

( ) ln(Pr(0))

• ut

in

• ut

in in

• ut
• ut

ENF X N N N       = = =  = −

2 2

( 1) 1

• ut

in

• ut

ENF X    = +

Remark: ENF measurements in electronics: noise generator + noise power meter

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 23

σ/µ σ/µ

Key SiPM parameters in ENF metrics

Noise source Key parameter Typical values for SiPM ENF expression SiPM ENF PMT ENF Fluctuation of multiplication Mean gain and standard deviation of gain μ(gain) ~ 106 σ(gain) ~ 105

) ( ) ( 1

2 2

gain gain Fm   + =

1.01 1.2 Crosstalk Probability of crosstalk event Pct 5% – 40%

1 1 ln(1 ) Fct Pct = + −

1.05 – 2 1 Afterpulsing Probability of afterpulse Pap 5% – 20% 1 Fap Pap = + 1.05 – 1.2 1.01 Shot noise of dark counts Dark count rate DCR Signal events in Tpulse 105 – 106 cps

1 DCR Tpulse Fdcr Nph PDE  = + 

1.01 1 Noise / losses of photon detections Photon detection efficiency PDE 20% – 50%

1 Fpde PDE =

2 – 5 3 – 5 Total noise in linear dynamic range Total ENF_linear

_ ENF linear Fm Fct Fap Fdcr Fpde =    

3 – 12 4 – 6 Binomial nonlinearity Mean number of possible single pixel firings n

exp( ) 1 ; ; binomial distribution Nph PDE n n ENFpix Npix n  − = =

Dead time nonlinearity Mean rate of possible single pixel firings λ

; 1 ; nonparalizible model n ENFrec Trec Tpulse   = = + 

total nl ap ct dcr pde m total total ph ph

• ut
• ut

ENF DQE F F F F F F ENF ENF N N N N PNR 1 ) ( ) ( ) ( ) (

* *

=       = =    

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 24

The end

Thank you for your attention!

Questions? Objections? Opinions? … vin@lebedev.ru

Sergey Vinogradov Statistical Modeling of SiPM Noise ICASiPM 11-15 June 2018 Schwetzingen, Germany 25

FBK results on Generalized Poisson model

• A. Gola, A. Ferri, A. Tarolli, N. Zorzi, and C. Piemonte, “SiPM optical crosstalk amplification due to scintillator crystal: effects on timing

performance,” Phys. Med. Biol., vol. 59, no. 13, p. 3615, 2014. Sergey Vinogradov LIGHT – 2017 19 October 2017 26

DESY results on Generalized Poisson

◙

• E. Garutti group @ DESY: V. Chmill et al., NIMA 2017

Sergey Vinogradov LIGHT – 2017 19 October 2017 27