Application  Note  3  –  v1.0.1  –  March  2014  Understanding  SPC2:  SNR  and  DR  compared  to  analogue  imagers    

   

 

AN2  -­‐  1    

Understanding SPC2: SNR and DR compared to analogue imagersSPAD   imagers   are   single   photon   counters   that  work  in  a  quite  different  manner  of  conventional  imagers.   Being   a   photon   counter,   dark   noise   is  usually   expressed   in   terms   of   “counts   per  second”,   as   opposed   to   “dark   current”   o   rms  values.   Being   a   Geiger   mode   detector   with  internal  gain,  several  “analogue”  noises  typical  of  other   imagers  are  not  present.  These  facts  make  it   difficult   to   directly   compare   sensitivity  performances   on   the   base   of   these   numbers  alone,  without   taking   into  account  all   the  aspect  of   the   desired  measurement   and,   ultimately   the  requires  SNR.  On  the  other  hand,  a  peculiar  effect  of   these   detectors,   namely   the   “dead-­‐time”   [1]  between  two  successive  photon  detections,  must  be   taken   into   account,   as   opposed   to   the   “full-­‐well  capacity”,  when  dealing  with  dynamic  range  (DR).   This   different   behavior   of   SPAD   imagers  must   be   understood   clearly   for   a   proper  evaluation   of   the   DR,   especially   concerning   the  role  of  integration  time.  

SNR  calculation  In  general  SNR  in  a  measurement  is  a  function  of  several  parameters:  -­‐   Photon   Detection   Efficiency   (PDE)   of   the  detector  -­‐  Fill  factor  -­‐  Integration  Time  -­‐  Noise  sources  For   an   ideal   detector,   SNR   has   a   limit   that   is  imposed   by   the   intrinsic   shot   noise   of   the  impinging   photons.   Assuming   a   Poisson  distribution  of  the  photons,  this  noise  is  equal  to  the  square  root  of  the  number  of  photons  in  the  

signal   (since   for   a   Poisson   distribution   the  variance  is  equal  to  the  mean  value),  i.e.  for  ANY  detector  SNR  cannot  be  higher  than:    

𝑆𝑁𝑅!"#$% =𝑁!"#$%&𝑁!"#$%&

= 𝑁!"#$%& = 𝑇 ∙ 𝑛!"#$%&  

 where   N   represents   the   number   of   photon  acquired   in   the   integration   time,   T   is   the  integration   time   and   thus   n   represents   the  photon  flux  (photon/s).  A  real  detector  will  have  typically  a  non  ideal  PDE,  a  reduced  fill-­‐factor  and  some  additional  noise:    

𝑆𝑁𝑅!"#$ =𝑃𝐷𝐸 ∙ 𝐹𝐹 ∙ 𝑁!"#$!"

𝑃𝐷𝐸 ∙ 𝐹𝐹 ∙ 𝑁!"#$%& + 𝜎!!"#$%  

 where  𝜎!!"#$%  is   the   total   variance   of   the   noise  added  by  the  detection  chain.  For   a   SPAD   [1],   the   only   noise   source   is   the   so-­‐called   dark-­‐count   noise,   i.e.   spurious   triggerings  of   the  detector  caused  by   thermal  generation  or  tunneling  effects.  Since  also  dark-­‐counts  follows  a  Poisson   distribution,   the   SNR   expression,   when  the   role  of   the   integration   time   is  highlighted,   is  simply:    

𝑆𝑁𝑅!"#$ =𝑃𝐷𝐸 ∙ 𝐹𝐹 ∙ 𝑇 ∙ 𝑛!"#$%&

𝑃𝐷𝐸 ∙ 𝐹𝐹 ∙ 𝑇 ∙ 𝑛!"#$%& + 𝑇 ∙ 𝑛!"#$%  

 where  nnoise  is  the  dark-­‐counting  rate  of  the  SPAD.  It   should   be   noted   that   for   an   integration   time  shorter  than  1/  nnoise   the  contribution  to  SNR  of  the   detector   noise   is   negligible   as   compared   to  

Application  Note  3  –  v1.0.1  –  March  2014  Understanding  SPC2:  SNR  and  DR  compared  to  analogue  imagers    

   

 

AN2  -­‐  2    

photon  shot  noise.  For   a   conventional   analogue   detector,   several  noise   sources   must   be   considered.   Assuming  uncorrelated   all   noise   sources,   the   total   noise  variance  is  the  sum  of  the  contribution  from  each  source.  Typical  noise  sources  are:  

-­‐ dark  current  noise  -­‐ reset  noise  -­‐ read-­‐out   noise   (noise   from   amplifiers,   ADCs,  

charge  transfer  etc.)  -­‐ quantization  noise  

For   an   analog   detector   with   internal   gain  (typically  ICCD  and  EMCCD),  the  internal  gain  will  make  negligible  all  noise  sources  introduced  after  the   gain   stage,   i.e.   read-­‐out   noise   and  quantization   noise.   This   will   comes   at   the  expense   of   an   excess   noise   factor   produced   by  the   inherent   noise   of   the   gain   process.   For  instance,   for   an   EMCCD   used   at   high   gain,   the  SNR  becomes:    𝑆𝑁𝑅!"##$

≅𝑃𝐷𝐸 ∙ 𝐹𝐹 ∙ 𝑇 ∙ 𝑛!"#$%&

𝑘!"##$! (𝑃𝐷𝐸 ∙ 𝐹𝐹 ∙ 𝑇 ∙ 𝑛!"#$%& + 𝑇 ∙ 𝑛!"#$)  

 where   k   is   the   excess   noise   factor,   that   for   an  EMCCD   tends   to   2  for   high   gain.   This   means  that  when  the  contribution  from  the  dark  current  is  negligible  (as  usual  for  high  performance  CCDs),  the   excess   noise   has   the   same   effect   on   SNR   of  halving   the   PDE.   For   other   type   of   analog  detector   with   internal   gain   the   excess   noise  factor  is  usually  higher.  

Dynamic  Range  calculation  The   Dynamic   Range   is   normally   defined   as   the  ratio  between   the  maximum   recordable   and   the  minimum   detectable   signal   in   a   frame.   The  maximum   is   related   to   the   saturation   of   the  detector,  the  minimum  is  the  level  that  results  in  a   SNR=1   taking   into   account   the   noise   (see  

above).   Saturation   is   caused   by   different   effects  in   SPAD   and   in   traditional   detectors,   so   also  dynamic  range  calculation  is  a  bit  different.  In   traditional   detectors,   the   maximum  recordable  signal  in  a  frame  is  fixed  and  equal  to  the   capacity   of   the   potential   well   in   which  charges   are   accumulated.   Typically   the   full-­‐well  capacity  (FWC)  is  in  the  range  of  20,000-­‐200,000  photoelectrons.   It   is   usually   assumed   that   noise  level   is   so   low   that   dark   current   will   not  considerably   reduce   the   useful   well   capacity,  which   is  normally   true.  Note   that   this   saturation  level   is  not   related   to   the   integration   time:  each  frame   can   integrate   at   maximum   a   full-­‐well  capacity  of  photoelectron.  On  the  other  hand,  the  minimum  detectable   signal   is   equal   to   the  noise  of   the   detector,   which   is   usually   dominated   by  the  readout  noise,  and  thus  it  is  not  dependent  as  well   on   the   integration   time   (at   least   for  reasonably   short   integration   time).   It   is   also  assumed  that  the  ADC  is  able  to  resolve  at  least  a  bunch   of   photoelectrons   equal   to   the   readout  noise.   If   this   is   not   the   case,   the   minimum  detectable   signal   is   also   a   function   of   ADC  resolution.  In  the  more  usual  case,  dynamic  range  is  thus:    

𝐷𝑅!!" ≅𝐹𝑊𝐶

𝜎!"#$%&'  

 For   detectors   with   internal   gain,   one   has   to  consider   that   the   gain  will   reduce   the  maximum  detectable  signal,  since  the  well  will  fill  faster.  In   SPADs   there   is   no   well   to   fill.   Photons   are  detected   one   at   a   time   and   digitally   counted.  What  actually   limits   the  number  of  detections   is  the   dead-­‐time   between   events   (counter  saturation   is   usually   not   a   problem,   since   its  depth   can   be   chosen   taking   into   account   the  minimum  dead-­‐time  of  the  detector).  This  means  that   the   saturation   level,   in   terms  of  maximum  number  of  photons  per   frame,   is  dependent  on  the   integration   time.   Instead,   the   maximum  

Application  Note  3  –  v1.0.1  –  March  2014  Understanding  SPC2:  SNR  and  DR  compared  to  analogue  imagers    

   

 

AN2  -­‐  3    

photon   flux   (photon/s)   is   constant   for   a   given  dead-­‐time.   Concerning   the   minimum   detectable  signal,   again,   it   is   equal   to   the   noise   of   the  detector.   Since   SPADs   have   no   read-­‐out   noise,  one  has  now  to  consider   the  contribution  of   the  dark-­‐counting   rate.     For   very   short   integration  

time   (𝑻 < 𝟏 𝒏𝒅𝒂𝒓𝒌 )   the   minimum   detectable  

signal   is   simply   1   photon   per   frame,   i.e.   we  reached   the   limit   for   photon   counting.   As   a  result   of   the   dependence  of   the   saturation   level  on   the   integration   time,   dynamic   range   for   a  SPAD  increases  with  integration  time:    

𝐷𝑅!"#$ =𝑇/𝑡!"#!𝜎!"#$

=𝑇/𝑡!"#!𝑇 ∙ 𝑛!"#$

∝𝑇              𝑖𝑓  𝑇 < 1 𝑛!"#$  

𝑇        𝑖𝑓  𝑇 > 1 𝑛!"#$    

   

References  1. S.   Cova,   M.   Ghioni,   A.   Lacaita,   C.   Samori,   and   F.   Zappa,  

“Avalanche   photodiodes   and   quenching   circuits   for   single-­‐photon  detection.”  Appl  Opt,  vol.  35,  no.  12,  pp.  1956–1976,  1996.  

 

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