Discussion in 'Electronic Design' started by George Herold, Jun 24, 2013.

1. ### George HeroldGuest

I’m counting photons and at high count rates I see a drop off in the number counted. It seems this may be due to the dead time of the detector. (How long it takes to recover after detecting one event till it can see the next.)
So I found this in wiki,

My question is about the following equation,

(That probably won’t work.) So it says the real number of events
N~= Nm /(1 - Nm*tau*T) where Nm is number measured, tau is the dead time, and T is the measurement time. I’m wondering about the approximate sign. I figured it should be exactly equal to the above result. (But maybe I’m missing something.)

My reasoning is as follows,
(OK first I wrote my equations for the rate of counts.) So I’ll call R_rthe real rate of events, and R_m the measured rate. I’ll stick with taufor the dead time, and T for total time.)
Then the real number of events I should count is, N_r = R_r*T,
and the measured number is N_m = R_m *T.
Now the total dead time is N_m*tau, and the number of events I missed during that total dead time is, N_missed = N_m*tau*R_r = R_m*R_r*tau*T.
Then the real counts should be the number measured + the number missed.
Or,
R_r*T = R_m*T + R_m*R_r*tau*T

Dropping big T, I get
R_r = R_m + R_m*R_r*tau or R_r = R_m / (1 – Rm*tau)
which is the equation from wiki, without the approimate sign.

Oh..frick.. I just realized that my detector is paralyzable (to use the wiki language.)
(An event in the dead time resets the dead time.) That puts a monkey wrench in the works.
Well, I'll post anyway,

George H.

2. ### Martin BrownGuest

That is almost right provided that the number of counts in the dead time
correction remains small and roughly a Poisson distribution of N apart
from the fact that it should be dimensionless tau/T not tau*T.

It also assumes that there is a single tau that describes the detector
recovery time accurately which might not be entirely true if the photons
are not monochromatic and so have different energies. Basically the
pulse size out of your detector depends on the number of electrons in it
and the recovery time to recharge afterwards.

The correct answer usually lies between that which assumes
non-paralysable (which isn't true) and the worst case which assumes
every would be count event paralyses the detector for tau. This is
necessarily an over estimate of the correction since some will overlap.
It isn't worth the effort to solve a cubic though!

N ~ Nm/(1-N*tau/T)

Hence a quadratic equation for N

N^2(tau/T) - N + Nm = 0

with solution N ~ (1 - sqrt(1 - 4(tau/T)Nm))T/tau/2

approx N ~ 1/(1-3(tau/T)Nm)

Subject to typos and algebraic errors. The actual answer for any
detector subject to dead time is bounded by these two extremes.
ISTR An event inside the dead time doesn't always completely reset the
dead time. If deadtime correction is more than 10% it becomes less
accurate. It is very definitely a ~ correction equation although for
some detectors you can tweak it empirically to do a little better.

You also have to consider lifetime wear and tear on the photocathode.

3. ### George HeroldGuest

Grin.. yeah I saw my translation mistake.. but I figured I'd leave it be.. people like to have something to 'fix'.
In this case that's almost exactly true. We got in a few thousand LED's
for our Spad. (A lifetime supply.) See here,

(Ignore the personal stuff about me... written by my boss.)
I was spot checking the shipment and found one 'golden' LED. It shows one isolated channel with basically the same pulse height every time... really sweet. (rather than the more typical array of heights as seen in figure 2.)

Hmm OK maybe not as bad as I thought then. It's an RC reset time, but if Iset the threshold up high, then the RC is ~linear, then I could assume that on average the missed pulse was in the middle, so an addition 1/2 tau...
OK more thinking required on my part.
OK thanks.. I'll have to check your answer tonight.
Thanks Martin... I was getting ~20% type corrections at the highest counting rates... (~2.5 us dead time, 75k Hz count rate)
And then the LED has this nasty after-pulsing, that screws up the count forlow rates.

George H.

4. ### JamieGuest

I'll op for the latter!

Jamie

5. ### Martin BrownGuest

Any particular choice of LED best or do they all sort of work?
Potentially a useful pulse counting demo for showing to school teachers.

SPAD to me is another thing entirely.

UK safety critical work uses it for (railway) Signal Passed at Danger.
IOW a driver going through a red light with a train - often a precursor
to a major collision if there is no secondary safety system.
But mostly they presumably have multiple channels with different gain.
NB If you try to compute this directly it will suffer rounding error use

N ~ 2*tau*Nm/(1 + sqrt(1-4(tau/T)Nm))/T

Which is numerically stable (again subject to typos)
You can get an empirical fit using 1 <= alpha <= 3 for most sensors.

Any that might be seriously abused with high count rates you need to
defend against division by zero as well. Various makers actually use

N ~ Nm(1+alpha*(tau/T)*Nm)

to avoid an extra test. The error in this is usually insignificant
compared to other things that go wrong at very high count rates.
I wonder if there is any way to make the recovery a bit faster?

Once the channel has stopped conducting you might be able to have a
comparator that zaps it with a faster recharging rate between two very
carefully chosen limit voltages.

6. ### George HeroldGuest

Only one LED AND114-R works at 'low voltage'. I found that other LEDs would avalanche but up above 100V.. there seemed to be some photo-response but I didn't look too hard.
Each channel seems to be separate.. I haven't seen one channel effecting the other. (like causing more after pulsing in that channel.) Each channel has it's own breakdown voltage, and the pulse height is the difference between bais voltage and breakdown voltage... I don't know if I would characterize each channel with a 'gain'. A breakdown 'event' seems to discharge theentire capacitance of the LED. So in that regard there is some channel to channel interaction. And each channel seems to have it's own after pulsing probability. A few channels are really 'bad' in that regard and after breaking down once, (from maybe a photon) will just go on and on... Actually from a physics point of view the after pulsing is interesting. A non-random pulse stream... So here's some dark count data. Plotted is a histogram of the time between pulses vs the number of occurrences. Instead of the 'normal' log-linear plot, this is log-log.

https://www.dropbox.com/s/yb418p04zrgju2m/DARK-CNT.WMF
(I can re-post a a bit map if you can't see the wmf file)

I haven't looked much at the wavelength response. But the pulse height seems independent of the source... after pulsing looks the same as photon induced breakdown.
Sure! Reduce the 100k ohm resistor. So 10k ohm is ten times faster.. but then you've got to keep the bias voltage just above the breakdown voltage or the breakdown doesn't quench. I played around a bit with some active quenching circuits... but really not worth the trouble.
Search for "Avalanche photodiodes and quenching circuits" There's a articleon the web by S.Cova etal. in applied optics (vol. 35 #12, pg 1956) (1996)
(I copied a link.. but way too long.)

One issue with making it fast is you see more after pulsing. The log-log dark count data I posted falls off at short time (1 and 2 us) only because ofthe dead time.

George H.

8. ### Martin BrownGuest

The first one was OK apart from the "clever" WMF transparency and the
observational dots were a bit on the small side to see in my rendering.