Re: shot noise poisson distribution and central limit theorem
- From: Edward Green <spamspamspam3@xxxxxxxxxxx>
- Date: Thu, 25 Dec 2008 15:20:08 -0800 (PST)
On Dec 25, 3:42 pm, lanospam <lanos...@xxxxxxx> wrote:
Hi all,
It is said that for photo-electric detectors, the photon shot noise
increases as the square root of the intensity because arriving
photons follows a Poisson distribution.
That's an interesting assertion. If arriving photons follow a Poisson
distribution, as seems probable, than the number arriving in a fixed
period would... hmm... I was about to say "exponential", but maybe
that's wrong. So the mean number arriving goes with the square root
of the number arriving per second?
I still don't see this. Oh... I get it. The mean number arriving
per second goes with the intensity goes with ... "duh" ... the mean
number arriving per second. But, sense this give us the power, to get
a mean field, we have to take the square root.
OK.
But if we just consider the fact that the more photons are received
(accumulated) in the detector, the less is the standard deviation of
the mean, we reach the same result.
Actually, for every measurement device, like a weighing balance or a CCD,
we know that if we repeat the measurement then the signal/noise of the
averaged result increases as the square-root of the number of measurement..
In the case of photon detection, can't we consider that it is the same
measurement (unique photon detected) which is repeated many time when
intensity increases ? In this case, why do we need to state that photons
are following a Poisson distribution to reach the conclusion that
signal/noise increase as square-root of intensity ? For me it sounds
like a very general truth not related to the specific Poisson distribution.
I know there is a mistake somewhere in my reasoning but I can't find where.
You taught me something (the first, standard argument). Can I return
the favor?
I don't think the standard deviation of the number of photons received
per second is the "noise" here. Or maybe it is (wow, I'm being very
insightful ;-). Couldn't we take it provisionally that you've found
an alternate argument? Since the photons _do_ follow an Poisson
distribution coming in, that wouldn't mean that the first argument is
wrong, even though you point out (assuming you are correct in
identifying the standard deviation in the number of arrivals with
"noise") that the assumption is stronger than necessary to reach the
result. Nature would supply the assumption. The error would be to
reverse the reasoning, and conclude that the photon arrivals are
Poisson because the amplitude of the noise signal goes with the square
root of intensity.
The exact equivalence of these two square root dependencies escapes me
for the moment, though it does seem plausible.
Maybe following the square root of arrivals per period for _any_
period, with the same time constant, is sufficient to establish
Poisson arrivals from the square root dependence?
.
- References:
- shot noise poisson distribution and central limit theorem
- From: lanospam
- shot noise poisson distribution and central limit theorem
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