Please weigh in on photon noise

In photography we don't know what to expect exactly, so don't we measure the outcome during exposure, which comes with a mean and a standard deviation?

Good discussion JACS and John, I learned more about shot noise and its workings in this thread than at any other time in my photographic explorations. I might try to write this up for the less well versed.

I've explained why there is no need to invoke "well known". The Poisson distribution is a consequence.

Here's a very nice series of short lectures that might help some folk: MIT RES.6-012 Introduction to Probability, Spring 2018

Particularly L22.1..10. Or start earlier if more background is needed. L21.n covers (discrete time) Bernoulli processes.

Fun Fact: The idea of a Poisson process evolved about 100 years after Poisson's death. Poisson himself never studied them.

I think we've done this to death.

We will be arguing next what the meaning of the word "is" is.

Everything is a consequence of something else in a way. The (continuous) Poisson process is a consequence of the way light is created in the first place, depending on the source; then you may want to go down to QM to understand why, etc.

Ditto.

the red queen said in a commanding tone something like "a word means what your make it mean"; I agree that this thread has beeen enlightening.

p.

You are grossly distorting the meaning of expected value in statistics with this and the other comments you made. If you want to account for variable brightness across the scene (just so that you can keep arguing), you need to introduce two new dimensions but the expected value still exists and is a function of the position as well.

Yes, Eric Fossum has quite a few pertinent things to say on the DPReview thread: www.dpreview.com/forums/thread/4710831

Brief and concise are desirable. Here is a concise answer: Photon noise is "a spatially and temporally random phenomenon described by Bose-Einstein statistics". [1]

But that doesn't answer your questions unless you happen to be familiar with bosons.

The following quote from E. M. Purcell [2] is more informative. I hope you find it brief and concise.

“*Think then, of a stream of wave packets each about c/Δν  long, in a random sequence [c = speed of light, v = frequency]. There is a certain probability that two such trains accidentally overlap. When this occurs they interfere and one may find four photons, or none, or something in between as a result. It is proper to speak of interference in this situation because the conditions of the experiment are just such as will ensure that these photons are in the same quantum state. To such interference one may ascribe the ‘abnormal’ density fluctuations in any assemblage of bosons.”

E. M. Purcell 1956, Nature, 178, 1449.

Purcell referred to these density fluctuations as wave noise.

So, according to Purcell.

Yes, it is.

The answer is both. Light waves can interact with each other. When groups of light waves are in superposition (the definition of a wave packet) they interact with each other, constructive and, or destructive wave interference occurs. The Schrödinger equation applies [3] and the result is an amplitude (i.e. non-classical)  probability distribution (Purcell's "abnormal density fluctuations") in photoelectron production.

This means the distribution of photoelectron production we observe in digital photographs as noise is linked to the same mathematics that describes electron density in atoms. Of course, photoelectron production and electron probability density in atoms are completely dissimilar.

I have now strayed from brief and concise.

But Purcell's answer to your questions means speculation about fluctuations in photon creation are unnecessary. Wave packet interference occurs after photon creation. Fluctuations in photon arrival are unnecessary as well. Arrival is irrelevant since the fluctuations are amplitude density fluctuations.

Purcell's answer certainly means invoking Newtonian physics analogies such as raindrops or particles falling into buckets do not apply to photon noise.


1. James Janesick "Photon Noise", SPIE Press, 2007,  p21.
2. According to Wikopedia: "Edward Mills Purcell (August 30, 1912 – March 7, 1997) was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery (published 1946) of nuclear magnetic resonance in liquids and in solids.")

3. The Schrödinger Equation, Terence Tao, Department of Mathematics, UCLA, Los Angeles CA (see footnote 2)

4. A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf Introduction to Quantum Noise, Measurement and Amplification Rev. Mod. Phys. 82, 1155 – Published 15 April 2010. Eqn 2.1

This is a common description, but I have a problem with it since watching this video called "How big is a photon?":
www.youtube.com/watch?v=SDtAh9IwG-I&t

He sets up an experiment where he splits the light from a HeNe laser and then combines the beams to get interference. The light intensity is sow low that statistically there is only one photon in the beam paths at at any given time, and the lengths of the two paths are also different. If photons are wave packets localized in space, and if we increase the path length difference enough, we should see a reduction in the interference pattern, but this doesn't seem to be the case. The length difference doesn't matter at all. If the size of these packets are infinite, are they really "packets"?

Instead he argues that light is an electromagnetic field which is not quantized. Quantization happens when we interact with the field, and the probability of detection of a photon is proportional to the intensity of the field.

So there are no photons. Only localized field interactions which we call photons. I'm not a phycisist so I don't know if this is correct, but to me it is at least intuitive. It makes it very easy to explain the experiment in the video. Does this also apply to other particles? I have no clue, but it sounds similar to the question in quantum mechanics "does the world exist when we're not looking?".

Oh, no. This type of "wave noise" is largely irrelevant for photography.

See the link posted here: www.dpreview.com/forums/thread/4710831?page=3#forum-post-67026232. The wave packet theory is in the past, it seems.

Interesting post, I can't comment on wave/particle duality because I understand the issue has not been satisfactorily resolved yet. But with regards to Purcell's comment above, perhaps they are not Newtonian physics analogies, rather observations that can be explained statistically at pixel scale thanks to Planck/Einstein and others.

For instance if one were to assume that the raindrops falling in one bucket arrived at a certain rate but with random timing without collisions, they would arrive with the observed Poisson distribution. Just like photons from a source landing on a pixel's photodiode. And we can actually measure the relative counts.

Jack

Jack's post above made me take another look at this post of AShortUserName. There is so much misinformation even in that paragraph, I do not even know where to start. "Light waves can interact with each other" - not in the linear model mentioned later. "When groups of light waves are in superposition (the definition of a wave packet)" : no, this is not a definition of a wave packet, you are thinking of a superposition of wave packets, as Tao says. "They interact with each other": actually, a superposition means linearity (a sum) which is on the opposite side of interaction. "The Schrödinger equation applies [3]": it does not, it is the QED for photons but even if we forget that, the (linear) Schrödinger equation excludes interaction, and we do have a superposition. "The result is an amplitude (i.e. non-classical)  probability distribution (Purcell's "abnormal density fluctuations") in photoelectron production": this is not even wrong, it just does not mean anything.

A stimulating video. However, he loses me when he says

On the contrary, the spatial/time probability of detecting a photon is indeed quantized, as we have been able to observe for quite some time once we got clean detectors, see for instance recently Eric Fossum's QIS work:.

As he explains, the Gaussian looking shapes around each peak are the result of the convolution of the detector's read noise with impulses representing the discrete Poisson distribution of actual detected counts. As detector read noise gets smaller and smaller, the resulting probability distribution tends to the discrete Poisson probability mass function.

So if the resulting probability (light*detection) is quantized with a Poisson distribution, what can we say about the field from the source?

We know from an earlier post that Poisson*Binomial = Poisson, therefore the field being quantized would match observation.

What class of continuous field probability distributions times Binomial detection would result in a Poisson distribution out?

Jack