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Shot noise is generally a white noise. For an imaging system based on detectors the shot noise variance that scalers linearly with flux and time (details here and here): $$\sigma^2=I\cdot QE\cdot\Delta t$$

My question is, if optics color that noise. Consider the fact that the point spread function may have a relatively long tail. If I use a laser to illuminate, at the laser output shot noise is indeed white. However, after light is reflected towards my detector, don't the optics PSF color the noise?

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Going back to the physical problem:

You send $N$ photons within some fixed timeframe $\Delta t$. The number of these photons has the variance as you describe it in your question.

However, after light is reflected towards my detector, don't the optics PSF color the noise?

As long as you're not exciting the material in your optics to emit photons in a fashion that has some memory, e.g. through photoluminescense or phosphorence, no? You might reducing the $N$ that gets through, but that's just going to be a multiplicative factor: If there was no photon before in a smaller time interval, there will not be a new one after going through your optics. And, since the processes that your optics will have don't exhibit memory, there's also no correlation introduced.

The additional delay that your photon might induce through any PSF applies to each photon individually; however, for every time frame of fixed length, the occurrence of a photon stays independent identically distributed. And independent -> white.

In conclusion: no, the noise remains white.

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    $\begingroup$ In the case of photoluminescence or phosphorescence the temporal impulse response of the system to intensity variations would be spectrally weighted. However, with steady input illumination the system would still reach a steady state where the flux on each pixel is constant, and the noise at each pixel is white -- there would just be different fluxes at each pixel than without photouminescence or phosphorescence. $\endgroup$
    – TimWescott
    Feb 8 at 7:04
  • $\begingroup$ @TimWescott absolutely, but at the point where we reach a steady state in terms of the timescales involved, we can model the distribution of arrived Photons simply as normal distribution – Poisson approximates Normal pretty well for large intensities. $\endgroup$ Feb 8 at 10:37
  • $\begingroup$ No DSP in my argument :) I'm not sure what we actually disagree on: we both say the noise remains white, right, with or without memory-giving mechanisms. $\endgroup$ Feb 8 at 15:33
  • $\begingroup$ I totally misread your answer -- and I don't think it was because it was because it was vague in any way. Gonna delete a comment or two :O . $\endgroup$
    – TimWescott
    Feb 8 at 15:38
  • $\begingroup$ oh that happens to me all the time! It's also a super confusing thing to think about Poisson noise processes in the quantum regime, especially considering Heisenberg uncertainty doesn't seem very compatible to the idea that we're taking exact time frames and put count probabilities on them. $\endgroup$ Feb 8 at 15:56

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