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I'm somehow struggling with the definition of the SNR (S/N) of optical detectors when it comes to shot-noise. I found some literature where it is defined as follows

$$SNR = N/\sqrt{\bar{N}} = \sqrt{\bar{N}}$$ like here.

and the squared version of this equation in other sources [2]

$$SNR = n^2/\bar{n} = \bar{n}$$.

where n is the number of photons. Which one is the correct one now?

[2]: Reider, G.A., 2016. Photonics. Springer.

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When capturing images on detectors shot noise shows up as a variance in the number of photocounts detected on each pixel. Let $N$ be the incident photon flux (constant). What you should know is that both the mean photocount level and the variance in the photocount level will both be proportional the the incident photon flux $N$. This means the standard deviation (square root of variance) of the photocount level is proportional to $\sqrt{N}$. This is because photocounts are distributed like a Poisson process.

The question then is how to define the signal to noise for this process. Let's consider a different statistical problem for the moment. Consider we have a random variable $H$ which represents the height of 10,000 people measured in $cm$. We can think about the mean of this random variable, $\langle H \rangle$. The mean will have units of $cm$. However, the variance of this random variable, $\text{Var}(H) = \langle H^2 \rangle - \langle H \rangle^2$ has units of $cm^2$. This means that we would not want to define the "signal to noise" of this random variable as the ratio of the mean to the variance as that quantity would have units of $cm^{-1}$ and $SNR$ would ideally be a unitless measure of the spread of a statistical sample.

The natural next step is to either 1) take the square root of the variance, $\sigma_H = \sqrt{\text{Var}(H)}$ which has units of $cm$ to compare this to the mean or 2) to square the mean to get units of $cm^2$ for comparison with the variance.

The takeaway is that you should think of variance as having units of "signal squared" and standard deviation as having units of "signal".

Back to your question. The first formula you post is the ratio of the mean photocount level to the standard deviation of the photocount level.

$$ SNR = \frac{N}{\sqrt{N}} = \sqrt{N} $$

The second formula you post is the ratio of the photocount level squared to the variance of the photocount level:

$$ SNR = \frac{N^2}{N} = N $$

Both formulas are valid. Authors may have various reasons for defining the $SNR$ in one way or another. What you should remember is that the mean in proportional to $N$, the variance is also proportional to $N$ and the standard deviation is proportional to $\sqrt{N}$.

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Shot noise increases as $\sqrt{n}$, assuming that the signal is proportional to the detected photons as in CCD or CMOS photodetctors. So the SNR is given by the first equation you gave, when the detected photon count is large enough. For limiting sensitivty, other noise contributions become significant, including read noise, dark current, etc. All uncorrelated noise contributions are then added in quadrature (RMS sum) to obtain the SNR. A good reference on sensor noise and characterization is J. R. Janesic, Photon Transfer.

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