Can there be multiple shot noise sources? For example: laser & camera

Question

Suppose you have a system consisting of a laser, a filter (with transmission $T$), and a camera. To my understanding, both the laser and the detector produce shot noise. Is that correct?

For a camera pixel this would mean that if there arrive $N_{\text{pixel}}$ photons/s and the camera is exposed for $t_{\text{exposure}}$ [s] the average number of photons is $N_{\text{pixel}} t_{\text{exposure}}$ with shot noise $\sqrt{N_{\text{pixel}} t_{\text{exposure}}}$. The laser spot incidents on multiple pixels unevenly so $N_{\text{pixel}}$ is not the same for all pixels.

But the laser also produces a shot noise. How do you observe this noise in the final detected number of photons on the pixels? Taking into account the filter between the laser and the camera.

Is anybody able to clear this up for me?

Answer 1

So for a given time interval the laser will have a mean photon number, that is the average number of photons present in a pulse that is of that interval. There will be a probability distribution around this though, Poissonian in this case. So within your exposure time you will have a probability distribution of detecting $n$ photons as $P(n)$.

I think what you're asking is how does the photon shot noise translate into electrical signal noise? You will have the responsivity of the pixel $R$ in units of Amps/Watt (Tells you how many amps of photocurrent you get for a watt of optical power) then you will have your optical power in that exposure time which is $N_{\text{photons}}hf/t_{\text{exposure}}$. Your photocurrent is just the product of the two. However $N_{\text{photons}}$ will we probabilistic, i.e. given by the probability of having $n$ photons given by $P(N_{\text{photons}})$. This gives you the probability distribution of photocurrent.

Answer 2

The probability to detect $n$ photons emitted from a laser source - namely a coeherent state generator - is Poissonian. Therefore, the pixels of your camera will detect $n\pm \sqrt n$ photons. This photon noise adds up to the electronic noise, generating a total noise with the following magnitude $$ \sigma_{tot} = \sqrt{ \sigma^2_e + n } $$ where $\sigma_e$ is the electron noise. In practice, since the numbers of photons is expected to be different pixel-by-pixel on your camera, the noise will have a spatial dependency.