Poisson photon counting (${\sigma }_N=\sqrt{⟨N⟩}$)

For independent photon detections with a fixed mean rate, counts in a fixed exposure have a Poisson distribution, so the variance equals the mean. Multiplying by a representative photon energy gives ${\sigma }_E=h\nu \sqrt{N}$. A source-only signal-to-noise threshold $q$ therefore needs an expected $N=q^2$ photons. Background, detector noise, variable illumination and photon correlations can invalidate that source-only formula.