On the Poisson Approximation to Photon Distribution for Faint Lasers

It is proved, that for a certain kind of input distribution, the strongly binomially attenuated photon number distribution can well be approximated by a Poisson distribution. This explains why we can adopt poissonian distribution as the photon number statistics for faint lasers. The error of such an approximation is quantitatively estimated. Numerical tests are carried out, which coincide with our theoretical estimations. This work lays a sound mathematical foundation for the well-known intuitive idea which has been widely used in quantum cryptography.