  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.
