The statistical distribution of switch-on delay in single-mode semiconductor lasers I: equilibrium off-condition

The theory of stochastic processes as applied to photon emission and absorption events is used to calculate the distribution of delay in switch-on from a sub-threshold condition in directly modulated semiconductor lasers down to a probability of 10^-10. This involves the derivation of the relative probability distribution of photon number in the laser late enough in the switch-on process such that deterministic relations can be applied thereafter. This distribution, assumed constant in some treatments, is found to change only a little from its initial form, which is a negative binomial, From this one deduces a delay distribution whose width is proportional to the period of the switch-on transient, relatively independently of the precise starting point, but which can be narrowed by injection of additional spontaneous emission. Experiment satisfactorily supports the theory