The generation of picosecond pulses in passively mode-locked solid-state lasers. III. Intensity maxima and their probability distribution

Abstract

This is the third in a series of papers analysing the generation of picosecond pulses in passively mode-locked solidstate lasers. The growth of intensity fluctuations inside the cavity and the corresponding development of a train of pulses are carefully considered. This paper is particularly concerned with intensity maxima and their probability distribution.It is shown that the hitherto accepted method of uniform sampling of the intensity profile leads to errors and that it is better to obtain the required probability distribution directly from the power spectrum of the intensity fluctuations inside the cavity. It is assumed that below threshold the electromagnetic radiation inside the cavity has the character of narrow-band noise. Above threshold the amplitude of each mode ceases to be a random quantity, stochastic properties of the system being expressed by phase fluctuations alone, the envelope of the wave now acquiring the form R = R (φ1…, φJ; t), where φ1 is the phase of the jth mode, the total number of modes being J. One can then derive a probability density function F(p), which defines the probability of an intensity maximum (or a peak) lying within an interval dP centred on P. The knowledge of F(P) leads to the derivation of a probability distribution function f(Y), where the random variable Y = P ₁ /P ₂ is the ratio of the largest p₁ to the second largest maximum p₂. The actual probability of double-pulse generation is finally expressed in terms of a related random variable Z = Y ^d , where d is the So-called discrimination parameter which expresses the degree of non-linearity introduced into the system by the saturuble absorber. Numerical calculation based on the new approach indicate that earlier results, which assume uniform sampling of the intensity profile, in general underestimated the probability of double-pulse generation by 10–20%