Stochastic and probabilistic equations for three- and four-level lasers: tutorial

In this tutorial, Langevin stochastic equations and Markov probability equations are used to model electron- and photon-number fluctuations in three- and four-level lasers. Equations are derived for the moments of the electron and photon numbers (means, variances and correlations). Both approaches produce the same moment equations. In the Langevin approach, the moments of the noise terms must be specified by other means, whereas in the Markov approach, they are determined self-consistently and satisfy the shot-noise rule: For each process that is modeled by the rate equations, the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative, depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. Formulas are derived for the relative intensity noise and its spectrum. The consequences of these results for three- and four-level lasers are discussed. (C) 2020 Optical Society of America