The Single-Mode Homogeneously Broadened Laser II: Phase-Space Analysis

Abstract

We now set about the task of reducing the laser phase-space equation of motion (7.100) to the Fokker—Planck equation (7.71a). There are two steps to be taken. We must eliminate derivatives beyond second order, and we must eliminate the explicit appearance of the variables v, v*, and m describing the laser medium. Actually, we are not quite going to pass directly from (7.100) to (7.71a). We eliminate derivatives beyond second order using van Kampen’s system size expansion. But in Sect. 5.1.3 we discussed the fact that a systematic “small noise” expansion generally leads directly to a linear Fokker—Planck equation. The laser Fokker—Planck equation (7.71a) is nonlinear. It is possible to arrive at this nonlinear equation from (7.100) by dropping derivatives and performing the adiabatic elimination of atomic variables. This approach, however, does not treat the fluctuations in a systematic way. Equation (7.71a) retains terms of the same order as terms that are dropped; at least it does so in certain operating regions. A systematic system size expansion leads directly to the linearized version of (7.71). This expansion should tell us if, and when, the linearization breaks down. We will therefore first seek a self-consistent laser theory, including fluctuations, analogous to the theory of the radiatively damped two-level medium developed in Sect. 6.3.5; we seek a set of macroscopic equations like (6.167a)–(6.167c) and a linear Fokker—Planck equation describing fluctuations about the macroscopic state. We will find that this linearized theory holds below threshold.