Time-domain master equation for pulse evolution and laser mode-locking

Using the well-known analogy between the space and time domains we derive a temporal master equation (ME) operator which can be applied to any cavity containing dispersive and filtering elements, phase or amplitude modulators, and one nonlinear element. The cavity properties are described in terms of 2 × 2 `KIJL' matrices. We show that this ME correctly reproduces the cavity mode structure in the linear limit. Numerical simulation of an actively mode-locked Fabry–Perot laser with the nonlinear medium at an end mirror gives results in excellent agreement with those found using the more conventional Huygens' integral method. Using a simple perturbation approach based on the nonlinear Schrödinger equation (NLS) we also show that the field in this laser is soliton-like, and give analytic expressions for the soliton parameters.