Master equation for spatio-temporal beam propagation and Kerr lens mode-locking

We present a novel spatio-temporal master equation (ME) for describing the evolution of optical fields in laser cavities. Our ME introduces a new type of propagation operator explicitly dependent on the ABCD elements of the cavity. We derive this and show that it correctly reproduces the cavity mode structure in the linear limit. We apply our ME to the problem of Kerr lens mode-locking (KLM) and show that our numerical results, in one dimension (x), are in excellent agreement with those found using the more conventional Huygens' integral method. Dispersion and other fast-time effects are then added to give a full spatio-temporal ME. Again this is applied to KLM and we show that stable soliton-like pulses result.