Ultra-short pulses in linear and nonlinear media

We consider the evolution of ultra-short optical pulses in linear and nonlinear media. For the linear case, we first show that the initial-boundary value problem for Maxwell's equations in which a pulse is injected into a quiescent medium at the left endpoint can be approximated by a linear wave equation which can then be further reduced to the linear short-pulse equation (SPE). A rigorous proof is given that the solution of the SPE stays close to the solutions of the original wave equation over the time scales expected from the multiple scales derivation of the SPE. For the nonlinear case we compare the predictions of the traditional nonlinear Schrödinger equation (NLSE) approximation with those of the SPE. We show that both equations can be derived from Maxwell's equations using the renormalization group method, thus bringing out the contrasting scales. The numerical comparison of both equations with Maxwell's equations shows clearly that as the pulse length shortens, the NLSE approximation becomes steadily less accurate, while the SPE provides a better and better approximation.