Ultra-short pulses in linear and nonlinear media

TLDR

We consider the evolution of ultra‑short optical pulses in linear and nonlinear media. The study aims to show that in the linear case the Maxwell initial‑boundary problem can be approximated by a linear wave equation and further reduced to the linear short‑pulse equation, and in the nonlinear case to compare predictions of the NLSE and the short‑pulse equation. The authors derive both the linear short‑pulse equation and the nonlinear Schrödinger equation from Maxwell’s equations, using a linear wave approximation for the linear case and the renormalization group method for both, highlighting the differing scales. The short‑pulse equation closely tracks the Maxwell solutions over expected time scales, and numerically outperforms the NLSE as pulse length shortens.

Abstract

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. A rigorous proof is given that the solution of the short pulse equation stays close to the solutions of the original wave equation over the time scales expected from the multiple scales derivation of the short pulse equation. For the nonlinear case we compare the predictions of the traditional nonlinear Schr\"odinger equation (NLSE) approximation which those of the short pulse equation (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 to Maxwell's equations shows clearly that as the pulse length shortens, the NLSE approximation becomes steadily less accurate while the short pulse equation provides a better and better approximation.