Ultrashort Optical Pulse Propagation in terms of Analytic Signal

Sh Amiranashvili,A Demircan

doi:10.1155/2011/989515

Abstract

We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field. To this end a second-order nonlinear wave equation is first simplified using a unidirectional approximation. Then the analytic signal is introduced, and all nonresonant nonlinear terms are eliminated. The derived propagation equation accounts for arbitrary dispersion, resonant four-wave mixing processes, weak absorption, and arbitrary pulse duration. The model applies to the complex electric field and is independent of the slowly varying envelope approximation. Still the derived propagation equation posses universal structure of the generalized nonlinear Schrödinger equation (NSE). In particular, it can be solved numerically with only small changes of the standard split-step solver or more complicated spectral algorithms for NSE. We present exemplary numerical solutions describing supercontinuum generation with an ultrashort optical pulse.

Highlights

Complex envelope adequately describes linear and nonlinear propagation of a wave packet with many field cycles [1]
We demonstrate that ultrashort optical pulses propagating in a nonlinear dispersive medium are naturally described through incorporation of analytic signal for the electric field
In the following we study the nonlinear propagation of femtosecond pulses in the anomalous dispersion regime of a microstructured fiber, where complex and comprehensive behavior can be observed

Summary

Introduction

Complex envelope adequately describes linear and nonlinear propagation of a wave packet with many field cycles [1]. Several simplified unidirectional propagation equations have been derived for special dispersion profiles Such models do not use the pulse envelope and apply directly to the pulse field (see [16,17,18,19,20] and a review paper [21]). The generalized NSE was successfully applied to many propagation problems where the pulse envelope evolves as fast as the pulse field and SVEA cannot be used. The sliding time average cannot be used for a fewcycle pulse for which we recall that Ψ is as fast as E itself (see [34] for a critical review of several possible definitions of the envelope) This contradiction is addressed in the present paper. The analytic signal approach is illustrated by exemplary numerical solutions

Basic Equations

Unidirectional Approximation

Resonances

Analytic Signal

Numerical Solutions

Conclusions