Generalized envelope equation for studying sub-cycle dynamics and multiple-harmonic spectral broadening in highly nonlinear waveguides

The numerical modelling of ultrashort pulse propagation in highly nonlinear waveguides such as photonic crystal fibers or photonic nanowires typically uses some form of generalized nonlinear Schrodinger equation (Dudley et al., 2006). However, when considering propagation of ultrabroad bandwidth fields with temporal structure approaching the single-cycle regime, there is a clear need to examine the validity of such envelope-based approaches. In this paper, we present a generalized nonlinear envelope equation (GNEE) that we demonstrate to be valid for regimes where the pulse bandwidth spans many times the optical carrier frequency, and where the temporal electric field contains structure on a sub-cycle 50 attosecond timescale.