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 [1]. 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. The key elements of our approach compared to previous few-cycle propagation models are the explicit identification of a third harmonic generation (THG) term in the nonlinear polarization, and the integration of carrier evolution effects directly onto the complex pulse envelope itself. This allows us lift the "slowly varying envelope" and "slowly evolving wave" approximations that have previously limited the application of envelope-approaches to sub-cycle regimes. The use of a GNEE model also allows efficient numerical solution and ready integration of realistic effects such as frequency-dependent mode area and Raman scattering. Our GNEE simulations have been compared in detail with the direct numerical integration of Maxwell's equations. For propagation in a dispersionless χ{sup}(3) medium, Fig. 1(a) presents results for optical shock formation, showing (i) temporal field profile, (ii) detail of the optical shock, and (iii) corresponding spectrum. GNEE simulations (solid lines) are in excellent quantitative agreement with Maxwell's equations (circles) even for this extreme case where nonlinear spectral broadening extends over 10 times the carrier frequency, and the temporal carrier exhibits shock dynamics on a sub-50 attosecond timescale. GNEE modelling is particularly relevant to experiments in highly nonlinear waveguides such as silica nanowires where extreme confinement can lead to new supercontinuum (SC) generation regimes. Fig. 1(b) shows results simulating SC generation in a 600 nm diameter nanowire, illustrating broadening over multiple octaves. Significantly, simulations without the inclusion of the THG term in the nonlinearity show significantly reduced broadening, illustrating its crucial role in this regime. Additional simulations show that the GNEE approach can also accurately model carrier envelope offset (CEO) phase effects and CEO-dependent nonlinear interactions.
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