General framework for ultrafast nonlinear photonics: unifying single and multi-envelope treatments [Invited]

Numerical modeling of ultrashort pulse propagation is important for designing and understanding the underlying dynamical processes in devices that take advantage of highly nonlinear interactions in dispersion-engineered optical waveguides. Once the spectral bandwidth reaches an octave or more, multiple types of nonlinear polarization terms can drive individual optical frequencies. This issue is particularly prominent in χ⁽²⁾ devices where all harmonics of the input pulse are generated and there can be extensive spectral overlap between them. Single-envelope approaches to pulse propagation have been developed to address these complexities; this has led to a significant mismatch between the strategies used to analyze moderate-bandwidth devices (usually involving multi-envelope models) and those used to analyze octave-spanning devices (usually involving models with one envelope per waveguide mode). Here we unify the different strategies by developing a common framework, applicable to any optical bandwidth, that allows for a side-by-side comparison between single- and multi-envelope models. We include both χ⁽²⁾ and χ⁽³⁾ interactions in these models, with emphasis on χ⁽²⁾ interactions. We show a detailed example based on recent supercontinuum generation experiments in a thin-film LiNbO₃ on sapphire quasi-phase-matching waveguide. Our simulations of this device show good agreement between single- and multi-envelope models in terms of the frequency comb properties of the electric field, even for multi-octave-spanning spectra. Building on this finding, we explore how the multi-envelope approach can be used to develop reduced models that help build physical insights about new ultrafast photonics devices enabled by modern dispersion-engineered waveguides, and discuss practical considerations for the choice of such models. More broadly, we give guidelines on the pros and cons of the different modeling strategies in the context of device design, numerical efficiency, and accuracy of the simulations.