Few-cycle pulse propagation

We present a comprehensive framework for treating the nonlinear interaction of few-cycle pulses using an envelope description that goes beyond the traditional slowly varying envelope approximation method. This is applied to a range of simulations that demonstrate how the effect of a ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ nonlinearity differs between the many-cycle and few-cycle cases. Our approach, which includes diffraction, dispersion, multiple fields, and a wide range of nonlinearities, builds upon the work of Brabec and Krausz [T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997)] and Porras [M. A. Porras, Phys. Rev. A 60, 5069 (1999)]. No approximations are made until the final stage when a particular problem is considered.