Mathematical Aspects of Modeling the Propagation of Intense, Ultrashort Long-Wavelength Laser Pulses

Abstract

This thesis consists of three main contributions to the theoretical and practical pursuit of modeling the nonlinear propagation of ultrashort, long-wavelength laser pulses through transparent media. We organize these contributions into three main chapters. First, we derive a novel pseudo-spectral shooting method to model intense, ultrashort optical pulses incident on remote targets. The method naturally extends the unidirectional pulse propagation equation (UPPE) to include backward propagating radiation either due to nonlinear coupling with the ``forward'' field or reflections generated at material boundaries. We derive explicit convergence conditions and highlight the applicability of the method through several realistic examples. We also propose an alternative pseudo-spectral iteration scheme for modeling counter-propagating pulses and perform relevant simulations. Second, we derive and analyze a carrier-wave-resolved, asymptotic PDE model for the propagation of intense, ultrashort long-mid-IR laser pulses through weakly dispersive bulk materials. The derived model is essentially an extension of the well-known Kadomtsev-Petviashvili equation (gKPE). We give an original proof of sufficient conditions for the solutions of this equation to become singular over a finite propagation path from first principles. We thoroughly analyze the properties of the predicted mathematical collapse (critical versus supercritical) and find rigorous conditions on the ambient material properties which leads to a nonzero, finite amount of energy being deposited into the collapsing sub-cycle filament. We also compare the (3+1) dynamics with reduced, lower-dimensional models and find (1+1)-soliton solutions to the resulting equation. Finally, we observe the emergence of novel attosecond-duration localizations of the electric on the route to the predicted supercritical collapse in our gKPE model and show that these ultra-intense spatiotemporal structures are robust and persist even in full UPPE simulation for a 10 micron laser pulse propagating in air.