| Summary: | This dissertation examines models, methods, and applications of electric field pulse propagation in nonlinear optics. Standard nonlinear optical propagation models such as the NLS equation are derived using a procedure invoking a slowly-varying wave approximation which amounts to discarding second order derivatives in the propagation direction. This work follows a more intuitive procedure emphasizing unidirectionality, the core trait of laser light propagation, by projecting a nonlinear wave system onto a unidirectional subspace. The projection method is discussed as a general theory and then applied to a series of different electric field configurations. Two important full-field propagation models are examined. The unidirectional pulse propagation equations (UPPE's) are generated from Maxwell's equations with the sole approximation being that of unidirectionality. The second model studied is the MKP equation which is a canonical full-field propagation equation particularly amenable to mathematical analysis due to its status as a conserved system. Applications unique to full-field propagation including electric field shock and harmonic walk-off induced collapse arrest are studied through numerical simulations. An emphasis is placed on the mid-infrared to long-infrared wavelength regime where significant differences between envelope models and electric field models manifest as a result of extremely weak dispersion. Presented are the first embedded Runge-Kutta exponential time-differencing (RKETD) methods of fourth order with third order embedding and fifth order with third order embedding for non-Rosenbrock type nonlinear systems. A procedure for constructing RKETD methods that accounts for both order conditions and stability is outlined. In the stability analysis, the fast time scale is represented by a full linear operator in contrast to particular scalar cases considered before. An effective time-stepping strategy based on reducing both ETD function evaluations and rejected steps is described. Comparisons of performance with adaptive-stepping integrating factor (IF) are carried out on a set of canonical partial differential equations including the standard z-propagated UPPE.
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