Optical bullets in (2+1)D photonic structures and their interaction with localized defects

This dissertation studies light propagation in Kerr-nonlinear two dimensional waveguides with a Bragg resonant, periodic structure in the propagation direction. The model describing evolution of the electric field envelopes is the system of 2D Nonlinear Coupled Mode Equations (2D CME). The periodic structure induces a range of frequencies (frequency gap) in which linear waves do not propagate. It is shown that, similarly to the ID case of a fiber grating, the 2D nonlinear system supports localized solitary wave solutions, referred to as 2D gap solitons, which have frequencies inside the linear gap and can travel at, any speed smaller than or equal to the speed of light in the corresponding homogeneous medium. Such solutions are constructed numerically via Newton's iteration. Convergence is obtained only near the upper edge of the gap. Gap solitons with a nonzero velocity are constructed by numerically following a bifurcation curve parameterized by the velocity v. It is shown that gap solitons are saddle points of the corresponding Hamiltonian functional and that no (constrained) local minima of the Hamiltonian exist. The linear stability problem is formulated and reasons for the failure of the standard Hamiltonian PDE approach for determining linear stability are discussed.; In the second part of the dissertation interaction of 2D gap solitons with localized defects is studied and trapping of slow enough 2D gap solitons is demonstrated. This study builds on [JOSA B 19, 1635 (2002)], where such trapping of 1D gap solitons is considered. Analogously to this 1D problem trapping in the 2D model is explained as a resonant energy transfer into one or more defect modes existent for the particular defect. For special localized defects exact linear modes are found explicitly via the separation of variables. Numerical computation of linear defect modes is used for more general defects. Corresponding nonlinear modes are then constructed via Newton's iteration by following a bifurcation curve as their total power is increased. Trapping of a given gap soliton is then predicted if a nonlinear defect mode with the same frequency and smaller or equal total power exists. Finally, a finite dimensional ODE model is devised to describe the dynamics after trapping takes place. The system is given explicitly in the case the defect supports two linear defect modes. Its validity is verified numerically.; The PDE dynamics are approximated via numerical simulations employing a finite difference time domain method with an implicit-explicit 'ESDIRK' type time integration scheme and perfectly matched boundary layers to treat outgoing radiation waves.