Finite-difference time-domain model of resonator coupling and nonstationary nonparaxial spatial optical soliton focusing and scattering

The emergence of lightwave communication and computing, and applications in nanophotonics require an understanding of complex electromagnetic wave phenomena in linear and nonlinear materials having frequency-dependent and intensity-dependent polarizations. Both the complexity of the underlying physics and the problems required for modern engineering applications necessitate understanding the vector-nature of the electromagnetic field, and therefore solving Maxwell's equations directly.; Electromagnetic waves propagating in homogeneous nonlinear materials can self-focus to form optical spatial solitons, stable optical beams that propagate without supporting waveguide structures. Understanding their behavior requires modeling both Maxwell's equations and the linear and nonlinear polarizations in the materials supporting the beam propagation. This behavior is often described by the Nonlinear Schrodinger (NLS) equation. However, the use of the NLS equations requires both the scalar and paraxial approximations and does not model the full Maxwell's equations. These approximations place severe limitations on the regime where the NLS equation is valid. Furthermore, the NLS equation does not directly apply to nonintegrable models which describe realistic materials.; The finite-difference time-domain (FDTD) method provides a direct time-domain solution of the full-vector Maxwell's equations without any assumptions of the direction of propagation or plane wave solutions and is applicable to arbitrary material geometries. Auxiliary differential equation (ADE) methods extend the FDTD method to incorporate polarization by time-stepping auxiliary differential equations with Maxwell's curl equations. In this research, ADE methods are extended to solve nonlinear optics problems where the electric field has two or three orthogonal vector electric field components in a realistic material characterized by a three-pole Sellmeier linear dispersion, an instantaneous Kerr nonlinearity, and a dispersive Raman nonlinearity. This technique is designated the general vector auxiliary differential equation (GVADE) FDTD method. First, this method is validated through comparisons to published results for temporal soliton propagation in a dispersive nonlinear material. The method is then validated for the spatial soliton case by reproducing known results for both fundamental spatial soliton propagation of wide beams and the periodic focusing and defocusing of narrow beams.; The GVADE FDTD method is then used to explore the behavior of over-powered spatial solitons, a new class of solitons where the transverse profile of the beam is a solution of the NLS equation, but with twice the amplitude. The collisions of over-powered solitons into compact air holes is explored and an exciting result is found: the scattering of the electromagnetic field energy and subsequent coalescence into lower-power solitons. This computational technique is general and should permit future investigations and design of devices exploiting spatial soliton interactions in background media having submicron air holes and dielectric metal inclusions.; Finally, this dissertation addresses three-dimensional modeling of optical resonators and the vertical coupling to bus waveguides. Optical resonators have applications in wavelength filtering, routing, switching, modulation, and multiplexing, and demultiplexing and they have been studied both experimentally and numerically. This research extends previous studies of waveguide-to-resonator coupling and presents the first published fully three-dimensional FDTD model of vertical waveguide coupling and subsequent optical beam confinement in a photonic racetrack.