| In this dissertation, we consider two problems. Both problems deal with diffractive optics and design of grating structures.; The first is known as the resonance design problem. In this case, we encounter one of the most interesting new developments in diffractive optics which is the integration of a zero-order grating and a planar waveguide to create a resonance. Such structures are able to yield ultra narrow bandwidth filters which potentially have many extraordinary applications. The main step in the design of such a grating structure is to find the resonant wavelength. For any fixed grating structure, calculation of the resonant wavelength is found by solving, a nonlinear eigenvalue problem.; The second of the two design problems focuses on nonlinear diffraction gratings. Here, we will consider a plane wave of frequency ω1 incident on a grating composed of some nonlinear optical material. The effects of the nonlinear material interacting with field give rise to diffracted waves with frequencies ω1 and ω2 = 2ω 1. The creation of waves with the doubled frequency is a phenomenon unique to nonlinear optics known as second harmonic generation (SHG). The design problem in this context is to create a grating structure which enhances the very weak nonlinear optical effects. |
