The realization of wavelength scale and sub-wavelength scale fabrication of integrated optical devices has led to a concurrent need for computational design tools that can accurately model electromagnetic phenomena on these length scales. This dissertation describes the physical, analytical, numerical, and software developments utilized for practical implementation of two particular frequency domain design tools: the modal method for multilayer waveguides and one-dimensional lamellar gratings and the Rigorous Coupled Wave Analysis (RCWA) for 1D, 2D, and 3D periodic optical structures and integrated optical devices. These design tools, including some novel numerical and programming extensions developed during the course of this work, were then applied to investigate the design of a few unique integrated waveguide and grating structures and the associated physical phenomena exploited by those structures. The properties and design of a multilayer, multimode waveguide-grating, guided mode resonance (GMR) filter are investigated. The multilayer, multimode GMR filters studied consist of alternating high and low refractive index layers of various thicknesses with a binary grating etched into the top layer. The separation of spectral wavelength resonances supported by a multimode GMR structure with fixed grating parameters is shown to be controllable from coarse to fine through the use of tightly controlled, but realizable, choices for multiple layer thicknesses in a two material waveguide; effectively performing the simultaneous engineering of the wavelength dispersion for multiple waveguide grating modes. This idea of simultaneous dispersion band tailoring is then used to design a multilayer, multimode GMR filter that possesses broadened angular acceptance for multiple wavelengths incident at a single angle of incidence. The effect of a steady-state linear loss or gain on the wavelength response of a GMR filter is studied. A linear loss added to the primary guiding layer of a GMR filter is shown to produce enhanced resonant absorption of light by the GMR structure. Similarly, linear gain added to the guiding layer is shown to produce enhanced resonant reflection and transmission from a GMR structure with decreased spectral line width. A combination of 2D and 3D modeling is utilized to investigate the properties of an embedded waveguide grating structure used in filtering/reflecting an incident guided mode. For the embedded waveguide grating, 2D modeling suggests the possibility of using low index periodic inclusions to create an embedded grating resonant filter, but the results of 3D RCWA modeling suggest that transverse low index periodic inclusions produce a resonant lossy cavity as opposed to a resonant reflecting mirror. A novel concept for an all-dielectric unidirectional dual grating output coupler is proposed and rigorously analyzed. A multilayer, single-mode, high and graded-index, slab waveguide is placed atop a slightly lower index substrate. The properties of the individual gratings etched into the waveguide's cover/air and substrate/air interfaces are then chosen such that no propagating diffracted orders are present in the device superstrate and only a single order is present outside the structure in the substrate. The concept produces a robust output coupler that requires neither phase-matching of the two gratings nor any resonances in the structure, and is very tolerant to potential errors in fabrication. Up to 96% coupling efficiency from the substrate-side grating is obtained over a wide range of grating properties.

Communication using fiber optics is an integral part of modern societies and one of the most important parts of this is the grating filter of a laser. In this report we introduce both the periodic and the non-periodic grating filter and discuss how there can be resonance in these structures. We then provide an exact method for calculating the spectrum of these grating filters and study three different methods to calculate this approximately. The first one is the \emph{Fourier approximation} which is very simple. For the studied filters the fundamental form of the results for this method is correct, even though the details are not. The second method consists of calculating the spectrum exactly for some values and then use interpolation by splines. This method gives satisfactory results for the types of gratings analysed. Finally a method of perturbation is provided for the periodic grating filter as well as an outline for how this can be extended to the non-periodic grating filter. For the studied filters the results of this method are very promising. The method of perturbations may also give a deeper understanding of how a filter works and we therefore conclude that it would be of interest to study the method of perturbations further, while all the studied methods can be useful for computation of the spectrum depending on the required precision. / Fiberoptisk kommunikation utgör en viktig del i moderna samhällen och en av de grudläggande delarna av detta är Bragg-filter i lasrar. I den här rapporten introducerar vi både det periodiska och det icke-periodiska Bragg-filtret och diskuterar hur resonans kan uppstå i dessa. Vi presenterar sedan en exakt metod för att beräkna spektrumet av dessa filter samt studerar tre approximativa metoder för att beräkna spektrumet. Den första metoden är \emph{Fourier-approximationen} som är väldigt enkel. För de studerade filtrena blir de grundläggande formerna korrekta med Fourier-approximationen, medan detaljerna är fel. Den andra metoden består av att räkna ut spektrumet exakt för några punkter och sedan interpolera med hjälp av splines. Den här metoden ger mycket bra resultat för de studerade filtrena. Till sist presenteras en metod baserad på störningsteori för det periodiska filtret, samt en översikt över hur det här kan utökas till det icke-periodiska filtret. Denna metod ger mycket lovande resulat och den kan även ge djupare insikt i hur ett filter fungerar. Vi sluter oss därför till att det vore intressant att vidare studera metoder med störningar, men även att alla studerade metoder kan vara användabara för beräkningen av spektra beroende på vilken precision som krävs.