In this thesis, we restrict ourself to two dimensional settings. While for specific configurations one could regard the present two dimensional model as an approximate description of realistic devices in terms of effective indices, in other cases simulations in three spatial dimensions are certainly necessary, e.g. for vertically coupled resonators. Therefore our model is formulated such that an extension to three dimensions is straightforward. We treat the circular microcavities as traveling wave resonators in the framework of a pure frequency domain description.
The most common resonator model, discussed in Chapter 1, permits a basic understanding of the functioning of these devices. The resonators are functionally represented in terms of two bent-straight waveguide couplers with appropriate connections using bent and straight waveguides. The abstract scattering matrices of these couplers, and the propagation constants of the cavity bends allow to compute the spectral responses of the resonators. Generally, these quantities are treated as free parameters. One of the objectives of this work is to present a systematic approach to compute these free parameters for given resonator configurations. Another objective is to characterize the response of the resonators systematically for various geometrical parameters (e.g. the radius of the cavity, the widths of the waveguides, the separation distances) and material parameters (e.g. the refractive indices).
A rigorous classical analytic model of confined optical wave propagation along 2-D bent slab waveguides and curved dielectric interfaces is investigated in Chapter 2. This frequency domain model is based on ansatz of piecewise continuous bend mode profiles in terms of Bessel and Hankel functions. This approach provides a clear picture of the behaviour of bend modes, concerning their decay for large radial arguments or effects of varying bend radius. For the numerical implementation of this model, fast and reliable routines are required to evaluate Bessel functions with large complex orders and large arguments. Using the ``uniform asymptotic expansions'' of Bessel/Hankel functions, we found that with present standard computers it is not a problem to carry out the rigorous analytic evaluation of the problem. Our implementation enabled detailed studies of bent waveguide properties, including higher order bend modes and whispering gallery modes, their interference patterns, and issues related to bend mode normalization and orthogonality properties. Also a perturbational expression is derived for the shift in the propagation constant due to changes in the core refractive index.
Capitalizing on the availability of rigorous analytical modal solutions for 2-D bent waveguides, Chapter 3 presents a model of bent-straight waveguide couplers using a frequency domain spatial coupled mode formalism, derived by means of a variational principle. The formulation is consistent with standard physical notions; it takes into account that multiple modes in each of the cores may turn out to be relevant for the functioning of the resonators supplemented with such couplers. Simulation results for the response of 2-D couplers for monomode and multimode settings for varying separation distances, radii, and different wavelengths are discussed. The resulting scattering matrices show reciprocity as expected according to the symmetry of the coupler structures, which also provides a useful means of assessing the reliability of the simulations.
Having explained how to compute the required cavity mode propagation constants and the scattering matrices, Chapter 4 presents simulation results for the entire resonator devices. We also discuss a few procedures for the faster calculation of the spectral response. The examples cover the spectral response and field for microresonators with mono- and multi-modal cavities for TE and TM polarizations. Comparisons with finite difference time domain simulations show very good overall agreement. A detailed analysis of effect of the separation distances on the resonator spectral response is carried out, which leads to a useful criterion that should be satisfied by the numerical simulations. Also, a perturbational approach for the evaluation of tuning of resonators by slight changes of the cavity core refractive index is presented.
The present work about coupled mode theory based modeling and analysis of 2-D circular integrated optical microresonators paved the way for analogous simulations of realistic microresonators in three spatial dimensions.