The present work remains in the scope of classical optics, on the basis of the Maxwell equations. Referring to the classification of resonator types given in [53], in this thesis we treat the circular microcavities as traveling wave resonators in the framework of a pure frequency domain description. For modeling purposes, we adopt the functional decomposition as outlined by the ``standard model'' in Section 1.4. The advantage of such a decomposition is that the analysis of microresonators reduces to the modeling of straight waveguides and bent waveguides, and the modeling of bent-straight waveguide couplers. Out of this, models for straight waveguides are quite well established, and sophisticated numerical tools for their analysis exist already 1.4.
The decomposition approach of the ``standard model'' is well known in the literature [35,46,47], where it is typically presented as a parametrical model. For qualitative study of the spectral response of microresonators, such a parametrical model is sufficient. But from the design point of view, it is rather inadequate. The numerical implementation of such model requires the propagation constants of the cavity segment modes, and the coupling coefficients of the bent-straight waveguide couplers.
The real valued frequency domain model of the wave propagation along bent waveguides or curved interfaces (i.e. cavity segments) is well known [10,63,64]. For recent work on this topic, see Refs. [65,66,67,68]. But due to the difficulties associated with complex order Bessel functions, this approach is hardly ever pursued in detail. Important issues like computing bend modes and their propagation constants (especially for very small bent radii), or the mode normalization still require proper attention.
To some degree, the time domain and the frequency domain viewpoints for propagation of the cavity modes are equivalent, and results obtained with one model can be interpreted in the framework of the other model [40,44]. But this switching of viewpoints is not straightforward, and it leads to a few rather obscure issues like the nature of cavity fields at infinity, or the issue of ``phase matching'' for the coupling between the cavity fields and the straight waveguide fields. As commented in Section 1.5, a model of bent-straight waveguide couplers is lacking, which is consistent in terms of physical notions and based on rigorous mathematical concepts.
A major part of the work presented in this thesis is centered around topics related to the wave propagation along curved interfaces. We establish a model for bent-straight waveguide couplers with a sound mathematical foundation, that is consistent with the physical notions. We derive the governing equations from first principles. Going beyond the mere abstract theoretical model, another objective of this work is the reliable and efficient numerical implementation of the approach. We will also address this issue in the present work.
To investigate feasibility and scope of the above approach, in this thesis we restrict ourself to the modeling of 2-D microresonators. Naturally, we discuss only horizontally coupled devices. The theory, however, has been formulated such that an extension to the 3-D setting (including the vertically coupled case) can follow the same line of arguments [69].