Proper understanding of the propagation of electromagnetic fields along the cavity is very important, because the behavior of these fields is mainly responsible for the resonances in the spectral response. This is systematically explained by a time domain model of isolated (circular) cavities in terms of integer valued angular mode number and complex valued eigenfrequencies [39,40,41]. But in case of the present resonators, the cavity is coupled to the external straight waveguides. These waveguides are usually modeled in terms of given real valued frequency [42,43]. Thus, to study a cavity coupled to straight waveguides, a complex eigenfrequency model of (isolated) cavities is not the most suitable choice. Problems for the description of the coupling arise from the radially growing field solutions of the time domain cavity modes [39,40]. Nevertheless, attempts are made to study analytically the above coupling by combining a complex eigenfrequency model of the cavities with a real valued frequency model of the straight waveguides [41,44].
To avoid such juggling of viewpoints, either parametrical or ab initio frequency domain models are used. While very few studies exist about ab initio models, parametrical models, such as discussed in Section 1.4, are quite popular, e.g. see [33,45,46,47]. Treating the (complex valued) cavity mode propagation constants and the coupler scattering matrices (which parameterize the interaction between the cavity and the straight waveguide in terms of coupling coefficients) as free parameters, one can qualitatively analyze the effects of the above parameters on the spectral response. Under simplifying assumptions of a lossless coupler and unidirectional monomode wave propagation, universal relations for the coupling of optical power between a cavity ring and one bus waveguide are derived in [48]. In Ref. [38] a detailed procedure of fitting these free parameters in the ring-resonator model to experimental measurements is outlined. But when it comes to the design of microresonators, one must know how to determine systematically these free parameters, given the geometrical and material properties of the device.
The coupling between a circular cavity and a straight waveguide has been modeled with phenomenologically derived expressions for the coupling coefficient [22], with integral equations based on Green's functions [49], and with different versions of coupled mode theory. Coupled mode theory [50,51,52] proved to be a quite useful tool for the analysis of the interaction between straight waveguides. Motivated by this success, in Ref. [44] space dependent coupled mode theory has been used to model the coupling between a curved waveguide and a straight waveguide. That approach is based on a complex eigenfrequency model of the cavities. Also the coupled mode equations are derived phenomenologically. As another variant, time dependent coupled mode theory was also attempted [22,53].
Concerning the modeling of three dimensional microresonators, by using the effective index method certain 3-D settings can be reduced to 2-D, which are then analyzed by means of phenomenologically derived expressions for the coupling coefficients [33] or by conformal mapping method [54].
Apart from the above analytical and parametrical methods, pure numerical methods like Finite Difference Time Domain (FDTD) [55,56] are also used for the simulation of microresonators. In Ref. [57] a finite difference based Helmholtz solver is used to compute the spectral response of 2-D microdisk resonators. Even for the 2-D setting, pure numerical methods turn out to be time consuming, and these become prohibitively time consuming in case of the 3-D setting. These numerical models are generally reserved for benchmarking of the results obtained with other techniques, not for practical design work.
Finally one should emphasize that specific 2-D configurations can be treated in an accurate and highly efficient analytical way in terms of integral equations [58]. This concerns eigenvalue [59,60] and scattering problems [61,62] for micro-ring and disk cavities with regular deformations, in the vicinity of one straight dielectric waveguide or half-block. Unfortunately, the extension to 3-D appears to be far from obvious.