Conclusions

A functional decomposition of the resonators, outlined in Chapter 1, in terms of two bent-straight waveguide couplers with appropriate straight waveguide and bent waveguide connections, led to expressions for their spectral responses in terms of the constituent cavity mode propagation constants and the coupler scattering matrices. We presented a systematic approach to compute these parameters for given resonators.

To determine the cavity mode propagation constants, in Chapter 2 we turn to a classical analytic frequency domain model for 2-D optical bent slab waveguides and curved dielectric interfaces with piecewise constant refractive index profiles. A field ansatz in terms of complex order Bessel and Hankel functions led to an eigenvalue equation that is to be solved for the complex valued mode propagation constants. Unlike the fields in a complex frequency model of full cavities [39,40], in the present case the asymptotic expansions of the relevant Hankel functions show that the modal solutions decay according to $1/\sqrt{r}$ for growing radial coordinates $r$. For the normalization of bend modes, we derived quite compact expressions for the angular modal power. We also discussed orthogonality properties of nondegenerate, directional, and polarized modal solutions of the bent waveguide problem. For the later discussion of tuning of resonators by changes of the core refractive index, a perturbational analysis for shifts in the propagation constants of bend modes is discussed.

Our implementation of ``uniform asymptotic expansions'' for Bessel functions of complex order facilitated the computational evaluation of the present analytic model of bent waveguides. In Chapter 2 we have presented a series of detailed (benchmark) examples of fundamental and higher order modes of bent slabs and whispering gallery modes. These include the computation of propagation constants (in view of the arbitrariness in the definition of the bend radius), bend mode profiles, and the spatial evolution of the related physical fields. A few illustrative examples for interferences of bend modes have been shown, that exhibit a periodic angular beating pattern (apart from the mode decay) in the guiding regions of the bends, and tangential, ray-like bundles of outgoing waves in the exterior regions. The validity of the perturbational expression for shifts in the propagation constants of moderately lossy modes has been verified.

To obtain the required coupler scattering matrices, in Chapter 3 we have proposed a spatial coupled mode theory based model of 2-D bent-straight waveguide couplers. In this pure frequency domain approach, the coupled mode equations are rigorously derived by a variational principle. Leveraged by the availability of the analytic bent modes on unbounded radial intervals, we could implement this model in consistent standard physical notions. By solving the coupled mode equations numerically, and projecting the resulting coupled mode field on the modes of the straight waveguide, we obtained the scattering matrices of the couplers. In symmetrical coupler settings, the scattering matrices satisfy a reciprocity property, that permits to assess the validity of the simulation results.

With the above coupled mode theory model, a detailed study of the effects of separation distance, the radius of the bent waveguide, and the wavelength on the scattering matrices has been carried out. For couplers involving bent waveguides that support multiple whispering gallery modes, with the present approach we have systematically investigate the significance of the individual modes. This feature of the present CMT formulation provides good insight for the characterization of resonances of the entire device.

With bent modes and coupler scattering matrices being available, in Chapter 4 we further elaborated the resonator model discussed in Chapter 1. We have formulated it for the multimodal setting. Also, approximations for fast-yet-reliable spectrum computation have discussed. The resonator model has been assessed for several examples of monomodal/multimodal structures with ring or disk cavities. In case of a resonator with multimodal cavity, it turns out that only a few most relevant cavity basis fields are required to construct approximate solutions to the scattering problems that are sufficient for purposes of practical resonator design. With the present CMT approach, one can associate different resonance extrema in the spectral response with specific cavity modes. The comparison of CMT results with rigorous FDTD simulations shows good agreement. Moreover, the computational effort for the CMT analysis combined with the interpolation technique of Chapter 4 is substantially lower. The resonator field plots obtained with the CMT model provide useful qualitative impressions of the functioning of the resonators.

An analysis of effect of the separation distances on the spectral response shows that for the identical couplers setting, at resonance the drop power is maximum and the through power is minimum. The constrain of invariance of the drop power for interchanging the gaps is very well satisfied by the present CMT simulations. The perturbational analysis of shifts in the resonances due to slight changes in the cavity core refractive index agrees quite well with direct calculations.