Concluding remarks

In this chapter we have reconsidered a classical rigorous analytic model for 2-D optical bent slab waveguides and curved dielectric interfaces with piecewise constant refractive index profiles. A frequency domain ansatz in terms of complex order Bessel and Hankel functions leads to an eigenvalue equation (transverse resonance condition) that is to be solved for the complex valued angular mode number.

According to the asymptotic expansions of the relevant Hankel functions, it is shown that the modal solutions decay according to $1/\sqrt{r}$ for growing radial coordinates $r$, i.e. specific products of the profile components are integrable along the radial axis. For purposes of bend mode normalization, we could derive quite compact expressions for the angular modal power. A complex valued product of two general fields in the polar coordinate system has been defined, which is suitable to express orthogonality properties of nondegenerate, directional, and polarized modal solutions of the bent waveguide problem. Perturbational analysis for the effect of core refractive index changes is presented.

A series of detailed (benchmarking) examples complements the former abstract reasoning. Concerning propagation constants, these emphasize the arbitrariness in the definition of the bend radius. Examples for profiles of bend modes and for the spatial evolution of the related physical fields are given, for fundamental and higher order modes of bent slabs with relatively small refractive index contrast, as well as for whispering gallery modes supported by high-contrast curved interfaces. 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 (real part of) propagation constants of moderately lossy modes is also verified.

With the present results, a sound analytical basis for (2-D) coupled-mode-theory modeling of resonator devices involving microrings or microdisks as cavities has been established. We expect that many of the notions discussed in this chapter are directly transferable to the case of 3-D configurations involving bent channels with 2-D cross sections.