Existing approaches for modeling of bent waveguides

Initial frequency domain models of optical bent waveguides can be found in Ref. [10,63]. Since then, various different techniques were applied to the task. In Refs. [76,77] a method for the calculation of bend mode losses is presented, based on ray path notions from geometrical optics. By conformal mapping [78,79], the bent waveguide problem can be transformed into equations for a (leaky) straight waveguide. In Ref. [80] conformal mapping with perfectly matched layer (PML) boundary conditions is used to analyze bent waveguides. In a perturbational approach [81] the curvature is treated as a perturbation of a straight waveguide, and the bent waveguide modes are expressed in terms of straight waveguide fields. Other techniques of a more analytical character (e.g. WKB approximations [82], a transfer matrix approach [83]), or of a numerical nature (e.g. beam propagation [84], the method of lines [85], finite difference [86] or finite element discretizations [87]) are applied as well.

The pure analytic approach for modeling of optical bent slab waveguides is quite well known [10,63,64,65,66,88,43], though apparently hardly ever evaluated rigorously, especially for bent waveguides with small radii. When trying to do so, a major obstacle is encountered in the form of the necessity to compute Bessel functions of large complex order and large argument; we experienced that efficient facilities for these function evaluations are not provided by the standard numerical libraries. To overcome that hindrance, most authors resorted to approximations of the problem, such that reliable results for bent slab waveguides, e.g. for the purpose of a bend mode solver benchmark, still seem to be rare.

Alternatively, one can consider time-domain resonances that are supported by full circular cavities. In that viewpoint, the field solutions are parameterized by an integer azimuthal index; the frequency takes the role of a complex valued eigenvalue. Any difficulties with the complex order Bessel functions are avoided in that way, and the values for frequencies and propagation constants can be largely translated between the two viewpoints [39]. However, the field solutions obtained in the latter way are not directly useful for applications, where one is interested in pieces of bent waveguides only. Also, for a microresonator model that combines modal solutions for bent and straight waveguides by means of coupled mode theory integrals, the fixed-frequency bend mode profiles as discussed in this chapter are required.

The present discussion is concerned with a frequency domain model, where the (real valued) frequency or vacuum wavelength is regarded as a given parameter, and one is interested in solutions of the Maxwell equations for wave propagation along angular segments of the curved structures, that are characterized by a complex valued propagation constant.