Abstract

Bends play a central role in the design of integrated optical systems, for example in isolation of the output arms of a directional coupler, modulator, or switch. Introduction of a bend in a guide results in a redistribution of energy toward the outside of the bend as well as continuous radiation from the structure. There are also transition losses at input and output planes of the bend. Approximations for the radiation losses for bends, described geometrically as arcs of circles, have been obtained previously for arcs of large radius of curvature.¹ More recently it has been shown² that the application of conformal transformations to the analysis of waveguide bends can lead to relatively simple approximate expressions for guide parameters for arcs of arbitrary radius of curvature. In this paper we present numerical results obtained from the conformal-transformation based analysis on a guide formed by a step increase in refractive index over a region of arbitrary thickness, and on a guide generated by a single step change in refractive index when the surface of the higher index region is convex. As a result of a logarithmic conformal transformation W=RℓnZ/R, the boundaries of a guide with arcs as boundaries in the x-y plane become parallel straight lines in the u-v plane. The index of refraction varies as n exp(u/R) in the transverse direction (u) and remains constant along the coordinate v which is the direction of propagation in the transformed planar configuration, and nor mal to u. The terms W and Z in the above transformation represent the complex values u+iv and x+iy respectively and R is the radius of some circle in the x-y plane.

© 1976 Optical Society of America