Transverse Coupling in Fiber Optics Part III: Bending Losses

Open waveguides support modes whose phase velocity is smaller than the velocity of plane waves in the surrounding medium. Thus, no radiation takes place under normal conditions. If the fiber is bent, however, the phase velocity increases in proportion to the distance from the curvature center. At some radius, it exceeds the velocity of plane waves in the medium and a radiation loss is suffered. This effect is of great practical importance in fiber communication because it sets a limit on how sharp bends can be made without resulting in untolerable loss. For most single-mode optical-glass fibers, a radius of curvature of the order of 1 cm can be tolerated. For gas lenses and weakly guiding millimeter-wave systems, the minimum radius is sometimes as large as 100 m. The relative insensitivity of glass fibers to bends results from the rather large change in refractive index in the cross section that they provide. It constitutes their main advantage compared to other guiding systems. 1379 Different methods have been used to evaluate the radiation losses of curved waveguides. For simple geometries, it is possible to solve the b o u n d a r y value problem using a cylindrical coordinate system, t h e loss being given by t h e imaginary part of the propagation constant. This method was used by Miller and Talanov 1 for a reactive surface, and by Marcatili 3 for dielectric slabs with various imbedding materials and for waveguides with rectangular cross section. Another method consists in evaluating the power radiated at the radius where t h e phase velocity of t h e guided mode becomes equal to t h e velocity in t h e surrounding medium.