This paper introduces a simple, analytical method for generalizing the behavior of bent, weakly-guided fibers and waveguides. It begins with a comprehensive study of the modes of the bent step-index fiber, which is later extended to encompass a wide range of more complicated waveguide geometries. The analysis is based on the introduction of a scaling parameter, analogous to the V-number for straight step-index fibers, for the bend radius. When this parameter remains constant, waveguides of different bend radii, numerical apertures and wavelengths will all propagate identical mode field distributions, except scaled in size. This allows the behavior of individual waveguides to be broadly extended, and is especially useful for generalizing the results of numerical simulations. The technique is applied to the bent step-index fiber in this paper to arrive at simple analytical formulae for the propagation constant and mode area, which are valid well beyond the transition to whispering-gallery modes. Animations illustrating mode deformation with respect to bending and curves describing polarization decoupling are also presented, which encompass the entire family of weakly-guided, step-index fibers.
© 2007 Optical Society of America
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