Hyperbolic-secant mode coupling

A solution of the coupled mode equations for a hyperbolic secant spatial variation of the coupling between two modes is presented. An analytic expression is given for the transmission coefficient for arbitrary complex differential propagation constant and coupling strength. The expression is particularly simple in the case when the differential attenuation between the modes is negligible. Design curves are presented in terms of normalized parameters. The hyperbolic secant coupling may be truncated and still yield virtually the same transmission as for infinite coupling length. The required coupling length is indicated by a comparison of the ideal expression with the results of numerical integration of the coupled mode equations. Hyperbolic secant coupling can be particularly useful for the design of low-loss bends and twists in overmoded waveguide.