Nonperturbative calculation of bend loss for a pulse in a bent planar waveguide

Conformally transforming the scalar Helmholtz equation for a step-index planar waveguide with a constant bend radius and assuming a bend radius large in comparison to the core half-width yields a differential equation whose solutions can be written in terms of Airy functions. A correct expansion of the leaky mode field in terms of these Airy functions and matching the field and its derivative across boundaries yields an eigenvalue equation whose complex roots give the leaky mode propagation constant and bend loss directly. This allows a second-order Taylor series expansion for the bend loss as a function of wavelength over the spectrum of a pulse to be determined. A comparison between bend loss for a cw wave and a pulse with the same carrier frequency is made.