Approximate analysis of circular bends in nonlinear planar waveguides

The modal properties of a bent planar waveguide with a nonlinear core and cladding are investigated. We assume weak nonlinearity: the self-induced refractive index change is small compared to the difference of refractive index at the core-cladding interface. We develop a first-order perturbation theory to describe the effects of the bend and weak nonlinearity, giving rise to a set of coupled equations for the modal fields. We give exact solutions to the resultant perturbation equations for the modal field in terms of Mathieu functions of fractional order. We identify a narrow region of power for which bound modes exist but have no linear equivalent. We also employ a Gaussian approximation and find that it provides simple and accurate analytic expressions for the modal parameters such as, field distribution, correction to the propagation constant and the induced shift of the field.