Spatial integration and differentiation of optical beams in a slab waveguide by a dielectric ridge supporting high-Q resonances.

We show that a very simple structure consisting of a single subwavelength dielectric ridge on the surface of a slab waveguide enables spatial integration and differentiation of the profile of optical beams propagating in the waveguide. The integration and differentiation operations are performed in reflection and in transmission, respectively, at oblique incidence of the beam impinging on the ridge. The implementation of these operations is associated with the resonant excitation of a cross-polarized eigenmode of the ridge. We demonstrate that the quality factor of the resonances strongly varies along the dispersion curves and allows one to achieve the required tradeoff between the integration (or differentiation) quality and the amplitude of the resulting beam. The presented rigorous numerical simulation results confirm high-quality integration and differentiation. The proposed integrated structure may find application in ultrafast all-optical analog computing and signal processing systems.