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Photonic devices performing required temporal and spatial transformations of optical signals are of great interest for a wide range of applications including all-optical information processing and analog optical computing. Among the most important operations of analog optical processing are the operations of temporal and spatial differentiation. Various types of resonant photonic structures performing these operations were previously proposed, such as phase-shifted Bragg gratings and other multilayer structures, resonant diffraction gratings, and nanoresonators.
In the current work, we present an overview of our recent results dedicated to the design of resonant nanophotonic structures for optical implementation of various differential operators including integrated structures for Bloch surface waves and guided modes. A special attention is paid to a simple planar (integrated) optical differentiator consisting of two identical grooves on the surface of a dielectric slab waveguide (the details are presented in our recently published work [L. L. Doskolovich, E. A. Bezus, N. V. Golovastikov, D. A. Bykov, and Victor A. Soifer, “Planar two-groove optical differentiator in a slab waveguide,” Opt. Express 25(19), 22328–22340 (2017)]). The studied planar differentiator operates in reflection and enables temporal and spatial differentiation of optical pulses and beams propagating in the waveguide. The differentiation is associated with the excitation of an eigenmode localized at the ridge cavity located between the grooves. We show that by changing the groove length one can choose the required quality factor of the resonance (and, consequently, the linearity interval of the transfer function of the differentiator) in accordance with the width of the frequency or spatial (angular) spectrum of the incident pulse or beam. The presented numerical simulation results demonstrate high-quality spatial, temporal and the so-called spatiotemporal differentiation. The proposed differentiator may find application in the design of on-chip all-optical analog computing and signal processing systems.
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