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.
We propose a horizontally symmetrical three-layer dielectric structure composed of a high-index central (core) layer surrounded by two identical low-index cladding layers, which acts as an optical differentiator in reflection. If the refractive index of the surrounding medium is greater that the refractive index of the cladding layers, the spectra of the considered structure may exhibit resonant features associated with the excitation of a leaky mode localized at the central layer. At resonant conditions, the reflection coefficient will vanish at certain values of frequency and angle of incidence, which enables the differentiation of the incident optical pulse. We theoretically justify that this three-layer structure can perform temporal differentiation (differentiation of an incident optical pulse envelope), spatial differentiation (differentiation of an optical beam profile) and the so-called “spatiotemporal differentiation” (differentiation of an optical signal envelope along a certain direction in the (x,t)-plane). Rigorous numerical simulation results demonstrate high quality of differentiation. It is shown that the resonance quality factor increases with the increase in the thickness of the cladding layers, which makes it possible to achieve a required linearity interval of the differentiating filter. The proposed differentiator is more compact than Fourier correlators containing graded-index lenses and substantially easier to fabricate than metasurface-based devices incorporating periodically arranged nanoresonators and may find application in ultrafast analogue computing and signal processing systems.
In this work, we study numerically and theoretically phase-shifted Bragg gratings (PSBG) for Bloch surface waves (BSW) propagating along the interfaces between a 1D photonic crystal and a homogeneous medium. The studied on-chip structure consists of a set of dielectric ridges located on the photonic crystal surface constituting two symmetrical onchip Bragg gratings separated by a defect layer. Rigorous simulation results demonstrate that the surface wave diffraction on the proposed on-chip PSBG is close to the diffraction of plane electromagnetic waves on conventional PSBG. For the considered examples, the correlation coefficient between the spectra of conventional PSBG and on-chip PSBG exceeds 0.99 near the resonance corresponding to the excitation of the eigenmodes localized in the defect layer. Conventional PSBG are widely used for spectral filtering as well as for temporal and spatial transformations of optical pulses and beams including differentiation and integration of pulse envelope or beam profile. In the present work, we discuss the capability of on-chip PSBG to implement the operations of temporal and spatial differentiation of BSW pulses and beams. The presented examples demonstrate the possibility of using the proposed structure for high-quality differentiation. The obtained results can be applied for the design of the prospective integrated systems for on-chip alloptical analog computing.