Nikita V. Golovastikov

Time-domain differentiation of optical pulses in reflection and in transmission using the same resonant grating

Propagation of an optical pulse through a diffraction grating that has a resonance near the pulse central wavelength is discussed. We show that under definite conditions the envelopes of reflected and transmitted pulses may represent the derivative of the incident pulse envelope. Based on the resonance representation of the reflection and transmission spectra, the comparative analysis of the energy efficiency and accuracy of grating-based differentiators operating in reflection and transmission is conducted. A resonant diffraction grating for differentiation of ps-pulses in reflection and in transmission is designed within the framework of the rigorous electromagnetic theory.

Spatiotemporal pulse shaping using resonant diffraction gratings.

We propose a new theoretical model describing spatiotemporal transformations of two-dimensional optical pulses by resonant diffraction gratings. The diffraction of the pulse is described in terms of a linear system. Simple analytical approximations for the transfer function and the impulse response of the system are derived. The derived approximations contain five independent parameters, which can be estimated using the rigorous coupled-wave analysis. The presented numerical simulation results demonstrate that the resonant grating can perform complex pulse transformations, such as the simultaneous spatial and temporal differentiation of the optical pulse envelope.

Spatiotemporal optical pulse transformation by a resonant diffraction grating

The diffraction of a spatiotemporal optical pulse by a resonant diffraction grating is considered. The pulse diffraction is described in terms of the signal (the spatiotemporal incident pulse envelope) passage through a linear system. An analytic approximation in the form of a rational function of two variables corresponding to the angular and spatial frequencies has been obtained for the transfer function of the system. A hyperbolic partial differential equation describing the general form of the incident pulse envelope transformation upon diffraction by a resonant diffraction grating has been derived from the transfer function. A solution of this equation has been obtained for the case of normal incidence of a pulse with a central frequency lying near the guided-mode resonance of a diffraction structure. The presented results of numerical simulations of pulse diffraction by a resonant grating show profound changes in the pulse envelope shape that closely correspond to the proposed theoretical description. The results of the paper can be applied in creating new devices for optical pulse shape transformation, in optical information processing problems, and analog optical computations.

Analytical description of 3D optical pulse diffraction by a phase-shifted Bragg grating.

Diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating (PSBG) is considered. The pulse diffraction is described in terms of signal transmission through a linear system with a transfer function determined by the reflection or transmission coefficient of the PSBG. Resonant approximations of the reflection and transmission coefficients of the PSBG as functions of the angular frequency and the in-plane component of the wave vector are obtained. Using these approximations, a hyperbolic partial differential equation (Klein-Gordon equation) describing a general class of transformations of the incident 3D pulse envelope is derived. A solution to this equation is found in the form of a convolution integral. The presented rigorous simulation results fully confirm the proposed theoretical description. The obtained results may find application in the design of new devices for spatiotemporal pulse shaping and for optical information processing and analog optical computing.

Differentiating space–time optical signals using resonant nanophotonics structures

A theoretical description of the space–time transformations of an optical signal, which passes through resonant gratings and Bragg gratings with a defect, is proposed. The problem of differentiating a space–time optical signal using a resonant grating has been solved. The strict solution to the Maxwell equations using the Fourier modal method is involved to determine the parameters of the transfer function of the resonant diffraction structure and to carry out numerical modeling, which has confirmed the proposed theoretical description.

Planar two-groove optical differentiator in a slab waveguide

We propose a simple planar optical differentiator consisting of two grooves on the surface of a slab waveguide. The studied 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 located between the grooves. The presented numerical simulation results demonstrate high-quality spatial, temporal and the so-called spatiotemporal differentiation. The proposed differentiator may find application in ultrafast analog computing and signal processing systems.

A simple three-layer dielectric structure for spatiotemporal differentiation of optical signals

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.

Resonant integrated nanophotonic structures for analog differentiation of optical signals (Conference Presentation)

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.

On-chip phase-shifted Bragg gratings and their application for spatiotemporal transformation of Bloch surface waves

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.

3D PULSE DIFFRACTION ON A PHASE-SHIFTED BRAGG GRATING

We consider diffraction of a three-dimensional (3D) spatiotemporal optical pulse by a phase-shifted Bragg grating. Resonant approximations of the reflection and transmission coefficients of a phase-shifted Bragg grating as functions of the angular frequency and the in-plane components of the wave vector are obtained. Using these approximations, analytical expressions for re- flected and transmitted 3D pulse envelopes are derived. The presented rigorous simulation results fully confirm the proposed theoretical description.