Analyzing stability of a delay differential equation involving two delays

Analysis of the systems involving delay is a popular topic among applied scientists. In the present work, we analyze the generalized equation $D^{\alpha} x(t) = g\left(x(t-\tau_1), x(t-\tau_2)\right)$ involving two delays viz. $\tau_1\geq 0$ and $\tau_2\geq 0$. We use the the stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.Comment: 10 pages, 7 figure

Theory of Coupled Resonator Optical Waveguides (CROW) Exhibiting High Order Exceptional Points of Degeneracy

We present a novel approach and a theoretical framework for generating high order exceptional points of degeneracy (EPD) in photonic structures based on periodic coupled resonators optical waveguides (CROWs). Such EPDs involve the coalescence of Floquet-Bloch eigenwaves in CROWs, without the presence of gain and loss, which is in contrast to the requirement of Parity-Time (PT) symmetry to develop exceptional points based on gain and loss balance. The EPDs arise here by introducing symmetry breaking in a conventional chain of coupled resonators through coupling the chain of resonators to an adjacent uniform optical waveguide, which leads to unique modal characteristics that cannot be realized in conventional CROWs. Such remarkable characteristics include high quality factors (Q-factor) and strong field enhancement, even without any mirrors at the two ends of a cavity. We show for the first time the capability of CROWs to exhibit EPDs of various order; including the degenerate band edge (DBE) and the stationary inflection point (SIP). The proposed CROW of finite length shows enhanced quality factor when operating near the DBE, and the Q-factor exhibits an anomalous scaling with the CROW's length. We develop the theory of EPDs in such unconventional CROW using coupled-wave equations, and we derive an analytical expression for the dispersion relation. The proposed unconventional CROW concepts have various potential applications including Q-switching, nonlinear devices, lasers, and extremely sensitive sensors.Comment: 16 pages, 11 figure

Theory of coupled resonator optical waveguides exhibiting high-order exceptional points of degeneracy

We present an approach and a theoretical framework for generating high-order exceptional points of degeneracy (EPDs) in photonic structures based on periodic coupled resonator optical waveguides (CROWs). Such EPDs involve the coalescence of Floquet-Bloch eigenwaves in CROWs, without the presence of gain and loss, which contrasts with the parity-time symmetry required to develop exceptional points based on gain and loss balance. The EPDs arise here by introducing symmetry breaking in a conventional chain of coupled resonators through periodic coupling to an adjacent uniform optical waveguide, which leads to unique modal characteristics that cannot be realized in conventional CROWs. Such remarkable characteristics include high quality factors (Q factors) and strong field enhancement, even without any mirrors at the two ends of a cavity. We show for the first time the capability of CROWs to exhibit EPDs of various orders, including the degenerate band edge (DBE) and the stationary inflection point. The proposed CROW of finite length shows an enhanced quality factor when operating near the DBE, and the Q factor exhibits an unconventional scaling with the CROW's length. We develop the theory of EPDs in such unconventional CROW using coupled-wave equations, and we derive an analytical expression for the dispersion relation. The proposed unconventional CROW concepts have various potential applications including Q switching, nonlinear devices, lasers, and extremely sensitive sensors

Wannier-Stark resonances in optical and semiconductor superlattices

In this work, we discuss the resonance states of a quantum particle in a periodic potential plus a static force. Originally this problem was formulated for a crystal electron subject to a static electric field and it is nowadays known as the Wannier-Stark problem. We describe a novel approach to the Wannier-Stark problem developed in recent years. This approach allows to compute the complex energy spectrum of a Wannier-Stark system as the poles of a rigorously constructed scattering matrix and solves the Wannier-Stark problem without any approximation. The suggested method is very efficient from the numerical point of view and has proven to be a powerful analytic tool for Wannier-Stark resonances appearing in different physical systems such as optical lattices or semiconductor superlattices.Comment: 94 pages, 41 figures, typos corrected, references adde

On the analysis of mixed-index time fractional differential equation systems

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalised Mittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.Comment: 21 pages, 6 figures (some are made up of sub-figures - there are 15 figures or sub-figures

Topological Photonics

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio