We report a new type of phononic crystals with topologically nontrivial band gaps for both longitudinal and transverse polarizations, resulting in protected one-way elastic edge waves. In our design, gyroscopic inertial effects are used to break the time-reversal symmetry and realize the phononic analogue of the electronic quantum (anomalous) Hall effect. We investigate the response of both hexagonal and square gyroscopic lattices and observe bulk Chern numbers of 1 and 2, indicating that these structures support single and multimode edge elastic waves immune to backscattering. These robust one-way phononic waveguides could potentially lead to the design of a novel class of surface wave devices that are widely used in electronics, telecommunication, and acoustic imaging.

Topological Phononic Crystals with One-Way Elastic Edge Waves

Nonlinear topological photonics, which explores topics common to the fields of topological phases and nonlinear optics, is expected to open up a new paradigm in topological photonics. Here, we demonstrate second-harmonic generation (SHG) via nonlinear interaction of double topological valley-Hall kink modes in all-dielectric photonic crystals (PhCs). We first show that two topological frequency band gaps can be created around a pair of frequencies, ${\ensuremath{\omega}}_{0}$ and $2{\ensuremath{\omega}}_{0}$, by gapping out the corresponding Dirac points in two-dimensional honeycomb PhCs. Valley-Hall kink modes along a kink-type domain wall interface between two PhCs placed together in a mirror-symmetric manner are generated within the two frequency band gaps. Importantly, through full-wave simulations and mode dispersion analysis, we demonstrate that tunable, bidirectional phase-matched SHG via nonlinear interaction of the valley-Hall kink modes inside the two band gaps can be achieved. In particular, by using Stokes parameters associated with the magnetic part of the valley-Hall kink modes, we introduce the concept of SHG directional dichroism, which is employed to characterize optical probes for sensing chiral molecules. Our work opens up avenues toward topologically protected nonlinear frequency mixing and active photonic devices implemented in all-dielectric material platforms.

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Second-harmonic generation via double topological valley-Hall kink modes in all-dielectric photonic crystals

https://doi.org/10.1016/j.revip.2022.100076

A brief review of topological photonics in one, two, and three dimensions

On-machine electric discharge truing of small ball-end fine diamond grinding wheels

The effects of gain and loss on the band structures of a bulk topological dielectric photonic crystal (PC) with ${C}_{6v}$ symmetry and the PC-air-PC interface are studied based on first-principle calculation. To illustrate the importance of parity-time (PT) symmetry, three systems are considered, namely the PT-symmetric, PT-asymmetric, and lossy systems. We find that the system with gain and loss distributed in a PT symmetric manner exhibits a phase transition from a PT exact phase to a PT broken phase as the strength of the gain and loss increases, while for the PT-asymmetric and lossy systems, no such phase transition occurs. Furthermore, based on the Wilson loop calculation, the topology of the PT-symmetric system in the PT exact phase is demonstrated to keep unchanged as the Hermitian system. At last, different kinds of edge states in Hermitian systems under the influences of gain and loss are studied and we find that while the eigenfrequencies of nontrivial edge states become complex conjugate pairs, they keep real for the trivial defect states.

Comparative study of Hermitian and non-Hermitian topological dielectric photonic crystals

As an important figure of merit for characterizing the quantized collective behaviors of the wavefunction, Chern number is the topological invariant of quantum Hall insulators. Chern number also identifies the topological properties of the photonic topological insulators (PTIs), thus it is of crucial importance in PTI design. In this paper, we develop a first principle computatioal method for the Chern number of 2D gyrotropic photonic crystals (PCs), starting from the Maxwell's equations. Firstly, we solve the Hermitian generalized eigenvalue equation reformulated from the Maxwell's equations by using the full-wave finite-difference frequency-domain (FDFD) method. Then the Chern number is obtained by calculating the integral of Berry curvature over the first Brillouin zone. Numerical examples of both transverse-electric (TE) and transverse-magnetic (TM) modes are demonstrated, where convergent Chern numbers can be obtained using rather coarse grids, thus validating the efficiency and accuracy of the proposed method.

First-principle calculation of Chern number in gyrotropic photonic crystals.

A novel symplectic algorithm is proposed to solve the Maxwell–Schrodinger (M–S) system for investigating light–matter interaction. Using the fourth-order symplectic integration and fourth-order collocated differences, M–S equations are discretized in temporal and spatial domains, respectively. The symplectic finite-difference time-domain (SFDTD) algorithm is developed for accurate and efficient study of coherent interaction between electromagnetic fields and artificial atoms. Particularly, the Dirichlet boundary condition is adopted for modeling the Rabi oscillation problems under the semiclassical framework. To implement the Dirichlet boundary condition, image theory is introduced, tailored to the high-order collocated differences. For validating the proposed SFDTD algorithm, three-dimensional numerical studies of the population inversion in the Rabi oscillation are presented. Numerical results show that the proposed high-order SFDTD(4, 4) algorithm exhibits better numerical performance than the conventional FDTD(2, 2) approach at the aspects of accuracy and efficiency for the long-term simulation. The proposed algorithm opens up a promising way toward a high-accurate energy-conservation modeling and simulation of complex dynamics in nanoscale light–matter interaction.

Simulating Maxwell–Schrödinger Equations by High-Order Symplectic FDTD Algorithm

A unified 3-D ADI-FDTD algorithm with one-step leapfrog approach for modeling frequency-dependent dispersive media

In this article, an efficient Z-transform-based finite-difference time-domain (Z-FDTD) is developed to implement and analyze electromagnetic scatterings in the 3D biaxial anisotropy. In terms of the Z-transform technique, we first discuss the conversion relationship between time- or frequency-domain derivative operators and the corresponding Z-domain operator, then build up the Z-transform-based iteration from the electric flux D converted to the electric field E based on dielectric tensor ε (and from the magnetic flux B converted to the magnetic field H in line with permeability tensor μ) by combining the constitutive formulations about the biaxial anisotropy. As a result, the iterative process about the Z-FDTD implementation can be smoothly carried out by means of combining with the Maxwell’s equations. To our knowledge, it is inevitably necessary for the absorbing boundary condition (ABC) to be considered in the electromagnetic scattering; hence, we utilize the unsplit-field complex-frequency-shifted perfectly matched layer (CFS-PML) to truncate the Z-FDTD’s physical region, and then capture time- and frequency-domain radiation with the electric dipole. In the 3D simulations, we select two different biaxial anisotropic models to validate the proposed formulations by using the popular commercial software COMSOL. Moreover, it is certain that those results are effective and available for electromagnetic scattering problems under the oblique incidence executed by the Z-FDTD method.

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Guoda Xie

Papers

First-principle calculation of Chern number in gyrotropic photonic crystals.

Simulating Maxwell–Schrödinger Equations by High-Order Symplectic FDTD Algorithm

A unified 3-D ADI-FDTD algorithm with one-step leapfrog approach for modeling frequency-dependent dispersive media

Z-Transform-Based FDTD Implementations of Biaxial Anisotropy for Radar Target Scattering Problems

A unified 3-D simulating framework for Debye-type dispersive media and PML technique based on recursive integral method