Classical Wave Systems Research Articles

Topological photonics in metamaterials

Originally a pure mathematical concept, topology has been vigorously developed in various physical systems in recent years, and underlies many interesting phenomena such as the quantum Hall effect and quantum spin Hall effect. Its widespread influence in physics led the award of the 2016 Nobel Prize in Physics to this field. Topological photonics further expands the research field of topology to classical wave systems and holds promise for novel devices and applications, e.g., topological quantum computation and topological lasers. Here, we review recent developments in topological photonics but focus mainly on their realizations based on metamaterials. Through artificially designed resonant units, metamaterials provide vast degrees of freedom for realizing various topological states, e.g., the Weyl point, nodal line, Dirac point, topological insulator, and even the Yang monopole and Weyl surface in higher-dimensional synthetic spaces, wherein each specific topological nontrivial state endows novel metamaterial responses that originate from the feature of some high-energy physics.

Advances in pseudo-magnetic field in artificial structures

<p indent=0mm>Pseudo-magnetic fields (PMFs) are firstly proposed in stretched graphene systems, in which the electrons behave as they do in a real magnetic field. Physically, the mechanical strain results in a lattice deformation and changes the hopping strength between carbon atoms, and thus induces an effective vector potential by shifting the Dirac points at the Brillouin zone. The PMF opens the door to control quantum transport by mechanical means and to explore unprecedented physics in a high-field regime, thus has drawn widespread attention recently. Inspired by the realization of strain-induced PMF in graphene, the concept of PMF is introduced into classical wave (e.g., optical and acoustic) systems using similar ways, because the strain tensor can also be written in the form of the vector potential within the tight binding approximation in classical wave systems. For classical wave systems that do not (or weakly) respond to external magnetic fields, the introduction of PMFs provides a new mechanism for controlling waves, and brings many physical properties analogous to those in real magnetic fields, such as Landau level quantization and quantum-Hall-like effect. Meanwhile, the macroscopic controllability enables the (optical or acoustic) artificial structure to provide an excellent macroscopic platform to explore PMFs and associated phenomena. In this review article, we introduce the recent progress of PMFs in artificial structures, especially on the construction routes of PMFs and related physical properties. The first classical PMF is realized in a triaxially deformed honeycomb photonic lattice, which consists of long dielectric waveguide arrays. Similarly, the PMF can be achieved in the other classical wave systems like microwave and acoustic systems by applying a triaxial strain field on the artificial structures or using the simulated strain effect by displacing the lattice sites. To simplify the experimental realization, a uniaxial strain field or deformation is proposed to realize the PMF in classical wave systems. Particularly, the microscopic controllability of artificial structures provides us a more simple approach to experimentally generate the PMF by engineering the individual unit-cell structure. By patterning the gradually varied unit-cell structures, the linearly varying vector potential can be realized in the gradient artificial structure, which results in a large uniform PMF. In such a system, the frequency-resolved spectrum is measured to demonstrate the Landau Levels induced by the PMF, and the quantum-Hall-like edge states are also observed. Benefiting from the macroscopic controllability of artificial structures, the PMF can be extended to 3D classical systems by engineering gradient artificial structures, in which the shifting of Weyl points effectively induces a gauge field. Particularly, in contrast to the zero-order Landau plateau in a 2D system with PMF, the zero-order chiral Landau level with linear dispersion is observed in 3D artificial structures with PMF. The zero-order chiral Landau level is characterized by one-way propagation, and the direction of group velocity is determined by the chirality of the Weyl point and the direction of PMF. The introduction of PMFs in 2D and 3D classical systems not only provides us a possibility to explore novel physics associated to the magnetic field without real magnetic field, but also opens the door to a wide range of potential applications, such as enhancing light or sound emission and nonlinear wave mixing due to the high density of states of Landau levels, optical or sound laser, and topological transports of the edge states under PMFs.

Moiré circuits: Engineering magic-angle behavior

Moir\'e superlattices in the twisted bilayer graphene provide an unprecedented platform to investigate a wide range of exotic quantum phenomena. Recently, the twist degree of freedom has been introduced into various classical wave systems, giving rise to new ideas for the wave control. The question is whether twistronics and moir\'e physics can be extended to electronics with potential applications in the twist-enabled signal processing. Here, we demonstrate both in theory and experiment that lots of fascinating moir\'e physics can be engineered using electric circuits with extremely high degrees of freedom. By suitably designing the interlayer coupling and biasing of one sublattice for the twisted bilayer circuit, the low-energy flat bands with large bandgaps away from other states can be realized at various twist angles. Based on the moir\'e circuit with a fixed twist angle, we experimentally demonstrate the effect of band narrowing as well as the localization of electric energy when a magic value of the interlayer coupling is applied. Furthermore, the topological edge states, which originate from the moir\'e potential induced pseudomagnetic field, are also observed for the first time. Our findings suggest a flexible platform to study twistronics beyond natural materials and other classical wave systems, and may have potential applications in the field of intergraded circuit design.

Conserved quantities, exceptional points, and antilinear symmetries in non-Hermitian systems

Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general -symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples.

Four dimensional second-order topological insulator based on a synthetic plasmonic metasurface.

It is possible to explore higher dimensional topological properties in lower dimensional structures by introducing additional synthetic dimensions. In this Letter, we construct a four-dimensional (4D) second-order topological insulator using gradient nanoparticle arrays arranged in a periodic lattice. The nanoparticle array has spatially varying inter-particle distance along x and y directions, which can be regarded as two synthetic dimensions. Different from higher-order topological insulators in classical wave systems, the higher-order topological states in this 4D system are protected by a pair of first Chern numbers in two-dimensional (2D) subspaces instead of by the quantized 2D Zak phases. It is shown that there exist (4-1)- and (4-2)-dimensional boundary states for both transverse and longitudinal collective resonant modes, which provides new, to the best of our knowledge, mechanisms for light confinement and control in such a plasmonic superlattice.

Dielectric Characterization of Liquid Mixtures Using EIT-like Transmission Window

Electromagnetically induced transparency (EIT) is a destructive interference phenomenon which has been widely reported in both quantum and classical wave systems. In an EIT-like system, sharp transmission windows appear within stop bands, indicating potentials in sensor applications. In this work, we design a planar microwave circuit with an EIT-like spectrum for dielectric characterization of liquid mixtures. The sharp EIT-like transmission window relies on the strong coupling between an open-circuited stub and a split ring resonator, which are sensitive to surrounding environment. Binary liquid mixtures of water-ethanol and water-methanol are injected into the container fixed on the surface of the sensitive area. Complex permittivity of the liquid under test affects both the center frequency and the quality factor of the EIT-like transmission window. Thus, complex permittivity of liquid mixtures can be further retrieved from measured S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</sub> . Measurements together with theoretical and numerical analysis are discussed in detail and results match well with Debye function. We also analyze the achievable high sensitivity (5%) of the proposed EIT-like sensor when measuring solid samples with low permittivity dynamic range and conduct experiments. The proposed design is easy to fabricate and can be used to test various materials.

Topological Corner Modes Induced by Dirac Vortices in Arbitrary Geometry.

Recently, higher-order topologies have been experimentally realized, featuring topological corner modes (TCMs) between adjacent topologically distinct domains. However, they have to comply with specific spatial symmetries of underlying lattices, hence their TCMs only emerge in very limited geometries, which significantly impedes generic applications. Here, we report a general scheme of inducing TCMs in arbitrary geometry based on Dirac vortices from aperiodic Kekulé modulations. The TCMs can now be constructed and experimentally observed in square and pentagonal domains incompatible with underlying triangular lattices. Such bound modes at arbitrary corners do not require their boundaries to run along particular lattice directions. Our scheme allows an arbitrary specification of numbers and positions of TCMs, which will be important for future on-chip topological circuits. Moreover, the general scheme developed here can be extended to other classical wave systems. Our findings reveal rich physics of aperiodic modulations, and advance applications of TCMs in realistic scenarios.

Robust one-way edge state in convection-diffusion systems

Topological insulators are insulated in the bulk but can support edge states on the surface. Since the discovery of edge states in quantum mechanics systems, the related physics has also been extended to classical wave systems. Here, we manage to reveal that edge states are not necessarily limited to wave systems but can also exist in convection-diffusion systems that are essentially different from wave systems. For this purpose, we study heat transfer in a graphene-like (or honeycomb) lattice to demonstrate thermal edge states with robustness against defects and disorders. Convection is compared to electron cyclotron, which breaks space-reversal symmetry and determines the direction of thermal edge propagation. Diffusion leads to interference-like behavior between opposite convection, preventing temperature propagation in the bulk. We also display thermal unidirectional interface states between two lattices with opposite convection. These results extend the physics of edge states beyond wave systems.

Corner states and topological transitions in two-dimensional higher-order topological sonic crystals with inversion symmetry

Macroscopic two-dimensional sonic crystals with inversion symmetry are studied to reveal higher-order topological physics in classical wave systems. By tuning a single geometry parameter, the band topology of the bulk and the edges can be controlled simultaneously. The bulk band gap forms an acoustic analog of topological crystalline insulators with edge states which are gapped due to symmetry reduction on the edges. In the presence of mirror symmetry, the band topology of the edge states can be characterized by the Zak phase, illustrating the band topology in a hierarchy of dimensions, which is at the heart of higher-order topology. Moreover, the edge band gap can be closed without closing the bulk band gap, revealing an independent topological transition on the edges. The rich topological transitions in both bulk and edges can be well-described by the symmetry eigenvalues at the high-symmetry points in the bulk and surface Brillouin zones. We further analyze the higher-order topology in the shrunken sonic crystals where slightly different physics but richer corner and edge phenomena are revealed. In these systems, the rich, multidimensional topological transitions can be exploited for topological transfer among zero-, one- and two- dimensional acoustic modes by controlling the geometry.

Near- and Far-Field Excitation of Topological Plasmonic Metasurfaces

The breathing honeycomb lattice hosts a topologically non-trivial bulk phase due to the crystalline-symmetry of the system. Pseudospin-dependent edge states, which emerge at the interface between trivial and non-trivial regions, can be used for the directional propagation of energy. Using the plasmonic metasurface as an example system, we probe these states in the near- and far-field using a semi-analytical model. We provide the conditions under which directionality was observed and show that it is source position dependent. By probing with circularly-polarised magnetic dipoles out of the plane, we first characterise modes along the interface in terms of the enhancement of source emissions due to the metasurface. We then excite from the far-field with non-zero orbital angular momentum beams. The position-dependent directionality holds true for all classical wave systems with a breathing honeycomb lattice. Our results show that a metasurface in combination with a chiral two-dimensional material, could be used to guide light effectively on the nanoscale.

Topological states in twisted pillared phononic plates

In recent years, the advances in topological insulator in the fields of condensed matter have been extended to classical wave systems such as acoustic and elastic waves. However, the quantitative robustness study of topological states which is indispensable in practical realization is rarely reported. In this work, we proposed topologically protected edge states with zigzag, bridge and armchair interfaces in a new twisted phononic plate. The robustness of non-trivial band gap in bulk structure is clearly presented versus twisted angles, revealing a threshold of 5 degrees which is the key fundamental information for the robustness of topological edge states. We further defined a localized displacement ratio as an efficient parameter to characterize edge states. Due to the different orientation of the three interfaces, zigzag and bridge edge states show higher quantitative robustness in their localized displacement ratio. A map of robustness as a function of both frequency and twisted angle highlights the better performance of the topological zigzag edge state. Robustness is evaluated for twisted angle and for all possible types of interfaces for the first time, which benefits for the design and fabrication of solid functional devices with great potential applications.

Elastic phononic plates with first-order and second-order topological phases

Topologically protected 1D edge states and the 2D surface states have been observed in variant classical wave systems. By extending the quantized polarization, Berry phase, to higher multipole moments, the higher-order topological phases, expressed as 0D corner states, have been theoretically predicted and recently experimentally observed in electric, electromagnetic and acoustic systems. In this paper, we design an elastic phononic plate consisting of a bi-layered hexagonal lattice. The valley-polarized gapless 1D edge state is observed in the elastic phononic plate, by lifting the Dirac cones at the Brillouin zone (BZ) corners. When the bi-layered phononic plates are stacked one by one, the chirality of the oblique columns along the z axis guarantees this 2D elastic phononic plate to be a 3D chiral phononic system. Weyl cones and topologically protected gapless 2D surface states are yielded in this 3D chiral phononic system. The above topological phases are first-order ones with gapless (d-1)-dimensional edge/surface states. By defining a composite cell consisting of three-unit cells and modulating the intra- and inter-coupling within/among the composite cells, the folded double Dirac cone at the BZ center will be gapped and the topological phase of the elastic phononic plate will evolve to a second-order one, expressed as the robust 0D corner state, instead of the gapless 1D edge state. The realization of the first-order edge/surface states and second-order corner states in a unified elastic phononic platform opens up a new path for the design of on-chip topological devices for advanced elastic wave manipulation among the bulk, edge/surface and corner modes.

Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal

Three-dimensional (3D) gapless topological phases can be classified by the dimensionality of the band degeneracies, including zero-dimensional (0D) nodal points, one-dimensional (1D) nodal lines, and two-dimensional (2D) nodal surfaces. Both nodal points and nodal lines have been realized recently in photonics and acoustics. However, a nodal surface has never been observed in any classical-wave system. Here, we report on the experimental observation of a twofold symmetry-enforced nodal surface in a 3D chiral acoustic crystal. In particular, the demonstrated nodal surface carries a topological charge of 2, constituting the first realization of a higher-dimensional topologically-charged band degeneracy. Using direct acoustic field measurements, we observe the projected nodal surface and its Fermi-arc-like surface states and demonstrate topologically-induced robustness of the surface states against disorders. This discovery of a higher-dimensional topologically-charged band degeneracy paves the way toward further explorations of the physics and applications of new topological semimetal phases.

Transfer matrix method for the analysis of space-time-modulated media and systems

Space-time modulation adds another powerful degree of freedom to the manipulation of classical wave systems. It opens the door for complex control of wave behavior beyond the reach of stationary systems, such as nonreciprocal wave transport and realization of gain media. Here we generalize the transfer matrix method and use it to create a general framework to solve wave propagation problems in time-varying acoustic, electromagnetic, and electric circuit systems. The proposed method provides a versatile approach for the study of general space-time varying systems, which allows any number of time-modulated elements with arbitrary modulation profile, facilities the investigation of high order modes, and provides an interface between space-time modulated systems and other systems.

Observation of topological gravity-capillary waves in a water wave crystal

The discovery of topological phases of matter, initially driven by theoretical advances in quantum condensed matter physics, has been recently extended to classical wave systems, reaching out to a wealth of novel potential applications in signal manipulation and energy concentration. Despite the fact that wave propagation in many realistic media (metals at optical frequencies, polymers at ultrasonic frequencies) is inherently dispersive, topological wave transport in photonic and phononic crystals has so far been limited to ideal situations and proof-of-concept experiments involving dispersionless media. Here, we report the first experimental demonstration of topological edge states in a classical water wave system supporting highly dispersive wave propagation, in the intermediate regime of gravity-capillary waves. We use a stochastic method to rigorously take into account the inherent dispersion and devise a water wave crystal insulator supporting valley-selective transport at topological domain walls. Our measurements, performed with a high-speed camera under stroboscopic illumination, unambiguously demonstrate the possibility of valley-locked transport of water waves.

Experimental observation of valley-Hall edge states in elastic waveguides based on diatomic-graphene-like phononic crystals

Inspired by recent discoveries of topological phases of matter in quantum physics, there has been a rapidly growing research effort in creating their analogs in other classical wave systems, including acoustics. Achieving robust wave transmission even in the presence of disorder and defects could have a profound impact on many practical applications and devices. In this study, we report on the design and experimental validation of a fully continuous and load-bearing phononic structural waveguide capable of one-directional guided modes along the walls of topologically distinct domains. The lattice structure of the waveguide is inspired by diatomic graphene which allows realizing an elastodynamic analog of the quantum valley Hall effect (QVHE). Despite the fact that individual bulk properties are topologically trivial (i.e., associated with a zero Chern number), the dynamic behavior acquires topological significance in the neighborhood of the high symmetry points in momentum space; the so-called valleys. Our theoretical and experimental results confirm the existence of protected edge states traveling along the walls of domains having broken space inversion symmetry.

Experimental Demonstration of Acoustic Chern Insulators.

We report the experimental realization of an acoustic Chern insulator (ACI), by using an angular-momentum-biased resonator array with the broken Lorentz reciprocity. High Q-factor resonance of the constituent rotors is leveraged to reduce the required rotation speed. ACI is a new topological acoustic system analogous to the electronic quantum Hall insulator, based on an effective magnetic field. Experimental results show that the ACI featured with a stable and uniform metafluid flow bias supports one-way nonreciprocal transport of sound at its edges, which is topologically immune to various types of defects. Our work opens up opportunities for exploring unique observable topological phases and developing topological-insulator-based nonreciprocal devices in acoustics.

Tunable THz generalized Weyl points.

Weyl points, as linearly double degenerated point of band structures, have been extensively researched in electronic and classical wave systems. However, Weyl points' realization is always accompanied with delicate "lattice structures". In this work, frequency-tunable terahertz (THz) generalized Weyl points inside the parameter space have been investigated and displayed by a specially designed photonic crystal with polydimethylsiloxane (PDMS) immersed in 4-cyano'-pentylbipenyl (5CB) liquid crystals (LCs). The reflective phase vortices as a signature of the generalized Weyl points are observed through our numerically simulations. Besides, interface states between photonic crystals and any reflective substrates are fulfilled too. Meanwhile, we could also change the orientation of LC molecule by the external magnetic field so as to tune the frequency of the first two bands' Weyl point from 0.27698THz to 0.30013THz. This band lies in the short-range wireless communication. Thus, our proposal may be beneficial to the investigation and application of Weyl points' properties and strongly localized states.

Observation of higher-order topological acoustic states protected by generalized chiral symmetry.

Topological systems are inherently robust to disorder and continuous perturbations, resulting in dissipation-free edge transport of electrons in quantum solids, or reflectionless guiding of photons and phonons in classical wave systems characterized by topological invariants. Recently, a new class of topological materials characterized by bulk polarization has been introduced, and was shown to host higher-order topological corner states. Here, we demonstrate theoretically and experimentally that 3D-printed two-dimensional acoustic meta-structures can possess nontrivial bulk topological polarization and host one-dimensional edge and Wannier-type second-order zero-dimensional corner states with unique acoustic properties. We observe second-order topological states protected by a generalized chiral symmetry of the meta-structure, which are localized at the corners and are pinned to 'zero energy'. Interestingly, unlike the 'zero energy' states protected by conventional chiral symmetry, the generalized chiral symmetry of our three-atom sublattice enables their spectral overlap with the continuum of bulk states without leakage. Our findings offer possibilities for advanced control of the propagation and manipulation of sound, including within the radiative continuum.

Manipulation and Characterization of the Valley-Polarized Topological Kink States in Graphene-Based Interferometers.

Valley polarized topological kink states, existing broadly in the domain wall of hexagonal lattice systems, are identified in experiments. Unfortunately, only very limited physical properties are given. Using an Aharanov-Bohm interferometer composed of domain walls in graphene systems, we study the periodical modulation of a pure valley current in a large range by tuning the magnetic field or the Fermi level. For a monolayer graphene device, there exists one topological kink state, and the oscillation of the transmission coefficients has a single period. The π Berry phase and the linear dispersion relation of kink states can be extracted from the transmission data. For a bilayer graphene device, there are two topological kink states with two oscillation periods. Our proposal provides an experimentally feasible route to manipulate and characterize the valley-polarized topological kink states in classical wave and electronic graphene-type crystalline systems.