Bloch Wave Research Articles

Analytical expression for the magnetic susceptibility in terms of magnetic Bloch states

Analytical expression for the magnetic susceptibility in terms of magnetic Bloch states

Study on Bandgap of Two‐Dimensional Square Lattice Acoustic Metamaterial with Multiembedded Cores

Acoustic metamaterials have been widely studied for their excellent ability of vibration isolation. In this article, a 2D square lattice acoustic metamaterial structure with multiembedded cores is proposed. Based on Bloch's theorem and finite element method, the band structures and bandgap properties of the structure are calculated and analyzed, and the vibration isolation ability of the structure is investigated. The effects of structural parameters on the bandgap property are analyzed. Thus, the anisotropy of the structure is also analyzed by the contours of phased velocity and group velocity. The results show that the structure has significant bandgap properties and vibration isolation. The second‐order similarity ratio has significant effects on the bandgap property. Via increasing the second‐order similarity ratio, the total bandgap width and low‐frequency bandgap width are widened. In addition, the structure has significant anisotropy, and the energy of elastic wave propagates at the direction of 45° and 135°. The research in this article supplements the design and research of 2D lattice acoustic metamaterials and makes a beneficial exploration for future engineering applications.

The edge states of one-dimensional spin-$\frac{1}{2}$-like noninteracting periodic systems under the most compactly supported Wannier-type functions

Abstract A hallmark of topological insulators is topologically protected edge states, which are determined by topological invariants defined under Bloch functions. Of course, not all edge states are stable, and they are independent of topological invariants. And in a recent study, it was found that even stable protected gapless edge states cannot be explained by topological invariants in two-dimensional (2D) systems. Motivated by this idea, we carried out a thorough study in one-dimensional (1D) spin-$\frac{1}{2}$-like noninteracting periodic systems and found that all edge states can be explained by the most compactly supported Wannier-type functions (MCWTFs). All the answers are contained in the flat-band Hamiltonian defined by MCWTFs. Edge states can be obtained by solving special equations using the parameters of MCWTFs. Symmetries are the special solutions of these equations, and we have proved the relationship between zero-energy edge states and symmetry analytically in partial cases, which was only a well-known result in the past. By solving the equations, we also find that not all zero-energy edge states are determined by symmetry, which is beyond the current framework. Through numerical verification, we show that these zero-energy edge states, like those protected by symmetry, are not affected by the energy spectrum of the system, so they are only related to the Bloch functions. We also derive special properties of zero-energy edge states protected by chiral symmetry from the perspective of MCWTFs, which are consistent with the previous literature. Finally, we do a test in the 2D case and find the connection between the Chern number and the Wannier functions. Our results show that compared with topological invariants, the origin of the edge state can be understood more comprehensively from the perspective of Wannier functions.

Topological linear response of hyperbolic Chern insulators

We establish a connection between the electromagnetic Hall response and band topological invariants in hyperbolic Chern insulators by deriving a hyperbolic analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By generalizing the Kubo formula to hyperbolic lattices, we show that the Hall conductivity is quantized to -e^2C_{ij}/h−e2Cij/h, where C_{ij}Cij is the first Chern number. Through a flux-threading argument, we provide an interpretation of the Chern number as a topological invariant in hyperbolic band theory. We demonstrate that, although it receives contributions from both Abelian and non-Abelian Bloch states, the Chern number can be calculated solely from Abelian states, resulting in a tremendous simplification of the topological band theory. Finally, we verify our results numerically by computing various Chern numbers in the hyperbolic Haldane model.

Transfer of solitons and half-vortex solitons via adiabatic passage

We show that the transfer of matter-wave solitons and half-vortex solitons in a spin-orbit coupled Bose-Einstein condensate between two (or more) arbitrarily chosen sites of an optical lattice can be implemented using the adiabatic passage. The underlying linear Hamiltonian has a flat band in its spectrum, so that even sufficiently weak interatomic interactions can sustain well-localized Wannier solitons which are involved in the transfer process. The adiabatic passage is assisted by properly chosen spatial and temporal modulations of the Rabi frequency. Within the framework of a few-mode approximation, the mechanism is enabled by a dark state created by coupling the initial and target low-energy solitons with a high-energy extended Bloch state, like in the conventional stimulated Raman adiabatic passage used for the coherent control of quantum states. In real space, however, the atomic transfer between initial and target states is sustained by the current carried by the extended Bloch state, which remains populated during the whole process. The full description of the transfer is provided by the Gross-Pitaevskii equation. Protocols for the adiabatic passage are described for one- and two-dimensional optical lattices, as well as for splitting and subsequent transfer of an initial wave packet simultaneously to two different target locations. Published by the American Physical Society 2024

The closure of derivative average radial integrable spaces in the Bloch space

A description of the closure in the Bloch space of the Bloch functions that are in the derivative average radial integrable space is given. As a byproduct, we get some new characterizations for the Bloch space B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{B}$\\end{document}, the little Bloch space B0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathcal{B}_{0}$\\end{document}, and the closure of the space H2∩B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H^{2}\\cap \\mathcal{B}$\\end{document} in the Bloch norm.

Water wave interactions with surface-piercing vertical barriers in a rectangular tank: Connections with Bloch waves and quasimodes

Water wave interactions with surface-piercing vertical barriers in a rectangular tank: Connections with Bloch waves and quasimodes

Tracking Rayleigh–Bloch waves swapping between Riemann sheets

Rayleigh–Bloch waves are modes localized to periodic arrays of scatterers with unbounded unit cells. Here, Rayleigh–Bloch waves are studied for line arrays of sound-hard circular scatterers embedded in a two-dimensional acoustic medium, for which it has recently been shown that Rayleigh–Bloch waves exist for higher frequencies than previously thought. Moreover, it was shown that Rayleigh–Bloch waves can cut off (disappear) and cut on (reappear), and additional Rayleigh–Bloch waves can cut on and interact with the existing ones. These complicated behaviours are reconsidered using a family of quasi-periodic Green’s functions that allow particular plane-wave components to become unbounded away from the array. The Green’s function formulation is combined with the block Sakurai–Sugiura method to trace the trajectories of the Rayleigh–Bloch wavenumbers as they swap between Riemann sheets that are categorized according to the unbounded plane wave(s). A detailed analysis is presented for three different scatterer radius values, and contrasting qualitative behaviours are identified. The findings are consistent with those published previously, extend to higher frequencies than allowed by the previous approach, and provide new understanding of Rayleigh–Bloch waves around the critical frequency intervals where they cut on/cut off/interact.

Dielectric Surface-Based Biosensors for Enhanced Detection of Biomolecular Interactions: Advances and Applications

Surface plasmon resonance (SPR) biosensors are extensively utilized for analyzing molecular interactions due to their high sensitivity and label-free detection capabilities. Recent innovations in surface-sensitive biosensors with dielectric surfaces address the inherent limitations associated with traditional gold surfaces, such as thermal effects and biocompatibility issues, which can impede broader applications. This review examines state-of-the-art biosensor configurations, including total internal reflection, optical waveguide, photonic crystal resonators, Bloch surface wave biosensors, and surface electrochemical biosensors, which can enhance analyte signals and augment the molecular detection efficiency at the sensor interface. These technological advancements not only improve the resolution of binding kinetics analysis and single-molecule detection but also extend the analytical capabilities of these systems. Additionally, this review explores prospective advancements in augmenting field enhancement and incorporating multimodal sensing functionalities, emphasizing the significant potential of these sophisticated biosensing technologies to profoundly enhance our understanding of molecular interactions.

Coupling light into a guided Bloch surface wave using an inversely designed nanophotonic cavity

Controlling the propagation of light in the form of surface modes on miniaturized platforms is crucial for multiple applications. For dielectric multilayers that sustain Bloch surface waves at their interface to an isotropic dielectric medium, a conventional approach to manipulate them exploits shallow surface topographies fabricated on top of the truncated stack. However, such structures typically exhibit low index contrasts, making it challenging to confine, steer, and guide the Bloch surface waves. Here, we theoretically and experimentally demonstrate a device for a Bloch surface wave platform that resonantly couples light from a cavity to a straight waveguide. The structure is designed using topology optimization in a 2D geometry under the effective index approximation. In particular, the cavity–waveguide coupling efficiency of the radiation emitted by an individual source in the cavity center is optimized. The cavity is experimentally found to exhibit a narrow resonant peak that can be tuned by scaling the structure. The waveguide is shown to guide only light that resonates in the cavity. Fully three-dimensional simulations of the entire device validate the experimental observations.

Anomalous frozen evanescent phonons

Evanescent Bloch waves are eigensolutions of spatially periodic problems for complex-valued wavenumbers at finite frequencies, corresponding to solutions that oscillate in time and space and that exponentially decay in space. Such evanescent waves are ubiquitous in optics, plasmonics, elasticity, and acoustics. In the limit of zero frequency, the wave “freezes” in time. We introduce frozen evanescent waves as the eigensolutions of the Bloch periodic problem at zero eigenfrequency. Elastic waves, i.e., phonons, in metamaterials serve as an example. We show that, in the complex plane, the Cauchy-Riemann equations for analytical functions connect the minima of the phonon band structure to frozen evanescent phonons. Their exponential decay length becomes unusually large if a minimum in the band structure tends to zero and thereby approaches a soft mode. This connection between unusual static and dynamic behaviors allows to engineer large characteristic decay lengths in static elasticity. For finite-size samples, the static solutions for given boundary conditions are linear combinations of frozen evanescent phonons, leading to interference effects. Theory and experiment are in excellent agreement. Anomalous behavior includes the violation of Saint Venant’s principle, which means that large decay-length frozen evanescent phonons can potentially be applied in terms of remote mechanical sensing.

Signature of Correlated Insulator in Electric Field Controlled Superlattice.

On a two-dimensional crystal, a "superlattice" with nanometer-scale periodicity can be imposed to tune the Bloch electron spectrum, enabling novel physical properties inaccessible in the original crystal. While creating 2D superlattices by means of nanopatterned electric gates has been studied for band structure engineering in recent years, evidence of electron correlations─which drive many problems at the forefront of physics research─remains to be uncovered. In this work, we demonstrate signatures of a correlated insulator phase in Bernal-stacked bilayer graphene modulated by a gate-defined superlattice potential, manifested as resistance peaks centered at integer multiples of single electron per superlattice unit cell carrier densities. The observation is consistent with the formation of a stack of flat low-energy bands due to the superlattice potential combined with inversion symmetry breaking. Our work paves the way to custom-designed superlattices for studying band structure engineering and strongly correlated electrons in 2D materials.

Band gaps of elastic waves in 1-D dielectric phononic crystal with the flexoelectric and strain gradient effects consideration

The propagation of Bloch waves in one dimensional phononic crystal consisting of dielectric elastic solids is studied with consideration of the strain gradient, inertial gradient and flexoelectric effects. The transfer matrixes for single layer and one-cell of phononic structure are derived based on the constitutive equations and the governing equations of dielectric elastic solids. The dispersion equation for Bloch waves is obtained by the application of periodic conditions for the generalized displacements and tractions considered within strain gradient theory of electro-elasticity. Based on the numerical solution of the derived dispersion equation, the influences of the micro-stiffness length scale, micro-inertial length scale and flexoelectric coefficients on the dispersion and the bandgap are discussed.

Tunable wave coupling in periodically rotated Miura-ori tubes.

Origami folding structures are vital in shaping programmable mechanical material properties. Of particular note, tunable dynamical properties of elastic wave propagation in origami structures have been reported. Despite the promising features of origami metamaterials, the influence of the kinematics of tessellated origami structures on elastic wave propagation remain unexplored. This study proposes elastic metamaterials using connected Miura-ori tubes, the kinematics of which are coupled by folding and unfolding motions in a tubular axis; achieved by periodically connecting non-rotated and rotated Miura-ori tubes. The kinematics generate wave modes with localized deformations within the unit cell of the metamaterials, affecting the global elastic deformation of Miura-ori tubes via the coupling of wave modes. Dispersion analysis, using the generalized Bloch wave framework based on bar-and-hinge models, verifies the influence of kinematics in the connected tubes on elastic wave propagation. Furthermore, folding the connected tubes changes the coupling strength of wave modes between the kinematics and global elastic deformation of the tubes by breaking the ideal kinematics. The coupling of wave modescontributes to the formation of the band gaps and their tunability. These findings enable adaptive and in situ tunability of band structures to prohibit elastic waves in the desired frequency ranges.This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Anomalous Non-Hermitian Open-Boundary Spectrum.

For a long time, it was presumed that continuum bands could be readily encompassed by open-boundary spectra, irrespective of the system's modest dimensions. However, our findings reveal a nuanced picture: under open-boundary conditions, the proliferation of complex eigenvalues progresses in a sluggish, oscillating manner as the system expands. Consequently, even in larger systems, the overlap between continuum bands and open-boundary eigenvalues becomes elusive, with the surprising twist that the count of these complex eigenvalues may actually diminish with increasing system size. This counterintuitive trend underscores that the pursuit of an ideal, infinite-sized system scenario does not necessarily align with enlarging the system size. Notably, despite the inherent non-Hermiticity of our system, the eigenstates distribute themselves in a manner reminiscent of Bloch waves. These discoveries hold potential significance for both theoretical explorations and experimental realizations of non-Hermitian systems.

Ultrasonic signatures of multiphase architectured media: Application to biomechanics

Assessing the mechanical characteristics of periodic architectured scaffolds employed in bone tissue repair, while preserving their structural integrity, poses a significant challenge from both theoretical and experimental viewpoints. By leveraging concepts arising from the phononic crystals community, here we investigate the reflection and transmission of elastic waves propagating through water-immersed biphasic architectured samples in the MHz regime, which exhibit a periodic organization at a length scale of a few hundred micrometers. Experimental outcomes are systematically compared with modeled predictions, spanning a range of scaffold-like samples engineered to mimic the expected variations that take place in a biological environment. A particular attention is given to critical modeling considerations such as the behavior in reflection, the interaction of Bloch waves with the viscoelastic properties of the constituent phases, as well as the role of modal conversion at the interface between the homogeneous incident medium and the architectured one. Altogether, the reported results suggest that specific ultrasonic signatures associated with elastic wave propagation in periodic media, encompassing phenomena such as bandgaps and dispersion, offer valuable insights towards the nondestructive monitoring of micro-architectured media like scaffolds.

Optical Activities of Organic Crystals: Crystal Orbital Formulation of Anisotropic Gyration.

To describe the optical activities of crystals, gyration tensors are necessary for all wavelengths of incident light. To date, few studies on direct calculations of gyration tensors from Bloch states have been conducted. Herein, a practical procedure to obtain gyration tensors of organic crystals is presented using the crystal orbital method. The extended Hückel method was adopted to evaluate the gyration tensors, epitomizing the one-electron formulation. To confirm the validity of the formulation, the optical rotatory power of alanine and γ-glycine was examined. The reproduced profiles of the optical rotatory power were consistent with the results of recent experiments. This is a general formulation of the one-electron theory of optical activities for three-dimensional crystals. In principle, the optical rotatory strength tensor is not invariant with translation. For systems with small unit cells, however, the formalism is quasi-invariant with respect to translation. The quasi origin-independent formalism is sufficiently substantial to be applicable to modern crystal optics.

Enhanced high-frequency continualization scheme for inertial beam-lattice metamaterials

A multifield continualization technique is introduced that offers a thermodynamically consistent description of the constitutive and dispersive properties of beam-lattice inertial metamaterials with periodic microstructures. The balance equations governing the mechanics of the discrete Lagrangian system are appropriately handled using an innovative continualization scheme to derive an equivalent integral-type continuum model. Based on formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions incorporating shift operators and appropriate pseudo-differential downscaling laws, the proposed multifield enhanced continualization scheme allows the derivation of a gradient-type continuum model of given rank and equivalent to lattices. Two different resolution techniques are proposed. Firstly, the corresponding infinite-order average differential equations are tackled using a perturbative approach to describe the forced Bloch wave propagation in the metamaterial. Secondly, higher-order continuum models are employed through proper differential equation truncation to characterize the dispersive properties of the metamaterial in both high- and low-frequency regimes. Moreover, an energetically consistent generalized equivalent Micropolar continua, with non-local inertial terms, are here identified. The multifield continualization procedure is applied to two-dimensional periodic microstructures with tetrachiral, hexachiral, and hexa-tetrachiral topologies. Illustrative examples highlight the ability of the equivalent continuum model to accurately describe the effective constitutive properties of inertial metamaterials with periodic microstructures and to define a dynamic response consistent with the discrete Lagrangian model, validated and tested through virtual experimental verification under free and forced wave conditions.

Attenuation of Lamb waves in coupled-resonator viscoelastic waveguide

Guidance of elastic waves is one of the main applications of artificial crystal structures. The attenuation of the guided waves is, however, often overlooked, as most of the proposed waveguides only comprise ideal elastic materials. In this work, we study the propagation of evanescent Lamb waves guided in coupled-resonator viscoelastic waveguide (CRVW), with special attention to attenuation. CRVW is defined by considering a linear chain of coupled defect cavities in a phononic plate made of epoxy. The viscoelastic behavior of epoxy is characterized numerically by the Kelvin–Voigt (K–V) model. Based on finite element analysis, the complex band structure and the spectrum of frequency response function (FRF) are obtained. Due to viscosity, guided Lamb waves are spatially damped. Two theoretical models are devised to predict the displacement distributions inside and outside a bandgap for guided waves, respectively, considering either the first or the first two least evanescent Bloch waves identified in the complex band structure. A CRVW sample is fabricated and characterized experimentally by laser vibrometry. Evanescent Lamb waves are observed to be strongly confined along the waveguide and at the same time to decay rapidly along the waveguide axis. Experiments and numerical simulations are found to be in fair agreement. The present work is expected to inspire practical applications of highly confined viscoelastic phononic waveguides.

Changing magnetic ground state by an electric field in a single-layer FeCl2

Abstract Employing the electric field has been performed to investigate the magnetic ground state in a single-layer iron dichloride (FeCl2). We provided some lattice parameters to see the ground state dependency on the electric field. To minimize the computational time, we applied the generalized Bloch theorem via density functional theory to avoid the supercell approach. As we varied the electric field, we saw the magnetic transition for some lattice parameters. In addition, the magnetic moments were nearly stable for all those states, indicating the high spin state. As a concluding remark, employing the electric field is very useful to manipulate the ground state that can be applied to magnetism-based nanotechnology.