Nonreciprocal Wave Transmission Research Articles

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass–spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass–spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. This novel approach, the combination of the temporal Floquet theory for time-modulated systems and generalized Brillouin zone, enables the prediction of eigenvalue distribution under open boundary conditions and also quantitative characterization of non-Hermitian skin modes. The proposed theory is verified by some numerical experiments.

Programming nonreciprocity and harmonic beam steering via a digitally space-time-coded metamaterial antenna

Recent advancement in digital coding metasurfaces incorporating spatial and temporal modulation has enabled simultaneous control of electromagnetic (EM) waves in both space and frequency domains by manipulating incident EM waves in a transmissive or reflective fashion, resulting in time-reversal asymmetry. Here we show in theory and experiment that a digitally space-time-coded metamaterial (MTM) antenna with spatiotemporal modulation at its unit cell level can be regarded as a radiating counterpart of such digital metasurface, which will enable nonreciprocal EM wave transmission and reception via surface-to-leaky-wave transformation and harmonic frequency generation. Operating in the fast wave (radiation) region, the space-time-coded MTM antenna is tailored in a way such that the propagation constant of each programmable unit cell embedded with varactor diodes can toggle between positive and negative phases, which is done through providing digital sequences by using a field-programmable gate array (FPGA). Owing to the time-varying coding sequence, harmonic frequencies are generated with different main beam directions. Furthermore, the space time modulation of the digitally coded MTM antenna allows for nonreciprocal transmission and reception of EM waves by breaking the time-reversal symmetry, which may enable many applications, such as simultaneous transmitting and receiving, unidirectional transmission, radar sensing, and multiple-input and multiple-output (MIMO) beamformer.

Energy trapping in a phononic crystal cavity enhanced by nonreciprocal acoustic wave transmission

Defect mode induced energy trapping at the bandgap frequency of a phononic crystal has been widely explored. Unlike this extensively used mechanism, this work reports the use of nonreciprocity in the transmission band to trap energy inside a phononic crystal cavity. Passive nonreciprocity is due to natural viscosity of the background liquid (water) and asymmetry of aluminum scatterers. The level of non-resonant energy trapping was compared for three cavities with different symmetry. Enhancement of energy trapping at a frequency of 624 kHz was observed experimentally for the cavity where nonreciprocity suppresses acoustic radiation into environment. Experimental results were further investigated and confirmed using finite element numerical analysis.

Non-reciprocal wave propagation in time-modulated elastic lattices with inerters

Non-reciprocal wave propagation in acoustic and elastic media has received much attention of researchers in recent years. This phenomenon can be achieved by breaking the reciprocity through space- and/or time-dependent constitutive material properties, which is an important step in overcoming the limitations of conventional acoustic- and phononic-like mechanical lattices. A special class of mechanical metamaterials with non-reciprocal wave transmission are latices with time-modulated mass and stiffness properties. Here, we investigate the non-reciprocity in elastic locally resonant and phononic-like one-dimensional lattices with inerter elements where mass and stiffness properties are simultaneously modulated through inerters and springs as harmonic functions of time. By considering the Bloch theorem and Fourier expansions, the frequency-band structures are determined for each configuration while asymmetric band gaps are found by using the weighting and threshold method. The reduction in frequency due to introduced inerters was observed in both phononic and locally resonant metamaterials. Dynamic analysis of finite-length lattices by the finite difference method revealed a uni-directional wave propagation. Special attention is given to phononic-like lattice based on a discrete-continuous system of multiple coupled beams. Moreover, the existence of edge modes in the discrete phononic lattice is confirmed through the bulk-edge correspondence and their time evolution quantified by the topologically invariant Chern number. The proposed methodology used to investigate non-reciprocal wave transmission in one-dimensional inerter-based lattices can be extended to study more complex two-dimensional lattices.

Engineering nonreciprocal wave dispersion in a nonlocal micropolar metabeam

Active metamaterials with electronic control schemes can exhibit nonreciprocal and/or complex elastic coefficients that result in non-Hermitian wave phenomena. Here, we investigate theoretically and experimentally a non-Hermitian micropolar metabeam with piezoelectric elements and electronic nonlocal feed-forward control. Since the nonlocal feed-forward control breaks spatial reciprocity, the proposed metabeam supports nonreciprocal flexural wave propagation, featuring unidirectional amplification/attenuation and non-Hermitian skin effect. Theoretical homogenization modeling is developed to consider the nonlocal effect into an effective complex bending stiffness. The unidirectional wave amplification/attenuation is attributed to the energy conversion between electrical power and mechanical work. The non-Hermitian skin effect, characterized by a winding number, is the manifestation of the flexural nonreciprocity and admits an extensive number of localized bulk eigenmodes on open boundaries. The nonlocal metabeam is also employed to engineer the anomalous wave dispersion such as tunable roton-like dispersion and band tilting. The nonlocal micropolar metabeam could pave the ways for designing non-Hermitian topological mechanical metamaterials featuring programmable nonreciprocal wave transmission and engineering roton-like wave dispersion relations under ambient environments.

Nonreciprocal wave transmission through discrete complex saturable Ginzburg–Landau dimer

Nonreciprocal wave transmission through discrete complex saturable Ginzburg–Landau dimer

Nonreciprocal Sound Propagation via Cascaded Time-Modulated Slab Resonators

In this work, we demonstrate nonreciprocal acoustic wave transmission in two-cascaded Fabry-Perot-like slab resonators undergoing time modulation of their effective density. A phase difference is introduced into the time modulation between the two coupled resonators to produce spatial bias to access unidirectional wave propagation. A theoretical model based on plane-wave expansion and the transfer-matrix method is developed to study the Fabry-Perot-based system. The theoretical model allows rapid and precise characterization of the acoustic dispersion, with results in excellent agreement with finite-element-based numerical simulations. We demonstrate acoustic nonreciprocity behavior for the fundamental acoustic mode and further show that, with proper optimization, the coupled Fabry-Perot-based device can achieve nonreciprocal acoustic frequency conversion. This work could inspire the design of simple and compact nonreciprocal acoustic devices for efficient wave control.

Time scale disparity yielding acoustic nonreciprocity in a two-dimensional granular-elastic solid interface with asymmetry.

We study nonreciprocal wave transmission across the interface of two dissimilar granular media separated by an elastic solid medium. Specifically, a left, larger-scale and a right smaller-scale granular media composed of two-dimensional, initially uncompressed hexagonally packed granules are interfacing with an intermediate linearly elastic solid, modeled either as a thin elastic plate or a linear Euler-Bernoulli beam. The granular media are modeled by discrete elements and the elastic solid by finite elements assuming a plane stress approximation for the thin plate. Accounting for the combined effects of Hertzian, frictional and rotational interactions in the granular media, as well as the highly discontinuous interfacial effects between the (discrete) granular media and the (continuous) intermediate elastic solid, the nonlinear acoustics of the integrated system is computationally studied subject to a half-sine shock excitation applied to a boundary granule of either the left or right granular medium. The highly discontinuous and nonlinear interaction forces coupling the granular media to the elastic solid are accurately computed through an algorithm with interrelated iteration and interpolation at successive adaptive time steps. Numerical convergence is ensured by monitoring the (linearized) eigenvalues of a nonlinear map of interface forces at each (variable) time step. Due to the strong nonlinearity and hierarchical asymmetry of the left and right granular media, time scale disparity occurs in the response of the interface which breaks acoustic reciprocity. Specifically, depending on the location and intensity of the applied shock, propagating wavefronts are excited in the granular media, which, in turn, excite either (slow) low-frequency vibrations or (fast) high-frequency acoustics in the intermediate elastic medium. This scale disparity is due to the size disparity of the left and right granular media, which yields drastically different wave speeds in the resulting propagating wavefronts. As a result, the continuum part of the interface responds with either low-frequency vibrations-when the shock is applied to the larger-scale granular medium, or high-frequency waves-when the shock is applied to the smaller-scale granular medium. This provides the fundamental mechanism for breaking reciprocity in the interface. The nonreciprocal interfacial acoustics studied here apply to a broad class of asymmetric hybrid (discrete-continuum) nonlinear systems and can inform predictive designs of highly effective granular shock protectors or granular acoustic diodes.

Tunable nonlocal purely active nonreciprocal acoustic media

Previously engineered wave-bearing materials including metamaterials generate their acoustic responses from a combination of passive geometrical structures possibly augmented by active elements. The passive geometrical features typically translate to uncontrolled properties outside the band of interest and are the source of adverse effects such as narrowing the operating band significantly or generating difficult-to-control instabilities due to interactions with the active elements. This paper proposes a nonlocal active metamaterial (NAM) architecture that employs a purely active approach to tailoring the acoustic field utilizing spatially separated sensors and actuators. A significant benefit of this approach is that the acoustic properties are described by analytical closed-form equations exposing the complex interplay between the various components of the material. The method is demonstrated by engineering extremely nonreciprocal wave transmission in an otherwise completely transparent subwavelength NAM. We experimentally demonstrate the remarkable flexibility of the system by electronically tuning the acoustic isolation characteristics and discuss the massive design space uncovered by the NAM approach.

Nonreciprocal Dzyaloshinskii-Moriya Magnetoacoustic Waves.

We study the interaction of surface acoustic waves with spin waves in ultrathin CoFeB/Pt bilayers. Because of the interfacial Dzyaloshinskii-Moriya interaction (DMI), the spin wave dispersion is nondegenerate for oppositely propagating spin waves in CoFeB/Pt. In combination with the additional nonreciprocity of the magnetoacoustic coupling itself, which is independent of the DMI, highly nonreciprocal acoustic wave transmission through the magnetic film is observed. We systematically characterize the magnetoacoustic wave propagation in a thickness series of CoFeB(d)/Pt samples as a function of magnetic field magnitude and direction, and at frequencies up to 7GHz. We quantitatively model our results to extract the strength of the DMI and magnetoacoustic driving fields.

Switchable phonon diodes using nonlinear topological Maxwell lattices

Recent progress in topological mechanics have revealed a family of Maxwell lattices that exhibit topologically protected floppy edge modes. These modes lead to a strongly asymmetric elastic wave response. In this paper, we show how topological Maxwell lattices can be used to realize non-reciprocal transmission of elastic waves. Our design leverages the asymmetry associated with the availability of topological floppy edge modes and the geometric nonlinearity built in the mechanical systems response to achieve the desired non-reciprocal behavior, which can be further turned into strongly one-way phonon transport via the addition of on-site pinning potentials. Moreover, we show that the non-reciprocal wave transmission can be switched on and off via topological phase transitions, paving the way to the design of cellular metamaterials that can serve as tunable topologically protected phonon diodes.

Non-Reciprocal Wave Transmission in a Bilinear Spring-Mass System

Abstract Significant amplitude-independent and passive non-reciprocal wave motion can be achieved in a one-dimensional (1D) discrete chain of masses and springs with bilinear elastic stiffness. Some fundamental asymmetric spatial modulations of the bilinear spring stiffness are first examined for their non-reciprocal properties. These are combined as building blocks into more complex configurations with the objective of maximizing non-reciprocal wave behavior. The non-reciprocal property is demonstrated by the significant difference between the transmitted pulse displacement amplitudes and energies for incidence from opposite directions. Extreme non-reciprocity is realized when almost-zero transmission is achieved for the propagation from one direction with a noticeable transmitted pulse for incidence from the other. These models provide the basis for a class of simple 1D non-reciprocal designs and can serve as the building blocks for more complex and higher dimensional non-reciprocal wave systems.

Performance mechanisms and stability criteria for highly sub-wavelength, broadband nonreciprocal non-local acoustic metamaterials

Active acoustic metamaterials have received considerable attention recently due in large part to their ability to generate nonreciprocal wave transmission. We have shown previously how the spatial separation of acoustic sensors and sources can be leveraged to generate highly sub-wavelength, broadband nonreciprocal behavior in an acoustic waveguide. We showed that by transmitting pressure signals from one location along the waveguide to a loudspeaker at another location, we could achieve a peak isolation of 35 dB with a Q10dB factor of just over 0.25. Here, we discuss the mechanisms by which this nonreciprocity is achieved and the conditions required for the system to be stable. We show how conclusions regarding the performance and stability characteristics of an idealized plane wave acoustic model of our system must be adjusted when accounting for the controlling electronics and loudspeaker dynamics used in the experimental testing.Active acoustic metamaterials have received considerable attention recently due in large part to their ability to generate nonreciprocal wave transmission. We have shown previously how the spatial separation of acoustic sensors and sources can be leveraged to generate highly sub-wavelength, broadband nonreciprocal behavior in an acoustic waveguide. We showed that by transmitting pressure signals from one location along the waveguide to a loudspeaker at another location, we could achieve a peak isolation of 35 dB with a Q10dB factor of just over 0.25. Here, we discuss the mechanisms by which this nonreciprocity is achieved and the conditions required for the system to be stable. We show how conclusions regarding the performance and stability characteristics of an idealized plane wave acoustic model of our system must be adjusted when accounting for the controlling electronics and loudspeaker dynamics used in the experimental testing.

Non-reciprocal wave phenomena in energy self-reliant gyric metamaterials.

This work presents a mechanism by which non-reciprocal wave transmission is achieved in a class of gyric metamaterial lattices with embedded rotating elements. A modulation of the device's angular momentum is obtained via prescribed rotations of a set of locally housed spinning motors and is then used to induce space-periodic, time-periodic, as well as space-time-periodic variations, which influence wave propagation in distinct ways. Owing to their dependence on gyroscopic effects, such systems are able to break reciprocal wave symmetry without stiffness perturbations rendering them consistently stable as well as energy self-reliant. Dispersion patterns, band gap emergence, as well as non-reciprocal wave transmission in the space-time-periodic gyric metamaterials are predicted both analytically from the gyroscopic system dynamics as well as numerically via time-dependent full wave simulations. In addition to breaking reciprocity, the authors show that the energy content of a frictionless gyric metamaterial is conserved over one temporal modulation cycle enabling it to exhibit a stable response irrespective of the pumping frequency.

Broadband passive nonlinear acoustic diode

In dynamical and acoustical systems, breaking reciprocity is achievable by employing external biases, spatial temporal variations of material properties, or nonlinearities. In this present work, we propose, theoretically analyze, and experimentally demonstrate the first passive broad-band mechanical diode, by adding an asymmetric local nonlinear interface to an otherwise linear waveguide. It is shown that for a broad range of input energies and frequencies, three different types of nonreciprocal behavior are obtained, (i) different transmitted energies depending on the direction of wave propagation, (ii) transmission of acoustic waves allowed in one direction but not in the reverse direction, and (iii) arrest of propagating waves at the nonlinear interface only in one (preferred) direction. A unique feature of the proposed acoustic device compared with previous designs is its capability to achieve passive nonreciprocity without altering or distorting the frequency content of the sending signal. The proposed system can pave the way for designing passive acoustic or thermal diodes, and adaptable nonreciprocal wave transmission devices of enhanced robustness that are tunable with, and passively adaptive to energy.

A piezo-metastructure with bistable circuit shunts for adaptive nonreciprocal wave transmission

In this paper, we present a piezo-metastructure shunted with bistable circuits to achieve adaptive nonreciprocal elastic wave transmission. Static properties of the bistable circuit are first investigated, followed by numerical investigation of wave transmission characteristics on the nonlinear piezo-metastructure. Both the local-resonance bandgap and the bandgap transmission phenomenon, also known as supratransmission, are explored and investigated. By introducing circuit asymmetry, the supratransmission thresholds, critical excitation amplitudes to enable bandgap transmission, are found at different levels for the different wave transmission directions and hence creating an excitation amplitude range within which nonreciprocal wave transmission can be facilitated. Effect of the asymmetry factor on nonreciprocity properties is subsequently analyzed and the trade-offs between the forward transmission amplitude and the range of excitation with nonreciprocity are identified. Additionally, it is demonstrated that wave transmission characteristics of the proposed nonlinear piezo-metastructure can be adaptively tuned by conveniently adjusting stable equilibria of the bistable circuit. Lastly, the observed wave transmission properties are further corroborated by experimental investigations. Overall, the results illustrate a novel means to manipulate unidirectional elastic wave transmission using a nonlinear piezo-metastructure.

Acoustic diode: Wave non-reciprocity in nonlinearly coupled waveguides

The paper describes a passive time-independent setting for non-reciprocal wave transmission in mechanical and acoustic systems with strong nonlinearities. In the proposed system vibro-impact elements with pre-defined clearances are used to couple two non-dispersive waveguides. The asymmetry necessary for the non-reciprocal behavior is realized through unequal grounding springs of the vibro-impact elements. We show that, for appropriate selection of the parameters, the proposed system acts as a mechanical diode, allowing the transmission of acoustic waves in one direction and completely preventing reverse transmission. Two different designs of the coupling elements are suggested, with the possibility of single-sided or double-sided impacts. A unique feature of the proposed non-reciprocal acoustic system is that minimal distortion of the harmonic content of the transmitted wave occurs, in contrast to current designs where nonlinear non-reciprocity is achieved at the expense of a rather strong distortion of the transmitted signals. For both designs, we derive exact solutions for propagation and reflection of the harmonic waves, and demonstrate the possibility for strong non-reciprocity. Stability properties of the observed solutions in the space of parameters are also explored.

Application of magnetoelastic materials in spatiotemporally modulated phononic crystals for nonreciprocal wave propagation

In this paper, a physical platform is proposed to change the properties of phononic crystals in space and time in order to achieve nonreciprocal wave transmission. The utilization of magnetoelastic materials in elastic phononic systems is studied. Material properties of magnetoelastic materials change significantly with an external magnetic field. This property is used to design systems with a desired wave propagation pattern. The properties of the magnetoelastic medium are changed in a traveling wave pattern, which changes in both space and time. A phononic crystal with such a modulation exhibits one-way wave propagation behavior. An extended transfer matrix method (TMM) is developed to model a system with time varying properties. The stop band and the pass band of a reciprocal and a nonreciprocal bar are found using this method. The TMM is used to find the transfer function of a magnetoelastic bar. The obtained results match those obtained via the theoretical Floquet–Bloch approach and numerical simulations. It is shown that the stop band in the transfer function of a system with temporal varying property for the forward wave propagation is different from the same in the backward wave propagation. The proposed configuration enables the physical realization of a class of smart structures that incorporates nonreciprocal wave propagation.

Nonreciprocal wave transmission through an extended discrete nonlinear Schrödinger dimer.

We analyze a one-dimensional extended discrete nonlinear Schrödinger (DNLS) dimer model for nonreciprocal wave transmission. The extension corresponds to the addition of a nonlocal or intersite nonlinear response in addition to a purely cubic local (on-site) nonlinear response, which refines the purely cubic model and aligns to more realistic situations. We observe that a diodelike action persists in the extended case; however, the inclusion of nonlocal response tends to reduce the diode action. We show that this extension results in achieving the diode effect at lower incoming intensities as compared to the purely cubic case. We also report that a nearly perfect diode action is possible in the extended case for a higher level of asymmetry between on-site potentials than its cubic counterpart. Moreover, we vary different site-dependent parameters to probe for regimes of a better diode effect within this extended model. We also present the corresponding stability analysis for the exact stationary solutions to the extended DNLS equation, we discuss the bifurcation behavior in detail, and we explicitly give the regions of stability.

Diamond Configuration for Non-Reciprocal Transmission

A system scheme is presented which allows non-reciprocal wave transmission or directional amplification of electromagnetic signals, using a boxed four-node method. Edges represent strong hopping interactions and diagonals stand for weak parametric interactions. Using careful optimization of values for design parameters, we are able to obtain non-reciprocity in excess of 12dB and 130dB for intrinsic and extrinsic configurations at identical input/output frequencies. For the directional amplification, an isolation as high as 40dB is demonstrated with forward/backward gains of $\pm$20dB. Cascading two such systems potentially can offer high isolations at high gains.