Valley Pseudospin Polarized Evanescent Coupling between Microwave Ring Resonator and Waveguide in Phononic Topological Insulators

A coupled ring-waveguide structure is at the core of bosonic wave-based information processing systems, enabling advanced wave manipulations such as filtering, routing, and multiplexing. However, its miniaturization is challenging due to momentum conservation issues in rings with larger curvature that induce significant backscattering and radiation leakage and hampering stable operation. Here, we address it by taking an alternative approach of using topological technology in wavelength-scale and microwave ring-waveguide coupled systems built in nanoengineered phononic crystals. Our approach, which leverages pseudospin conservation in valley topological systems, eliminates phonon backscattering and achieves directional evanescent coupling. The resultant hypersonic waves in the tiny ring exhibit robust transport and resonant circulation. Furthermore, the ring-waveguide hybridization enables critical coupling, where valley-dependent ring-waveguide interference blocks the transmission. Our findings reveal the capability of topological phenomena for managing ultrahigh-frequency phonons in nano/microscale structures and pave the way for advanced phononic circuits in classical and quantum signal processing applications.

Reinforcement hybridization in staggered composites enhances wave attenuation performance

Advanced composites with superior wave attenuation or vibration isolation capacity are in high demand in engineering practice. In this study, we develop the hybrid dynamic shear-lag model with Bloch's theorem to investigate the hybrid effect of reinforcement on wave attenuation in bioinspired staggered composites. We present for the first time the relationship between macroscopic wave filtering and hybridization of building blocks in staggered composites. Viscoelasticity was taken into account for both reinforcement and matrix to reflect the damping effect on wave transmission. Our findings indicate that reinforcement hybridization significantly enhances wave attenuation performance through two critical parameters: the linear stiffness and linear density of reinforcements. For purely elastic constituents, reinforcement hybridization consistently improves wave attenuation by reducing the initial frequency of the first bandgap and broadening it. For viscoelastic constituents, increasing the heterogeneity of reinforcements can benefit wave attenuation, particularly in ultralow frequency regimes, due to the strengthening of the damping effect. Our case study demonstrates that controlling the difference in linear density can result in up to a 59 % reduction in energy transmission. Our analysis suggests that hybridizing reinforcements could provide a new approach to designing and synthesizing advanced composites with exceptional wave attenuation performance.

Graphene-based phononic crystal lenses: Machine learning-assisted analysis and design

The advance of artificial intelligence and graphene-based composites brings new vitality into the conventional design of acoustic lenses which suffers from high computation cost and difficulties in achieving precise desired refractive indices. This paper presents an efficient and accurate design methodology for graphene-based gradient-index phononic crystal (GGPC) lenses by combing theoretical formulations and machine learning methods. The GGPC lenses consist of two-dimensional phononic crystals possessing square unit cells with graphene-based composite inclusions. The plane wave expansion method is exploited to obtain the dispersion relations of elastic waves in the structures and then establish the data sets of the effective refractive indices in structures with different volume fractions of graphene fillers in composite materials and filling fractions of inclusions. Based on the database established by the theoretical formulation, genetic programming, a superior machine learning algorithm, is introduced to generate explicit mathematical expressions to predict the effective refractive indices under different structural information. The design of GGPC lenses is conducted with the assistance of the machine learning prediction model, and it will be illustrated by several typical design examples. The proposed design method offers high efficiency, accuracy as well as the ability to achieve inverse design of GGPC lenses, thus significantly facilitating the development of novel phononic crystal lenses and acoustic energy focusing.

Imaging phonon eigenstates and elucidating the energy storage characteristics of a honeycomb-lattice phononic crystal cavity

We extend gigahertz time-domain imaging to a wideband investigation of the eigenstates of a phononic crystal cavity. Using omnidirectionally excited phonon wave vectors, we implement an ultrafast technique to experimentally probe the two-dimensional acoustic field inside and outside a hexagonal cavity in a honeycomb-lattice phononic crystal formed in a microscopic crystalline silicon slab, thereby revealing the confinement and mode volumes of phonon eigenstates-some of which are clearly hexapole in character-lying both inside and outside the phononic-crystal band gap. This allows us to obtain a quantitative measure of the spatial acoustic energy storage characteristics of a phononic crystal cavity. We also introduce a numerical approach involving toneburst excitation and the monitoring of the acoustic energy decay together with the integral of the Poynting vector to calculate the Q factor of the principal in-gap eigenmode, showing it to be limited by ultrasonic attenuation rather than by phonon leakage to the surrounding region.

Refraction, beam splitting and dispersion of GHz surface acoustic waves by a phononic crystal

We exploit a time-resolved ultrafast optical technique to study the propagation of point-excited surface acoustic waves on a microscopic two-dimensional phononic crystal in the form of a square lattice of holes in a silicon substrate. Constant-frequency images and the dispersion relation are extracted, and the latter measured in detail in the region around the phononic band gap. Mode conversion and refraction at the interface between the phononic crystal and surrounding non-structured silicon substrate is studied at constant frequencies. Symmetric phonon beam splitting, for example, is shown to lead to a striking Maltese-cross pattern when phonons exit a square region of phononic crystal excited near its center.

Acoustic sorting of airborne particles by a phononic crystal waveguide

A two-dimensional phononic crystal linear defect waveguide is utilized for size-based sorting of millimeter-sized solid particles in the air through acoustic radiation force. The waveguide channels ultrasonic waves at 20 kHz, as calculated through Finite-Element Method simulations. Spherical solid particles released from rest at the top of the vertically aligned waveguide experience the combined effect of the acoustic radiation, gravity, and drag forces. When the particles are released from the symmetry plane of the waveguide, they follow straight paths where the ones with radii smaller than a threshold value are trapped at the waveguide nodal planes, whereas larger particles are let pass through. This requires input sound pressure levels between 173 dB and 177 dB. Moreover, such particles can also be differentiated with respect to density. Alternatively, the release of particles with a slight offset from the symmetry center induces unbalanced acoustic radiation potential, and thus uneven radiation force, resulting in the initiation of horizontal displacement whose extent depends on particle radius. Thus, both simulation results and experimental findings suggest that this scheme can be employed in size-based particle separation. Sorting of spherical glass particles with 0.5 mm and 1.0 mm radii are experimentally demonstrated for low ultrasonic transducer acoustic power output up to 90 W. The proposed approach can be utilized in applications where contact-free separation of airborne particles is required.

L-shape triple defects in a phononic crystal for broadband piezoelectric energy harvesting

This study proposes a phononic crystal (PnC) with triple defects in an L-shape arrangement for broadband piezoelectric energy harvesting (PEH). The incorporation of defects in PnCs has attracted significant attention in PEH fields owing to properties such as energy localization and amplification near the defect. Several studies have been conducted to enhance output electric power of PnC-based PEH systems with single defects. However, it is susceptible to the limitations of narrow bandwidth. Recently, double-defect-incorporated systems have been proposed to widen the PEH bandwidth via defect-band splitting. Nevertheless, the PEH performance rapidly decreases in the frequency range between the split defect bands. The limitations of single- and double-defect-incorporated systems can be resolved by the incorporation of the proposed design concept, called the L-shape triple defects in a PnC. The isolated single defect at the top vertex of the letter 'L' compensates for the limitations of double-defect-incorporated systems, whereas the double defects at the bottom vertices compensate for the limitations of the single-defect-incorporated systems. Hence, the proposed design can effectively confine and harvest elastic-wave energy over broadband frequencies while enhancing the application of single and double defects. The effectiveness of the proposed design concept is numerically validated using the finite element method. In the case of a circular hole-type PnC, it is verified that the PnC with L-shape triple defects broadens the bandwidth, and improves the output voltage and electric power compared with those of single- and double-defect-incorporated systems. This study expands the design space of defect-incorporated PnCs and might shed light on other engineering applications of the frequency detector and elastic wave power transfer.

Double defects-induced elastic wave coupling and energy localization in a phononic crystal

This study aims to investigate elastic wave localization that leverages defect band splitting in a phononic crystal with double defects through in-depth analysis of comparison of numerical and experimental results. When more than one defect is created inside a phononic crystal, these defects can interact with each other, resulting in a distinctive physical phenomenon from a single defect case: defect band splitting. For a phononic crystal consisting of circular-hole type unit cells in a thin aluminum plate, under A0 (the lowest antisymmetric) Lamb waves, both numerical simulations and experiments successfully confirm the defect band splitting phenomenon via frequency response functions for the out-of-plane displacement calculated/measured at the double defects within a finite distance. Furthermore, experimental visualization of in-phase and out-of-phase defect mode shapes at each frequency of the split defect bands is achieved and found to be in excellent agreement with the simulated results. Different inter-distance combinations of the double defects reveal that the degree of the defect band splitting decreases with  the increasing distance due to weaker coupling between the defects. This work may shed light on engineering applications of a multiple-defect-introduced phononic crystal, including broadband energy harvesting, frequency detectors, and elastic wireless power transfer.

Effects of initial stress on band gap of Love waves in a layered domain-inverted phononic crystal structure

Love wave propagation in periodically domain-inverted lithium niobate substrate with a homogeneous zinc oxide waveguide layer is theoretically investigated. Much attention is devoted to the effect of initial stress on the band gap and dispersion behavior of Love wave in the layered structure. The results reveal a strong dependency of the band structure on the waveguide layer. In particular, as the waveguide layer gradually thickens, the band gap shifts into a lower frequency range remarkably. Additionally, the initial stress has a significant influence on the location of the band gap, but a small effect on the bandgap width. This separate effect provides the potential to actively tune the surface acoustic waves by the stress.

Optimization on ultrasonic plastic welding systems based on two-dimensional phononic crystal

Large-sized tools are needed for plastic welding of large objects. However, the displacement distribution at the radiation surface of the tools is not uniform due to the strong lateral vibration. In this paper, two-dimensional (2D) phononic crystal (PnC) is introduced by processing periodically arranged grooves in large-sized tools. This method is different from the traditional experience-based grooving method. The lateral vibration is suppressed due to the characteristic of PnC. In this paper, six kinds of ultrasonic plastic welding systems based on 2D PnC with different groove width and height are designed. The vibration characteristics of ultrasonic plastic welding systems were simulated and experimented. The band gaps of large-sized tools are analyzed. The results show that the lateral vibration is suppressed when the operating frequency of the ultrasonic plastic welding system is within the band gap of the large tool. The displacement distribution at the radiation surface of the tools is more uniform. A theoretical support is provided for the design of large-sized ultrasonic plastic welding systems.

One-step polymeric phononic crystal manufacture

A versatile system to construct bulk polymeric phononic crystals by using acoustic waves is described. In order to fabricate this material, a customised cavity device fitted with a ∼2 MHz acoustic transducer and an acoustic reflector is employed for the acoustic standing wave creation in the device chamber. The polymer crystal is formed when the standing waves are created during the polymerisation process. The resulting crystals are reproduced into the shape of the tunable device cavity with a unique periodic feature. The separation is related to the applied acoustic wave frequency during the fabrication process and each unit cell composition was found to be made up to two material phases. To assess the acoustic properties of the polymer crystals their average acoustic velocity is measured relative to monomer solutions of different concentrations. It is demonstrated that one of the signature characteristics of phononic crystal, the slow wave effect, was expressed by this polymer. Furthermore the thickness of a unit cell is analysed from images obtained with microscope. By knowing the thickness the average acoustic velocity is calculated to be 1538 m/s when the monomer/cross-linker concentration is 1.5 M. This numerical calculation closely agrees with the predicted value for this monomer/cross-linker concentration of 1536 m/s. This work provides a methodology for rapid accessing a new type of adaptable phononic crystal based on flexible polymers.

Designing a tunable acoustic resonator based on defect modes, stimulated by selectively biased PZT rods in a 2D phononic crystal

Reconfigurable phononic crystals (PnCs) and related devices are highly attractive because of their flexibility for different applications. We present the design procedure for a tunable acoustic resonator based on a 2D PnC, consisting of a periodic array of piezoelectric rods of radii 175 μm as inclusions arranged in air background. A single point defect devised by a rod of radius 161 μm, replacing one of the inclusions, plays the role of the acoustic resonator, leading to a defect frequency in the phononic band gap (f_d ≈ 432 kHz). Applying a ∼1% strain to the defect rod, via an external voltage, tunes the defect resonant frequency within the phononic band gap. It is shown that the maximum tunability and the frequency shift depends on the defect size, and is achieved about Δf_d = 440 Hz for the defect with the expense of descending quality factor. Considering the pattern of the localized pressure field, we introduce a multi-defect structure with five symmetric defect rods, corresponding to the maxima of field distribution. It is shown that maximum frequency shift of the dominant defect frequency is achieved about Δf_d = 1.14 kHz for defect radius of 161 μm, when all five defect rods are strained. The proposed tunable filter based on multi-defect structure results in an enhancement of about 2.6 times in the maximum frequency shift, in comparison with the single defect structure, and introduces a promising approach for realizing tunable acoustic devices.

Effects of thermo-order-mechanical coupling on band structures in liquid crystal nematic elastomer porous phononic crystals

Liquid crystal nematic elastomers are one kind of smart anisotropic and viscoelastic solids simultaneously combing the properties of rubber and liquid crystals, which is thermal sensitivity. In this paper, the wave dispersion in a liquid crystal nematic elastomer porous phononic crystal subjected to an external thermal stimulus is theoretically investigated. Firstly, an energy function is proposed to determine thermo-induced deformation in NE periodic structures. Based on this function, thermo-induced band variation in liquid crystal nematic elastomer porous phononic crystals is investigated in detail. The results show that when liquid crystal elastomer changes from nematic state to isotropic state due to the variation of the temperature, the absolute band gaps at different bands are opened or closed. There exists a threshold temperature above which the absolute band gaps are opened or closed. Larger porosity benefits the opening of the absolute band gaps. The deviation of director from the structural symmetry axis is advantageous for the absolute band gap opening in nematic state whist constrains the absolute band gap opening in isotropic state. The combination effect of temperature and director orientation provides an added degree of freedom in the intelligent tuning of the absolute band gaps in phononic crystals.

Achieving directional propagation of elastic waves via topology optimization

This paper presents a study on topology optimization of novel material microstructural configurations to achieve directional elastic wave propagation through maximization of partial band gaps. A waveguide incorporating a periodic-microstructure material may exhibit different propagation properties for the plane elastic waves incident from different inlets. A topology optimization problem is formulated to enhance such a property with a gradient-based mathematical programming algorithm. For alleviating the issue of local optimum traps, the random morphology description functions (RMDFs) are introduced to generate random initial designs for the optimization process. The optimized designs finally converge to the orderly material distribution and numerical validation shows improved directional propagation property as expected. The utilization of linear two-dimension phononic crystal with efficient partial band gap is suitable for directional propagation with a broad frequency range.

Band structure calculation of SH waves in nanoscale multilayered piezoelectric phononic crystals using radial basis function method with consideration of nonlocal interface effects

In this paper, the radial basis function (RBF) collocation method based on the nonlocal Eringen piezoelectricity theory is developed to compute the band structures of nanoscale multilayered piezoelectric phononic crystals taking account of nonlocal interface effects. Detailed calculations are performed for anti-plane transverse waves propagating obliquely or vertically in the system. The correctness of the present method is verified by comparing the numerical results with those obtained by applying the transfer matrix method in the case of nonlocal perfect interfaces. The effects of nonlocal interface imperfections are considered by comparing with the nonlocal perfect interfaces. In addition, the influences of the piezoelectric constant, the nanoscale size, the impedance ratio and the incidence angle on the cut-off frequency and band structures are investigated and discussed in detail. Numerical results show that the nonlocal interface discontinuity has more obvious effect on the low-frequency band structures at the microscopic scale than at the macroscopic scale. Furthermore, at the macroscopic scale, the nonlocal interface imperfection has an obvious effect on the high frequency waves, but the effect on the low frequency waves is not obvious, and the nonlocal interface imperfection has no effect on the cut-off frequency at the microscopic scale.

Experiments of wave cancellation with elastic phononic crystal

The objective of this work is to experimentally demonstrate that two incident beams of ultrasonic waves can be cancelled by using an elastic phononic crystal (PC) prism. Although PCs are known to be used for wave cancellation, there appears no experimental demonstration especially for elastic waves. Here, we use an elastic PC prism embedded in an aluminum plate, which can split an input incident beam into multiple output beams. Two signals of different incident angles are reversely sent to the prism for the wave cancellation experiment. For successful wave cancellation experiments, the magnitudes and phase difference of the input sources were carefully tuned. The experimental results were found to agree well with the predictions from numerical simulations.

Level repulsion of GHz phononic surface waves in quartz substrate with finite-depth holes

This paper presents numerical and experimental results on the level repulsion of gigahertz surface acoustic waves in an air/ST-cut quartz phononic structure with finite-depth holes. The colorful dispersion with the parameter of the in-plane (sagittal plane) ratio of polarization was adopted to determine the Rayleigh wave bandgap induced by the level repulsion. The results of numerical analyses showed that the frequency and width of the bandgap induced by the level repulsion strongly depend on the geometry of the air holes in the phononic structure. In the experiment, a pair of slanted interdigital transducers with frequency in the gigahertz range was designed and fabricated to generate and receive broadband Rayleigh waves, whereas the reactive ion etching process with electron-beam lithography was used to fabricate submicrometer phononic structures. The measured results of the bandgap induced by the level repulsion agreed favorably with the numerical prediction.

Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with functionally graded interlayers

The effects of functionally graded interlayers on dispersion relations of elastic waves in a one-dimensional piezoelectric/piezomagnetic phononic crystal are studied in this paper. First, the state transfer equation of the functionally graded interlayer is derived from the motion equation by the reduction of order (from second order to first order). The transfer matrix of the functionally graded interlayer is obtained by solving the state transfer equation with the spatial-varying coefficient. Based on the transfer matrixes of the piezoelectric slab, the piezomagnetic slab and the functionally graded interlayers, the total transfer matrix of a single cell is obtained. Further, the Bloch theorem is used to obtain the resultant dispersion equations of in-plane and anti-plane Bloch waves. The dispersion equations are solved numerically and the numerical results are shown graphically. Five kinds of profiles of functionally graded interlayers between a piezoelectric slab and a piezomagnetic slab are considered. It is shown that the functionally graded interlayers have evident influences on the dispersion curves and the band gaps.

Dispersion relations of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with initial stresses

The dispersion relations of elastic waves in a one-dimensional phononic crystal formed by periodically repeating of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are studied in this paper. The influences of initial stress on the dispersive relation are considered based on the incremental stress theory. First, the incremental stress theory of elastic solid is extended to the magneto-electro-elasto solid. The governing equations, constitutive equations, and boundary conditions of the incremental stresses in a magneto-electro-elasto solid are derived with consideration of the existence of initial stresses. Then, the transfer matrices of a pre-stressed piezoelectric slab and a pre-stressed piezomagnetic slab are formulated, respectively. The total transfer matrix of a single cell in the phononic crystal is obtained by the multiplication of two transfer matrixes related with two adjacent slabs. Furthermore, the Bloch theorem is used to obtain the dispersive equations of in-plane and anti-plane Bloch waves. The dispersive equations are solved numerically and the numerical results are shown graphically. The oblique propagation and the normal propagation situations are both considered. In the case of normal propagation of elastic waves, the analytical expressions of the dispersion equation are derived and compared with other literatures. The influences of initial stresses, including the normal initial stresses and shear initial stresses, on the dispersive relations are both discussed based on the numerical results.

Phononic crystal surface mode coupling and its use in acoustic Doppler velocimetry

It is numerically shown that surface modes of two-dimensional phononic crystals, which are Bloch modes bound to the interface between the phononic crystal and the surrounding host, can couple back and forth between the surfaces in a length scale determined by the separation of two surfaces and frequency. Supercell band structure computations through the finite-element method reveal that the surface band of an isolated surface splits into two bands which support either symmetric or antisymmetric hybrid modes. When the surface separation is 3.5 times the lattice constant, a coupling length varying between 30 and 48 periods can be obtained which first increases linearly with frequency and, then, decreases rapidly. In the linear regime, variation of coupling length can be used as a means of measuring speeds of objects on the order of 0.1m/s by incorporating the Doppler shift. Speed sensitivity can be improved by increasing surface separation at the cost of larger device sizes.

Investigation of complete bandgaps in a piezoelectric slab covered with periodically structured coatings

The propagation of elastic waves in a piezoelectric slab covered with periodically structured coatings or the so-called stubbed phononic crystal slab is investigated. Four different models are selected and the effects of distribution forms and geometrical parameters of the structured coatings on complete bandgaps are discussed. The phononic crystal slab with symmetric coatings can generate wider complete bandgaps while that with asymmetric coatings is favorable for the generation of multi-bandgaps. The complete bandgaps, which are induced by locally resonant effects, change significantly as the geometry of the coatings changes. Moreover, the piezoelectric effects benefit the opening of the complete bandgaps.

Systematic topology optimization of solid-solid phononic crystals for multiple separate band-gaps with different polarizations

Phononic crystals (PnCs) have attracted considerable interest due to their unique and outstanding band-gap characteristics. In many applications, it is desirable to have a unit cell with specific band-gaps. The distribution of elastic materials within a unit cell has significant effect on the band-gaps, which is extremely difficult to be determined without systematic synthesis method. In this paper, topology optimization techniques are utilized to obtain two-dimensional (2D) square lattice PnCs with maximized relative band-gaps between multiple consecutive bands. The optimization follows two-stage design process using Genetic algorithms (GAs) in combination with finite element method (FEM). Three numerical examples are given to optimize 2D steel/epoxy PnCs in one-eighth symmetry for coupled mode, shear mode and mixed mode respectively. The results show that the optimized PnCs with different band-gaps, which can easily be found by the developed method, have different materials layout, and the PnCs with the lowest order band-gap are simple lattice and have the highest value of application in noise reduction and vibration isolation. Some optimized PnCs with higher order band-gaps have the same lattice as those with the lowest order band-gap, and whose absolute band-gaps are inversely proportional to the minimum feature size of primitive cells.

Numerical study and topology optimization of 1D periodic bimaterial phononic crystal plates for bandgaps of low order Lamb waves

The optimum topology of bimaterial phononic crystal (PhCr) plates with one-dimensional (1D) periodicity to attain maximum relative bandgap width of low order Lamb waves is computationally investigated. The evolution of optimized topology with respect to filling fraction of constituents, alternatively stiff scattering inclusion, is explored. The underlying idea is to develop PhCr plate structures with high specific bandgap efficiency at particular filling fraction, or further with multiscale functionality through gradient of optimized PhCr unitcell all over the lattice array. Multiobjective genetic algorithm (GA) is employed in this research in conjunction with finite element method (FEM) for topology optimization of silicon-tungsten PhCr plate unitcells. A specialized FEM model is developed and verified for dispersion analysis of plate waves and calculation of modal response. Modal band structure of regular PhCr plate unitcells with centric scattering layer is studied as a function of aspect ratio and filling fraction. Topology optimization is then carried out for a few aspect ratios, with and without prescribed symmetry, over various filling fractions. The efficiency of obtained solutions is verified as compared to corresponding regular centric PhCr plate unitcells. Moreover, being inspired by the obtained optimum topologies, definite and easy to produce topologies are proposed with enhanced bandgap efficiency as compared to centric unitcells. Finally a few cases are introduced to evaluate the frequency response of finite PhCr plate structures produced by achieved topologies and also to confirm the reliability of calculated modal band structures. Cases made by consecutive unitcells of different filling fraction are examined in order to attest the bandgap efficiency and multiscale functionality of such graded PhCr plate structures.

Multiple beam splitting in elastic phononic crystal plates

This work presents an experimental evidence for triple beam splitting in an elastic plate with an embedded elastic phononic crystal (PC) prism and elaborates on its working mechanism. While there were reports on negative refraction and double beam splitting with PCs, no experimental evidence on the splitting of triple or more ultrasonic elastic beams through PCs has been shown yet. After the experimental results are presented in case of triple beam splitting, further analysis is carried out to explain how triple or more beams can be split depending on elastic PC prism angles.

Research on bandgaps in two-dimensional phononic crystal with two resonators

In this paper, the bandgap properties of a two-dimensional phononic crystal with the two resonators is studied and embedded in a homogenous matrix. The resonators are not connected with the matrix but linked with connectors directly. The dispersion relationship, transmission spectra, and displacement fields of the eigenmodes of this phononic crystal are studied with finite-element method. In contrast to the phononic crystals with one resonators and hollow structure, the proposed structures with two resonators can open bandgaps at lower frequencies. This is a very interesting and useful phenomenon. Results show that, the opening of the bandgaps is because of the local resonance and the scattering interaction between two resonators and matrix. An equivalent spring-pendulum model can be developed in order to evaluate the frequencies of the bandgap edge. The study in this paper is beneficial to the design of opening and tuning bandgaps in phononic crystals and isolators in low-frequency range.

Hybrid phononic crystal plates for lowering and widening acoustic band gaps

We propose hybrid phononic-crystal plates which are composed of periodic stepped pillars and periodic holes to lower and widen acoustic band gaps. The acoustic waves scattered simultaneously by the pillars and holes in a relevant frequency range can generate low and wide acoustic forbidden bands. We introduce an alternative double-sided arrangement of the periodic stepped pillars for an enlarged pillars' head diameter in the hybrid structure and optimize the hole diameter to further lower and widen the acoustic band gaps. The lowering and widening effects are simultaneously achieved by reducing the frequencies of locally resonant pillar modes and prohibiting suitable frequency bands of propagating plate modes.

Focusing and waveguiding of Lamb waves in micro-fabricated piezoelectric phononic plates

This paper presents results on the numerical and experimental studies of focusing and waveguiding of the lowest anti-symmetric Lamb wave in micro-fabricated piezoelectric phononic plates. The phononic structure was based on an AT-cut quartz plate and consisted of a gradient-index phononic crystal (GRIN PC) lens and a linear phononic plate waveguide. The band structures of the square-latticed AT-cut quartz phononic crystal plates with different filling ratios were analyzed using the finite element method. The design of a GRIN PC plate lens which is attached with a linear phononic plate waveguide is proposed. In designing the waveguide, propagation modes in square-latticed PC plates with different waveguide widths were studied and the results were served for the experimental design. In the micro-fabrication, deep reactive ion etching (Deep-RIE) process with a laboratory-made etcher was utilized to fabricate both the GRIN PC plate lens and the linear phononic waveguide on an 80 μm thick AT-cut quartz plate. Interdigital transducers were fabricated directly on the quartz plate to generate the lowest anti-symmetric Lamb waves. A vibro-meter was used to detect the wave fields and the measured results on the focusing and waveguiding of the piezoelectric GRIN PC lens and waveguide are in good accordance with the numerical predictions. The results of this study may serve as a basis for developing an active micro plate lens and related devices.

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger's equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.