Two-dimensional Phononic Crystals Research Articles

Phase-control in two-dimensional phononic crystals

A theoretical model is developed to ascertain the necessary band structure and equi-frequency contour (EFC) features of two-dimensional phononic crystals (PCs) for the realization of phase control between propagating acoustic waves. Two different PCs, a square array of cylindrical polyvinylchloride inclusions in air and a triangular array of cylindrical steel inclusions in methanol, offer band structures and EFCs with highly dissimilar features. We demonstrate that PCs with EFCs showing non-collinear wave and group velocity vectors are ideal systems for controlling the phase between propagating acoustic waves. Finite-difference time-domain simulations are employed to validate theoretical models and demonstrate the control of phase between propagating acoustic waves in PC structures.

Probing Thermomechanics at the Nanoscale: Impulsively Excited Pseudosurface Acoustic Waves in Hypersonic Phononic Crystals

High-frequency surface acoustic waves can be generated by ultrafast laser excitation of nanoscale patterned surfaces. Here we study this phenomenon in the hypersonic frequency limit. By modeling the thermomechanics from first-principles, we calculate the system’s initial heat-driven impulsive response and follow its time evolution. A scheme is introduced to quantitatively access frequencies and lifetimes of the composite system’s excited eigenmodes. A spectral decomposition of the calculated response on the eigemodes of the system reveals asymmetric resonances that result from the coupling between surface and bulk acoustic modes. This finding allows evaluation of impulsively excited pseudosurface acoustic wave frequencies and lifetimes and expands our understanding of the scattering of surface waves in mesoscale metamaterials. The model is successfully benchmarked against time-resolved optical diffraction measurements performed on one-dimensional and two-dimensional surface phononic crystals, probed using light at extreme ultraviolet and near-infrared wavelengths.

Dispersion curves of surface acoustic waves in a two-dimensional phononic crystal

Reliable numerical simulations of band structure for surface acoustic waves propagating in a two-dimensional phononic crystal are reported. Through an efficient finite element method and specific boundary conditions, a theoretical approach allowing a direct computation of surface acoustic wave’s band structure for phononic crystal is proposed. Three types of phononic structures are investigated; fluid/solid, solid/solid, and air connected stubbed substrate. Using sound cone limitation, calculated results show the propagation of surface acoustic waves in the nonradiative region of the substrate. In addition, the modal displacements of the original guided surface modes supported by the studied structures are computed showing their original characteristics.

Effective shear speed in two-dimensional phononic crystals

The quasistatic limit of the antiplane shear-wave speed (effective speed) $c$ in 2D periodic lattices is studied. Two new closed-form estimates of $c$ are derived by employing two different analytical approaches. The first proceeds from a standard background of the plane wave expansion (PWE). The second is a new approach, which resides in $\mathbf{x}$ space and centers on the monodromy matrix (MM) introduced in the 2D case as the multiplicative integral, taken in one coordinate, of a matrix with components being the operators with respect to the other coordinate. On the numerical side, an efficient PWE-based scheme for computing $c$ is proposed and implemented. The analytical and numerical findings are applied to several examples of 2D square lattices with two and three high-contrast components, for which the new PWE and MM estimates are compared with the numerical data and with some known approximations. It is demonstrated that the PWE estimate is most efficient in the case of densely packed stiff inclusions, especially when they form a symmetric lattice, while in general it is the MM estimate that provides the best overall fitting accuracy.

Features of propagation of surface acoustic waves in two-dimensional phononic crystals on the surface of a lithium niobate crystal

Two-dimensional phononic crystals are fabricated on the surface of a lithium niobate crystal. A numerical method is developed for calculating the frequency dependences of the transmission and reflection coefficients of a surface acoustic wave for a rectangular region with two-dimensional periodic structure on the surface of an elastic body. Measurements of the frequency dependences of the reflection and transmission coefficients are performed for propagation in the part of the lithium niobate crystal surface on which two-dimensional phononic crystals were fabricated. Forbidden zones are found in the transmission spectra of the waves transmitted through phononic crystals. It is shown that, if the width of the incident wave beam is less than the width of the region with the two-dimensional periodic structure (the two-dimensional phononic crystal), then a resonant increase in the transmission coefficient arises near the center of the rejection band.

Low-Frequency Forbidden Bands in Phononic Crystal Plates with Helmholtz Resonators

A theoretical investigation of Lamb wave propagation in two-dimensional phononic crystals composed of an array of solid Helmholtz resonators (HRs) on a thin plate is presented. The dispersion relations, power transmission spectra, and spectra of resonances are calculated by finite-element analysis. Owing to the simultaneous mechanisms of local resonances and Bragg scattering, the structure exhibits low-frequency forbidden bands and Bragg band gaps that can be effectively shifted by changing the geometry of the HRs. As a result, low-frequency band gaps within the audible range can be achieved. Furthermore, the calculated power transmission and resonance spectra for a finite phononic crystal structure show an evident resonance nature and are directly related to the formation of low-frequency band gaps. The spectra of the monolayer HR structure show that the resonances can either induce high reflection or intensify transmission. The effects of different excitation conditions for generating different Lamb wave modes on the transmission of the wave energy are also studied. The calculated results show that the transmission varies with the incident Lamb waves of different modes.

The investigation of point defect modes of phononic crystal for high Q resonance

Point defect cavity was constructed in a two-dimensional phononic crystal plate. It was excited by a small piezo chip in the cavity, and the vibrations of each point defect mode were detected by an optical interferometer. Point defect modes on 2D phononic crystal plate in vacuum, air, and with water loaded were investigated theoretically and experimentally. It is shown that the Q factors of the point defect modes are determined by inner attenuation, bandgap effects, and medium. The SSS mode (breathing mode) has highest Q factor in vacuum and air among nine modes thanks to low inner attenuation and low energy leakage. By selectively loading the PC with water on one side, the point defect modes with shear movement surfaces suffer lower attenuation and still have rather high Q factors. These conclusions will help to design a new kind of resonator or sensor.

Observation of band gaps in the gigahertz range and deaf bands in a hypersonic aluminum nitride phononic crystal slab

We report on the observation of elastic waves propagating in a two-dimensional phononic crystal composed of air holes drilled in an aluminum nitride membrane. The theoretical band structure indicates the existence of an acoustic band gap centered around 800 MHz with a relative bandwidth of 6.5% that is confirmed by gigahertz optical images of the surface displacement. Further electrical measurements and computation of the transmission reveal a much wider attenuation band that is explained by the deaf character of certain bands resulting from the orthogonality of their polarization with that of the source.

Tunable phononic crystals with anisotropic inclusions

We present a theoretical study on the tunability of phononic band gaps in two-dimensional phononic crystals consisting of various anisotropic cylinders in an isotropic host. A two-dimensional plane-wave expansion method was used to analyze the band diagrams of the phononic crystals; the anisotropic materials used in this work include cubic, hexagonal, trigonal, and tetragonal crystal systems. By reorienting the anisotropic cylinders, we show that phononic band gaps for bulk acoustic waves propagating in the phononic crystal can be opened, modulated, and closed. The methodology presented here enables enhanced control over acoustic metamaterials which have applications in ultrasonic imaging, acoustic therapy, and nondestructive evaluation.

A Novel Micromechanical Resonator Using Two-Dimensional Phononic Crystal Slab

Two-dimensional (2-D) Silicon phononic crystal (PnC) slab of a square array of cylindrical air holes in a 10μm thick free-standing silicon plate with line defects is characterized as a cavity-mode PnC resonator. Piezoelectric aluminum nitride (AlN) film is deployed as the inter-digital transducers (IDT) to transmit and detect acoustic waves, thus making the whole microfabrication process CMOS-compatible. Both the band structure of the PnC and the transmission spectrum of the proposed PnC resonator are analyzed and optimized using finite element method (FEM). The measured quality factor (Q factor) of the microfabricated PnC resonator is over 1,000 at its resonant frequency of 152.46MHz. The proposed PnC resonator shows promising acoustic resonance characteristics for RF communications and sensing applications.

Acoustic Waves in Two-Dimensional Phononic Crystals With Reticular Geometric Structures

Previous studies on photonic crystals raise the exciting topic of phononic crystals. This paper presents the results of tunable band gaps in the acoustic waves of two-dimensional phononic crystals with reticular geometric structures using the 2D and 3D finite element methods. This paper calculates and discusses the band gap variations of the bulk modes due to different sizes of reticular geometric structures. Results show that adjusting the orientation of the reticular geometric structures can increase or decrease the total elastic band gaps for mixed polarization modes. The band gap phenomena of elastic or acoustic waves can potentially be utilized to achieve vibration-free, high-precision mechanical systems, and sound insulation.

Elastic Wave Localization in Two-Dimensional Phononic Crystals with One-Dimensional Aperiodicity

The band structures of in-plane elastic waves propagating in two-dimensional phononic crystals with one-dimensional aperiodicity are analyzed in this paper. The localization of wave propagation is discussed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method. By treating the aperiodicity as the deviation from the periodicity in a special way, two kinds of aperiodic phononic crystals that have Thue-Morse and Rudin-Shapiro sequence in one direction and translational symmetry in the other direction are considered. The transmission coefficients based on eigenmode match theory are also calculated and the results show the same behaviors as the localization factor does. In the case of Thue-Morse and Rudin-Shapiro structures, the band structures of Thue-Morse sequence exhibit similarities with quasi-periodic sequence not present in the results of Rudin-Shapiro sequence.

Localization of In-Plane Guided Elastic Waves in a Two-Dimensional Solid Waveguiding Phononic Crystal

Combined with the supercell technique, the plane wave expansion method is used to calculate the band structures for the in-plane wave of the two-dimensional solid-solid phononic crystals with line defects and the random disorders in either radius or location of the scatterers. The influences of the random disorders on the band structures and guided waves will be discussed. Propagation of the wave with one certain frequency in the waveguiding phononic crystals with different disorder degree is studied.

Application of Dirichlet-to-Neumann Map to Calculation of Band Gaps for Scalar Waves in Two-Dimensional Phononic Crystals

This paper is aimed at applying Dirichlet-to-Neumann (DtN) map to calculate the band gaps of scalar waves in a two-dimensional phononic crystal which is composed of circular cylinders embedded in a host medium in either square or triangular lattice. Detailed computation is performed for both solid/solid and fluid/fluid systems with emphasis on comparison to the plane wave expansion method. The numerical results show some merits of the DtN method. Its computing cost to obtain the results of high accuracy is very low, especially for higher frequency computation or for systems with large acoustic mismatch. Another feature of the DtN method is that it can yield band structures at an arbitrary frequency interval which does not necessarily start from zero.

Negative refraction of longitudinal waves in a two-dimensional solid-solid phononic crystal

Negative refraction of longitudinal waves in a two-dimensional solid-solid phononic crystal

Material loss influence on the complex band structure and group velocity in phononic crystals

The influence of material loss on the complex band structure of two-dimensional phononic crystals is investigated. A viscoelasticity model is added to the extended plane-wave expansion (EPWE) method, with viscosity proportional to the frequency. It is found that losses have a stronger influence on the real than on the imaginary part of Bloch waves, in contrast with propagation in homogeneous media. Flat bands, i.e., bands initially showing low group velocity without losses, acquire an enhanced damping as compared to bands with larger group velocities. Losses are also found to limit the appearance of large group slownesses, or conversely small group velocities.

Influence of Random Disorders on Anti-Plane Waveguiding Modes in a Two-Dimensional Phononic Crystal

In this paper, combined with the supercell technique, the plane wave expansion method is used to calculate the band structures of the two-dimensional phononic crystals with line defects and the random disorders in either radius or location of the scatterers. Phononic systems with plumbum scatterers embedded in an epoxy matrix are calculated in detail. The influences of the random disorder on the band structures of anti-plane waveguiding modes will be discussed. The displacement distributions are calculated to show the wave localization phenomenon. Propagation of the guided wave in the phononic crystals with different disordered degree is studied. The analysis is relevant to the assessment of the influences of manufacture errors on wave behaviors in waveguiding phononic crystals as well as the possible control of wave propagation by intentionally introducing disorders into the systems.

Studies on Band Structures and Wave Localization Phenomenon of the Randomly Disordered Two-Dimensional Solid-Fluid Phononic Crystals

The supercell based plane wave expansion method is used to study the effects of random disorders on the band structures of a two-dimensional (2D) solid-fluid phononic crystal. Phononic systems with steel scatterers embedded in a water matrix are calculated in detail. The radius disorder and location disorder are concerned. The influences of the disorder degree on the first band gap are investigated. The localization phenomenon is discussed by computing the displacement fields in the supercell.

Local resonances-induced low-frequency band gaps in two-dimensional phononic crystal slabs with periodic stepped resonators

In this paper, the numerical investigation of Lamb wave propagation in two-dimensional phononic crystals composed of an array of stepped resonators on a thin slab is presented. The dispersion relations, power transmission spectra and spectra of resonances are studied using a finite-element method. Because of the simultaneous mechanisms of the local resonances and Bragg scattering, the structures exhibit low-frequency forbidden bands and Bragg band gaps, which can be effectively shifted by changing the resonator geometries as well as the lattice symmetries of the resonator array. As a result, a low-frequency gap within the audible regime can be demonstrated. Furthermore, for a finite phononic crystal slab, the calculated transmission and resonance spectra show an evident resonance nature which can be directly related to the formation of the low-frequency gaps. Based on the spectra of elastic waves through the single-layer stepped resonators, the resonances of the stepped resonators are found to either induce high reflection or intensify the transmission. The effects of different excitation conditions for generating specific slab modes with different polarization states on the acoustic energy transmission and attenuation are also studied. The results show that the polarization states of the incident slab modes influence the spectra of resonances, power transmission and attenuation.

Lamb waves in two-dimensional phononic crystal plate with anisotropic inclusions

An analysis is given to the band structure of the two-dimensional phononic crystal plate constituted of a square array of elastic anisotropic, circular Pb cylinders embedded in elastic isotropic epoxy. The numerical results show that the band gap can be tuned by rotating the anisotropic material orientation. It is found that the influence of anisotropy on band gap of Lamb wave is clearly different from that on the band gap of bulk waves. The thickness of the system under study is a sensitive parameter to affect the influence of anisotropic materials on the normalized gap width.