Waves In Two-dimensional Phononic Crystals Research Articles

Coupled topological rainbow trapping of elastic waves in two-dimensional phononic crystals

Rainbow trapping, observed in elastic waves, has attracted considerable scientific interest owing to its potential applications in energy harvesting, buffering, and wavelength-division multiplexing devices. However, previous approaches have often necessitated complex geometric modifications to the scatterer, such as altering dimensions or shifting along diagonals to corners, limiting practical utility. Here, we realize the coupled topological edge states (CTESs) of elastic waves in a two-dimensional (2D) solid phononic crystal (PC) with inversion center changes. Changing the inversion center along the x or y directions by a specific distance can induce the topological phase transition. The topological edge states (TESs) arise at the interface by combining PCs with different topologies positioned adjacent to each other. Furthermore, it is demonstrated that TES exhibits topological robustness against defects. By introducing a gradient into the PC structure by altering the geometrical parameters of scatterers along the interface, the topological rainbow trapping of elastic waves is achieved. Finally, the CTES are generated by the interaction between TESs of different interfaces, which can lead to coupled topological rainbow trapping in phononic heterostructures with different displacement parameters along the multiple interface gradient. Our results pave the way for manipulating the symmetric and antisymmetric topological modes of elastic waves in topologically coupled waveguides, which offers potential applications in selective filtering and multiband waveguiding.

Polarization of Acoustic Waves in Two-Dimensional Phononic Crystals Based on Fused Silica.

The two-dimensional square-lattice phononic crystal is one of the recently proposed acoustic metamaterials. Strong anisotropic propagation of elastic waves makes the material promising for various potential applications in acoustics and acousto-optics. This paper presents a study of the propagation of elastic waves in two-dimensional phononic crystals based on fused silica. The band structures of a phononic crystal are obtained by solving the wave equation in its variational form by the finite element method. The main phononic crystal acoustic characteristics that are of practical interest in acousto-optics are calculated based on the analysis of the dispersion relations. It is shown that the choice of the phononic crystal geometry makes it possible to control the distributions of both the inverse phase velocities and the energy walk-off angles of acoustic modes. The calculations of the acoustic modes' polarization are in a particular focus. It is demonstrated that under certain conditions, there are exactly three acoustic modes propagating in a phononic crystal, the averaged polarization vectors of which are mutually orthogonal for any directions of the acoustic wave's propagation. It is argued that the acoustic properties of phononic crystals meet the requirements of acousto-optics.

Singular boundary method for band structure calculations of in-plane waves in 2D phononic crystals

In this study, the singular boundary method, which is a boundary collocation method, is first employed to calculate the band structures of in-plane waves in two-dimensional phononic crystals. When a unit cell of a phononic crystal is considered, the whole boundaries, including the continuity and periodic boundaries are discretized by the singular boundary method. Then, a linear eigenvalue equation is derived. For a given frequency, an eigenvalue that involves the Block wave can be calculated. Therefore, by sweeping the frequencies, the band structure can be obtained. The accuracy, efficiency and convergence of the proposed method are tested by several numerical examples, and the results demonstrate that the singular boundary method can provide stable and efficient results for band structure calculations of in-plane waves in two-dimensional phononic crystals.

Simultaneous multi-band valley-protected topological edge states of shear vertical wave in two-dimensional phononic crystals with veins

The introduction of the concept of valley pseudospin to phononic crystals has made a remarkable topologically protected interface transport of sound, which opens a novel research area referred to as valley Hall topological insulators. Here, we demonstrate the simultaneous multi-band edge states of shear vertical waves in two-dimensional phononic crystals with veins. The multi-band edge states are topologically valley-protected and are obtained by simultaneously gapping multiple Dirac points at K (or K′) under the inversion symmetry breaking. As the relative radius of the two adjacent steel columns varies, the band diagram undergoes a topological transition which can be characterized by topological charge distributions and opposite valley Chern numbers. Subsequently, the vortex chirality of the bulk valley modes is unveiled. With numerical simulations, simultaneous multi-band valley dependent edge states and the associated valley-protected backscattering suppression around the curved waveguide are further demonstrated. Our work could become a promising platform for applications of multi-functional topological acoustic devices.

Propagation of elastic waves in two-dimensional phononic crystals composed of viscoelastic materials

We have investigated the transmission characteristics of acoustic waves through two-dimensional phononic crystals composed of viscoelastic materials such as silicon rubber. In the present work, we improve a previously proposed method in which it was assumed that the Poisson ratio of silicon rubber is independent of time. The improved method removes the assumption and independently handles two parameters that describe a viscoelastic material: time-dependent bulk modulus and shear rate. We find that the time-dependent shear rates can modulate the frequency dependences of the transmission rate to various degrees. Therefore, in acoustic wave transmission, we can obtain large gaps in the audible domain by choosing the substrate material appropriately. In addition, we find that the viscoelasticity of the substrate makes the fine structure for the transmission rates smooth.

Acoustic beam splitting in two-dimensional phononic crystals using self-collimation effect

We propose two models of self-collimation-based beam splitters in phononic crystals. The finite element method is used to investigate the propagation properties of acoustic waves in two-dimensional phononic crystals. The calculated results show that the efficiency of the beam splitter can be controlled systematically by varying the radius of the rods or by changing the orientation of the square rods in the line defect. The effect of changing the side length of the square rods on acoustic wave propagation is discussed. The results show that the total transmission/reflection range decreases/increases as the side length increases. We also find that the relationship between the orientation of the transflective point and the side length of the square rods is quasi-linear.

Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique

Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique

Simulations of acoustic waves bandgaps in a surface of silicon with a periodic hole structure in a thin nickel film

We have performed simulations of dispersion relations for surface acoustic waves in two-dimensional phononic crystal by the finite elements method (FEM) and by the plane wave method (PWM). Considered medium is a thin nickel layer on a silicon single crystal (001) surface. The nickel film is decorated with cylindrical holes of the depth equal to the nickel film thickness arranged in a square lattice. We have obtained full bandgaps for the surface waves propagating in the medium of particular range of filling factor and layer thickness. The width of the bandgap had reached over 500[MHz] for the sample of the lattice constant 500[nm] and is sufficient for experimental design.

Local resonances in phononic crystals and in random arrangements of pillars on a surface

The propagation of surface acoustic waves in two-dimensional phononic crystals of pillars on a surface is investigated experimentally for hexagonal and honeycomb lattice symmetries. A random array of pillars is also compared to the periodic phononic crystals. Taking into account that the geometrical and physical characteristics of the pillars are the same in all cases, it is shown that the locally resonant band gap in the low frequency range is almost independent of periodicity and resilient to randomness. In contrast, the Bragg band gap disappears with the random array.

Love waves in two-dimensional phononic crystals with depth-dependent properties

We calculate subsonic spectra of the Love waves, i.e., of the shear horizontal waves in the coated substrate, using developed analytical approach. Coating or substrate or both are two-dimensional heterogeneous in the sagittal plane and uniform along the out-of-plane direction. Slow coating permits multiple subsonic dispersion branches which are folded due to lateral periodicity. It is observed that low-frequency branches may either cross or repulse each other, the latter giving rise to low-frequency band gaps inside the Brillouin zone. Such behavior is likelier when the periodic inclusion occurs within the coating close enough to its free surface.

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

A numerical method, the so-called multiple monopole (MMoP) method, based on the generalized multipole technique (GMT) is proposed to calculate the band structures of in-plane waves in two-dimensional phononic crystals, which are composed of arbitrarily shaped cylinders embedded in a solid host medium. To find the eigenvalues (eigenfrequencies) of the problem, besides the sources used to expand the wave fields, an extra monopole source is introduced which acts as the external excitation. By varying the excitation frequency, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and the boundary of the irreducible first Brillouin zone (FBZ), the band structures can be obtained. Some typical numerical examples with different acoustic impedance ratios and with inclusions of various shapes are presented to validate the proposed method.

All-angle negative refraction for surface acoustic waves in pillar-based two-dimensional phononic structures

We describe an all-angle negative refraction effect for surface acoustic waves in two-dimensional phononic crystals made of cylindrical pillars assembled in a square lattice and deposited on the surface of a semi-infinite substrate. The convexity of the iso-frequency contours of some branches leads to a negative refraction effect despite the fact that the effective index is not negative. It occurs for the frequency range where the group velocity is never in opposite direction to the phase velocity. In addition, the use of cylindrical pillars acting as acoustic resonant elements on the surface permits us to achieve this phenomenon with a sub-wavelength feature size structure; therefore, the effect of all-angle negative refraction can be shifted down to low frequencies, which is highly desirable for high-resolution superlensing applications. A flat lens for surface acoustic waves has also been designed. It demonstrates the focusing of an acoustic source into an image on the other side of the finite-size structure with an image resolution of (λ/4), which overcomes the Rayleigh diffraction limit. The numerical simulations are based on the efficient finite element method and analyze pillars and the substrate of lithium niobate.

1P1-6 Time-resolved imaging of negative refraction of surface acoustic waves in two-dimensional phononic crystals(Poster Session)

1P1-6 Time-resolved imaging of negative refraction of surface acoustic waves in two-dimensional phononic crystals(Poster Session)

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.

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.

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.

Experimental demonstration of surface acoustic waves in two-dimensional phononic crystals with fluid background

The surface acoustic waves of a two-dimensional phononic crystal in water are investigated experimentally. By measuring the field distribution on the surface of the phononic crystal, a surface acoustic mode is clearly demonstrated. By introducing a periodical corrugation onto the surface of the phononic crystal and by a pointlike stimulation on the surface, a collimated acoustic beam with a small divergent angle is observed, which results from the coupling of the point source to the surface modes. This experimental realization may be of significance in the applications of surface acoustic waves in phononic crystals.

Calculation of band structures for surface waves in two-dimensional phononic crystals with a wavelet-based method

A wavelet-based method is developed to calculate the band structures of surface modes in two-dimensional phononic crystals including mixed fluid/solid systems and solid/solid systems with small or large acoustic mismatch. The defect modes of the surface waves are also calculated by using the supercell technique. The method is validated by recomputing the samples already studied in literatures. The results show some merits of the present method. In addition, the present method is applied to some new samples to show more properties of the surface modes. The influences of various factors, especially the acoustic mismatch, on the surface modes are discussed in detail. The present method may serve as an alternative method for studying the surface waves in general phononic lattices.

Wavelet method for calculating the defect states of two-dimensional phononicx crystals

Based on the variational theory, a wavelet-based numerical method is developed to calculate the defect states of acoustic waves in two-dimensional phononic crystals with point and line defects. The supercell technique is applied. By expanding the displacement field and the material constants (mass density and elastic stiffness) in periodic wavelets, the explicit formulations of an eigenvalue problem for the plane harmonic bulk waves in such a phononic structure are derived. The point and line defect states in solid-liquid and solid-solid systems are calculated. Comparisons of the present results with those measured experimentally or those from the plane wave expansion method show that the present method can yield accurate results with faster convergence and less computing time.

Negative refraction and focusing of ultrasound in two-dimensional phononic crystals

We present experimental demonstrations of negative refraction and focusing of ultrasonic waves in two-dimensional phononic crystals made of stainless steel rods assembled in a triangular lattice and immersed in a liquid. Negative refraction is achieved for the range of frequencies in the second band, where the directions of the wave vector and group velocity are antiparallel to each other due to circular equifrequency contours. Negative refraction is unambiguously observed using a prism-shaped crystal. By exploiting the circular equifrequency contours in the second band, focusing of the ultrasonic field emitted by a pointlike source was demonstrated using a flat phononic crystal filled with and immersed in water. During these experiments, the importance of imaging in the regime of all angle negative refraction (AANR) was established for obtaining high-quality images. The regime of AANR was achieved in a similar flat crystal, in which the liquid inside the crystal (methanol) was different from the outside medium (water). This design resulted in matching circular equifrequency contours at the frequency of $0.55\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$, implying that a flat ultrasonic lens with an effective refractive index of $\ensuremath{-}1$ was realized. By imaging a subwavelength line source with this crystal, a resolution of $0.55\ensuremath{\lambda}$ was observed, which is just above the diffraction limit.