Two-dimensional Phononic Crystals Research Articles

An improved method of eigen-mode matching theory in two-dimensional phononic crystal

The eigen-mode matching theory has been improved to be able to calculate not only the dynamic transmission coefficient but also the phononic band structure at the same time. The system with rectangle elastic isotropic lead rods embedded in uniform epoxy has been investigated as a practical example. The numerical results reveal that the method has a good convergency in calculating the phononic band structure and is also much more efficient to calculate the transmission coefficient. This new method can be extensively used in the propagation problems of the layered phononic crystal systems.

Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals

A wavelet-based method is developed to calculate elastic band gaps of two-dimensional phononic crystals. The wave field is expanded in the wavelet basis and an equivalent eigenvalue problem is derived in a matrix form involving the adaptive computation of integrals of the wavelets. The method is applied to a binary system. We first compute the band gaps of Au cylinders in an epoxy host. The advantages of the wavelet-based method are discussed in regard with the well-known plane-wave expansion method. Then the method is used to compute the band gaps of Au cylinders in a soft rubber with elastic constant ${10}^{6}$ times lower than that of Au. The convergence tests show that the wavelet-based method can reduce the Gibbs effect to a certain extent. These advantages make it possible for easy calculations of band structures of mixed solid-fluid phononic crystals where the traditional plane wave method encounters difficulties. In addition, the adaptability of wavelets makes the method possible for efficient band gap computations of more complex phononic structures.

The directional propagation characteristics of elastic wave in two-dimensional thin plate phononic crystals

Abstract The directional propagation characteristics of elastic wave during pass bands in two-dimensional thin plate phononic crystals are analyzed by using the lumped-mass method to yield the phase constant surface. The directions and regions of wave propagation in phononic crystals for certain frequencies during pass bands are predicted with the iso-frequency contour lines of the phase constant surface, which are then validated with the harmonic responses of a finite two-dimensional thin plate phononic crystals with 16 × 16 unit cells. These results are useful for controlling the wave propagation in the pass bands of phononic crystals.

The two-dimensional phononic band gaps tuned by the position of the additional rod

The plane-wave expansion method is used to numerically study the band structure of the two-dimensional phononic crystal with two inclusion rods in the unit cell. The numerical results show that the band gaps can be tuned by changing the position of the additional rod. The maximum quality factor of the desired band gap can be obtained by carefully tuning the position of the additional rod.

Design of a highly magnified directional acoustic source based on the resonant cavity of two-dimensional phononic crystals

The authors report a design of a highly magnified directional acoustic source based on the planar resonant cavity of two-dimensional phononic crystals. The authors demonstrate that the order of the resonant mode and the reflective ratio of the double phononic crystal slab are the key factors to the magnified ratio of the directional acoustic amplifier. With properly designed mode and cavity width of the asymmetric phononic structure, the optimal magnified amplitude can be achieved by more than 86.5 times in comparison with the amplitude of the original line source freely radiating in water.

Complete band gaps in two-dimensional phononic crystal slabs

The propagation of acoustic waves in a phononic crystal slab consisting of piezoelectric inclusions placed periodically in an isotropic host material is analyzed. Numerical examples are obtained for a square lattice of quartz cylinders embedded in an epoxy matrix. It is found that several complete band gaps with a variable bandwidth exist for elastic waves of any polarization and incidence. In addition to the filling fraction, it is found that a key parameter for the existence and the width of these complete band gaps is the ratio of the slab thickness, d, to the lattice period, a. Especially, we have explored how these absolute band gaps close up as the parameter d/a increases. Significantly, it is observed that the band gaps of a phononic crystal slab are distinct from those of bulk acoustic waves propagating in the plane of an infinite two-dimensional phononic crystal with the same composition. The band gaps of the slab are strongly affected by the presence of cutoff frequency modes that cannot be excited in infinite media.

Acoustic directional radiation and enhancement caused by band-edge states of two-dimensional phononic crystals

The authors numerically investigate the radiation of a line acoustic source placed inside a phononic crystal of square lattice. Their results show that it is possible to obtain a highly directional acoustic source with a large radiation enhancement, operating at the band-edge frequency of the phononic crystal. The angular distributions of the radiation power and the radiation enhancement factor are strongly dependent on the position of the acoustic source relative to the unit cell in which it is placed. A simple realization of a highly directional acoustic source with a half-power angular width of about 5.2° is presented.

Extreme acoustic band gaps obtained under high symmetry in 2D phononic crystals

Using the plane wave expansion method we have studied the effect of symmetry on the complete acoustic band gaps in two-dimensional binary phononic crystals including five types of straight rod arranged in hexagonal lattices. For each type of rod, two cases are considered: water rods in mercury (low-density scatters in high-density background) and mercury rods in water (high-density scatters in low-density background). The crystals' symmetry can be changed by rotating the noncircular rods, and the results show that the extreme (both the maximum and minimum) gaps at any given filling factor appear under high symmetry. So the above maximum gaps can be obtained by adjusting the rods' orientation. And the peaks of those maximum gaps are relative to the rods' rotational symmetry, the relations of which are different for the two cases with different density contrast.

Negative refraction imaging of acoustic waves by a two-dimensional three-component phononic crystal

Negative refraction imaging of acoustic waves by a two-dimensional three-component phononic crystal

Formation mechanism of the low-frequency locally resonant band gap in the two-dimensional ternary phononic crystals

The low-frequency band gap and the corresponding vibration modes in two-dimensional ternary locally resonant phononic crystals are restudied successfully with the lumped-mass method. Compared with the work of C. Goffaux and J. Sanchez-Dehesa (Phys. Rev. B 67 14 4301(2003)), it is shown that there exists an error of about 50% in their calculated results of the band structure and one band is missing in their results. Moreover, the in-plane modes shown in their paper are improper, which results in the wrong conclusion on the mechanism of the ternary locally resonant phononic crystals. Based on the lumped-mass method and better description of the vibration modes according to the band gaps, the locally resonant mechanism in forming the subfrequency gaps is thoroughly analysed. The rule used to judge whether a resonant mode in the phononic crystals can result in a corresponding subfrequency gap is also verified in this ternary case.

Locally resonant phononic crystals made of hollow spheres or cylinders

The acoustic properties of three- and two-dimensional phononic crystals consisting of hollow spheres or cylinders are studied by means of numerical calculations by the layer-multiple-scattering and the finite-difference--time-domain methods. The frequency band structure of these crystals is analyzed in conjunction with relevant transmission spectra, and the nature and physical origin of the different modes is elucidated. The building units of these crystals exhibit localized resonance states which can lead to almost dispersionless bands within an otherwise absolute gap. A feature that makes such systems potentially useful in the way of practical applications as efficient frequency filters. The influence of different geometric and material parameters on the position of these resonances is systematically examined. Such an optimized system made of a polymer material is proposed and studied in detail.

Defect states of homogeneity dislocation structures in two-dimensional phononic crystal

The defect states in two-dimensional phononic crystal homogeneity dislocation structures has been studied by means of the plane-wave expansion method in combination with a supercell technique. Transverse dislocation structures and longitudinal dislocation structures are investigated respectively. The results show that, acting as a line defect, transverse dislocation structures can mold the flow of sound in the bandgap along the dislocation channel and form the so called waveguides; while longitudinal dislocation structures, which act just as point defects, can form a cavitylike void surrounded by the three nearest cylinders around the dislocation line to create a localized state. In addition, the number and the position of defect bands strongly depend on the transverse and longitudinal dislocation displacement, so we can artificially control the defect bands by adjusting the transverse and longitudinal dislocations of lattices in a homogeneity islocation structures.

Study of the band-structure and the uncoupled modes in two-dimensional phononic crystals with the multiple-scattering theory

The multi-scattering theory (MST) method is applied to study the band structure of the two-dimensional phononic band-gap material. A comprehensive study is performed for the coupling between various order components of the cylindrical scattered wave. It has been found that there are decoupling effects at the edge of the band gap. Therefore, a self-contained self-consistent multiple scattering model of the zeroth-order scattered wave is obtained at the edge of the first band gap. A Mie scattering factor and a lattice structure factor are introduced to give a physical insight into the variation of the gap’s starting frequency by graphic analytic method.

Application of genetic algorithm in optimization of band gap of two-dimensional phononic crystals

A genetic algorithm (GA) with adaptive crossover and mutation probability is developed to design a two-dimensional phononic crystal of a square lattice with maximal absolute band gap. The composites of Pb cylinders in epoxy matrix are studied with GA method connection with plane-wave expansion approach. We find that GA method is effective to find out excellent topology of unit cell with a widest absolute band gap.

VIBRATION PROPERTY OF TWO DIMENSION PHONONIC CRYSTALS THIN PLATE

Based on the vibrational equation of the thin plate, the band structure for the two-dimensional phononic crystals thin plate composed of two kind of material is provided by using plane wave expansion method. The composite materials thin plate exhibiting vibration band gap is found. But it is different from the xy mode band structure of two-dimensional phononic crystals. For the finite periodic structure thin plate, the vibration frequency response by finite element method is simulated. The frequency ranges of sharp drop in frequency response curve show good agreement with the band gaps. The wave field distribution of the frequency both in band gap and out of band gap is calculated in order to describe the vibration property of the phononic crystal thin plate. The existence of band gaps in the periodic structures thin plate gives a new idea in vibration control application of phononic crystals.

Three-dimensional phononic band gap calculations using the FDTD method and a PC cluster system

This paper aims at studying the band gap phenomena of three-dimensional phononic crystals using the finite difference time domain (FDTD) method and a PC cluster system. In the paper, Bloch's theorem is applied to the wave equation and to the boundary conditions of the periodic structure. We calculate the variations of displacements and take discrete Fourier transform to acquire the resonances of the structures. Then, the dispersion relations of the bulk acoustic wave can be obtained and the band gaps are predicted accordingly. On the other hand, because of larger data calculation in three-dimensional phononic crystals, we introduce the PC cluster system and parallel FDTD programs written with respect to the architecture of a PC cluster system. Finally, we discuss the numerical calculation of two-dimensional and three-dimensional phononic crystals consisting of steel/epoxy and draw conclusions regarding the band gap phenomena between these phononic crystals.

Phononic Band Gap Width Control through Structural and Material Parameters in Two-Dimensional Phononic Crystals

The phononic band structures of two-dimensional solid phononic crystals with different lattice and scatterer symmetry are studied numerically, with three types of lattice (square, triangular, and rectangular) and four different scatterer shapes (circle, hexagon, square, and rectangle) considered. XY and Z vibration modes are investigated separately. Two types of phononic crystal are considered: one composed of high-density rods embedded in a low-density matrix, the other of low-density rods in a high-density matrix. In the former case, lattice type and polarization being fixed, the broadest gaps are obtained when the symmetry of the rods corresponds to that of the lattice (the shape of a rod is identical with that of the first Brillouin zone); the largest gap width values are observed in triangular lattice-based crystals (compared to those based on the square and rectangular lattices), the shape of the corresponding first Brillouin zone being closest to a circle. These rules do not apply to structures in which the density of the rod material is lower than that of the matrix. In this case, when the symmetry of the rods corresponds to that of the lattice, gaps either fail to appear at all, or are much narrower than in other configurations. The effect of other material parameter values (such as the longitudinal and transversal velocity values) on the relation between the energy gap width and the scatterer symmetry is found to be much lesser.

Erratum: Two-dimensional phononic crystals studied using a variational method: Application to lattices of locally resonant materials [Phys. Rev. B67, 144301 (2003)

Erratum: Two-dimensional phononic crystals studied using a variational method: Application to lattices of locally resonant materials [Phys. Rev. B<b>67</b>, 144301 (2003)

Mode-selecting acoustic filter by using resonant tunneling of two-dimensional double phononic crystals

In this letter, we investigated the resonant tunneling of elastic waves through double phononic potential barriers formed by two slabs of two-dimensional phononic crystals consisting of pure solid components. It is found that the resonant tunneling longitudinal waves can be distinguished easily from the resonant tunneling transverse waves. Thus, such double-barrier structures can be served as a mode-selecting acoustic filter, used to pick out the single longitudinal wave component or transverse wave component at a certain frequency.

Homogenization of Two-Dimensional Phononic Crystals at LowFrequencies

Effective velocities of elastic waves propagating intwo-dimensional phononic crystal at low frequencies are analysedtheoretically, and exact analytical formulas for effective velocitiesof elastic waves are derived according to the method presented byKrokhin et al. [Phys. Rev. Lett. 91 (2003) 264302]. Numericalcalculations for phononic crystals consisted of array of Pb cylindersembedded in epoxy show that the composites have distinct anisotropy atlow filling fraction. The anisotropy increases as the filling fractionincreases, while as the filling fraction closes to the limitation, theanisotropy decreases.