Two-dimensional Phononic Crystals Research Articles

Elastic wave localization in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity

The band structures of in-plane elastic waves propagating in two-dimensional phononic crystals with one-dimensional random disorder and aperiodicity are analyzed in this paper. The localization of wave propagation is discussed by introducing the concept of the localization factor, which is calculated by the plane-wave-based transfer-matrix method. By treating the random disorder and aperiodicity as the deviation from the periodicity in a special way, three kinds of aperiodic phononic crystals that have normally distributed random disorder, Thue–Morse and Rudin–Shapiro sequence in one direction and translational symmetry in the other direction are considered and the band structures are characterized using localization factors. Besides, as a special case, we analyze the band gap properties of a periodic planar layered composite containing a periodic array of square inclusions. The transmission coefficients based on eigen-mode matching theory are also calculated and the results show the same behaviors as the localization factor does. In the case of random disorders, the localization degree of the normally distributed random disorder is larger than that of the uniformly distributed random disorder although the eigenstates are both localized no matter what types of random disorders, whereas, for the case of Thue–Morse and Rudin–Shapiro structures, the band structures of Thue–Morse sequence exhibit similarities with the quasi-periodic (Fibonacci) sequence not present in the results of the Rudin–Shapiro sequence.

Influence of defect states on band gaps of the 4340 steel in epoxy in two-dimensional phononic crystal

The band gaps of a two-dimensional phononic crystal are studied by using the plane-wave expansion method and the supercell calculations. The two-dimensional phononic crystal is formed by square-shape arrangement of 4340 steel inserted into epoxy resin. The band gaps of different structures are calculated such as defect-free, 45 crystal defect states, 90 crystal defect states, 135 crystal defect states and 180 crystal defect states. It is found that the bandwidth of defect state is about 31 times larger than that of the defect-free crystal; with F=0.10.9 the band gaps occur in different defect structures and increase with defect state angle increasing. In addition, the influences of filling fraction on the band gap number and the relative width of the minimum band gap are also discussed.

Analysis of elastic waves transmitted through a 2-D phononic crystal exhibiting negative refraction

A two-dimensional phononic crystal (PC) made of a square lattice of air holes in an aluminum matrix is studied. The band structure calculated in the irreducible Brillouin zone of the PC exhibits a branch with a negative slope that allows negative refraction. This phenomenon has been numerically verified using a prism-shaped PC for plane waves entering the PC with two different incidences. A detailed study of the waves at the exit of the PC shows that the plane wave is reconstructed after several wavelengths. Finally, the description of the refracted waves is interpreted using a point source array, giving information about the angular spreading and the relative amplitude of each refracted beam.

1307 フォノニック結晶による超音波の音響レンズ効果とエネルギー輸送効率(OS13.フォトニック・フォノニック構造の設計とシミュレーション(2),OS・一般セッション講演)

Lens effect of acoustic waves in a two-dimensional (2D) phononic crystal is studied by numerical simulation based on the finite-difference time-domain (FDTD) method. We calculate the phonon band structure of 2D phononic crystals, consisting of metal cylinders placed periodically in water. Lens effect is observed for far-field wave in the phononic crystal with graded-index structure. These results indicate that a novel energy-transmission device can be designed using phononic crystals.

Band structure in two-dimensional fiber–air phononic crystals

A two-dimensional phononic crystal (PC) composed of textile fiber and air is initially discussed in this paper, which is different from the previous PCs with rigid inclusions. The plain wave expansion method is used to calculate band structure of different PCs by altering fiber material properties and structure parameters. Numerical results show that the effect of material properties of soft fiber on band structure of phononic crystal can be ignored, while the effect of structural parameters is obvious.

One-way mode transmission in one-dimensional phononic crystal plates

We investigate theoretically the band structures of one-dimensional phononic crystal (PC) plates with both antisymmetric and symmetric structures, and show how unidirectional transmission behavior can be obtained for either antisymmetric waves (A modes) or symmetric waves (S modes) by exploiting mode conversion and selection in the linear plate systems. The theoretical approach is illustrated for one PC plate example where unidirectional transmission behavior is obtained in certain frequency bands. Employing harmonic frequency analysis, we numerically demonstrate the one-way mode transmission for the PC plate with finite superlattice by calculating the steady-state displacement fields under A modes source (or S modes source) in forward and backward direction, respectively. The results show that the incident waves from A modes source (or S modes source) are transformed into S modes waves (or A modes waves) after passing through the superlattice in the forward direction and the Lamb wave rejections in the backward direction are striking with a power extinction ratio of more than 1000. The present structure can be easily extended to two-dimensional PC plate and efficiently encourage practical studies of experimental realization which is believed to have much significance for one-way Lamb wave mode transmission.

Research on the Parameters of Phononic Crystals Suitable for Automotive Noise Control

Phononic crystal is a novel type of periodic composites which is presented near years. When the elastic wave frequency falls within its band gap of the phononic crystal, the propagation of vibration and sound are forbidden. This property provides a new research method for the control of low-frequency interior noise. MSC. Nastran simulation method is applied to compare the propagation of two-dimensional Bragg phononic crystal and the locally resonant phononic crystal. The impact of the parameters on the propagation is analyzed by using MSC.Nastran simulation method in order to fabricate the photonic crystal structure suitable for the automotive interior noise control.

Band structure of phononic crystals with general damping

In this paper, we present theoretical formalisms for the study of wave dispersion in damped elastic periodic materials. We adopt the well known structural dynamics techniques of modal analysis and state-space transformation and formulate them for the Bloch wave propagation problem. First, we consider a one-dimensional lumped parameter model of a phononic crystal consisting of two masses in the unit cell whereby the masses are connected by springs and dashpot viscous dampers. We then extend our analysis to the study of a two-dimensional phononic crystal, modeled as a dissipative elastic continuum, and consisting of a periodic arrangement of square inclusions distributed in a matrix base material. For our damping model, we consider both proportional damping and general damping. Our results demonstrate the effects of damping on the frequency band structure for various types and levels of damping. In particular, we reveal two intriguing phenomena: branch overtaking and branch cut-off. The former may result in an abrupt drop in the relative band gap size, and the latter implies an opening of full or partial wavenumber (wave vector) band gaps. Following our frequency band structure analysis, we illustrate the concept of a damping ratio band structure.

Negative refraction of zero order flexural Lamb waves through a two-dimensional phononic crystal

We investigate experimentally the refraction of antisymmetric Lamb waves at frequencies in the second band of a two-dimensional phononic crystal. The heterostructure is made of a square lattice of air inclusions in a silicon plate. Our experimental scheme is based on the selective generation of narrow band elastic waves at a few megahertz. Both positive and negative refractions are observed. The comparison of our data with computed dispersion curves shows very good agreement.

Negative refraction of surface acoustic waves in the subgigahertz range

We used the picosecond ultrasonic technique to experimentally demonstrate the negative refraction of surface acoustic waves in a two-dimensional phononic crystal. The sample is made of a square lattice of circular voids drilled at the surface of a thick silica substrate. The lattice parameter is of a few micrometers. Broad band surface acoustic waves with Fourier components in the gigahertz range were excited. The negative refraction is observed for Rayleigh waves with frequencies in the second band and propagating along the direction $\ensuremath{\Gamma}\text{M}$ in the reduced Brillouin zone. We then used a plane-wave expansion method to calculate the dispersion of the waves in our sample and to analyze our experimental data.

Symmetric and Anti-Symmetric Lamb Waves in a Two-Dimensional Phononic Crystal Plate

It is well known that Lamb waves in a plate with a mirror plane can be separated into two uncoupled sets: symmetric and anti-symmetric modes. Based on this property, we present a revised plane wave expansion method (PWE) to calculate the band structure of a phononic crystal (PC) plate with a mirror plane. The developed PWE method can be used to calculate the band structure of symmetric and anti-symmetric modes separately, by which the depending relationship between the partial acoustic band gap (PABG), which belongs to the symmetric and anti-symmetric modes alternatively, and the position of the scatterers can be determined. As an example of its application, the band structure of the Lamb modes in a two-dimensional PC plate with two layers of void circular inclusions is investigated. The results show that the band structure for the symmetric and anti-symmetric modes can be changed by the position of the scatterers drastically, and larger PABGs will be opened when the scatterers are inserted into the area of the plate, where the elastic potential energy is concentrated.

Locally resonant surface acoustic wave band gaps in a two-dimensional phononic crystal of pillars on a surface

We investigate theoretically the propagation of acoustic waves in a two-dimensional array of cylindrical pillars on the surface of a semi-infinite substrate. Through the computation of the band structure of the periodic array and of the transmission of waves through a finite length array, we show that the phononic crystal can support a number of surface propagating modes in the nonradiative region of the substrate, or sound cone, as limited by the slowest bulk acoustic wave. The modal shape and the polarization of these guided modes are more complex than those of classical surface waves propagating on a homogeneous surface. Significantly, an in-plane polarized wave and a transverse wave with sagittal polarization appear that are not supported by the free surface. In the band structure, guided modes define band gaps that appear at frequencies markedly lower than those expected from the Bragg interference condition. We identify them as originating from local resonances of the individual cylindrical pillars and show their dependence on the geometrical parameters, in particular with the height of the pillars. The transmission of surface acoustic waves across a finite array of pillars shows the signature of the locally resonant band gaps for surface modes and their dependence on the symmetry of the source and its polarization. Numerical simulations are performed by using the finite element method and considering silicon pillars on a silicon substrate.

Breaking bands degeneracy in two-dimensional hybrid triangular lattice phononic crystal: A theoretical and experimental study

Using symmetry reduction concept, the band gap properties of a two-dimensional hybrid triangular lattice phononic crystal were investigated both theoretically and experimentally. The band structure was calculated with the plane wave expansion (PWE) method. The results revealed that the band degeneracy at high symmetry point in the first Brillouin zone was lifted due to reducing symmetry, and hence a new low-frequency absolute band gap between the first and the second bands appeared. The measured transmission spectra were found to be in good agreement with the numerical results.

A postprocessing method based on chirp Z transform for FDTD calculation of point defect states in two-dimensional phononic crystals

In this paper, a new postprocessing method for the finite difference time domain (FDTD) calculation of the point defect states in two-dimensional (2D) phononic crystals (PNCs) is developed based on the chirp Z transform (CZT), one of the frequency zooming techniques. The numerical results for the defect states in 2D solid/liquid PNCs with single or double point defects show that compared with the fast Fourier transform (FFT)-based postprocessing method, the method can improve the estimation accuracy of the eigenfrequencies of the point defect states significantly when the FDTD calculation is run with relatively few iterations; and furthermore it can yield the point defect bands without calculating all eigenfrequencies outside the band gaps. The efficiency and accuracy of the FDTD method can be improved significantly with this new postprocessing method.

Collimation and enhancement of elastic transverse waves in two-dimensional solid phononic crystals

In this Letter, we numerically investigate the propagation characteristics of elastic transverse waves emitted by line sources embedded inside two-dimensional (2D) solid phononic crystals (PCs). The results show that collimation and enhancement of elastic transverse waves can be achieved at the band edge frequencies. We find that the collimation effect originates from the flat equifrequency contours (EFCs) at the band edge of appropriately designed 2D solid PCs. It is shown that, in addition to geometric symmetry, appropriate constituent material combination is essential to obtain flat EFCs at the band edge. A highly directional and enhanced elastic transverse wave source with a half power angular width of only 5.6° and an enhancement factor of 530 is realized simply by utilizing a finite-size 2D solid PC structure.

Negative group velocity from resonances in two-dimensional phononic crystals

We investigate two routes to obtain negative group velocity bands in two-dimensional phononic crystal structures. The negative dispersion originates from the resonances of sub-wavelength building blocks and as such, the system should be regarded as acoustic metamaterials. The first kind of acoustic metamaterial exhibits effectively negative bulk modulus and negative mass density simultaneously. Monopolar and dipolar Mie resonances are combined to achieve an effective medium with negative refractive index. In particular, we present a double negative metamaterial for airborne sonic waves. We then show that we can obtain negative group velocity from quadrupole resonances, and the result is explained using the quasi-static approximation. The negative dispersion in quadrupole bands cannot be described by standard effective medium theories even though the frequency can be very low.

Polarization state and level repulsion in two-dimensional phononic crystals and waveguides in the presence of material anisotropy

We investigate the polarization of Bloch waves in two-dimensional piezoelectric phononic crystals and phononic crystal waveguides managed therein. It is found that in addition to the strong coupling induced for waves polarized in the plane of the periodic structuration, a weaker but non-negligible coupling of polarization components originates from material anisotropy. Numerical illustrations are given for an array of air holes in lithium niobate arranged according to a square lattice. It is observed that when a band mostly polarized in-plane gets close to a band mostly polarized out-of-plane, a phenomenon of repelling can occur, that in some instances introduces a local band gap. This interaction is accompanied by a transfer of the polarization state from one band to the other.

Experimental investigation of shell modes in two-dimensional phononic crystal consisting of hollow cylinders

We experimentally demonstrate the flexural shell modes of thin-walled hollow steel cylinders immersed in water. For a two-dimensional square-arranged phononic crystal consisting of such hollow cylinders, two flat bands are created because of the coupling of the shell modes among neighboring cylinders. The narrow bands associated with shell-modes are first validated by experiment. In particular, by analyzing the transmission spectra, band-structure, and eigen-pressure field distribution, we give detailed discussion on the anisotropic coupling behavior of the two flat bands with external incident plane waves.

Experimental demonstration of the negative refraction of a transverse elastic wave in a two-dimensional solid phononic crystal

The negative refraction of transverse elastic waves is demonstrated experimentally in a two-dimensional phononic crystal (PC) made of a square lattice of cylindrical air cavities in an aluminum matrix. Dispersion curves of elastic waves in this PC exhibit a unique branch with phase and group velocities of opposite signs in a broad frequency range. Measurement of refraction angles through prismatic PC included in an aluminum block demonstrates negative refraction of elastic transverse wave.

Two-dimensional phononic crystals with time-varying properties: a multiple scatteringanalysis

Multiple scattering theory is a versatile two- and three-dimensional method for characterizingthe acoustic wave transmission through many scatterers. It provides analytical solutions towave propagation in scattering structures, and its computational complexity growslogarithmically with the number of scatterers. In this paper we show how the 2D method canbe adapted to include the effects of time-varying material parameters. Specifically, a newT-matrix is defined to include the effects of frequency modulation that occurs intime-varying phononic crystals. Solutions were verified against finite differencetime domain (FDTD) simulations and showed excellent agreement. This newmethod enables fast characterization of time-varying phononic crystals without theneed to resort to lengthy FDTD simulations. Also, the method of combiningT-matrices toform the T-supermatrix remains unchanged provided that the new matrix definitions are used. Themethod is quite compatible with existing implementations of multiple scattering theoryand could be readily extended to three-dimensional multiple scattering theory.