Volume 220, Issue 9-10 - Phononic Crystals – Sonic Band-Gap Materials

In the study of classical waves, i.e., electromagnetic and elastic waves, wave localization and spectral gap materials are two of the most important developments over the past two decades. While the main subject matter of this volume is on spectral gap materials, it is the purpose of this article to attempt a qualitative overview of the two subjects, in terms of the underlying physics.

We present a review, through selected illustrative examples, of the physics of classical vibrational modes in phononic lattices, which elaborates on the theory, the formalism, the methods, and mainly on the numerical and experimental results related to phononic crystals. Most of the topics addressed here, are written in a self-consistent way and they can be read as independent individual parts.

We have examined the propagation of surface acoustic waves across structures with periodically varying elastic properties. These so-called phononic lattices include a) alternating layers of aluminum and polymer in a multilayer structure, b) a 2-d hexagonal lattice of holes drilled in an aluminum substrate and filled with polymer, and c) a 2-d hexagonal array of Al posts surrounded by polymer. A simple method is employed for probing the elastic properties of these periodic structures with ultrasound in the 0.5–5 MHz range. Ultrasonic surface waves are introduced and detected by water-immersion transducers, cylindrically focused to a line on the sample surface. Transmission and reflection patterns are observed by continuously scanning the wavevector angle with respect to the symmetry axes of the lattice. By Fourier transforming the transmitted signals using a broadband source, we obtain the frequency responses of the lattices as a function of propagation angle. Phononic band gaps, waveguide channeling and Scholte-like interface modes are observed. The results are compared to models of the propagation of transverse waves through analogous bulk structures, for which theories are available. These preliminary models explain the basic features of the experimental results, such as frequency gaps and wave channeling. A number of observations, however, remain to be explained. The angle-scanning technique provides a powerful probe of these modulated structures.

Experimental measurements, using an impulse response technique, are reported on a one-dimensional acoustic band gap system composed of a waveguide with a series of regularly spaced resonant structures. The amplitude data of the experimental results demonstrate the frequency and extent of the forbidden transmission bands and the phase information is analysed to determine the acoustic dispersion — the band structure — in the one-dimensional array. The results exhibit generally good agreement with previous theoretical analysis both in terms of the location of the forbidden transmission frequencies and in the form of the band structure. This simple system is a good candidate for the exploration of a number of postulated acoustic band gap effects.

Transmission of acoustic waves through a two-dimensional composite material made of PVC cylinders surrounded by air is measured experimentally. The spectrum presents a very large absolute band gap in the audible frequency range. A waveguide created inside this phononic crystal by removing a row of cylinders can transmit very efficiently the waves falling inside the stop band. We show the existence of deaf modes in the band structure of the linear waveguide. Resonant filtering is also demonstrated experimentally by coupling the waveguide to a side branch resonator of variable length. Frequency filtering is observed in the form of narrow dips in the transmission spectrum of the waveguide. Most of these observations compare favorably with theoretical calculations of dispersion curves and transmission coefficients of model structures using the plane wave expansion and the finite difference time domain methods. Narrow dips similar to those of the guide with resonator are also observed in the transmission spectrum of a waveguide with a sharp bend.

We present a combined experimental and theoretical study of the guiding, bending and filtering of acoustic waves in an ultrasonic crystal. The crystal consists of a two-dimensional periodical array of steel rods immersed in water, for wich a complete acoustic band gap extending from 240 to 325 kHz is found experimentally. Waveguides for acoustic waves are further created by removing a line defect, on which stubs can be added by removing rods from the side-walls of the waveguide. Full transmission is observed for a one-period-wide straight waveguide within the full-band-gap of the perfect phononic crystal, i.e. for a waveguide aperture smaller than one acoustic wavelength. Waveguiding over a wide frequency range is also obtained for a one-period-wide waveguide with two sharp 90° bends. Finite-difference time-domain computations are found to be in good agreement with the measurements in all experimental configurations.

In this paper, we investigate the band gaps of micro-machined air/silicon phononic structures both theoretically and experimentally. Based on the plane wave expansion method, dispersion relations of the surface and bulk modes with square lattices in air/silicon two-dimensional phononic structures are calculated and discussed. Band gap widths due to the filling fraction and temperature variation are also analyzed. On the experimental side, generation and reception of high frequency surface acoustic wave in this band structure are realized by a pair of interdigital transducers (IDT) with frequency around 200 MHz. Details of the fabricating process of the phononic structure and the high frequency surface wave generating and receiving IDTs are given. The results demonstrate clearly the existence of SAW band gap in the micro-machined phononic structure and this study may serve as a basis for studying the band gap of SAW in micro-machined phononic structure with the dimension in the order of micrometer and find applications in the RF SAW devices.

After a brief description of the layer multiple scattering method as applied to phononic crystals, we present some results obtained by this method, relating to: crystals of polystyrene spheres in water; crystals of silica spheres in air; and crystals of steel spheres in polyester. We relate the transmission characteristics of slabs of these ma terials to the complex band structure of the corresponding infinite crystals. We emphasize aspects of the underlying physics which have not been discussed previously.

Tunneling and dispersion of ultrasonic pulses is investigated in 3D phononic crystals consisting of 0.8 mm-diameter tungsten carbide beads that are close packed in a fcc crystal array embedded in either water or epoxy. Pulsed ultrasonic techniques allow us to measure the phase velocity and group velocity, i.e. the dynamics of wave propagation, as well as the transmission coefficient. Our experimental data are well interpreted using multiple scattering theory (MST). In the tungsten carbide/water crystals, dispersion phenomena were studied at frequencies in and around the gap in the ΓL direction. A strong suppression of the group velocity, and large variations of the group velocity dispersion (GVD) were found at frequencies around the band edges. By contrast, fast group velocities and nearly constant GVD with values around zero were observed at gap frequencies, indicating that tunneling in phononic crystals is essentially dispersionless. In the tungsten carbide/epoxy crystals a wide gap ( to our knowledge, largest measured 3D band gap ) was measured covering a frequency range from 1.2 MHz to 4.3 MHz along the ΓL crystal direction. The agreement between the theory and experiments gives strong evidence of the existence of a large complete gap (1.5 MHz to 3.9 MHz), which is theoretically predicted from the band structure calculations.

Using multiple-scattering theory, we studied the transmission properties of a slab of composite material that have sonic band gaps due to local resonances. Thin slabs of such material have transmission properties that are apparently different from conventional band gap material. For example, there can be transmission peaks in the frequency range inside the bulk sonic band gap. If the slab thickness is changed, we found that the top of band gap shifts while the bottom of band gap, being pinned by the resonance frequency, does not. By changing the slab thickness, the “effective band gap” may be narrowed or broadened, depending on the filling fraction of the locally resonant units. In order to provide an intuitive understanding of the phenomena, we constructed a simple model to understand the phenomena by comparing its transmission and band structure with that of the locally resonant sonic materials.

We find that a phononic gap in a periodic or quasiperiodic elastic composite can be significantly enhanced by inserting air inclusions into the systems. The positions of the insertion are chosen to suppress the shear potential energy of the acoustical branches and lower their frequencies. This is demonstrated in two dimensions. Gap positions and sizes as functions of the radii of the air cylinders for systems of aluminum cylinders in epoxy and steel cylinders in epoxy are presented for both triangular and 12-fold quasiperiodic lattices.

This work presents a finite-difference scheme to study the scattering of sound in the time-domain. It can be applied to heterogeneous fluids in any dimensionality. The scheme is based on the split-operator technique introduced in quantum mechanics to solve the time-dependent Schrödinger equation. This scheme presents advantages in comparison with the usual finite-difference time-domain algorithms. Its main characteristics are: a) its precision is of the order (Δ t ) 3 ; b) it can work with non-homogeneous space-discretizations; and c) it satisfies the energy conservation law. Here, the scheme is also applied to solve four simple tests cases in one-dimensional space.

Band gaps, i.e frequency ranges for which waves cannot propagate, can be found in most elastic structures if the material or structure has a specific periodic modulation of material properties. In this paper, we maximize phononic band gaps for infinite periodic beams modelled by Timoshenko beam theory, for infinite periodic, thick and moderately thick plates, and for finite thick plates. Parallels are drawn between the different optimized crystals and structures and several new designs obtained using the topology optimization method.

We consider in-plane elastic waves propagating through a doubly periodic array of cylinders of Tantalum (with both circular and elliptical cross-sections) which are embedded in a matrix of fused silica. We find some sonic gap for fairly small filling fractions of the cylinders which eventually vanish in the limit of high-filling fraction. In the case of a doubly periodic array of elliptical cylinders, removal of a cylinder within a macro-cell leads to two localised eigenstates.