Indeed, bands can have their iso-frequency contours convex near the M point of the Brillouin zone. Indeed, the band folding resulting from the periodicity can induce a negative slope above the first band exhibiting opposition between the group velocity and the phase velocity, which is an effect known as a backward wave .įinally, in the band diagram of some phononic crystal the negative refraction is not only achieved by employing a negative effective index or backward waves. In this case, the physical constants are locally positive everywhere inside the crystal and negative refraction is achieved because of crystal anisotropy. Another way of achieving the negative refraction is by using a periodic heterostructure where the effective index is controlled through the band structure. This implies that waves (acoustic or electromagnetic) propagating in such media will have their Poynting vector and wave vector pointing in opposite directions. One can obtain the negative refraction by using an effective medium showing a negative index related to the simultaneous negativity of permittivity and permeability for electromagnetic waves and the mass density and compressibility for acoustic waves . Furthermore, in the case of phononic crystals, Sukhovich and co-workers demonstrated a sub-wavelength imaging of acoustic waves using a structure consisting of triangular lattice steel cylinders in methanol and all surrounded by water .ĭifferent kinds of negative refraction phenomena have been reported for electromagnetic waves and acoustic waves . However, focusing a source through a photonic crystal leads to a diffraction-limited image . These features allow considerable control over wave propagation and open the door to new approaches for a variety of applications such as design of superlenses that are suitable to overcome the diffraction limit and therefore to focus a point source theoretically into a perfect image in the metamaterial regime. In particular, these structures exhibit, under specific geometrical and composition conditions, unusual acoustic wave phenomena such as the negative refraction whereby the wave arrows are bending in the wrong way, which corresponds to an inverted Snell's law . Beyond the existence of the acoustic or elastic band gaps, there has been a great deal of interest in studying the wave propagating properties in these artificial crystals. The main advantage is that the local resonance can be altered by using shape design as in the case of Helmholtz resonators and consequently tune the band gaps. In the case of surface acoustic waves, locally resonant band gaps for surface guided modes were demonstrated theoretically and observed experimentally in a square array of cylindrical pillars deposited on a semi-infinite substrate. The second mechanism is based on locally resonant elements with a lattice constant several orders smaller than the relevant wavelength . In addition, wave trapping and guiding or even demultiplexing through discrete defects have also been reported. Several authors reported band gaps for acoustic or elastic waves in two-dimensional (2D) and three-dimensional phononic crystals . The band gaps can first occur by using the Bragg interference of waves scattered by inclusions. In this case the transmission spectra exhibit a strong attenuation for which the acoustic or elastic waves are strictly prohibited in all propagating directions. The propagation of acoustic and elastic waves in periodic structures with spatially modulated elastic moduli and mass density possesses a number of important properties such as the occurrence of frequency band gaps . The numerical simulations are based on the efficient finite element method and analyze pillars and the substrate of lithium niobate. 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. A flat lens for surface acoustic waves has also been designed. 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. It occurs for the frequency range where the group velocity is never in opposite direction to the phase velocity. 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. 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.
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