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An introduction to phononic crystals (3)


What is a phononic crystal?

The phononic crystal concept owes much to crystallography. This branch of physics deals with the study of the most ordered of solid matter. The atoms in crystals are arranged following a periodical three dimensional arrays. Within a crystal, the inter atomic distances are typically of the order of the Angstrom (10-10 meters). The dimensions involved in phononic crystals, which are artificial handcrafted structures, are much larger. They range from a few meters down to a hundred of nanometers or less. At this scale, matter appears as continuous and the laws of classical mechanics apply and can be employed with trust, as a general rule. The idea behind the phononic crystal is to manufacture a periodically structured artificial material, for instance by assembling at least two different materials. Intuitively, the more the involved acoustic properties are contrasted, the more likely it is to observe phenomena linked to wave interference.

Square lattice

Square lattice

Triangular lattice

Triangular lattice

A phononic crystal is essentially made of inclusions arranged periodically in a propagating medium (or matrix). For a two-dimensional phononic crystal, the inclusions are cylinders that can be arranged as in a square or triangular lattice for example. They need to be composed of a material different from the one the matrix is made of, but holes drilled in this latter can work as well. The key requirement is that the elastic wave scattering on these inclusions is very efficient.

The concept of band gaps can be understood by considering the interference of waves multiply scattered within a phononic crystal. Let us first consider the case of an isolated scatterer: the fraction of the incident wave that impinges on the scatterer is dispersed throughout space but seems to originate from a unique source. When a set of scatterers is positioned periodically, waves are strongly dispersed from one obstacle to the other, and end up filling all available space and propagating in every possible direction. They interfere constructively or destructively depending on the wave frequency and on the phononic crystal geometry. A band gap appears when the scattered waves interfere destructively in a given direction, such that their superposition decreases exponentially when traversing the crystal.

Millimeter-sized phononic crystal

Millimeter-sized phononic crystal

The fundamental idea proposed by Kushwaha and colleagues at the University of Lille in 1993 is that the band gap can exist whatever the propagation direction, which is termed a complete band gap. A phononic crystal possessing a complete band gap would be a perfect mirror, reflecting back all incident waves. Indeed, the waves impinging on the phononic crystal could not possibly penetrate it. Following the same line of thought, an acoustic source or detector (what is typically called a transducer) completely surrounded by a thick phononic crystal would remain perfectly deaf to all external sources. Anyway, these extraordinary properties are only true for the frequencies that fall within the complete band gap. For other frequencies, destructive interferences are balanced by constructive ones and waves are transmitted at least partially.

3D phononic crystal

3D phononic crystal

A number of rules have been derived by researchers over the last fifteen years to specify the conditions under which band gap phenomena can be observed. As a general rule, they appear when the wave length (the period of spatial repetition of the wave) is of the order of the spatial period of the phononic crystal. The acoustic velocity contrast and the contrast of density between the scattering inclusions and the propagation medium are the parameters that mainly govern the width of band gaps. Inclusions with shapes that are beneficial for isotropic diffusion are preferred, for instance cylinders or spheres. It is in addition important to adjust the size of the inclusions with respect to the pitch of the array. Furthermore, all periodical arrays are not born equivalent. For instance, for two dimensional structures such as the Sempere's sculpture introduced above, the square lattice turns out to provide larger complete band gaps than the triangular lattice does. In the case of three dimensional arrays, piles of heavy spheres (made of steel or lead) embedded in a light matrix (air, water, or epoxy) and mimicking the structure of diamond (i.e., carbon atoms arranged according to a face centered cubic lattice) yield wide complete band gaps. However, the exploration of all possibilities offered by the choice of the periodical arrangement, of the materials composing the matrix and the scattering inclusions, and of the shape of these inclusions, is far from finished and will keep researchers active for yet a few years. Another essential point is that the principles of phononic crystals are expressed in the same manner whatever the scale chosen for their realization, though the operating frequencies change. This fundamental property has two important consequences. The first is that the concepts behind phononic crystals can be demonstrated to scale, that is to say via structures with rather large dimensions, hence easily accessed. The second consequence is that along with the development of micro and nano technologies, smaller and smaller phononic crystals can be manufactured using processes similar to those of microelectronics, employed for instance for the microprocessors in computers.