Omnidirectional elastic band gap in finite lamellar structures

This paper presents a comprehensive theoretical analysis of the occurrence of omnidirectional reflection in one-dimensional phononic crystal structures. We discuss the conditions for a one-dimensional layered structure, made of elastic materials, to exhibit total reflection of acoustic incident waves in a given frequency range, for all incident angles and all polarizations. The property of omnidirectional reflection can be fulfilled with a simple finite superlattice if the substrate from which the incident waves are launched is made of a material with high acoustic velocities (this is very similar to the case of omnidirectional optical mirror where the incident light is generated in vacuum). However, if the substrate is made of a material with low acoustic velocities, we propose two solutions to obtain an omnidirectional band gap, namely, the cladding of a superlattice with a layer of high acoustic velocities, which acts like a barrier for the propagation of phonons, or the association in tandem of two different superlattices in such a way that the superposition of their band structures exhibits an absolute acoustic band gap. We discuss the appropriate choices of the material and geometrical properties to realize such structures. The behavior of the transmission coefficients are discussed in relation with the dispersion curves of the finite structure embedded between two substrates. Both transmission coefficients and densities of states (from which we derive the dispersion curves) are calculated in the framework of a Green's function method.