Abstract
We investigate modal conversion at the boundary between a homogeneous incident medium and a phononic crystal, with consideration of the impact of symmetry on the excitation of Bloch waves. We give a quantitative criterion for the appearance of deaf Bloch waves, which are antisymmetric with respect to a symmetry axis of the phononic crystal, in the frame of generalized Fresnel formulas for reflection and transmission at the phononic crystal boundary. This criterion is used to index Bloch waves in the complex band structure of the phononic crystal, for directions of incidence along a symmetry axis. We argue that within deaf frequency ranges transmission is multi-exponential, as it is within frequency band gaps.
Highlights
As periodic composite media, phononic crystals (PC) support elastic or acoustic waves whose properties are prescribed through the Bloch-Floquet theorem.[1,2] The eigenmodes of the Helmholtz equation for monochromatic wave propagation are called Bloch waves and are each associated with a (ω, k) pair, with ω the angular frequency and k the Bloch wavevector
We have investigated modal conversion at the boundary between a homogeneous incident medium and a phononic crystal
Bloch waves separate between symmetric and antisymmetric provided the wavevector varies along a symmetry axis of the phononic crystal
Summary
Phononic crystals (PC) support elastic or acoustic waves whose properties are prescribed through the Bloch-Floquet theorem.[1,2] The eigenmodes of the Helmholtz equation for monochromatic wave propagation are called Bloch waves and are each associated with a (ω, k) pair, with ω the angular frequency and k the Bloch wavevector. Deaf Bloch waves were originally observed in two-dimensional (2D) PC of rigid rods in air arranged according to a square lattice.[9,10,11,12] They were soon to be found in three-dimensional (3D) PC of solid spheres in a solid[13,14] or a fluid matrix,[15] and in 2D PC of steel rods in water with hexagonal-type lattices.[16] Recently, they have been observed in phononic crystal slabs.[17,18,19] Deaf Bloch waves manifest themselves as dips in the transmission, much as band gaps do, and lead to discrepancies between the theoretical band structure and the experimental result This discrepancy is soon explained by observing that a plane wave incident normally on the PC is symmetric with respect to the propagation direction; if a particular Bloch wave is antisymmetric with respect to the same axis, it cannot be excited and it cannot contribute to transmission through the PC. This procedure permits the separation of band foldings occurring at symmetry points of the Brillouin zone from bands becoming propagating above a given cut-off frequency
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