META Conference, META'12

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Dispersion in media containing resonant inclusions: where does it come from?
Geoffroy Lerosey, Fabrice Lemoult, Mathias Fink

Last modified: 2012-01-02


In this talk, we use a very simplified model in order to grasp the physics of an array of resonators in a homogeneous medium. We study a quasi one dimensional system, that is, a waveguide filled with a linear array of resonators. We first prove that, to a large extent, the response of such 1D medium is governed by a far field coupling between the individual elements. This coupling can be understood as Fano interferences between the incoming plane waves and the field reemitted by the resonators. We give a phenomenological description of this effect in terms of the frequency response of an oscillator, and confront it to the typical results obtained from near field coupling analysed in the tight binding approach. We develop a simple formalism based on the transmission matrix of the system that permits us to obtain the dispersion relation of quasi 1D metamaterials using solely the far field transmission coefficient of a single unit cell and the period of the medium. This approach is valid for finite size media, does not rely on any assumption regarding the scale of the period with regards to the wavelength, and give a clear physical picture of both the hybridization effects observed in acoustics, the effective properties of many metamaterials and also the surface waves supported by structured metals. We verify our approach and the obtained formalism on various designs. Finally we prove that it also gives the shape of the hybridization band gaps, or equivalently of the negative effective properties of the media, and discuss their relation to the period of the medium and the response of the considered resonator.


metamaterials, general theory, hybridization band gaps, phononic crystals