Homogenization of resonant metafilms

Abstract

In this talk, we will illustrate with three different examples the modeling of resonant metasurfaces using asymptotic homogenization techniques. We will first present the method on a periodic array of Helmholtz resonators (open cavities) in acoustics. In this case, the combination of bulk homogenization and interface homogenization allows to derive an approximated model in the time domain where the array of resonators is replaced by an anisotropic effective medium with effective jump conditions at boundaries. These conditions encapsulate all the evanescent contribution of the fields near the opening of the resonators and account for the transition between a quarter wavelength resonator to a Helmholtz resonator as the opening shrinks. In a second example we will focus on Mie resonance in elastodynamic for anti-plane shear waves. We will consider an array of sub-wavelength soft inclusions embedded in an elastic matrix. Due to the contrast of material properties between the constituents, the wavelength inside the soft inclusions become comparable to its size, hence leading to Mie resonances. We will show how asymptotic homogenization allows to capture the resonant behavior in an effective model consisting in frequency-dependent jump conditions across the array. Finally we will extend these results to a non-linear example in acoustic consisting in a screen of air bubbles inside water subjected to large pressure variations which is known to exhibit subwavelength Minnaert resonance. Homogenization theory applied to Navier-Stokes equations allows in this case to derive non-linear jump conditions where the behavior of the bubbles radii is governed by a non-linear equation of the Rayleigh-Plesset’s type.