Multiple Scattering Formulation of Two Dimensional Acoustic and Electromagnetic Metamaterials

This work presents a multiple scattering formulation of two dimensional acoustic metamaterials. It is shown that in the low frequency limit multiple scattering allows us to define frequency-dependent effective acoustic parameters for arrays of both ordered and disordered cylinders. This formulation can lead to both positive and negative acoustic parameters, where the acoustic parameters are the scalar bulk modulus and the tensorial mass density and, therefore, anisotropic wave propagation is allowed with both positive or negative refraction index. It is also shown that the surface fields on the scatterer are the main responsible of the anomalous behavior of the effective medium, therefore complex scatterers can be used to engineer the frequency response of the effective medium, and some examples of application to different scatterers are given. Finally, the theory is extended to electromagnetic wave propagation, where Mie resonances are found to be the responsible of the metamaterial behavior. As an application, it is shown that it is possible to obtain metamaterials with negative permeability and permittivity tensors by arrays of all-dielectric cylinders and that anisotropic cylinders can tune the frequency response of these resonances

The Inverse Grating Problem: Efficient Design of Anomalous Flexural Wave Reflectors and Refractors

We present an extensive formulation of the inverse grating problem for exural waves, in which the energy of each diffracted mode is selected and the grating configuration is then obtained by solving a linear system of equations. The grating is designed as a lineal periodic repetition of a unit cell comprising a cluster of resonators attached at points whose physical properties are directly derived by inversion of a given matrix. Although both active and passive attachments can be required in the most general case, it is possible to find configurations with only passive, i.e. damped, solutions. This inverse design approach presents an alternative to the design of metasurfaces for exural waves overcoming the limitations of gradient phase metasurfaces, which require a continuous variation of the surface's impedance. When the grating is designed in such a way that all the energy is channeled to a single diffracted mode, it behaves as an anomalous refractor or re ector. The negative refractor is analyzed in depth, and it is shown that with only three scatterers per unit cell is it possible to build such a device with unitary efficiency

Acoustic anomalous reflectors based on diffraction grating engineering

We present an efficient method for the design of anomalous reflectors for acoustic waves. The approach is based on the fact that the anomalous reflector is actually a diffraction grating in which the amplitude of all the modes is negligible except for the one traveling towards the desired direction. A supercell of drilled cavities in an acoustically rigid surface is proposed as the basic unit cell, and analytical expressions for an inverse diffraction problem are derived. It is found that the the number of cavities required for the realization of an anomalous reflector is equal to the number of diffracted modes to cancel, and this number depends on the relationship between the incident and reflected angles. Then, the “retroreflection” effect is obtained by just one cavity per unit cell; also, with only two cavities it is possible to change the reflection angle of a normally incident wave, and five cavities are enough to design a general retroreflector changing the incident and reflected angles at oblique incidence. Finally, the concept of Snell’s law violation is extended not only to the incident and reflected angles, but also to the plane in which it happens, and a device based on a single cavity in a square lattice is designed in such a way that the reflection plane is rotated π/4 with respect to the plane of incidence. Numerical simulations are performed to support the predictions of the analytical expressions, and an excellent agreement is found

Strong spatial dispersion in time-modulated dielectric media

We present an effective medium description of time-modulated dielectric media. By taking the averaged fields over one modulation period, the relationship between them is derived, therefore defining the different constitutive parameters. In the most general situation, it is found that the effective material is described by means of a spatially and temporally dispersive transverse dielectric function and a constant longitudinal dielectric function. It has been also found that the frequency dependence in the former is weak, in comparison with its wavenumber dependence (spatial dispersion). Different physical consequences of this spatial dispersion are discussed, with special emphasis on the weak dispersion approximation and the limit in which it is found that the effective material behaves as a resonant and isotropic magnetodielectric medium with no additional longitudinal mode, as it is commonly found in spatially dispersive materials. Time-dependent media therefore opens an alternative way of designing dynamically tunable metamaterial

Loss Compensation in Time-Dependent Elastic Metamaterials

Materials with properties that are modulated in time are known to display wave phenomena showing energy increasing with time, with the rate mediated by the modulation. Until now there has been no accounting for material dissipation, which clearly counteracts energy growth. This paper provides an exact expression for the amplitude of elastic or acoustic waves propagating in lossy materials with properties that are periodically modulated in time. It is found that these materials can support a special propagation regime in which waves travel at constant amplitude, with temporal modulation compensating for the normal energy dissipation. We derive a general condition under which amplification due to time-dependent properties offsets the material dissipation. This identity relates band-gap properties associated with the temporal modulation and the average of the viscosity coefficient, thereby providing a simple recipe for the design of loss-compensated mechanical metamaterials