High-frequency homogenization of zero-frequency stop band photonic and phononic crystals

We present an accurate methodology for representing the physics of waves, in periodic structures, through effective properties for a replacement bulk medium: this is valid even for media with zero-frequency stop bands and where high-frequency phenomena dominate. Since the work of Lord Rayleigh in 1892, low-frequency (or quasi-static) behaviour has been neatly encapsulated in effective anisotropic media; the various parameters come from asymptotic analysis relying upon the ratio of the array pitch to the wavelength being sufficiently small. However, such classical homogenization theories break down in the high-frequency or stop band regime whereby the wavelength to pitch ratio is of order one. Furthermore, arrays of inclusions with Dirichlet data lead to a zero-frequency stop band, with the salient consequence that classical homogenization is invalid. Higher-frequency phenomena are of significant importance in photonics (transverse magnetic waves propagating in infinite conducting parallel fibres), phononics (anti-plane shear waves propagating in isotropic elastic materials with inclusions) and platonics (flexural waves propagating in thin-elastic plates with holes). Fortunately, the recently proposed high-frequency homogenization (HFH) theory is only constrained by the knowledge of standing waves in order to asymptotically reconstruct dispersion curves and associated Floquet-Bloch eigenfields: it is capable of accurately representing zero-frequency stop band structures. The homogenized equations are partial differential equations with a dispersive anisotropic homogenized tensor that characterizes the effective medium. We apply HFH to metamaterials, exploiting the subtle features of Bloch dispersion curves such as Dirac-like cones, as well as zero and negative group velocity near stop bands in order to achieve exciting physical phenomena such as cloaking, lensing and endoscope effects. These are simulated numerically using finite elements and compared to predictions from HFH. An extension of HFH to periodic supercells enabling complete reconstruction of dispersion curves through an unfolding technique is also introduced.