Abstract
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.
Highlights
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
One wishes to reflect light of any polarization at any angle and for Dirichlet media such a gap occurs at zero frequency
Classical homogenization is constrained to low frequencies and long waves in moderate contrast PCs (Silveirinha and Fernandes 2005) and so-called high-contrast homogenization only captures the essence of stop bands in PCs when the permittivity inside the inclusions is much higher than that of the surrounding matrix
Summary
A two-dimensional structure composed of a doubly periodic, i.e. periodic in both x and y directions, square array of cells with, not necessarily circular, identical holes inside them is considered (see figure 3). Due to periodic (or anti-periodic) boundary conditions of u and its first derivatives with respect to ξi on ∂ S1, and due to the homogeneous Dirichlet boundary conditions of the ui ’s on ∂ S2, the left-hand side of equation (13) vanishes. It follows that 1 = 0, which is an important deduction as it implies that the asymptotic behaviour of dispersion curves near isolated eigenfrequencies is at least quadratic. One primary aim of the present paper is to deduce formulae for the local behaviour of dispersion curves near standing wave frequencies as these shed light upon the physical effects observed. These locally quadratic dispersion curves are completely described by 0 and the tensor Ti j
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