Metamaterial eigenmodes beyond homogenization

Abstract

Metamaterial homogenization theories usually start with crude approximations that are valid in certain limits in zero order, such as small frequencies, wave vectors and material fill fractions. In some cases they remain surprisingly robust exceeding their initial assumptions, such as the well-established Maxwell-Garnett theory for elliptical inclusions that can produce reliable results for fill fractions far above its theoretical limitations. We here present a rigorous solution of Maxwell’s equations in binary periodic materials employing a combined Greens-Galerkin procedure to obtain a low-dimensional eigenproblem for the evanescent Floquet eigenmodes of the material. In its general form, our method provides an accurate solution of the multi-valued complex Floquet bandstructure, which currently cannot be obtained with established solvers. It is thus shown to be valid in regimes where homogenization theories naturally break down. For small frequencies and wave numbers in lowest order, our method simplifies to the Maxwell-Garnett result for 2D cylinder and 3D sphere packings. It therefore provides the missing explanation why Maxwell-Garnett works well up to extremely high fill fractions of approximately 50% depending on the constituent materials, provided the inclusions are arranged on an isotropic lattice.

Broadband circular dichroism in chiral plasmonic woodpiles

“…If a periodic optical material contains base materials with loss, such as plasmonic metals, even the fundamental Floquet modes do not fit into the standard band structure picture with real-valued wave vectors [ 3 ]. Instead, a complex band structure picture has to be adopted [ 40 , 41 ]. Calculating these modes for general geometries remains challenging and is currently not possible with established software packages (based on for example finite differences or finite elements).…”

“…While the calculation of evanescent Floquet modes is not implemented in standard simulation tools based on the finite element or finite difference methods, they can be naturally obtained through a plane-wave approach [ 40 ] for photonic crystals. For plasmonic materials, such an approach is, however, very inefficient as the generally poor convergence behavior of the plane-wave basis becomes a particularly big problem at metal-dielectric interfaces [ 41 ], where the permittivity changes its sign. In simple geometries, which are homogeneous in the propagation direction, such as the lamellar grating [ 43 ], the fishnet structure [ 42 ], or hyperbolic aligned wire media [ 44 , 45 ], the computation simplifies to an essentially one- or two-dimensional problem.…”

“…The simulated scattering of an outside source, however, always corresponds to an a-priory unknown superposition of self-consistent modal solutions within the PC. A physically inspired set of such modal solutions is provided through evanescent Floquet states [40][41][42]. These states can be thought of as a natural complex generalization of the Bloch modes discussed in the canonical band structure picture employed in the photonic crystal theory [3].…”