Collaborative Research: Computational Framework for Non-asymptotic Homogenization with Applications to Metamaterials

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The project is aimed at developing a non-asymptotic homogenization theory of Maxwell's equations in artificial periodic composites (metamaterials). Currently, there is a consensus that sufficiently large lattice cell sizes are necessary for some nontrivial physical effects to occur in such structures. (The most intriguing of these effects is high-frequency magnetism.) Classical homogenization theories work well in the zero-cell-size limit but are difficult to apply to large metamaterial cells. In contrast, the proposed theory is non-asymptotic and does not involve any series expansions with respect to the cell size. The electromagnetic field in the material is approximated by a finite set of functions (modes) usually but not necessarily Trefftz functions such as Bloch waves. The coarse-grained fields and flux densities are defined via curl-conforming and div-conforming interpolations, respectively. A linear map between these interpolants is established and defines an extended material tensor. In a certain canonical basis, this extended tensor has a classical block of 36 local parameters and a novel block quantifying nonlocal effects. From the differential-geometric perspective, this constitutive relationship can be viewed as a realization of Bossavit-Hiptmair's discrete Hodge operators (linear maps between discretized 1-forms that correspond to vector fields and 2-forms that correspond to fluxes). Over the last decade, metamaterials have attracted unprecedented attention due to a variety of potential applications that include superlensing, electromagnetic cloaking, electromagnetically-induced transparency, efficient antennas, and more. Experimental demonstrations of these effects have been limited to proofs of principle and at optical frequencies have so far been incomplete. Subwavelength optical imaging has been achieved only in the quasi-static (near-field) regime, standard for the more conventional near-field optics; the so-called "carpet cloak" conceals surface bumps rather than 3D objects, and so on. Moreover, applications that do not depend critically on the effective medium description of metamaterials appear to be more easily achievable than the ones that do. The latter group includes, notably, superlensing and cloaking. This suggests that, to make further progress, theoretical and mathematical issues at the heart of metamaterial science must be unambiguously resolved. The main problem can be stated as follows. Given the composition of a metamaterial cell and the operating frequency, determine whether this metamaterial can be reasonably described as a continuous medium with some effective parameters, just like any natural optical material; if the answer is positive, develop a rigorous methodology for such a description. The proposed research is aimed at solving this problem in the most difficult case when the cell size of the composite is an appreciable fraction of the wavelength of light. The methodology, once developed, will allow the scientific community to delineate the possible from the impossible in the field of metamaterials. The intellectual merit of the proposed research is in the development of a new paradigm of non-asymptotic homogenization, of new computational methods related to it, and in the application of the proposed methodology to electromagnetic metamaterials, allowing one to gain a much deeper understanding of their properties and limitations. As a new area of research, non-asymptotic homogenization will also have a broader technical impact in other areas of applied physics and engineering, such as acoustics, heat transfer and possibly elasticity.