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

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The project is aimed at developing a non-asymptotic homogenizationtheory of Maxwell's equations in artificial periodic composites(metamaterials). Currently, there is a consensus that sufficientlylarge lattice cell sizes are necessary for some nontrivial physicaleffects to occur in such structures. (The most intriguing of theseeffects is high-frequency magnetism.) Classical homogenizationtheories work well in the zero-cell-size limit but are difficult toapply to large metamaterial cells. In contrast, the proposed theory isnon-asymptotic and does not involve any series expansions with respectto the cell size. The electromagnetic field in the material isapproximated by a finite set of functions (modes) usually but notnecessarily Trefftz functions such as Bloch waves. The coarse-grainedfields and flux densities are defined via curl-conforming anddiv-conforming interpolations, respectively. A linear map betweenthese interpolants is established and defines an extended materialtensor. In a certain canonical basis, this extended tensor has aclassical block of 36 local parameters and a novel block quantifyingnonlocal effects. From the differential-geometric perspective, thisconstitutive relationship can be viewed as a realization ofBossavit-Hiptmair's discrete Hodge operators (linear maps betweendiscretized 1-forms that correspond to vector fields and 2-forms thatcorrespond to fluxes).Over the last decade, metamaterials have attracted unprecedentedattention due to a variety of potential applications that includesuperlensing, electromagnetic cloaking, electromagnetically-inducedtransparency, efficient antennas, and more. Experimentaldemonstrations of these effects have been limited to proofs ofprinciple and at optical frequencies have so far been incomplete.Subwavelength optical imaging has been achieved only in thequasi-static (near-field) regime, standard for the more conventionalnear-field optics; the so-called "carpet cloak" conceals surface bumpsrather than 3D objects, and so on. Moreover, applications that do notdepend critically on the effective medium description of metamaterialsappear to be more easily achievable than the ones that do. The lattergroup includes, notably, superlensing and cloaking. This suggeststhat, to make further progress, theoretical and mathematical issues atthe heart of metamaterial science must be unambiguously resolved. Themain problem can be stated as follows. Given the composition of ametamaterial cell and the operating frequency, determine whether thismetamaterial can be reasonably described as a continuous medium withsome effective parameters, just like any natural optical material; ifthe answer is positive, develop a rigorous methodology for such adescription. The proposed research is aimed at solving this problem inthe most difficult case when the cell size of the composite is anappreciable fraction of the wavelength of light. The methodology, oncedeveloped, will allow the scientific community to delineate thepossible from the impossible in the field of metamaterials.The intellectual merit of the proposed research is in the developmentof a new paradigm of non-asymptotic homogenization, of newcomputational methods related to it, and in the application of theproposed methodology to electromagnetic metamaterials, allowing one togain a much deeper understanding of their properties and limitations.As a new area of research, non-asymptotic homogenization will alsohave a broader technical impact in other areas of applied physics andengineering, such as acoustics, heat transfer and possibly elasticity.