Superlenses achieve resolution finer than convential lenses, and have some startling properties, such as making polarizable dipoles essentially invisible if they lie within a critical distance of the lens. This project improves our understanding of superlenses and invisibility, particularly for spherical superlenses, which not only superresolve but also magnify. It also seeks to characterize exotic electromagnetic behaviors that composite materials formed from constituent materials with extreme properties can exhibit in the quasistatic limit, where the wavelength of the time-harmonic radiation is much larger than the microstructure. The investigator also studies a new class of material named "massnetic materials." These have in their microstructure collections of spinning tops, each situated in a cavity in the body, and each weighted on one side by a mass that allows one to increase or decrease the spin of each top by appropriately oscillating the body. These materials should be able to store and release energy on the microscale. The investigator also explores new equations of elastodynamics, and studies novel microstructures with the unusual property that their average momentum depends not just on their overall velocity, but also on how they are deformed, i.e. on their strain.

Metamaterials, i.e. composite materials with properties unachievable in ordinary materials, have attracted a great deal of interest and are beginning to revolutionize our understanding of materials and the properties they can exhibit. More technogical applications of these materials are now possible due to advances in our ability to tailor the microstructure of substances, for instance through nanotechnology. This project studies how composites can be constructed from high contrast materials to exhibit elastic and electromagnetic properties far richer than existing materials. In the defence, automotive, aerospace, electronics, and other manufacturing and telecommunication industries there is a constant need for new materials. The impact of such new materials is likely to be greatest when their properties are radically different from any material we know. The project could lead to the development of whole classes of radically different new materials with novel properties. Also it should give a much needed firm theoretical foundation to Pendry's work on spherical superlenses, and enhance our understanding of invisibility.

Project Report

The intellectual merit of the proposal is reflected in the fact that it resulted in 33 peer reviewed journal publications that advanced scientific knowledge. One publication, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance" had the most (about 12,800) downloads in 2007 among all papers published by the the Optical Society of America. A new method of cloaking was discovered. It has the advantage that an object is cloaked over a broad range of frequencies. Active sources create a "quiet zone" where the object to be cloaked is placed. The method of cloaking due to anomalous localized resonance, associated with superlenses, was shown to hold for continuous sources and not just discrete sources. This sort of cloaking was found to associated with other "folded geometries" where space is folded back on itself, and then unfolded to map to a physically realistic geometry. A complete characterization was obtained of all possible dynamic responses of networks of springs with masses situated at the internal nodes. This is important since an understanding of what is possible in such discrete networks may be key to understanding what can be achieved in elastic metamaterials with continuous microstructure. A metamaterial was discovered which realizes the "Willis equations", with behavior quite unlike that of conventional materials: the momentum depends on both the velocity and the strain, while the stress, depends not only on strain but also on the acceleration. New energy minimization, or rather power minimization, variational principles were obtained for electrodynamics, acoustics and elastodynamics for the case where the fields oscillate with time at fixed frequency. These should provide an important tool in the analysis of such equations. They generalize the variational principle for static conductivity that Dirichlet discovered over a hundred years ago. Connections were made between the problem of bounding the effective conductity or effective elasticity tensor of composites and the problem of obtaining upper and lower limits on the volume occupied by an inclusion in a body from measurements of the fields and potentials at the boundary of the body. These connections led to new sharp bounds on the volume occupied by an inclusion that can easily be calculated given appropriate boundary measurements. The broader impacts of the proposal were that it helped in the interdisciplinary training of two postdocs and four graduate students. Also the award helped excite the general public about scientific research, through interviews I gave which resulted in two LiveScience articles, an online CNN article, a TV segment on KSL new, and also through a large public lecture on cloaking in the prominent "Frontiers of Science" series at the University of Utah. The work on bounding the volume occupied by an inclusion could turn out to have important medical applications, in the detection of breast cancers and in organ screening.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Michael H. Steuerwalt
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University of Utah
Salt Lake City
United States
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