Analysis and Computation of Electromagnetic Transport in Composite Materials

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The interaction of an electromagnetic field with aninhomogeneous material is a pervasive problem arising in a broadarray of applications. Often a wave interacts with a random orstructured medium, and one is interested in the effective behavioras the wave propagates through or is localized by the medium. Moreover, systems as varied as photonic crystals and band gapstructures, random resistor netorks, sea ice and other porousmedia, and electrorheological fluids exhibit critical behavior intheir effective electromagnetic properties as some parameter isvaried. The investigators conduct fundamental, mathematicalstudies of the interaction of electromagnetic fields with suchcomposites, in conjunction with state of the art numericalexperiments on model problems. In particular, a method based onspectral theory and analytic continuation has been developed toobtain rigorous bounds on the effective complex permittivity inthe quasistatic limit. More recently, this approach has led tothe finding that the effective transport properties oftwo-component media share the same analytic properties as theorder parameters in statistical mechanics, such as themagnetization in an Ising model. These powerful relations haveonly been exploited in the static case and raise many importantquestions, yet the ideas of statistical mechanics form a naturalframework for analyzing critical behavior of wave phenomena aswell. Much of the analytic structure displayed in the static casecarries over to the Helmholtz equation in a composite. Thisobservation, as well as its implications for using statisticalmechanics in this context, is investigated analytically andnumerically. These investigations may yield fundamental advancesin the mathematics, computation, and physics of electromagneticfields and their interactions with composite systems. Graduateand undergraduate students are involved in key projects combiningmathematical and computational analysis. In a broad range of problems across many disciplines,electromagnetic fields such as light, radar, or microwavesinteract with inhomogeneous materials such as semiconductors,oil-filled rocks, bone or heart tissue, radar absorbing coatings,or shipping containers. Examples arise in physics, materialsscience, electrical and bio-engineering, chemistry, biology,geophysics, and astrophysics, and are central to communicationsand medical technologies. The investigators conduct fundamentalmathematical studies of electromagnetic fields interacting withcomposite media, in conjunction with state of the art numericalexperiments on model problems. In many important examples, theeffective electromagnetic properties depend critically on someparameter in the system. For example, whether or not a wave canpropagate through some types of structured media, called photoniccrystals, depends critically on the ratio of the wavelength to thescale of the structure or other properties of the medium. Recently it was found that the mathematics underlying electricaltransport in composites is almost identical to the mathematicsunderlying statistical mechanics. Statistical mechanics dealswith phase transitions such as the freezing of water at thecritical temperature of zero degrees Centigrade, and provides anatural framework for posing key questions about these systems. The investigators develop and apply the ideas of statisticalmechanics to electromagnetic systems using mathematical analysisand sophisticated computations. They seek fundamental newinsights, as well as novel ways of understanding and predictinghow electromagnetic fields interact with composites. The projectactively involves graduate and undergraduate students engaged inanalytical and computational projects that are central to theoverall goals. Results, particularly concerning criticalproperties, could potentially affect a broad range of applicationareas, including media structured on the nanometer scale.