Research Projects

Understanding electromagnetic wave propagation in a medium is crucial to the advancement of various technological innovations. Maxwell's equations have proven to be an accurate model for describing electromagnetic scattering over a large range of length scales. This project aims to analyze important questions arising in homogenization and controllability of the linear Maxwell's equations (with a particular emphasis on the anisotropic case) and apply this analysis to the design of negative-index materials and cloaking. Much of existing work on homogenization of Maxwell's equations has been in the time-harmonic setting, and typically the medium is assumed to be non-conducting. This choice can largely be attributed to non-zero conductivity coefficients in Maxwell's equations resulting in frequency-dependent effective parameters for the homogenized equations, thereby leading to a history dependent term in the homogenized problem. A similar phenomenon is also observed in other contexts such as neutron transport equations. The ``extended phase space method" addresses the history-dependence by increasing the number of independent variables in the effective equation in order to get rid of the non-local term. We propose to adapt this method for the time-dependent Maxwell's equations, and derive quantitative rates for the homogenization process. Moreover, existing work in this area assume that the charge density is a square integrable function. Unfortunately, in many practical contexts, this assumption is invalid. We propose to relax this assumption, clarify the role of charge density in the upscaling process and to characterise the effective charge density in the homogenized equation. The homogenization analysis of the time-dependent Maxwell's equations will play a vital role in the design of effective meta-materials. Two applications include the design of approximate cloaking devices, and developing negative index materials. There is a vast literature on the material properties required to achieve the above design goals, but alas such materials are not naturally occurring. The homogenization theory allows for constructing such artificial materials using composite media with different small-scale structures, effectively making it an inverse problem for determining the fine structures. Similar work has already been done in the time-harmonic case. We propose to pursue its extension to the time-dependent problem with conducting media wherein the different frequencies of the time-harmonic setting are coupled due to the memory dependence of the limiting homogenized problem. The study of the controllability of partial differential equations with rapidly varying coefficients was initiated by Jacques-Louis Lions in the 1980's. Inspired by this and related works, we would like to kick-start a program which blends techniques from control theory and the theory of homogenization to understand the behaviour of control sequences in the homogenization process. It has been known that the conductivity coefficient being non-zero in a subset of non-zero Lebesgue measure suffices to get the decay of solutions to Maxwell's equations with perfectly conducting boundary conditions. We would like to explore the quantitative aspects of these results while also studying the interplay between the geometry of the support of the conductivity and the stability (Geometric control). Finally, while there is a rich body of analytical work in the theory of homogenization for the time-harmonic Maxwell's equation, the related numerical machinery is relatively underdeveloped. We propose to develop high-order accurate and scalable integral equation methods towards designing effective meta-materials. The two-scale nature of the problem combined with the requirement of many solutions of the forward problem in the design loop underscores the need for accurate and fast solvers.