Electromagnetic Signatures of Inhomogeneities: Visibility vs. Invisibility

This mathematics research project aims to improve understanding of the relation between the regularity, the geometry, and the material contents of an inhomogeneity and its visibility/invisibility properties when probed by electromagnetic waves. The investigator plans to develop tools to enhance image resolution and to provide analysis that determines which geometric features are more visible or less visible. It is anticipated that the work will have direct implications for electromagnetic imaging techniques, whether they concern radar detection of unknown objects, nondestructive testing of material components, or biomedical imaging of live organs. The same tools and analysis will also be brought to bear on electromagnetic cloaking. The project will examine to what extent one may create approximate cloaks that work at broad bands of frequencies and at the same time use materials that are physically realistic. This will provide a better understanding of the limitations of passive cloaking devices and their physical realizability. The investigator will mentor a postdoctoral researcher as an integral part of the research. The research is intended to elucidate the relation between the regularity, the geometry, and the material contents of an inhomogeneity and its visibility/invisibility properties. Such understanding can be used to better detect inhomogeneities or hide them. The project also aims to develop novel asymptotic formulas for the effect of a change in boundary conditions on small boundary sets, a topic of importance for optimal design. The project investigates four sub-topics: (1) study of regularity of (the boundary of) inhomogeneities that exhibit non-scattering wave numbers, (2) study of geometry of (the boundary of) inhomogeneities that exhibit in finitely many non-scattering wave numbers, (3) comparison of recent results about the infeasibility of perfect cloaking and earlier results about the feasibility of approximate cloaking, and (4) development of uniformly valid asymptotic formulas for the field effects of "small" internal inhomogeneities and "small" boundary (condition) inhomogeneities. The mathematical techniques will include sharp estimates of the effects of small boundary condition inhomogeneities of extreme contrast and hodograph transform techniques with associated elliptic PDE estimates applied to examine the regularity properties of non-scattering inhomogeneities. To develop the relevant measure of the smallness of boundary condition inhomogeneities of extreme contrast, the investigator will introduce and examine various novel notions of capacity. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.