NameHansen, Derek J. (author), Vogelius, Michael (chair), Nussbaum, Roger (internal member), Falk, Richard (internal member), Weinstein, Michael (outside member), Rutgers University, Graduate School - New Brunswick, ,
DescriptionThis thesis is a study of the asymptotic perturbation formulas that result from electromagnetic (or acoustic) wave scattering by small, penetrable objects. The ultimate purpose of these formulas is to aid in solving the inverse problem of reconstructing small inhomogeneities embedded within an otherwise known background medium.
For simplicity, we consider the time-harmonic, transverse magnetic setting, in which case the scalar electric field satisfies a two-dimensional Helmholtz equation.
We first derive, in the case of fixed frequency, a rigorous asymptotic formula for the boundary field perturbation caused by small inhomogeneities of arbitrary shape within a bounded domain.
We then derive formal asymptotic formulas in the
case where frequency is allowed to grow as the size of a single, smooth inhomogeneity tends to zero.
For high frequencies, we use the technique of geometric optics to derive an integral formula for the scattered field, which we then simplify by a stationary phase analysis. The resulting asymptotic formula is ripe with geometric information to aid in solving the inverse problem.
In a step toward a rigorous proof of this high frequency asymptotic formula, we prove an estimate of a Sobolev norm of the scattered field in the case of a penetrable, though conducting, circular scatterer.
NoteIncludes bibliographical references (p. 183-189).
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
RightsThe author owns the copyright to this work.