The blow-up of electromagnetic fields in 3-dimensional invisibility cloaking for Maxwell's equations

Abstract: Transformation optics constructions have allowed the design of cloaking devices that steer electromagnetic, acoustic and quantum waves around a region without penetrating it, so that this region is hidden from external observations. The proposed material parameters are anisotropic, and singular at the interface between the cloaked region and the cloaking device. The presence of these singularities causes various mathematical problems and physical effects on the interface surface. In this paper, we analyze the 3-dimensional cloaking for Maxwell's equations when there are sources or sinks present inside the cloaked region. In particular, we consider nonsingular approximate invisibility cloaks based on the truncation of the singular transformations. We analyze the limit of solutions when the approximate cloaking approaches the ideal cloaking in the sense of distributions. We show that the solutions in the approximate cloaks converge to a distribution that contains Dirac's delta distribution supported on the interface surface. In particular, this implies that the limit of solutions are not measurable functions, making them outside of those classes of functions that have earlier been used in the models of the ideal invisibility cloaks. Also, we give a rigorous meaning for the "extraordinary surface voltage effect" considered in physical literature of invisibility cloaks.