Localized gradient enhancement near anisotropic electromagnetic scatterers

Abstract

This work investigates time-harmonic electromagnetic scattering governed by the Maxwell system, where bounded anisotropic scatterers are embedded in a homogeneous electromagnetic background. We focus on the localized enhancement of the gradients of the total electric and magnetic fields in small boundary-attached neighborhoods of finitely many prescribed points near boundaries of anisotropic electromagnetic scatterers. We show that, through a suitable construction of incident electromagnetic waves, the gradients of both the total electric field and the total magnetic field can be made arbitrarily large in these neighborhoods. The main strategy is based on the introduction of auxiliary boundary-attached electromagnetic neighborhoods and the associated electric and magnetic fields, which exhibit strong gradient variation near the prescribed points. Using the approximation property of Maxwell Herglotz wave functions, these auxiliary fields are then approximated by physically admissible incident waves in the neighborhood of the scatterers. Together with the well-posedness and continuous dependence of the anisotropic scattering problem, this implies that the corresponding scattered field can be controlled to be sufficiently weak in the relevant region. Consequently, the total field is dominated by the incident field near the prescribed points and inherits its large-gradient behavior. The result provides a theoretical mechanism for localized gradient enhancement in anisotropic electromagnetic scattering and may have implications for field concentration, high-resolution probing, and sensitivity analysis of electromagnetic responses in complex media.