Electromagnetic wave scattering by subwavelength hole structures has received significant research interest in the past two decades, due to the unusual physical phenomena that arise in these media when external radiation is present, such as the extraordinary optical transmission (EOT) and strongly localized optical field at the hole apertures. It turns out that resonances, which are broadly defined as complex eigenvalues of the underlying Maxwell’s operator, play a major role in EOT and anomalous field enhancement for such media. These resonances can be induced by the geometry (such as tiny holes) or the medium parameter (such as permittivity values) of the problem, or their collaborative interactions. In this paper, we survey the mathematical theory that has been developed to understand the various resonances in subwavelength hole structures, along with quantitative analyses of their resonant scattering and the induced EOT phenomena. We also review computational methods proposed for modeling resonant wave scattering in these multiscale media and the mathematical frameworks established for applications in sensing and imaging. Finally, we discuss open problems and outstanding mathematical challenges in the field. The mathematical investigation of the resonances for this class of problems provides the fundamental theory as well as computational algorithms for the design of more efficient subwavelength optical devices and their applications. It also sheds light on the studies of other related spectral problems with the differential operators defined over multiscale media.
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