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This work presents a mathematical theory for electromagnetic scatteringresonances in a subwavelength annular hole embedded in a metallic slab, withthe annulus width $h\ll1$. The model is representative among many 3Dsubwavelength hole structures, which are able to induce resonant scattering ofelectromagnetic wave and the so-called extraordinary optical transmission. Wedevelop a multiscale framework for the underlying scattering problem based upona combination of the integral equation in the exterior domain and the waveguidemode expansion inside the tiny hole. The matching of the electromagnetic fieldover the hole aperture leads to a sequence of decoupled infinite systems, whichare used to set up the resonance conditions for the scattering problem. Byperforming rigorous analysis for the infinite systems and the resonanceconditions, we characterize all the resonances in a bounded domain over thecomplex plane. It is shown that the resonances are associated with the TE andTEM waveguide modes in the annular hole, and they are close to the real axiswith the imaginary parts of order ${\cal O}(h)$. We also investigate theresonant scattering when an incident wave is present. It is proved that theelectromagnetic field is amplified with order ${\cal O}(1/h)$ at the resonantfrequencies that are associated with the TE modes in the annular hole. On theother hand, one particular resonance associated with the TEM mode can not beexcited by a plane wave but can be excited with a near-field electric dipolesource, leading to field enhancement of order ${\cal O}(1/h)$.

Mathematical theory for electromagnetic scattering resonances and field enhancement in a subwavelength annular gap