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