We investigate the Fano resonance in grating structures using coupled resonators. The grating consists of a perfectly conducting slab with periodically arranged subwavelength slit holes, where inside each period, a pair of slits sit very close to each other. The slit holes act as resonators and are strongly coupled. It is shown rigorously that there exist two groups of resonances corresponding to poles of the scattering problem. One sequence of resonances has imaginary part in the order of ε, where ε is the size of the slit aperture, while the other sequence has imaginary part in the order of ε2. When coupled with the incident wave at resonant frequencies, the narrow-band resonant scattering induced by the latter will interfere with the broader background resonant radiation induced by the former. The interference of these two resonances generates the Fano-type transmission anomaly, which persists in the whole radiation continuum of the grating structure as long as the slit aperture size is small compared to the incident wavelength.

The scattering cross section is the fundamental measure of the strength of a scattering interaction. All scattering in electron microscopy arises from the Coulomb interaction, and scattering cross sections, whether elastic or inelastic, will therefore all have common features. Simple forms of both elastic and inelastic cross sections are reviewed in the context of high resolution and analytical microscopy. Some recent developments, such as the calculation of Fano resonances in electron energy loss spectra of transition metals and rare earth elements are also discussed.