  The anomalous features in diffraction patterns first observed by Wood over a
century ago have been the subject of many investigations, both experimental and
theoretical. The sharp, narrow structures - and the large resonances with which
they are sometimes associated - arise in numerous studies in optics and
photonics. In this paper we present an analytical method to study diffracted
fields of optically thin gratings that highlights the nonanalyticities
associated with the anomalies. Using this approach we can immediately derive
diffracted fields for any polarization in a compact notation. While our
equations are approximate, they fully respect energy conservation in the
electromagnetic field, and describe the large exchanges of energy between
incident and diffracted fields that can arise even for thin gratings.
