  We propose a self-consistent vectorial method, based on a Green's function
technique, to describe the Fano resonances that appear in guided mode resonance
gratings. The model provides intuitive expressions of the reflectivity and
transmittivity matrices of the structure, involving coupling integrals between
the modes of a planar reference structure and radiative modes. These
expressions are used to derive a physical analysis in configurations where the
effect of the incident polarization is not trivial. We provide numerical
validations of our model. On a technical point of view, we show how the Green's
tensor of our planar reference structure can be expressed as two scalar Green's
functions, and how to deal with the singularity of the Green's tensor.
