  Grazing-angle scattering (GAS) is a type of Bragg scattering of waves in
slanted non-uniform periodic gratings, when the diffracted order satisfying the
Bragg condition propagates at a grazing angle with respect to the boundaries of
a slab-like grating. Rigorous analysis of GAS of bulk TE electromagnetic waves
is undertaken in holographic gratings by means of the enhanced T-matrix
algorithm. A comparison of the rigorous and the previously developed
approximate theories of GAS is carried out. A complex pattern of numerous
previously unknown resonances is discovered and analysed in detail for gratings
with large amplitude, for which the approximate theory fails. These resonances
are associated not only with the geometry of GAS, but are also typical for wide
transmitting gratings. Their dependence on grating amplitude, angles of
incidence and scattering, and grating width is investigated numerically.
Physical interpretation of the predicted resonances is linked to the existence
and the resonant generation of special new eigenmodes of slanted gratings. Main
properties of these modes and their field structure are discussed.
