  Extremely asymmetrical scattering (EAS) is a new type of Bragg scattering in
thick, slanted, periodic gratings. It is realised when the scattered wave
propagates parallel to the front boundary of the grating. Its most important
feature is the strong resonant increase in the scattered wave amplitude
compared to the amplitude of the incident wave: the smaller the grating
amplitude, the larger the amplitude of the scattered wave. In this paper,
rigorous numerical analysis of EAS is carried out by means of the enhanced
T-matrix algorithm. This includes investigation of harmonic generation inside
and outside the grating, unusually strong edge effects, fast oscillations of
the incident wave amplitude in the grating, etc. Comparison with the previously
developed approximate theory is carried out. In particular, it is demonstrated
that the applicability conditions for the two-wave approximation in the case of
EAS are noticeably more restrictive than those for the conventional Bragg
scattering. At the same time, it is shown that the approximate theory is
usually highly accurate in terms of description of EAS in the most interesting
cases of scattering with strong resonant increase of the scattered wave
amplitude. Physical explanation of the predicted effects is presented.
