  We closely study the local amplifications of visible light on a thin
dielectric slab presenting a sub-wavelength array of small, rectangular,
bottom-closed holes. The high-quality Fabry-Perot resonances of eigen modes
which vertically oscillate, and their corresponding near-field maps, especially
inside the voids, are numerically quantified with RCWA and analytically
interpreted through a quasi-exact modal expansion. This last method gives
explicit opto-geometrical rules allowing to finely understand the general
trends in 1D and 2D. In more advanced examples, we show that multi-cavity
and/or slightly thicker two-dimensional gratings may generate anomalously
frequency-susceptible surfaces over a broad spectral range. Also, dielectric
membranes a few nanometers thick only, can catch light, with tremendous
enhancements of the electric field intensity ($>10^6$) that largely extends in
the surrounding space.
