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.
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