  We propose a simple design of all-dielectric silicon-based planar
metamaterial manifested an extremely sharp resonant reflection and transmission
in the wavelength of about 1550 nm due to both low dissipative losses and
involving a trapped mode operating method. The quality factor of the resonance
exceeds in tens times the quality factor of resonances in known plasmonic
structures. The designed metamaterial is envisioned for aggregating with a
pumped active medium to achieve an enhancement of luminescence and to produce
an all-dielectric analog of a "lasing spaser". We report that an essential
enhancement (more than 500 times) of luminescence of layer contained pumped
quantum dots may be achieved by using the designed metamaterial. This value
exceeds manyfold the known luminescence enhancement by plasmonic planar
metamaterials.
