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We present a theory which explains how to achieve an enhancement of nonlineareffects in a thin layer of nonlinear medium by involving a planar periodicstructure specially designed to bear a trapped-mode resonant regime. Inparticular, the possibility of a nonlinear thin metamaterial to produce thebistable response at a relatively low input intensity due to a large qualityfactor of the trapped-mode resonance is shown. Also a simple design of anall-dielectric low-loss silicon-based planar metamaterial which can provide anextremely sharp resonant reflection and transmission is proposed. The designedmetamaterial is envisioned for aggregating with a pumped active medium toachieve an enhancement of quantum dots luminescence and to produce anall-dielectric analog of a 'lasing spaser'.

Electromagnetic wave diffraction by periodic planar metamaterials with nonlinear constituents