Extremely narrow sharply peaked resonances at the edge of the continuum

We report a critical narrowing of resonances of a driven potential well, when their eigenfrequencies approach the edge of the continuum. The resonances also obtain unusual sharp-peak shapes at the continuum boundary. The situation can be realized for the electromagnetic wave propagating across the dielectric thin films with a periodically modulated interface(s). We show the general phenomenon semi-analytically on a simplified model of a driven quantum potential well, also by rigorous numerical analysis of Maxwell equations for the wave propagation across the thin film with a modulated interface(s). We justify the phenomenon experimentally, by the measurements of light reflection from the dielectric thin film deposited on a periodically modulated surface. The narrow and sharp-peak resonances can be used for an efficient narrow-band frequency- and spatial filtering of light.