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For the explanation of the enhanced transmission through the perforated metal plate with below-cutoff holes, the eigenoscillations of the double-periodic screen are studied. It is found that the origin of the recently discovered effects is the existence of the eigenoscillations of the interface between the free halfspace and the metal halfspace perforated with double-periodic set of channels and the formation of the corresponding symmetrically and anti-symmetrically coupled ldquosupermodesrdquo when two such interfaces form a screen of finite thickness. The amplitude and polarization structure of the eigenfields and the behavior of the complex-valued eigenfrequencies depending on the geometry parameters are studied by the mode-matching technique (MMT) both for a finite-thickness screen and for an interface. The symmetrical oscillation of the perforated screen is shown to be the cause of both the well-known phenomenon of total transmission through frequency selective surfaces (FSS), and the recently discovered enhanced transmission through cutoff holes. When the holes shrink, the field pattern of this eigenoscillation transforms from the one concentrated within the holes to that concentrated in the form of Floquet-waves above and below the screen, i.e., out of the holes. In contrast, the antisymmetrical oscillations are always located in frequency near the Floquet-waves grazing points at any hole dimensions and cause the so-called Wood anomalies.