Grating structures adapted to support cavity modes ("CMs"), including CMs produced by waveguide modes (WGs) of TE -polarized radiation; and those produced by WGs or vertically-oriented surface plasmons (VSPs) on the groove walls of incident TM- polarized radiation are provided. Such grating structures include those that provide enhanced transmission for a predetermined polarization state at a predetermined wavelength, simultaneous TM and TE transmission, and those that provide light circulation and weaving. The grating structures can include wires, or arrays of holes in thin (metallic) films, and include multiple-groove-per-period structures. Methods for optimizing such grating structures are also provided.