We study the propagation of waves in waveguide gratings with a double negative cover. The waveguide grating is considered as a small perturbation to a perfect waveguide and the effects of index variations on the propagating modes in the grating region are studied. To describe the energy transfer between the modes of the original structure, two coupled-mode equations are derived using the perturbation method. General formulas for both longitudinal and transverse coupling coefficients of thin-film waveguide gratings are obtained, which can be used for both conventional and left-handed thin-film waveguide gratings. To emphasize the unusual properties of the left-handed waveguide gratings, we numerically obtain the coupling coefficients of waveguide gratings with a double negative cover and compare them with those of conventional waveguides. We show that the coupling coefficients of the left-handed waveguide gratings enhance more than 3 orders of magnitude. While the longitudinal coupling coefficient of a conventional waveguide is zero, we report that it is nonzero for a waveguide grating with a double negative cover. In particular, we show that for especial ratios of the substrate to cover permittivity, the coupling coefficient of the fundamental mode (𝑚=0) only exists within a small frequency range.
