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We investigate numerically transmittance of polarized electromagnetic wave through periodic and aperiodic binary multilayered nanostructures made of left- and right-handed materials. The transmittance is calculated as a function of light wavelength and incidence angle as well as ordering, refractive indices and thicknesses of superlattice layers. Thanks to extension of the transfer matrix formalism over the case of complex refraction angles, such effects as tunnelling, absorption and strong dispersion in left-handed metamaterials are taken into account. We find that these effects cannot be neglected and cause new transmission bands to appear in the areas where total internal reflection in a single metamaterial layer occurs.