  We study the propagation of light in a three-dimensional double-periodic
Ag/TiO2 multilayer metamaterial composed of coupled plasmonic waveguides
operating in the visible and UV spectral range. For these frequencies, light
propagation in the plane of the waveguides is described by a negative phase
velocity, while for the orthogonal direction light propagation is described by
a Bloch wave composed of a large number of harmonics. As a result, the material
cannot generally be described by a single phase index: decomposing the Bloch
wave into different harmonics we show that for the wavelength range of interest
the positive index m=1 harmonic dominates the propagation of light in the
orthogonal direction. These results are corroborated by numerical simulations
and optical refraction experiments on a double-periodic Ag/TiO2 multilayer
metamaterial prism in the 380-600 nm spectral range, which show that positive
refraction associated with right-handed harmonics dominates. Studying the
isofrequency contours we find that despite the occurrence of multiple harmonics
the double-periodic structure can act as a flat lens: for a slab consisting of
an integer number of unit cells all harmonics are degenerate and constructively
interfere at the image plane. This work identifies important considerations
relevant to the design of many three dimensional periodic metamaterials.
