Light propagation in a fishnet metamaterial

We derive the dispersion relation of Bloch modes in a fishnet metamaterial in the optical frequency domain. The dependence of the longitudinal wave vector component on the transverse one and the frequency, which governs diffraction and dispersive spreading of localized light excitations, respectively, are analyzed. We show that this type of metamaterial exhibits an involved anisotropic behavior with diffraction changing the sign in passing even a zero diffraction point. For completeness we derive furthermore a formal angle-dependent effective refractive index which exhibits discontinuities. Thus we conclude that an effective refractive index tends to get meaningless and that only the dispersion relation predicts reliably light propagation in metamaterials. The results are double checked with those obtained from a retrieval algorithm based on angular resolved reflection/transmission data of a finite slab. Excellent agreement is observed.