Dispersion requirements of low-loss negative refractive index materials and their realisability

A necessary dispersion requirement is given for the effective relative permeability mu_r(omega) and permittivity epsiv_r(omega) as a function of frequency for low-loss homogeneous dispersive materials and negative refractive index materials, in particular, if they exist at some dimensional scale D. The requirements are valid both for isotropic and uniaxial materials, the latter is defined by material properties that are invariant under a rotation about a principal axis. Using asymptotic requirements at zero frequency and Foster´s reactance theorem, the general bandgap structure is given for low-loss materials. It is shown that such materials can only be achieved if the material exhibits finite bandgaps separating regions of positive and/or negative refractive index regions. In degenerate cases, the bandgaps shrink to zero but low-loss propagation remains impossible in the neighbourhood of the critical frequencies. The effect of small losses are considered and compared with Stockman´s criterion. Losses may be arbitrarily small within a band provided there are no bounds to the size of the real parts of the relative permittivity and permeability at the band edges. A numerical example is given showing the effect of small losses.