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Negative refractive index in dielectric crystals containing stoichiometric rare-earth ions

We investigate the prospect of achieving negative permittivity and permeability at optical frequencies in a dielectric crystal containing stoichiometric rare-earth ions. We derive the necessary transition linewidth, ion density and electric and magnetic oscillator strengths using a simplified model of non-interacting dipoles. We identify Erbium crystals in a magnetically ordered phase as the most promising material to meet these conditions, and describe initial optical measurements of two potential candidates, \ercl{} and ${}^7$\lierf{}, which display linewidths of 3~GHz and 250~MHz, respectively. The properties of ${}^7$\lierf{} satisfied our criterion for negative permeability.