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Increasing the refractive index available for optical and nanophotonicsystems opens new vistas for design: for applications ranging from broadbandmetalenses to ultrathin photovoltaics to high-quality-factor resonators, higherindex directly leads to better devices with greater functionality. Althoughstandard transparent materials have been limited to refractive indices smallerthan 3 in the visible, recent metamaterials designs have achieved refractiveindices above 5, accompanied by high losses, and near the phase transition of aferroelectric perovskite a broadband index above 26 has been claimed. In thiswork, we derive fundamental limits to the refractive index of any material,given only the underlying electron density and either the maximum allowabledispersion or the minimum bandwidth of interest. The Kramers--Kronig relationsprovide a representation for any passive (and thereby causal) material, and awell-known sum rule constrains the possible distribution of oscillatorstrengths. In the realm of small to modest dispersion, our bounds are closelyapproached and not surpassed by a wide range of natural materials, showing thatnature has already nearly reached a Pareto frontier for refractive index anddispersion. Surprisingly, our bound shows a cube-root dependence on electrondensity, meaning that a refractive index of 26 over all visible frequencies islikely impossible. Conversely, for narrow-bandwidth applications, nature doesnot provide the highly dispersive, high-index materials that our bounds suggestshould be possible. We use the theory of composites to identify metal-basedmetamaterials that can exhibit small losses and sizeable increases inrefractive index over the current best materials.

Fundamental limits to the refractive index of transparent optical materials