Increasing the refractive index available for optical and nanophotonic systems opens new vistas for design, for applications ranging from broadband metalenses to ultrathin photovoltaics to high-quality-factor resonators. In this work, fundamental limits to the refractive index of any material are derived, given only the underlying electron density and either the maximum allowable dispersion or the minimum bandwidth of interest. In the realm of small to modest dispersion, the bounds are closely approached and not surpassed by a wide range of natural materials, showing that nature has already nearly reached a Pareto frontier for refractive index and dispersion. Conversely, for narrow-bandwidth applications, nature does not provide the highly dispersive, high-index materials that the bounds suggest should be possible. The theory of composites to identify metal-based metamaterials that can exhibit small losses and sizeable increases in refractive index over the current best materials is used. Moreover, if the "elusive lossless metal" can be synthesized, it is shown that it would enable arbitrarily high refractive index in the high-dispersion regime, nearly achieving the bounds even at refractive indices of 100 and beyond at optical frequencies.
Keywords: classical optics; light-matter interactions; metamaterials; optical materials; refraction.
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