Increasing the refractive index available for optical and nanophotonic systems opens new vistas for optical design, for applications from metalenses to high-quality-factor resonators. In this work, we derive fundamental limits to the refractive index of any material, 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, our bounds are closely approached by a wide range of natural materials, showing that nature has already 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 our bounds suggest should be possible. We identify metal-based metamaterials that can exhibit small losses and sizeable increases in refractive index over the current best materials.
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