Fundamental limits to the refractive index of transparent optical\n materials

Increasing the refractive index available for optical and nanophotonic\nsystems opens new vistas for design: for applications ranging from broadband\nmetalenses to ultrathin photovoltaics to high-quality-factor resonators, higher\nindex directly leads to better devices with greater functionality. Although\nstandard transparent materials have been limited to refractive indices smaller\nthan 3 in the visible, recent metamaterials designs have achieved refractive\nindices above 5, accompanied by high losses, and near the phase transition of a\nferroelectric perovskite a broadband index above 26 has been claimed. In this\nwork, we derive fundamental limits to the refractive index of any material,\ngiven only the underlying electron density and either the maximum allowable\ndispersion or the minimum bandwidth of interest. The Kramers--Kronig relations\nprovide a representation for any passive (and thereby causal) material, and a\nwell-known sum rule constrains the possible distribution of oscillator\nstrengths. In the realm of small to modest dispersion, our bounds are closely\napproached and not surpassed by a wide range of natural materials, showing that\nnature has already nearly reached a Pareto frontier for refractive index and\ndispersion. Surprisingly, our bound shows a cube-root dependence on electron\ndensity, meaning that a refractive index of 26 over all visible frequencies is\nlikely impossible. Conversely, for narrow-bandwidth applications, nature does\nnot provide the highly dispersive, high-index materials that our bounds suggest\nshould be possible. We use the theory of composites to identify metal-based\nmetamaterials that can exhibit small losses and sizeable increases in\nrefractive index over the current best materials.\n