Fundamental limits to the refractive index of transparent optical materials (2103.15697v1)

Abstract: 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, higher index directly leads to better devices with greater functionality. Although standard transparent materials have been limited to refractive indices smaller than 3 in the visible, recent metamaterials designs have achieved refractive indices above 5, accompanied by high losses, and near the phase transition of a ferroelectric perovskite a broadband index above 26 has been claimed. 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. The Kramers--Kronig relations provide a representation for any passive (and thereby causal) material, and a well-known sum rule constrains the possible distribution of oscillator strengths. In the realm of small to modest dispersion, our 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. Surprisingly, our bound shows a cube-root dependence on electron density, meaning that a refractive index of 26 over all visible frequencies is likely impossible. Conversely, for narrow-bandwidth applications, nature does not provide the highly dispersive, high-index materials that our bounds suggest should be possible. We use 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.

• The paper derives fundamental limits on the refractive index by applying the Kramers–Kronig relations and sum rules.
• It reveals a cube-root dependency on electron density and shows that natural materials are near a Pareto frontier in optical performance.
• The study indicates that low-loss composite metamaterials might achieve refractive indices far beyond current natural limits.

Fundamental Limits to the Refractive Index of Transparent Optical Materials

The paper, authored by Hyungki Shim, Francesco Monticone, and Owen D. Miller, presents a rigorous investigation into the upper bounds of the refractive index attainable by transparent optical materials. This research is crucial for elevating the performance and versatility of various optical and nanophotonic systems, including metalenses, photovoltaics, and resonators.

Overview

The central objective of this work is to derive fundamental limits for the refractive index of transparent materials, constrained by the material's electron density and either its maximum allowable dispersion or minimum bandwidth of interest. The methodology leverages the Kramers--Kronig (KK) relations to construct constraints from the principle of causality and applies a well-known sum rule that limits the distribution of oscillator strengths available in the material.

Key Findings

1. Cube-root Dependency: A surprising outcome of the analysis is the cube-root dependency of the refractive index on the electron density. This suggests that materials cannot achieve exceedingly high refractive indices simply by increasing electron density or allowing high dispersion. For example, a refractive index of 26 across all visible frequencies is theoretically impossible given typical electron densities.
2. Constraints from Natural Materials: It was found that natural materials already come close to achieving a Pareto frontier in the trade-off between refractive index and dispersion. This implies little room for improvement (typically in the range of 1.1–1.5 times the current best) without departing from natural constraints.
3. Composite Materials: The paper utilizes the theory of composites to identify metal-based metamaterials capable of surpassing current material limits with small losses and significant refractive index increases. Specifically, they note that if a "lossless metal" could be synthesized, indices beyond 100 at optical frequencies might be possible.

Implications

The implications of this work are significant for both theoretical physics and practical applications in nanotechnology and optics. Theoretical advancements offer a deeper understanding of the limitations imposed by physical laws on material properties, while the practical aspect provides a path forward for designing new materials or metamaterials with elevated optical properties.

Future Outlook

Looking ahead, the research suggests potential in exploring new composite materials and metamaterials that cleverly utilize structural design to achieve high refractive indices within the bounds identified. Furthermore, advances in material science, particularly in synthesizing near-zero-loss metals, could revolutionize the field by making such high-refractive-index materials feasible in real-world applications.

In summary, this paper fundamentally advances our understanding of the capabilities and limitations of optical materials, providing a robust analytical framework and predictive insights that can invigorate future research and material innovation in optical engineering.