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# Zero-refractive-index materials and topological photonics

## Abstract

The refractive index is one of the most basic optical properties of a material and its interaction with light. Modern materials engineering—particularly the concept of metamaterials—has made it necessary to consider its subtleties, including anisotropy and complex values. Here we re-examine the refractive index and find a general way to calculate the direction-dependent refractive index and the condition for zero index in a given direction. By analogy with linear versus circular polarization, we show that when the zero-index direction is complex-valued, a material supports waves that can propagate in only one sense, for example, clockwise. We show that there is an infinite family of both time-reversible and time-irreversible homogeneous electromagnetic media that support unidirectional propagation for a particular polarization. As well as extending the concept of the refractive index, shedding new light on our understanding of topological photonics and providing new sets of material parameters, our simple picture also reproduces many of the findings derived using topology.

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## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source data are provided with this paper.

## Code availability

The figures were generated using Mathematica and Python, and full wave simulations were performed using COMSOL Multiphysics 5.4. Mathematica and Python code and COMSOL models are available from the corresponding author upon reasonable request.

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## Acknowledgements

S.A.R.H. acknowledges financial support from a Royal Society TATA University Research Fellowship (RPG-2016-186). M.W. acknowledges funding from an EPSRC vacation bursary. S.A.R.H. acknowledges useful conversations with W. L. Barnes and I. R. Hooper, as well as I. R. Hooper’s numerical expertise.

## Author information

Authors

### Contributions

S.A.R.H. devised the theory and wrote the manuscript. M.W. contributed to the theory, commented on the manuscript and wrote the numerical code to produce Fig. 1.

### Corresponding author

Correspondence to S. A. R. Horsley.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Supplementary information

### Supplementary Information

Supplementary calculations and figures.

## Source data

### Source Data Fig. 2

Numerical data for Cartesian plots.

### Source Data Fig. 3

Numerical data for Cartesian plots.

### Source Data Fig. 4

Numerical data for Cartesian plots.

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Horsley, S.A.R., Woolley, M. Zero-refractive-index materials and topological photonics. Nat. Phys. 17, 348–355 (2021). https://doi.org/10.1038/s41567-020-01082-2

• Accepted:

• Published:

• Issue Date:

• DOI: https://doi.org/10.1038/s41567-020-01082-2

• ### Realizing quasi-monochromatic switchable thermal emission from electro-optically induced topological phase transitions

• Nitish Kumar Gupta
• Sapireddy Srinivasu
• S. Anantha Ramakrishna

Scientific Reports (2022)