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.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 /Â 30Â days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$209.00 per year
only $17.42 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
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.
References
Mark Smith, A. From Sight to Light: The Passage from Ancient to Modern Optics (Univ. Chicago Press, 2015).
Young, T. A Course of Lectures on Natural Philosophy and the Mechanical Arts Vol. 1 (Franklin Classics Trade Press, 2018).
Kadic, M., Milton, G. W., van Hecke, M. & Wegener, M. 3D metamaterials. Nat. Rev. Phys. 1, 198–210 (2019).
Guo, A. et al. Observation of \({\mathcal{PT}}\)-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009).
Horsley, S. A. R., Artoni, M. & La Rocca, G. C. Spatial Kramers–Kronig relations and the reflection of waves. Nat. Photon. 9, 436–439 (2015).
Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966–3969 (2000).
Silveirinha, M. & Engheta, N. Tunneling of electromagnetic energy through subwavelength channels and bends. Phys. Rev. Lett. 97, 157403 (2006).
Leonhardt, U. & Philbin, T. G. Geometry and Light: The Science of Invisibility (Dover, 2010).
Edwards, B., AlĂą, A., Young, M. E., Silveirinha, M. G. & Engheta, N. Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide. Phys. Rev. Lett. 100, 033903 (2008).
Liberal, I., Mahmoud, A. M., Li, Y., Edwards, B. & Engheta, N. Photonic doping of epsilon-near-zero media. Science 355, 1058–1062 (2017).
Davoyan, A. R. & Engheta, N. Theory of wave propagation in magnetized near-zero-epsilon metamaterials: evidence for one-way photonic states and magnetically switched transparency and opacity. Phys. Rev. Lett. 111, 257401 (2013).
Silveirinha, M. G. Chern invariants for continuous media. Phys. Rev. B 92, 125153 (2015).
Horsley, S. A. R. Unidirectional wave propagation in media with complex principal axes. Phys. Rev. A 97, 023834 (2018).
Horsley, S. A. R. Topology and the optical Dirac equation. Phys. Rev. A 98, 043837 (2018).
Thaller, B. The Dirac Equation (Springer, 2013).
Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).
Barnett, S. M. Optical Dirac equation. New J. Phys. 16, 093008 (2014).
Horsley, S. A. R. Indifferent electromagnetic modes: bound states and topology. Phys. Rev. A 100, 053819 (2019).
Mechelen, T. V. & Jacob, Z. Photonic Dirac monopoles and skyrmions: spin-1 quantization. Opt. Mater. Express 9, 95–111 (2019).
Barnes, W. L., Horsley, S. A. R. & Vos, W. L. Classical antennas, quantum emitters and densities of optical states. J. Opt. 22, 073501 (2020).
Silveirinha, M. G. \({\mathcal{P}}\cdot {\mathcal{T}}\cdot {\mathcal{D}}\) symmetry protected scattering anomaly in optics. Phys. Rev. B 95, 035153 (2017).
Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).
Liu., F. & Li, J. Gauge field optics with anisotropic media. Phys. Rev. Lett. 114, 103902 (2015).
Olver, F. W. J. et al. NIST Digital Library of Mathematical Functions Release 1.0.23 (NIST, 2019); http://dlmf.nist.gov/
Aharonov, Y. & Casher, A. Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field. Phys. Rev. A 19, 2461–2462 (1978).
Atiyah, M. F. & Singer, I. M. The index of elliptic operators on compact manifolds. Bull. Am. Math. Soc. 69, 422–433 (1969).
Rechtsman, M. C. et al. Photonic Floquet topological insulators. Nature 496, 196–200 (2013).
Gao, W. et al. Topological photonic phase in chiral hyperbolic metamaterials. Phys. Rev. Lett. 114, 037402 (2015).
Bliokh, K. Y., Smirnova, D. & Nori, F. Quantum spin Hall effect of light. Science 348, 1448–1451 (2015).
COMSOL Multiphysics v. 4.4 (COMSOL, 2018); http://www.comsol.com
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 and Affiliations
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
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567-020-01082-2
This article is cited by
-
Electromagnetic metamaterials to approach superconductive-like electrical conductivity
Scientific Reports (2023)
-
Coherent multipolar amplification of chiroptical scattering and absorption from a magnetoelectric nanoparticle
Communications Physics (2023)
-
Tutorial: Topology, Waves, and the Refractive Index
International Journal of Theoretical Physics (2023)
-
Realizing quasi-monochromatic switchable thermal emission from electro-optically induced topological phase transitions
Scientific Reports (2022)