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# Ghost hyperbolic surface polaritons in bulk anisotropic crystals

## Abstract

Polaritons in anisotropic materials result in exotic optical features, which can provide opportunities to control light at the nanoscale1,2,3,4,5,6,7,8,9,10. So far these polaritons have been limited to two classes: bulk polaritons, which propagate inside a material, and surface polaritons, which decay exponentially away from an interface. Here we report a near-field observation of ghost phonon polaritons, which propagate with in-plane hyperbolic dispersion on the surface of a polar uniaxial crystal and, at the same time, exhibit oblique wavefronts in the bulk. Ghost polaritons are an atypical non-uniform surface wave solution of Maxwell’s equations, arising at the surface of uniaxial materials in which the optic axis is slanted with respect to the interface. They exhibit an unusual bi-state nature, being both propagating (phase-progressing) and evanescent (decaying) within the crystal bulk, in contrast to conventional surface waves that are purely evanescent away from the interface. Our real-space near-field imaging experiments reveal long-distance (over 20 micrometres), ray-like propagation of deeply subwavelength ghost polaritons across the surface, verifying long-range, directional and diffraction-less polariton propagation. At the same time, we show that control of the out-of-plane angle of the optic axis enables hyperbolic-to-elliptic topological transitions at fixed frequency, providing a route to tailor the band diagram topology of surface polariton waves. Our results demonstrate a polaritonic wave phenomenon with unique opportunities to tailor nanoscale light in natural anisotropic crystals.

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## Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## Code availability

The code that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We acknowledge support from T. Wang for micro-FTIR measurements, and Y. F. Hao and Y. S. Wang for antenna fabrication. P.L. acknowledges the support from the National Natural Science Foundation of China (grant no. 62075070). C.-W.Q. acknowledges support from the National Research Foundation, Prime Minister’s Office, Singapore, under its Competitive Research Programme (CRP award NRF CRP22-2019-0006) and from grant R-261-518-004-720 from the Advanced Research and Technology Innovation Centre (ARTIC). G.H. acknowledges support from A*STAR under its AME Young Individual Research Grants (YIRG, no. A2084c0172). A.A. acknowledges support from the Office of Naval Research (grant no. N00014-19-1-2011), the Simons Foundation, the Air Force Office of Scientific Research MURI program, and the Department of Defense Vannevar Bush Faculty Fellowship. Q.D. acknowledges support from the National Natural Science Foundation of China (grant no. 51925203). D.H. acknowledges support from the National Natural Science Foundation of China (grant no. 11704085). X.Z. acknowledges support from the National Key Research and Development Program of China (grant no. 2018YFA0704403).

## Author information

Authors

### Contributions

C.-W.Q. and P.L. conceived the study. W.M. fabricated the samples. G.H. performed the theory analysis coordinated by C.-W.Q. and A.A. D.H. and W.M. performed the s-SNOM measurements with the help of T.S. and Y.Z. G.H., R.C. and W.M. performed the simulations. P.L., C.-W.Q., A.A., Q.D. and X.Z. coordinated and supervised the work. W.M., G.H., A.A., C.-W.Q. and P.L. wrote the manuscript with input from all co-authors.

### Corresponding authors

Correspondence to Xinliang Zhang or Qing Dai or Andrea Alù or Cheng-Wei Qiu or Peining Li.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Thomas Folland and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Centimetre-sized calcite substrates with different optic axis-orientations and corresponding XRD patterns, angle-resolved far-field reflectance.

ac, Optical microscope images of calcite substrates for θ = 90°, 23.3°, 48.5° (the crystal surface is along plane (001), (100), (104), respectively). d, Schematics of the characteristic planes and the corresponding angle θ with respect to optic axis. Black dashed arrow represents optic axis. Green, red and blue lines indicate the characteristic plane (100), (001), (104), respectively. e, X-ray diffraction (XRD) data of calcite substrates with different angles with respect to optic axis. XRD patterns of three calcite substrates with different θ exhibited strong diffraction peaks at 31.4°, 64.7°, 29.4° indicating the characteristic plane (001), (100) and (104) of calcite, respectively. f, Fourier-transform infrared reflection spectra of calcite substrates shown in ac for different polarization angles of the incident light. The 0° polarization defined here is parallel with direction of red arrow in x–y plane. g, Theoretically fitted spectra using the dielectric permittivities according to Methods. The results for the calcite substrates for θ = 90° (the surface is along plane (001)), θ = 23.3° (the surface is along plane (100)) and θ = 48.5° (the surface is along plane (104)), respectively.

### Extended Data Fig. 2 Comparison of the modal patterns of the g-HPs, s-HPs and v-HPs.

ac, Transverse cross-section of the simulated near-field distributions of a g-HPs mode (a), s-HPs mode (b) and the fundamental v-HPs waveguide mode M0 (c). These results are shown in Fig. 1. d, e, Near-field distributions of the higher order v-HPs waveguide modes, M1 (d) and M2 (e). The s-HPs possess pure imaginary-valued kz and thus exponentially decay inside the crystal. The v-HPs exhibiting real-valued wave vector kz in the material. They thus can accumulate the phase variations in the vertical direction to form Fabry–Pérot interferences between the two interfaces, resulting in the different order waveguide modes. By contrast, as a consequence of the oblique wavefronts, the g-HPs exhibit an unusual propagation feature: their electric fields exponentially attenuate with sinusoidal phase oscillations inside calcite.

### Extended Data Fig. 3 Comparison of in-plane polariton dispersion for s-HPs, g-HPs and v-HPs.

ac, Schematic illustration of bulk calcite crystal with θ = 0°, 23.3°, 90°. df, Natural crystals can exhibit extreme dielectric anisotropy, arising when the permittivity tensor elements along orthogonal principal axes have opposite signs (for example, either type I: Re(ε) > 0, Re $$({\varepsilon }_{\parallel })$$ < 0, or type II: Re(ε) < 0, Re$$({\varepsilon }_{\parallel })$$ > 0, for uniaxial materials4,5). These features result in polaritons—light–matter hybrid electromagnetic excitations—with a hyperbolic dispersion, that is, the polariton wavevector k can support the hyperbolic isofrequency contours. Because of the two types of anisotropic dielectric permittivity, the hyperbolic dispersions are accordingly in the form of two types of open hyperboloids, which are the solutions of the equation of the wavevector k given by$$\,{\text{k}}_{z}^{2}/{{\rm{\varepsilon }}}_{\perp }+({\text{k}}_{x}^{2}+{\text{k}}_{y}^{2})/{{\rm{\varepsilon }}}_{\parallel }=\,{\text{k}}_{0}^{2}$$, in which k0 is the free-space wavevector. As a result, the polaritons in strongly anisotropic materials are called hyperbolic polaritons1,2,3. The figure shows schematic illustrations of three-dimensional isofrequency and projected in-plane isofrequency contours (represented by the blue dashed line) at the kx–ky plane for s-HPs (d), g-HPs (e) and v-HPs (f) at the corresponding angle θ = 0°, 23.3°, 90°. The black dashed arrow represents the optic axis. g, A false-colour map showing the Fourier transform results of dipole-launched g-HPs. Dashed red, green and white lines correspond to the theoretical IFCs of in-plane wave vectors for s-HPs, g-HPs and v-HPs respectively.

### Extended Data Fig. 4 Numerical simulations of g-HPs considering the material loss.

ac, Transverse cross-section of the simulated near-field distribution of a g-HP mode for different losses. It is clear that the g-HPs still exist in the presence of the loss. However, the propagation length is reduced when adding the loss. For all cases we use $${\varepsilon }_{\parallel }$$ = 2.34.

### Extended Data Fig. 5 Evaluation of the effective wavevectors of g-HP rays.

a, Near-field image of antenna-launched g-HPs, shown in Fig. 2b. b, Magnified image of the g-HP ray, taken from the area marked in a. c, Fourier transform of b, indicating the composite of the super-composed mode.

### Extended Data Fig. 6 Comparison of experimental and simulated near-field images of disk-launched g-HPs.

ac, Disk-launched g-HPs at three different frequencies: ω = 1,450 cm−1 (a), ω = 1,460 cm−1 (b) and ω = 1,470 cm−1 (c). Left, experimental near-field images of g-HPs. Middle, simulated near-field images of disk-launched g-HPs. Right, Fourier transform of the experimental near-field images shown in the left panels. Green and white lines are theoretical IFCs of in-plane wave vectors by considering the interference factors (±k0cosφ) according to Supplementary Fig. 3. Considering that the metallic s-SNOM tip is not included in the calculations, we thus assign the experimental near-field distribution to disk-launched polaritons.

## Supplementary information

### Supplementary Information

This file contains Supplementary Information Sections 1-4 as follows: the theory to determine the g-HP propagation features; details of numerical simulations; Supplementary Figs 1-9 and Supplementary References.

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Ma, W., Hu, G., Hu, D. et al. Ghost hyperbolic surface polaritons in bulk anisotropic crystals. Nature 596, 362–366 (2021). https://doi.org/10.1038/s41586-021-03755-1

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