# Wave Front Tuning of Coupled Hyperbolic Surface Waves on Anisotropic Interfaces

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## Abstract

**:**

## 1. Introduction

## 2. Symmetric Isotropic System (Metal-Dielectric-Metal)

## 3. Symmetric Anisotropic System (Uniaxial Metal-Isotropic Dielectric-Uniaxial Metal)

## 4. Asymmetric Anisotropic System (Uniaxial Metal-Isotropic Dielectric-Isotropic Metal)

## 5. Wave Front Tuning of Hyperbolic Surface Waves on Anisotropic Interfaces

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HSW | Hyperbolic Surface Wave |

HMM | Hyperbolic Metamaterial |

TE | Transverse Electric |

TM | Transverse Magnetic |

## References

- Polo, J.; Lakhtakia, A. Surface electromagnetic waves: A review. Laser Photonics Rev.
**2011**, 5, 234–246. [Google Scholar] [CrossRef] - Takayama, O.; Bogdanov, A.A.; Lavrinenko, A.V. Photonic surface waves on metamaterial interfaces. J. Phys. Condens. Matter
**2017**, 29, 463001. [Google Scholar] [CrossRef] - Barnes, W.L.; Dereux, A.; Ebbesen, T.W. Surface plasmon subwavelength optics. Nature
**2003**, 424, 824–830. [Google Scholar] [CrossRef] [PubMed] - Maier, S. Plasmonics: Fundamentals and Applications; Springer: New York, NY, USA, 2007. [Google Scholar]
- Han, Z.; Bozhevolnyi, S.I. Radiation guiding with surface plasmon polaritons. Rep. Prog. Phys.
**2012**, 76, 016402. [Google Scholar] [CrossRef] [PubMed] - Yeh, P.; Yariv, A.; Cho, A.Y. Optical surface waves in periodic layered media. Appl. Phys. Lett.
**1978**, 1978 32, 104–105. [Google Scholar] [CrossRef][Green Version] - Liscidini, M.; Sipe, J.E. Analysis of Bloch-surface-wave assisted diffraction-based biosensors. J. Opt. Soc. Am. B
**2009**, 26, 279–289. [Google Scholar] [CrossRef] - Descrovi, E.; Sfez, T.; Quaglio, M.; Brunazzo, B.; Dominici, L.; Michelotti, F.; Herzig, H.P.; Martin, O.J.F.; Giorgis, F. Bloch surface waves on ultrathin polymeric ridges. Nano Lett.
**2010**, 10, 2087–2091. [Google Scholar] [CrossRef][Green Version] - Sinibaldi, A.; Danz, N.; Descrovi, E.; Munzert, P.; Schulz, U.; Sonntag, F.; Dominici, L.; Michelotti, F. Direct comparison of the performance of Bloch surface wave and surface plasmon polariton sensors. Sens. Actuators B Chem.
**2012**, 174, 292–298. [Google Scholar] [CrossRef] - Yu, L.; Barakat, E.; Sfez, T.; Hvozdara, L.; Francesco, J.D.; Herzig, H.P. Manipulating Bloch surface waves in 2D: A platform concept-based flat lens. Light. Sci. Appl.
**2014**, 3, e124. [Google Scholar] [CrossRef] - Dyakonov, M. New type of electromagnetic wave propagating at an interface. Sov. Phys. JETP
**1988**, 67, 714–716. [Google Scholar] - Takayama, O.; Crasovan, L.-C.; Johansen, S.K.; Mihalache, D.; Artigas, D.; Torner, L. Dyakonov surface waves: A review. Electromagnetics
**2008**, 28, 126–145. [Google Scholar] [CrossRef] - Takayama, O.; Crasovan, L.; Artigas, D.; Torner, L. Observation of Dyakonov surface waves. Phys. Rev. Lett.
**2009**, 102, 043903. [Google Scholar] [CrossRef] [PubMed] - Takayama, O.; Nikitin, A.Y.; Martin-Moreno, L.; Torner, L.; Artigas, D. Dyakonov surface wave resonant transmission. Opt. Express
**2011**, 19, 6339–6347. [Google Scholar] [CrossRef] [PubMed][Green Version] - Takayama, O.; Artigas, D.; Torner, L. Coupling plasmons and dyakonons. Opt. Lett.
**2012**, 37, 1983–1985. [Google Scholar] [CrossRef] [PubMed] - Takayama, O.; Artigas, D.; Torner, L. Practical dyakonons. Opt. Lett.
**2012**, 37, 4311–4313. [Google Scholar] [CrossRef] [PubMed] - Takayama, O.; Artigas, D.; Torner, L. Lossless directional guiding of light in dielectric nanosheets using Dyakonov surface waves. Nat. Nanotechnol.
**2014**, 9, 419–424. [Google Scholar] [CrossRef] - Crasovan, L.-C.; Takayama, O.; Artigas, D.; Johansen, S.K.; Mihalache, D.; Torner, L. Enhanced localization of Dyakonov-like surface waves in left-handed materials. Phys. Rev. B
**2006**, 74, 155120. [Google Scholar] [CrossRef][Green Version] - Gao, J.; Lakhtakia, A.; Lei, M. On Dyakonov-Tamm waves localized to a central twist defect in a structurally chiral material. J. Opt. Soc. Am. B
**2009**, 26, 1615–1621. [Google Scholar] [CrossRef] - Pulsifer, D.P.; Faryad, M.; Lakhtakia, A. Observation of the Dyakonov-Tamm wave. Phys. Rev. Lett.
**2013**, 111, 243902. [Google Scholar] [CrossRef][Green Version] - Pulsifer, D.P.; Faryad, M.; Lakhtakia, A.; Hall, A.S.; Liu, L. Experimental excitation of the Dyakonov–Tamm wave in the grating-coupled configuration. Opt. Lett.
**2014**, 39, 2125–2128. [Google Scholar] [CrossRef] - Abbas, F.; Lakhtakia, A.; Naqvi, Q.A.; Faryad, M. An optical-sensing modality that exploits Dyakonov–Tamm waves. Photon. Res.
**2015**, 3, 5–8. [Google Scholar] [CrossRef] - Mackay, T.G.; Zhou, C.; Lakhtakia, A. Dyakonov–Voigt surface waves. Proc. R. Soc. A
**2019**, 475, 20190317. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jacob, Z.; Narimanov, E.E. Optical hyperspace for plasmons: Dyakonov states in metamaterials. Appl. Phys. Lett.
**2008**, 93, 221109. [Google Scholar] [CrossRef] - Repän, T.; Takayama, O.; Lavrinenko, L.V. Hyperbolic surface waves on anisotropic materials without hyperbolic dispersion. Opt. Express
**2020**. submitted. [Google Scholar] - Smith, D.R.; Schurig, D. Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors. Phys. Rev. Lett.
**2003**, 90, 077405. [Google Scholar] [CrossRef] - Krishnamoorthy, H.N.S.; Jacob, Z.; Narimanov, E.; Kretzschmar, I.; Menon, V.M. Topological transitions in metamaterials. Science
**2012**, 336, 205–209. [Google Scholar] [CrossRef][Green Version] - Cortes, C.L.; Newman, W.; Molesky, S.; Jacob, Z. Quantum nanophotonics using hyperbolic metamaterials. J. Opt.
**2012**, 14, 063001. [Google Scholar] [CrossRef][Green Version] - Drachev, V.P.; Podolskiy, V.A.; Kildishev, A.V. Hyperbolic metamaterials: New physics behind a classical problem. Opt. Express
**2013**, 21, 15048–15064. [Google Scholar] [CrossRef] - Poddubny, A.; Iorsh, I.; Belov, P.; Kivshar, Y. Hyperbolic metamaterials. Nat. Photonics
**2013**, 7, 948–957. [Google Scholar] [CrossRef] - Shekhar, P.; Atkinson, J.; Jacob, Z. Hyperbolic metamaterials: Fundamentals and applications. Nano Converg.
**2014**, 1, 14. [Google Scholar] [CrossRef] - Ferrari, L.; Wu, C.; Lepage, D.; Zhang, X.; Liu, Z. Hyperbolic metamaterials and their applications. Prog. Quantum Electron.
**2015**, 40, 1–40. [Google Scholar] [CrossRef] - Takayama, O.; Lavrinenko, A.V. Optics with hyperbolic materials [Invited]. J. Opt. Soc. Am. B
**2019**, 36, F38–F48. [Google Scholar] [CrossRef][Green Version] - Kildishev, A.V.; Boltasseva, A.; Shalaev, V.M. Planar photonics with metasurfaces. Science
**2013**, 339, 1232009. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kapitanova, P.V.; Ginzburg, P.; Rodríguez-Fortuño, F.J.; Filonov, D.S.; Voroshilov, P.M.; Belov, P.A.; Poddubny, A.N.; Kivshar, Y.S.; Wurtz, G.A.; Zayats, A.V. Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes. Nat. Commun.
**2014**, 5, 3226. [Google Scholar] [CrossRef] [PubMed] - High, A.A.; Devlin, R.C.; Dibos, A.; Polking, M.; Wild, D.S.; Perczel, J.; de Leon, N.P.; Lukin, M.D.; Park, H. Visible-frequency hyperbolic metasurface. Nature
**2015**, 522, 192–196. [Google Scholar] [CrossRef] [PubMed] - Takayama, O.; Shkondin, E.; Bodganov, A.; Panah, M.E.A.; Golenitskii, K.; Dmitriev, P.; Repän, T.; Malureanu, R.; Belov, P.; Jensen, F.; et al. Mid-infrared surface waves on a high aspect ratio nanotrench platform. ACS Photon.
**2017**, 4, 2899–2907. [Google Scholar] [CrossRef][Green Version] - Takayama, O.; Dmitriev, P.; Shkondin, E.; Yermakov, O.; Panah, M.; Golenitskii, K.; Jensen, F.; Bogdanov, A.; Lavrinenko, A. Experimental observation of Dyakonov plasmons in the mid-infrared. Semiconductors
**2018**, 52, 442–446. [Google Scholar] [CrossRef][Green Version] - Li, P.; Dolado, I.; Alfaro-Mozaz, F.J.; Casanova, F.; Hueso, L.E.; Liu, S.; Edgar, J.H.; Nikitin, A.Y.; Vélez, S.; Hillenbrand, R. Infrared hyperbolic metasurface based on nanostructured van der Waals materials. Science
**2018**, 359, 892–896. [Google Scholar] [CrossRef][Green Version] - Ma, W.; Alonso-González, P.; Li, S.; Nikitin, A.Y.; Yuan, J.; Martín-Sánchez, J.; Taboada-Gutiérrez, J.; Amenabar, I.; Li, P.; Vélez, S.; et al. In-plane anisotropic and ultra-low-loss polaritons in a natural van der Waals crystal. Nature
**2018**, 562, 557–562. [Google Scholar] [CrossRef][Green Version] - Shin, H.; Fan, S. All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure. Phys. Rev. Lett.
**2006**, 96, 073907. [Google Scholar] [CrossRef] - Sarid, D. Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications; Cambridge University Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Chen, C.-L. Foundations for Guided-Wave Optics; Wiley-Interscience: Hoboken, NJ, USA, 2007. [Google Scholar]
- Malitson, I.H. A redetermination of some optical properties of calcium fluoride. Appl. Opt.
**1963**, 2, 1103–1107. [Google Scholar] [CrossRef] - Rodríguez-de Marcos, L.V.; Larruquert, J.I.; Méndez, J.A.; Aznárez, J.A. Self-consistent optical constants of MgF2, LaF3, and CeF3 films. Opt. Mater. Express
**2017**, 7, 989–1006. [Google Scholar] [CrossRef][Green Version] - Repän, T.; Novitsky, A.; Willatzen, M.; Lavrinenko, A.V. Pseudocanalization regime for magnetic dark-field hyperlenses. Phys. Rev. B
**2017**, 96, 195166. [Google Scholar] [CrossRef][Green Version] - Gomez-Diaz, J.S.; Tymchenko, M.; Alu, A. Hyperbolic plasmons and topological transitions over uniaxial metasurfaces. Phys. Rev. Lett.
**2015**, 114, 233901. [Google Scholar] [CrossRef][Green Version] - Yermakov, O.Y.; Ovcharenko, A.I.; Song, M.; Bogdanov, A.A.; Iorsh, I.V.; Kivshar, Y.S. Hybrid waves localized at hyperbolic metasurfaces. Phys. Rev. B
**2015**, 91, 235423. [Google Scholar] [CrossRef][Green Version] - Samusev, A.; Mukhin, I.; Malureanu, R.; Takayama, O.; Permyakov, D.V.; Sinev, I.V.; Baranov, D.; Yermakov, O.; Iorsh, I.V.; Bogdanov, A.A.; et al. Polarization-resolved characterization of plasmon waves supported by an anisotropic metasurface. Opt. Express
**2017**, 26, 32631–32639. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**(

**a**) Geometry for isotropic three-layer system under consideration, metal-dielectric-metal layers. A dielectric (spacer) layer with thickness, h, is sandwiched between two metal layers. (

**b**) Propagation constant of modes, $\beta $ in three-layer system for various spacer layer thicknesses, h. Blue lines indicate even modes, while green solid (dotted) lines indicate odd modes with normal (reversed) phase propagation direction.

**Figure 2.**Even (green) and odd modes (blue lines) in symmetric three layer system (uniaxial metal-isotropic dielectric-uniaxial metal), for various spacer (isotropic dielectric layer) thicknesses. Both (

**a**) type-1 ${\epsilon}_{o}=-2.37$, ${\epsilon}_{e}=-1$ and (

**b**) type-2 ${\epsilon}_{o}=-1$, ${\epsilon}_{e}=-7$ systems are shown. The black dotted line shows solution from corresponding two-layer system of uniaxial metal and isotropic dielectric interface. The black dashed line shows single-interface solution.

**Figure 3.**(

**a**) Real and imaginary parts of the propagation constant ${k}_{z}$ for the odd mode in the three layer system (details in the text) for losses $\gamma =0.01$ (blue) and $\gamma =0.1$ (orange). Dotted black line indicates corresponding dispersion of single interface. (

**b**,

**c**) Semi-analytically calculated field profiles for losses $\gamma =0.01$ (

**b**) and $\gamma =0.1$ (

**c**). (

**d**) Full-wave simulations of the field profile for losses $\gamma =0.1$. Insets show Fourier transformed fields, showing dispersion of the propagating waves.

**Figure 4.**(

**a**) Real and imaginary parts of the propagation constant ${k}_{z}$ for the even mode in the three layer system (details in the text) for losses $\gamma =0.01$ (blue) and $\gamma =0.1$ (orange). The black line shows corresponding HMM dispersion. (

**b**) Semi-analytically calculated field profiles for losses $\gamma =0.01$. (

**c**) Full-wave simulations of the field profile for losses $\gamma =0.1$. Insets show Fourier transformed fields, showing dispersion of the propagating waves.

**Figure 5.**(

**a**) Propagation constant for the symmetric three layer system (dashed black line, uniaxial metal-isotropic dielectric-uniaxial metal) and for asymmetric (uniaxial metal-isotropic dielectric-isotropic metal) system, plotted for various different ${\epsilon}_{m}$. Orange line indicates ${\epsilon}_{m}=-1.89$ chosen for the simulations. (

**b**) Semi-analytically calculated fields for low-loss ($\gamma =0.01$) system. (

**c**) Simulated fields as per FEM simulations with $\gamma =0.1$.

**Figure 6.**Geometry for the two pseudocanalizing systems, with symmetric (

**a**) and nonsymmetric (

**b**) three layer system, with anisotropic media indicated by (1) and (2) and the metal for asymmetric system shown with (3). Insets indicate placement of dipole sources to excite the waves. In (

**a**) the sources are aligned such to only excite the odd mode. (+) and (-) indicate regions with normal and reversed phase propagation, respectively.

**Figure 7.**Comparison of surface wave implementations of a pseudocanalizing system. Full-wave simulation results of a symmetric system are shown in (

**a**), with corresponding semi-analytical calculations with reduced losses ($\gamma =0.01$) in (

**b**) and full losses ($\gamma =0.1$) in (

**c**). Similarly (

**d**) shows FEM results of an asymmetric system, with corresponding semi-analytically calculated fields for $\gamma =0.01$ in (

**e**) and $\gamma =0.1$ in (

**f**). (+) and (-) indicate regions with normal and reversed phase propagation [with geometry and material parameters specified Figure 6a for (a)–(c) and Figure 6b for (d)–(f)]. The dashed line shows the distance for which waves travelled equal distance through the two regions with opposite phase propagation properties. With ideal pseudocanalization the original source fields would be restored here.

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**MDPI and ACS Style**

Repän, T.; Takayama, O.; Lavrinenko, A.V.
Wave Front Tuning of Coupled Hyperbolic Surface Waves on Anisotropic Interfaces. *Photonics* **2020**, *7*, 34.
https://doi.org/10.3390/photonics7020034

**AMA Style**

Repän T, Takayama O, Lavrinenko AV.
Wave Front Tuning of Coupled Hyperbolic Surface Waves on Anisotropic Interfaces. *Photonics*. 2020; 7(2):34.
https://doi.org/10.3390/photonics7020034

**Chicago/Turabian Style**

Repän, Taavi, Osamu Takayama, and Andrei V. Lavrinenko.
2020. "Wave Front Tuning of Coupled Hyperbolic Surface Waves on Anisotropic Interfaces" *Photonics* 7, no. 2: 34.
https://doi.org/10.3390/photonics7020034