Abstract

The existence of surface waves at the boundary of a hyperbolic-gyromagnetic metamaterial is studied. The surface waves, which are analytically formulated in terms of the eigenfields, appear in the spatial gap between two elliptically polarized bulk modes of the metamaterial. The surface waves are chiral in the sense that they propagate unidirectionally along the edge and reverse the propagation direction upon changing sign of the gyrotropic parameter. The topological feature of the chiral surface waves can be characterized by the Berry phases of the bulk modes, showing the bulk-edge correspondence for the underlying medium. The unidirectionality of the chiral surface waves and their immunity to disorder are further demonstrated by the propagation of electromagnetic waves around sharp corners.

© 2017 Optical Society of America

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References

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2017 (1)

A. Slobozhanyuk, S. H. Mousavi, X. Ni, D. Smirnova, Y. S. Kivshar, and A. B. Khanikaev, “Three-dimensional all-dielectric photonic topological insulator,” Nat. Photonics 11, 130–136 (2017).
[Crossref]

2016 (5)

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U. S. A. 113, 4924–4928 (2016).
[Crossref] [PubMed]

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15, 542–548 (2016).
[Crossref] [PubMed]

A. P. Slobozhanyuk, A. B. Khanikaev, D. S. Filonov, D. A. Smirnova, A. E. Miroshnichenko, and Y. S. Kivshar, “Experimental demonstration of topological effects in bianisotropic metamaterials,” Sci. Rep. 6, 22270 (2016).
[Crossref]

B. Yang, M. Lawrence, W. Gao, Q. Guo, and S. Zhang, “One-way helical electromagnetic wave propagation supported by magnetized plasma,” Sci. Rep. 6, 21461 (2016).
[Crossref] [PubMed]

V. I. Fesenko, I. V. Fedorin, and V. R. Tuz, “Dispersion regions overlapping for bulk and surface polaritons in a magnetic-semiconductor superlattice,” Opt. Lett. 41, 2093–2096 (2016).
[Crossref] [PubMed]

2015 (7)

M. G. Silveirinha, “Chern invariants for continuous media,” Phys. Rev. B 92, 125153 (2015).
[Crossref]

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

V. R. Tuz, “Gyrotropic-nihility state in a composite ferrite-semiconductor structure,” J. Opt. 17, 035611 (2015).
[Crossref]

T. Ma, A. B. Khanikaev, S. H. Mousavi, and G. Shvets, “Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides,” Phys. Rev. Lett. 114, 127401 (2015).
[Crossref] [PubMed]

L.-H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114, 223901 (2015).
[Crossref] [PubMed]

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114, 037402 (2015).
[Crossref] [PubMed]

W.-L. Gao, F.-Z. Fang, Y.-M. Liu, and S. Zhang, “Chiral surface waves supported by biaxial hyperbolic metamaterials,” Light-Sci. Appl. 4, e328 (2015).
[Crossref]

2014 (4)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

O. Takayama, D. Artigas, and L. Torner, “Lossless directional guiding of light in dielectric nanosheets using Dyakonov surface waves,” Nat. Nanotechnol. 9, 419–424 (2014).
[Crossref] [PubMed]

P.-H. Chang, C.-Y. Kuo, and R.-L. Chern, “Wave propagation in bianisotropic metamaterials: angular selective transmission,” Opt. Express 22, 25710–25721 (2014).
[Crossref] [PubMed]

2013 (6)

C. J. Zapata-Rodríguez, J. J. Miret, S. Vuković, and M. R. Belić, “Engineered surface waves in hyperbolic metamaterials,” Opt. Express 21, 19113–19127 (2013).
[Crossref] [PubMed]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

A. B. Khanikaev, S. Hossein Mousavi, W.-K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12, 233–239 (2013).
[Crossref]

M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496, 196–200 (2013).
[Crossref] [PubMed]

L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljacic, “Weyl points and line nodes in gyroid photonic crystals,” Nat. Photonics 7, 294–299 (2013).
[Crossref]

2012 (3)

W. Li, Z. Liu, X. Zhang, and X. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 161108 (2012).
[Crossref]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[Crossref]

O. Takayama, D. Artigas, and L. Torner, “Practical dyakonons,” Opt. Lett. 37, 4311–4313 (2012).
[Crossref] [PubMed]

2011 (3)

J. A. Polo and A. Lakhtakia, “Surface electromagnetic waves: a review,” Laser Photonics Rev. 5, 234–246 (2011).
[Crossref]

V. Yannopapas, “Gapless surface states in a lattice of coupled cavities: a photonic analog of topological crystalline insulators,” Phys. Rev. B 84, 195126 (2011).
[Crossref]

X. L. Qi and S. C. Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
[Crossref]

2010 (3)

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[Crossref]

J. Gao, A. Lakhtakia, and M. Lei, “Dyakonov-Tamm waves guided by the interface between two structurally chiral materials that differ only in handedness,” Phys. Rev. A 81, 013801 (2010).
[Crossref]

V. Boucher and D. Ménard, “Effective magnetic properties of arrays of interacting ferromagnetic wires exhibiting gyromagnetic anisotropy and retardation effects,” Phys. Rev. B 81, 174404 (2010).
[Crossref]

2009 (1)

Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102, 093901 (2009).
[Crossref] [PubMed]

2008 (3)

Z. Jacob and E. E. Narimanov, “Optical hyperspace for plasmons: Dyakonov states in metamaterials,” Appl. Phys. Lett. 93, 221109 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref] [PubMed]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljačić, “Reflection-free one-way edge modes in a gyromagnetic photonic crystal,” Phys. Rev. Lett. 100, 013905 (2008).
[Crossref] [PubMed]

2006 (2)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

C. Wu, B. A. Bernevig, and S.-C. Zhang, “Helical liquid and the edge of quantum spin Hall systems,” Phys. Rev. Lett. 96, 106401 (2006).
[Crossref] [PubMed]

2005 (3)

T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances,” J. Phys. Soc. Japan 74, 1674–1677 (2005).
[Crossref]

C. L. Kane and E. J. Mele, “Z2 topological order and the quantum spin Hall effect,” Phys. Rev. Lett. 95, 146802 (2005).
[Crossref] [PubMed]

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[Crossref] [PubMed]

2003 (2)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys.-Condes. Matter 15, 6345 (2003).
[Crossref]

1988 (2)

F. D. M. Haldane, “Model for a quantum Hall effect without landau levels: condensed-matter realization of the ’parity anomaly’,” Phys. Rev. Lett. 61, 2015–2018 (1988).
[Crossref] [PubMed]

M. I. D’yakonov, “New type of electromagnetic wave propagating at the interface,” Sov. Phys. JETP 94, 119–123 (1988).

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

1982 (1)

B. I. Halperin, “Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Phys. Rev. B 25, 2185–2190 (1982).
[Crossref]

1980 (1)

K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance,” Phys. Rev. Lett. 45, 494–497 (1980).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and D. L. Mills, Surface Polaritons: Electromagnetic Waves at Surfaces and Interfaces, vol. 1 (Elsevier, 1982).

Artigas, D.

O. Takayama, D. Artigas, and L. Torner, “Lossless directional guiding of light in dielectric nanosheets using Dyakonov surface waves,” Nat. Nanotechnol. 9, 419–424 (2014).
[Crossref] [PubMed]

O. Takayama, D. Artigas, and L. Torner, “Practical dyakonons,” Opt. Lett. 37, 4311–4313 (2012).
[Crossref] [PubMed]

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[Crossref] [PubMed]

Bai, B.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Belic, M. R.

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Béri, B.

W. Gao, M. Lawrence, B. Yang, F. Liu, F. Fang, B. Béri, J. Li, and S. Zhang, “Topological photonic phase in chiral hyperbolic metamaterials,” Phys. Rev. Lett. 114, 037402 (2015).
[Crossref] [PubMed]

Bernevig, B. A.

C. Wu, B. A. Bernevig, and S.-C. Zhang, “Helical liquid and the edge of quantum spin Hall systems,” Phys. Rev. Lett. 96, 106401 (2006).
[Crossref] [PubMed]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[Crossref]

Boucher, V.

V. Boucher and D. Ménard, “Effective magnetic properties of arrays of interacting ferromagnetic wires exhibiting gyromagnetic anisotropy and retardation effects,” Phys. Rev. B 81, 174404 (2010).
[Crossref]

Chan, C. T.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102, 093901 (2009).
[Crossref] [PubMed]

Chang, P.-H.

Chen, H.

Y. Lai, H. Chen, Z.-Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102, 093901 (2009).
[Crossref] [PubMed]

Chen, W.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, X.

L. Huang, X. Chen, B. Bai, Q. Tan, G. Jin, T. Zentgraf, and S. Zhang, “Helicity dependent directional surface plasmon polariton excitation using a metasurface with interfacial phase discontinuity,” Light-Sci. Appl. 2, e70 (2013).
[Crossref]

Chen, X.-D.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014).
[Crossref] [PubMed]

Chen, Y.

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U. S. A. 113, 4924–4928 (2016).
[Crossref] [PubMed]

Chen, Y.-F.

C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, Y. Chen, L. Feng, and Y.-F. Chen, “Photonic topological insulator with broken time-reversal symmetry,” Proc. Natl. Acad. Sci. U. S. A. 113, 4924–4928 (2016).
[Crossref] [PubMed]

Cheng, X.

X. Cheng, C. Jouvaud, X. Ni, S. H. Mousavi, A. Z. Genack, and A. B. Khanikaev, “Robust reconfigurable electromagnetic pathways within a photonic topological insulator,” Nat. Mater. 15, 542–548 (2016).
[Crossref] [PubMed]

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Supplementary Material (2)

NameDescription
» Visualization 1: AVI (838 KB)      Visualization of Fig. 6(a)
» Visualization 2: AVI (913 KB)      Visualization of Fig. 6(b)

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Figures (6)

Fig. 1
Fig. 1 Equifrequency surfaces of the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and |κ| = 0.8 and (b) εt = −2, εz = 1, μ = −1, and |κ| = 1.2. Thick black curves are equifrequency contours at ky = 0.
Fig. 2
Fig. 2 Dispersion curves of the surface waves for the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and |κ| = 0.8 and (b) εt = −2, εz = 1, μ = −1, and |κ| = 1.2. Black curves stand for bulk waves at ky = 0. Blue and red curves stand for surface waves for positive and negative values of κ, respectively. Gray regions correspond to spatial gaps between bulk modes. Arrows indicate the propagation directions of surface waves.
Fig. 3
Fig. 3 (a) Profile of the normalized tangential electric fields along the distance from the interface for the surface wave in Fig. 2(a) at kz/k0 = 1.2 (marked by blue dot) and (b) Trajectory of the directions of Poynting vectors by varying the distance from the interface for the surface wave in Fig. 2(b) at kz/k0 = 1.2 (marked by red dot). Dashed line indicates the direction of average Poynting vector.
Fig. 4
Fig. 4 Polarization handedness of the surface and bulk waves for the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and κ = 0.8 and (b) εt = −2, εz = 1, μ = −1, and κ = −1.2. Blue and red curves denote right- and left-handed elliptical polarizations, respectively. Gray dashed circles are dispersion curves of (a) vacuum and (b) negative index medium.
Fig. 5
Fig. 5 Berry curvatures and Berry phases of the bulk eigenmodes for the hyperbolic-gyromagnetic metamaterial with (a) εt = 2, εz = −1, μ = 1, and κ = 0.8 and (b) εt = −2, εz = 1, μ = −1, and κ = −1.2. Blue and red arrows denote the outward and inward Berry curvatures, respectively.
Fig. 6
Fig. 6 Numerical simulation of the chiral surface wave at kz/k0 = 1.2 excited at the interface (marked by asterisk symbol) between (a) vacuum and the hyperbolic-gyromagnetic metamaterial with εt = 2, εz = −1, μ = 1, and κ = 0.8 ( Visualization 1) and (b) negative index medium and the metamaterial with εt = −2, εz = 1, μ = −1, and κ = −1.2 ( Visualization 2). The coordinates are scaled by λ0 = 2π/k0.

Equations (21)

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ε _ = ε 0 [ ε t 0 0 0 ε t 0 0 0 ε z ] , μ _ = μ 0 [ μ i κ 0 i κ μ 0 0 0 μ ] ,
[ ( k × I _ ) ε _ 1 ( k × I _ ) + k 0 2 μ _ ] H = 0 ,
[ μ k 0 2 k z 2 ε t k y 2 ε z i κ k 0 2 + k x k y ε z k x k z ε t i κ k 0 2 + k x k y ε z μ k 0 2 k z 2 ε t k x 2 ε z k y k z ε t k x k z ε t k y k z ε t μ k 0 2 k x 2 ε t k y 2 ε t ] [ H x H y H z ] = 0 .
μ ( k t 2 + k z 2 ) ( ε t k t 2 + ε z k z 2 ) ε t [ ε t μ 2 k t 2 + ε z ( μ ˜ 2 k t 2 + 2 μ 2 k z 2 ) ] k 0 2 + ε t 2 ε z μ μ ˜ 2 k 0 4 = 0 ,
k z 2 = ε t μ k 0 2 ε + k t 2 2 ε z ± ε 2 k t 4 4 ε z 2 ε t κ 2 k t 2 k 0 2 μ + ε t 2 κ 2 k 0 4 ,
( k t 2 + k z 2 ε t μ k 0 2 ) ( k t 2 ε z + k z 2 ε t μ k 0 2 ) = 0 ,
H x = ( k t 2 ε t μ ω 2 ) ( ε t k x 2 + ε z k z 2 ε t ε z μ ω 2 ) ε z k y 2 k z 2 ,
H y = ε t ( k x k y i κ ε z ω 2 ) ( k t 2 ε t μ ω 2 ) + ε z k x k y k z 2 ,
H z = k z [ ε t k x 3 i κ ε t ε z k y ω 2 + k x ( ε t k y 2 + ε z k z 2 ε t ε z μ ω 2 ) ] .
E x = η 0 ε t k 0 ( k z H y k y H z ) ,
E y = η 0 ε t k 0 ( k x H z k z H x ) ,
E z = η 0 ε z k 0 ( k y H x k x H y ) ,
H 1 = ( k z , 0 , k x ) , E 1 = η 0 k 0 ( k x k y , k x 2 k z 2 , k y k z ) ,
H 2 = ( k y , k x , 0 ) , E 2 = η 0 k 0 ( k x k z , k y k z , k x 2 + k y 2 ) ,
H ± = H ( k x , k y ± , k z ) , E ± = E ( k x , k y ± , k z ) ,
k y ± = { 1 2 μ ε t [ α + ε t k 0 2 μ ε + k z 2 ± μ 2 ε 2 k z 4 + ε t k 0 2 ( α 2 ε t k 0 2 2 β μ k z 2 ) ] k x 2 } 1 / 2 ,
C 1 H x , z 1 + C 2 H x , z 2 = C + H x , z + + C H x , z ,
C 1 E x , z 1 + C 2 E x , z 2 = C + E x , z + + C E x , z ,
| k x k y k x k z 1 ε t ( k z H y + k y + H z + ) 1 ε t ( k z H y k y H z ) k y k z k x 2 + k y 2 1 ε z ( k y + H x + k x H y + ) 1 ε z ( k y H x k x H y ) k z k y H x + H x k x 0 H z + H z | = 0 .
F ( k x , k z ) = 0 ,
γ n = i S × n | n d a ,

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