Many properties of photonic structures rely on band topology characterized by the integer invariants that can change during the topological transitions and give rise to the disorder-robust topological edge, corner, or interface states. Typically, the periods of such structures are comparable to the wavelength. However, in many cases, the unit cell becomes deeply subwavelength and hence the entire metamaterial can be described in terms of the effective material parameters. Here, focusing on subwavelength topological metamaterials, we identify the behavior of permittivity and permeability accompanying the topological transition on the example of the two structures possessing D6 symmetry.
References
1.
D.
Xiao
, M.-C.
Chang
, and Q.
Niu
, “Berry phase effects on electronic properties
,” Rev. Mod. Phys.
82
, 1959
–2007
(2010
).2.
M. Z.
Hasan
and C. L.
Kane
, “Colloquium: Topological insulators
,” Rev. Mod. Phys.
82
, 3045
–3067
(2010
).3.
L.
Lu
, J. D.
Joannopoulos
, and M.
Soljačić
, “Topological photonics
,” Nat. Photonics
8
, 821
–829
(2014
).4.
T.
Ozawa
, H. M.
Price
, A.
Amo
, N.
Goldman
, M.
Hafezi
, L.
Lu
, M. C.
Rechtsman
, D.
Schuster
, J.
Simon
, O.
Zilberberg
, and I.
Carusotto
, “Topological photonics
,” Rev. Mod. Phys.
91
, 015006
(2019
).5.
T.
Karzig
, C.-E.
Bardyn
, N. H.
Lindner
, and G.
Refael
, “Topological polaritons
,” Phys. Rev. X
5
, 031001
(2015
).6.
S.
Klembt
, T. H.
Harder
, O. A.
Egorov
, K.
Winkler
, R.
Ge
, M. A.
Bandres
, M.
Emmerling
, L.
Worschech
, T. C. H.
Liew
, M.
Segev
, C.
Schneider
, and S.
Höfling
, “Exciton-polariton topological insulator
,” Nature
562
, 552
–556
(2018
).7.
Z.
Yang
, F.
Gao
, X.
Shi
, X.
Lin
, Z.
Gao
, Y.
Chong
, and B.
Zhang
, “Topological acoustics
,” Phys. Rev. Lett.
114
, 114301
(2015
).8.
S. D.
Huber
, “Topological mechanics
,” Nat. Phys.
12
, 621
–623
(2016
).9.
W. A.
Benalcazar
, T.
Li
, and T. L.
Hughes
, “Quantization of fractional corner charge in Cn-symmetric higher-order topological crystalline insulators
,” Phys. Rev. B
99
, 245151
(2019
).10.
M. G.
Silveirinha
, “Chern invariants for continuous media
,” Phys. Rev. B
92
, 125153
(2015
).11.
M. G.
Silveirinha
, “2 topological index for continuous photonic materials
,” Phys. Rev. B
93
, 075110
(2016
).12.
M. G.
Silveirinha
, “Bulk-edge correspondence for topological photonic continua
,” Phys. Rev. B
94
, 205105
(2016
).13.
S.
Pakniyat
, S. A. H.
Gangaraj
, and G. W.
Hanson
, “Chern invariants of topological continua: A self-consistent nonlocal hydrodynamic model
,” Phys. Rev. B
105
, 035310
(2022
).14.
L.-H.
Wu
and X.
Hu
, “Scheme for achieving a topological photonic crystal by using dielectric material
,” Phys. Rev. Lett.
114
, 223901
(2015
).15.
S.
Yves
, R.
Fleury
, T.
Berthelot
, M.
Fink
, F.
Lemoult
, and G.
Lerosey
, “Crystalline metamaterials for topological properties at subwavelength scales
,” Nat. Commun.
8
, 16023
(2017
).16.
Y.
Li
, Y.
Sun
, W.
Zhu
, Z.
Guo
, J.
Jiang
, T.
Kariyado
, H.
Chen
, and X.
Hu
, “Topological LC-circuits based on microstrips and observation of electromagnetic modes with orbital angular momentum
,” Nat. Commun.
9
, 4598
(2018
).17.
Y.
Yang
, Y. F.
Xu
, T.
Xu
, H.-X.
Wang
, J.-H.
Jiang
, X.
Hu
, and Z. H.
Hang
, “Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials
,” Phys. Rev. Lett.
120
, 217401
(2018
).18.
S.
Barik
, A.
Karasahin
, C.
Flower
, T.
Cai
, H.
Miyake
, W.
DeGottardi
, M.
Hafezi
, and E.
Waks
, “A topological quantum optics interface
,” Science
359
, 666
–668
(2018
).19.
J.
Noh
, W. A.
Benalcazar
, S.
Huang
, M. J.
Collins
, K. P.
Chen
, T. L.
Hughes
, and M. C.
Rechtsman
, “Topological protection of photonic mid-gap defect modes
,” Nat. Photonics
12
, 408
–415
(2018
).20.
M. A.
Gorlach
, X.
Ni
, D. A.
Smirnova
, D.
Korobkin
, D.
Zhirihin
, A. P.
Slobozhanyuk
, P. A.
Belov
, A.
Alù
, and A. B.
Khanikaev
, “Far-field probing of leaky topological states in all-dielectric metasurfaces
,” Nat. Commun.
9
, 909
(2018
).21.
D.
Smirnova
, S.
Kruk
, D.
Leykam
, E.
Melik-Gaykazyan
, D.-Y.
Choi
, and Y.
Kivshar
, “Third-harmonic generation in photonic topological metasurfaces
,” Phys. Rev. Lett.
123
, 103901
(2019
).22.
N.
Parappurath
, F.
Alpeggiani
, L.
Kuipers
, and E.
Verhagen
, “Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing
,” Sci. Adv.
6
, eaaw4137
(2020
).23.
A. M.
Nicolson
and G. F.
Ross
, “Measurement of the intrinsic properties of materials by time-domain techniques
,” IEEE Trans. Instrum. Meas.
19
, 377
–382
(1970
).24.
W.
Weir
, “Automatic measurement of complex dielectric constant and permeability and microwave frequencies
,” Proc. IEEE
62
, 33
–36
(1974
).25.
M. A.
Gorlach
and M.
Lapine
, “Boundary conditions for the effective-medium description of subwavelength multilayered structures
,” Phys. Rev. B
101
, 075127
(2020
).26.
K. Y.
Bliokh
, D.
Leykam
, M.
Lein
, and F.
Nori
, “Topological non-Hermitian origin of surface Maxwell waves
,” Nat. Commun.
10
, 580
(2019
).27.
S.
Wu
, B.
Jiang
, Y.
Liu
, and J.-H.
Jiang
, “All-dielectric photonic crystal with unconventional higher-order topology
,” Photonics Res.
9
, 668
(2021
).28.
B.
Lv
, T.
Qian
, and H.
Ding
, “Angle-resolved photoemission spectroscopy and its application to topological materials
,” Nat. Rev. Phys.
1
, 609
–626
(2019
).29.
P. A.
Belov
, R.
Marqués
, S. I.
Maslovski
, I. S.
Nefedov
, M.
Silveirinha
, C. R.
Simovski
, and S. A.
Tretyakov
, “Strong spatial dispersion in wire media in the very large wavelength limit
,” Phys. Rev. B
67
, 113103
(2003
).30.
M. G.
Silveirinha
, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters
,” Phys. Rev. B
75
, 115104
(2007
).31.
A.
Alù
, “First-principles homogenization theory for periodic metamaterials
,” Phys. Rev. B
84
, 075153
(2011
).32.
M. A.
Gorlach
and P. A.
Belov
, “Nonlocality in uniaxially polarizable media
,” Phys. Rev. B
92
, 085107
(2015
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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