Topological photonic crystals: a review

Abstract

The field of topological photonic crystals has attracted growing interest since the inception of optical analog of quantum Hall effect proposed in 2008. Photonic band structures embraced topological phases of matter, have spawned a novel platform for studying topological phase transitions and designing topological optical devices. Here, we present a brief review of topological photonic crystals based on different material platforms, including all-dielectric systems, metallic materials, optical resonators, coupled waveguide systems, and other platforms. Furthermore, this review summarizes recent progress on topological photonic crystals, such as higher-order topological photonic crystals, non-Hermitian photonic crystals, and nonlinear photonic crystals. These studies indicate that topological photonic crystals as versatile platforms have enormous potential applications in maneuvering the flow of light.

Keywords

topological photonic crystals / topological phase transitions / non-Hermitian photonics / higher-order topological photonic crystals

References

[1] Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics. Physical Review Letters, 1987, 58(20): 2059–2062

[2] John S. Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987, 58(23): 2486–2489

[3] Wang B, Cappelli M A. A plasma photonic crystal bandgap device. Applied Physics Letters, 2016, 108(16): 161101

[4] Akahane Y, Asano T, Song B S, Noda S. High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature, 2003, 425(6961): 944–947

[5] Shelby R A, Smith D R, Schultz S. Experimental verification of a negative index of refraction. Science, 2001, 292(5514): 77–79

[6] Shalaev V M, Cai W, Chettiar U K, Yuan H K, Sarychev A K, Drachev V P, Kildishev A V. Negative index of refraction in optical metamaterials. Optics Letters, 2005, 30(24): 3356–3358

[7] Klitzing K, Dorda G, Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Physical Review Letters, 1980, 45(6): 494–497

[8] Thouless D J, Kohmoto M, Nightingale M P, den Nijs M. Quantized hall conductance in a two-dimensional periodic potential. Physical Review Letters, 1982, 49(6): 405–408

[9] Kohmoto M. Topological invariant and the quantization of the Hall conductance. Annals of Physics, 1985, 160(2): 343–354

[10] Kane C L, Mele E J. Quantum spin Hall effect in graphene. Physical Review Letters, 2005, 95(22): 226801

[11] Bernevig B A, Zhang S C. Quantum spin Hall effect. Physical Review Letters, 2006, 96(10): 106802

[12] Bernevig B A, Hughes T L, Zhang S C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 2006, 314(5806): 1757–1761

[13] König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X L, Zhang S C. Quantum spin hall insulator state in HgTe quantum wells. Science, 2007, 318(5851): 766–770

[14] Haldane F D, Raghu S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Physical Review Letters, 2008, 100(1): 013904

[15] Wang Z, Chong Y D, Joannopoulos J D, Soljacić M. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Physical Review Letters, 2008, 100(1): 013905

[16] Wang Z, Chong Y, Joannopoulos J D, Soljacić M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature, 2009, 461(7265): 772–775

[17] Hafezi M, Demler E A, Lukin M D, Taylor J M. Robust optical delay lines with topological protection. Nature Physics, 2011, 7(11): 907–912

[18] Umucalılar R O, Carusotto I. Artificial gauge field for photons in coupled cavity arrays. Physical Review A, 2011, 84(4): 043804

[19] Khanikaev A B, Mousavi S H, Tse W K, Kargarian M, MacDonald A H, Shvets G. Photonic topological insulators. Nature Materials, 2013, 12(3): 233–239

[20] Nalitov A V, Malpuech G, Terças H, Solnyshkov D D. Spin-orbit coupling and the optical spin Hall effect in photonic graphene. Physical Review Letters, 2015, 114(2): 026803

[21] Wu L H, Hu X. Scheme for achieving a topological photonic crystal by using dielectric material. Physical Review Letters, 2015, 114(22): 223901

[22] Cheng X, Jouvaud C, Ni X, Mousavi S H, Genack A Z, Khanikaev A B. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nature Materials, 2016, 15(5): 542–548

[23] Dong J W, Chen X D, Zhu H, Wang Y, Zhang X. Valley photonic crystals for control of spin and topology. Nature Materials, 2017, 16(3): 298–302

[24] Yang Y, Xu Y F, Xu T, Wang H X, Jiang J H, Hu X, Hang Z H. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials. Physical Review Letters, 2018, 120(21): 217401

[25] Fang K, Yu Z, Fan S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photonics, 2012, 6(11): 782–787

[26] Lumer Y, Plotnik Y, Rechtsman M C, Segev M. Self-localized states in photonic topological insulators. Physical Review Letters, 2013, 111(24): 243905

[27] Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A. Photonic Floquet topological insulators. Nature, 2013, 496(7444): 196–200

[28] Titum P, Lindner N H, Rechtsman M C, Refael G. Disorder-induced Floquet topological insulators. Physical Review Letters, 2015, 114(5): 056801

[29] Leykam D, Rechtsman M C, Chong Y D. Anomalous topological phases and unpaired dirac cones in photonic Floquet topological insulators. Physical Review Letters, 2016, 117(1): 013902

[30] Maczewsky L J, Zeuner J M, Nolte S, Szameit A. Observation of photonic anomalous Floquet topological insulators. Nature Communications, 2017, 8(1): 13756

[31] Mukherjee S, Spracklen A, Valiente M, Andersson E, Öhberg P, Goldman N, Thomson R R. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nature Communications, 2017, 8(1): 13918

[32] Mukherjee S, Chandrasekharan H K, Öhberg P, Goldman N, Thomson R R. State-recycling and time-resolved imaging in topological photonic lattices. Nature Communications, 2018, 9(1): 4209

[33] Zhu B, Zhong H, Ke Y, Qin X, Sukhorukov A A, Kivshar Y S, Lee C. Topological Floquet edge states in periodically curved waveguides. Physical Review A, 2018, 98(1): 013855

[34] Nathan F, Abanin D, Berg E, Lindner N H, Rudner M S. Anomalous Floquet insulators. Physical Review B, 2019, 99(19): 195133

[35] Ma T, Shvets G. All-Si valley-Hall photonic topological insulator. New Journal of Physics, 2016, 18(2): 025012

[36] Wu X, Meng Y, Tian J, Huang Y, Xiang H, Han D, Wen W. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals. Nature Communications, 2017, 8(1): 1304

[37] Slobozhanyuk A, Mousavi S H, Ni X, Smirnova D, Kivshar Y S, Khanikaev A B. Three-dimensional all-dielectric photonic topological insulator. Nature Photonics, 2017, 11(2): 130–136

[38] Yang Y, Gao Z, Xue H, Zhang L, He M, Yang Z, Singh R, Chong Y, Zhang B, Chen H. Realization of a three-dimensional photonic topological insulator. Nature, 2019, 565(7741): 622–626

[39] Young S M, Zaheer S, Teo J C, Kane C L, Mele E J, Rappe A M. Dirac semimetal in three dimensions. Physical Review Letters, 2012, 108(14): 140405

[40] Yang B J, Nagaosa N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nature Communications, 2014, 5(1): 4898

[41] Liu Z K, Zhou B, Zhang Y, Wang Z J, Weng H M, Prabhakaran D, Mo S K, Shen Z X, Fang Z, Dai X, Hussain Z, Chen Y L. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science, 2014, 343(6173): 864–867

[42] Yang B, Guo Q, Tremain B, Barr L E, Gao W, Liu H, Béri B, Xiang Y, Fan D, Hibbins A P, Zhang S. Direct observation of topological surface-state arcs in photonic metamaterials. Nature Communications, 2017, 8(1): 97

[43] Li F, Huang X, Lu J, Ma J, Liu Z. Weyl points and Fermi arcs in a chiral phononic crystal. Nature Physics, 2018, 14(1): 30–34

[44] Burkov A A, Hook M D, Balents L. Topological nodal semimetals. Physical Review B, 2011, 84(23): 235126

[45] Yan Z, Wang Z. Tunable Weyl points in periodically driven nodal line semimetals. Physical Review Letters, 2016, 117(8): 087402

[46] He H, Qiu C, Ye L, Cai X, Fan X, Ke M, Zhang F, Liu Z. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature, 2018, 560(7716): 61–64

[47] Adair R, Chase L L, Payne S A. Nonlinear refractive index of optical crystals. Physical Review B, 1989, 39(5): 3337–3350

[48] Berger V. Nonlinear photonic crystals. Physical Review Letters, 1998, 81(19): 4136–4139

[49] Mingaleev S F, Kivshar Y S. Self-trapping and stable localized modes in nonlinear photonic crystals. Physical Review Letters, 2001, 86(24): 5474–5477

[50] Soljačić M, Luo C, Joannopoulos J D, Fan S. Nonlinear photonic crystal microdevices for optical integration. Optics Letters, 2003, 28(8): 637–639

[51] Fleischer J W, Segev M, Efremidis N K, Christodoulides D N. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature, 2003, 422(6928): 147–150

[52] Soljačić M, Joannopoulos J D. Enhancement of nonlinear effects using photonic crystals. Nature Materials, 2004, 3(4): 211–219

[53] Haddad L H, Weaver C M, Carr L D. The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices. New Journal of Physics, 2015, 17(6): 063033

[54] Hadad Y, Khanikaev A B, Alù A. Self-induced topological transitions and edge states supported by nonlinear staggered potentials. Physical Review B, 2016, 93(15): 155112

[55] Leykam D, Chong Y D. Edge solitons in nonlinear-photonic topological insulators. Physical Review Letters, 2016, 117(14): 143901

[56] Roushan P, Neill C, Megrant A, Chen Y, Babbush R, Barends R, Campbell B, Chen Z, Chiaro B, Dunsworth A, Fowler A, Jeffrey E, Kelly J, Lucero E, Mutus J, O’Malley P J J, Neeley M, Quintana C, Sank D, Vainsencher A, Wenner J, White T, Kapit E, Neven H, Martinis J. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nature Physics, 2017, 13(2): 146–151

[57] Tai M E, Lukin A, Rispoli M, Schittko R, Menke T, Dan Borgnia , Preiss P M, Grusdt F, Kaufman A M, Greiner M. Microscopy of the interacting Harper-Hofstadter model in the two-body limit. Nature, 2017, 546(7659): 519–523

[58] Zhou X, Wang Y, Leykam D, Chong Y D. Optical isolation with nonlinear topological photonics. New Journal of Physics, 2017, 19(9): 095002

[59] Dobrykh D A, Yulin A V, Slobozhanyuk A P, Poddubny A N, Kivshar Y S. Nonlinear control of electromagnetic topological edge states. Physical Review Letters, 2018, 121(16): 163901

[60] Rajesh C, Georgios T. Self-induced topological transition in phononic crystals by nonlinearity management. 2019, arXiv:1904. 09466v1

[61] Bender C M, Boettcher S. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters, 1998, 80(24): 5243–5246

[62] Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N, Peschel U. Parity-time synthetic photonic lattices. Nature, 2012, 488(7410): 167–171

[63] Yang Y, Peng C, Liang Y, Li Z, Noda S. Analytical perspective for bound states in the continuum in photonic crystal slabs. Physical Review Letters, 2014, 113(3): 037401

[64] Zhen B, Hsu C W, Lu L, Stone A D, Soljačić M. Topological nature of optical bound states in the continuum. Physical Review Letters, 2014, 113(25): 257401

[65] Malzard S, Poli C, Schomerus H. Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry. Physical Review Letters, 2015, 115(20): 200402

[66] Zeuner J M, Rechtsman M C, Plotnik Y, Lumer Y, Nolte S, Rudner M S, Segev M, Szameit A. Observation of a topological transition in the bulk of a non-Hermitian system. Physical Review Letters, 2015, 115(4): 040402

[67] Zhen B, Hsu C W, Igarashi Y, Lu L, Kaminer I, Pick A, Chua S L, Joannopoulos J D, Soljačić M. Spawning rings of exceptional points out of Dirac cones. Nature, 2015, 525(7569): 354–358

[68] Cerjan A, Raman A, Fan S. Exceptional contours and band structure design in parity-time symmetric photonic crystals. Physical Review Letters, 2016, 116(20): 203902

[69] Bulgakov E N, Maksimov D N. Topological bound states in the continuum in arrays of dielectric spheres. Physical Review Letters, 2017, 118(26): 267401

[70] Feng L, El-Ganainy R, Ge L. Non-Hermitian photonics based on parity-time symmetry. Nature Photonics, 2017, 11(12): 752–762

[71] Kodigala A, Lepetit T, Gu Q, Bahari B, Fainman Y, Kanté B. Lasing action from photonic bound states in continuum. Nature, 2017, 541(7636): 196–199

[72] Weimann S, Kremer M, Plotnik Y, Lumer Y, Nolte S, Makris K G, Segev M, Rechtsman M C, Szameit A. Topologically protected bound states in photonic parity-time-symmetric crystals. Nature Materials, 2017, 16(4): 433–438

[73] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N. Non-Hermitian physics and PT symmetry. Nature Physics, 2018, 14(1): 11–19

[74] Kawabata K, Shiozaki K, Ueda M. Anomalous helical edge states in a non-Hermitian Chern insulator. Physical Review B, 2018, 98(16): 165148

[75] Kunst F K, Edvardsson E, Budich J C, Bergholtz E J. Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Physical Review Letters, 2018, 121(2): 026808

[76] Lieu S. Topological phases in the non-Hermitian Su-Schrieffer-Heeger model. Physical Review B, 2018, 97(4): 045106

[77] Pan M, Zhao H, Miao P, Longhi S, Feng L. Photonic zero mode in a non-Hermitian photonic lattice. Nature Communications, 2018, 9(1): 1308

[78] Qi B, Zhang L, Ge L. Defect states emerging from a non-Hermitian flatband of photonic zero modes. Physical Review Letters, 2018, 120(9): 093901

[79] Shen H, Zhen B, Fu L. Topological band theory for non-Hermitian Hamiltonians. Physical Review Letters, 2018, 120(14): 146402

[80] Wang H F, Gupta S K, Zhu X Y, Lu M H, Liu X P, Chen Y F. Bound states in the continuum in a bilayer photonic crystal with TE-TM cross coupling. Physical Review. B, 2018, 98(21): 214101

[81] Yao S, Song F, Wang Z. Non-Hermitian Chern bands. Physical Review Letters, 2018, 121(13): 136802

[82] Yao S, Wang Z. Edge states and topological invariants of non-Hermitian systems. Physical Review Letters, 2018, 121(8): 086803

[83] Chen X D, Deng W M, Shi F L, Zhao F L, Chen M, Dong J W. Direct observation of corner states in second-order topological photonic crystal slabs. 2018, arXiv:1812.08326

[84] Ezawa M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Physical Review Letters, 2018, 120(2): 026801

[85] Ezawa M. Minimal models for Wannier-type higher-order topological insulators and phosphorene. Physical Review B, 2018, 98(4): 045125

[86] Ezawa M. Magnetic second-order topological insulators and semimetals. Physical Review B, 2018, 97(15): 155305

[87] Ezawa M. Higher-order topological electric circuits and topological corner resonance on the breathing kagome and pyrochlore lattices. Physical Review B, 2018, 98(20): 201402

[88] Geier M, Trifunovic L, Hoskam M, Brouwer P W. Second-order topological insulators and superconductors with an order-two crystalline symmetry. Physical Review B, 2018, 97(20): 205135

[89] Khalaf E. Higher-order topological insulators and superconductors protected by inversion symmetry. Physical Review B, 2018, 97(20): 205136

[90] Kunst F K, van Miert G, Bergholtz E J. Lattice models with exactly solvable topological hinge and corner states. Physical Review B, 2018, 97(24): 241405

[91] Peterson C W, Benalcazar W A, Hughes T L, Bahl G. A quantized microwave quadrupole insulator with topologically protected corner states. Nature, 2018, 555(7696): 346–350

[92] Schindler F, Cook A M, Vergniory M G, Wang Z, Parkin S S, Bernevig B A, Neupert T.Higher-order topological insulators. Science Advances, 2018, 4(6): eaat0346

[93] van Miert G, Ortix C. Higher-order topological insulators protected by inversion and rotoinversion symmetries. Physical Review B, 2018, 98(8): 081110

[94] Xie B Y, Wang H F, Wang H X, Zhu X Y, Jiang J H, Lu M H, Chen Y F. Second-order photonic topological insulator with corner states. Physical Review B, 2018, 98(20): 205147

[95] Xie B Y, Su G X, Wang H F, Su H, Shen X P, Zhan P, Lu M H, Wang Z L, Chen Y F. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals. Physical Review Letters, 2019, 122(23): 233903

[96] Yasutomo O, Feng L, Ryota K, Katsuyuki W, Katsunori W, Yasuhiko A, Satoshi I. Photonic crystal nanocavity based on a topological corner state. 2018, arXiv:1812.10171

[97] Călugăru D, Juričić V, Roy B. Higher-order topological phases: a general principle of construction. Physical Review B, 2019, 99(4): 041301

[98] Hu H, Huang B, Zhao E, Liu W V. Dynamical singularities of Floquet higher-order topological insulators. 2019, arXiv:1905. 03727v1

[99] Armstrong J A, Bloembergen N, Ducuing J, Pershan P S. Interactions between light waves in a nonlinear dielectric. Physical Review, 1962, 127(6): 1918–1939

[100] Kleinman D A. Nonlinear dielectric polarization in optical media. Physical Review, 1962, 126(6): 1977–1979

[101] Adler E. Nonlinear optical frequency polarization in a dielectric. Physical Review, 1964, 134(3A): A728–A733

[102] Miller R C. Optical second harmonic generation in piezoelectric crystals. Applied Physics Letters, 1964, 5(1): 17–19

[103] Fejer M M, Magel G, Jundt D H, Byer R L. Quasi-phase-matched second harmonic generation: tuning and tolerances. IEEE Journal of Quantum Electronics, 1992, 28(11): 2631–2654

[104] Yamada M, Nada N, Saitoh M, Watanabe K. First-order quasi-phase matched LiNbO3waveguide periodically poled by applying an external field for efficient blue second–harmonic generation. Applied Physics Letters, 1993, 62(5): 435–436

[105] Celebrano M, Wu X, Baselli M, Großmann S, Biagioni P, Locatelli A, De Angelis C, Cerullo G, Osellame R, Hecht B, Duò L, Ciccacci F, Finazzi M. Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation. Nature Nanotechnology, 2015, 10(5): 412–417

[106] Rubin M H, Klyshko D N, Shih Y H, Sergienko A V. Theory of two-photon entanglement in type-II optical parametric down-conversion. Physical Review A, 1994, 50(6): 5122–5133

[107] Monken C H, Ribeiro P S, Pádua S. Transfer of angular spectrum and image formation in spontaneous parametric down-conversion. Physical Review A, 1998, 57(4): 3123–3126

[108] Arnaut H H, Barbosa G A. Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion. Physical Review Letters, 2000, 85(2): 286–289

[109] Howell J C, Bennink R S, Bentley S J, Boyd R W. Realization of the Einstein-Podolsky-Rosen paradox using momentum- and position-entangled photons from spontaneous parametric down conversion. Physical Review Letters, 2004, 92(21): 210403

[110] Harder G, Bartley T J, Lita A E, Nam S W, Gerrits T, Silberhorn C. Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics. Physical Review Letters, 2016, 116(14): 143601

[111] Carriles R, Schafer D N, Sheetz K E, Field J J, Cisek R, Barzda V, Sylvester A W, Squier J A. Imaging techniques for harmonic and multiphoton absorption fluorescence microscopy. Review of Scientific Instruments, 2009, 80(8): 081101

[112] Grinblat G, Li Y, Nielsen M P, Oulton R F, Maier S A. Enhanced third harmonic generation in single germanium nanodisks excited at the anapole mode. Nano Letters, 2016, 16(7): 4635–4640

[113] Sipe J E, Moss D J, van Driel H. Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals. Physical Review B, 1987, 35(3): 1129–1141

[114] Zhu S, Zhu Y, Ming N. Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice. Science, 1997, 278(5339): 843–846

[115] Soavi G, Wang G, Rostami H, Purdie D G, De Fazio D, Ma T, Luo B, Wang J, Ott A K, Yoon D, Bourelle S A, Muench J E, Goykhman I, Dal Conte S, Celebrano M, Tomadin A, Polini M, Cerullo G, Ferrari A C. Broadband, electrically tunable third-harmonic generation in graphene. Nature Nanotechnology, 2018, 13(7): 583–588

[116] Slusher R E, Hollberg L W, Yurke B, Mertz J C, Valley J F. Observation of squeezed states generated by four-wave mixing in an optical cavity. Physical Review Letters, 1985, 55(22): 2409–2412

[117] Deng L, Hagley E W, Wen J, Trippenbach M, Band Y, Julienne P S, Simsarian J, Helmerson K, Rolston S, Phillips W D. Four-wave mixing with matter waves. Nature, 1999, 398(6724): 218–220

[118] Bencivenga F, Cucini R, Capotondi F, Battistoni A, Mincigrucci R, Giangrisostomi E, Gessini A, Manfredda M, Nikolov I P, Pedersoli E, Principi E, Svetina C, Parisse P, Casolari F, Danailov M B, Kiskinova M, Masciovecchio C. Four-wave mixing experiments with extreme ultraviolet transient gratings. Nature, 2015, 520(7546): 205–208

[119] Singh S K, Abak M K, Tasgin M E. Enhancement of four-wave mixing via interference of multiple plasmonic conversion paths. Physical Review B, 2016, 93(3): 035410

[120] Zhang H, Virally S, Bao Q, Ping L K, Massar S, Godbout N, Kockaert P. Z-scan measurement of the nonlinear refractive index of graphene. Optics Letters, 2012, 37(11): 1856–1858

[121] Alam M Z, De Leon I, Boyd R W. Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region. Science, 2016, 352(6287): 795–797

[122] Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M C, Schuster D, Simon J, Zilberberg O, Carusotto I. Topological photonics. Reviews of Modern Physics, 2019, 91(1): 015006

[123] Berry M V. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, 1802, 1984(392): 45–57

[124] Pancharatnam S. Generalized theory of interference and its applications. Proceedings of the Indian Academy of Sciences, Section A, Physical Sciences, 1956, 44(6): 398–417

[125] Skirlo S A, Lu L, Igarashi Y, Yan Q, Joannopoulos J, Soljačić M. Experimental observation of large Chern numbers in photonic crystals. Physical Review Letters, 2015, 115(25): 253901

[126] Lu L, Wang Z, Ye D, Ran L, Fu L, Joannopoulos J D, Soljačić M. Experimental observation of Weyl points. Science, 2015, 349(6248): 622–624

[127] Xiao M, Lin Q, Fan S. Hyperbolic Weyl point in reciprocal chiral metamaterials. Physical Review Letters, 2016, 117(5): 057401

[128] Lin Q, Xiao M, Yuan L, Fan S. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension. Nature Communications, 2016, 7(1): 13731

[129] Fang C, Weng H, Dai X, Fang Z. Topological nodal line semimetals. Chinese Physics B, 2016, 25(11): 117106

[130] Lu L, Fu L, Joannopoulos J D, Soljačić M. Weyl points and line nodes in gyroid photonic crystals. Nature Photonics, 2013, 7(4): 294–299

[131] Yang B, Guo Q, Tremain B, Liu R, Barr L E, Yan Q, Gao W, Liu H, Xiang Y, Chen J, Fang C, Hibbins A, Lu L, Zhang S. Ideal Weyl points and helicoid surface states in artificial photonic crystal structures. Science, 2018, 359(6379): 1013–1016

[132] Chen W J, Jiang S J, Chen X D, Zhu B, Zhou L, Dong J W, Chan C T. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nature Communications, 2014, 5(1): 5782

[133] Slobozhanyuk A P, Khanikaev A B, Filonov D S, Smirnova D A, Miroshnichenko A E, Kivshar Y S. Experimental demonstration of topological effects in bianisotropic metamaterials. Scientific Reports, 2016, 6(1): 22270

[134] Shalaev M I, Walasik W, Tsukernik A, Xu Y, Litchinitser N M. Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nature Nanotechnology, 2019, 14(1): 31–34

[135] Chen X D, Zhao F L, Chen M, Dong J W. Valley-contrasting physics in all-dielectric photonic crystals: orbital angular momentum and topological propagation. Physical Review B, 2017, 96(2): 020202

[136] Chen X D, Shi F L, Liu H, Lu J C, Deng W M, Dai J Y, Cheng Q, Dong J W. Tunable electromagnetic flow control in valley photonic crystal waveguides. Physical Review Applied, 2018, 10(4): 044002

[137] He M, Zhang L, Wang H. Two-dimensional photonic crystal with ring degeneracy and its topological protected edge states. Scientific Reports, 2019, 9(1): 3815

[138] Ma T, Khanikaev A B, Mousavi S H, Shvets G. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides. Physical Review Letters, 2015, 114(12): 127401

[139] Chen W J, Xiao M, Chan C T. Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states. Nature Communications, 2016, 7(1): 13038

[140] Chen Y, Chen H, Cai G. High transmission in a metal-based photonic crystal. Applied Physics Letters, 2018, 112(1): 013504

[141] El-Kady I, Sigalas M, Biswas R, Ho K, Soukoulis C. Metallic photonic crystals at optical wavelengths. Physical Review B, 2000, 62(23): 15299–15302

[142] Gao F, Gao Z, Shi X, Yang Z, Lin X, Xu H, Joannopoulos J D, Soljačić M, Chen H, Lu L, Chong Y, Zhang B. Probing topological protection using a designer surface plasmon structure. Nature Communications, 2016, 7(1): 11619

[143] Gao W, Yang B, Tremain B, Liu H, Guo Q, Xia L, Hibbins A P, Zhang S. Experimental observation of photonic nodal line degeneracies in metacrystals. Nature Communications, 2018, 9(1): 950

[144] Gao F, Xue H, Yang Z, Lai K, Yu Y, Lin X, Chong Y, Shvets G, Zhang B. Topologically protected refraction of robust kink states in valley photonic crystals. Nature Physics, 2018, 14(2): 140–144

[145] Karch A. Surface plasmons and topological insulators. Physical Review B, 2011, 83(24): 245432

[146] Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M. Imaging topological edge states in silicon photonics. Nature Photonics, 2013, 7(12): 1001–1005

[147] Mittal S, Ganeshan S, Fan J, Vaezi A, Hafezi M. Measurement of topological invariants in a 2D photonic system. Nature Photonics, 2016, 10(3): 180–183

[148] Harari G, Bandres M A, Lumer Y, Rechtsman M C, Chong Y D, Khajavikhan M, Christodoulides D N, Segev M.Topological insulator laser: theory. Science, 2018, 359(6381): eaar4003

[149] Bandres M A, Wittek S, Harari G, Parto M, Ren J, Segev M, Christodoulides D N, Khajavikhan M. Topological insulator laser: experiments. Science, 2018, 359(6381): eaar4005

[150] Midya B, Zhao H, Feng L. Non-Hermitian photonics promises exceptional topology of light. Nature Communications, 2018, 9(1): 2674

[151] Barik S, Karasahin A, Flower C, Cai T, Miyake H, DeGottardi W, Hafezi M, Waks E. A topological quantum optics interface. Science, 2018, 359(6376): 666–668

[152] Blanco-Redondo A, Bell B, Oren D, Eggleton B J, Segev M. Topological protection of biphoton states. Science, 2018, 362(6414): 568–571

[153] Piper J R, Fan S. Total absorption in a graphene monolayer in the optical regime by critical coupling with a photonic crystal guided resonance. ACS Photonics, 2014, 1(4): 347–353

[154] Gan X, Mak K F, Gao Y, You Y, Hatami F, Hone J, Heinz T F, Englund D. Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity. Nano Letters, 2012, 12(11): 5626–5631

[155] Heeger A J, Kivelson S, Schrieffer J R, Su W P. Solitons in conducting polymers. Reviews of Modern Physics, 1988, 60(3): 781–850

[156] Su W P, Schrieffer J R, Heeger A J. Solitons in Polyacetylene. Physical Review Letters, 1979, 42(25): 1698–1701

[157] Miri M-A, Alù A. Exceptional points in optics and photonics. Science, 2019, 363(6422): eaar7709

[158] Gupta S K, Zou Y, Zhu X Y, Lu M H, Zhang L, Liu X P, Chen Y F. Parity-time symmetry in Non-Hermitian complex media. 2018, arXiv:1803.00794

[159] Lee T E. Anomalous edge state in a non-Hermitian lattice. Physical Review Letters, 2016, 116(13): 133903

[160] Ghatak A, Das T. New topological invariants in non-Hermitian systems. Journal of Physics Condensed Matter, 2019, 31(26): 263001

[161] St-Jean P, Goblot V, Galopin E, Lemaître A, Ozawa T, Le Gratiet L, Sagnes I, Bloch J, Amo A. Lasing in topological edge states of a one-dimensional lattice. Nature Photonics, 2017, 11(10): 651–656

[162] Parto M, Wittek S, Hodaei H, Harari G, Bandres M A, Ren J, Rechtsman M C, Segev M, Christodoulides D N, Khajavikhan M. Edge-mode lasing in 1D topological active arrays. Physical Review Letters, 2018, 120(11): 113901

[163] Zhao H, Miao P, Teimourpour M H, Malzard S, El-Ganainy R, Schomerus H, Feng L. Topological hybrid silicon microlasers. Nature Communications, 2018, 9(1): 981

[164] Ota Y, Katsumi R, Watanabe K, Iwamoto S, Arakawa Y. Topological photonic crystal nanocavity laser. Communications on Physics, 2018, 1(1): 86

[165] Haldane F D M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Physical Review Letters, 1988, 61(18): 2015–2018

[166] Schmidt J, Marques M R G, Botti S, Marques M A L. Recent advances and applications of machine learning in solid-state materials science. NPJ Computational Materials, 2019, 5(1): 83

[167] Pilozzi L, Farrelly F A, Marcucci G, Conti C. Machine learning inrerse problem for topological photonics. Communications Physics, 2018, 1(1): 57

[168] Long Y, Ren J, Li Y, Chen H. Inverse design of photonic topological state via machine learning. Applied Physics Letters, 2019, 114(18): 181105

[169] Barth C, Becker C. Machine learning classification for field distributions of photonic modes. Communications on Physics, 2018, 1(1): 58

[170] Fano U. Effects of configuration interaction on intensities and phase shifts. Physical Review, 1961, 124(6): 1866–1878

[171] Limonov M F, Rybin M V, Poddubny A N, Kivshar Y S. Fano resonances in photonics. Nature Photonics, 2017, 11(9): 543–554

[172] Miroshnichenko A E, Flach S, Kivshar Y S. Fano resonances in nanoscale structures. Reviews of Modern Physics, 2010, 82(3): 2257–2298

[173] Luk’yanchuk B S, Miroshnichenko A E, Kivshar Y S. Fano resonances and topological optics: an interplay of far- and near-field interference phenomena. Journal of Optics, 2013, 15(7): 073001

[174] Gao W, Hu X, Li C, Yang J, Chai Z, Xie J, Gong Q. Fano-resonance in one-dimensional topological photonic crystal heterostructure. Optics Express, 2018, 26(7): 8634–8644

[175] Zangeneh-Nejad F, Fleury R. Topological Fano resonances. Physical Review Letters, 2019, 122(1): 014301

[176] Liang G Q, Chong Y D. Optical resonator analog of a two-dimensional topological insulator. Physical Review Letters, 2013, 110(20): 203904

Acknowledgements

This work was supported by the National Key R&D Program of China (Nos. 2018YFA0306200, and 2017YFA0303702) and the National Natural Science Foundation of China (Grant Nos. 11625418, 51732006, and 11890700), as well as the Academic Program Development of Jiangsu Higher Education (PAPD).

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