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 higherorder 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
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).
Footnotes
Hongfei Wang is a Ph.D. candidate in the Department of Materials Science and Engineering at Nanjing University. He spent his bachelor time at Anhui University during 2011–2015. His research topics include topological photonics, non-Hermitian photonics, and computational physics.
Dr. Samit Kumar Gupta received his Ph.D. in 2016 from the Department of Physics, Indian Institute of Technology Guwahati, India. Afterward, in 2017 he joined the College of Engineering and Applied Sciences, National Laboratory of Solid State Microstructures, Nanjing University as a Postdoc Fellow. His research interests include fundamental and applied aspects of nonlinear optics, nonlinear waves, non-Hermitianp hysics, and topological photonics.
Dr. Biye Xie spent his bachelor time at the University of Science and Technology of China. He received his Ph.D. degree in Physics from the University of Hong Kong, China. His research interest includes topological photonics, topological phononics, metamaterials, and quantum information.
Prof. Minghui Lu received his Ph.D. degree from Nanjing University in 2007. He is an Associate Professor at Nanjing University since 2009 and a Professor in 2013. He had been a visiting scholar at SIMES, Stanford University during 2011–2012. His current research interests mainly focus on fundamental study of photonic and acoustic artificial structures and metamaterials as well as their related applications.
References
- 1.Yablonovitch E. Inhibited spontaneous emission in solid-state physics and electronics. Physical Review Letters. 1987;58(20):2059–2062. doi: 10.1103/PhysRevLett.58.2059. [DOI] [PubMed] [Google Scholar]
- 2.John S. Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters. 1987;58(23):2486–2489. doi: 10.1103/PhysRevLett.58.2486. [DOI] [PubMed] [Google Scholar]
- 3.Wang B, Cappelli M A. A plasma photonic crystal bandgap device. Applied Physics Letters. 2016;108(16):161101. [Google Scholar]
- 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. doi: 10.1038/nature02063. [DOI] [PubMed] [Google Scholar]
- 5.Shelby R A, Smith D R, Schultz S. Experimental verification of a negative index of refraction. Science. 2001;292(5514):77–79. doi: 10.1126/science.1058847. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1364/ol.30.003356. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 9.Kohmoto M. Topological invariant and the quantization of the Hall conductance. Annals of Physics. 1985;160(2):343–354. [Google Scholar]
- 10.Kane C L, Mele E J. Quantum spin Hall effect in graphene. Physical Review Letters. 2005;95(22):226801. doi: 10.1103/PhysRevLett.95.226801. [DOI] [PubMed] [Google Scholar]
- 11.Bernevig B A, Zhang S C. Quantum spin Hall effect. Physical Review Letters. 2006;96(10):106802. doi: 10.1103/PhysRevLett.96.106802. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.1133734. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.1148047. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.100.013904. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.100.013905. [DOI] [PubMed] [Google Scholar]
- 16.Wang Z, Chong Y, Joannopoulos J D, Soljacić M. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature. 2009;461(7265):772–775. doi: 10.1038/nature08293. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 18.Umucalılar R O, Carusotto I. Artificial gauge field for photons in coupled cavity arrays. Physical Review A. 2011;84(4):043804. [Google Scholar]
- 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. doi: 10.1038/nmat3520. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.114.026803. [DOI] [PubMed] [Google Scholar]
- 21.Wu L H, Hu X. Scheme for achieving a topological photonic crystal by using dielectric material. Physical Review Letters. 2015;114(22):223901. doi: 10.1103/PhysRevLett.114.223901. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/nmat4573. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/nmat4807. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.120.217401. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 26.Lumer Y, Plotnik Y, Rechtsman M C, Segev M. Self-localized states in photonic topological insulators. Physical Review Letters. 2013;111(24):243905. doi: 10.1103/PhysRevLett.111.243905. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/nature12066. [DOI] [PubMed] [Google Scholar]
- 28.Titum P, Lindner N H, Rechtsman M C, Refael G. Disorder-induced Floquet topological insulators. Physical Review Letters. 2015;114(5):056801. doi: 10.1103/PhysRevLett.114.056801. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.117.013902. [DOI] [PubMed] [Google Scholar]
- 30.Maczewsky L J, Zeuner J M, Nolte S, Szameit A. Observation of photonic anomalous Floquet topological insulators. Nature Communications. 2017;8(1):13756. doi: 10.1038/ncomms13756. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1038/ncomms13918. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1038/s41467-018-06723-y. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 34.Nathan F, Abanin D, Berg E, Lindner N H, Rudner M S. Anomalous Floquet insulators. Physical Review B. 2019;99(19):195133. [Google Scholar]
- 35.Ma T, Shvets G. All-Si valley-Hall photonic topological insulator. New Journal of Physics. 2016;18(2):025012. [Google Scholar]
- 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. doi: 10.1038/s41467-017-01515-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/s41586-018-0829-0. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.108.140405. [DOI] [PubMed] [Google Scholar]
- 40.Yang B J, Nagaosa N. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nature Communications. 2014;5(1):4898. doi: 10.1038/ncomms5898. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.1245085. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/s41467-017-00134-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 44.Burkov A A, Hook M D, Balents L. Topological nodal semimetals. Physical Review B. 2011;84(23):235126. [Google Scholar]
- 45.Yan Z, Wang Z. Tunable Weyl points in periodically driven nodal line semimetals. Physical Review Letters. 2016;117(8):087402. doi: 10.1103/PhysRevLett.117.087402. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/s41586-018-0367-9. [DOI] [PubMed] [Google Scholar]
- 47.Adair R, Chase L L, Payne S A. Nonlinear refractive index of optical crystals. Physical Review B. 1989;39(5):3337–3350. doi: 10.1103/physrevb.39.3337. [DOI] [PubMed] [Google Scholar]
- 48.Berger V. Nonlinear photonic crystals. Physical Review Letters. 1998;81(19):4136–4139. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.86.5474. [DOI] [PubMed] [Google Scholar]
- 50.Soljačić M, Luo C, Joannopoulos J D, Fan S. Nonlinear photonic crystal microdevices for optical integration. Optics Letters. 2003;28(8):637–639. doi: 10.1364/ol.28.000637. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/nature01452. [DOI] [PubMed] [Google Scholar]
- 52.Soljačić M, Joannopoulos J D. Enhancement of nonlinear effects using photonic crystals. Nature Materials. 2004;3(4):211. doi: 10.1038/nmat1097. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 55.Leykam D, Chong Y D. Edge solitons in nonlinear-photonic topological insulators. Physical Review Letters. 2016;117(14):143901. doi: 10.1103/PhysRevLett.117.143901. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 57.Tai M E, Lukin A, Rispoli M, Schittko R, Menke T, Dan B, 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. doi: 10.1038/nature22811. [DOI] [PubMed] [Google Scholar]
- 58.Zhou X, Wang Y, Leykam D, Chong Y D. Optical isolation with nonlinear topological photonics. New Journal of Physics. 2017;19(9):095002. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.121.163901. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/nature11298. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.113.037401. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.113.257401. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.115.200402. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.115.040402. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/nature14889. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.116.203902. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.118.267401. [DOI] [PubMed] [Google Scholar]
- 70.Feng L, El-Ganainy R, Ge L. Non-Hermitian photonics based on parity-time symmetry. Nature Photonics. 2017;11(12):752–762. [Google Scholar]
- 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. doi: 10.1038/nature20799. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/nmat4811. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 74.Kawabata K, Shiozaki K, Ueda M. Anomalous helical edge states in a non-Hermitian Chern insulator. Physical Review B. 2018;98(16):165148. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.121.026808. [DOI] [PubMed] [Google Scholar]
- 76.Lieu S. Topological phases in the non-Hermitian Su-Schrieffer-Heeger model. Physical Review B. 2018;97(4):045106. [Google Scholar]
- 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. doi: 10.1038/s41467-018-03822-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.120.093901. [DOI] [PubMed] [Google Scholar]
- 79.Shen H, Zhen B, Fu L. Topological band theory for non-Hermitian Hamiltonians. Physical Review Letters. 2018;120(14):146402. doi: 10.1103/PhysRevLett.120.146402. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 81.Yao S, Song F, Wang Z. Non-Hermitian Chern bands. Physical Review Letters. 2018;121(13):136802. doi: 10.1103/PhysRevLett.121.136802. [DOI] [PubMed] [Google Scholar]
- 82.Yao S, Wang Z. Edge states and topological invariants of non-Hermitian systems. Physical Review Letters. 2018;121(8):086803. doi: 10.1103/PhysRevLett.121.086803. [DOI] [PubMed] [Google Scholar]
- 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 [DOI] [PubMed]
- 84.Ezawa M. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Physical Review Letters. 2018;120(2):026801. doi: 10.1103/PhysRevLett.120.026801. [DOI] [PubMed] [Google Scholar]
- 85.Ezawa M. Minimal models for Wannier-type higher-order topological insulators and phosphorene. Physical Review B. 2018;98(4):045125. [Google Scholar]
- 86.Ezawa M. Magnetic second-order topological insulators and semimetals. Physical Review B. 2018;97(15):155305. doi: 10.1103/PhysRevLett.120.026801. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.120.026801. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 89.Khalaf E. Higher-order topological insulators and superconductors protected by inversion symmetry. Physical Review B. 2018;97(20):205136. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/nature25777. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/sciadv.aat0346. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 93.van Miert G, Ortix C. Higher-order topological insulators protected by inversion and rotoinversion symmetries. Physical Review B. 2018;98(8):081110. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.122.233903. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 98.Hu H, Huang B, Zhao E, Liu W V. Dynamical singularities of Floquet higher-order topological insulators. 2019, arXiv:1905. 03727v1 [DOI] [PubMed]
- 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. [Google Scholar]
- 100.Kleinman D A. Nonlinear dielectric polarization in optical media. Physical Review. 1962;126(6):1977–1979. [Google Scholar]
- 101.Adler E. Nonlinear optical frequency polarization in a dielectric. Physical Review. 1964;134(3A):A728–A733. [Google Scholar]
- 102.Miller R C. Optical second harmonic generation in piezoelectric crystals. Applied Physics Letters. 1964;5(1):17–19. [Google Scholar]
- 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. [Google Scholar]
- 104.Yamada M, Nada N, Saitoh M, Watanabe K. First-order quasiphase matched LiNbO3waveguide periodically poled by applying an external field for efficient blue second-harmonic generation. Applied Physics Letters. 1993;62(5):435–436. [Google Scholar]
- 105.Celebrano M, Wu X, Baselli M, Großmann S, Biagioni P, Locatelli A, De Angelis C, Cerullo G, Osellame R, Hecht B, Duó L, Ciccacci F, Finazzi M. Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation. Nature Nanotechnology. 2015;10(5):412–417. doi: 10.1038/nnano.2015.69. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/physreva.50.5122. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.85.286. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.92.210403. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.116.143601. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1063/1.3184828. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1021/acs.nanolett.6b01958. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/physrevb.35.1129. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/s41565-018-0145-8. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.55.2409. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/nature14341. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1364/OL.37.001856. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.aae0330. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.115.253901. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.aaa9273. [DOI] [PubMed] [Google Scholar]
- 127.Xiao M, Lin Q, Fan S. Hyperbolic Weyl point in reciprocal chiral metamaterials. Physical Review Letters. 2016;117(5):057401. doi: 10.1103/PhysRevLett.117.057401. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/ncomms13731. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 129.Fang C, Weng H, Dai X, Fang Z. Topological nodal line semimetals. Chinese Physics B. 2016;25(11):117106. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1126/science.aaq1221. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/ncomms6782. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/srep22270. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1038/s41565-018-0297-6. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/s41598-019-40677-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.114.127401. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/ncomms13038. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 140.Chen Y, Chen H, Cai G. High transmission in a metal-based photonic crystal. Applied Physics Letters. 2018;112(1):013504. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1038/ncomms11619. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1038/s41467-018-03407-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 144.Gao F, Xue H, Yang Z, Lai K, Yu Y, Lin X, Chong Y, Shvets G, Zhang B. Topologically protected refraction ofrobustkinkstates in valley photonic crystals. Nature Physics. 2018;14(2):140–144. [Google Scholar]
- 145.Karch A. Surface plasmons and topological insulators. Physical Review B. 2011;83(24):245432. [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1126/science.aar4003. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.aar4005. [DOI] [PubMed] [Google Scholar]
- 150.Midya B, Zhao H, Feng L. Non-Hermitian photonics promises exceptional topology of light. Nature Communications. 2018;9(1):2674. doi: 10.1038/s41467-018-05175-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 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. doi: 10.1126/science.aaq0327. [DOI] [PubMed] [Google Scholar]
- 152.Blanco-Redondo A, Bell B, Oren D, Eggleton B J, Segev M. Topological protection of biphoton states. Science. 2018;362(6414):568–571. doi: 10.1126/science.aau4296. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1021/nl302746n. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 156.Su W P, Schrieffer J R, Heeger A J. Solitons in Polyacetylene. Physical Review Letters. 1979;42(25):1698–1701. [Google Scholar]
- 157.Miri M-A, Alù A. Exceptional points in optics and photonics. Science. 2019;363(6422):eaar7709. doi: 10.1126/science.aar7709. [DOI] [PubMed] [Google Scholar]
- 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 [DOI] [PubMed]
- 159.Lee T E. Anomalous edge state in a non-Hermitian lattice. Physical Review Letters. 2016;116(13):133903. doi: 10.1103/PhysRevLett.116.133903. [DOI] [PubMed] [Google Scholar]
- 160.Ghatak A, Das T. New topological invariants in non-Hermitian systems. Journal of Physics Condensed Matter. 2019;31(26):263001. doi: 10.1088/1361-648X/ab11b3. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.120.113901. [DOI] [PubMed] [Google Scholar]
- 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. doi: 10.1038/s41467-018-03434-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 164.Ota Y, Katsumi R, Watanabe K, Iwamoto S, Arakawa Y. Topological photonic crystal nanocavity laser. Communications on Physics. 2018;1(1):86. [Google Scholar]
- 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. doi: 10.1103/PhysRevLett.61.2015. [DOI] [PubMed] [Google Scholar]
- 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. [Google Scholar]
- 167.Pilozzi L, Farrelly F A, Marcucci G, Conti C. Machine learning inrerse problem for topological photonics. Communications Physics. 2018;1(1):57. [Google Scholar]
- 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. [Google Scholar]
- 169.Barth C, Becker C. Machine learning classification for field distributions of photonic modes. Communications on Physics. 2018;1(1):58. [Google Scholar]
- 170.Fano U. Effects of configuration interaction on intensities and phase shifts. Physical Review. 1961;124(6):1866–1878. [Google Scholar]
- 171.Limonov M F, Rybin M V, Poddubny A N, Kivshar Y S. Fano resonances in photonics. Nature Photonics. 2017;11(9):543–554. [Google Scholar]
- 172.Miroshnichenko A E, Flach S, Kivshar Y S. Fano resonances in nanoscale structures. Reviews of Modern Physics. 2010;82(3):2257–2298. [Google Scholar]
- 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. [Google Scholar]
- 174.Gao W, Hu X, Li C, Yang J, Chai Z, Xie J, Gong Q. Fano-resonance in one-dimensional topological photonic crystal hetero-structure. Optics Express. 2018;26(7):8634–8644. doi: 10.1364/OE.26.008634. [DOI] [PubMed] [Google Scholar]
- 175.Zangeneh-Nejad F, Fleury R. Topological Fano resonances. Physical Review Letters. 2019;122(1):014301. doi: 10.1103/PhysRevLett.122.014301. [DOI] [PubMed] [Google Scholar]
- 176.Liang G Q, Chong Y D. Optical resonator analog of a two-dimensional topological insulator. Physical Review Letters. 2013;110(20):203904. doi: 10.1103/PhysRevLett.110.203904. [DOI] [PubMed] [Google Scholar]
