Please wait a minute...

Frontiers of Optoelectronics

Front. Optoelectron.    2020, Vol. 13 Issue (1) : 50-72
Topological photonic crystals: a review
Hongfei WANG1, Samit Kumar GUPTA1, Biye XIE1, Minghui LU1,2,3()
1. National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, China
2. Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
3. Jiangsu Key Laboratory of Artificial Functional Materials, Nanjing University, Nanjing 210093, China
Download: PDF(5403 KB)   HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

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     
Corresponding Authors: Minghui LU   
Just Accepted Date: 22 October 2019   Online First Date: 13 January 2020    Issue Date: 03 April 2020
 Cite this article:   
Hongfei WANG,Samit Kumar GUPTA,Biye XIE, et al. Topological photonic crystals: a review[J]. Front. Optoelectron., 2020, 13(1): 50-72.
E-mail this article
E-mail Alert
Articles by authors
Hongfei WANG
Samit Kumar GUPTA
Biye XIE
Minghui LU
Fig.1  Direct analogy between electrons (fermionic) and photons (bosonic) systems
Fig.2  (a) Gyromagnetic photonic crystal used in experiments. The blue rods indicate the gyromagnetic material and 0.2T magnetic field is applied along the z direction; (b) top view of actual waveguides; (c), (e) unidirectional and non-reciprocal propagation; (d) robust propagation against backscattering; (f) reciprocal transmission measured using the bulk photonic crystal and the projected dispersion including bulk and edge states; (g) non-reciprocal transmission via chiral edge states. Reproduced from Ref. [16]
Fig.3  (a) Schematic diagram of triangular photonic crystals; (b) projected dispersion of 2D topological photonic crystals; (c) electromagnetic field distributions ( Ez) in different pseudospins; (d) schematics of the vertical domain composed of bianisotropic metacrystal; (e) band structures of two kinds of unit cells in (d); (f) dispersion relation of surface states with k|| and k directions. Reproduced from Refs. [21,37]
Fig.4  (a) BCC unit cell of gyroid photonic crystals and corresponding Brillouin zone; (b) band structures of nodal lines with two air spheres on two gyroids; (c) schematic of metallic inclusion which includes the saddle shape, and helicoid surface states; (d) Brillouin zone of metallic inclusions and the band structures with Weyl points (rad/blue points). Reproduced from Refs. [130,131]
Fig.5  (a) Arrangement of metallic rods and put into the parallel plate waveguide with different topology characteristics, and the corresponding band structures; (b) schematic plot of the topological switch and the transmission performance with switch operation; (c) schematic of metacrystals with copper cut-wire and their band structures; (d) band structures at kz=0?(2π /a) and kz=0.1 ?(2π /a) planes, and the measured result with the Fourier transform. Reproduced from Refs. [22,143]
Fig.6  (a) Two coupled resonators descripted by Hamiltonian with spin freedom and the 2D array of resonators; (b) edge states in different spins and the transmission in the presence of disorder perturbation; (c) experimental set-up for the measurement; (d) unit coupled resonators including four link and four site resonators, and the scanning electron microscope image (SEM) of the resonant array; (e) topological edge states that propagating around the defect in the experiment and simulation. Reproduced from Refs. [17,146]
Fig.7  (a) Schematics of the helical waveguides comprising the honeycomb lattice; (b) projected dispersion of straight waveguides (R =0? μm) and helical waveguides (R =8? ?μm); (c) microscope image of the photonic waveguide array; (d) light propagation at different distance, z means the length of distances; (e) four different bonds existed in the lattice with coupling constants J1,2, 3,4 and the sketch to achieve it; (f) experiment measured with chiral edge states along different paths. Reproduced from Refs. [27,31]
Fig.8  (a) 2D square lattices made of DBR cavities and connected by phase elements; (b) transmission spectrum (blue line) of two cavities in (a) and the phase in two cavities (red dashed line); (c) schematic of a graphene layer combined with photonic crystals, and their absorption; (d) photonic crystals integrated with graphene. Their optical image of microscope and SEM image; (e) fundamental resonant mode of three-hole defect cavities. Reproduced from Refs. [18,153,154]
Fig.9  (a) 2D lattice of photonic crystal, where d1 and d2 corresponding to the coupling expressed in distance; (b) band structures of trivial, gapless, and nontrivial situations; (c) diagram of 3D structure; (d) photograph of higher-order topological insulator surrounded by ordinary insulators; (e) eigenfrequencies of bulk, edge and corner states; (f) simulation of corner states; (g) experimental measurement of the corner states. Reproduced from Ref. [95]
Fig.10  (a) Nonlinear SSH model, two resonators in every unit cell; (b) band structures of trivial case when the mode intensity I =0; (c) width of band gap changed by intensity; (d) winding number tuned by intensity; (e) time evolution of excitation probability for single-photon state in three qubits Q1,2, 3; (f) probability for a two-photon case. Reproduced from Refs. [54,56]
Fig.11  (a) Dispersion of non-Hermitian system near the exceptional points; (b) array of passive waveguides and realized structure of fused silica glass; (c) SEM image of photonic crystal slabs and their band structure measured by experiment. Reproduced from Refs. [67,72,79]
1 E Yablonovitch. Inhibited spontaneous emission in solid-state physics and electronics. Physical Review Letters, 1987, 58(20): 2059–2062 pmid: 10034639
2 S John. Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987, 58(23): 2486–2489 pmid: 10034761
3 B Wang, M A Cappelli. A plasma photonic crystal bandgap device. Applied Physics Letters, 2016, 108(16): 161101
4 Y Akahane, T Asano, B S Song, S Noda. High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature, 2003, 425(6961): 944–947 pmid: 14586465
5 R A Shelby, D R Smith, S Schultz. Experimental verification of a negative index of refraction. Science, 2001, 292(5514): 77–79 pmid: 11292865
6 V M Shalaev, W Cai, U K Chettiar, H K Yuan, A K Sarychev, V P Drachev, A V Kildishev. Negative index of refraction in optical metamaterials. Optics Letters, 2005, 30(24): 3356–3358 pmid: 16389830
7 K Klitzing, G Dorda, M Pepper. 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 D J Thouless, M Kohmoto, M P Nightingale, M den Nijs. Quantized hall conductance in a two-dimensional periodic potential. Physical Review Letters, 1982, 49(6): 405–408
9 M Kohmoto. Topological invariant and the quantization of the Hall conductance. Annals of Physics, 1985, 160(2): 343–354
10 C L Kane, E J Mele. Quantum spin Hall effect in graphene. Physical Review Letters, 2005, 95(22): 226801 pmid: 16384250
11 B A Bernevig, S C Zhang. Quantum spin Hall effect. Physical Review Letters, 2006, 96(10): 106802 pmid: 16605772
12 B A Bernevig, T L Hughes, S C Zhang. Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science, 2006, 314(5806): 1757–1761 pmid: 17170299
13 M König, S Wiedmann, C Brüne, A Roth, H Buhmann, L W Molenkamp, X L Qi, S C Zhang. Quantum spin hall insulator state in HgTe quantum wells. Science, 2007, 318(5851): 766–770 pmid: 17885096
14 F D Haldane, S Raghu. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry. Physical Review Letters, 2008, 100(1): 013904 pmid: 18232766
15 Z Wang, Y D Chong, J D Joannopoulos, M Soljacić. Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Physical Review Letters, 2008, 100(1): 013905 pmid: 18232767
16 Z Wang, Y Chong, J D Joannopoulos, M Soljacić. Observation of unidirectional backscattering-immune topological electromagnetic states. Nature, 2009, 461(7265): 772–775 pmid: 19812669
17 M Hafezi, E A Demler, M D Lukin, J M Taylor. Robust optical delay lines with topological protection. Nature Physics, 2011, 7(11): 907–912
18 R O Umucalılar, I Carusotto. Artificial gauge field for photons in coupled cavity arrays. Physical Review A, 2011, 84(4): 043804
19 A B Khanikaev, S H Mousavi, W K Tse, M Kargarian, A H MacDonald, G Shvets. Photonic topological insulators. Nature Materials, 2013, 12(3): 233–239 pmid: 23241532
20 A V Nalitov, G Malpuech, H Terças, D D Solnyshkov. Spin-orbit coupling and the optical spin Hall effect in photonic graphene. Physical Review Letters, 2015, 114(2): 026803 pmid: 25635557
21 L H Wu, X Hu. Scheme for achieving a topological photonic crystal by using dielectric material. Physical Review Letters, 2015, 114(22): 223901 pmid: 26196622
22 X Cheng, C Jouvaud, X Ni, S H Mousavi, A Z Genack, A B Khanikaev. Robust reconfigurable electromagnetic pathways within a photonic topological insulator. Nature Materials, 2016, 15(5): 542–548 pmid: 26901513
23 J W Dong, X D Chen, H Zhu, Y Wang, X Zhang. Valley photonic crystals for control of spin and topology. Nature Materials, 2017, 16(3): 298–302 pmid: 27893722
24 Y Yang, Y F Xu, T Xu, H X Wang, J H Jiang, X Hu, Z H Hang. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials. Physical Review Letters, 2018, 120(21): 217401 pmid: 29883132
25 K Fang, Z Yu, S Fan. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nature Photonics, 2012, 6(11): 782–787
26 Y Lumer, Y Plotnik, M C Rechtsman, M Segev. Self-localized states in photonic topological insulators. Physical Review Letters, 2013, 111(24): 243905 pmid: 24483665
27 M C Rechtsman, J M Zeuner, Y Plotnik, Y Lumer, D Podolsky, F Dreisow, S Nolte, M Segev, A Szameit. Photonic Floquet topological insulators. Nature, 2013, 496(7444): 196–200 pmid: 23579677
28 P Titum, N H Lindner, M C Rechtsman, G Refael. Disorder-induced Floquet topological insulators. Physical Review Letters, 2015, 114(5): 056801 pmid: 25699461
29 D Leykam, M C Rechtsman, Y D Chong. Anomalous topological phases and unpaired dirac cones in photonic Floquet topological insulators. Physical Review Letters, 2016, 117(1): 013902 pmid: 27419570
30 L J Maczewsky, J M Zeuner, S Nolte, A Szameit. Observation of photonic anomalous Floquet topological insulators. Nature Communications, 2017, 8(1): 13756 pmid: 28051080
31 S Mukherjee, A Spracklen, M Valiente, E Andersson, P Öhberg, N Goldman, R R Thomson. Experimental observation of anomalous topological edge modes in a slowly driven photonic lattice. Nature Communications, 2017, 8(1): 13918 pmid: 28051060
32 S Mukherjee, H K Chandrasekharan, P Öhberg, N Goldman, R R Thomson. State-recycling and time-resolved imaging in topological photonic lattices. Nature Communications, 2018, 9(1): 4209 pmid: 30310062
33 B Zhu, H Zhong, Y Ke, X Qin, A A Sukhorukov, Y S Kivshar, C Lee. Topological Floquet edge states in periodically curved waveguides. Physical Review A, 2018, 98(1): 013855
34 F Nathan, D Abanin, E Berg, N H Lindner, M S Rudner. Anomalous Floquet insulators. Physical Review B, 2019, 99(19): 195133
35 T Ma, G Shvets. All-Si valley-Hall photonic topological insulator. New Journal of Physics, 2016, 18(2): 025012
36 X Wu, Y Meng, J Tian, Y Huang, H Xiang, D Han, W Wen. Direct observation of valley-polarized topological edge states in designer surface plasmon crystals. Nature Communications, 2017, 8(1): 1304 pmid: 29101323
37 A Slobozhanyuk, S H Mousavi, X Ni, D Smirnova, Y S Kivshar, A B Khanikaev. Three-dimensional all-dielectric photonic topological insulator. Nature Photonics, 2017, 11(2): 130–136
38 Y Yang, Z Gao, H Xue, L Zhang, M He, Z Yang, R Singh, Y Chong, B Zhang, H Chen. Realization of a three-dimensional photonic topological insulator. Nature, 2019, 565(7741): 622–626 pmid: 30626966
39 S M Young, S Zaheer, J C Teo, C L Kane, E J Mele, A M Rappe. Dirac semimetal in three dimensions. Physical Review Letters, 2012, 108(14): 140405 pmid: 22540776
40 B J Yang, N Nagaosa. Classification of stable three-dimensional Dirac semimetals with nontrivial topology. Nature Communications, 2014, 5(1): 4898 pmid: 25222476
41 Z K Liu, B Zhou, Y Zhang, Z J Wang, H M Weng, D Prabhakaran, S K Mo, Z X Shen, Z Fang, X Dai, Z Hussain, Y L Chen. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science, 2014, 343(6173): 864–867 pmid: 24436183
42 B Yang, Q Guo, B Tremain, L E Barr, W Gao, H Liu, B Béri, Y Xiang, D Fan, A P Hibbins, S Zhang. Direct observation of topological surface-state arcs in photonic metamaterials. Nature Communications, 2017, 8(1): 97 pmid: 28733654
43 F Li, X Huang, J Lu, J Ma, Z Liu. Weyl points and Fermi arcs in a chiral phononic crystal. Nature Physics, 2018, 14(1): 30–34
44 A A Burkov, M D Hook, L Balents. Topological nodal semimetals. Physical Review B, 2011, 84(23): 235126
45 Z Yan, Z Wang. Tunable Weyl points in periodically driven nodal line semimetals. Physical Review Letters, 2016, 117(8): 087402 pmid: 27588882
46 H He, C Qiu, L Ye, X Cai, X Fan, M Ke, F Zhang, Z Liu. Topological negative refraction of surface acoustic waves in a Weyl phononic crystal. Nature, 2018, 560(7716): 61–64 pmid: 30068954
47 R Adair, L L Chase, S A Payne. Nonlinear refractive index of optical crystals. Physical Review B, 1989, 39(5): 3337–3350 pmid: 9948635
48 V Berger. Nonlinear photonic crystals. Physical Review Letters, 1998, 81(19): 4136–4139
49 S F Mingaleev, Y S Kivshar. Self-trapping and stable localized modes in nonlinear photonic crystals. Physical Review Letters, 2001, 86(24): 5474–5477 pmid: 11415279
50 M Soljačić , C Luo, J D Joannopoulos, S Fan. Nonlinear photonic crystal microdevices for optical integration. Optics Letters, 2003, 28(8): 637–639 pmid: 12703925
51 J W Fleischer, M Segev, N K Efremidis, D N Christodoulides. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature, 2003, 422(6928): 147–150 pmid: 12634781
52 M Soljačić, J D Joannopoulos. Enhancement of nonlinear effects using photonic crystals. Nature Materials, 2004, 3(4): 211–219 pmid: 15034564
53 L H Haddad, C M Weaver, L D Carr. 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 Y Hadad, A B Khanikaev, A Alù. Self-induced topological transitions and edge states supported by nonlinear staggered potentials. Physical Review B, 2016, 93(15): 155112
55 D Leykam, Y D Chong. Edge solitons in nonlinear-photonic topological insulators. Physical Review Letters, 2016, 117(14): 143901 pmid: 27740799
56 P Roushan, C Neill, A Megrant, Y Chen, R Babbush, R Barends, B Campbell, Z Chen, B Chiaro, A Dunsworth, A Fowler, E Jeffrey, J Kelly, E Lucero, J Mutus, P J J O’Malley, M Neeley, C Quintana, D Sank, A Vainsencher, J Wenner, T White, E Kapit, H Neven, J Martinis. Chiral ground-state currents of interacting photons in a synthetic magnetic field. Nature Physics, 2017, 13(2): 146–151
57 M E Tai, A Lukin, M Rispoli, R Schittko, T Menke, Dan Borgnia, P M Preiss, F Grusdt, A M Kaufman, M Greiner. Microscopy of the interacting Harper-Hofstadter model in the two-body limit. Nature, 2017, 546(7659): 519–523 pmid: 28640260
58 X Zhou, Y Wang, D Leykam, Y D Chong. Optical isolation with nonlinear topological photonics. New Journal of Physics, 2017, 19(9): 095002
59 D A Dobrykh, A V Yulin, A P Slobozhanyuk, A N Poddubny, Y S Kivshar. Nonlinear control of electromagnetic topological edge states. Physical Review Letters, 2018, 121(16): 163901 pmid: 30387643
60 C Rajesh, T Georgios. Self-induced topological transition in phononic crystals by nonlinearity management. 2019, arXiv:1904. 09466v1
61 C M Bender, S Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Physical Review Letters, 1998, 80(24): 5243–5246
62 A Regensburger, C Bersch, M A Miri, G Onishchukov, D N Christodoulides, U Peschel. Parity-time synthetic photonic lattices. Nature, 2012, 488(7410): 167–171 pmid: 22874962
63 Y Yang, C Peng, Y Liang, Z Li, S Noda. Analytical perspective for bound states in the continuum in photonic crystal slabs. Physical Review Letters, 2014, 113(3): 037401 pmid: 25083664
64 B Zhen, C W Hsu, L Lu, A D Stone, M Soljačić. Topological nature of optical bound states in the continuum. Physical Review Letters, 2014, 113(25): 257401 pmid: 25554906
65 S Malzard, C Poli, H Schomerus. Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity-time symmetry. Physical Review Letters, 2015, 115(20): 200402 pmid: 26613422
66 J M Zeuner, M C Rechtsman, Y Plotnik, Y Lumer, S Nolte, M S Rudner, M Segev, A Szameit. Observation of a topological transition in the bulk of a non-Hermitian system. Physical Review Letters, 2015, 115(4): 040402 pmid: 26252670
67 B Zhen, C W Hsu, Y Igarashi, L Lu, I Kaminer, A Pick, S L Chua, J D Joannopoulos, M Soljačić. Spawning rings of exceptional points out of Dirac cones. Nature, 2015, 525(7569): 354–358 pmid: 26352476
68 A Cerjan, A Raman, S Fan. Exceptional contours and band structure design in parity-time symmetric photonic crystals. Physical Review Letters, 2016, 116(20): 203902 pmid: 27258869
69 E N Bulgakov, D N Maksimov. Topological bound states in the continuum in arrays of dielectric spheres. Physical Review Letters, 2017, 118(26): 267401 pmid: 28707917
70 L Feng, R El-Ganainy, L Ge. Non-Hermitian photonics based on parity-time symmetry. Nature Photonics, 2017, 11(12): 752–762
71 A Kodigala, T Lepetit, Q Gu, B Bahari, Y Fainman, B Kanté. Lasing action from photonic bound states in continuum. Nature, 2017, 541(7636): 196–199 pmid: 28079064
72 S Weimann, M Kremer, Y Plotnik, Y Lumer, S Nolte, K G Makris, M Segev, M C Rechtsman, A Szameit. Topologically protected bound states in photonic parity-time-symmetric crystals. Nature Materials, 2017, 16(4): 433–438 pmid: 27918567
73 R El-Ganainy, K G Makris, M Khajavikhan, Z H Musslimani, S Rotter, D N Christodoulides. Non-Hermitian physics and PT symmetry. Nature Physics, 2018, 14(1): 11–19
74 K Kawabata, K Shiozaki, M Ueda. Anomalous helical edge states in a non-Hermitian Chern insulator. Physical Review B, 2018, 98(16): 165148
75 F K Kunst, E Edvardsson, J C Budich, E J Bergholtz. Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Physical Review Letters, 2018, 121(2): 026808 pmid: 30085697
76 S Lieu. Topological phases in the non-Hermitian Su-Schrieffer-Heeger model. Physical Review B, 2018, 97(4): 045106
77 M Pan, H Zhao, P Miao, S Longhi, L Feng. Photonic zero mode in a non-Hermitian photonic lattice. Nature Communications, 2018, 9(1): 1308 pmid: 29615630
78 B Qi, L Zhang, L Ge. Defect states emerging from a non-Hermitian flatband of photonic zero modes. Physical Review Letters, 2018, 120(9): 093901 pmid: 29547321
79 H Shen, B Zhen, L Fu. Topological band theory for non-Hermitian Hamiltonians. Physical Review Letters, 2018, 120(14): 146402 pmid: 29694133
80 H F Wang, S K Gupta, X Y Zhu, M H Lu, X P Liu, Y F Chen. Bound states in the continuum in a bilayer photonic crystal with TE-TM cross coupling. Physical Review. B, 2018, 98(21): 214101
81 S Yao, F Song, Z Wang. Non-Hermitian Chern bands. Physical Review Letters, 2018, 121(13): 136802 pmid: 30312068
82 S Yao, Z Wang. Edge states and topological invariants of non-Hermitian systems. Physical Review Letters, 2018, 121(8): 086803 pmid: 30192628
83 X D Chen, W M Deng, F L Shi, F L Zhao, M Chen, J W Dong. Direct observation of corner states in second-order topological photonic crystal slabs. 2018, arXiv:1812.08326
84 M Ezawa. Higher-order topological insulators and semimetals on the breathing kagome and pyrochlore lattices. Physical Review Letters, 2018, 120(2): 026801 pmid: 29376716
85 M Ezawa. Minimal models for Wannier-type higher-order topological insulators and phosphorene. Physical Review B, 2018, 98(4): 045125
86 M Ezawa. Magnetic second-order topological insulators and semimetals. Physical Review B, 2018, 97(15): 155305
87 M Ezawa. Higher-order topological electric circuits and topological corner resonance on the breathing kagome and pyrochlore lattices. Physical Review B, 2018, 98(20): 201402
88 M Geier, L Trifunovic, M Hoskam, P W Brouwer. Second-order topological insulators and superconductors with an order-two crystalline symmetry. Physical Review B, 2018, 97(20): 205135
89 E Khalaf. Higher-order topological insulators and superconductors protected by inversion symmetry. Physical Review B, 2018, 97(20): 205136
90 F K Kunst, G van Miert, E J Bergholtz. Lattice models with exactly solvable topological hinge and corner states. Physical Review B, 2018, 97(24): 241405
91 C W Peterson, W A Benalcazar, T L Hughes, G Bahl. A quantized microwave quadrupole insulator with topologically protected corner states. Nature, 2018, 555(7696): 346–350 pmid: 29542690
92 F Schindler, A M Cook, M G Vergniory, Z Wang, S S Parkin, B A Bernevig, T. NeupertHigher-order topological insulators. Science Advances, 2018, 4(6): eaat0346
93 G van Miert, C Ortix. Higher-order topological insulators protected by inversion and rotoinversion symmetries. Physical Review B, 2018, 98(8): 081110
94 B Y Xie, H F Wang, H X Wang, X Y Zhu, J H Jiang, M H Lu, Y F Chen. Second-order photonic topological insulator with corner states. Physical Review B, 2018, 98(20): 205147
95 B Y Xie, G X Su, H F Wang, H Su, X P Shen, P Zhan, M H Lu, Z L Wang, Y F Chen. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals. Physical Review Letters, 2019, 122(23): 233903
96 O Yasutomo, L Feng, K Ryota, W Katsuyuki, W Katsunori, A Yasuhiko, I Satoshi. Photonic crystal nanocavity based on a topological corner state. 2018, arXiv:1812.10171
97 D Călugăru, V Juričić, B Roy. Higher-order topological phases: a general principle of construction. Physical Review B, 2019, 99(4): 041301
98 H Hu, B Huang, E Zhao, W V Liu. Dynamical singularities of Floquet higher-order topological insulators. 2019, arXiv:1905. 03727v1
99 J A Armstrong, N Bloembergen, J Ducuing, P S Pershan. Interactions between light waves in a nonlinear dielectric. Physical Review, 1962, 127(6): 1918–1939
100 D A Kleinman. Nonlinear dielectric polarization in optical media. Physical Review, 1962, 126(6): 1977–1979
101 E Adler. Nonlinear optical frequency polarization in a dielectric. Physical Review, 1964, 134(3A): A728–A733
102 R C Miller. Optical second harmonic generation in piezoelectric crystals. Applied Physics Letters, 1964, 5(1): 17–19
103 M M Fejer, G Magel, D H Jundt, R L Byer. Quasi-phase-matched second harmonic generation: tuning and tolerances. IEEE Journal of Quantum Electronics, 1992, 28(11): 2631–2654
104 M Yamada, N Nada, M Saitoh, K Watanabe. 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 M Celebrano, X Wu, M Baselli, S Großmann, P Biagioni, A Locatelli, C De Angelis, G Cerullo, R Osellame, B Hecht, L Duò, F Ciccacci, M Finazzi. Mode matching in multiresonant plasmonic nanoantennas for enhanced second harmonic generation. Nature Nanotechnology, 2015, 10(5): 412–417 pmid: 25895003
106 M H Rubin, D N Klyshko, Y H Shih, A V Sergienko. Theory of two-photon entanglement in type-II optical parametric down-conversion. Physical Review A, 1994, 50(6): 5122–5133 pmid: 9911514
107 C H Monken, P S Ribeiro, S Pádua. Transfer of angular spectrum and image formation in spontaneous parametric down-conversion. Physical Review A, 1998, 57(4): 3123–3126
108 H H Arnaut, G A Barbosa. 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 pmid: 10991264
109 J C Howell, R S Bennink, S J Bentley, R W Boyd. 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 pmid: 15245267
110 G Harder, T J Bartley, A E Lita, S W Nam, T Gerrits, C Silberhorn. Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics. Physical Review Letters, 2016, 116(14): 143601 pmid: 27104708
111 R Carriles, D N Schafer, K E Sheetz, J J Field, R Cisek, V Barzda, A W Sylvester, J A Squier. Imaging techniques for harmonic and multiphoton absorption fluorescence microscopy. Review of Scientific Instruments, 2009, 80(8): 081101 pmid: 19725639
112 G Grinblat, Y Li, M P Nielsen, R F Oulton, S A Maier. Enhanced third harmonic generation in single germanium nanodisks excited at the anapole mode. Nano Letters, 2016, 16(7): 4635–4640 pmid: 27331867
113 J E Sipe, D J Moss, H van Driel. Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals. Physical Review B, 1987, 35(3): 1129–1141 pmid: 9941520
114 S Zhu, Y Zhu, N Ming. Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice. Science, 1997, 278(5339): 843–846
115 G Soavi, G Wang, H Rostami, D G Purdie, D De Fazio, T Ma, B Luo, J Wang, A K Ott, D Yoon, S A Bourelle, J E Muench, I Goykhman, S Dal Conte, M Celebrano, A Tomadin, M Polini, G Cerullo, A C Ferrari. Broadband, electrically tunable third-harmonic generation in graphene. Nature Nanotechnology, 2018, 13(7): 583–588 pmid: 29784965
116 R E Slusher, L W Hollberg, B Yurke, J C Mertz, J F Valley. Observation of squeezed states generated by four-wave mixing in an optical cavity. Physical Review Letters, 1985, 55(22): 2409–2412 pmid: 10032137
117 L Deng, E W Hagley, J Wen, M Trippenbach, Y Band, P S Julienne, J Simsarian, K Helmerson, S Rolston, W D Phillips. Four-wave mixing with matter waves. Nature, 1999, 398(6724): 218–220
118 F Bencivenga, R Cucini, F Capotondi, A Battistoni, R Mincigrucci, E Giangrisostomi, A Gessini, M Manfredda, I P Nikolov, E Pedersoli, E Principi, C Svetina, P Parisse, F Casolari, M B Danailov, M Kiskinova, C Masciovecchio. Four-wave mixing experiments with extreme ultraviolet transient gratings. Nature, 2015, 520(7546): 205–208 pmid: 25855456
119 S K Singh, M K Abak, M E Tasgin. Enhancement of four-wave mixing via interference of multiple plasmonic conversion paths. Physical Review B, 2016, 93(3): 035410
120 H Zhang, S Virally, Q Bao, L K Ping, S Massar, N Godbout, P Kockaert. Z-scan measurement of the nonlinear refractive index of graphene. Optics Letters, 2012, 37(11): 1856–1858 pmid: 22660052
121 M Z Alam, I De Leon, R W Boyd. Large optical nonlinearity of indium tin oxide in its epsilon-near-zero region. Science, 2016, 352(6287): 795–797 pmid: 27127238
122 T Ozawa, H M Price, A Amo, N Goldman, M Hafezi, L Lu, M C Rechtsman, D Schuster, J Simon, O Zilberberg, I Carusotto. Topological photonics. Reviews of Modern Physics, 2019, 91(1): 015006
123 M V Berry. 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 S Pancharatnam. Generalized theory of interference and its applications. Proceedings of the Indian Academy of Sciences, Section A, Physical Sciences, 1956, 44(6): 398–417
125 S A Skirlo, L Lu, Y Igarashi, Q Yan, J Joannopoulos, M Soljačić. Experimental observation of large Chern numbers in photonic crystals. Physical Review Letters, 2015, 115(25): 253901 pmid: 26722920
126 L Lu, Z Wang, D Ye, L Ran, L Fu, J D Joannopoulos, M Soljačić. Experimental observation of Weyl points. Science, 2015, 349(6248): 622–624 pmid: 26184914
127 M Xiao, Q Lin, S Fan. Hyperbolic Weyl point in reciprocal chiral metamaterials. Physical Review Letters, 2016, 117(5): 057401 pmid: 27517792
128 Q Lin, M Xiao, L Yuan, S Fan. Photonic Weyl point in a two-dimensional resonator lattice with a synthetic frequency dimension. Nature Communications, 2016, 7(1): 13731 pmid: 27976714
129 C Fang, H Weng, X Dai, Z Fang. Topological nodal line semimetals. Chinese Physics B, 2016, 25(11): 117106
130 L Lu, L Fu, J D Joannopoulos, M Soljačić. Weyl points and line nodes in gyroid photonic crystals. Nature Photonics, 2013, 7(4): 294–299
131 B Yang, Q Guo, B Tremain, R Liu, L E Barr, Q Yan, W Gao, H Liu, Y Xiang, J Chen, C Fang, A Hibbins, L Lu, S Zhang. Ideal Weyl points and helicoid surface states in artificial photonic crystal structures. Science, 2018, 359(6379): 1013–1016 pmid: 29326117
132 W J Chen, S J Jiang, X D Chen, B Zhu, L Zhou, J W Dong, C T Chan. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide. Nature Communications, 2014, 5(1): 5782 pmid: 25517229
133 A P Slobozhanyuk, A B Khanikaev, D S Filonov, D A Smirnova, A E Miroshnichenko, Y S Kivshar. Experimental demonstration of topological effects in bianisotropic metamaterials. Scientific Reports, 2016, 6(1): 22270 pmid: 26936219
134 M I Shalaev, W Walasik, A Tsukernik, Y Xu, N M Litchinitser. Robust topologically protected transport in photonic crystals at telecommunication wavelengths. Nature Nanotechnology, 2019, 14(1): 31–34 pmid: 30420760
135 X D Chen, F L Zhao, M Chen, J W Dong. Valley-contrasting physics in all-dielectric photonic crystals: orbital angular momentum and topological propagation. Physical Review B, 2017, 96(2): 020202
136 X D Chen, F L Shi, H Liu, J C Lu, W M Deng, J Y Dai, Q Cheng, J W Dong. Tunable electromagnetic flow control in valley photonic crystal waveguides. Physical Review Applied, 2018, 10(4): 044002
137 M He, L Zhang, H Wang. Two-dimensional photonic crystal with ring degeneracy and its topological protected edge states. Scientific Reports, 2019, 9(1): 3815 pmid: 30846836
138 T Ma, A B Khanikaev, S H Mousavi, G Shvets. Guiding electromagnetic waves around sharp corners: topologically protected photonic transport in metawaveguides. Physical Review Letters, 2015, 114(12): 127401 pmid: 25860770
139 W J Chen, M Xiao, C T Chan. Photonic crystals possessing multiple Weyl points and the experimental observation of robust surface states. Nature Communications, 2016, 7(1): 13038 pmid: 27703140
140 Y Chen, H Chen, G Cai. High transmission in a metal-based photonic crystal. Applied Physics Letters, 2018, 112(1): 013504
141 I El-Kady, M Sigalas, R Biswas, K Ho, C Soukoulis. Metallic photonic crystals at optical wavelengths. Physical Review B, 2000, 62(23): 15299–15302
142 F Gao, Z Gao, X Shi, Z Yang, X Lin, H Xu, J D Joannopoulos, M Soljačić, H Chen, L Lu, Y Chong, B Zhang. Probing topological protection using a designer surface plasmon structure. Nature Communications, 2016, 7(1): 11619 pmid: 27197877
143 W Gao, B Yang, B Tremain, H Liu, Q Guo, L Xia, A P Hibbins, S Zhang. Experimental observation of photonic nodal line degeneracies in metacrystals. Nature Communications, 2018, 9(1): 950 pmid: 29507346
144 F Gao, H Xue, Z Yang, K Lai, Y Yu, X Lin, Y Chong, G Shvets, B Zhang. Topologically protected refraction of robust kink states in valley photonic crystals. Nature Physics, 2018, 14(2): 140–144
145 A Karch. Surface plasmons and topological insulators. Physical Review B, 2011, 83(24): 245432
146 M Hafezi, S Mittal, J Fan, A Migdall, J M Taylor. Imaging topological edge states in silicon photonics. Nature Photonics, 2013, 7(12): 1001–1005
147 S Mittal, S Ganeshan, J Fan, A Vaezi, M Hafezi. Measurement of topological invariants in a 2D photonic system. Nature Photonics, 2016, 10(3): 180–183
148 G Harari, M A Bandres, Y Lumer, M C Rechtsman, Y D Chong, M Khajavikhan, D N Christodoulides, M. SegevTopological insulator laser: theory. Science, 2018, 359(6381): eaar4003
149 M A Bandres, S Wittek, G Harari, M Parto, J Ren, M Segev, D N Christodoulides, M Khajavikhan. Topological insulator laser: experiments. Science, 2018, 359(6381): eaar4005
150 B Midya, H Zhao, L Feng. Non-Hermitian photonics promises exceptional topology of light. Nature Communications, 2018, 9(1): 2674 pmid: 29991729
151 S Barik, A Karasahin, C Flower, T Cai, H Miyake, W DeGottardi, M Hafezi, E Waks. A topological quantum optics interface. Science, 2018, 359(6376): 666–668 pmid: 29439239
152 A Blanco-Redondo, B Bell, D Oren, B J Eggleton, M Segev. Topological protection of biphoton states. Science, 2018, 362(6414): 568–571 pmid: 30385574
153 J R Piper, S Fan. 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 X Gan, K F Mak, Y Gao, Y You, F Hatami, J Hone, T F Heinz, D Englund. Strong enhancement of light-matter interaction in graphene coupled to a photonic crystal nanocavity. Nano Letters, 2012, 12(11): 5626–5631 pmid: 23043452
155 A J Heeger, S Kivelson, J R Schrieffer, W P Su. Solitons in conducting polymers. Reviews of Modern Physics, 1988, 60(3): 781–850
156 W P Su, J R Schrieffer, A J Heeger. Solitons in Polyacetylene. Physical Review Letters, 1979, 42(25): 1698–1701
157 M-A Miri, A Alù. Exceptional points in optics and photonics. Science, 2019, 363(6422): eaar7709
158 S K Gupta, Y Zou, X Y Zhu, M H Lu, L Zhang, X P Liu, Y F Chen. Parity-time symmetry in Non-Hermitian complex media. 2018, arXiv:1803.00794
159 T E Lee. Anomalous edge state in a non-Hermitian lattice. Physical Review Letters, 2016, 116(13): 133903 pmid: 27081980
160 A Ghatak, T Das. New topological invariants in non-Hermitian systems. Journal of Physics Condensed Matter, 2019, 31(26): 263001 pmid: 30893649
161 P St-Jean, V Goblot, E Galopin, A Lemaître, T Ozawa, L Le Gratiet, I Sagnes, J Bloch, A Amo. Lasing in topological edge states of a one-dimensional lattice. Nature Photonics, 2017, 11(10): 651–656
162 M Parto, S Wittek, H Hodaei, G Harari, M A Bandres, J Ren, M C Rechtsman, M Segev, D N Christodoulides, M Khajavikhan. Edge-mode lasing in 1D topological active arrays. Physical Review Letters, 2018, 120(11): 113901 pmid: 29601765
163 H Zhao, P Miao, M H Teimourpour, S Malzard, R El-Ganainy, H Schomerus, L Feng. Topological hybrid silicon microlasers. Nature Communications, 2018, 9(1): 981 pmid: 29515127
164 Y Ota, R Katsumi, K Watanabe, S Iwamoto, Y Arakawa. Topological photonic crystal nanocavity laser. Communications on Physics, 2018, 1(1): 86
165 F D M Haldane. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Physical Review Letters, 1988, 61(18): 2015–2018 pmid: 10038961
166 J Schmidt, M R G Marques, S Botti, M A L Marques. Recent advances and applications of machine learning in solid-state materials science. NPJ Computational Materials, 2019, 5(1): 83
167 L Pilozzi, F A Farrelly, G Marcucci, C Conti. Machine learning inrerse problem for topological photonics. Communications Physics, 2018, 1(1): 57
168 Y Long, J Ren, Y Li, H Chen. Inverse design of photonic topological state via machine learning. Applied Physics Letters, 2019, 114(18): 181105
169 C Barth, C Becker. Machine learning classification for field distributions of photonic modes. Communications on Physics, 2018, 1(1): 58
170 U Fano. Effects of configuration interaction on intensities and phase shifts. Physical Review, 1961, 124(6): 1866–1878
171 M F Limonov, M V Rybin, A N Poddubny, Y S Kivshar. Fano resonances in photonics. Nature Photonics, 2017, 11(9): 543–554
172 A E Miroshnichenko, S Flach, Y S Kivshar. Fano resonances in nanoscale structures. Reviews of Modern Physics, 2010, 82(3): 2257–2298
173 B S Luk’yanchuk, A E Miroshnichenko, Y S Kivshar. Fano resonances and topological optics: an interplay of far- and near-field interference phenomena. Journal of Optics, 2013, 15(7): 073001
174 W Gao, X Hu, C Li, J Yang, Z Chai, J Xie, Q Gong. Fano-resonance in one-dimensional topological photonic crystal heterostructure. Optics Express, 2018, 26(7): 8634–8644 pmid: 29715828
175 F Zangeneh-Nejad, R Fleury. Topological Fano resonances. Physical Review Letters, 2019, 122(1): 014301 pmid: 31012649
176 G Q Liang, Y D Chong. Optical resonator analog of a two-dimensional topological insulator. Physical Review Letters, 2013, 110(20): 203904 pmid: 25167412
Full text