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Photonic topological insulators are artificial electromagnetic materials that support topologically non-trivial, unidirectional states of light.JOURNAL, Lu, Ling, Joannopoulos, John D., Soljačić, Marin, Topological photonics, Nature Photonics, November 2014, 8, 11, 821–829, 10.1038/nphoton.2014.248, en, 1749-4893, 1408.6730, 2014NaPho...8..821L, 119191655, Photonic topological phases are classical electromagnetic wave analogues of electronic topological phases studied in condensed matter physics. Similar to their electronic counterparts, they, can provide robust unidirectional channels for light propagation.JOURNAL, Ozawa, Tomoki, Price, Hannah M., Amo, Alberto, Goldman, Nathan, Hafezi, Mohammad, Lu, Ling, Rechtsman, Mikael C., Schuster, David, Simon, Jonathan, Zilberberg, Oded, Carusotto, Iacopo, Topological photonics, Reviews of Modern Physics, 25 March 2019, 91, 1, 015006, 10.1103/RevModPhys.91.015006, 1802.04173, 2019RvMP...91a5006O, 10969735, The field that studies these phases of light is referred to as topological photonics, even though the working frequency of these electromagnetic topological insulators may fall in other parts of the electromagnetic spectrum such as the microwave range.

History

Topological order in solid state systems has been studied in condensed matter physics since the discovery of integer quantum Hall effect. But topological matter attracted considerable interest from the physics community after the proposals for possible observation of symmetry-protected topological phases (or the so-called topological insulators) in graphene,JOURNAL, Kane, C. L., Mele, E. J., Quantum Spin Hall Effect in Graphene, Physical Review Letters, 23 November 2005, 95, 22, 226801, 10.1103/PhysRevLett.95.226801, 16384250, cond-mat/0411737, 2005PhRvL..95v6801K, 6080059, and experimental observation of a 2D topological insulator in CdTe/HgTe/CdTe quantum wells in 2007.JOURNAL, Bernevig, B. Andrei, Hughes, Taylor L., Zhang, Shou-Cheng, Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells, Science, 15 December 2006, 314, 5806, 1757–1761, 10.1126/science.1133734, 17170299, en, 0036-8075, cond-mat/0611399, 2006Sci...314.1757B, 7295726, JOURNAL, Hasan, M. Z., Kane, C. L., Colloquium: Topological insulators, Reviews of Modern Physics, 8 November 2010, 82, 4, 3045–3067, 10.1103/RevModPhys.82.3045, 1002.3895, 2010RvMP...82.3045H, 16066223,weblink In 2008, Haldane and Raghu proposed that unidirectional electromagnetic states analogous to (integer) quantum Hall states can be realized in nonreciprocal magnetic photonic crystals.JOURNAL, Haldane, F. D. M., Raghu, S., Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Physical Review Letters, 10 January 2008, 100, 1, 013904, 10.1103/PhysRevLett.100.013904, 18232766, cond-mat/0503588, 2008PhRvL.100a3904H, 44745453, This prediction was first realized in 2009 in the microwave frequency regime.JOURNAL, Wang, Zheng, etal, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature, 2009, 461, 7265, 772–775, 10.1038/nature08293, 19812669, 2009Natur.461..772W, 1721.1/88469, 4427789, free, This was followed by the proposals for analogous quantum spin Hall states of electromagnetic waves that are now known as photonic topological insulators.JOURNAL, Hafezi, Mohammad, Demler, Eugene A., Lukin, Mikhail D., Taylor, Jacob M., Robust optical delay lines with topological protection, Nature Physics, November 2011, 7, 11, 907–912, 10.1038/nphys2063, en, 1745-2481, 1102.3256, 2011NatPh...7..907H, 2008767, JOURNAL, Khanikaev, Alexander B., Hossein Mousavi, S., Tse, Wang-Kong, Kargarian, Mehdi, MacDonald, Allan H., Shvets, Gennady, Photonic topological insulators, Nature Materials, March 2013, 12, 3, 233–239, 10.1038/nmat3520, 23241532, en, 1476-4660, 1204.5700, 2013NatMa..12..233K, It was later found that topological electromagnetic states can exist in continuous media as well--theoretical and numerical study has confirmed the existence of topological Langmuir-cyclotron waves in continuous magnetized plasmas.JOURNAL, Qin, Hong, Fu, Yichen, 2023-03-31, Topological Langmuir-cyclotron wave, Science Advances, en, 9, 13, eadd8041, 10.1126/sciadv.add8041, 2375-2548, 10065437, 37000869, JOURNAL, Fu, Yichen, Qin, Hong, 2021-06-24, Topological phases and bulk-edge correspondence of magnetized cold plasmas, Nature Communications, en, 12, 1, 3924, 10.1038/s41467-021-24189-3, 34168159, 2041-1723, 8225675,

Platforms

Photonic topological insulators are designed using various photonic platforms including optical waveguide arrays,JOURNAL, Rechtsman, Mikael, etal, Photonic Floquet Topological Insulators, Nature, April 10, 2013, 496, 7444, 196–200, 10.1038/nature12066, 23579677, 1212.3146, 2013Natur.496..196R, 4349770, coupled ring resonators,JOURNAL, Hafezi, M., Mittal, S., Fan, J., Migdall, A., Taylor, J. M., Imaging topological edge states in silicon photonics, Nature Photonics, December 2013, 7, 12, 1001–1005, 10.1038/nphoton.2013.274, en, 1749-4893, 1302.2153, 2013NaPho...7.1001H, 14394865, bi-anisotropic meta-materials, and photonic crystals.JOURNAL, Wu, Long-Hua, Hu, Xiao, Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material, Physical Review Letters, 3 June 2015, 114, 22, 223901, 10.1103/PhysRevLett.114.223901, 26196622, 1503.00416, 2015PhRvL.114v3901W, More recently, they have been realized in 2D dielectricJOURNAL, Gorlach, Maxim A., Ni, Xiang, Smirnova, Daria A., Korobkin, Dmitry, Zhirihin, Dmitry, Slobozhanyuk, Alexey P., Belov, Pavel A., Alù, Andrea, Khanikaev, Alexander B., Far-field probing of leaky topological states in all-dielectric metasurfaces, Nature Communications, 2 March 2018, 9, 1, 909, 10.1038/s41467-018-03330-9, 29500466, 5834506, 2018NatCo...9..909G, en, 2041-1723, free, and plasmonicJOURNAL, Honari-Latifpour, Mostafa, Yousefi, Leila, Topological plasmonic edge states in a planar array of metallic nanoparticles, Nanophotonics, 2019, 8, 5, 799–806, 10.1515/nanoph-2018-0230, 2019Nanop...8..230H, 2192-8614, free, meta-surfaces. Despite the theoretical prediction, no experimental demonstration of photonic topological insulator in continuous media has been reported.

Chern number

As an important figure of merit for characterizing the quantized collective behaviors of the wavefunction, Chern number is the topological invariant of quantum Hall insulators. Chern number also identifies the topological properties of the photonic topological insulators (PTIs), thus it is of crucial importance in PTI design. The full-wave finite-difference frequency-domain (FDFD) method based MATLAB program for computing the Chern number has been written.JOURNAL, Zhao, Ran, Xie, Guo-Da, Chen, Menglin L. N., Lan, Zhihao, Huang, Zhixiang, Sha, Wei E. I., 2020-02-17, First-principle calculation of Chern number in gyrotropic photonic crystals,weblink Optics Express, EN, 28, 4, 4638–4649, 10.1364/OE.380077, 32121697, 1094-4087, 2001.08913, 2020OExpr..28.4638Z, 210911652, Recently, the finite-difference method has been extended to analyze the topological invariant of non-Hermitian topological dielectric photonic crystals by first-principle Wilson loop calculation. JOURNAL, Chen, Menglin L. N., Jiang, Li Jun, Zhang, Shuang, Zhao, Ran, Lan, Zhihao, Sha, Wei E. I., 2021-09-01, Comparative study of Hermitian and non-Hermitian topological dielectric photonic crystals,weblink Physical Review A, EN, 104, 3, 033501, 10.1103/PhysRevA.104.033501, 2109.05498, 2021PhRvA.104c3501C, 237492242, All MATLAB codes can be found at GitHub website. {{Citation|title=Topological-Invariant-Optics|url=https://github.com/Sha-Group/Topological-Invariant-Optics}}

See also

• Symmetry-protected topological order
• Metamaterial

References

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