Photonic topological insulator
Photonic topological insulators are artificial electromagnetic materials that support topologically non-trivial, unidirectional states of light.<ref>Lu, Ling; Joannopoulos, John D.; Soljačić, Marin (November 2014). "Topological photonics". Nature Photonics. 8 (11): 821–829. arXiv:1408.6730. Bibcode:2014NaPho...8..821L. doi:10.1038/nphoton.2014.248. ISSN 1749-4893. S2CID 119191655.</ref> 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.<ref>Ozawa, Tomoki; Price, Hannah M.; Amo, Alberto; Goldman, Nathan; Hafezi, Mohammad; Lu, Ling; Rechtsman, Mikael C.; Schuster, David; Simon, Jonathan; Zilberberg, Oded; Carusotto, Iacopo (25 March 2019). "Topological photonics". Reviews of Modern Physics. 91 (1): 015006. arXiv:1802.04173. Bibcode:2019RvMP...91a5006O. doi:10.1103/RevModPhys.91.015006. S2CID 10969735.</ref>
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.<ref name="khanikaev" />
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,<ref>Kane, C. L.; Mele, E. J. (23 November 2005). "Quantum Spin Hall Effect in Graphene". Physical Review Letters. 95 (22): 226801. arXiv:cond-mat/0411737. Bibcode:2005PhRvL..95v6801K. doi:10.1103/PhysRevLett.95.226801. PMID 16384250. S2CID 6080059.</ref> and experimental observation of a 2D topological insulator in CdTe/HgTe/CdTe quantum wells in 2007.<ref>Bernevig, B. Andrei; Hughes, Taylor L.; Zhang, Shou-Cheng (15 December 2006). "Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells". Science. 314 (5806): 1757–1761. arXiv:cond-mat/0611399. Bibcode:2006Sci...314.1757B. doi:10.1126/science.1133734. ISSN 0036-8075. PMID 17170299. S2CID 7295726.</ref><ref>Hasan, M. Z.; Kane, C. L. (8 November 2010). "Colloquium: Topological insulators". Reviews of Modern Physics. 82 (4): 3045–3067. arXiv:1002.3895. Bibcode:2010RvMP...82.3045H. doi:10.1103/RevModPhys.82.3045. S2CID 16066223.</ref>
In 2008, Haldane and Raghu proposed that unidirectional electromagnetic states analogous to (integer) quantum Hall states can be realized in nonreciprocal magnetic photonic crystals.<ref>Haldane, F. D. M.; Raghu, S. (10 January 2008). "Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry". Physical Review Letters. 100 (1): 013904. arXiv:cond-mat/0503588. Bibcode:2008PhRvL.100a3904H. doi:10.1103/PhysRevLett.100.013904. PMID 18232766. S2CID 44745453.</ref> This prediction was first realized in 2009 in the microwave frequency regime.<ref>Wang, Zheng; et al. (2009). "Observation of unidirectional backscattering-immune topological electromagnetic states". Nature. 461 (7265): 772–775. Bibcode:2009Natur.461..772W. doi:10.1038/nature08293. hdl:1721.1/88469. PMID 19812669. S2CID 4427789.</ref> This was followed by the proposals for analogous quantum spin Hall states of electromagnetic waves that are now known as photonic topological insulators.<ref>Hafezi, Mohammad; Demler, Eugene A.; Lukin, Mikhail D.; Taylor, Jacob M. (November 2011). "Robust optical delay lines with topological protection". Nature Physics. 7 (11): 907–912. arXiv:1102.3256. Bibcode:2011NatPh...7..907H. doi:10.1038/nphys2063. ISSN 1745-2481. S2CID 2008767.</ref><ref name="khanikaev">Khanikaev, Alexander B.; Hossein Mousavi, S.; Tse, Wang-Kong; Kargarian, Mehdi; MacDonald, Allan H.; Shvets, Gennady (March 2013). "Photonic topological insulators". Nature Materials. 12 (3): 233–239. arXiv:1204.5700. Bibcode:2013NatMa..12..233K. doi:10.1038/nmat3520. ISSN 1476-4660. PMID 23241532.</ref> 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.<ref name=":0">Qin, Hong; Fu, Yichen (2023-03-31). "Topological Langmuir-cyclotron wave". Science Advances. 9 (13): eadd8041. doi:10.1126/sciadv.add8041. ISSN 2375-2548. PMC 10065437. PMID 37000869.</ref><ref name=":1">Fu, Yichen; Qin, Hong (2021-06-24). "Topological phases and bulk-edge correspondence of magnetized cold plasmas". Nature Communications. 12 (1): 3924. doi:10.1038/s41467-021-24189-3. ISSN 2041-1723. PMC 8225675.</ref>
Platforms
Photonic topological insulators are designed using various photonic platforms including optical waveguide arrays,<ref>Rechtsman, Mikael; et al. (April 10, 2013). "Photonic Floquet Topological Insulators". Nature. 496 (7444): 196–200. arXiv:1212.3146. Bibcode:2013Natur.496..196R. doi:10.1038/nature12066. PMID 23579677. S2CID 4349770.</ref> coupled ring resonators,<ref>Hafezi, M.; Mittal, S.; Fan, J.; Migdall, A.; Taylor, J. M. (December 2013). "Imaging topological edge states in silicon photonics". Nature Photonics. 7 (12): 1001–1005. arXiv:1302.2153. Bibcode:2013NaPho...7.1001H. doi:10.1038/nphoton.2013.274. ISSN 1749-4893. S2CID 14394865.</ref> bi-anisotropic meta-materials, and photonic crystals.<ref>Wu, Long-Hua; Hu, Xiao (3 June 2015). "Scheme for Achieving a Topological Photonic Crystal by Using Dielectric Material". Physical Review Letters. 114 (22): 223901. arXiv:1503.00416. Bibcode:2015PhRvL.114v3901W. doi:10.1103/PhysRevLett.114.223901. PMID 26196622.</ref> More recently, they have been realized in 2D dielectric<ref>Gorlach, Maxim A.; Ni, Xiang; Smirnova, Daria A.; Korobkin, Dmitry; Zhirihin, Dmitry; Slobozhanyuk, Alexey P.; Belov, Pavel A.; Alù, Andrea; Khanikaev, Alexander B. (2 March 2018). "Far-field probing of leaky topological states in all-dielectric metasurfaces". Nature Communications. 9 (1): 909. Bibcode:2018NatCo...9..909G. doi:10.1038/s41467-018-03330-9. ISSN 2041-1723. PMC 5834506. PMID 29500466.</ref> and plasmonic<ref>Honari-Latifpour, Mostafa; Yousefi, Leila (2019). "Topological plasmonic edge states in a planar array of metallic nanoparticles". Nanophotonics. 8 (5): 799–806. Bibcode:2019Nanop...8..230H. doi:10.1515/nanoph-2018-0230. ISSN 2192-8614.</ref> meta-surfaces. Despite the theoretical prediction,<ref name=":0" /><ref name=":1" /> 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.<ref>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". Optics Express. 28 (4): 4638–4649. arXiv:2001.08913. Bibcode:2020OExpr..28.4638Z. doi:10.1364/OE.380077. ISSN 1094-4087. PMID 32121697. S2CID 210911652.</ref> 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. <ref>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". Physical Review A. 104 (3): 033501. arXiv:2109.05498. Bibcode:2021PhRvA.104c3501C. doi:10.1103/PhysRevA.104.033501. S2CID 237492242.</ref>
