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一维微米尺度的激光器一般是采用一维的Su-Schrieffer-Heeger(SSH)模型的光子晶格[42, 45, 49],它是一种最简单构建拓扑结构的模型,最初是用来描述聚乙炔分子的电子输运所提出来的一种双周期紧束缚模型[50]。其结构简单,但具有丰富的物理现象。图2(a)、图2(b)为SSH模型的原理图[45],图2(c)、图2(d)为px,py轨道子空间在晶格链[45],图2(e)为环形谐振晶格[49],图2(f)为一维微环阵列[42]。在SSH模型中,每个原胞由两个不等价的格点an,bn组成,多个原胞组成一条晶格常数为a的一维二聚体晶格链,一维二聚体晶格链的胞内跃迁强度t与胞间跃迁强度t′可以调节。在动量空间,以每个原胞的an,bn格点为基础,哈密顿量可以写成[51]:
$$ H(k)=\left(\genfrac{}{}{0pt}{}{\begin{array}{cc}0& t+{t}^{\text{′}}\mathrm{exp}\left(ika\right)\end{array}}{\begin{array}{cc}t+{t}^{\text{′}}\mathrm{exp}\left(-ika\right)& 0\end{array}}\right) $$ 通过求解得到两个具有能隙的能带,能隙宽度为2|t-t′|。可以看到,当一维二聚体晶格链的胞内跃迁强度t与胞间跃迁强度t′不相等时,系统将产生能隙。当胞内跃迁强度t大于胞间跃迁强度t′时,系统的缠绕数为0,当胞内跃迁强度t小于胞间跃迁强度t′时,系统缠绕数为1,这两个缠绕数对应于系统两个不同的拓扑相,导致在这两种情况下系统具有完全不同的拓扑态,分别为拓扑平庸和拓扑非平庸。拓扑非平庸的主要特征是在链与真空的边界处会发生拓扑相变,从而出现边界态,在能带中的表现是在能隙中出现了边缘态。因此可以根据SSH模型来设计具有拓扑边界态的拓扑结构[52]。
第一个用拓扑边缘态产生激光的一维SSH模型光子晶格是使用无机半导体制成的,其中量子阱作为增益介质。2017年,Philippe St-Jean等人采取了微柱的zigzag极化晶格实现了第一个基于具有拓扑保护的边缘态的激光器[45](见图1(a))。在微柱的zigzag极化晶格中,SSH哈密顿量是通过px,py轨道的耦合实现的。图2(c)中,微柱间的交替排列导致了模式重叠,由此可以看出px轨道在重叠时表现为强的胞内耦合强度,弱的胞间耦合强度;而对于py轨道则表现为相反的耦合排列。虽然这两种轨道子空间具有相同的能隙,但对于晶格链的边缘,(见图2(d)),只有py轨道子空间在晶格链的边缘拥有一个弱链接,呈现出一个拓扑边缘态,而px轨道子空间在晶格链的边缘拥有一个强链接。由于SSH模型的手性对称性,对于弱的增益/损耗,体模的增益区域和损耗区域重叠,使得体模的净增益消失。而局域在晶格链的终端处或者畴壁处的拓扑模式,通过泵浦产生非零的净增益。为了让拓扑模式中获得比体模更大的增益,他们对晶格链的边缘进行了光泵浦。由于能隙保持打开的状态,泵浦晶格链的边缘产生了拓扑界面态的单模激光[53]。
同年,Parto M等人[49]以及Zhao H等人[42]使用以 InGaAsP、InP量子阱作为增益介质的环形谐振晶格实现了微环激光器。通过相邻环间的交错分离来实现胞内耦合强度t和胞间耦合强度t′。两者工作的区别是:Parto M等人[49]将拓扑态局域在了晶格链的边缘处,在晶格链的边缘设计了一个弱耦合,见图2(e);而Zhao H等人[42]将拓扑态局域在了两个具有不同拓扑数的晶格链的界面上,见图2(f)。通过设计不同的泵浦方式,使得相应的拓扑模式得到增益,从而发出激光。
对于上述工作,比较有趣的是拓扑激光器的模式对无序具有鲁棒性,晶格的哈密顿量的手性对称性使得拓扑模式对胞内耦合强度t和胞间耦合强度t′的扰动不敏感,只要拓扑模式的能量能很好地隔离在能隙中间,所产生的激光并不会受到很大的影响。St-Jean P[45]和Zhao H等人[42]的实验都很好地证实了这一点。
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Haldane FDM等人第一次提出通过打破时间反演对称性来打开光子系统拓扑能隙[12],并且在微波频段实现[60],其中光子晶体结构由铁氧体棒构成。在这些开创性的工作中,由于系统的时间反演对称性被打破,在具有不同拓扑性质的结构之间会出现手性边缘模式,它具有对缺陷、无序的鲁棒性。通常,在传统的光子晶体腔中缺陷和无序的存在会导致散射损失,使得腔的品质因子下降以及输出功率降低,最终使得激光器的性能下降,而拓扑手性边缘模式对于这些是具有鲁棒性的,可以很好地提高激光器的性能。2017年,Bahari B等第一次用实验实现了二维拓扑边缘态激光器[47],该课题组使用嵌着YIG(Yttrium iron garnet)结构的光子晶体,通过磁光效应来打破了时间反演对称性,从而在Γ点打开了拓扑带隙[47]。在静态磁场的作用下,用1 060 nm波长脉冲光去泵浦整个结构,增益强烈地局域在边缘态上,边界的形状对其是没有影响的,如图4(a)所示。通过改变静态磁场的方向,可以验证拓扑手性边缘模式的单向性。但是由于光学波段地磁光效应非常弱,使得打开的拓扑能隙非常小(42 pm),如图4(b)所示。然而,拓扑边缘模式所发出来的光需要进一步使用二阶强度相关的实验证明[35]。在后续的工作中,该团队设计了环形拓扑非平庸-平庸界面来产生携带轨道角动量的相干光束[47]。该设计成功产生了任意大的拓扑电荷,并将不同的轨道角动量多路复用发射到单个样品上。
另一个二维激光器的实现是基于蜂窝晶格的激子-极化子与强磁场的结合[61]。极化子的自旋-轨道耦合与磁场的导致的Zeeman位移的结合得到由Chern相位来描述的拓扑能隙[62-63]。由于能到的拓扑能隙很小,使得边缘态的强局域变得困难。随后,Y. V. Kartashov和D. V. Skryabin通过理论证实了非线性稳定的激光模式是可以实现的[64],他们还发现,当达到激光阈值之后,自作用项会导致边缘模式的频率向体带边缘移动。
由此可以看到,文中所讨论为二维拓扑激光器提供了一种新型的光源,通过打破时间反演对称,使得它可以以任意形状发射激光,且具有手性度。有趣的是,极化子晶格呈现显著的非线性[65],因此提供了一个很好的平台来研究激光机制中的拓扑性质与非线性相互作用[66-67]。这类打破时间反演对称性的拓扑激光器的实验研究有很大的应用前景。
除此之外,还有一种二维拓扑激光器,即没有打破时间反演对称性的激光器[68-69]。此类激光器是基于Harper-Hofstadter模型所构建的,最初使用的是无源的光子晶体平台的环形谐振器[11, 32]。
Harper-Hofstadter模型是一个在垂直于二维平面上施加一个均匀磁场的二维方形晶格模型。Hafezi M等人通过二维静态谐振硅波导环构造光子等效磁场的方案,实现了Harper-Hofstadter模型。该结构有一个特点是光子从一个谐振器跳到另一个谐振器的路径长度与反方向跳跃的路径长度不同,这种路径差有效的引入了非零的相位。除非这个非零的相位是一个2Π的整数倍或者半整数倍,否则Harper-Hofstadter模型的时间反演对称性会被打破。而在Hafezi M等人文章中,在时间反演对称性没有被打破的同时,Si的静态谐振器微环支持两个简并的模式,分别是顺时针传输以及逆时针传输的谐振模式[11, 32]。可以把这两个模式理解成光学体系中的赝自旋,其中顺时针谐振模式理解成自旋向上,遵循磁场强度为B的Harper-Hofstadter模型,逆时针谐振模式理解成自旋向下,遵循磁场强度为-B的Harper-Hofstadter模型。整个系统保持了时间反演对称性,系统整体的哈密顿量如下:
$$ {H}_{overall}=\left[\begin{array}{cc}H\left(B\right)& 0\\ 0& H(-B)\end{array}\right] $$ 这个系统类似于电子系统中的保持时间反演对称的量子自旋霍尔哈密顿量[70-72]。
图4(c)为构造Harper-Hofstadter模型的环形谐振器波导结构[68];图4(d)为激光模式对于拓扑列阵中的缺陷具有鲁棒性[68]。Bandres M等人使用由InGaAsP制作的环形谐振器波导构造了B=2Π/4的Harper-Hofstadter模型[68],其中InGaAsP是带着光学增益的,见图4(c)。该团队通过在结构边缘增加增益,得到拓扑边缘态的激光。通过对拓扑Harper-Hofstadter模型和一个简单的正方晶格的激光比较可知,拓扑激光器具有单模、鲁棒等优越性。其中,简单的正方晶格是拓扑平庸的,对于拓扑非平庸情况下,所发出的激光光谱上仅仅只有一个峰,然而拓扑平庸的情况下的激光光谱的峰很宽,同时拓扑激光器对结构的无序具有鲁棒性,见图4(d)。
Research progress of topological lasers (Invited)
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摘要: 随着拓扑光子学的发展,对缺陷、微扰等具有鲁棒性的拓扑边缘态、拓扑角态的发现革新了半导体激光器,并且推动了拓扑激光器的发展。首先,回顾了近几年拓扑激光器的发展历程,以及不同拓扑激光器的原理;其次,分析了各类拓扑激光器的最新实现,并且解释了拓扑边缘态、拓扑角态的基础物理。在这些实验中,拓扑激光器的模式是由介质结构决定,并通过光学增益来激发激光。分析表明拓扑角态相比于拓扑边缘态具有强局域性、小模式体积,因此基于拓扑角态的拓扑激光器更高效,阈值更低,这为未来的光子集成芯片提供了可能;最后,展望了拓扑激光器面临挑战和潜在的应用方向,有助于探索更实用的拓扑激光器。Abstract: With the development of topological photonics, topological lasers and semiconductor lasers are promoted by the discovery of the topological edge states and corner states with robustness against defects and perturbations. Firstly, the development history of the topological lasers and the principles of the various kinds of topological lasers was reviewed; Secondly recent realizations of various topological lasers were analyzed and the basic physics about topological edge states and topological corner states was explained. In these experiments, the modes of topological laser were decided by the dielectric structure. The laser was excited by pumping the photonic gain. The analysis show that topological lasers based on topological corner states have higher efficiency and lower threshold than those based on topological edge state, due to the high quality factor and small mode volume of topological corner state, which provides the possibility for future photonic integrated chip. Finally, the challenge and potential applications in the future were outlooked, which was beneficial to explore practical topological laser.
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[1] Chalcraft A R A, Lam S, O’Brien D, et al. Mode structure of the L 3 photonic crystal cavity [J]. Applied Physics Letters, 2007, 90(24): 241117. doi: 10.1063/1.2748310 [2] Safavi-Naeini A H, Alegre T P M, Winger M, et al. Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity [J]. Applied Physics Letters, 2010, 97(18): 181106. doi: 10.1063/1.3507288 [3] Vučković J, Lončar M, Mabuchi H, et al. Design of photonic crystal microcavities for cavity QED [J]. Physical Review E, 2001, 65(1): 016608. doi: 10.1103/PhysRevE.65.016608 [4] Yamamoto T, Pashkin Y A, Astafiev O, et al. Demonstration of conditional gate operation using superconducting charge qubits [J]. Nature, 2003, 425(6961): 941-944. doi: 10.1038/nature02015 [5] Yoshie T, Vučković J, Scherer A, et al. High quality two-dimensional photonic crystal slab cavities [J]. Applied Physics Letters, 2001, 79(26): 4289-4291. doi: 10.1063/1.1427748 [6] Faraon A, Waks E, Englund D, et al. Efficient photonic crystal cavity-waveguide couplers [J]. Applied Physics Letters, 2007, 90(7): 073102. doi: 10.1063/1.2472534 [7] Goyal A K, Pal S. Design and simulation of high sensitive photonic crystal waveguide sensor [J]. Optik, 2015, 126(2): 240-243. doi: 10.1016/j.ijleo.2014.08.174 [8] Gu L, Jiang W, Chen X, et al. High speed silicon photonic crystal waveguide modulator for low voltage operation [J]. Applied Physics Letters, 2007, 90(7): 071105. doi: 10.1063/1.2475580 [9] Barik S, Karasahin A, Flower C, et al. A topological quantum optics interface [J]. Science, 2018, 359(6376): 666-668. doi: 10.1126/science.aaq0327 [10] Tambasco J L, Corrielli G, Chapman R J, et al. Quantum interference of topological states of light [J]. Science Advances, 2018, 4(9): 3187. doi: 10.1126/sciadv.aat3187 [11] Hafezi M, Mittal S, Fan J, et al. Imaging topological edge states in silicon photonics [J]. Nature Photonics, 2013, 7(12): 1001-1005. doi: 10.1038/nphoton.2013.274 [12] Haldane F D M, Raghu S. Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry [J]. Physical Review Letters, 2008, 100(1): 013904. doi: 10.1103/PhysRevLett.100.013904 [13] Khanikaev A B, Mousavi S H, Tse W K, et al. Photonic topological insulators [J]. Nature Materials, 2013, 12(3): 233-239. doi: 10.1038/nmat3520 [14] Lu L, Joannopoulos J D, Soljačić M. Topological photonics [J]. Nature Photonics, 2014, 8(11): 821-829. doi: 10.1038/nphoton.2014.248 [15] Mittal S, Goldschmidt E A, Hafezi M. A topological source of quantum light [J]. Nature, 2018, 561(7724): 502-506. doi: 10.1038/s41586-018-0478-3 [16] Ozawa T, Price H M, Amo A, et al. Topological photonics [J]. Reviews of Modern Physics, 2019, 91(1): 015006. doi: 10.1103/RevModPhys.91.015006 [17] Rechtsman M C, Zeuner J M, Plotnik Y, et al. Photonic Floquet topological insulators [J]. Nature, 2013, 496(7444): 196-200. doi: 10.1038/nature12066 [18] Wang Y, Lu Y H, Mei F, et al. Direct observation of topology from single-photon dynamics [J]. Physical Review Letters, 2019, 122(19): 193903. doi: 10.1103/PhysRevLett.122.193903 [19] Wang Z, Chong Y, Joannopoulos J D, et al. Observation of unidirectional backscattering-immune topological electromagnetic states [J]. Nature, 2009, 461(7265): 772-775. doi: 10.1038/nature08293 [20] Wu L H, Hu X. Scheme for achieving a topological photonic crystal by using dielectric material [J]. Physical Review Letters, 2015, 114(22): 223901. doi: 10.1103/PhysRevLett.114.223901 [21] Chen W J, Jiang S J, Chen X D, et al. Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide [J]. Nature Communications, 2014, 5(1): 5782. doi: 10.1038/ncomms6782 [22] Yang Y, Xu Y F, Xu T, et al. Visualization of a unidirectional electromagnetic waveguide using topological photonic crystals made of dielectric materials [J]. Physical Review Letters, 2018, 120(21): 217401. doi: 10.1103/PhysRevLett.120.217401 [23] Gao Y F, Sun J P, Xu N, et al. Manipulation of topological beam splitter based on honeycomb photonic crystals [J]. Optics Communications, 2021, 483: 126646. doi: 10.1016/j.optcom.2020.126646 [24] He L, Ji H Y, Wang Y J, et al. Topologically protected beam splitters and logic gates based on two-dimensional silicon photonic crystal slabs [J]. Optics Express, 2020, 28(23): 34015-34023. doi: 10.1364/OE.409265 [25] Qi L, Wang G L, Liu S, et al. Engineering the topological state transfer and topological beam splitter in an even-sized Su-Schrieffer-Heeger chain [J]. Physical Review A, 2020, 102(2): 022404. doi: 10.1103/PhysRevA.102.022404 [26] Yang Y, Gao Z, Xue H, et al. Realization of a three-dimensional photonic topological insulator [J]. Nature, 2019, 565(7741): 622-626. doi: 10.1038/s41586-018-0829-0 [27] Ji C Y, Liu G B, Zhang Y, et al. Transport tuning of photonic topological edge states by optical cavities [J]. Physical Review A, 2019, 99(4): 043801. doi: 10.1103/PhysRevA.99.043801 [28] Kim H R, Hwang M S, Smirnova D, et al. Multipolar lasing modes from topological corner states [J]. Nature Communications, 2020, 11(1): 5758. doi: 10.1038/s41467-020-19609-9 [29] Li Y, Yu Y, Liu F, et al. Topology-controlled photonic cavity based on the near-conservation of the valley degree of freedom [J]. Physical Review Letters, 2020, 125(21): 213902. doi: 10.1103/PhysRevLett.125.213902 [30] Ota Y, Liu F, Katsumi R, et al. Photonic crystal nanocavity based on a topological corner state [J]. Optica, 2019, 6(6): 786-789. doi: 10.1364/OPTICA.6.000786 [31] Raghu S, Haldane F D M. Analogs of quantum-Hall-effect edge states in photonic crystals [J]. Physical Review A, 2008, 78(3): 033834. doi: 10.1103/PhysRevA.78.033834 [32] Hafezi M, Demler E A, Lukin M D, et al. Robust optical delay lines with topological protection [J]. Nature Physics, 2011, 7(11): 907-912. doi: 10.1038/nphys2063 [33] Gao F, Xue H, Yang Z, et al. Topologically protected refraction of robust kink states in valley photonic crystals [J]. Nature Physics, 2018, 14(2): 140-144. doi: 10.1038/nphys4304 [34] Ma T, Shvets G. All-Si valley-hall photonic topological insulator [J]. New Journal of Physics, 2016, 18(2): 025012. doi: 10.1088/1367-2630/18/2/025012 [35] Noh J, Huang S, Chen K P, et al. Observation of photonic topological valley Hall edge states [J]. Physical Review Letters, 2018, 120(6): 063902. doi: 10.1103/PhysRevLett.120.063902 [36] Shalaev M I, Walasik W, Tsukernik A, et al. Robust topologically protected transport in photonic crystals at telecommunication wavelengths [J]. Nature Nanotechnology, 2019, 14(1): 31-34. doi: 10.1038/s41565-018-0297-6 [37] Yoshimi H, Yamaguchi T, Katsumi R, et al. Experimental demonstration of topological slow light waveguides in valley photonic crystals [J]. Optics Express, 2021, 29(9): 13441-13450. doi: 10.1364/OE.422962 [38] Gong Y, Wong S, Bennett A J, et al. Topological insulator laser using valley-Hall photonic crystals [J]. ACS Photonics, 2020, 7(8): 2089-2097. doi: 10.1021/acsphotonics.0c00521 [39] Harder T H, Sun M, Egorov O A, et al. Coherent topological polariton laser [J]. ACS Photonics, 2021, 8(5): 1377-1384. doi: 10.1021/acsphotonics.0c01958 [40] Pilozzi L, Conti C. Topological cascade laser for frequency comb generation in PT-symmetric structures [J]. Optics Letters, 2017, 42(24): 5174-5177. doi: 10.1364/OL.42.005174 [41] Smirnova D, Tripathi A, Kruk S, et al. Room-temperature lasing from nanophotonic topological cavities [J]. Light: Science & Applications, 2020, 9(1): 127. [42] Zhao H, Miao P, Teimourpour M H, et al. Topological hybrid silicon microlasers [J]. Nature Communications, 2018, 9(1): 981. doi: 10.1038/s41467-018-03434-2 [43] Qian Z, Li Z, Hao H, et al. Absorption reduction of large purcell enhancement enabled by topological state-led mode coupling [J]. Physical Review Letters, 2021, 126(2): 023901. doi: 10.1103/PhysRevLett.126.023901 [44] Xie X, Zhang W, He X, et al. Cavity quantum electrodynamics with second-order topological corner state [J]. Laser & Photonics Reviews, 2020, 14(8): 1900425. [45] St-Jean P, Goblot V, Galopin E, et al. Lasing in topological edge states of a one-dimensional lattice [J]. Nature Photonics, 2017, 11(10): 651-656. doi: 10.1038/s41566-017-0006-2 [46] Han C, Lee M, Callard S, et al. Lasing at topological edge states in a photonic crystal L3 nanocavity dimer array [J]. Light: Science & Applications, 2019, 8(1): 40. [47] Bahari B, Ndao A, Vallini F, et al. Nonreciprocal lasing in topological cavities of arbitrary geometries [J]. Science, 2017, 358(6363): 636-640. doi: 10.1126/science.aao4551 [48] Zhang W, Xie X, Hao H, et al. Low-threshold topological nanolasers based on the second-order corner state [J]. Light: Science & Applications, 2020, 9(1): 109. [49] Parto M, Wittek S, Hodaei H, et al. Edge-mode lasing in 1D topological active arrays [J]. Physical Review Letters, 2018, 120(11): 113901. doi: 10.1103/PhysRevLett.120.113901 [50] Fang K, Yu Z, Fan S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation [J]. Nature Photonics, 2012, 6(11): 782-787. doi: 10.1038/nphoton.2012.236 [51] Asbóth J K, Oroszlány L, Pályi A. A short course on topological insulators [J]. Lecture Notes in Physics, 2016, 919: 166. [52] Ota Y, Takata K, Ozawa T, et al. Active topological photonics [J]. Nanophotonics, 2020, 9(3): 547-567. doi: 10.1515/nanoph-2019-0376 [53] Weimann S, Kremer M, Plotnik Y, et al. Topologically protected bound states in photonic parity–time-symmetric crystals [J]. Nature Materials, 2017, 16(4): 433-438. doi: 10.1038/nmat4811 [54] Ota Y, Katsumi R, Watanabe K, et al. Topological photonic crystal nanocavity laser [J]. Communications Physics, 2018, 1(1): 86. doi: 10.1038/s42005-018-0083-7 [55] Xiao M, Zhang Z Q, Chan C T. Surface impedance and bulk band geometric phases in one-dimensional systems [J]. Physical Review X, 2014, 4(2): 021017. doi: 10.1103/PhysRevX.4.021017 [56] Alpeggiani F, Andreani L C, Gerace D. Effective bichromatic potential for ultra-high Q-factor photonic crystal slab cavities [J]. Applied Physics Letters, 2015, 107(26): 261110. doi: 10.1063/1.4938395 [57] Simbula A, Schatzl M, Zagaglia L, et al. Realization of high-Q/V photonic crystal cavities defined by an effective Aubry-André-Harper bichromatic potential [J]. APL Photonics, 2017, 2(5): 056102. doi: 10.1063/1.4979708 [58] Alpeggiani F, Kuipers L. Topological edge states in bichromatic photonic crystals [J]. Optica, 2019, 6(1): 96-103. doi: 10.1364/OPTICA.6.000096 [59] Pilozzi L, Conti C. Topological lasing in resonant photonic structures [J]. Physical Review B, 2016, 93(19): 195317. doi: 10.1103/PhysRevB.93.195317 [60] Wang Z, Chong Y D, Joannopoulos J D, et al. Reflection-free one-way edge modes in a gyromagnetic photonic crystal [J]. Physical Review Letters, 2008, 100(1): 013905. doi: 10.1103/PhysRevLett.100.013905 [61] Klembt S, Harder T H, Egorov O A, et al. Exciton-polariton topological insulator [J]. Nature, 2018, 562(7728): 552-556. doi: 10.1038/s41586-018-0601-5 [62] Nalitov A V, Solnyshkov D D, Malpuech G. Polariton Z topological insulator [J]. Physical Review Letters, 2015, 114(11): 116401. doi: 10.1103/PhysRevLett.114.116401 [63] Karzig T, Bardyn C E, Lindner N H, et al. Topological polaritons [J]. Physical Review X, 2015, 5(3): 031001. doi: 10.1103/PhysRevX.5.031001 [64] Kartashov Y V, Skryabin D V. Two-dimensional topological polariton laser [J]. Physical Review Letters, 2019, 122(8): 083902. doi: 10.1103/PhysRevLett.122.083902 [65] Carusotto I, Ciuti C. Quantum fluids of light [J]. Reviews of Modern Physics, 2013, 85(1): 299. doi: 10.1103/RevModPhys.85.299 [66] Kartashov Y V, Skryabin D V. Modulational instability and solitary waves in polariton topological insulators [J]. Optica, 2016, 3(11): 1228-1236. doi: 10.1364/OPTICA.3.001228 [67] Kartashov Y V, Skryabin D V. Bistable topological insulator with exciton-polaritons [J]. Physical Review Letters, 2017, 119(25): 253904. doi: 10.1103/PhysRevLett.119.253904 [68] Bandres M A, Wittek S, Harari G, et al. Topological insulator laser: Experiments [J]. Science, 2018, 359(6381): 4005. [69] Harari G, Bandres M A, Lumer Y, et al. Topological insulator laser:Theory [J]. Science, 2018, 359(6381): 4003. [70] Kane C L, Mele E J. Z 2 topological order and the quantum spin Hall effect [J]. Physical Review Letters, 2005, 95(14): 146802. doi: 10.1103/PhysRevLett.95.146802 [71] Kane C L, Mele E J. Quantum spin Hall effect in graphene [J]. Physical Review Letters, 2005, 95(22): 226801. doi: 10.1103/PhysRevLett.95.226801 [72] Bernevig B A, Hughes T L, Zhang S C. Quantum spin Hall effect and topological phase transition in HgTe quantum wells [J]. Science, 2006, 314(5806): 1757-1761. doi: 10.1126/science.1133734 [73] Chen X D, Deng W M, Shi F L, et al. Direct observation of corner states in second-order topological photonic crystal slabs [J]. Physical Review Letters, 2019, 122(23): 233902. doi: 10.1103/PhysRevLett.122.233902 [74] Dutt A, Minkov M, Williamson I A D, et al. Higher-order topological insulators in synthetic dimensions [J]. Light: Science & Applications, 2020, 9(1): 131. [75] Imhof S, Berger C, Bayer F, et al. Topolectrical-circuit realization of topological corner modes [J]. Nature Physics, 2018, 14(9): 925-929. doi: 10.1038/s41567-018-0246-1 [76] Langbehn J, Peng Y, Trifunovic L, et al. Reflection-symmetric second-order topological insulators and superconductors [J]. Physical Review Letters, 2017, 119(24): 246401. doi: 10.1103/PhysRevLett.119.246401 [77] Liu T, Zhang Y R, Ai Q, et al. Second-order topological phases in non-hermitian systems [J]. Physical Review Letters, 2019, 122(7): 076801. doi: 10.1103/PhysRevLett.122.076801 [78] Serra-Garcia M, Peri V, Süsstrunk R, et al. Observation of a phononic quadrupole topological insulator [J]. Nature, 2018, 555(7696): 342-345. doi: 10.1038/nature25156 [79] Mittal S, Orre V V, Zhu G, et al. Photonic quadrupole topological phases [J]. Nature Photonics, 2019, 13(10): 692-696. doi: 10.1038/s41566-019-0452-0 [80] Noh J, Benalcazar W A, Huang S, et al. Topological protection of photonic mid-gap defect modes [J]. Nature Photonics, 2018, 12(7): 408-415. doi: 10.1038/s41566-018-0179-3 [81] Peterson C W, Benalcazar W A, Hughes T L, et al. A quantized microwave quadrupole insulator with topologically protected corner states [J]. Nature, 2018, 555(7696): 346-350. doi: 10.1038/nature25777 [82] Benalcazar W A, Bernevig B A, Hughes T L. Quantized electric multipole insulators [J]. Science, 2017, 357(6346): 61-66. doi: 10.1126/science.aah6442 [83] Xie B Y, Su G X, Wang H F, et al. Visualization of higher-order topological insulating phases in two-dimensional dielectric photonic crystals [J]. Physical Review Letters, 2019, 122(23): 233903. doi: 10.1103/PhysRevLett.122.233903 [84] Zhang X, Wang H X, Lin Z K, et al. Second-order topology and multidimensional topological transitions in sonic crystals [J]. Nature Physics, 2019, 15(6): 582-588. doi: 10.1038/s41567-019-0472-1 [85] Schindler F, Cook A M, Vergniory M G, et al. Higher-order topological insulators [J]. Science Advances, 2018, 4(6): 0346. doi: 10.1126/sciadv.aat0346 [86] Franca S, van den Brink J, Fulga I C. An anomalous higher-order topological insulator [J]. Physical Review B, 2018, 98(20): 201114. doi: 10.1103/PhysRevB.98.201114 [87] Kempkes S N, Slot M R, van Den Broeke J J, et al. Robust zero-energy modes in an electronic higher-order topological insulator [J]. Nature Materials, 2019, 18(12): 1292-1297. doi: 10.1038/s41563-019-0483-4 [88] Park M J, Kim Y, Cho G Y, et al. Higher-order topological insulator in twisted bilayer graphene [J]. Physical Review Letters, 2019, 123(21): 216803. doi: 10.1103/PhysRevLett.123.216803 [89] Xue H, Yang Y, Gao F, et al. Acoustic higher-order topological insulator on a kagome lattice [J]. Nature Materials, 2019, 18(2): 108-112. doi: 10.1038/s41563-018-0251-x [90] Xue H, Yang Y, Liu G, et al. Realization of an acoustic third-order topological insulator [J]. Physical Review Letters, 2019, 122(24): 244301. doi: 10.1103/PhysRevLett.122.244301 [91] Zhong H, Kartashov Y V, Szameit A, et al. Theory of topological corner state laser in Kagome waveguide arrays [J]. APL Photonics, 2021, 6(4): 040802. doi: 10.1063/5.0042975 [92] Atala M, Aidelsburger M, Barreiro J T, et al. Direct measurement of the Zak phase in topological Bloch bands [J]. Nature Physics, 2013, 9(12): 795-800. doi: 10.1038/nphys2790 [93] Liu X J, Ren M, Pan Q, et al. The Zak phase calculation of one-dimensional photonic crystals with classical and quantum theory [J]. Physica E: Low-dimensional Systems and Nanostructures, 2021, 126: 114415. doi: 10.1016/j.physe.2020.114415 [94] Arakawa Y, Sakaki H. Multidimensional quantum well laser and temperature dependence of its threshold current [J]. Applied Physics Letters, 1982, 40(11): 939-941. doi: 10.1063/1.92959 [95] Yoshida H, Yamashita Y, Kuwabara M, et al. Demonstration of an ultraviolet 336 nm AlGaN multiple-quantum-well laser diode [J]. Applied Physics Letters, 2008, 93(24): 241106. doi: 10.1063/1.3050539 [96] Qian C, Wu S, Song F, et al. Two-photon Rabi splitting in a coupled system of a nanocavity and exciton complexes [J]. Physical Review Letters, 2018, 120(21): 213901. doi: 10.1103/PhysRevLett.120.213901 [97] Yang J, Qian C, Xie X, et al. Diabolical points in coupled active cavities with quantum emitters [J]. Light: Science & Applications, 2020, 9(1): 6.