中国光学十大进展:反手性拓扑光子态(特邀)‡【增强内容出版】
1 引言
光子是信息的理想载体,具有宽带大、功耗低、响应速度快等优异特性。如今,电子器件由于逐渐逼近其物理极限,面临着日益增加的散热和隧穿效应问题。以光子代替电子作为信息和能量载体已经成为科学界的共识。然而,如何设计对光信息精准操控和处理的光子器件、光子集成芯片是目前研究中的重点和难点。光子晶体是由不同折射率材料按周期排列而成的人工微结构[1-4],它的出现为控制电磁波传输以及光与物质间相互作用提供了强大而有效的平台,有望实现全光集成,以助力新一代全光通信、光纤激光器以及超敏光学传感等领域飞速发展。然而,受制于当前微纳制造工艺极限以及加工环境严苛要求,光子晶体制备过程中不可避免的加工误差以及杂质都会降低光子学器件的良品率,导致光在传输过程中产生巨大的背向散射。背向散射的存在严重阻碍了光的高效传输,这在弯曲波导和慢光波导中尤为显著。光学传输过程的互易性定律决定了人们无法在现有体系中解决背向散射问题。因此,如何从物理根本上提出新机制以解决传统光子晶体中存在的背向散射问题成为了当前最热门的研究课题之一,而拓扑光子晶体的出现为解决这一问题提供了有效途径。
在过去的十多年里,拓扑光子学已成为一个快速发展的前沿领域。随着拓扑光子学的蓬勃发展,人们在拓扑光子晶体中发现具有抗背向散射、免疫缺陷以及单向传输特性的拓扑光子态[5-13]。这是因为拓扑光子态受到拓扑带隙的保护,局部缺陷、扰动、无序不足以使拓扑带隙消失,因此与传统光子晶体相比,拓扑光子晶体中光的传输表现出极强鲁棒性,甚至可以绕过远大于光波长的金属障碍物向前传输而不产生任何背向散射。磁光光子晶体是最早用于实现拓扑光子态的光学结构,同时也是研究拓扑光子态的产生、相互作用以及新颖拓扑光学现象最常用的平台。2008年,Haldane和Raghu[14-15]首次创造性地将拓扑概念引入光子晶体中。他们提出在具有时间反转对称性破缺的光子晶体中可以实现手性单向边界态。根据体边对应关系[16-18],手性单向边界态可以存在于具有不同陈数的材料或结构的边界上,同时可以免受缺陷和背向散射的影响。随后,美国麻省理工学院Soljačić课题组[19-20]、中国科学院物理研究所李志远课题组[21-22]以及香港科技大学陈子亭课题组[23-24]先后在磁化磁光光子晶体中观测到了手性单向边界态。这些开创性的理论和实验工作开启了拓扑光子学研究热潮,截至目前,人们已经在光学[5,10,12,25-26]、声学[27-28]、热学[29-31]、机械[32-33]、电路[34-37]等系统中演示了各类拓扑现象(手性边界态[21-24,38-39]、自旋边界态[40-43]、谷边界态[44-46]、高阶拓扑态[47-48]),展示了非厄米[49-51]、非线性[51-53]、非阿贝尔[54-55]系统中拓扑态的独特色散及传输行为,并开发了诸如拓扑波导[56-58]、拓扑激光[59-62]、拓扑光纤[63-66]等功能器件。近期已有一系列的综述文章详细阐述了这些前沿方向的最新进展[5-8,10-13,25,67],本文不再进行赘述。
手性拓扑光子态作为最早被预测与实现的拓扑态,能够无背向散射地绕过尖锐拐角、障碍物,并可以沿着任意几何形状的路径传输,展现出真正的单向传输特性,也被称为手性拓扑光子态。截至目前,人们已经在不同类型的有序晶格中演示了手性拓扑光子态的单向传输现象,例如正方晶格[19-21,68]、蜂窝晶格[23,69]、三角晶格[38]、四-六晶格[70]等,同时也在无序[71-74]以及非晶[75]晶格中演示了手性拓扑光子态传输的强鲁棒性,并从电动力学、电磁学角度揭示了单向边界态的微观物理起源[76-77]。广泛的研究工作已经证明,手性拓扑光子态可用于设计实现新型拓扑光子学器件,例如单模/多模单向波导[47-48]、非互易拓扑激光[59]、任意轨道角动量拓扑天线[78-79]、零群速度色散拓扑延迟线[80-83]。除了单向传输及强传输鲁棒性外,手性拓扑光子态的另一个显著特征是在拓扑光子晶体的两个平行边界沿相反方向传输。2018年,Colomés等[84]在经典Haldane模型的基础上,通过修改次近邻跃迁方向提出了反手性Haldane模型,构建了在拓扑结构的两个平行边界沿相同方向单向传输的反手性拓扑态。此后,反手性拓扑态的概念被迁移到不同的物理体系,例如光学[85-87]、声学[88]以及电路[89]等体系。2020年,华南理工大学李志远团队首次在理论上提出,通过对蜂窝晶格磁光光子晶体的两套对顶三角子晶格施加相反的磁场可以实现反手性拓扑光子态,并设计了紧凑型多通道拓扑单向波导,展示了反手性拓扑光子态的独特传输性质[85]。随后,新加坡南洋理工大学张柏乐团队以及华南理工大学李志远团队在实验上分别观测到了反手性拓扑光子态[86,90],并设计实现了拓扑环形腔、拓扑分束器,为拓扑光子学物理以及器件研究开辟了全新的道路。
为展示反手性拓扑光子态的最新进展,本文将首先介绍四种典型的物理模型,包括Dirac模型、经典Haldane模型、反手性Haldane模型以及异质Haldane模型,并展示不同拓扑态的独特色散及传输行为。然后讨论手性单向边界态、反手性单向边界态以及单向体态在光子晶体中的实现。接着重点阐述磁光光子晶体中反手性拓扑光子态及单向体态的理论预测与实验观测,并展示这些拓扑态在拓扑光子学器件中的应用潜力。最后简述不同物理体系中反手性拓扑态的研究进展,并展望反手性拓扑态的未来发展以及机遇挑战。
2 物理模型
本节从二维Dirac模型出发,逐步推演经典Haldane模型、反手性Haldane模型以及异质Haldane模型。首先,对于具有蜂窝晶格的石墨烯条带,可以用Dirac模型来描述其中电子传输行为[91]。如图1(a1)所示,Dirac模型中A和B两套三角子晶格间仅存在最近邻跃迁,其哈密顿量表示为
式中:
图 1. 电子体系的模型示意图。(a)Dirac模型;(b)经典Haldane模型;(c)反手性Haldane模型;(d)异质Haldane模型
Fig. 1. Schematics of electronic models. (a) Dirac model; (b) Haldane model; (c) modified Haldane model; (d) heterogeneous Haldane model
普林斯顿大学Haldane教授[92]基于Dirac模型,在三角子晶格中引入次近邻跃迁(t2)以系统地打破时间反演对称性,实现了手性单向边界态,称为经典Haldane模型。图1(b1)为经典Haldane模型的示意图,其中A和B代表两套不同的三角子晶格。当在A和B上施加大小相同但方向相反的磁通,此时最近邻跃迁不受影响,而次近邻跃迁的存在会导致电子在跃迁过程获得一个附加相位φ。因此,经典Haldane模型的哈密顿量可以写为
式中:第一项代表电子的最近邻跃迁;第二项代表次近邻跃迁;t1和t2分别是最近邻和次近邻跃迁强度。当A子晶格和B子晶格之间的次近邻跃迁方向相反(
进一步,Colomés等[84]修改了经典Haldane模型中次近邻跃迁的方向,实现了反手性Haldane模型,如
最近,华南理工大学李志远课题组在反手性Haldane模型基础上,提出异质Haldane模型[93]。如
3 设计实现
基于Bloch理论,具有周期性结构的光子晶体可以产生光子能带,类似于固体中的电子能带[25]。类比紧束缚模型,通过设计合适的光子晶体结构,可以构建独特色散行为以实现对光传输的调控。例如,蜂窝晶格电介质光子晶体可以很好地类比Dirac模型,在布里渊区中构建成对的、由局域边界态相连接的简并狄拉克点[94-98]。光学Dirac模型已经被广泛研究,不再赘述。本节主要介绍如何在具有不同磁化配置的二维磁光光子晶体中实现光学经典Haldane模型、反手性Haldane模型以及异质Haldane模型。磁性光子晶体中的磁光介质柱以商用磁光材料钇铁石榴石晶体(YIG)为例,其相对介电常数为ε=14.5~16.0,在微波波段具有良好的磁光响应。在不施加外加磁场时,YIG的相对磁导率μ=1。而在面外方向(例如z轴方向)施加外部磁场时,YIG晶体会产生强烈旋磁各向异性,其磁导率[13]会变成张量形式
式中:
对于无磁化蜂窝晶格磁光光子晶体,由于系统受到时间和空间反演对称性保护,其色散行为类似于Dirac模型,在布里渊区中存在成对的简并的狄拉克点。而当对蜂窝晶格施加沿+z方向的均匀磁场时,外磁场的存在会打破磁光光子晶体的时间反演对称性,使得在高对称点K和K′处的狄拉克点简并被打破,两条能带相互分离,从而形成受拓扑保护的非平庸带隙,如
图 2. 磁光光子晶体模型示意图。(a)均匀磁化光子晶体;(b)交错磁化光子晶体;(c)异质磁光光子晶体(黄色矩形为金属障碍物)
Fig. 2. Schematic diagrams of magnetic photonic crystals (MPC). (a) Uniformly magnetized MPC; (b) cross-magnetized MPC; (c) heterogeneously magnetized MPC (yellow rectangle inside is a metal obstacle)
进一步,对蜂窝晶格磁光光子晶体中两套对顶三角子晶格A和B分别施加沿-z和+z方向的外加磁场(交错磁化),如
最后,通过将两种具有相反磁化方向的交错磁化光子晶体沿y方向周期排列可以构建异质磁化磁光光子晶体,如
进一步具体介绍实现反手性单向边界态和单向体态的实验装置。2020年,华南理工大学李志远课题组[85]在理论上首次提出了具有交错磁化的磁光光子晶体可以实现反手性单向边界态,同时也提出了可行的实验方案,如
图 3. 二维及三维反手性态的实现。(a)二维反手性态实验设计的单胞示意图[13];(b)(c)实验观测到反手性边界态的装置图以及测量到的反手性边界态[86,90];(d)单向体态的仿真以及实际测量场图[93];(e)三维反手性表面态的实验装置图以及测量到的投影能带图[87]
Fig. 3. Realization of antichiral states in 2D and 3D system. (a) Experimental design of a unit cell in 2D system[13]; (b) (c) experimental sample for measuring antichiral edge states and the measurement results[86,90]; (d) simulation and experimental results of one-way bulk states[93]; (e) experimental design of 3D antichiral surface states and the experimental result of project band[87]
理论和实验研究都表明,基于Haldane和反手性Haldane模型的单向边界态是实现真正单向抗背向散射能量传输最可靠的方案。然而,这种单向传输只能局限在条带的边缘,这一点使得高通量鲁棒的能量传输被大大限制。最近,李志远课题组[93]提出在异质Haldane模型中构建的二维磁光光子晶体可以实现单向体态。利用之前介绍的异质磁化光子晶体结构,仿真和实验测量上都证实了单向体态的存在。同时,这种单向体态可以绕过体内的金属障碍物无背向散射地继续向前传输,具有传输鲁棒性。如
最近,南方科技大学高振课题组与南洋理工大学张柏乐课题组[87]合作,在磁光外尔光子晶体中构建了三维反手性Haldane模型并且在实验上观察到反手性表面态。通过在三维蜂窝晶格中的不同子晶格位点A和B上实现相反的磁通并引入层间耦合,在时间反演和空间反演都被打破的情况下,两对外尔点将在频率上偏移,并且连接着两对外尔点的倾斜费米弧表面态色散表明系统具有反手性表面态。如
值得强调的是,在时间周期调制的Floquet体系中,当两个三角子晶格在调制方向相反时也可以实现反手性边界态[99]。另外,中山大学董建文课题组[100]在时间反演对称保护的三维光子晶体中实现了反手性表面态,该系统的交错磁通量是通过层间耦合来实现的。当kz不为零时,该三维系统可以简化为二维反手性Haldane模型,在色散上呈现一对频移的狄拉克点,从而产生反手性表面态。不过,因为整个系统是时间反演对称的,所以对于正kz和负kz,反手性表面态的传播方向是相反的。这一结果扩展了反手性拓扑光子态在三维体系的实现方案,也为拓扑相以及光学器件研究提供了崭新思路。
4 反手性拓扑光子态的应用
伴随着反手性拓扑光子态的实现,反手性拓扑光学功能及器件也被广泛展示,例如紧凑多通道单向波导[85]、拓扑单向环形器[86]、拓扑分束器[90]以及可重构光学成像[101]。本节将逐一介绍这些反手性拓扑光学功能及器件的结构配置和实现机制。
众所周知,手性拓扑光子态在拓扑光子晶体的两个平行界面是沿着相反方向传输的,展示出手性特征。当考虑单边界的能量传输时,由于拓扑光子态具有单向传播特性,光在传输路径上能够免疫金属障碍物和结构缺陷的影响。但是,在考虑双边界的能量传输时,由于手性特征,只有一个边界能进行能量传输(从左端口到右端口),这大大地限制了单向波导的利用率、减少了信道传输个数。因此,利用手性拓扑光子态构建多通道集成单向波导时,器件体积将会非常庞大。为解决这一问题,华南理工大学李志远团队[85]利用反手性拓扑光子态设计实现了结构更简单、更紧凑的多通道单向波导。
图 4. 反手性态的应用。(a)紧凑型多通道单向波导[13];(b)拓扑单向循环器[86];(c)可重构拓扑分束器[90];(d)可重构光成像[101]
Fig. 4. Application of antichiral states. (a) Compact three-channel one-way waveguide[13]; (b) topological unidirectional circulator[86]; (c) reconfigurable topological beam splitting[90]; (d) reconfigurable light imaging[101]
利用单向传输的特性,反手性拓扑光子态还可以用于设计高性能拓扑光子学原型器件,例如南洋理工大学张柏乐团队[86]设计实现了反手性拓扑单向环形腔。
华南理工大学李志远团队[90]进一步提出反手性拓扑光子晶体还可以用于构建其他类型拓扑光子晶体无法实现的同时具有大带宽、无串扰、多通道等特性的单向拓扑分束器。如
最近,南京大学的卢明辉团队[101]通过打破空间反演对称性,在反手性拓扑光子晶体中引入了谷拓扑光子态,实现了具有任意几何形状的可重构光学成像。当反手性拓扑光子晶体的两套对顶三角子晶格的磁光介质柱半径被改变时,反手性光子晶体的空间反演对称性会被打破,狄拉克点的简并被打开。当空间反演对称性破缺逐渐占据主导地位时,反手性拓扑光子晶体会转变为谷拓扑光子晶体以支持谷边界态的存在。为了抑制谷间散射,谷边界态需要在通过谷陈数相反的两个光子晶体构成的边界上实现,同时因为外加磁场的存在,谷边界态的色散还会有反手性的特性。除了边界态,在这种结构中还有高阶谷角态的存在。基于这种光子晶体设计可以进一步实现更为复杂的可调谐、可重构光成像,如
5 其他体系反手性拓扑态的研究进展
本节将简要介绍其他拓扑系统中反手性拓扑态的研究进展。除了在磁性光子晶体中实现反手性边界态以外,近年来人们也在其他物理体系中设计实现了反手性边界态,例如,声学超材料[88]、电路超材料[89]、激子极化激元[102]、莫尔石墨烯hBN异质结构[103]、具有Dzyaloshinskii-Moriya相互作用的海森堡铁磁材料[104]。
具体而言,南洋理工大学张柏乐课题组在由声学谐振器构成的蜂窝晶格中,通过对两套对顶三角子晶格分别引入顺时针和逆时针气流,促使简并狄拉克点在频率上发生偏移,在理论上证实了声学反手性边界态的存在[88]。在电路超材料领域,Yang等[89]通过互联电容和电感构建了编织电路晶格,在实验上展示了反手性边界态的传输。其中子晶格之间的最近邻和次近邻跃迁是通过电感和电容之间不同的互连方式实现的。而在激子极化激元体系中,Mandal等[102]利用激子极化凝聚体与偏振相关的相互作用,使沿y方向线性极化的极化子展现出反手性色散行为。另外,来自苏黎世联邦理工学院的Denner等[103]理论上提出在0.2°转角的双层石墨烯莫尔超晶格中间加入两层hBN夹层,在外加磁场作用下这种异质结构可以实现反手性态。研究表明,扭转石墨烯以及其中加入的hBN薄层能显著降低实现反手性态需要的外加磁场。来自南洋理工大学的Bhowmick等[104]基于海森堡铁磁材料,在蜂窝晶体子晶格中引入不相等的Dzyaloshinskii-Moriya相互作用,使得布里渊区的能带在K和K′点向相反方向偏移,从而也可以实现反手性边界态。
近期的一些工作还报道了基于反手性边界态的独特性质。例如,Wang等[105]研究发现在石墨烯-超导体结中,反手性系统中存在的倾斜能带结构可以实现非对称的安德列夫反射。再如,Mannaï等[106]研究发现晶格应变不仅可以反转反手性边界态的传播方向,还可以破坏反手性边界态的存在。另外,通过改变施加在石墨烯条带锯齿形边界的侧电势,也可以实现对反手性边界态的调制[107]。这些工作展示了反手性边界态的丰富调控手段与自由度,可以为反手性拓扑光子态的操控提供有益参考。
6 结束语
本文详细总结了近三年来反手性拓扑光子态的最新进展,包括物理模型的推演、特征色散的展示、传输现象的观测以及功能器件的实现,并简要阐述了不同物理系统中的反手性拓扑态。可以看到,尽管反手性拓扑光子态研究进展迅速,但该领域还处于起步阶段,仍有许多物理机制、现象及应用等待挖掘。首先,目前反手性拓扑光子晶体的工作频率主要在微波波段,导致光子晶体结构体积庞大,不利于实现集成化拓扑光子学器件。一方面是因为常见磁性材料在可见光波段的磁光响应非常微弱,无法为光子晶体系统提供足够大的时间反演破缺,从而难以实现光波段反手性拓扑光子态。另一方面对于无需打破时间反演对称性的拓扑光子晶体,通常需要设计结构复杂的三维光子晶体,这对纳米级加工工艺提出了巨大挑战。因此,如何在光学波段实现反手性拓扑光子态成为了亟待解决的问题,而基于Floquet系统[99,108-115]、激子-极化子系统[102,116-117]、磁性外尔半金属[118]、磁性半导体材料[119-120]的设计方案也许会为实现光学波段反手性拓扑光子态提供有效途径。其次,绝大多数工作聚焦在二维系统中的反手性拓扑光子态,使得反手性拓扑光子晶体对光传输的操控仅局限于单向边界态与单向面态,而三维反手性拓扑光子晶体可以提供更多自由度以实现对反手性拓扑光子态的多维度操控。一种方式是通过堆叠二维反手性拓扑光子晶体并调控层间耦合来实现三维反手性拓扑光子态,从而可以将一维边界态拓展到二维面态[87,100]。另一种方式是构建磁性三维木堆光子晶体,通过引入线缺陷或面缺陷波导来构建具有任意路径的三维拓扑网络以支持反手性拓扑光子态的传输。再次,当前反手性拓扑光子态研究主要聚焦于反手性边界态实现及其鲁棒性验证,与其他物理效应(人工赝磁场、非线性、非厄米、非阿贝尔等)及对称性(高阶拓扑、无序、位错、分形、转角等)协同作用的研究仍旧匮乏。可以预见,协同不同物理效应及对称性可以极大地丰富反手性拓扑光学系统的物理现象并产生独特功能器件。另外,目前反手性拓扑态的实现仅局限于光学[85-86]、凝聚态[102-104]、声学[88]、电路[89]等系统,若将反手性拓扑光子态的概念迁移到更为广泛的物质波和经典波系统(例如机械波[121]、热学[122-124]),将为不同物理体系的拓扑态操控提供强大手段。最后,尽管当前反手性拓扑光子态无法为实现集成化全光光路提供有效助力,但仍有望为关键微波光电子器件的改造和升级提供有希望的手段,以实现可重构、紧凑、强鲁棒性的拓扑光电子器件。
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Article Outline
纪子韬, 陈剑锋, 李志远. 中国光学十大进展:反手性拓扑光子态(特邀)‡[J]. 激光与光电子学进展, 2024, 61(15): 1500001. Zitao Ji, Jianfeng Chen, Zhiyuan Li. China's Top 10 Optical Breakthroughs: Antichiral Topological Photonic States (Invited)[J]. Laser & Optoelectronics Progress, 2024, 61(15): 1500001.