Topological phases in ring resonators: recent progress and future prospects
aa r X i v : . [ phy s i c s . op ti c s ] S e p Topological phases in ring resonators: recent progress and future prospects
Daniel Leykam
1, 2, a) and Luqi Yuan Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Korea Basic Science Program, Korea University of Science and Technology, Daejeon 34113, Korea b)3)
State Key Laboratory of Advanced Optical Communication Systems and Networks,School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China c) (Dated: 30 September 2020) Topological photonics has emerged as a novel paradigm for the design of electromagnetic systems from mi-crowaves to nanophotonics. Studies to date have largely focused on the demonstration of fundamental con-cepts, such as non-reciprocity and waveguiding protected against fabrication disorder. Moving forward, thereis a pressing need to identify applications where topological designs can lead to useful improvements in deviceperformance. Here we review applications of topological photonics to ring resonator-based systems, includingone- and two-dimensional resonator arrays, and dynamically-modulated resonators. We evaluate potentialapplications such as quantum light generation, disorder-robust delay lines, and optical isolation, as well asfuture research directions and open problems that need to be addressed.Keywords: Topological photonics; Silicon photonics; Ring resonator; Coupled resonator optical waveguide;Optical isolator
I. INTRODUCTION
Demand for miniaturised optical components such aswaveguides and lenses that can be incorporated into com-pact photonic devices is pushing fabrication techniquesto their limits. Continued progress will require new ap-proaches to minimise the detrimental influence of fabrica-tion imperfections and disorder. Topological photonics isa young sub-field of photonics which seeks to address thischallenge using novel design approaches inspired by ex-otic electronic condensed matter materials such as topo-logical insulators . Loosely speaking, topological sys-tems provide a systematic way to create disorder-robustmodes or observables using a collection of imperfect com-ponents or modes. For example, certain classes of “topo-logically nontrivial” systems exhibit special edge modeswhich can propagate reliably without backscattering evenin the presence of strongly scattering defects, forming thebasis for superior optical waveguides.One natural setting where this robustness can poten-tially be useful is in the design of integrated photonic cir-cuits , where the strong light confinement brings sensitiv-ity to nanometer-scale fabrication imperfections. How-ever, there is not yet any disruptive killer applicationwhere topological photonic devices have achieved supe-rior performance compared to mature design paradigms,despite growing interest in topological photonics sinceseminal works published in 2008 . To help bridgethis gap, several reviews have been published recently,some providing comprehensive surveys of topologicalphotonic systems , and others focusing on specificapplications such as incorporating topological concepts a) Present address: Centre for Quantum Technologies, NationalUniversity of Singapore, 3 Science Drive 2, Singapore 117543 b) Electronic mail: [email protected] c) Electronic mail: [email protected] into active devices such as lasers , nanophotonics , non-reciprocal devices , and nonlinear optical processes .The aim of this brief review is to complement theserecent surveys with a concise introduction to applica-tions of ideas from topological photonics to optical ringresonator-based systems. Ring resonators are a versatileand important ingredient of integrated photonic circuits,as they can be used as compact filters, sensors, and de-lay lines, and as a means of enhancing nonlinear opticaleffects . However, active tuning is typically requiredto compensate for resonance shifts induced by variousperturbations, increasing device complexity and energyconsumption . We will discuss some of the ways inwhich topological designs may lead to superior deviceswith improved reliability. We will focus on topologicalsystems formed by coupling together multiple resonatorsto form a lattice, or by considering coupling between mul-tiple resonances of a single ring using external modula-tion or nonlinearity. In both cases the lattice formed bythe coupled modes can be designed to have topologicalband structures and achieve protection against certainkinds of disorder.The outline of this article is as follows: Sec. II startswith a brief overview of the basic concepts underlyingtopological photonics and ring resonators. Next, we re-view the implementation of topological photonics usingarrays of coupled ring resonators in Sec. III. Sec. IV dis-cusses how we can use dynamic modulation as a novel de-gree of freedom for the engineering of topological effectsin ring resonators. We discuss future research directionsand promising potential applications of topological ringresonators in Sec. V, before concluding with Sec. VI. II. BACKGROUNDA. Topological photonics
The key idea underlying topological photonics is thebulk-boundary correspondence, which states that thetopological properties of the photonic band structure ofa bulk periodic medium can be related to the appearanceof robust modes localized to edges or domain walls of thesystem . These “topologically-protected” edge modeshave very different properties compared to conventionaldefect modes. For example, as they arise due to the topo-logical properties of the bulk photonic band structure,they are robust against certain classes of local perturba-tions at the edge or domain wall and can only be removedby large perturbations capable of closing the bulk bandgap. Thus, topology allows us to create special robustboundary modes protected by a higher-dimensional bulk.For the purposes of this review, two classes of topologi-cal modes are of interest. One-dimensional media can ex-hibit topologically-protected modes localized to the endsof the system. These modes have their frequencies pinnedto the middle of the band gap, even in the presence ofdisorder, as long as certain symmetries are preserved. Inthis case, topology provides a systematic way to createlocalized defect modes at a specific frequency.The second important class of topological modesare edge states of two-dimensional topological systems.These exhibit either unidirectional or spin-controlledpropagation along the edge. The former appear intopological systems with broken time-reversal symme-try (known as quantum Hall topological phases ), whilein photonics the latter require a combination of time-reversal symmetry and some other internal symmetryprotecting a spin-like degree of freedom (known as quan-tum spin-Hall topological phases ). In both cases the re-sulting topological edge states are of interest as a meansof creating disorder-robust optical waveguides.The generic approach to create a topological systemis to is to start with a simple periodic medium exhibit-ing some degeneracy in its photonic band structure, andthen break one of the symmetries protecting the degen-eracy to open a band gap. Breaking the symmetry in theright way will create a topologically nontrivial band gaphosting protected edge modes.Fig. 1 shows a simple one-dimensional example of thisidea, where one can reduce the translation symmetry ofa periodic lattice by staggering its site positions to createa lattice of dimers. Reducing the translation symmetryopens a mini-gap the lattice’s band structure, analogousto the Su-Schrieffer-Heeger tight binding Hamiltonian forelectron transport in polymers ,ˆ H SSH = X n (cid:16) J ˆ a † n − ˆ a n + J ˆ a † n ˆ a n +1 (cid:17) + c . c ., (1)where ˆ a n is the annihilation operator for the n th site,and J , are coupling strengths. Note that while ˆ H SSH is ACB
Regular (non-topological) Bulk modesTrivial dimerized latticeBulk modes Bulk modesNontrivial dimerized latticeBulk modes Bulk modesEndmodes
FIG. 1. Schematic of general procedure for creating a topolog-ical medium. (A) A regular periodic lattice can be modelled asa collection of resonant elements (shaded circles) with individ-ual resonant frequencies ω coupled together to form a bandof delocalized bulk modes. The coupling strength J is de-termined by the overlap between the modes of the individuallattice sites, sketched in red. (B,C) Reducing the translationsymmetry of the lattice by introducing staggered couplings J = J opens a mini-gap in the bulk spectrum of size 2 δJ ,where δJ = | J − J | and J = ( J + J ). (B) The trivialdimerized lattice J > J hosts two bands corresponding tosymmetric and antisymmetric dimer modes, separated by amini-gap. (C) In the nontrivial dimerized lattice J < J thedimers at the ends are broken, producing in mid-gap modeslocalized to the ends of the lattice. written using second quantization notation, it is equallyapplicable to classical states of light, where the eigenval-ues of ˆ H SSH give the frequency detunings of the collectivearray modes, as shown in Fig. 1.There are two different ways to dimerize a finite lattice;we can have either J > J or J < J , correspondingto two inequivalent topological phases distinguished bya quantized winding number. The trivial phase J > J forms a set of dimers and does not exhibit any end modes.On the other hand, in the nontrivial phase J < J thedimers are broken at the ends of the system, resulting ina pair of localized end modes with frequencies lying inthe middle of the band gap. The end modes are topolog-ically protected in the sense that introducing disorder tothe inter-site couplings does not shift their frequencies;they remain “pinned” to the middle of the band gap andlocalized to the ends as long as the disorder is sufficientlyweak that the bulk band gap remains open.Thanks to its simplicity and relative ease of implemen-tation, photonic analogues of the Su-Schrieffer-Heegermodel have been realized in a variety of platforms , in-cluding waveguide lattices , plasmonic and dielectricnanoparticle arrays , photonic crystals , and mi-croring resonators .To create two-dimensional topological phases there aretwo common approaches: using synthetic gauge fieldsor by perturbing honeycomb lattices. Synthetic gaugefield refers to complex (direction-dependent) coupling be-tween different sites of the photonic lattice, arising in sys-tems with non-reciprocity or broken time-reversal sym-metry, corresponding to tight binding coupling terms ofthe form Je iθ ˆ a † n +1 ˆ a n + Je − iθ ˆ a † n ˆ a n +1 , where θ is the cou-pling phase. Complex coupling is formally equivalent tothe effect of an electromagnetic vector potential on elec-tron transport. Using a suitable position-dependent syn-thetic gauge field θ = θ ( x, y ) allows one to create aneffective magnetic field for light , resulting in analoguesof the quantum Hall topological phase .The second approach to creating two-dimensionaltopological phases is based on the honeycomb lattice,which exhibits Dirac point degeneracies at the cornersof its Brillouin zone. Weak symmetry-breaking pertur-bations are capable of lifting the degeneracy to open atopological band gap. One can break either time-reversalsymmetry to create the quantum Hall phase , or otherinternal symmetries (e.g. related to sublattice or polar-ization degrees of freedom) to create quantum spin Hallphases .Other classes of topological models beyond the abovegapped topological insulating phases are attracting in-creasing attention. For example, higher-order topologi-cal phases can give rise to modes localized to the cor-ners of two-dimensional systems . Three-dimensionalWeyl topological phases exhibit protected degeneraciesin their bulk band structure . Non-Hermitian topo-logical phases can emerge in systems with structured gainor loss . For further discussion on photonic topolog-ical phases we direct the reader to Refs. 1, 2, 6, and 7.It is important to stress that these topological photonicsystems are only analogous to the topological tight bind-ing models used to describe electronic condensed mat-ter systems. Thus, while the electronic quantum Hallphase exhibits a Hall conductivity precisely quantizedto 1 part in 10 , in photonics various effects such asmaterial absorption, out-of-plane scattering, and imper-fect symmetries mean that the topological protection isonly approximate, so the edge modes are only protectedagainst certain classes of perturbations. Thus, it is cru-cial to identify systems where topology provides protec- tion against the most significant sources of disorder; moststudies to date rely on deliberately-introduced defectsto demonstrate topological protection. For example, theend states of the Su-Schrieffer-Heeger model are only pro-tected against the “off-diagonal” disorder in the inter-sitecoupling coefficients, and are not protected against the“diagonal” disorder in the individual sites’ resonant fre-quencies, which leads to random variations to the endmodes’ resonant frequencies. B. Ring resonators
Ring resonator generally refers to any optical waveg-uide forming a closed loop, regardless of its size orshape . Fig. 2 presents some examples of ring res-onators, including micrometer-scale integrated opticalresonators, millimeter-scale spoof plasmon resonators,and kilometer-scale fiber loops. Microwaves and fiberloops provide a convenient setting for studying noveldesign approaches as they are easier to fabricate, andin some cases can be assembled using off-the-shelf op-tical components, while integrated photonic circuits areof more interest for potential applications due to theircompactness and scalability.Resonances occur whenever the propagation phase ac-cumulated over a round trip forms a multiple of 2 π .Key characteristics of ring resonators are their resonancewidth, free spectral range (FSR; the spacing betweenneighbouring resonances), quality factor (resonance fre-quency divided by width), and finesse (free spectral rangedivided by resonance width). The resonance width is dic-tated both by the intrinsic losses due to waveguide bend-ing, scattering losses due surface roughness, absorption,and extrinsic losses introduced by coupling the resonatorto external waveguides. We emphasize that in passivesystems topological designs generally do not provide pro-tection against these sources of loss.The free spectral range is inversely proportional to theround trip path length. In systems with small FSR suchas long fiber loops one typically studies the propagationdynamics in the time domain. For on-chip signal pro-cessing applications it is desirable to have a large freespectral range, exceeding the signal bandwidth, demand-ing a high refractive index contrast to minimise waveg-uide bending losses. The resulting strong light confine-ment in turn makes the ring’s resonances highly sensi-tive to local perturbations to the refractive index, whichcan be both a strength and a weakness. For example,the sensitivity to perturbations allows ring resonators tobe employed as highly compact and efficient sensors andoptical switches . On the other hand, for spectral fil-tering applications active tuning is typically required tokeep the resonance fixed at the desired frequency .A high quality factor is desirable for nonlinear opticsapplications. For example, an intensity-dependent re-fractive index enables bistability as the input frequencyis tuned, which is useful for all-optical switching . An μ m300 μ m AC D Variable couplerAmpli fi er Ampli fi er fi ber 4 km fi ber B μ m 5.0 mm 3.5 mm50:5050:50Input Photodiodes FIG. 2. Examples of ring resonators in different platforms.(A) Silica microtoroid resonator, adapted from Ref. 42. (B)Coupled spoof plasmon ring resonators formed by subwave-length metal pillars, adapted from Ref. 43. (C) 14-ringCROW on silicon integrated with thermal tuners, adaptedfrom Ref. 44. (D) Coupled fiber loops, where photodiodesare used to monitor the propagation dynamics by trackingthe intensity within each loop, adapted from Ref. 45. Exper-iments in panels (A,C,D) were conducted at telecom wave-lengths ( λ ≈ ω ≈ . intensity-dependent refractive index can arise not onlydue to the optical Kerr effect, but also from thermalnonlinearities and free carrier dispersion . The differ-ing characteristic time scales of these nonlinear effectscan lead to complex pulsating dynamics . Ultra-fastKerr nonlinearities are also employed for frequency mix-ing applications, where the relatively uniformly spacingof the ring resonances are ideal for frequency comb gen-eration .The strong dispersion close to resonance allows ringresonators to be used to delay and store optical signals.For single rings there is a trade-off between the delaytime and the operating bandwidth (their product is afixed constant). Larger delays for a fixed bandwidthcan be achieved using arrays of coupled rings, knownas coupled resonator optical waveguides (CROWs) .In integrated photonics, however, disorder in the formof nanometer-scale variations to the rings’ height andthickness leads to significant misalignment of the indi-vidual rings’ resonance frequencies, severely degradingthe CROW performance . Therefore one requires eitherclever designs that are robust against these fabrication variations or active tuning to compensate for the dis-order .The thermo-optic effect is the most common way totune integrated photonic ring resonators, using micro-heaters placed on top of the individual rings. While ther-mal tuners offer a large tuning range, they are slow (op-erating on the µ s scale), have poor energy efficiency ,require careful design to minimise cross-talk between dif-ferent tuners , and inevitably introduce sensitivity toenvironmental temperature fluctuations . Electro-optictuning can operate much faster (sub ns) and with greaterenergy efficiency, however the tuning range is smaller andadditional absorption losses are introduced .There is growing interest in methods to improve the re-liability of ring resonator-based devices, such as creatingtemperature-insensitive resonators by combining materi-als with opposite thermo-optic coefficients , and intro-ducing tunable backscattering to cancel out backscatter-ing caused by fabrication imperfections via destructiveinterference . Topological designs are a promising alter-native exhibiting passive robustness against disorder. Forexample, the topological edge modes of two-dimensionallattices are robust against the “diagonal” disorder formedby misalignment in the rings’ resonance frequencies, pro-vided the disorder strength does not exceed the size ofthe topological band gap.For further in-depth discussion of the physics and ap-plications of photonic ring resonators we recommendRefs. 12, 55, and 65. III. TOPOLOGICAL COUPLED RESONATOR LATTICES
Arrays of coupled ring resonators provide a flexibleplatform for implementing topological lattice models. Inweakly coupled arrays light propagation is governed byeffective tight binding Hamiltonians which describe theevanescent coupling of light between neighbouring res-onators. The magnitude of the coupling coefficients canbe controlled simply by varying the separation betweenthe resonators. Furthermore, coupling resonant rings viaanti-resonant links allows one to tune the phase of thecoupling. One can effectively break time-reversal sym-metry by considering modes with a fixed “spin” (clock-wise or anti-clockwise circulation direction) and neglect-ing backscattering within the rings. On the other hand,taking this backscattering into account introduces an in-plane effective magnetic field . Together, these ingredi-ents enable the realization of a wide variety of topologicaltight binding models in one and two dimensions.Studies of topological ring resonator lattices beganwith the seminal theoretical work of Hafezi et al. , whichshowed that one can effectively break time-reversal sym-metry by exciting modes with a particular circulationdirection and then use asymmetric link rings to imple-ment an analogue of the quantum Hall lattice model.The asymmetric link rings result in a phase differencebetween the two coupling directions, equivalent to a vec- A InputOutput BC Normalized Intensity01
FIG. 3. Synthetic magnetic field in a ring resonator lattice.(A) Schematic of lattice, consisting of resonant “site” rings(blue) coupled via anti-resonant “link” rings (red). Light isinjected at the input into anticlockwise-circulating site modes,effectively breaking time-reversal symmetry. (B) Schematicof coupling terms in the tight binding Hamiltonian. Couplingin the vertical direction is symmetric, while coupling in thehorizontal direction is accompanied by a hopping phase e ± iθ due to the asymmetry of the link ring. An inhomogeneoushopping phase θ = αy induces a synthetic magnetic flux α .(C) Image of the first photonic topological resonator latticeand propagation of its edge states, adapted from Ref. 67. tor potential in the tight binding Hamiltonian. By mak-ing this hopping phase inhomogeneous (e.g. proportionalto the y coordinate) one can create a vector potentialformally equivalent to an out-of-plane effective magneticfield, which implements a lattice model of the quantumHall effect, as shown in Fig. 3(A,B) and described by thetight binding Hamiltonian ˆ H QH = J X x,y (cid:16) ˆ a † x +1 ,y ˆ a x,y e − iαy + ˆ a † x,y ˆ a x +1 ,y e iαy +ˆ a † x,y +1 ˆ a x,y + ˆ a † x,y ˆ a x,y +1 (cid:17) , (2)where α is the effective magnetic flux threading each pla-quette of the square lattice.The topology of the quantum Hall lattice is character- ized by the quantized Chern number, which determinesthe number of chiral backscattering-protected states atthe edge of the lattice. Specifically, for a given band gapone sums the Chern numbers of all the bulk bands belowthe gap to obtain the gap Chern number . The numberof chiral edge states at an interface between two mediais then given by the difference between the gap Chernnumbers of the two media. Note that as this resonatorlattice obeys time-reversal symmetry the opposite spinhosts counter-propagating edge states (forming a quan-tum spin Hall phase), and hence the protection againstbackscattering only holds as long as the two spins remaindecoupled.Following this proposal the first experiment was re-ported in 2013 . The experiment was performed in thetelecom band ( λ ≈ ≈ GHz and inter-site coupling strength J ≈
16 GHz,deep in the tight binding regime. The diagonal disorderin the rings’ resonant frequencies estimated to be ≈ . J ,smaller than the size of the topological band gap. Otherforms of disorder were found to be negligible for the sys-tem parameters considered: the strength of the inter-sitecoupling disorder and intra-site coupling between the twospins were both estimated to be 0 . J .In contrast to the ideal lossless case, in practice theindividual rings exhibited intrinsic losses κ in ≈ . The quantum Hall lattice model was also imple-mented using silicon nitride ring resonators, however thepropagation distance of the topological edge states waslimited by stronger intrinsic losses κ in ≈ J ≈
60 GHz .Other topological tight binding models have also beenstudied using ring resonator lattices. A higher ordertopological phase exhibiting protected corner states wasdemonstrated in 2019 . Next-nearest neighbour cou-pling was used to implement an analogue of the Haldanemodel , which exhibits quantum Hall edge states evenin the absence of a net effective magnetic flux . Zhuet al. have proposed a honeycomb lattice design host-ing topological edge states which co-exist with a nearlyflat bulk band . It is also possible to shrink these two-dimensional lattices down to quasi-one-dimensional delaylines, which maintain some resistance against disorder .In 2018 the ring resonator platform was used to imple-ment one- and two-dimensional topological laser mod-els by embedding a quantum well gain medium into theresonators . One-dimensional experiments werecarried out using the Su-Schrieffer-Heeger lattice, createdby staggering the separation between neighbouring rings.Pumping one of two sublattices comprising the array in-duced lasing of its mid-gap topological edge states .Two-dimensional lasing experiments utilised the quan-tum Hall lattice, where a pump localized to the latticeedges induced lasing in its chiral edge states . Inboth cases, the potential advantage of the topological ap-proach is the ability to induce lasing in collective arraymodes that are localized by the topological band gap. Forfurther information on topological lasing, we recommendthe recent reviews Refs. 8 and 11.One advantage of the silicon photonics platform is theability to implement actively-tunable devices, for exam-ple by incorporating thermo-optic phase shifters to tunethe resonant frequencies of the individual rings. Mittal etal. employed tunable phase shifters at the edge of thequantum Hall lattice to directly measure the topologicalwinding number of the edge states. Similar tuning of theeffective magnetic flux in ring-shaped lattices enables theobservation of the Hofstadter butterfly via the lattice’sscattering resonances . There are recent proposals toimplement phase shifters throughout the entire lattice inorder to tune its topological properties, thereby enablingone to switch the topological edge states on or off or re-route them between different output ports . Zhaoet al. demonstrated controllable re-routing of topolog-ical states in the quantum Hall resonator lattice usingstructured bulk gain .The above studies of topological resonator lattices fo-cused on the weak coupling limit described by tight bind-ing models. However, topological phases can also arisein the strongly coupled lattices, without requiring exter-nal modulation or anti-resonant link rings . These“anomalous Floquet” phases are not predicted by thetight binding approximation and only emerge when con-sidering the full transfer matrix description of the lightcoupling between neighbouring rings. A ring resonatorlattice implementing an anomalous Floquet topologicalphase was first demonstrated in 2016 using spoof plas-mons at microwave frequencies . Similar anomalousedge states can emerge in weakly coupled arrays withgain and loss, where the transfer matrix description isessential to account for the growth or decay of the opti-cal field as it circulates through each ring .The anomalous Floquet topological phase was scaledup to telecom wavelengths by Afzal et al. using the sil-icon photonic resonator lattice shown in Fig. 4 . Themain distinguishing feature compared to previous obser-vations of quantum Hall edge states is the presence ofedge states in all of the array’s band gaps. One advan-tage of the strongly-coupled anomalous Floquet phases ABC
FIG. 4. Anomalous Floquet topological resonator lattice,adapted from Ref. 83. (A) Microscope image of the siliconresonator lattice. (B) Image of scattered light when the chi-ral edge state is excited (left), compared with the simulatededge state intensity distribution (right). (C) Measured trans-mission spectrum of the lattice, with high transmission in allthree band gaps (shaded grey regions I, II, III) mediated bythe anomalous Floquet topological edge states. is that their bulk bands and edge states can have band-widths comparable to the rings’ free spectral range, incontrast to tight binding lattices which are typically re-stricted to small bandwidths. However, the stronger cou-pling implies a reduction in the rings’ quality factors andhence suppression of nonlinear effects. Thus, whether itis better to employ anomalous Floquet-type lattices ortopological tight binding lattices will depend on the par-ticular application.
IV. TOPOLOGY OF DYNAMICALLY-MODULATEDRESONATORS
Resonators undergoing the dynamic modulation of therefractive index provide a flexible way to construct ef-fective lattice models described by time-dependent tightbinding Hamiltonians. Such systems break time-reversal
FIG. 5. Dynamically-modulated resonators generating ef-fective magnetic fluxes. (A) Two resonators are con-nected through the dynamic modulation. (B) Topologically-protected one-way edge mode propagates around the defectin a two-dimensional resonator lattice, where the spatial dis-tribution of modulation phases gives an effective magneticfield. (C) An array of rings undergoing dynamical modula-tion, which creates a synthetic space (D) with one dimensionbeing the frequency dimension, where a topological one-wayedge mode can be excited. (A,B) are adapted from Ref. 28.(C,D) are adapted from Ref. 90. symmetry and provide an important platform for imple-menting various topological phenomena. This specificsubject was started with the pioneering paper by Fang etal. , who showed that in photonic systems where the re-fractive index is harmonically modulated, the modulationphase actually gives rise to an effective gauge potentialfor photons. Based on this idea, a photonic Aharonov-Bohm interferometer was proposed as a design for anoptical isolator.A subsequent work by Fang et al. proposed a schemefor generating an effective magnetic field for photons ,based on spatially inhomogeneous modulation phases.The effective magnetic field for photons breaks time-reversal symmetry and can be used to induce nontrivialquantum Hall phases in two-dimensional resonator lat-tices. Topologically-protected one-way edge modes canbe excited in this dynamically-modulated resonator lat-tice which are robust against defects, as shown in Fig. 5.In contrast to the lattices discussed in the previous sec-tion, these topological modes are also robust against spin-flipping disorders because time-reversal symmetry is bro-ken. However, they remain susceptible to intrinsic lossessuch as absorption.Experiments based on these ideas were implementedin a variety of platforms. The first proof-of-principledemonstration of a photonic Aharonov-Bohm interfer-ometer used an electrical network at radio frequencies .In 2014, the photonic Aharonov-Bohm effect has beendemonstrated at visible wavelengths by utilizing an ef-fective gauge potential induced by photon-phonon inter-actions . Later in the same year, the presence of an ef-fective gauge field has been constructed using the on-chipsilicon photonics technology, where the refractive indexof the silicon coupled waveguides was modulated by anapplied voltage . These works are important proofs-of-concepts for photonic gauge potentials induced by dy-namic modulation. These proposals for creating effective gauge poten-tials in dynamically-modulated resonators have triggeredmany follow-up studies on the manipulation of lightvia modulation phases. For example, spatially inho-mogeneous distribution of modulation phases in two-dimensional resonator lattices can be used to control theflow of light . A spatially homogeneous but time-dependent distribution of modulation phases in three di-mensions has also been considered, which results in prop-agation analogous to the dynamics of electrons in thepresence of a time-dependent electric field . By tem-porally modulating the effective electric field, one cantime-reverse the propagation of one-way quantum Halledge states .In the above studies the modulations under consider-ations were treated weakly, so that the dynamics sat-isfy the rotating-wave approximation. Topological phasetransitions have also been studied in the ultrastrong cou-pling regime, where the rotating wave approximationfails. In the ultrastrong coupling regime the topologi-cal edge modes have been shown to exhibit larger band-width and less susceptibility to losses . On the otherhand, experimental efforts are still ongoing to achieveultrastrong coupling using ring resonators. As an ex-perimental proof of concept, light guiding by an effectivegauge potential has been demonstrated in tilted waveg-uide arrays . Moreover, lithium niobate microring res-onators have been coupled and modulated by externalmicrowave excitation, which leads to an effective pho-tonic molecule . Various platforms have therefore beenshown as potential candidates for exploring resonatorsunder strong dynamic modulation.Besides studying topological physics in real space, dy-namically modulated resonators also provide a uniqueplatform to explore higher-dimensional topologicalphysics in lower dimensional physical systems, by incor-porating synthetic dimensions in photonics . Inspiredby earlier works of synthetic dimensions in lattice sys-tems , resonators supporting multiple degeneratemodes with different orbital angular momentum (OAM)have been used to simulate the topological physics, wherethe synthetic dimension is constructed by coupling modeswith different OAM using a pair of spatial light modula-tors .On the other hand, dynamically-modulated ring res-onators with the modulation frequency close to the res-onators’ free spectral range naturally gives rise to a syn-thetic dimension along the frequency axis of light .Using this idea, two-dimensional topologically-protectedone-way edge states have been proposed using one-dimensional resonator lattices [see Fig. 5(C)]. Such edgemodes convert the frequency of light unidirectionally to-wards higher (or lower) frequency components as shownin Fig. 5(D), which could form the basis for a topologicalfrequency converter . The four-dimensional quantumHall effect can also be studied using this approach, bycombining a three-dimensional resonator lattice with afourth synthetic frequency dimension .Synthetic dimensions in dynamically-modulated res-onators also provide a platform for exploring novel topo-logical phases that are difficult to implement usingpure spatial lattices. For example, using synthetic di-mensions it is possible to implement three-dimensionalWeyl and topological insulating phases usingtwo-dimensional arrays of rings. In two-layer two-dimensional ring lattices, higher-order topological phasesexhibiting corner states have also been designed .Based on the scheme of creating topological system in aone-dimensional array of ring resonators, a mode-lockedtopological insulator laser in synthetic dimensions hasbeen suggested, which triggers potential applications fordeveloping active photonic devices .One significant advantage of synthetic dimensions im-plemented using dynamically modulated resonators is theability to flexibly control the connectivity of the cou-plings in the synthetic space, which is difficult to achievein real space lattices . For example, one can in-troduce long-range couplings along the synthetic fre-quency dimension by using modulation frequencies thatare multiples of the FSR, enabling emulation of the two-dimensional Haldane model using three rings . More-over, in a single resonator, one can combine two internaldegrees of freedom of light such as frequency and OAMto construct a two-dimensional synthetic lattice . Insuch synthetic lattices, the effective magnetic field canbe naturally introduced through the additional couplingwaveguides, thereby creating topologically-protected oneway edge states. This may enable the robust manipula-tion of entanglement between multiple degrees of freedomof light.Besides topological physics, many other interestinganalogies with quantum and condensed matter physicscan be demonstrated using dynamically-modulated res-onators, including Bloch oscillations , parity-timesymmetric systems , and flatband lattices .Furthermore, the creation of local nonlinearity in thesynthetic frequency dimension are under study, whichcould significantly broadens the range of Hamiltoniansinvolving local interactions that can be considered in thephotonic synthetic space in dynamically-modulated res-onators .We conclude this Section by discussing some recentexperimental demonstrations of synthetic dimensions inphotonics. The first photonic topological insulator in asynthetic dimension was demonstrated using an array ofmultimode waveguides, where modulation of the refrac-tive index along the waveguide axis played the role of thedynamic modulation . The dynamically-modulatedresonator has also been implemented in the fiber-basedring experiments incorporating commercial electro-opticmodulators , where band structures associated withone-dimensional synthetic lattices along the frequencyaxis of light have been measured . Based on this ex-perimental setup, one can use the clockwise/counter-clockwise modes of a single ring as another degree offreedom to construct a synthetic Hall ladder with two FIG. 6. Topology in synthetic space, adapted from Ref. 128.(A) Schematic of a single ring formed by a fiber loop underthe dynamic modulation. The CW and CCW modes form aspin degree of freedom and are coupled via connecting waveg-uides of differing lengths. (B) The corresponding lattice givesa synthetic Hall ladder threaded by an effective magnetic flux φ with two independent synthetic dimensions. (C) Measur-ing the transmission of an input CW excitation as a functionof the detuning ∆ ω reveals two bands with opposite chirality.(D) The chiral current j C , a measure of the spin-sensitive di-rection of frequency conversion, is opposite for the two bands.(E) Steady state normalized photon number P ( m ) of the CWmode at the m -th resonance, where m L is the resonant modecloset to the input laser, indicating preferential conversion tohigher frequencies. independent physical synthetic dimensions, as shown inFig. 6 . An effective magnetic flux was generated inthe experiment and signatures of topological chiral one-way edge modes were observed. Recently, integratedlithium niobate resonators under dynamic modulationprovide another potential experimental platform to con-struct synthetic dimensions and explore topological pho-tonics in a synthetic space, which is potentially significantfor on-chip device applications . V. FUTURE DIRECTIONS
Having reviewed some of the seminal works on imple-menting topological effects using ring resonators, we nowdiscuss some promising directions for future research, in-cluding fundamental studies of topological phenomenaand practical problems which must be solved to maketopological ring resonators viable for device applications.Most studies of topological ring resonators to datehave focused on Hermitian topological lattice models,in which strictly speaking the topological protectiononly holds in the absence of any gain or loss. Thestudy of non-Hermitian topological phases induced byappropriately-structured gain or loss is a topic attractingenormous interest nowadays . Non-Hermitian cou-pling, which can induce novel non-Hermitian topologicalphases, can be implemented in coupled resonator latticeseither by introducing asymmetric backscattering to thesite rings or adding gain or loss into the links to in-duce a hopping direction-dependent amplification or at-tenuation . The latter has been implemented usingvariable-gain amplifiers in a microwave network andcoupled fiber loops . Experiments with microring res-onators remain limited to topological laser experimentsbased on adding gain to existing Hermitian topologicalphases , making this an interesting direction forfurther studies. Can we use ideas from non-Hermitiantopological phases to exploit or minimize the scatteringlosses present in integrated photonic ring resonators?Ring resonators also provide an ideal platform forstudying nonlinear topological systems , due to the en-hancement of nonlinear effects provided by high qualityfactor microresonators. Recent experiments have har-nessed nonlinearities to generate frequency combs fromsingle resonators . It will be interesting considertopological band structure effects in this context. Thehigh flexibility in controlling the inter-ring coupling inresonator lattices also provides an opportunity to im-plement exotic forms of nonlinearity, such as modelswith nonlinear coupling . Systems with nonlinear cou-pling can exhibit nonlinearity-induced topological tran-sitions , which were so far limited to electronic circuitexperiments . Fiber loops are another promising plat-form for exploring nonlinear effects due to their long ac-cessible propagation lengths and ability to compensatefor losses using fiber amplifiers .Recently, several studies have proposed the use of topo-logical modes supported by domain walls of topologi-cal photonic crystals as a means of constructing novelclasses of ring resonators . Light confinement istypically weaker than the standard approach based onintegrated photonic ridge waveguides, meaning larger res-onator sizes are required. A potential benefit of topolog-ical photonic crystal-based ring resonators is their abilityto support sharp corners without bending losses. How-ever, it remains to be seen whether they will be competi-tive with existing ring resonators. For example, topolog-ical photonic crystals have been shown to exhibit largelosses ( >
100 dB/cm) compared to conventional photoniccrystal waveguides (5 dB/cm), due to out-of-plane scat-tering losses .Research on topological photonics has so far largelyfocused on the fundamental science and demonstrationof novel topological effects. There remains a large gapbetween these studies and potential applications whichmust be bridged. While new kinds of topological phe-nomena such as higher order topological phases con-tinue to attract fundamental interest, the need for morechallenging ingredients such as high dimensional lattices or protecting symmetries makes any useful applicationsa far off prospect at this stage. Moving forward, wewill require better optimization of existing topologicaldesigns to make them more competitive with standardcomponents, moving from a paradigm of demonstratingtopological robustness by deliberately introducing defects(as is the case in most experiments utilising photoniccrystals, waveguide arrays, and metamaterials), to onewhere there is topological protection against the actualimperfections which limit the performance of real devices.Topological ring resonator lattices using silicon photonicsare noteworthy as they are perhaps the only platform todate in which the topology imbues protection against thedominant form of intrinsic disorder, i.e. the misalignmentin the rings’ resonant frequencies.Many applications of ring resonators employ small sys-tems consisting of up to a few coupled rings. Generallyfor topological protection to hold, we need to have a bulk,requiring a large system size. Intuitively, it is the pres-ence of a bulk which allows signals to route around im-perfections on the edge. So another important directionis to determine how to use topological ideas to improvethe performance of small systems of a few coupled res-onators. This is a direction where concepts such as syn-thetic dimensions will likely play a key role.Finally, one of the most exciting potential near-termapplications of topological resonator lattices is as reli-able delay lines or light sources in large scale quantumphotonic circuits. For example, topological edge statesmay be useful as disorder-robust delay lines for entangledstates of light . In 2018, Mittal et al. demonstratedexperimentally the generation of correlated photon pairsvia spontaneous four wave mixing in a topological edgemode . They observed better reproducibility of thephoton spectral statistics over several devices comparedto regular CROWs, which is promising for the scaling upand mass production of quantum photonic circuits. Veryrecently this idea was generalized to dual pump sponta-neous four wave mixing, which allows one to tune theresulting two photon correlations by changing the pumpfrequencies . VI. CONCLUSION
We have presented an overview of how ring resonatorsprovide a highly flexible platform for studying topologicalband structure effects in photonics. Ring resonators havenot only enabled the implementation of seminal topolog-ical lattice models from condensed matter physics, buthave also been used for some of the first observations ofnovel topological phases in any platform, such as higherorder topological corner states. There is now strong the-oretical and experimental evidence that topological ideasmay useful for designing superior delay lines or frequency-converters in integrated photonic circuits. As the basicconcepts are now well-established, future research willneed to shift focus towards optimization of existing topo-0logical ring resonator systems to improve their perfor-mance and make their figures of merit more competitivewith conventional ring resonator-based components.
ACKNOWLEDGEMENTS
We thank M. Hafezi for useful discussions. This re-search was supported by the Institute for Basic Science inKorea (IBS-R024-Y1, IBS-R024-D1), the National Nat-ural Science Foundation of China (11974245), and theNatural Science Foundation of Shanghai (19ZR1475700).
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