Floquet Mode Resonance: Trapping light in the bulk mode of a Floquet topological insulator by quantum self-interference
FFloquet Mode Resonance: Trapping light in the bulk mode of a Floquet topologicalinsulator by quantum self-interference
Shirin Afzal ∗ and Vien Van Department of Electrical and Computer Engineering,University of Alberta, Edmonton, AB, T6G 2V4, Canada
Floquet topological photonic insulators characterized by periodically-varying Hamiltonians areknown to exhibit much richer topological behaviors than static systems. In a Floquet insulator,the phase evolution of the Floquet-Bloch modes plays a crucial role in determining its topologicalbehaviors. Here we show that by perturbing the driving sequence, it is possible to manipulate thecyclic phase change of the system over each evolution period to induce quantum self-interference ofa bulk mode, leading to a new topological resonance phenomenon called Floquet Mode Resonance(FMR). The FMR is fundamentally different from other types of optical resonances in that it iscavity-less since it does not require physical boundaries. Its spatial localization pattern is insteaddictated by the driving sequence and can thus be used to probe the topological characteristics ofthe system. We demonstrated excitation of FMRs by edge modes in a Floquet octagon lattice onsilicon-on-insulator, achieving extrinsic quality factors greater than 10 . Imaging of the scatteredlight pattern directly revealed the hopping sequence of the Floquet system and confirmed the spatiallocalization of FMR in a bulk-mode loop. The new topological resonance effect could enable newapplications in lasers, optical filters and switches, nonlinear cavity optics and quantum optics. I. INTRODUCTION
Topological photonic insulators (TPIs) provide a richplayground for exploring both the physics of periodicsystems as well as applications of their exotic proper-ties [1, 2]. In particular, Floquet TPIs characterized byperiodically-varying Hamiltonians have recently gainedmuch attention as they can exhibit richer topological be-haviors than static undriven systems, such as the exis-tence of anomalous Floquet insulator (AFI) edge modesin lattices with topologically trivial energy bands [3–9]. In a periodically-driven system, the evolution of thephase bands of the Floquet-Bloch modes over each pe-riod plays a crucial role in determining its topologicalbehaviors [10, 11]. Here we investigate the possibility ofmanipulating the cyclic phase change of a Floquet modeto induce quantum interference in the lattice bulk, lead-ing to a new resonance phenomenon as well as a newmechanism for probing the topological behaviors of Flo-quet TPIs.The ability to form robust high quality factor res-onators in a topological lattice is of practical interest asit would significantly broaden the range of applications ofTPIs such as in lasers, filters, nonlinear cavity optics andquantum optics [12–17]. In 2D TPI lattices, travelling-wave resonators can be realized by exploiting the confine-ment of edge modes at the interface between topologicallytrivial and nontrivial insulators to form ring cavities, al-though these tend to have very long cavity lengths asthey require many lattice periods [18–22]. Topologicalresonators can also be realized by creating line defects[23] or point defects in the lattice bulk, e.g., by spatiallyshifting air holes in a photonic crystal to create a Dirac- ∗ [email protected] vortex topological cavity [24]. The resonance mode ispinned to the midgap and can be regarded as the 2Dcounterpart of 1D resonance modes in distributed feed-back lasers [25] and vertical cavity surface emitting lasers(VCSELs) [26]. In another variant of defect mode cavities[27], resonant confinement occurs due to band inversion-induced reflections from the interface walls as a result ofthe different parity modes inside and outside the cavity.The cavity mode is a bulk mode located at the Γ pointof the energy band diagram very close to the edge of thetopological bandgap. Recently, it was shown that topo-logical corner states with zero energy can also be usedto form resonances in a TPI [28–31]. However, the modeis not tunable and can only be formed at the corners ofthe lattice. Indeed all the topological photonic resonatorsreported to date are not continuously tunable and haveonly been realized for static TPI systems.Here we report a new resonance mechanism in a Flo-quet TPI whereby light is trapped in a Floquet bulk modedue to quantum self-interference. The resonance effect,which we refer to as Floquet Mode Resonance (FMR), isachieved by adiabatically tuning the drive-induced phaseof a Floquet mode to achieve constructive interference.This has the concomitant effect of shifting its quasienergyinto a topological bandgap to form an isolated flat-bandstate that is spatially localized in a bulk-mode resonantloop. The FMR is fundamentally different from othertypes of topological resonances in that it is cavity-lesssince it does not require creating physical boundaries inthe lattice. The lack of scattering from interface disconti-nuities means that FMRs can potentially have very highQ factors. The resonance can be formed anywhere inthe lattice bulk and can be continuously tuned across atopological bandgap. In addition, the spatial localizationpattern of the FMR is dictated by the driving sequenceof the Floquet TPI and can thus be used to probe thetopological behaviors of the system. We demonstrated a r X i v : . [ phy s i c s . op ti c s ] F e b FIG. 1. Driving sequence of a 2D Floquet microring lattice. (a) Schematic of the lattice showing a unit cell with fourmicrorings { A, B, C, D } and coupling angles θ a > θ b . (b) Equivalent coupled-waveguide array representation of the microringlattice, obtained by cutting the microrings at the points indicated by the open circles in (a) and unrolling them into straightwaveguides. The system evolves periodically in the direction of light propagation z in each microring, with each period consistingof four coupling steps j = { , , , } . Also shown is a phase detune ∆ φ applied to microring C in step j = 1 to perturb thedrive sequence. (c) Spatial localization of a bulk mode in a loop (red arrows): starting from step j = 1 in microring A (yellowstar), the hopping sequence of the lattice guarantees that light returns to its initial point after 3 evolution periods. FMR in a Floquet octagon lattice realized in silicon-on-insulator (SOI), using thermo-optic heater to excite andtune the resonance. Imaging of the scattered light inten-sity pattern reveals the hopping sequence of the Floquetmode and provides direct evidence of FMR localized ina bulk-mode loop. Our work not only introduces a ver-satile way for forming high-Q resonances in a Floquetlattice, but also provides a new method for probing thetopological behaviors of bulk modes in Floquet systems.
II. THEORETICAL ORIGIN OF FMR
The Floquet TPI we consider is a 2D square latticeof coupled microring resonators with identical resonancefrequencies. The lattice is characterized by two differentcoupling angles ( θ a > θ b ) in each unit cell (Fig. 1(a)),where θ a ( θ b ) represents the strong (weak) coupling be-tween resonator A ( D ) and its neighbors. Although 2Dmicroring lattices have been shown to exhibit Chern in-sulator behavior associated with static systems [32–35],the existence of FMRs can only be predicted by treat-ing the system as a Floquet insulator with periodically-varying Hamiltonian. As light circulates around eachmicroring, it couples periodically to its neighbors, sothat the Bloch modes of the lattice evolve in a cyclicalmotion with a period equal to the microring circumfer-ence L . The quasienergy spectrum of the lattice thushas a periodicity of 2 π/L . Within each Floquet Bril-louin zone, the microring lattice in general has threebandgaps (Fig. 2(a)), which can exhibit different topo-logical phases, including AFI behavior, depending on thecoupling angles ( θ a , θ b ) [36]. To better elucidate theFloquet nature of the lattice, we transform it into an equivalent 2D array of periodically-coupled waveguides(Fig. 1(b)) [36], with each period consisting of 4 couplingsteps between different pairs of adjacent waveguides. Inthe limit of perfect coupling ( θ a = π/ , θ b = 0), the hop-ping sequence guarantees that light starting from site A in a unit cell will return to its position after 3 periods,tracing out a bulk-mode loop depicted in Fig. 1(c). How-ever, in a uniform Floquet lattice, such a bulk mode doesnot exist in a bandgap since its phase change around theloop is not equal to an integer multiple of 2 π .Suppose that we now perturb the driving sequence byintroducing a phase shift ∆ φ in coupling step j of a mi-croring in the lattice (Fig. 1(b)). Taking a block of N × N unit cells with the perturbed microring located near itscenter, for sufficiently large N , we can treat this blockas a supercell of an infinite periodic lattice. Using thecoupled-waveguide array model, we can write the equa-tion of motion of the supercell as i ∂∂z | ψ ( k , z ) i = H ( k , z ) | ψ ( k , z ) i = X j =1 [ H ( j ) F B ( k , z ) + H ( j ) D ] | ψ ( k , z ) i (1)where k is the crystal momentum in the x - y plane, H ( j ) F B is the Floquet-Bloch Hamiltonian of the unperturbed su-percell in step j [36], and H ( j ) D is the perturbed Hamil-tonian in the same step. The perturbed Hamiltonianmatrix is zero everywhere except for a term of − ∆ φ/L in its k th diagonal element corresponding to the de-tuned microring k . Any state of the system evolves as | ψ ( k , z ) i = U ( k , z ) | ψ ( k , i , where U ( k , z ) = T e − i z R H ( k ,z ) dz (2) FIG. 2. Frequency and spatial localizations of energy-shifted Floquet states. (a) One Floquet Brillouin zone of the quasienergyband diagram of an AFI microring lattice consisting of 5 × θ a = 0 . π and θ b = 0 . π . The blue bands arecomposite transmission bands of Floquet states separated by three topological bandgaps (labeled I, II and III). The red bandsare the flat bands of Floquet bulk modes which are lifted from the transmission band manifolds due to a phase detune ∆ φ = π applied to step j = 1 of microring C . (b)-(d) Intensity distributions of the energy-shifted bulk modes Φ s obtained when phasedetune ∆ φ = π is applied to segment j = 1 of microring A , C and D , respectively. is the evolution operator. The evolution over eachroundtrip period of the microrings is given by the Flo-quet operator, U F ( k ) = U ( k , L ), whose eigenstates arethe Floquet modes | Φ n ( k , i with eigenvalues e − i(cid:15) n ( k ) L .In the absence of detuning (∆ φ = 0), the quasienergybands (cid:15) n ( k ) of the Floquet modes form composite trans-mission band manifolds, each containing 4 N degeneratebulk modes and separated by bandgaps. The effect ofthe phase detune ∆ φ is to break the degeneracy and liftone Floquet mode into the bandgap, forming an isolatedsingle band (Fig. 2(a)). Moreover, this energy-shiftedband becomes increasingly flattened as the phase detuneis increased, implying that the field distribution becomesmore strongly localized spatially. Figures 2(b)-(d) showthe field distributions of the isolated Floquet mode wheneach of microrings A , C and D , respectively, is detunedduring step j = 1. When microring A is detuned, thefield is localized in two coupled bulk-mode loops sharingthe common segment j = 1. By contrast, detuning mi-croring C results in the field strongly localized in onlya single bulk-mode loop traced out by the hopping se-quence. A similar mode pattern is also observed whensegment j = 2 of microring B is detuned. When theweakly-coupled microring D is detuned, light does notfollow the driving sequence but instead remains trappedin the same site resonator. This point-defect mode isalso observed in a topologically trivial lattice (see Sup-plemental Material). Thus by selectively applying phasedetunes to specific steps in the driving sequence, distinctmode patterns can be excited to probe the topological behavior of the Floquet lattice.The strong field localization in a bulk-mode loop is ef-fectively a resonance effect caused by the energy-shiftedFloquet mode constructively interfering with itself aftercompleting each roundtrip around the loop. Starting outeach cycle at z = 0, the shifted Floquet mode | Φ s ( k , i evolves as | Ψ s ( k , z ) i = U ( k , z ) | Φ s ( k , i . According toFloquet theorem, the state | Ψ s ( k , z ) i can also be ex-pressed as [11] | Ψ s ( k , z ) i = e − i(cid:15) s ( k ) z | Φ s ( k , z ) i (3)where | Φ s ( k , z ) i = e i(cid:15) s ( k ) z U ( k , z ) | Φ s ( k , i is the peri-odic z -evolved Floquet state satisfying | Φ s ( k , z + L ) i = | Φ s ( k , z ) i . Thus, the state | Ψ s ( k , z ) i will constructivelyinterfere with itself after every period L if | Ψ s ( k , z + L ) i = e − i(cid:15) s ( k )( z + L ) | Φ s ( k , z + L ) i = e − i(cid:15) s ( k ) z | Φ s ( k , z ) i (4)from which we obtain the condition for constructive inter-ference as (cid:15) s ( k ) L = 2 mπ , m ∈ Z . Using the quasienergyfor a stationary Floquet mode at k = , we can calculatethe shift in the resonant frequency of the FMR relativeto a microring resonance as ∆ ω s = (cid:15) s ( ) L ∆ ω F SR / π ,where ∆ ω F SR is the free spectral range (FSR) of the mi-crorings. Figure 3(a) plots the dependence of the cyclicphase change (cid:15) s ( ) L on the phase detune ∆ φ , showingthat the resonant frequency of an FMR can be continu-ously tuned across a topological bandgap. In the mod-ern theory of electric polarization in crystals, the Berry FIG. 3. Effects of the phase detune on the resonant frequencyand spatial localization of FMR. (a) Dependence of the cyclicphase change (cid:15) s ( ) L and resonant frequency shift of the FMRin each bandgap of an AFI lattice ( θ a = 0 . π , θ b = 0 . π )on the phase detune ∆ φ . The blue lines are the quasienergiesof the transmission bands, which remain largely unchangedwith phase detuning. (b) Variation of the average inverseparticipation ratio of FMR (in bandgap III) with phase detune∆ φ for Floquet lattices with coupling angle θ a varied from0 . π to 0 . π and θ b fixed at 0 . π . phase of a Bloch mode (divided by 2 π ) represents theaveraged displacement from a lattice site of the corre-sponding Wannier state in real space [37]. By the sameanalogy, the cyclic phase (cid:15) s ( k ) L , which includes both thedynamical and geometric phases of the system, can be in-terpreted as the average position of the Floquet mode inthe frequency domain [11, 38]. Here we show that bytuning the cyclic phase of a Floquet-Bloch mode, we canvary its frequency position to yield a resonance localizedin both spatial and frequency domains in an otherwisehomogeneous topological lattice.We note that this resonance effect is cavity-less since itdoes not require physical boundaries between the latticeand another medium but instead relies on an adiabaticchange in the Hamiltonian via a phase detune. Sinceno interface scattering takes place, FMRs can in prin-ciple have very high Q factors. Importantly, since thephase detune ∆ φ represents a local adiabatic change tothe Hamiltonian H F B , the energy-shifted band still re-tains the topological properties of the unperturbed lat-tice. This is evident from the fact that the FMR mode(Fig. 2(c)) retains the same spatial distribution of a bulkmode in a homogeneous lattice as we increase the phasedetune. Also, the bandgaps above and below the FMRstill support edge modes, implying that the topologicalbehavior of the lattice is not altered by the adiabaticphase detuning. In particular, simulations show that theFMR remains robust to random variations in the lattice(see Supplemental Material).We can quantify the degree of spatial localization of anFMR by computing its inverse participation ratio (IPR)[39]. For a z -evolved Floquet mode with normalization h Φ s ( , z ) | Φ s ( , z ) i = 1, we can define the average IPRover one evolution period as IP R = 1 L N X k =1 L Z | Φ ( k ) s ( , z ) | dz (5) where Φ ( k ) s is the field in site resonator k in the lat-tice. Figure 3(b) shows the average IPR of an FMR (inbandgap III) as a function of the phase detune for dif-ferent coupling angles of the lattice. It is seen that themode becomes more strongly localized as it is pusheddeeper into the bandgap. Thus, in general, we can ex-pect to achieve the strongest intensity enhancement forFMRs located near the center of the bandgap. The de-gree of localization is also higher for lattices with largercontrast between the coupling angles θ a and θ b . We notethat the maximum IP R achievable for FMR is 1/3 be-cause at any given position z in an evolution cycle, thefield is localized in three separate microrings in the bulk-mode loop. III. EXPERIMENTAL DEMONSTRATION OFFMR
We demonstrated FMR in a Floquet TPI lattice con-sisting of a square array of coupled octagon resonatorson an SOI substrate. Each unit cell consisted of 4 iden-tical octagons evanescently coupled to their neighborsvia identical coupling gaps g (Fig. 4(a)). Octagon res-onators were used to realize dissimilar coupling angles θ a and θ b in each unit cell by exploiting the differencebetween synchronous and asynchronous couplings. Thisis achieved by designing the octagons to have sides withequal lengths L s but alternating widths W and W , withoctagon D rotated by 45 o with respect to the other 3 oc-tagons. Octagon A can thus be strongly coupled to itsneighbors via synchronous coupling between waveguidesof the same width W , while octagon D is weakly cou-pled to its neighbors via asynchronous coupling betweenwaveguides of dissimilar widths W and W . We designedthe coupling angles to be θ a = 0 . π and θ b = 0 . π ,so that the lattice exhibits AFI behavior for TE polarizedlight in its three bandgaps over one FSR at the telecom-munication wavelengths [9] (see Supplemental Materialfor details of the lattice design). The fabricated latticeconsisted of 10 ×
10 unit cells (Fig. 4(b)). An input waveg-uide was coupled to resonator A of a unit cell on the leftboundary of the lattice to excite AFI edge modes andan output waveguide was coupled to resonator B on theright boundary to measure the transmission spectrum.Figure 5(a) (red trace) shows the transmission spec-trum measured for input TE light over one FSR ( ∼ .
833 nm wavelength (in bandgapIII) using an NIR camera (Fig. 5(b)) confirms the for-mation of an edge mode propagating along the latticeboundary from the left input waveguide to the right out-put waveguide.
FIG. 4. Design and implementation of the Floquet octagon lattice in SOI for demonstrating FMR. (a) Schematic of a unit cellof the Floquet octagon lattice, with octagon D rotated by 45 o with respect to octagons A , B and C to realize synchronous andasynchronous coupling angles θ a and θ b . (b) Microscope image of the fabricated octagon lattice in SOI showing the input andoutput waveguides used to measure the transmission spectrum, and the heater used to tune the phase of an octagon C on theleft boundary to excite FMR.FIG. 5. Experimental observation of FMR. (a) Measured transmission spectra of the Floquet octagon lattice over one FSRwhen there was no phase detune (red trace) and when a phase detune of ∆ φ = 1 . π was applied to microring C on the leftboundary (blue trace). (b) NIR camera image of scattered light intensity at 1612.833 nm wavelength in bandgap III withno phase detune, showing an AFI edge mode propagating along the left and bottom edges of the lattice. (c) NIR image at1612.833 nm wavelength with phase detune of ∆ φ = 1 . π , showing FMR localized in a bulk-mode loop. The edge mode is notvisible due to its much weaker intensity compared to the FMR. Inset (i) shows a map of scattered light intensity reconstructedfrom raw camera data superimposed on the octagon lattice; inset (ii) shows the simulated intensity distribution of the FMRfor comparison. Since the FMR exists in a bulk bandgap, we can cou-ple light into it using an AFI edge mode in the samebandgap. To excite an FMR near the left boundaryof the lattice, we thermo-optically tuned the phase ofan octagon C on the left boundary (Fig. 4(b)) using atitanium-tungsten heater fabricated on top of the res- onator (detuning the whole resonator C at the edge alsoexcites only a single bulk-mode loop). Figure 5(a) (bluetrace) shows the transmission spectrum when a phase de-tune of 1 . π (corresponding to electrical heating power P = 34 . FIG. 6. Tuning of FMR across the topological bandgap: (a) transmission spectra of FMR in bandgap III at various phasedetunes. Top horizontal scale indicates the phase detunes ∆ φ corresponding to the resonance dips. (b) Dependence of theresonant wavelength shift of the FMR (relative to the microring resonance at zero phase detune) on the phase detune (bottomhorizontal axis) and heating power (top horizontal axis). Blue circles are measurement data; red line is the linear best fit. (c)Variations of the extrinsic Q factor and the coupling rate µ of the FMR with phase detune ∆ φ . Black circles are measured Q;red line is the simulated Q of FMR in a lattice with θ a = 0 . π , θ b = 0 . π and roundtrip loss of 0.59 dB in each octagon. observe that the spectrum is almost identical to the spec-trum without phase detune (red trace), except for thepresence of two sharp dips located in bulk bandgaps I andIII. These dips indicate the presence of an FMR excitedin each bulk bandgap by the edge mode. To obtain visualconfirmation of the spatial localization of the FMR, weperformed NIR imaging of the scattered light intensity atthe resonance wavelength 1612 .
833 nm (in bandgap III)(Fig. 5(c)). The image clearly shows that light is localizedand trapped in a bulk-mode loop, which is not present inFig. 5(b) when no phase detune was applied. The bulkmode pattern directly captures the hopping sequence ofthe Floquet lattice as predicted in Fig. 1(c). Strikingly,the edge mode does not “go around” the detuned octagon C as when it encounters a defect, but instead excites theFMR and couples to it. We also note that transmissiondips occurring in the bulk transmission bands of the lat-tice, which appear with and without phase detuning, arecaused by random interference of light propagating deepinto the lattice bulk. Imaging of light intensity patternsat these wavelengths in the transmission band does notshow light localized in FMR loops [9].Focusing on the FMR in bandgap III, we measuredthe resonance spectrum for different phase detune values.The spectra are plotted in Fig. 6(a), showing that as thephase detune is increased, the FMR spectrum is pusheddeeper into the bandgap. The resonance linewidth alsobecomes narrower while the extinction ratio reaches amaximum of almost −
40 dB near the bandgap center. InFig. 6(b) we plotted the resonant wavelength shift ∆ λ as a function of the phase detune, with the correspond-ing heater power shown on the top horizontal axis. Thelinear relationship between ∆ λ and ∆ φ is in agreementwith the theoretically predicted dependence of the FMRquasienergy on the phase detune (Fig. 3(a)).The dependence of the Q factor of the FMR on the phase detune is shown in Fig. 6(c) (black circles). Weobtained Q values in the range 1 . × − . × ,with a slight increasing trend as the FMR moves deeperinto the bandgap. For comparison, the intrinsic Q fac-tor of a single resonator obtained from measurementof a stand-alone octagon (see Supplemental Material)was only slightly higher at 2 . × (corresponding toroundtrip loss of 0.35 dB). Using the designed couplingvalues ( θ a = 0 . π , θ b = 0 . π ) for the lattice anda slightly higher roundtrip loss of 0.59 dB in each oc-tagon, we simulated FMR spectra for various phase de-tunes and obtained the corresponding extrinsic Q factors(red line in Fig. 6(c)), which show good agreement withthe measured values. Larger discrepancies between sim-ulated and measured Q factors are observed for smallerphase detunes, which can be attributed to the fact thatthe FMR and edge mode are less localized near the bandedge and are thus more susceptible to lattice imperfec-tions. From the measured extrinsic Q factors Q ex , we cancalculate the effective coupling ( µ ) between the FMR andthe edge mode as µ = ω (1 /Q ex − /Q ), where ω is theresonant frequency and Q = 1 . × is the intrinsicQ factor. The results are also plotted in Fig. 6(c) (bluecircles). The coupling rate µ depends on the overlappingbetween the field distributions of the AFI edge mode andthe FMR. This dependence is seen to correlate with thevariation in the degree of spatial localization of the FMRas indicated by the plot of IP R vs. ∆ φ in Fig. 3(b). Asthe FMR is pushed deeper into the bandgap, it becomesmore strongly localized spatially so that its coupling tothe edge mode is weaker, which results in higher Q factor.Using the designed coupling angle values, simulations ofthe lattice showed that the theoretical extrinsic Q fac-tor (without loss) can exceed 10 , suggesting that theexperimental Q factor can be improved by reducing theroundtrip loss in the FMR loop, for example, by reducingscattering from the octagon corners and using materialswith lower absorption. IV. DISCUSSION AND CONCLUSION
The reported FMR represents a new mechanism fortrapping light in a TPI lattice by adiabatically tuningthe cyclic phase of a Floquet mode to induce quantumself-interference. The spatial localization pattern of theFMR is shown to be dictated by the driving sequenceof the Floquet lattice and is thus a manifestation of itstopological nature. Our work thus introduces a simpleway for directly exciting and isolating a Floquet bulkmode for probing its topological properties. The non-trivial behavior of the lattice can be deduced from thebulk-mode hopping pattern, in contrast to the approachused in previous works on TPIs, which typically rely onimaging edge modes at the lattice boundaries to verifyits topological nature. We note that it is also possibleto perturb the coupling angles in the driving sequence, which may provide additional mechanism for probing thebulk modes and controlling the FMRs.Compared to other topological resonators, the FMRis cavity-less, tunable, and can be formed anywhere inthe lattice bulk. The lack of physical cavity bound-aries suggests that very high Q factors can potentiallybe achieved. The resonance can also be dynamicallyswitched on and off, which could be useful for realizingoptical switches and modulators. In addition, our pre-liminary experimental results have shown that multipleadjacent FMRs could be excited to form coupled cavitysystems, which could open up new applications such ashigh-order coupled-cavity filters, optical delay lines, andlight transport in a bandgap through the lattice bulk byhopping between adjacent localized bulk modes.
ACKNOWLEDGMENTS
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I. COMPARISON BETWEEN FMR AND DEFECT STATE IN A TOPOLOGICALLY TRIVIALFLOQUET LATTICE
Here we provide a contrasting example of the different behaviors of an FMR in a topologically nontrivial bandgapand a conventional defect state in a trivial bandgap, both existing in the same Floquet lattice. We consider a microringlattice with coupling angles θ a = 0 . π and θ b = 0 . π , which behaves as a Floquet Chern insulator in bandgaps Iand III and as a normal insulator in bandgap II (see Ref. [1] for the topological characterization of Floquet microringlattices). These behaviors can be verified by the projected band diagram of a semi-infinite lattice (with 5 unit cellsin the y direction and infinite extent in the x direction) in Fig. S1(a), which shows edge states existing in bandgapsI and III but not in bandgap II. We apply a phase detune ∆ φ in step j = 1 of a microring C in the lattice, whichshifts the quasienergy of a bulk mode from each transmission band manifold into the bandgap below, as shown inFig. S1(b). Although the trivial bandgap II hosts an energy-shifted bulk state, the spatial field distribution of themode is markedly different from those in the nontrivial bandgaps I and III. For instance, for the same phase detuneof ∆ φ = 0 . π , the intensity distributions of the shifted states in bandgaps III and II are shown in Figs. S1(c) andS1(d), respectively. The shifted bulk mode in the nontrivial bandgap is not localized in the detuned resonator but alsohops to neighbor sites, forming a loop defined by the hopping sequence which gives rise to the nontrivial topologicalbehavior of the Floquet lattice. On the other hand, the shifted bulk mode in the trivial bandgap is localized inthe same detuned site resonator, which is similar to a point-defect mode in a static lattice. Thus it is possible todetermine the topological behaviour of a lattice by probing the spatial distribution of the energy-shifted bulk modein its bandgap. This technique complements the conventional approach of exciting edge modes along the latticeboundaries to verify its nontrivial topological behavior. FIG. S1. Comparison between FMR and point-defect state in a Floquet microring lattice. (a) Projected quasienergy banddiagram of a microring lattice with coupling angles θ a = 0 . π and θ b = 0 . π , with 5 unit cells in the y direction and infiniteextent in x ( a is the spacing between adjacent microrings). Bandgaps I and III are nontrivial with winding number w = 1;bandgap II is trivial with w = 0. (b) Quasienergies of the Floquet states as functions of phase detune ∆ φ applied to step j = 1of a microring C in a lattice with 5 × φ = 0 . π isapplied to the lattice. These states are indicated by the green and red dots in (b). a r X i v : . [ phy s i c s . op ti c s ] F e b II. ROBUSTNESS OF FMR
We investigated the robustness of FMR in the presence of random variations in the Floquet microring latticeshown in Fig. 4(b) of the main text, with 10 ×
10 unit cells and coupling angles θ a = 0 . π , θ b = 0 . π . Weformed an FMR near the left boundary by applying a phase detune of ∆ φ = 1 . π to a microring C on the leftboundary. We coupled light into the FMR using AFI edge mode, which is excited through the input waveguide.Figure S2(a) shows the simulated spectral response of light intensity inside the FMR loop (at the location indicatedby the yellow star in the inset diagram in Fig. S2(a)). The red trace is the ideal case with no random variations inthe lattice, showing three resonant peaks appearing in the 3 bandgaps over one FSR of the microrings. The greyarea indicates the variations in the intensity due to uniformly-distributed random deviations of up to ±
10% in thecoupling angles and roundtrip phases of the microrings in the lattice. It is seen that the FMR peaks still appear inthe 3 bandgaps at approximately the same quasienergies, implying that the frequency position of an FMR is robustto random variations. Figure S2(b) compares the spatial distributions of light intensity of the FMR in bandgap IIIwithout and with the random variations. It is seen that while random variations cause light to be spread out more tothe resonators surrounding the FMR loop, most of the light is still strongly localized in a bulk-mode loop. Thus thespatial localization of an FMR is also robust to random variations.
FIG. S2. Effects of random variations in the Floquet microring lattice on FMR: (a) Simulated spectral responses of lightintensity inside the FMR (at the location indicated by yellow star in the inset figure), when a phase detune ∆ φ = 1 . π isapplied to microring C on the left boundary. Red trace is the ideal lattice with no perturbation; the grey area shows thevariations in the intensity due to ±
10% random perturbations in the coupling angles and microring roundtrip phases obtainedfrom 100 simulations. (b) Intensity distributions of the FMR in bandgap III without and with the random perturbations(obtained from 20 simulations). The FMR loop appears in dark red color, which is also clearly visible for the case with randomperturbations, suggesting that the spatial localization of the FMR is also robust to variations.
III. DESIGN AND MEASUREMENT OF FLOQUET OCTAGON LATTICE
The Floquet octagon lattice was realized on an SOI substrate with 220 nm-thick Si layer lying on a 2 µ m-thickSiO buffer layer with a 2.2 µ m-thick SiO overcladding layer. The octagon resonators had sides of length L s = 16 . µ m and alternating widths W = 400 nm and W = 600 nm. The corners were rounded using arcs of radius R = 5 µ m to reduce scattering. The coupling gaps between adjacent octagons were fixed at g = 225 nm. From numericalsimulations using the Finite-Difference Time-Domain solver in Lumerical software [2], we obtained θ a = 0 . π and θ b = 0 . π for the synchronous and asynchronous coupling angles, respectively, around 1615 nm wavelength. Toexcite AFI edge mode and measure the transmission spectrum, input and output waveguides with 400 nm width werecoupled to their respective octagon resonators on the left and right boundaries of the lattice via the same couplinggap g = 225 nm. The 10 ×
10 lattice was fabricated using the Applied Nanotools SOI process [3].Transmission spectra of the lattice were measured using a tunable laser source (Santec TSL-510 1510 − × objective lens and anInGaAs NIR camera. The image sensor of the camera had 640 ×
512 pixels with a 15 µ m pitch.We excited an FMR and tuned its resonant frequency by varying the phase of an octagon C on the left boundaryof the lattice. A TiW heater covering the octagon perimeter (Fig. 4(b) in the main text) was fabricated on top ofthe resonator and current was applied to the heater to tune its phase. Figure 6(b) in the main text shows that themeasured resonant wavelength shift ∆ λ of the FMR varies approximately linearly with the applied heater power P .Using fact that the roundtrip phase detune ∆ φ of the octagon resonator also varies linearly with the heater power,we can correlate the measured ∆ λ vs. P plot with the simulated ∆ λ vs. ∆ φ plot across the bandgap. This allowsus to calibrate the heater efficiency and deduce the linear correspondence between the phase detune and the heaterpower. The relationship between ∆ φ and P is explicitly shown on the top and bottom horizontal axes of Fig. 6(b) inthe main text. IV. LOSS MEASUREMENT OF A STAND-ALONE OCTAGON RESONATOR
To determine the loss in the octagon resonators in the Floquet lattice, we fabricated a stand-alone octagon resonatorwith the same dimensions as the octagons in the lattice. The octagon was evanescently coupled to two waveguidesof 400 nm width at the top and bottom sides, as shown in Fig. S3(a). The coupling gaps between the resonator andthe waveguides were g = 225 nm, which is the same as those between adjacent octagon resonators in the Floquetlattice. We measured the transmission spectrum of the resonator at the through port by scanning input TE-polarizedlight over the 1510 nm - 1630 nm wavelength range. Two sample resonance spectra around 1515 nm and 1615 nmwavelengths are shown in Figs. S3(b) and S3(c), respectively. To determine the coupling coefficients and loss of theresonator, we fit each resonance spectrum using the equation for the power transmission at the through port [4]: T t = T min + F sin ( φ/ F sin ( φ/