We show that 2D photonic lattices consisting of coupled microring resonators can emulate quantum systems driven by periodic Hamiltonians and can thus be used to realize photonic Floquet topological insulators. By transforming a 2D microring lattice into an equivalent array of coupled waveguides with periodic boundary conditions, we explicitly derive the Floquet-Bloch Hamiltonian of the system and determine the winding numbers characterizing the band topology and bulk-edge correspondence of the lattice. By varying the coupling strengths between adjacent resonators, we show that a 2D microring lattice can support both anomalous Floquet insulator edge modes and Chern insulator edge modes over a wide range of coupling parameters.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The study of the Quantum Hall effect  revealed the existence of a new class of insulator materials whose band structures are characterized by nontrivial topological invariants. A distinguishing feature of these topological insulators is the existence of topologically-protected edge states at the material boundaries that are immune to defect scattering. This discovery has inspired recent efforts to study topological structures in the optics domain, with the aim of realizing photonic devices that can transport light via edge modes that are insensitive to fabrication imperfections [2–4]. Photonic topological insulators have been proposed and demonstrated in various structures such as waveguide arrays, photonic crystals, metamaterials, and coupled microring resonators [5–8].
It is now recognized that topological insulator properties can be observed in systems with either static or periodically-driven Hamiltonians [9,10]. Topological insulators with static Hamiltonians, also known as Chern insulators, are characterized by nontrivial integer Chern numbers of the bulk energy bands, with the sum of the Chern numbers below a bandgap corresponding exactly to the number of chiral edge modes that can be excited in that bandgap. On the other hand, in Floquet topological insulators with periodically-driven Hamiltonians, chiral edge modes can arise even though the Chern numbers of the energy bands within a periodic Floquet quasi-energy zone are trivial. The existence of these anomalous edge modes implies that the bulk-edge correspondence in a Floquet insulator cannot be determined by the Chern numbers. Instead, as shown in , these systems must be characterized by a different topological invariant - the winding number associated with the bulk evolution operator - whose value depends on the complete time history of the system during each driving period.
Photonic Floquet insulator behaviors have been demonstrated in optical resonator lattices whose coupling coefficients are modulated harmonically in time . Recently, it was shown that Floquet edge modes can also be observed in periodically-coupled waveguide arrays [13,14], whose system Hamiltonian varies periodically along the direction of light propagation rather than in time. Liang and Chong showed that edge modes also exist in a square lattice of strongly-coupled microring resonators in which the rings have the same direction of field propagation and are linked through coupling loops . Using network theory to analyze the system, Pasek and Chong later showed that the lattice exhibits characteristics of a Floquet topological insulator, although a complete characterization of the microring lattice in terms of its topological invariants was still lacking since the system evolution operator could not be obtained from the method .
In this paper, we formalize the description of a microring lattice as a Floquet topological insulator by showing that it can be described by a Hamiltonian that is periodic in the direction of wave propagation. Using a topological transformation (in real space) to convert the microring lattice into an equivalent coupled waveguide array, we can explicitly derive the Floquet-Bloch (FB) Hamiltonian of the system and construct its evolution operator. Knowledge of the system state at every point in each driving period allows us to determine the winding numbers and establish the bulk-edge correspondence of the lattice. To our knowledge, the topological characterization of a microring lattice in terms of the winding number of a periodically-driven system has not been shown before. In contrast to the work in  in which the microring resonators are assumed to have unidirectional field propagation, we will focus on the more general 2D microring lattice in which adjacent microrings have alternate directions of field propagation. This type of lattices is more natural to implement on an integrated optics platform and, as we will show, exhibits much richer topological phases than unidirectional microring lattices. More specifically, we show that Floquet topological characteristics can be observed over a wider range of coupling strengths and the lattice can support both Chern and anomalous Floquet edge modes.
This paper is organized as follows. We begin in Section 2 with the derivation of the FB Hamitonian of a 2D microring lattice. Section 3 outlines the method employed to calculate the winding numbers and Chern numbers of a Floquet insulator, followed by a detailed study of the topological characteristics and edge modes of the microring lattice. The paper ends with a discussion and summary in Section 5.
2. Floquet-Bloch Hamiltonian of a 2D microring lattice
Figure 1(a) shows a schematic of a 2D square microring lattice in which the microring resonators are assumed to have the same radius R and identical resonant frequencies. Light in each microring propagates in either the clockwise or counter-clockwise direction and scattering into the counter-propagating mode is assumed to be negligible. Coupling between neighbor microrings is achieved via evanescent field coupling between the microring waveguides, with the coupling strength defined by a coupling angle θ such that the fraction of power transfer between the two waveguides is equal to sin2 θ. The direction of light propagation in each evanescent wave coupler dictates that the fields in adjacent microrings propagate in alternate directions. For such a lattice, the smallest unit cell consists of four resonators, which are labeled A, B, C and D in the figure. In this paper we will focus on the particular coupling configuration where the coupling strength between microring A and its neighbors can be different from the coupling strength between the other resonators, since this system is found to exhibit richer topological characteristics than other coupling configurations. The lattice in this case can be characterized by two coupling parameters: coupling angle θa between microring A and its neighbors and coupling angle θb between microring D and its neighbors, as shown in Fig. 1(a). We also note that the lattice is unchanged if we exchange the values of θa and θb.
Within each resonator, the propagating field executes a periodic motion around the microring, with a spatial period equal to 2πR along the direction of propagation, which we will denote as the z direction. The microring lattice can thus be regarded as a periodically-driven system, although the driving is along z rather than in time. We note, however, that in the limit of weak couplings, the field amplitude in each resonator is approximately uniform along the microring waveguide, so that the lattice can be approximated as a static system with a Hamiltonian that is independent of z (also known as the energy coupling or tight binding approximation). In this case, it can be shown that the lattice is characterized by a topologically trivial Chern number.
The situation changes dramatically for strong couplings, which cause the field amplitude in each resonator to vary spatially along the microring waveguide. To determine the system Hamiltonian in this case, we employ the approach in  to transform the microring lattice into an equivalent coupled waveguide array as shown in Fig. 1(b). This is accomplished by “cutting” each microring at the point indicated by the small open circle on each microring in Fig. 1(a) and unrolling it to form a straight waveguide. In this manner we obtain a 2D array of coupled waveguides, each having equal length L = 2πR with periodic boundary conditions at the two ends. The waveguide lattice is also periodic in the transverse (x and y) directions with periodicity of 2a ≈ 4R, where we have neglected the evanescent coupling gaps between the microrings. In Fig. 1(b), a connection between two adjacent waveguides indicates coupling between two corresponding microring resonators. The figure also suggests that the evolution of the fields in the waveguide array over each period can be divided into four steps, with the couplings between pairs of waveguides in each step depicted in Fig. 1(c). Although evanescent coupling between two adjacent microrings occurs only over a small segment of the ring waveguides, we will assume in the following analysis that the coupling is constant over the entire quarter length of the waveguide array, so that the coupling angle θ between a pair of waveguides can be expressed as θ = κL/4, where κ is the coupling strength per unit length. This assumption greatly simplifies the computation of the evolution operator of the system and, as long as the coupling is adiabatic, does not change the topological characteristics of the lattice.
The field evolution along the waveguide array can be described using the Coupled Mode Equations . These equations can be cast in a similar form as the Schrodinger equation in which the direction of propagation (the z-axis) takes the role of time. Specifically, the equation of motion for the field in waveguide A at lattice position (m, n) is given byEq. (1) and together with the equations for fields B, C and D, we can write the equation of motion for the state vector |ψ〉 of the unit cell in the form
We note that for the unidirectional microring square lattice with uniform coupling strength κ = κa considered in [15,16], the FB Hamiltonian is simply given by Haa. This system has been shown to support edge modes in the bulk bandgap for coupling angle θa > π/4. It is interesting to note that the same Hamiltonian Haa also describes the photonic Floquet insulator based on an array of periodically-coupled waveguides demonstrated in . In fact, by transforming the unidirectional microring lattice into an equivalent coupled waveguide array, it can be shown that these two systems are topologically equivalent in both real and reciprocal spaces.
3. Topological invariants of a 2D microring lattice
Knowledge of the FB Hamiltonian allows us to construct the evolution operator of the microring lattice, which is given byEq. (3) can be simply evaluated for each interval. Specifically, in the first interval, 0 ≤ z ≤ L/4, we have Fig. 1(a)). The Floquet operator U(k, L) of the system, which describes the stroboscopic evolution of the fields at discrete intervals of L (or every microring roundtrip) is given by
Since U(k, L) is a unitary operator, its eigenvalues are complex with unit modulus. The phase angles of the eigenvalues give the quasi-energy bands εn(k) of the microring lattice, which are periodic with period of 2π/L. Each quasi-energy band εn can be characterized by a topological invariant similar to that of a static system, which is given by the Chern number 
As discussed in , the Floquet operator is insufficient for characterizing the topological characteristics of a periodically-driven system. Instead, a complete knowledge of the system history as given by the evolution operator U(k, z) is required to determine the topological invariant of a Floquet insulator, which is given by the winding number of U(k, z) at quasi-energy ε in a bulk bandgap. Furthermore, the computation of the winding number in  requires that the Floquet operator be trivial, i.e., it must be equal to the identity (U(k, L) = I). Since the Floquet operator of the microring lattice in Eq. (7) is generally nontrivial, we periodize the evolution operator of the system by modifying it as19] 11]
In the next section, we investigate the bulk-edge correspondence of the microring lattice and show that the system exhibits both Chern and Floquet topological characteristics for certain ranges of the coupling angles θa and θb.
4. Edge states in a 2D microring lattice
Over the quasi-energy range 0 ≤ ε < 2π/L, an infinite microring lattice in general can have three bandgaps: a bandgap centered at ε = π/L (labeled II), and two symmetric bandgaps (labeled I) above and below bandgap II, as shown in the projected bulk band diagram in Fig. 2(a) for a lattice with coupling angles θa = 0.45π, θb = 0.05π. In general, bandgap II is open for all values of the coupling angles except for θa = θb ≤ π/4, while bandgap I increases with larger contrast between θa and θb. The map of coupling angles θa and θb for which bandgaps I and II are open is shown in Fig. 2(b). For convenience, we also label the quasi-energy bands of the lattice n = 0, 1, 2 in the 0 ≤ ε < 2π/L range, as shown in Fig. 2(a). From the Floquet operator of the lattice, we computed the Chern number associated with each quasi-energy band. We found that in general, the Chern number is always trivial for the band n = 0. However, for the cases where bandgap I is open, it can take on nontrivial values for the upper bands n = 1 and 2, indicating that the microring lattice can behave as a Chern insulator (CI).
We next investigated the Floquet topological characteristics of the microring lattice by computing the winding numbers WI and WII associated with the two bandgaps. Specifically, WII is computed for ε = π/L, while WI is computed at the quasi-energy ε coinciding with the middle of bandgap I when it is open. To highlight the different possible topological characteristics that can be observed in the microring lattice, we considered four different sets of coupling angles (θa, θb) : (0.2π, 0.1π), (0.3π, 0), (0.45π, 0.2π), and (0.45π, 0.05π). Figures 3(a)–3(d) show the corresponding band diagrams of microring lattice strips with boundaries in the y direction and infinite length in the x direction. The number of unit cells in the y direction is 10 for each structure. The band diagrams of the microring strips are computed using the method in Appendix. In Fig. 3, we also show the winding number of each bandgap and the Chern numbers of the Floquet bands for each lattice. We observe that while all four lattices have an open bandgap II, lattices (a) and (b) have trivial winding number (WII = 0) and thus do not support edge states in this bandgap. On the other hand, lattices (c) and (d) have nontrivial winding numbers (WII = 1) and thus support edge modes, as can be verified by the modes crossing bandgap II in the corresponding band diagrams. However, as indicated in the figures, the Chern numbers associated with the bulk bands of both lattices are all trivial, so the edge modes in bandgap II of these lattices cannot be of the Chern type but must be classified as anomalous Floquet insulator (AFI) edge modes.
With respect to bandgap I, we observe that it is closed for lattices (a) and (c) but open for lattices (b) and (d). For the latter two cases, the band diagrams show the existence of edge modes in the bandgap, which confirms the nontrivial winding numbers (WI = 1) obtained in bandgap I of these lattices. However, for lattice (b), the Chern number associated with bulk bands n = 1 and 2 are nontrivial, indicating that the edge modes in bandgap I are of the Chern type. On the other hand, the Chern numbers of the bands in lattice (d) are all trivial, so the edge modes in bandgap I must be classified as AFI edge modes.
We summarize the topological characteristics of the microring lattice by showing in Fig. 4 the map of coupling angles θa and θb for which CI and AFI edge modes are supported in bandgaps I and II. The four examples in Fig. 3 are also indicated on the map. We note that the special case θb = 0 (or θa = 0), which corresponds to the removal of microring D (or A) from the lattice, represents the microring lattice investigated in [15, 16], where the existence of edge states in bandgap II was predicted for coupling angle θa > π/4. However, the map in Fig. 4 shows that AFI edge modes exist only for coupling angles θa greater than 0.36π for θb = 0, although this threshold value of θa can be lowered if θb is increased above zero. We attribute this discrepancy between our results and those reported in [15, 16] to the fact that in these previous works, the microring lattice was assumed to support only one direction of field propagation and the phase responses of the microring resonators serving as “linking loops” were neglected in the analysis. This also explains why the existence of the AFI and CI edge modes in bandgap I were overlooked in these works. Our analysis shows that a microring lattice with alternate directions of field propagation exhibits much richer topological characteristics than a unidirectional lattice, as seen in Fig. 4.
To verify the existence of the edge modes in the microring lattice, we used the method in  to compute the field distribution in a lattice consisting of 10 × 10 unit cells with coupling angles θa = 0.45π, θb = 0.05π, which corresponds to the structure in Fig. 3(d). Light enters the lattice via an input waveguide evanescently coupled to microring A of a unit cell on the left boundary and exits the lattice via an output waveguide coupled to microring B of a unit cell on the right boundary. A diagram of the input and output coupling schemes is shown in Fig. 5 (a). Figures 5(b) and 5(c) show the field distributions for two chiral edge modes: an AFI edge mode excited by input light with quasi-energy ε = 0.85π/L in bandgap II, and a CI edge mode excited with quasi-energy ε = 0.4π/L in bandgap I. Both modes are seen to have similar field distributions, which show light propagating along the boundaries of the lattice with the field penetrating only into the outermost layer of unit cells.
5. Discussion and Conclusion
By transforming a 2D microring lattice into an equivalent array of coupled waveguides with periodic boundary conditions, we have shown that the microring lattice can emulate a quantum system driven by a periodic Hamiltonian in the direction of light propagation. Knowledge of the FB Hamiltonian allows us to obtain the complete evolution history of the system over each driving period, which is necessary for determining the topological invariants and the bulk-edge correspondence of the system. In particular, we showed that the lattice exhibits nontrivial winding numbers for certain ranges of the coupling strengths, and correlated these values to the existence of anomalous Floquet edge modes in a microring strip with boundaries.
The analysis presented in this paper can also be used to determine the FB Hamiltonians and compute the winding numbers of other periodic microring lattice configurations in both one and two dimensions. One example is the hexagonal microring lattice studied in , which was shown to also support both Chern type and AFI edge modes, although the topological invariants of the lattice as a Floquet system have not been determined. It is also possible to apply our method to study microring lattices with complex coupling coefficients, whose phases can be used to simulate an artificial gauge, as first proposed by Hafezi et. al. in . Such a lattice has only been studied as a static system using the tight binding approximation. By treating it as a periodically-driven system in a gauge potential, it may be possible to observe richer Floquet topological behaviors in such a lattice.
Given a 2D microring lattice, it is always possible to construct a coupled waveguide array that is topologically equivalent in both real and reciprocal spaces, although the converse is not necessarily true. For the cases where such equivalence exists between a microring lattice and a waveguide array, the topological properties of one system can be predicted by studying the other. For example, the 2D periodically-coupled waveguide array in Fig. 1(b) has the same topological phase map shown in Fig. 4 as the equivalent microring lattice. From the experimental point of view, however, microring lattices are better candidates for demonstrating Floquet insulator behaviors since it is easier to fabricate a microring lattice on an integrated optics platform than a 2D waveguide array .
The Floquet microring lattices proposed in this paper may be realized using 2D coupled micro-racetracks, similar to the configurations used to experimentally demonstrate 2D microring optical filters  and topological insulators . Lattices of coupled racetracks offer the flexibility in realizing the coupling angles θa and θb by varying both the coupling gaps and lengths of the coupling sections. In particular, the coupling angle between two racetracks is given by θ = κLc, where the per-unit-length coupling strength κ can be realized by adjusting the evanescent coupling gap between the racetrack waveguides and Lc is the effective length of the coupling segment. It should be noted that due to the time-reversal symmetry of the microring lattices, there is inevitable scattering into the counter-propagating modes of the microrings, which may compromise the topological characteristics of the lattice. Back scattering into the counter-propagating modes generally arises due to the sidewall roughness of the microring waveguides and mode transitions at the coupling junctions. It is possible to minimize this back scattering by careful design of the coupling junctions and optimization of the fabrication process to reduce sidewall roughness.
Appendix: Computation of the band diagram of a microring lattice strip
The band diagram of a microring lattice strip with Ny unit cells in the y direction can be computed in a simple manner using the coupled waveguide array representation. Figure 6 shows the couplings between pairs of waveguides in one column of unit cells for each of the four coupling steps. The quasi-energy bands of the microring strip are given by the eigenvalues of the Floquet operatorFig. 6, we find that the evolution operators for steps j = 1, 2, and 3 have the simple block diagonal form
Natural Sciences and Engineering Research Council (NSERC) of Canada.
References and links
1. D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48(22), 1559 (1982). [CrossRef]
2. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7(11), 907 (2011). [CrossRef]
3. N. H. Lindner, G. Refael, and V. Galitski, “Floquet topological insulator in semiconductorquantum wells,” Nat. Phys. 7(6), 490 (2011). [CrossRef]
4. A. B. Khanikaev, S. H. Mousavi, W-k Tse, M. Kargarian, A. H. Macdonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233 (2013). [CrossRef]
5. J. M. Zeuner, M. C. Rechtsman, Y. Plotnik, Y. Lumer, S. Nolte, M. S. Rudner, M. Segev, and A. Szameit, “Observation of a topological transition in the bulk of a non-hermitian system,” Phys. Rev. Lett. 115(4), 040402 (2015). [CrossRef] [PubMed]
6. S. Barik, H. Miyake, W. DeGottardi, E. Waks, and M. Hafezi, “Two-dimensionally confined topological edge states in photonic crystals,” New J. Phys. 18(11), 113013 (2016). [CrossRef]
7. W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5, 5782 (2014). [CrossRef] [PubMed]
8. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics. 7(12), 1001 (2013). [CrossRef]
9. A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78(19), 195125 (2008). [CrossRef]
10. T. Kitagawa, E. Berg, M. Rudner, and E. Demler, “Topological characterization of periodically driven quantum systems,” Phys. Rev. B 82(23), 235114 (2010). [CrossRef]
11. M. S. Rudner, N. H. Lindner, E. Berg, and M. Levin, “Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems,” Phys. Rev. X 3(3), 031005 (2013).
12. K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photonics. 6(11), 782 (2012). [CrossRef]
13. M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, “Photonic Floquet topological insulators,” Nature 496(7444), 196 (2013). [CrossRef] [PubMed]
16. M. Pasek and Y. D. Chong, “Network models of photonic floquet topological insulators,” Phys. Rev. B 89(7), 075113 (2014). [CrossRef]
17. A. Tsay and V. Van, “Analytic theory of strongly-coupled microring resonators,” IEEE J. Quantum Electron. 47(7), 997–1005 (2011). [CrossRef]
18. A. Yariv, Optical electronics, 4th ed. (Saunders, 1991).
19. D. Carpentier, P. Delplace, M. Fruchart, and K. Gawedzki, “Topological index for periodically driven time-reversal invariant 2D systems,” Phys. Rev. Lett. 114(10), 106806 (2015) [CrossRef] [PubMed]