Trapping light in a Floquet topological photonic insulator by Floquet defect mode resonance

Topological photonic insulators (TPIs) provide a rich playground for exploring both the physics of periodic systems and applications of their exotic properties.^1,2 In particular, Floquet TPIs characterized by periodically varying Hamiltonians^3,4 have recently gained much attention as they can exhibit richer topological behaviors than static undriven systems, such as the existence of anomalous Floquet insulator (AFI) edge modes even though the energy bands of the lattice have a trivial Chern number.^5–9 In a periodically driven system, the evolution of the phase bands of the Floquet–Bloch modes, defined as the phases of the eigenvalues of the system’s evolution operator, plays a crucial role in determining its topological behaviors.^10,11 Here, we investigate the possibility of manipulating the cyclic phase change in a Floquet mode to induce self-interference and resonance effects in the lattice bulk.

The ability to form robust high quality factor resonators in a topological lattice is of practical interest as it would significantly broaden the range of applications of TPIs such as in lasers, filters, nonlinear cavity optics, and quantum optics.^12–17 In 2D TPI lattices, travelling-wave resonators can be realized by exploiting the confinement of edge modes at the interface between topologically trivial and nontrivial insulators to form ring cavities, although these tend to have very long cavity lengths as they require many lattice periods.^18–22 Topological resonators can also be realized by creating line defects²³ or point defects in the lattice bulk, e.g., by spatially shifting air holes in a photonic crystal to create a Dirac-vortex topological cavity.²⁴ The resonance mode is pinned to the midgap and can be regarded as the 2D counterpart of 1D resonance modes in distributed feedback lasers²⁵ 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 of the different parity modes inside and outside the cavity. The cavity mode is a bulk mode located at the Γ point of the energy band diagram very close to the edge of the topological bandgap. Recently, it was shown that topological corner states with zero energy can also be used to form resonances in a TPI.^28–35 However, the mode is not tunable and can only be formed at the corners of the lattice. Indeed, all the topological photonic resonators reported to date are not continuously tunable and have only been realized for static TPI systems.

Here, we report a new mechanism for forming resonance in a Floquet TPI by adiabatically tuning the cyclic phase of a Floquet mode to achieve constructive interference. This has the concomitant effect of shifting its quasienergy into a topological bandgap to form an isolated flat-band state that is spatially localized in a bulk-mode resonant loop, which we refer to as Floquet Defect Mode Resonance (FDMR). The resulting resonance effect can be regarded as a Floquet counterpart of the defect-mode cavity in static systems, with the main difference in that here we perturb the driving sequence rather than introducing a static defect such as a point or a line in the lattice.^{23,24,36–46} In particular, the perturbation in our lattice is drive-dependent and varies periodically with the Floquet system evolution. The ability to modify the driving sequence locally provides an additional degree of freedom for controlling the resonance mode that is not afforded in static systems. Notably, we found that the spatial localization pattern of the FDMR is determined by the perturbed driving sequence of the Floquet TPI and is distinctly different for trivial and nontrivial topological lattices. We also note that while drive-dependent defects have been used to investigate the robustness of edge modes in 2D Floquet TPIs based on coupled waveguides,⁴⁷ the perturbation of the Floquet–Bloch Hamiltonian to manipulate both the cyclic phase change and spatial localization of a Floquet mode to create a resonance has not been reported before.

A notable feature of the FDMR is that it is cavity-less since it does not require creating physical boundaries in the lattice. The lack of scattering from interface discontinuities means that FDMRs can potentially have very high Q factors. The resonance can be formed anywhere in the lattice bulk and can be continuously tuned across a topological bandgap. We demonstrated FDMR in a Floquet octagon lattice realized in silicon-on-insulator (SOI) using a thermo-optic heater to excite and tune the resonance. Imaging of the scattered light intensity pattern reveals the hopping sequence of the Floquet mode and provides direct evidence of FDMR localized in a bulk-mode loop. We achieved an extrinsic Q-factor of ∼1.7 × 10⁴, which is among the highest reported to date for 2D topological photonic resonators.⁴⁸ Our work thus introduces a new, versatile method for forming high-Q resonances in a Floquet lattice, which could have a wide range of applications in cavity optics.

The Floquet TPI we consider is a 2D square lattice of coupled microring resonators with identical resonance frequencies. The lattice is characterized by two different coupling angles (θ_a > θ_b) in each unit cell [Fig. 1(a)], where θ_a (θ_b) represents the strong (weak) coupling between resonator A (D) and its neighbors. Although 2D microring lattices have been shown to exhibit Chern insulator behavior associated with static systems,^49–52 the existence of FDMRs can only be predicted by accounting for the internal circulation of light in each microring and treating the system as a Floquet insulator with periodically varying Hamiltonian. As light circulates around each microring, it couples periodically to its neighbors so that the Bloch modes of the lattice evolve in a cyclical motion with a period equal to the microring circumference L. The quasienergy spectrum of the lattice thus has a periodicity of 2π/L. Within each Floquet–Brillouin zone, the microring lattice, in general, has three bandgaps [Fig. 2(a)], which can exhibit different topological phases, including AFI behavior, depending on the coupling angles (θ_a, θ_b).⁵³ To better elucidate the Floquet nature of the lattice, we transform it into an equivalent 2D array of periodically coupled waveguides⁵³ [Fig. 1(b)], with each period consisting of four coupling steps between different pairs of adjacent waveguides. In the limit of perfect coupling (θ_a = π/2, θ_b = 0), the hopping sequence guarantees that light starting from site A in a unit cell will return to its position after three periods, tracing out a bulk-mode loop depicted in Fig. 1(c). However, in a uniform Floquet lattice, such a bulk mode does not exist in a bandgap since its phase change around the loop is not equal to an integer multiple of 2π. We also note that although our analysis here is for a system periodically driven in space (along z) as emulated by the microring lattice, the same treatment and observations can also be applied to Floquet systems periodically driven in time.

Suppose that we now perturb the driving sequence by introducing a phase shift Δϕ in coupling step j of a microring in the lattice [Fig. 1(b)]. Taking a block of N × N unit cells with the perturbed microring located near its center, for sufficiently large N, we can treat this block as a supercell of an infinite periodic lattice. Using the coupled-waveguide array model, we can write the equation of motion of the supercell as

where k is the crystal momentum in the x–y plane, $HFB(j)$ is the Floquet–Bloch Hamiltonian of the unperturbed supercell in step j,⁵³ and $HD(j)$ is the perturbed Hamiltonian in the same step (see the supplementary material for the derivation of the Hamiltonian). The perturbed Hamiltonian matrix is zero everywhere except for a term of −4Δϕ/L in its kth diagonal element corresponding to the detuned microring k. Any state of the system evolves as |ψ(k, z)⟩ = U(k, z)|ψ(k, 0)⟩, where

is the evolution operator with $T$ being the time-order operator defined as $Te−i∫H(z′)dz′=limδz→0e−iH(z)δze−iH(z−δz)δze−iH(z−2δz)δz…e−iH(0)δz$⁠. The evolution over each roundtrip period of the microrings is given by the Floquet operator, U_F(k) = U(k, L), whose eigenstates are the Floquet modes |Φ_n(k, 0)⟩ with eigenvalues $e−iεn(k)L$⁠. In the absence of detuning (Δϕ = 0), the quasienergy bands ε_n(k) of the Floquet modes form composite transmission band manifolds, each containing 4N² degenerate bulk modes and separated by bandgaps. The effect of the phase detune Δϕ is to break the degeneracy and lift one Floquet mode into the bandgap, forming an isolated single band [Fig. 2(a)] that is also periodic in the frequency domain. Moreover, this energy-shifted band becomes increasingly flattened as the phase detune is increased, implying that the field distribution becomes more strongly localized spatially. Importantly, the spatial localization pattern depends on how the driving sequence is perturbed. For example, Figs. 2(b)–2(d) show the field distributions of the isolated Floquet mode when each of microrings A, C, and D, respectively, is detuned during step j = 1. When microring A is detuned, the field is localized in two coupled bulk-mode loops sharing the common segment j = 1. By contrast, detuning microring C results in the field strongly localized in only a single bulk-mode loop traced out by the hopping sequence. A similar mode pattern is also observed when segment j = 2 of microring B is detuned. When the weakly coupled microring D is detuned, light does not follow the driving sequence but instead remains trapped in the same site resonator, forming a point-defect mode. Interestingly, in a topologically trivial lattice, such a point-defect mode is always excited regardless of which microring is detuned (see the supplementary material for a comparison between FDMR and point-defect state in a Floquet microring lattice). Thus, by selectively applying phase detunes to specific steps in the driving sequence, distinct mode patterns can be excited to form single or coupled resonant loops. This highlights a key difference between FDMR and conventional defect-mode resonance in a static system, where the defect is introduced as a constant perturbation, so the system still remains undriven. The ability to vary the perturbation along the path of system evolution provides an additional degree of flexibility for controlling the spatial localization of the resonance mode. In particular, we emphasize that although our microring lattice with a phase detune can be treated as a static system using a mean field theory in which the phase perturbation is averaged uniformly over the detuned microring, the defect mode in this case will appear only as a point defect, as shown in Fig. 2(d). The bulk-mode loop patterns in Figs. 2(b) and 2(c) can only be predicted by taking into account the exact details of how the driving sequence is perturbed. We also note that unlike conventional defect-mode resonance in static systems, the spatial localization pattern of the FDMR extends far beyond the location of the defect, spanning over many site resonators from the perturbed microring segment, with the field pattern defined by the hopping sequence.

The strong field localization in a bulk-mode loop can be regarded as a resonance effect caused by the energy-shifted Floquet mode constructively interfering with itself after completing each roundtrip around the loop. Starting out each cycle at z = 0, the shifted Floquet mode |Φ_s(k, 0)⟩ evolves as |Ψ_s(k, z)⟩ = U(k, z)|Φ_s(k, 0)⟩. According to the Floquet theorem, the state |Ψ_s(k, z)⟩ can also be expressed as¹¹ 

where $|Φs(k,z)〉=eiεs(k)zU(k,z)|Φs(k,0)〉$ is the periodic z-evolved Floquet state satisfying |Φ_s(k, z + L)⟩ = |Φ_s(k, z)⟩. The state |Ψ_s(k, z)⟩ will constructively interfere with itself after every period L if |Ψ_s(k, z + L)⟩ = |Ψ_s(k, z)⟩ or

Since |Φ_s(k, z + L)⟩ = |Φ_s(k, z)⟩, we obtain the condition for constructive interference as ε_s(k)L = 2mπ, $m∈Z$⁠. Using the quasienergy for a stationary Floquet mode at k = 0, we can calculate the shift in the resonant frequency of the FDMR relative to a microring resonance as Δω_s = ε_s(0)LΔω_FSR/2π, where Δω_FSR is the free spectral range (FSR) of the microrings. Figure 3(a) plots the dependence of the cyclic phase change ε_s(0)L on the phase detune Δϕ, showing that the resonant frequency of an FDMR can be continuously tuned across a topological bandgap. The above analysis supports the picture that the FDMR is formed by the constructive interference of a Floquet bulk mode with itself and that by tuning the cyclic phase of the mode, we can vary its quasi-energy to create a resonance localized in both spatial and frequency domains in an otherwise homogeneous topological lattice.

We note that this resonance effect is cavity-less since it does not require physical boundaries between the lattice and another medium but instead relies on an adiabatic change in the Hamiltonian via a phase detune. Since no interface scattering takes place, FDMRs can, in principle, have very high Q factors. Importantly, since the phase detune Δϕ represents a local adiabatic change to the Hamiltonian H_FB, the energy-shifted band still retains the topological properties of the unperturbed lattice. This is evident from the fact that the FDMR mode [Fig. 2(c)] retains the same spatial distribution of a bulk mode in a homogeneous lattice as we increase the phase detune. In addition, the bandgaps above and below the FDMR still support edge modes, implying that the topological behavior of the lattice is not altered by the adiabatic phase detuning. In particular, simulations show that the FDMR remains robust to random variations in the lattice (see the supplementary material).

We can quantify the degree of spatial localization of an FDMR by computing its inverse participation ratio (IPR).⁵⁴ For a z-evolved Floquet mode with normalization ⟨Φ_s(0, z)|Φ_s(0, z)⟩ = 1, we can define the average IPR over one evolution period as

where $Φs(k)$ is the field in site resonator k in the lattice. Figure 3(b) shows the average IPR of an FDMR (in bandgap III) as a function of the phase detune for different coupling angles of the lattice. It is seen that the mode becomes more strongly localized as it is pushed deeper into the bandgap. Thus, in general, we can expect to achieve the strongest intensity enhancement for FDMRs located near the center of the bandgap. The degree of localization is also higher for lattices with larger contrast between the coupling angles θ_a and θ_b. We note that the maximum $IPR̄$ achievable for FDMR is 1/3 because at any given position z in an evolution cycle, the field is localized in three separate microrings in the bulk-mode loop.

We demonstrated FDMR in a Floquet TPI lattice consisting of a square array of coupled octagon resonators on an SOI substrate. Each unit cell consisted of four identical octagons evanescently coupled to their neighbors via identical coupling gaps g [Fig. 4(a)]. Octagon resonators were used to realize dissimilar coupling angles θ_a and θ_b in each unit cell by exploiting the difference between synchronous and asynchronous couplings. This is achieved by designing the octagons to have sides with equal lengths L_s but alternating widths W₁ and W₂, with octagon D rotated by 45° with respect to the other three octagons. Octagon A can thus be strongly coupled to its neighbors via synchronous coupling between waveguides of the same width W₁, while octagon D is weakly coupled to its neighbors via asynchronous coupling between waveguides of dissimilar widths W₁ and W₂. In this way, all site resonators in each unit cell have identical resonance frequencies, but the hopping amplitudes between resonators A and D to their respective neighbors can be made different. We designed the coupling angles to be θ_a = 0.458π and θ_b = 0.025π so that the lattice exhibits AFI behavior for TE polarized light in its three bandgaps over one FSR at the telecommunication wavelengths⁹ (see Sec. V for the details of the lattice design). The fabricated lattice consisted of 10 × 10 unit cells [Fig. 4(b)]. An input waveguide was coupled to resonator A of a unit cell on the left boundary of the lattice to excite AFI edge modes, and an output waveguide was coupled to resonator B on the right boundary to measure the transmission spectrum.

Figure 5(a) (red trace) shows the transmission spectrum measured for input TE light over one FSR (∼5 nm) of the resonators around 1615 nm wavelength (see Sec. V for the measurement setup). Three distinct bands of high transmission (labeled I–III) due to AFI edge mode propagation can be seen, which correspond to the topologically nontrivial bulk bandgaps of the Floquet lattice. Imaging of the scattered light intensity distribution at the 1612.833 nm wavelength (in bandgap III) using a near-infrared (NIR) camera [Fig. 5(b)] confirms the formation of an edge mode propagating along the lattice boundary from the left input waveguide to the right output waveguide.

Since the FDMR exists in a bulk bandgap, we can couple light into it using an AFI edge mode in the same bandgap. To excite an FDMR near the left boundary of the lattice, we thermo-optically tuned the phase of an octagon C on the left boundary [Fig. 4(b)] using a titanium–tungsten (TiW) heater fabricated on top of the resonator (detuning the whole resonator C at the edge also excites only a single bulk-mode loop). Figure 5(a) (blue trace) shows the transmission spectrum when a phase detune of 1.45π (corresponding to electrical heating power P = 34.9 mW) was applied to the octagon (see Sec. V for the heater calibration method). We observe that the spectrum is almost identical to the spectrum without phase detune (red trace), except for the presence of two sharp dips located in bulk bandgaps I and III. These dips indicate the presence of an FDMR excited in each bulk bandgap by the edge mode. The absence of a dip in bandgap II is likely due to loss in the resonators and the fact that the coupling between the edge mode and the FDMR, which, in general, depends on the resonance spectrum of the detuned microring, is much weaker in this frequency range compared to bandgaps I and III. To obtain visual confirmation of the spatial localization of the FDMR, we performed NIR imaging of the scattered light intensity at the resonance wavelength of 1612.833 nm (in bandgap III) [Fig. 5(c)]. The image clearly shows that light is localized and trapped in a bulk-mode loop, which is not present in Fig. 5(b) when no phase detune was applied. The bulk mode pattern directly captures the hopping sequence of the 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 the FDMR and couples to it. We also note that transmission dips occurring in the bulk transmission bands of the lattice, which appear with and without phase detuning, are caused by random interference of light propagating deep into the lattice bulk. Imaging of light intensity patterns at these wavelengths in the transmission band does not show light localized in FDMR loops.⁹ 

Focusing on the FDMR in bandgap III, we measured the resonance spectrum for different phase detune values. The spectra are plotted in Fig. 6(a), showing that as the phase detune is increased, the FDMR spectrum is pushed deeper into the bandgap. The resonance linewidth also becomes narrower, while the extinction ratio reaches a maximum of almost −40 dB near the bandgap center. In Fig. 6(b), we plotted the resonant wavelength shift Δλ as a function of the phase detune, with the corresponding heater power shown on the top horizontal axis. The linear relationship between Δλ and Δϕ is in agreement with the theoretically predicted dependence of the FDMR quasienergy on the phase detune [Fig. 3(a)].

The dependence of the Q factor of the FDMR on the phase detune is shown in Fig. 6(c) (black circles). We obtained Q values in the range 1.2 × 10⁴–1.7 × 10⁴, with a slight increasing trend as the FDMR moves deeper into the bandgap. For comparison, the intrinsic Q factor of a single resonator obtained from the measurement of a stand-alone octagon (see the supplementary material) was only slightly higher at 2.6 × 10⁴ (corresponding to roundtrip loss of 0.35 dB). Using the designed coupling values (θ_a = 0.458π, θ_b = 0.025π) for the lattice and a slightly higher roundtrip loss of 0.59 dB in each octagon, we simulated FDMR spectra for various phase detunes and obtained the corresponding extrinsic Q factors [red line in Fig. 6(c)], which show good agreement with the measured values. Larger discrepancies between simulated and measured Q factors are observed for smaller phase detunes, which can be attributed to the fact that the FDMR and edge mode are less localized near the band edge and are thus more susceptible to lattice imperfections. From the measured extrinsic Q factors Q_ex, we can calculate the effective coupling (μ) between the FDMR and the edge mode as μ = ω₀(1/Q_ex − 1/Q₀), where ω₀ is the resonant frequency and Q₀ = 2.6 × 10⁴ is the intrinsic Q factor. The results are also plotted in Fig. 6(c) (blue circles). The coupling rate μ depends on the overlapping between the field distributions of the AFI edge mode and the FDMR. This dependence is seen to correlate with the variation in the degree of spatial localization of the FDMR as indicated by the plot of $IPR̄$ vs Δϕ in Fig. 3(b). As the FDMR is pushed deeper into the bandgap, it becomes more strongly localized spatially so that its coupling to the edge mode is weaker, which results in the higher Q factor. Using the designed coupling angle values, simulations of the lattice showed that the theoretical extrinsic Q factor (without loss) can exceed 10⁶, suggesting that the experimental Q factor can be improved by reducing the roundtrip loss in the FDMR loop, for example, by reducing scattering from the octagon corners and using materials with lower absorption.

We report a new method for trapping light in a Floquet TPI lattice by adiabatically tuning the cyclic phase of a Floquet mode to induce self-interference. The spatial localization pattern of the FDMR is shown to depend on how the driving sequence of the Floquet lattice is perturbed. The spatial pattern is also found to be distinctly different for topologically trivial and nontrivial bandgaps, suggesting that there may be a connection between the topological invariant of the bulk lattice and the spatial localization of the FDMR beyond the bulk-defect correspondence,^37,43,44 which only predicts the number of static defect modes but not the spatial patterns. We note that it is also possible to perturb the coupling angles in the driving sequence, which may provide an additional mechanism for engineering the FDMRs, for example, to control the coupling strength to the edge mode or between coupled FDMR loops.

Compared to other topological resonators, the FDMR is cavity-less and tunable and can be formed anywhere in the lattice bulk. The lack of physical cavity boundaries suggests that very high Q factors can potentially be achieved. The resonance can also be dynamically switched on and off, which could be useful for realizing optical switches and modulators. In addition, our preliminary experimental results have shown that multiple adjacent FDMRs could be excited to form coupled cavity systems, which could open up new applications such as high-order coupled-cavity filters, optical delay lines, and light transport in a bandgap through the lattice bulk by hopping between adjacent localized bulk modes.

The Floquet octagon lattice was realized on an SOI substrate with a 220 nm-thick Si layer lying on a 2 μm-thick SiO₂ buffer layer with a 2.2 μm-thick SiO₂ overcladding layer. The octagon resonators had sides of length L_s = 16.08 μm and alternating widths of W₁ = 400 nm and W₂ = 600 nm. The corners were rounded using arcs of radius R = 5 μm to reduce scattering. The coupling gaps between adjacent octagons were fixed at g = 225 nm. From numerical simulations using the finite-difference time-domain solver in Lumerical software,⁵⁵ we obtained θ_a = 0.458π and θ_b = 0.025π for the synchronous and asynchronous coupling angles, respectively, around 1615 nm wavelength, which correspond to AFI behavior of the microring lattice according to the topological phase map in Ref. 53. For a conservatively estimated variation of ±5 nm in the waveguide widths of the couplers due to fabrication, the corresponding deviation in the coupling angle is simulated to be ±0.042π for θ_a and ±0.015π for θ_b. To excite AFI edge mode and measure the transmission spectrum, input and output waveguides with 400 nm width were coupled to their respective octagon resonators on the left and right boundaries of the lattice via the same coupling gap g = 225 nm. The 10 × 10 lattice was fabricated using the Applied Nanotools SOI process.⁵⁶ 

Transmission spectra of the lattice were measured using a tunable laser source (Santec TSL-510 1510–1630 nm) with the polarization adjusted to TE before being coupled to the input waveguide through a lensed fiber. The transmitted light was also collected with a lensed fiber for measurement with a photodetector and powermeter. Imaging of the scattered light intensity patterns from the chip was performed using a 20× objective lens and an InGaAs NIR camera. The image sensor of the camera had 640 × 512 pixel² with a 15 μm pitch.

We excited an FDMR and tuned its resonant frequency by varying the phase of an octagon C on the left boundary of the lattice. A titanium–tungsten (TiW) heater covering the octagon perimeter [Fig. 4(b)] was fabricated on top of the resonator, and current was applied to the heater to tune its phase. Figure 6(b) shows that the measured resonant wavelength shift Δλ of the FDMR varies approximately linearly with the applied heater power P. Using the fact that the roundtrip phase detune Δϕ of the octagon resonator also varies linearly with the heater power, we can correlate the measured Δλ vs P plot with the simulated Δλ vs Δϕ plot across the bandgap. This allows us to calibrate the heater efficiency and deduce the linear correspondence between the phase detune and the heater power. The relationship between Δϕ and P is explicitly shown on the top and bottom horizontal axes of Fig. 6(b).

See the supplementary material for the Floquet–Bloch Hamiltonian of the microring lattice, comparison between FDMR and point-defect state in a Floquet microring lattice, numerical investigation of robustness of FDMR in the presence of disorder, and loss measurement of a stand-alone octagon resonator.

This work was supported by the Natural Sciences and Engineering Research Council of Canada.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.