Broadband resonance-enhanced frequency generation by four-wave mixing in a silicon Floquet topological photonic insulator

Floquet topological insulators^1,2 based on periodically driven quantum systems can exhibit much richer topological behaviors than static systems, such as the existence of an anomalous Floquet insulator (AFI) phase not observed in time-independent topological insulators. In the optics domain, Floquet topological photonic insulators (TPIs) have been demonstrated using coupled waveguide arrays^3–6 and two-dimensional (2D) microring lattices.^7–11 These materials can potentially have practical applications in realizing photonic devices that are robust to imperfections by exploiting the topological protection property of the edge modes.^12–14 In addition, it is possible to control the flow of light through the lattice by perturbing the drive sequence of the system, which can lead to localized defect modes whose spatial intensity distributions are dictated by the hopping sequence of the lattice. In particular, we have recently shown¹⁵ that introducing a phase detune to a part of the drive sequence of a 2D Floquet TPI microring lattice can excite a Floquet mode spatially localized in a loop pattern with quasi-energy located in a topologically nontrivial bulk bandgap, leading to a resonance effect called Floquet Defect Mode Resonance (FDMR). Here, we exploit the periodicity of the quasienergy of the Floquet defect mode and its low group velocity dispersion in a 2D microring lattice to demonstrate resonance-enhanced four-wave mixing (FWM) over several Floquet–Brillouin zones. While nonlinear self-phase modulation effects have been investigated for soliton propagation in a 2D Floquet TPI based on periodically coupled waveguide arrays,¹⁶ our work represents the first investigation of parametric nonlinear interactions of Floquet bulk modes in a 2D topological resonator with Kerr nonlinearity, leading to enhanced broadband frequency generation on a Floquet topological platform.

Parametric processes based on third-order nonlinearity have important applications such as frequency generation, wavelength conversion, and correlated photon-pair generation. These processes have been demonstrated in various conventional integrated optics devices, such as FWM in silicon waveguides,¹⁷ photon pair generation in photonic crystal waveguides,¹⁸ and third-harmonic generation in chalcogenide photonic crystal waveguides.¹⁹ With the emergence of nonlinear topological photonics, there has been growing interest in achieving these parametric processes on a TPI platform, which could exploit the topological protection of edge modes to realize robust wavelength sources and frequency converters.^{9,12,14,20–23} For example, one-dimensional arrays of silicon nanodisks emulating the generalized Su–Schrieffer–Heeger model have been used to obtain enhanced third-harmonic generation at telecom frequencies through Mie resonances and the topological localization of edge states.²² Plasmonic metasurfaces consisting of 2D hexagonal arrays of nanoholes in graphene sheets under static magnetic fields have been demonstrated to form Chern insulator edge states, which were used to achieve FWM in the 13 THz frequency range.²³ Spontaneous FWM using edge modes in a Chern insulator based on the 2D lattice of silicon ring resonators has also been demonstrated for correlated photon pair generation.⁹ More recently, entangled photon emitters have also been achieved by FWM in a silicon AFI microring lattice.¹⁴ It should be noted that in these studies, the nonlinear processes occur via the propagation of edge modes in topological insulators, which require long lattice lengths for efficient frequency generation.

Various methods for improving the conversion efficiency (CE) of FWM have been demonstrated, including slow-light waveguides,^24,25 resonant field enhancement,^26–29 and graphene-enhanced resonators.^30–32 Due to their compact sizes and high quality factors (Q-factors), microring resonators are particularly promising for achieving highly efficient frequency generation due to strong nonlinear interactions of the resonance modes, with reported CE’s of −26 dB in high-Q Hydex-in-silica resonators^27,28 and −25 dB in silicon-on-insulator (SOI) resonators.²⁹ In order to achieve maximum CE enhancement due to resonance, the pump, signal, and idler waves must be aligned with the cavity resonances, while simultaneously satisfying the phase matching condition 2β_p = β_s + β_i, where β_p, β_s, and β_i are the propagation constants of the pump, signal, and idler, respectively, in the resonator. However, dispersion of the resonant modes causes a phase mismatch, defined as Δβ = 2β_p − β_s − β_i, which increases with the separation between the signal and idler waves, causing a precipitous degradation in the CE.³³ For a fixed pump frequency ω_p, the phase mismatch can be estimated as Δβ ≈ −β₂(ω_p)Δω, where β₂ = d²β/dω² is the group velocity dispersion (GVD) and Δω is the spectral separation between the signal and the pump. Thus, a requirement for achieving broadband wavelength conversion is that the resonance modes must have low GVD.

In this paper, we exploit the high Q-factor and low dispersion of FDMRs in a silicon Floquet microring lattice to achieve efficient broadband wavelength conversion by FWM of Floquet bulk modes. A unique feature of the FDMR is that it is cavity-less, since its spatial localization pattern is defined not by any lattice boundaries or interfaces but instead by the hopping sequence of a Floquet bulk mode. The lack of physical interfaces enables the FDMR to have a very high Q-factor, with values as high as $∼105$ for our silicon structure, which is the largest reported Q-factor for a 2D topological resonator. Combined with the low group velocity dispersion of the Floquet bulk mode, we achieved broadband wavelength conversion over 10.1 nm (from signal to idler wave), with an average CE enhancement of 12.5 dB due to the resonance effect. Our work could lead to efficient broadband and tunable wavelength conversion, parametric amplification, and correlated photon pair generation directly on a topologically protected photonic platform.

Floquet defect mode resonance can be regarded as the Floquet counterpart of point-defect resonance in static lattices, except that here the resonance mode arises from a periodic perturbation applied to the drive sequence of the Floquet lattice.¹⁵ The Floquet TPI we consider here is a 2D square microring lattice, which has been recently explored,^10,34 with each unit cell consisting of four identical microrings characterized by two coupling angles (θ_a, θ_b) as shown in Fig. 1(a). The lattice exhibits Anomalous Floquet Insulator (AFI)^4,35 behavior for $θa2+θb2⪆π2/8$⁠. Here, we remove microring D by setting the coupling angle θ_b to 0, which helps in reducing the coupling loss and propagation loss in the lattice, allowing high-Q and strongly localized FDMRs to be realized for enhanced nonlinear interactions. The resulting lattice, also shown in Fig. 1(a), is thus characterized by a single coupling angle θ_a between three resonators in each unit cell and resembles a proposed AFI microring lattice.³⁶ 

The Floquet nature of the lattice arises from the periodic circulation of light in each microring, with the driving Hamiltonian describing the sequence of couplings to neighbor resonators over one roundtrip (see the supplementary material for details). We also note that unlike a similar microring lattice used to realize Chern insulator behavior,⁸ our lattice does not use off-resonance link rings as couplers between site resonators but, instead, employs direct evanescent coupling between neighbor resonators. This allows stronger coupling with lower dispersion to be achieved, so that our Floquet TPI exhibits nearly identical nontrivial bandgaps over many Floquet–Brillouin zones, which is necessary for achieving broadband wavelength conversion by FWM.

We designed our microring lattice to have coupling angle θ_a = 0.441π so that it exhibits AFI behavior in all its three bandgaps over each Floquet–Brillouin zone.³⁴ Figure 1(b) shows the projected quasienergy band diagram, plotted as ɛL, where ɛ is the quasienergy and L is the roundtrip length of the microrings, within one Floquet–Brillouin zone of a lattice with boundaries along the x-direction. Each Floquet–Brillouin zone also corresponds to one free spectral range (FSR) of the microrings. The black lines trace the quasienergies of the transmission bands and edge modes crossing three topologically nontrivial bandgaps. The simulated intensity distribution of the edge mode at quasienergy ɛ = 1.7π/L (located in bandgap III) is shown in Fig. 1(c), which confirms light spatially localized along the bottom lattice boundary.

The FDMR is induced by applying a phase detune Δϕ to the bottom three-quarters of a microring located on the bottom edge of the lattice [marked by a red star in Fig. 1(a)]. The phase detune introduces a shift in the on-site potential of the defect resonator during three-quarters of each evolution period. This perturbation has the same periodicity of the driving sequence, since light experiences this phase detune each time it completes a roundtrip around the defect microring. The effect of the perturbation is to push a Floquet bulk mode from each transmission band into the bandgap below, forming a flat-band state with a quasienergy shift proportional to the amount of phase detune. These flat bands are depicted by the horizontal color gradients in the bandgaps in Fig. 1(b). The simulated spatial intensity pattern of the FDMR at quasienergy ɛ = 1.7π/L is shown in Fig. 1(d), which shows light traveling in a loop pattern and constructively interfering with itself to form a strong resonance.¹⁵ We note that this loop pattern is not due to light confinement by any physical lattice interfaces but follows the hopping sequence of a Floquet bulk mode in the microring lattice and extends far beyond the defect site resonator. Due to the periodic nature of the Floquet TPI, the FDMRs are also periodic in quasienergy, which allows resonance-enhanced FWM to be achieved over multiple Floquet–Brillouin zones.

We fabricated a Floquet microring lattice with 10 × 10 unit cells on an SOI substrate, an image of which is shown in Fig. 2. The microring waveguides were designed for the fundamental transverse-electric (TE) mode with 450 nm width and 220 nm height, lying on a 2 μm-thick SiO₂ substrate and covered by a 3 μm-thick SiO₂ cladding. Each microring was designed to have a square shape to allow for strong coupling between adjacent resonators, which is necessary to realize the AFI phase. The side lengths of the square were 19.64 μm with a coupling gap of 180 nm between neighbor resonators, which provides a large coupling angle θ_a with low dispersion. The corners of the square were rounded with 5 μm radius to reduce scattering loss. The overall circumference of each microring was chosen to provide sufficiently large FSR (∼5 nm) to observe distinct topological bandgaps. A close-up image of a single unit cell is provided in Fig. 2. Simulations showed that the coupling angle remained relatively constant across multiple microring FSRs, with values of 0.431π, 0.441π, and 0.450π at the signal (1524.1 nm), pump (1529.1 nm), and idler (1534.2 nm) wavelengths, respectively, used to demonstrate FWM.

To excite an edge mode along the bottom boundary of the lattice, an input waveguide was coupled to a microring at the bottom left corner and the transmitted light was collected via an output waveguide coupled to a microring at the bottom right corner. A metallic heater with a resistance of R ∼ 300 Ω was used to apply a phase detune to three-quarters of a resonator on the bottom edge of the lattice via the thermo-optic effect, thereby generating an FDMR, which is coupled to the AFI edge mode propagating along the bottom boundary.

We first measured the transmission spectrum of the AFI edge mode by sweeping the wavelength of an input TE-polarized laser source with 3 μW of on-chip power over several FSRs of the microrings and measuring the transmitted power at the output waveguide (see the supplementary material for details of the measurement setup). The result is presented by the red trace in Fig. 3(a), which shows three distinct wavelength regions in each FSR with relatively high and flat transmission due to edge mode propagation. We identify these regions as the three topological bandgaps of the lattice, as predicted by the projected band diagram in Fig. 1(b). Figure 1(e) shows a near-infrared image of the scattered light intensity of the lattice at 1511.97 nm wavelength, which confirms the presence of an edge mode in this bandgap propagating along the bottom boundary. We next applied 17.8 mW of electrical power to the heater to induce a phase detune in the defect ring. The measured transmission spectrum at the output waveguide is presented by the blue trace in Fig. 3(a), which is identical to the red trace except for the appearance of three sharp resonance dips. These dips correspond to an FDMR mode formed in bandgap III of each microring FSR. We note that the FDMR mode in bandgap I is not visible due to weak coupling to the edge mode, while bandgap II is too narrow to support a well-localized resonance mode. Figure 1(f) shows the near-infrared image of the scattered light intensity taken at 1511.97 nm resonance wavelength with heating applied, which confirms the spatial localization of the resonance mode in a loop pattern. We note that the introduction of the FDMR does not alter the topological properties of the lattice, as evidenced by the high edge-mode transmission at off-resonant wavelengths in the bandgaps. In our FWM experiment below, idler waves were generated at the FDMR resonance wavelengths and measured at the output waveguide, providing further evidence of topologically protected edge modes propagating along the sample boundary after the FDMR was turned on.

Figure 3(b) shows the tuning of an FDMR across bandgap III by varying the heater power from 13.9 to 21.2 mW. The linewidth of the resonance is also found to vary across the bandgap, which is due mainly to the dependence of the coupling between the edge mode and the FDMR loop on the roundtrip phase of the defect microring.¹⁵ From the plot, we obtain the sharpest resonance dip near the mid gap at 17.8 mW heater power, yielding an FDMR with a Q-factor of 72 600. As previously shown,¹⁵ the FDMR also exhibits the strongest spatial localization as measured by its inverse participation ratio around the bandgap center, which is important for achieving strong nonlinear effects. By fitting the resonance spectrum of the FDMR using an all-pass ring resonator model,³⁷ we extracted the effective coupling rate between the FDMR and the edge mode of μ = 20.6 GHz and an effective propagation loss of 4.54 dB/cm in the FDMR loop.

From Fig. 3(a), we observe that the spectral distance separating the FDMR resonance dips is equal to the microring’s FSR at 5.1 nm, although the roundtrip length of the FDMR loop is three times the microring circumference. This is due to the fact that the resonance modes in bandgaps I and II are suppressed. Figure 3(c) plots the resonance mode number m (blue circles) as a function of the measured resonant wavelength λ_m from 1510 to 1580 nm. The resonance mode numbers are determined relative to the value m = 514 computed for the 1529.1 nm resonance from the formula m = 3L · n_eff/λ_m, where n_eff is the simulated effective index of the microring waveguide at 1529.1 nm and 3L = 329.9 μm is the effective FDMR roundtrip length. We observe that the FDMR exists over a broad wavelength range with nearly constant mode spacing, as indicated by the best linear fit line. To estimate the GVD of the FDMR mode, we first calculated its propagation constant β(λ_m) = 2πn_m/λ_m, where the effective index n_m is computed from the resonant wavelengths in Fig. 3(c) as n_m = mλ_m/3L. We then performed a least-square fit to the β vs ω dispersion curve using a 4th-degree polynomial and computed the second-order derivative β₂ = d²β/dω² to obtain the GVD.²⁸ Figure 3(c) plots the GVD curve in solid red, from which we obtain experimental GVD values of −495 ± 2, −535 ± 2, and −697 ± 2 ps²/km at the signal, pump, and idler wavelengths, respectively. For comparison, we also plotted in dashed red the simulated GVD of the FDMR loop obtained by averaging the GVDs of the SOI waveguide sections and the coupling regions comprising the loop. The simulated GVD of the FDMR is comparable to that of a straight SOI waveguide,³⁸ with GVD ∼−4733 ps²/km around the pump wavelength of 1529 nm, which is larger than the experimental value. It has been demonstrated that GVD is strongly sensitive to the cross-sectional area and aspect ratio of the SOI waveguide,³⁹ and this is likely the source of discrepancy between the simulated and experimental values. However, the fact that the experimental GVD is less than or comparable to that of a straight waveguide over the 1510–1580 nm wavelength range demonstrates the low dispersion and broadband nature of the FDMR.

In the FWM experiment, with the heater power set at 17.8 mW, we tuned the signal and pump wavelengths to coincide with the FDMR at 1524.1 and 1529.1 nm, respectively. The spectral shapes of these resonances were similar, with a Q-factor of 73 000 at the signal wavelength and 95 000 at the pump wavelength. However, at the 1534.2 nm resonance where the idler wave was generated, a split resonance spectrum was prominent, which was caused by coherent back-scattering into the counter-propagating mode in the FDMR loop and was not observed in neighbor resonances. As a result, we obtained a lower effective Q-factor of 36 000 for this resonance mode. By applying 2.55 mW of optical power for both the pump and signal into the input waveguide, we obtained the wavelength spectrum at the output waveguide in Fig. 4, which shows an idler generated at both 1519.1 and 1534.2 nm. However, due to a slight red shift in the FDMR resonances caused by the nonlinear thermo-optic effect at high optical powers (see the supplementary material), the CE, defined as the ratio of the idler power to the signal power, was relatively low, around −46 and −49 dB for the up-converted and down-converted idler wavelengths, respectively. To optimize the CE, we swept the pump and signal wavelengths over the corresponding FDMR resonances to obtain the maximum generated idler power. Figure 5 shows the variations in the generated idler I power and wavelength as the pump wavelength is swept [Fig. 5(a)] and as the signal wavelength is swept [Fig. 5(b)]. In each plot, the darker spectra in the foreground were recorded without the heating power applied, so that no FDMR was induced and the idler I wave was generated purely from FWM of the edge modes propagating along the bottom lattice boundary. The lighter spectra in the background show the idler I power enhancement resulting from the FDMR when the heating power was applied. The pump and signal wavelengths were sampled in 5 pm steps and the spectral shape of each idler I spectrum corresponded to the line shapes of the input lasers.

We observe that the idler power generated by edge mode FWM with the heating power turned off exhibits minimal wavelength dependence, which reflects the flat, wide-band transmission characteristics of the Floquet edge modes across the bandgaps. However, when the heating power was turned on, the peak optical power of the resonance-enhanced idler wave follows the shape of the resonance spectrum of the FDMR, approaching the edge mode results when either the pump or signal was tuned out of the resonance. The peak idler power generated due to FDMR is about 12 dB higher than that generated due to the edge modes alone, and the 3 dB bandwidth at which the idler power drops by a half is measured to be 51 and 45 pm for the pump- and signal-wavelength sweeps, respectively, which is in agreement with the linewidth of the linear idler FDMR spectrum.

Since the generated idler wavelength is given by 1/λ_i = 2/λ_p − 1/λ_s, we obtain the rates of change in the idler wavelength with respect to the pump and signal wavelengths to be $δλi/δλp=2(λi/λp)2$ and $δλi/δλs=−(λi/λs)2$⁠, indicating that the idler wavelength changes twice as fast as the pump wavelength changes and in the opposite direction as the signal wavelength changes. These trends are also confirmed by the wavelength sweeps in Figs. 5(a) and 5(b). The wavelength conversion bandwidth, defined as the wavelength separation from the signal to the idler waves, is 10.1 nm and spans over three Floquet–Brillouin zones of the Floquet microring lattice. Such a large conversion bandwidth is enabled by the low dispersion of the Floquet modes in the FDMR.

With the pump and signal wavelengths set at the 1529.1 and 1524.1 nm resonances, respectively, we swept the input optical powers of both the pump and signal from 102 μW to 1.23 mW in the input waveguide. At each input power setting, we optimized the spectral locations of the pump (1529.039–1529.121 nm) and signal (1523.929–1523.974 nm) wavelengths to maximize the idler power (1534.169–1534.227 nm). Figure 6 compares the CE of FWM enhanced by the FDMR and that due only to the edge modes, as a function of the pump (and signal) power. For comparison, the effective propagation length of the edge modes from input waveguide to output waveguide was 1119 μm, which is more than three times the effective circumference of the FDMR loop. For both cases, we observe that the CE increases with input powers, with the FDMR providing a relatively constant average enhancement of 12.5 dB over the edge mode. At the highest available input pump power of 1.23 mW in the input waveguide, we obtained an internal CE of −37.4 dB with the FDMR present and −48.3 dB without. We also note that both CE trends do not show the sign of leveling off at high powers, indicating that the detrimental effect of free carrier absorption (FCA) in the silicon microrings is negligible at these input power levels.

The 330 μm roundtrip length of the FDMR gives an effective loop radius of 52.5 μm, allowing our device to be directly compared to that of a 50 μm-radius SOI microring resonator.²⁹ In their single-ring device, 2 mW of input pump power was required to reach a CE of −35 dB, with the saturation effect due to FCA observed at 5 mW of input pump power and a CE of −28 dB. Projecting along the fitted curve in Fig. 6, an input pump power of 5 mW would result in a resonance-enhanced CE of −27.3 dB in our FDMR, which is comparable to the single-ring device performance. We note, however, that the FDMR performance is somewhat degraded by mode splitting due to coherent backscattering. By optimizing the device to suppress this effect and achieve optimum trade-off between field enhancement and phase detune caused by the nonlinear thermo-optic effect, it is possible to further improve the CE in the FDMR. The edge mode achieved a CE of −48.8 dB/mm at 1.23 mW pump power, which is comparable to the value of −43 dB/mm reported for a bare silicon waveguide²⁵ but at a much higher pump power of 15.8 mW. It has also been shown²⁵ that by using the slow-light effect in a coupled-resonator optical waveguide (CROW) consisting of 8 microrings, a CE improvement of 28 dB over a bare waveguide could be achieved. The same approach can also be used to further increase the CE of our structure by coupling many FDMR loops together. It should be emphasized that while our FDMR provides comparable CE’s to those reported for conventional silicon microrings, the main benefit of the FDMR is that it provides a light source directly on a topological platform, which can potentially be used to realize robust photonic devices due to the property of topological protection.

In summary, we report the first demonstration of resonance-enhanced FWM by localized bulk modes in a Floquet TPI lattice using a compact, cavity-less resonance effect induced through a periodic perturbation of the drive sequence. The strongly localized FDMR, with tunable quasienergy across the TPI bandgap, provides high Q-factors and low dispersion, which enable efficient wavelength conversion over a broad bandwidth. Combined with the topological protection of the Floquet edge modes, the system could provide a robust topological photonic platform for nonlinear optics applications such as wavelength conversion, parametric amplification, frequency comb, and correlated photon pair generation. One particular example where the robustness of topologically protected edge modes could have the most impact is in quantum photonic applications, which rely on single photon and entangled photon states that are susceptible to decoherence by device imperfections. Our FDMR can serve as a compact and efficient emitter of single photons that are directly coupled to the edge modes for robust device operations.

See the supplementary material for an in-depth description of the Floquet–Bloch Hamiltonian of a 2D microring lattice with periodic perturbations, the experimental setup details, and a description of the nonlinear thermo-optic effect in FDMR at high optical powers.

This work was supported, in part, by financial contributions from the Natural Sciences and Engineering Research Council of Canada (Grant No. CGSD3–558491–2021).

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material.