Hyperparametric solitons in nondegenerate optical parametric oscillators

Domain: chemistry
Published: 2026-02-28
Editorial outcome: minor_revision

Abstract
Dissipative solitons and their frequency combs hold great potential for applications in optical communications, spectroscopy, precision time-keeping and beyond. Recent demonstrations based on the combination of second-harmonic generation and degenerate optical parametric oscillators (OPOs) show the interest in shifting soliton spectra away from the telecom’s C-band pump sources. However, these approaches lack the tunability offered by nondegenerate OPOs. This work presents a proof-of-principle demonstration of solitons in a nondegenerate OPO system based on a silicon-nitride microresonator, with engineered dispersion and optimized coupling rates. By pumping a relatively low-Q resonance in the C-band, we excite a signal soliton comb centred around a far-detuned, high-Q, O-band resonance, as well as repetition-rate-locked combs at the pump and idler frequencies, with the latter occurring at a wavelength beyond 2 µm. The solitons supported by this platform — hyperparametric solitons — are distinct from other families of dissipative solitons, as they emerge when the narrow-band signal mode, phase-matched under negative pump detuning, reaches sufficient power to drive bistability in the parametric signal. We investigate the properties of hyperparametric solitons, including their parametrically generated background and multisoliton states, both experimentally and through theoretical modelling.

Introduction
Microresonator optical frequency combs (microcombs) have made a profound impact on modern-day nonlinear integrated photonics and revolutionised fundamental research in optical solitons [1–4]. The unique features of such microresonator combs include their compactness and high repetition rates, with potential for mass production. At the same time, dissipative solitons represent the most application-relevant class of low-noise microcombs [4]. Broadening and stabilising soliton spectra have produced some of the most exciting research directions in this area. Spectra of solitons, typically centred around the pump wavelength, can span an octave in dispersion-engineered resonators [5–7] and may be further broadened by applying a second pump that also serves to stabilise the comb repetition rates and offset frequencies [8–10], and to initiate multi-colour soliton generation [11].

The rigid attachment of the aforementioned solitons, known as dissipative Kerr solitons [4], to a specific pump wavelength, which also builds the soliton background, may also limit the comb spectral bandwidth. Expanding the spectral range of dissipative solitons into the mid-infrared region, as well as across other telecom bands, and into the visible and even UV parts of the spectrum, while still utilising the 1550 nm (C-band) pump, will enhance the impact of microcombs in telecommunications, spectroscopy, and metrology.

By utilizing nonlinear frequency conversion processes, e.g., Raman scattering, second and third harmonic generation, and parametric frequency conversion, it is possible to design microresonators that generate microcombs far away from the pump frequency. For example, Kerr solitons in silica glass microresonators that are pumped in the C-band have been shown to generate Stokes-pair solitons through the Raman effect, allowing tunability across the L and U communication bands of optical fibers [13, 14]. Multi-colour solitons and combs with two pumps in Kerr resonators have also been demonstrated [8, 9, 15]. Second-harmonic generation in thin-film lithium niobate (LN) microresonators, which exhibit a combination of strong Kerr and second-order nonlinearities, has been employed to convert C-band solitons into non-solitonic near-visible combs [16] and to demonstrate two-colour solitons [17].

Parametric frequency conversion is, however, of primary interest in the context of our work. By engineering phase-matched parametric frequency conversion relying on either χ(2) or Kerr nonlinearity, one can create optical parametric oscillators (OPOs) generating signals far detuned from the pump frequency [18–31], see Figs. 1a,b for a conceptual illustration of both degenerate and nondegenerate Kerr OPOs. In Kerr microresonator OPOs, the parametric gain is proportional to the coupled-in, i.e., intraresonator, pump power. As the gain increases, when the pump frequency is tuned closer to the resonance, the OPO signal can start coexisting with a stable non-OPO state, see Fig. 1b. In the case of degenerate OPOs, this coexistence has been a key prerequisite for demonstrations of parametric solitons making trains of signal pulses having a zero (non-OPO) background [32–34], see Figs. 1c,d.

In particular, parametric solitons have been demonstrated using χ(2) nonlinearity for parametric down-conversion, from 775 nm pump to a 1550 nm signal, in an AlN microresonator [32]. A similar soliton experiment was later conducted in a fibre loop resonator, which included a short piece of χ(2)-active fibre for parametric gain [34]. The reference [33] used an all-Kerr Si3N4 microresonator with the dispersion profile phase-matched for the degenerate four-wave mixing process, 2f_signal = f_pump,1 + f_pump,2, with the pumps close to 1.0 µm and 1.5 µm creating parametric gain for the degenerate or near-degenerate signal photons at 1.2 µm, see Figs. 1a-1d. Here and below, f’s with various subscripts are used to annotate optical frequencies.

Common features of the parametric solitons observed in [32–34] were device operation near the degeneracy point, providing a zero background for soliton pulses and the coexistence of solitons with identical intensity profiles and phases differing by π [33, 34]. The zero-background property also means that the soliton central frequency does not dominate the rest of the signal comb, see Fig. 1d. This is unlike when a soliton is spectrally centred on the pump frequency and is governed by the classic form of the Lugiato-Lefever model [4].

Degenerate optical parametric oscillators (OPOs) are efficient for frequency conversion of the pump light, but they lack the tunability of nondegenerate OPOs with significant spectral separation between signal and idler, which is achieved by controlling phase matching. Indeed, in degenerate OPOs or second-harmonic generation, adjusting the phase-matching condition only controls the conversion efficiency but not the frequency of the generated signal. In recent developments, nondegenerate OPOs operating in the continuous-wave (CW) regime within microresonators and waveguides have achieved improvements in both tunability and efficiency [18–31]. However, despite the wealth of results in the CW domain, no microresonator nondegenerate OPO that generates soliton microcombs has been demonstrated to date. Our current work fills this gap by presenting a new class of dissipative optical solitons and highlighting critical differences with solitons in degenerate OPOs.

In introducing the physical concept of our work, it is natural to start by mentioning that theoretical results on bright zero-background solitons in degenerate OPOs were reported in optics as early as two decades ago [35–38], and even earlier in the contexts of fluid mechanics and condensed matter physics [39–41].

Theoretical investigations of bright solitons in nondegenerate OPO systems—where the signal and idler fields are well separated spectrally—were also conducted during this early period [42, 43]. These studies were technically and conceptually linked to the degenerate case and, therefore, found only solitons localised on a zero background, again arising in regimes where an OPO state coexists with a stable non-OPO state. Notably, we have not seen such solitons in the experiments reported below.

Here, we report a class of solitons in microresonator OPOs that have not been theoretically predicted or experimentally demonstrated. We use a Si3N4 resonator designed and pumped in C-band to support a nondegenerate phase-matched four-wave mixing process, f_signal = 2f_pump - f_idler, with the O-band signal near 1.25 µm, and the idler near 2 µm, see Fig. 1e. The resonator operates in the over-coupled regime, where the pulley coupler is designed to ensure that the coupling rate for the pump exceeds that for the signal and idler. We demonstrate that such conditions ensure that the signal not only increases with the pump but also depletes it sufficiently for the OPO state to fold on itself, creating a bistable loop between two distinct OPO states operating in the same resonator modes, see OPO-1 and OPO-2 in Fig. 1f.

Then, we experimentally and numerically demonstrate that the OPO bistability triggers the formation of signal solitons. This implies that the parametric generation does not turn off at the soliton tails since OPO-2 remains excited. This characteristic sets hyperparametric solitons apart from the parametric ones observed in degenerate OPOs, as illustrated in Figs. 1c,d and 1g,h. The term ”hyperparametric solitons”, which we coin here, emphasizes that parametric processes play an even more crucial role than in traditional ”parametric” solitons. The latter are situated on a zero OPO background, while our solitons are based on a fully developed monochromatic parametric signal. It is worth noting that the term hyperparametric oscillations has been previously used in the context of microresonators [20, 44], but not in relation to solitons.

The frequency of parametric solitons in the degenerate χ(2) OPOs is restricted to half of the pump frequency [32, 34]. Similarly, the frequency of Kerr parametric solitons hits the mid-point between the two pumps [33]. Thus, both setups are tunable only by adjusting the pump frequencies. In contrast, nondegenerate OPOs offer tuning capabilities in the range of several tens to up to 100 THz by choosing resonator geometries that provide suitable phase matching conditions [21–28, 45, 46].

Results
In this work, we used a Si3N4 microring resonator with a cross-section of 770 nm × 2520 nm (thickness × ring width) and approximately 200 GHz repetition rate designed to provide phase-matching conditions between two identical pump TM photons with frequency around 191.4 THz and signal and idler photons close to 242.0 THz and 140.8 THz, respectively. The details for designing the phase-matching and choosing the present geometry condition can be found in the Supplementary Information (Supplementary Figs. 1a-1c). The resonators were fabricated via a DUV-based subtractive approach, with the capability of wafer-scale manufacturing of Si3N4 photonic integrated circuits with ultra-low propagation loss and precise control of dimensions [47], see Methods. A pulley coupler with a gap of 650 nm was used to provide over-coupling for the pump, see Figs. 2a and 2b. The waveguide segment with the same curvature as the ring is approximately 50 µm long, which yielded the anti-resonance coupling range around 154 THz corresponding to the drop in the coupling rate and a sharp peak in loaded quality factor (Qload) [48], see the simulated result (solid curve) shown in Fig. 2d.

Available light sources allowed us to carry out the transmission measurements in two regimes that cover C-band (pump) and O-band (signal). Two transmission spectra are presented in Supplementary Information (see Supplementary Fig. 1d), which confirm the over-coupling of the TM mode family in C-band with low extinction ratios (<1 dB). With the resonances identified and fitted, the experimental total quality factors (i.e., Qload) for pump and signal are extracted and depicted in Fig. 2d, showing good agreement with the simulation. Qload of the signal is 1.3×10^6, higher than that of the pump resonance (Qload=2.8×10^5) near 191.4 THz, see Figs. 2c and 2d, which facilitates comb generation to be triggered and dominated by the signal.

The resonant frequencies f_µ and the four-wave mixing phase matching parameter,
∆f_µ = f_µ + f_-µ - 2f_0, µ=0,±1,±2,..., (1)
were computed numerically and plotted in Figs. 2e and 2f, coinciding with the experimental data (circles) very well. The signal and idler frequencies are determined from the condition of the parametric gain achieving its threshold value, which is most readily satisfied for a spread of modes in the proximity of the mode best complying with ∆f_µ = 0. Achieving ∆f_µ = 0 without invoking nonlinear frequency shifts is possible for large values of µ, when there is a change between anomalous and normal dispersion. A plot of the phase-matching parameter vs mode numbers, µ, for our resonator is shown in Fig. 2f, see also prior results in Refs. [21–28]. Defining dispersion as D_2µ = f_µ+1 + f_µ-1 - 2f_µ, we find that the groups of modes around the pump and signal have weakly anomalous dispersions, D_2p = 1.7 MHz and D_2s = 2.9 MHz, while the idler dispersion is large and normal, D_2i = -22.7 MHz. The FSRs of the pump, signal, and idler are found to be D_1p = 198.99 GHz and D_1s = 199.75 GHz, and D_1i = 200.73 GHz, respectively.

The comb characterisation setup is schematically shown in Fig. 3a, where the amplified pump is coupled into the microresonator via a lensed fibre. For on-chip power of 250 mW, we observed the excitation of monochromatic signal and idler waves with the frequencies predicted by zeros of the phase-matching parameter, see Fig. 3b, broadly in agreement with prior work on Kerr OPOs [21–28]. We also tuned the signal and idler frequencies by tuning a range of various modes, see Supplementary Fig. 2. The output powers of the pump and signal in Fig. 3b are 17.4 dBm and 5.4 dBm, respectively, corresponding to an on-chip conversion efficiency of approximately 2%, which can be further improved by engineering the coupling rate and implementing temperature control. We note that since the signal is gaining energy from a product of the pump and idler amplitudes, with the latter being squared, the sufficiently high idler’s Q is as critical as sufficiently high signal Q’s for the generation of strong CW signals, see the Q versus mode number plot in Fig. 2d.

For on-chip pump powers ranging from 375 to 500 mW, we successfully observed hyperparametric soliton states corresponding to low-noise, three-colour frequency combs (see Figs. 3c-3g, 4, and 5), which we describe in detail below. The microresonator chip was mounted on a temperature-controlled copper stage to enable tuning of the phase-matching condition. Figure 3c presents comb power traces with the pump suppressed, as the pump wavelength was swept (0.5 nm/s) for six fixed on-chip pump power values. The green-shaded regions indicate soliton steps. A representative soliton spectrum (Fig. 3d) exhibits three triangular spectral bands centred at the idler, pump, and signal frequencies, with the signal band showing the highest comb power by a significant margin.

A pronounced feature of the soliton spectra is the dominance of the signal comb power over the pump and idler combs. Specifically, the total power of the pump comb accounts for only 4% of the signal comb power, excluding the central lines of both combs. Coherence of the soliton state was confirmed via heterodyne detection between the comb line at µ = -6 and a reference tunable semiconductor laser (TSL), yielding a narrow beat note, see Fig. 3d. To measure the soliton repetition rate, f_rep, we employed an electro-optic (EO) comb generator [52], see Supplementary Fig. 3. The comparison between spectra with and without modulation is shown in Fig. 3e. A spectral region near 191.2 THz was filtered and used for radio-frequency (RF) detection, see Figs. 3f and 3g. With a frequency offset ∆f = 419.5 MHz and a modulation frequency f_mod = 25 GHz, the repetition rate is f_rep = 8×f_mod - ∆f ≈ 199.58 GHz. This value closely matches the signal FSR in the linear regime, D_1s = 199.75 GHz, unambiguously indicating that the dominant signal pulse drives the repetition rate locking across the entire comb spectrum. By fitting the signal spectrum with a sech-squared function, we found that the soliton has a 3-dB bandwidth of 2 THz, giving a transform-limited pulse duration of 158 fs at full width half maximum.

The fact that the central signal mode at µ = 253 dominates the comb spectrum generated around it should also be highlighted. This property contrasts sharply with that of parametric solitons in degenerate OPOs [32–34], which exhibit zero background and a central frequency that blends smoothly into the rest of the comb, cf. Figs. 1d and 1h. The strong central sideband in the signal comb in Fig. 3d should be compared with no such feature in the parametric soliton data in Fig. 2a(iv) in [32], Fig. 4d in [34], and Figs. 4a-4b in [33]. This effect is even more striking in the idler field at µ = -253. Thus, we are dealing with multi-colour mode-locked pulses with a background shaped by the monochromatic parametrically generated signal and idler waves. This distinct behaviour motivates our use of the term hyperparametric soliton, as was already discussed in the previous section, see Fig. 1.

To test the tunability of hyperparametric solitons, we measured soliton generation across a range of resonators with varying geometries, each providing different phase-matching conditions. In particular, changing the ring width from 2530 to 2610 nm, while maintaining the pump at 191.1 THz, has led to a shift of the signal frequency by 2.3 THz, as shown in Supplementary Fig. 4a. We have further pumped a sequence of resonances in the interval from 191.4 THz to 193.6 THz and tuned the signal frequency from 242 THz to 249.7 THz, as shown in Fig. S4b. Correspondingly, the idler frequency was adjusted from 140.8 THz to 137.8 THz. The EDFA’s range was the main factor limiting the observed tuning range.

Since no present concept or theory explains our measurements, it appears instructive to outline our numerical results and their interpretations. The first step in our analysis of mechanisms of the hyperparametric soliton generation was to solve the problem of three-wave parametric interaction of the µ=0 mode with several pairs of the signal and idler in the proximity of the ∆f_µ = 0 point, see Methods. Numerically computed signal components of all OPO states found above the parametric threshold for a chosen input power are shown in Fig. 4a. The pump, µ=0 mode, becomes strongly depleted by the generated signal and idler pairs so that its dependence on the laser frequency deviates from the parametric-process-free nonlinear resonance shown by the dashed line in Fig. 4b. Notably, generating |µ| = 253, 254, and 255 states leads to the bistable OPOs. One should note that the negative-detuning range of the 253 state stays isolated from the range of detunings where multiple other parametric states coexist, see Fig. 4a and Fig. 1f. We name the upper and lower branches of the bistability loop made by this state as OPO-1 and OPO-2. However, the instabilities of OPO-1 and OPO-2 relative to the excitation of other modes (modulational instability) can not be excluded.

To check stability and instability scenarios, we modelled a system of coupled equations for the amplitudes a_µ of the resonator modes µ = -512, ..., 0, ..., 511, covering the spectrum from 90 to 290 THz, see Methods. We have found that the OPO-1 is typically modulationally unstable, whereas OPO-2 is typically stable over the detuning interval during which they coexist, a condition that leads to the generation of hyperparametric solitons. The spectral carpet shown in Fig. 4d represents a bifurcation diagram computed by varying the laser frequency from large negative detunings, δ = (f_0 - f_p)/κ_2, to positive ones, illustrating the modulational instability and soliton generation processes. Here, f_p is the laser frequency and κ_2 is the width of the pumped resonance. CW parametric generation begins at δ ≈ -2.16 and destabilises due to sideband generation at δ ≈ -1.6, forming three-colour frequency combs. These then evolve into hyperparametric solitons that exist for δ values in the (-0.65, -0.06) range.

A numerically computed spectrum of the hyperparametric soliton (see Fig. 4c) qualitatively matches the typical experimentally observed spectra in Fig. 3d. The reconstructed pulse envelope profiles for a hyperparametric soliton’s signal, pump, and idler components are shown in Fig. 4e. It shows that the pump and idler waves are quasi-CW fields, and only the signal is a pronounced bright soliton resting on a background with finite amplitude. The idler is not expected to make a pulse because its frequency lies in the interval of large normal dispersion. While the pump is in the anomalous dispersion range, it experiences a loss rate three times higher than the signal and also remains quasi-CW. For the pump to trigger soliton generation of its own, i.e., without the parametric process involved, requires the laser frequency to be tuned closer to the tip of its resonance (δ ≈ 4 on the dashed black line in Fig. 4b).

When spectra of the pump, signal, and idler combs remain well isolated from each other, one can split the full coupled-mode system into equations for three modal groups and introduce the respective envelope functions, A_p, A_s, and A_i, Fig. 4 caption and Methods. By checking the signal equation, see Eq. (9) in Methods, reveals that the signal soliton centred on a mode µ' is excited by the quasi-CW nonlinear polarisation wave induced by the four-wave mixing process A_p^2 A_i^* ≈ a_0^2 a_-µ'^*.

Thus, the signal equation can be considered the generalised Lugiato-Lefever equation, where the pump has a complex dependence on the detuning introduced by the parametric process. The latter is anticipated to be the OPO-2 state. Diamonds, squares, and crosses in Fig. 4e show perfect matching between the hyperparametric soliton tails and the power values of the signal, pump and idler of the OPO-2 state, see Fig. 4a. This finding unambiguously confirms that the hyperparametric soliton has the background wave matching the OPO-2 state.

By pumping different resonator modes, tuning pump power, and varying the phase matching parameter, we have further checked that conditions for the hyperparametric solitons to exist are that the first parametric state bifurcating from the pump for the negative detunings must become bistable and provide a range of parameters (pump power and detuning) where the lower (OPO-2) branch is stable and the upper (OPO-1) branch is modulationally unstable. These conditions are typically met when the net signal losses are sufficiently small and the power of the four-wave mixing term, a_0^2 a_-µ'^*, driving the signal mode a_µ', is sufficiently high.

For the soliton generation, we recorded a video (see Supplementary Video 1) showing how the signal and pump spectra evolve during manual frequency tuning across the hyperparametric soliton range. Initially, non-solitonic combs replace the monochromatic signal and pump, which then evolve into soliton generation. The strong, ≈ 8 dBm, central sideband in the signal component rising above the comb starting around -8 dBm, unambiguously points to the hyperparametric regime. The soliton existence on the negatively detuned tail of the resonance, where intraresonator fields and thermal effects are modest, enables the simplicity of the manual tuning. Tuning to the range of positive detunings boosts the circulating power, leading to the sudden loss of resonance due to thermal shifts. Experimental access to these regimes may be the subject of future work using one of the established techniques for observing positively detuned non-parametric solitons of the Lugiato-Lefever equation [4, 51].

We also observed the generation of spectrally broad signals and idler combs at relatively high pump powers, see Supplementary Fig. 5. For large detuning, a sparse Turing-pattern-like OPO comb has a line separation of 11 FSR, which then gradually transitions to combs with broader, densely filled spectra. The pump-only Turing-pattern-like comb with the 28 FSR line-to-line separation is seen at the end of this experimental sequence. This level of pump power already takes us outside the hyperparametric soliton range, while pure Kerr solitons can be found numerically at these powers for red detuning. They are, however, not observed experimentally due to the lack of measures to control thermal effects in our setup.

The measured spectra can vary from a single soliton to quasi-crystals or soliton crystals by slightly changing the pump power, as shown in Fig. 5a. In crystals containing N solitons, only every Nth mode is excited, while quasi-crystals exhibit weaker lines between the strong spectral peaks. With the same resonator, we successfully excited the soliton crystals with N=2, 3, 4, 5, 6, 7. Additionally, multi-soliton non-crystal states, where N pulses are situated far apart from the 2π/N separation distances, have also been frequently observed, see the two-soliton state (ii). A single beat note near 1.5 GHz in heterodyne measurements shown in Fig. 5b confirms that the two-, three-, and four-soliton states are completely coherent. However, the five-, six-, and seven-soliton crystals are typically the breather states [53, 54].

Experimentally, we have observed the OPO comb spectra with signal and idler tuned between 230 and 249 THz (1.3–1.2 µm) and 138–150 THz (2.2–2 µm), respectively. At the extremities of this range, a broadband OPO comb was generated but not a soliton state due to the unoptimized coupling rate. While dispersion engineering allows extremely broad S-band phase matching [23], the existence of multi-colour solitons imposes its own constraints on the balance of dispersion and quality factors for which a complete understanding should be the subject of future work.

As the number of hyperparametric solitons in a crystal increases, the magnitude of the central sidebands in the idler and pump fields remains dominant. Conversely, the sidebands in the signal field decrease significantly. This is because the signal background rapidly loses power as the ring gets filled with pulses. Meanwhile, the pump and idler fields remain quasi-cw, with their power being only weakly dependent on the number of solitons in the crystal. This behaviour has been observed in the experimental and numerical data, see Figs. 5a,b and 5c,d.

Soliton crystals have now emerged as a sub-field that connects photonics, condensed matter, and topological physics [55–60], while the multi-soliton and crystal studies included parametric systems [38, 61], the hyperparametric frequency conversion effect paves a new path for their investigation.

Methods
Device fabrication and characterisation. The Si3N4 microresonators were fabricated using a deep ultraviolet (DUV) based subtractive process on 4-inch silicon wafers. A 4 µm thermal oxide layer was first grown as the bottom cladding. To mitigate the high tensile stress in thick Si3N4 films, interconnected stress-release trenches (3.5 µm depth) were patterned into the SiO2 substrate using dry etching, spaced 10-30 µm from waveguide regions to ensure waveguide uniformity. A 770 nm thick stoichiometric Si3N4 layer was then deposited in a single low-pressure chemical vapour deposition (LPCVD) step, achieving 0.6% thickness uniformity across the wafer. Waveguides were patterned using KrF 248 nm DUV stepper lithography with 180 nm resolution, followed by anisotropic dry etching with a C4F8 chemistry. An LPCVD SiO2 hard mask was employed during etching to achieve smooth sidewalls with 87° sidewall angles. The devices underwent high-temperature annealing at 1200°C for 11 hours in N2 atmosphere to reduce hydrogen-related absorption losses. A 1.3 µm thick LPCVD SiO2 top cladding was then deposited and subjected to identical annealing conditions.

OFC measurements. To characterize the resonator’s transmission, we swept two separate tunable lasers across the 1480–1640 nm, and 1260-1360 nm wavelength ranges with a high resolution of 0.1 pm and recorded the output optical power. These transmission spectra enabled the extraction of key resonator parameters, including resonance linewidths, free spectral ranges (FSRs), and phase-matching conditions. For hyperparametric soliton generation, we used a C-band tunable laser source, amplified by an erbium-doped fiber amplifier (EDFA), which was coupled into the waveguide through a lensed fiber. Before the lensed fiber, a fiber polarization controller (FPC) was employed to optimize the pump polarization, and a bandpass filter (BPF) was inserted to suppress amplified spontaneous emission (ASE) noise.

The output light was collected using a second lensed fiber and split into two detection paths. One path, carrying the majority of the signal, was directed to optical spectrum analyzers (OSAs) for spectral monitoring. Simultaneously, the other path was routed either through a fiber Bragg grating (FBG) for comb power monitoring using a digital storage oscilloscope (DSO), or through a bandpass filter (BPF) and combined with an auxiliary tunable semiconductor laser (TSL) for radio-frequency (RF) noise analysis using an electrical spectrum analyzer (ESA). Access to optical parametric oscillation (OPO) sidebands and hyperparametric soliton states was achieved by gradually tuning the pump laser into resonance, either through continuous wavelength sweeping or manual adjustment. The characteristic soliton dynamics are illustrated in the Supplementary Video 1, which captures the manual tuning of the pump wavelength from 1566.37 nm to 1566.42 nm at an on-chip pump power of 425 mW.

Nonlinear modelling. Multimode intra-resonator field is expressed as
A = e^{iMϑ-iω_p t} A + c.c., A = ∑ a_µ(t) e^{iµϑ}.
Here, ϑ = (0, 2π] is the angular coordinate in the laboratory frame, t is time and a_µ(t) are mode amplitudes and µ=0,±1,±2,... is the relative mode number. Coupled-mode equations governing nonlinear interactions between a_µ are [63–66]
i ∂_t a_µ = -γ ∑_{µ1,µ2,µ3} δ_{µ,µ1+µ2-µ3} a_µ1 a_µ2 a_µ3^* + (f_µ - f_p) a_µ - i κ_µ/2 (a_µ - δ_{µ,0} √bW).
Here, δ_{µ,µ'} = 1 for µ = µ' and is zero otherwise. The resonator spectrum, f_µ = ω_µ/2π, and the mode number dependent linewidth, κ_µ, were computed using Lumerical software, see Fig. 2. Since we work with frequencies, f_µ, and not the angular frequencies, ω_µ, time t in Eq. (3) was redefined as t = 2πt. The pump laser frequency, f_p, is tuned around f_0 = 191.44 THz corresponding to the M = 820 resonance. W is the on-chip laser power, b = 179.5 is the resonator power build-up factor for f_p = f_0. γ = f_0 n_2 / S_eff n_0 is the nonlinear parameter [16]; n_2 = 2.6·10^-19 m^2/W (Kerr coefficient), S_eff = 1.32 · 10^-12 m^2 (mode area), n_0 = 2.022 (refractive index), γ = 18.65 MHz/W. |a_µ|^2 represents intraresonator powers of the comb lines scaled to have units of Watts [66]. Data shown in Figs. 3c-e, 4 are obtained by modelling Eq. (3). While numerically solving Eq. (3), we divided them by κ_0/2, κ_0 = 683 MHz. The scaling parameter applied to plot dimensionless modal powers in Figs. 4a,b,e is κ_0/2γ = 18.3 W. Linewidth parameters are κ_µ = f_µ/Q_load, see Fig. 2d and Supplementary Table I. Three-wave OPO states, represented by the pump mode and a signal-idler pair, see Figs. 4a,b, were computed from the three-mode reduction of Eq. (3).
i ∂_t a_0 = -2γ a_µ a_-µ a_0^* - γ(|a_0|^2 + 2|a_µ|^2 + 2|a_-µ|^2) a_0 + (f_0 - f_p) a_0 - i κ_0/2 (a_0 - √bW),
i ∂_t a_µ = -γ a_0^2 a_-µ^* - γ(|a_0|^2 + 2|a_µ|^2 + 2|a_-µ|^2) a_µ + (f_µ - f_p) a_µ - i κ_µ/2 a_µ,
i ∂_t a_-µ = -γ a_0^2 a_µ^* - γ(|a_0|^2 + 2|a_µ|^2 + 2|a_-µ|^2) a_-µ + (f_-µ - f_p) a_-µ - i κ_-µ/2 a_-µ.
Parametric terms are placed first after the equal signs in all three equations. The role of phase-matching parameter, ∆f_µ, see Eq. (1), in the three-wave model becomes explicit on the observation that κ_µ ± µ - f_p = 1/2 ∆f_µ + (f_0 - f_p) ± ζ_µ, while the ζ_µ = (f_µ - f_-µ)/2 term can be eliminated from Eq. (4) by substituting a_±µ = a_±µ exp(∓iζ_µ t). So that, Eq. (4) becomes
i ∂_t a_0 = -2γ a_µ a_-µ a_0^* - γ(|a_0|^2 + 2|a_µ|^2 + 2|a_-µ|^2) a_0 + (f_0 - f_p) a_0 - i κ_0/2 (a_0 - √bW),
i ∂_t a_µ = -γ a_0^2 a_-µ^* - γ(|a_0|^2 + 2|a_µ|^2 + 2|a_-µ|^2) a_µ + (1/2 ∆f_µ + f_0 - f_p) a_µ - i κ_µ/2 a_µ,
i ∂_t a_-µ = -γ a_0^2 a_µ^* - γ(|a_0|^2 + 2|a_µ|^2 + 2|a_-µ|^2) a_-µ + (1/2 ∆f_µ + f_0 - f_p) a_-µ - i κ_-µ/2 a_-µ.
Equation (6) explicitly shows that the OPO operation depends on the pump detuning, f_0 - f_p, and the phase-mismatch parameter, ∆f_µ.

Values of the relative linewidth parameters for the relevant signal-idler pairs are specified in Supplementary Table I. Uncertainties in the parameter selection for numerical modelling come from the absence of experimental data on the loss values in the proximity of the signal and idler fields and temperature-induced fluctuations of the refractive index impacting phase-matching conditions. Pulse envelopes, A_p, A_s and A_i, corresponding to the well-separated pump, signal and idler spectra can be reconstructed from the modal amplitudes as
A_p = ∑_{µ=-N}^N a_µ e^{iµθ}, A_s = e^{-iζ_µ0 t} A_s, A_s = ∑_{µ=µ0-N}^{µ0+N} a_µ e^{iµθ}, A_i = e^{iζ_µ0 t} A_i, A_i = ∑_{µ=-µ0-N}^{-µ0+N} a_µ e^{iµθ},
where µ_0 and N can be chosen, e.g., as µ_0 = 253 and N = 100. We have now replaced the angular variable ϑ with θ = ϑ - D_1s t rotating with repetition rate at the signal frequency and will approximate dispersions only up to the second-order terms. Equations for three envelopes, although their use for modelling comb dynamics is left for future work, allow for an understanding of the hyperparametric soliton generation at a qualitative level and contrast it with the previously reported parametric solitons [32–34]:
i ∂_t A_p = E - i(D_1p - D_1s) ∂_θ A_p - 1/2 D_2p ∂_θ^2 A_p - 2γ A_p A_s A_i^* + (∆_p - γ|A_p|^2) A_p - i κ_0/2 (A_p - √bW),
i ∂_t A_s = - 1/2 D_2s ∂_θ^2 A_s - γ A_p^2 A_i^* + (∆_s - γ|A_s|^2) A_s - i κ_s/2 A_s,
i ∂_t A_i = - i(D_1i - D_1s) ∂_θ A_i - 1/2 D_2i ∂_θ^2 A_i - γ A_p^2 A_s^* + (∆_i - γ|A_i|^2) A_i - i κ_i/2 A_i,
where we omitted higher-order dispersion, and introduced short-hand notations for effective detunings incorporating pump detuning, phase-mismatch, and nonlinear cross-phase modulation terms,
∆_p = f_0 - f_p - 2γ(|A_s|^2 + |A_i|^2),
∆_s = 1/2 ∆f_µ0 + f_0 - f_p - 2γ(|A_p|^2 + |A_i|^2),
∆_i = 1/2 ∆f_µ0 + f_0 - f_p - 2γ(|A_p|^2 + |A_s|^2).
Since, in the hyperparametric soliton regime, the pump and idler fields are quasi-CW, the ∆_s parameter can be approximately treated as a quasi-constant, ∂_θ ∆_s ≈ 0, and equations for A_p(t,θ) and A_i(t,θ) can be approximated by equations for a_0(t) and a_-µ0(t). Then, Eq. (9) takes the form of the generalised Lugiato-Lefever model, where the four-wave mixing term, A_p^2 A_i^* ≈ a_0^2 a_-µ0^*, ∂_θ (A_p^2 A_i^*) ≈ 0, play the role of an effective pump, implicitly parametrised by the actual pump detuning, f_0 - f_p, and power, W. It is important to note that, even in this approximation, the effective pump in the signal equation, Eq. (9), is not an independent parameter like the pump term in the LLE model [4]. Instead, it is a variable whose stationary values and evolution are determined by the pump, a_0(t), and idler amplitudes, a_-µ0(t), which satisfy additional equations. The normalised repetition rate differences and dispersion values are (D_1p - D_1s)/κ_2 = -2.2126, (D_1i - D_1s)/κ_2 = 2.8199, D_2p/κ_2 = 0.0052, D_2s/κ_2 = 0.0085 and D_2i/κ_2 = -0.0699.

Eqs. (8)-(10) are transformed to the equations for the degenerate Kerr OPO used in Ref. [33] to demonstrate parametric solitons, when A_p is replaced with A_s, while A_s and A_i become two pump fields A_p1 and A_p2, so that the external pump term, √bW, is moved from Eq. (8) to Eqs. (9), (10). A_p1 and A_p2 can be approximated as quasi-CW fields [33], ∂_θ (A_p1 A_p2) ≈ 0, so that Eq. (8) becomes a parametric Ginzburg-Landau equation,
i ∂_t A_s = - 1/2 D_2s ∂_θ^2 A_s - 2γ A_p1 A_p2 A_s^* + (∆_s - γ|A_s|^2 - i κ_s/2) A_s,
possessing explicit bright soliton solutions on zero background [33–36, 39–41].

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript entitled “Hyperparametric solitons in nondegenerate optical parametric oscillators” by Haizhong Weng et al. reports the demonstration of solitons in a silicon-nitride microresonator-based nondegenerate OPO system, which by pumping a relatively low-Q resonance at 191 THz, they were able to excite a signal soliton comb centered around 240 THz as well as an idler comb at 140THz. This work is of interest to the microcomb and optical frequency comb community as it provides a way of generating optical frequency combs at wavelengths away from the pump wavelength, where the signal/idler comb generation wavelength can be engineered by choosing phase matching parameters. While the study is very interesting and caried out carefully, I strongly recommend that the authors to include the experimental demonstration of the control of the signal/idler comb generation wavelength by adjusting the phase matching parameters as indicated in the simulation results in the supplementary information Fig. S1. This would be very helpful as it would highlight the control over of the signal/idler comb generation wavelength. That being said, I believe that the manuscript is highly relevant to the Nature Communications audience.\nSome minor comments are:\n1) Can the authors include information about the limitations on the wavelength range for which a signal/idler optical frequency comb could be generated with such a silicon-nitride microresonator-based nondegenerate OPO system?\n2) Can the authors include information on the trade-offs if one would want to generate broader signal/idler optical frequency combs?", "reviewer_2": "This manuscript reports the first experimental observation and theoretical description of hyperparametric solitons in a nondegenerate Kerr microresonator optical parametric oscillator (OPO). By pumping a Si₃N₄ microresonator in the C-band, the authors generate a three-color frequency comb, with a soliton state present in the far-detuned O-band signal. The key novelty is that these solitons at the signal frequency are generated from a nondegenerate Kerr OPO process, distinct from previously reported degenerate Kerr parametric solitons that require two pumps.\nWhile the main claims are well-supported, several aspects require clarification, additional quantitative data, and expanded discussion before the paper is ready for publication.\nOne notable feature in the results is that the signal power is consistently higher than the idler power. The authors should explain the physical origin of this asymmetry. Furthermore, could the device be designed such that the idler wave hosts the soliton instead of the signal? Such a demonstration would broaden the scope and applicability of the work.\nA key advantage of nondegenerate OPOs is their potential for wide spectral tunability. However, the present results show soliton generation only at a fixed signal/idler wavelength pair determined by the resonator geometry and pump mode. While Supplementary Fig. S2 demonstrates CW OPO tuning, soliton comb tunability is not demonstrated. If such data exist, they should be included; if not, the claims about demonstrated tunability should be tempered and framed as a potential capability for future exploration. Additionally, the statement that solitons were “reproducibly observed in different resonators” would be more impactful if quantified, for example by reporting a yield such as “observed in X out of Y tested devices with similar geometry.”\nThe manuscript should clarify whether the reported pulse duration was estimated from the optical spectrum—assuming a transform-limited pulse—or directly obtained from numerical simulations.\nFor practical applications, it would also be useful to report absolute optical powers in each band (pump, signal, idler) and the total conversion efficiency from pump to signal/idler, in both the CW OPO and soliton regimes.\nIn Fig. 5(b), cases (i)–(iv) show no RF beat notes, unlike case (v). The authors should explain why no RF features are visible —whether this reflects complete coherence, or is instead due to measurement limitations.\nDoes the mismatch in group velocity between the pump, signal, and idle beams affect soliton formation?\nRecent work on OPO soliton crystals (arXiv:2504.04788, 2025) reports multiple possible phase states of OPO solitons in crystals, which significantly influence comb structures.\nThe authors should discuss whether similar behavior was observed in their simulations and experiments. Has the author observed the biphasic soliton or crystals in this OPO configuration?\nA pump–detuning phase diagram, distinguishing different operational states such as CW OPO, hyperparametric soliton, multi-soliton, and modulation instability regimes, would greatly aid in understanding the stability landscape of the system.", "reviewer_3": "Weng et al., in their manuscript ”Hyperparametric solitons in nondegenerate optical parametric oscillators,” present a novel method to generate solitons by creating the driving pump field from a pump-degenerate OPO within the same microring resonator. Their theory is thorough, and I find this regime interesting and likely useful to the community. However, as detailed in my comments below, the authors overclaim their results. I do not believe this is a “new soliton regime” but rather a new way to generate the driving field that yields the same soliton dynamics as direct pumping. They also lack a thorough review of previous work that frames the state-of-the-art. Additionally, their experimental work, though of good quality, does not convincingly demonstrate the actual claimed hyperparametric soliton regime. A clear demonstration that the soliton originates from the idler, not the main pump, is missing. This is especially true given the differing comb regimes observed while detuning the main pump, which are different between experiment and theory and cannot be solely blamed on thermal bistability (in particular since the theory claims blue-detuning operation). Therefore, in its current form, I do not recommend this manuscript for publication in Nature Communications. I hope the authors find my comments useful to improve their manuscript and enhance their work, which, as stated, is of interest and could be suitable for publication in a revised form.\n1. “Our current work fills this gap by presenting a new class of dissipative optical solitons, and highlighting critical differences with solitons in degenerate OPOs.” and “’Here, we report a class of solitons in microresonator [...]” are not exactly correct statements, and overall goes with the sentiment that the paper overclaims what they are demonstrated. Surely, the authors generate the driving field parametrically, which is new. Yet, as shown theoretically, the driving force in the signal equation a 2 pa ∗ i (p = pump, i = idler) is mainly driven by a CW term from the pump-degenerated OPO. Thus, the signal soliton, whether generated by a direct laser or a parametric CW process, results in the same soliton: cascading from the driving force with a strong pedestal from the out-of-phase pump part, clearly observed theoretically and experimentally. The authors also show in the method that the signal-idler degenerated parametric χ (2) or χ (3) solitons in refs [31-33] differ significantly. In such case, the complex conjugate of the soliton wave acts in the driving force term, substantially modifying the soliton’s parametrically driven dynamics and properties, creating the granted new ”soliton regime.” Unfortunately, this does not apply to the authors’ system and should be reflected in their claim.\n2. The authors omit a large body of work using parametric interactions to generate pulses (soliton, dark soliton, simultons, etc.). Surprisingly, multi-pumped parametric interaction with DKS is cited but not properly linked to the authors’ work. As noted, the authors’ system resembles a multi-pump system, with the main pump and CW OPO, where parametric interaction expands the frequency comb (ref [8-9]), generates bright-dark states by cross-phase modulation binding (ref 11), or nonlinear mixes with the soliton to create a parametric optical trapping (Moille et al PRL 134 (19), 193802, 2025). A recent complete theoretical description by Menyuk et al. 33 (10), 21824-21835 Optics Express 2025 models a multi-color system similarly to the authors’ method but is uncited, making the claim ”Since no present concept or theory explains our measurements,” too strong, though I agree the system lacks thorough theoretical study. Linking this state-of-the-art with the authors’ work would aid readers and clarify the pump and idler comb spectra, also described by XPM-induced effects in Stone et al., Phys. Rev. Applied 17, 024038, 2022. The authors should also cite single-pump work on comb generation and parametric interaction, as their claim ”no microresonator nondegenerate OPO that generates soliton microcombs has been demonstrated to date” is at best inaccurate. They omit Yang et al., Communications Physics 3, 27 (2020) on satellite frequency combs (not soliton microcombs) and Anderson et al., Phys. Rev. X 13, 011040, 2023 (especially figure 7), where cavity geometry modulation enables other degree of freedom for phase matching enabling parametric interaction with dark solitons for coherent satellite comb generation. Though these systems differ from the authors’, these omissions bias the manuscript and fail to represent the field’s state-of-the-art.\n3. The experimental results do not convincingly show a hyperparametric soliton. Both the main pump and signal lie in the anomalous dispersion regime, which should allow either to generate a soliton, and it is rather unclear how the authors prove that the soliton is not a “main-pump soliton” which is then spectrally translated around the signal/idler through XPM.\nFrom their simulations, the hyperparametric soliton should appear in the main-pump blue-detuned regime. In their simulation, although not really described, Figure 4-d shows that further detuning the main pump reveals additional regimes at the main pump where, I believe, a main-pump soliton may appear while the pump-degenerated OPO may persists.\nExperimentally in Figure 3-c, they show the system passing through multiple nonlinear regimes, including modulation instability, before reaching a soliton step, which begs the question if the authors are still blue-detuned. More importantly, the simulation does not corroborate such MI state appearing before reaching the hyperparametric soliton. Since cross-phase modulation alone explains the creation and bonding of the secondary pumped wave (see ref 11 and Menyuk et al), how do the authors experimentally exclude generating a main-pump soliton simply shifted around the CW-OPO signal? Without detuning measurements or a clear theory comparison of the intermediate states, their experimental evidence does not prove the hyperparametric soliton.\n4. Related to the claim of experimental demonstration of hyperparametric soliton, figure 5 is not convincing. The multi-DKS simulation shows clear comb spectral asymmetry around the pump mode, absent experimentally except in the top case. Also, the soliton crystal simulation shows dark teeth that are not very dark, possibly from the hyperparametric soliton’s background bump. Yet, the authors show perfect soliton crystals, (where interestingly Cnoidal waves/Turing rolls, and soliton crystals can occur in the blue-detuned regime see Qi et al, Optica 6, 9, 2019). These discrepancies cast doubt that the soliton forms at the signal rather than at the pump and then is translated through XPM around the signal. The authors could compare direct (multi)soliton generation at the main pump with their observed spectra to help confirm a hyperparametric soliton.\n5. The noise measurements are not thorough enough and lack a lot of detail to be able to claim low-noise operation. The authors do not report any detected power. If no light or light below the NEP is sent to the photodiode, only the noise floor appears. Therefore, it would help to compare their measurement with a noisy state to show the (multi)-soliton regime low-noise operation. The authors also fail to explain the beat in the noise measurement, which is puzzling. The system is not described as a breather, and no spectral overlap among the three comb components appears (which is not stated but should not be on the same frequency grid, but rather offset from one another). Hence, without a better explanation of the noise measurement, the soliton regime demonstration remains unconvincing.\n6. The ring resonator’s wide width allows for multi-mode operation at the signal. Because the sidewalls are not straight, avoided mode crossings could occur. This has been shown to enable high-power multi-mode family OPO generation (see ref 21) with minimal XPM-mediated loss channels (see Stone et al., Phys. Rev. Applied 17, 024038, 2022). The spectrum in Figure 3b is notably clean, resembling the normal-dispersion regime main pump (see 24), despite the main pump being in the anomalous regime and at very high power. The authors do not measure the mode in the O-Band, where lasers are directly available. Their claim that they work with the same mode family as the main pump relies on their dispersion model, which, given Figure 2e and the high measurement noise, is not convincing. I believe it is a crucial point to clarify as this could lead to significant discrepancy between the theory and experiment.\nMinor remarks:\n• The authors claim the pump is quasi-CW based on the azimuthal field profile. However, a comb clearly forms from XPM interaction with the hyperparametric soliton, altering the pump’s azimuthal profile. It is mostly CW-driven but not quasi-CW; otherwise, no comb would appear.\n• It is unclear why the authors use a pulley configuration when the system’s width is limited. A straight waveguide could offer similar coupling performance. Unless system losses must balance perfectly, since the parametric interaction occurs on-chip, the signal’s Qc affects only extraction efficiency, not parametric interaction efficiency."}, "author_response": "We thank all reviewers for their careful consideration and for their comments on our manuscript. Below, we provide point-by-point replies to all comments, explaining the revisions made and the new work performed to address them. In particular, we have performed and presented new experiments on soliton tunability, soliton noise measurements, and repetition rate measurements, and added new spectral data and linewidth and dispersion measurements. All new additions to the main manuscript text and supplementary document are highlighted in red font. The manuscript text and supplementary document lines are numbered, which further facilitates referencing in the reply letter.\nReviewer #1 (Remarks to the Author):\n[...]\nResponse:\nWe thank the reviewer for the positive evaluation of our work and for highlighting its novelty and rigor. We agree that providing more experimental evidence of the tunability of the signal/idler comb generation wavelengths via phase-matching control is highly relevant and would significantly strengthen the manuscript. To address this point, we have conducted additional measurements and included the corresponding data in the revised manuscript.\nIn particular, we have updated Supplementary Fig. S4 to include new experimental results that directly address this point. In Fig. S4(b), we show hyperparametric single-solitons generated in the same resonator (i.e., the one used in the main text), where we tuned the pump wavelength to excite a sequence of different TM00 modes. These experiments confirmed, respectively, the tunability of the signal and idler frequencies by varying the phase-matching wavelengths. Furthermore, Fig. S4(a) shows results from three other resonators with different ring widths and the same thickness, pumped at approximately 191.1 THz, demonstrating that dispersion engineering enables frequency tuning of the hyperparametric solitons when the pump frequency remains constant.\nNew paragraphs in the main text (line 322) and in the supplementary material document (sub-section IV) describe these new results in detail for readers.\nMinor comments:\n1) Can the authors include information about the limitations on the wavelength range for which a signal/idler optical frequency comb could be generated with such a silicon-nitride microresonator-based nondegenerate OPO system?\nResponse:\nTo address this we have added a new paragraph in the main text, see line 475, “Experimentally, we have observed the OPO comb spectra with signal and idler tuned between 230 and 249 THz (1.3-1.2 μm) and 138-150 THz (2.2-2 μm), respectively. At the extremities of this range, a broadband OPO comb was generated but not a soliton state due to the unoptimized coupling rate. While dispersion engineering allows extremely broad-band phase matching [22], the existence of multi-colour solitons imposes its own constraints on the balance of dispersion and quality factors for which a complete understanding should be the subject of future work.”\nOverall, our revised manuscript provides new and solid evidence of the soliton’s signal/idler components' tunability via dispersion engineering or pump frequency tuning. At the same time, we refrain from making unjustified claims which could overestimate the tunability range of hyperparametric solitons in the devices we are dealing with in this work. In future work, a reverse-design approach can help identify an optimal range of Q factors controlled by the waveguide-resonator gap and by simultaneous broadband dispersion engineering.\n2) Can the authors include information on the trade-offs if one would want to generate broader signal/idler optical frequency combs?\nResponse:\nThis question echoes the previous one and the above response, but shifts emphasis to the comb bandwidth. A broader signal-idler frequency comb can be achieved by increasing the pump power.\nThis is now addressed by new experimental data in Fig. S5 (supplementary document, section V, line 174) and by the new paragraph starting in line 446 of the main text: ”We also observed the generation of spectrally broad signal and idler combs at relatively high pump powers, see Fig. S5 in the Supplementary Information. For large detuning, a sparse Turing-pattern-like OPO comb has a line separation of 11 FSR, which then gradually transitions to combs with broader, densely filled spectra. The pump-only Turing-pattern-like comb with a 28 FSR line-to-line separation is seen at the end of this experimental sequence. This level of pump power already takes us outside the hyperparametric soliton range, while pure Kerr solitons can be found numerically at these powers for red detuning. They are, however, not observed experimentally due to the lack of measures to control thermal effects in our setup.”\nReviewer #2 (Remarks to the Author):\n[...]\nResponse:\nThe asymmetry mainly arises from coupling-loss dispersion, leading to different total Q factors for the signal and idler. It is critical that neither signal nor idler Q factors are too low. In our case, the lowest Q is at the pump frequency. Since the signal is gaining energy from a product of the pump and idler squared, a sufficiently high Q for the idler and hence the idler’s power in the central component (not in the idler’s comb) is critical for the generation of high bandwidth signal combs and also the high-power monochromatic central signal component. One should bear in mind that in the original manuscript, we measured linewidth only for the pump comb; now we have added measurements for the signal, but a suitable laser to measure the idler linewidth is not available to us. This leaves some uncertainty about the actual value of the idler’s Q.\nIn the Supplementary Fig. S2b, the idler intensity shows an increasing trend with the increase of frequency, which is consistent with the increasing trend of the Q factors shown in Fig. 2d. In the present 200-GHz SiN system, we notice that the idler always has a normal dispersion (see Fig. S1b), which hinders the powers in its sidebands, but not of its central component. Future work will explore new designs with variations in the ring radius and cross section, or new platforms (such as AlN or Ta2O5), to determine whether a long-wavelength idler wave could exhibit anomalous dispersion and host soliton generation.\nTo comment on this, we have added new text in lines 259-269 of the main manuscript.\n2. A key advantage of nondegenerate OPOs is their potential for wide spectral tunability. However, the present results show soliton generation only at a fixed signal/idler wavelength pair determined by the resonator geometry and pump mode. While Supplementary Fig. S2 demonstrates CW OPO tuning, soliton comb tunability is not demonstrated. If such data exist, they should be included; if not, the claims about demonstrated tunability should be tempered and framed as a potential capability for future exploration. Additionally, the statement that solitons were “reproducibly observed in different resonators” would be more impactful if quantified, for example by reporting a yield such as “observed in X out of Y tested devices with similar geometry.”\nResponse:\nWe also thank this reviewer for the positive evaluation of our work and for highlighting its novelty and rigor. The tunability issue was also brought up by the 1st referee and to a significant degree addressed by us above. In particular, the revised Supplementary Fig. S4 demonstrates new experimental results on the tunability of the signal frequency by tuning the pump frequency and changing the resonator geometry, see for details and text edits our reply to referee 1 (line 322 of the main text and section IV of the supplementary).\n3. The manuscript should clarify whether the reported pulse duration was estimated from the optical spectrum—assuming a transform-limited pulse—or directly obtained from numerical simulations.\nResponse:\nThe captions of Figs. 4 and 5 state that they are simulated pulse profiles. The numerical spectra match the experimental ones, and therefore, one can indeed rely on the numerical predictions of pulse durations.\n4. For practical applications, it would also be useful to report absolute optical powers in each band (pump, signal, idler) and the total conversion efficiency from pump to signal/idler, in both the CW OPO and soliton regimes.\nResponse:\nLines 259 and 284 now state: “The output powers of the pump and signal in Fig. 3b are 17.4 dBm and 5.4dBm, respectively, corresponding to an on-chip conversion efficiency of approximately 2%, which can be further improved by engineering the coupling rate and implementing temperature control.” and “A pronounced feature of the soliton spectra is the dominance of the signal comb power over the pump and idler combs. Specifically, the total power of the pump comb accounts for only 4% of the signal comb power, excluding the central lines of both combs.”, respectively.\n5. In Fig. 5(b), cases (i)–(iv) show no RF beat notes, unlike case (v). The authors should explain why no RF features are visible—whether this reflects complete coherence, or is instead due to measurement limitations.\nResponse:\nThe absence of RF beat notes in cases (i)–(iv) in Fig. 5(b) indeed indicates high coherence of the generated OPO soliton states. To avoid potential ambiguity, we employed an alternative method to directly characterise the coherence of these solitons by performing new heterodyne beat measurements between a comb line and an independent tunable laser (TSL). The new noise characterisation setup is shown in Fig. 3a, and its description is included in the Methods.\nLine 470 now states: “A single beat note near 1.5 GHz (fbeat) in hyterodyne measurements shown in Fig. 5b confirms that the two-, three-, and four-soliton states are completely coherent. However, the five-, six-, and seven-soliton crystals are typically the breather states [53, 54].”\n6. Does the mismatch in group velocity between the pump, signal, and idler beams affect soliton formation?\nResponse:\nSignal solitons and signal combs are primarily supported by the central components of the pump and idler fields, which suggests that the GV mismatch is not a key parameter. Though, of course, since all three fields have finite spectra, the GV mismatch will play its detrimental role if it is taken too far.\n7. Recent work on OPO soliton crystals (arXiv:2504.04788, 2025) reports multiple possible phase states of OPO solitons in crystals, which significantly influence comb structures. The authors should discuss whether similar behavior was observed in their simulations and experiments. Has the author observed the biphasic soliton or crystals in this OPO configuration?\nResponse:\nThis mentioned arXiv manuscript work does not study the hyperparametric case (it studies degenerate OPO), thus we have not observed and reported these solitons. We have added this citation, see Ref [61].\n8. A pump–detuning phase diagram, distinguishing different operational states such as CW OPO, hyperparametric soliton, multi-soliton, and modulation instability regimes, would greatly aid in understanding the stability landscape of the system.\nResponse:\nSuch bifurcation diagrams are shown in Figs. 1a, 1b and 1d. In particular, Fig. 1d shows how CW-parametric states are excited at detunings below -2, then replaced by Turing non-solitonic combs for dimensionless detunings from -1.5 to -0.6, which then evolve into hyperparametric solitons, existing till just before zero detunings. Figs. 1a and 1b show detailed bifurcation diagrams for all existing (for given power) CW parametric states. This is explained in detail in lines 350-390, see line 368, in particular.\n-----------------------------------------------------\nReviewer #3 (Remarks to the Author):\n[...]\nResponse:\nWe have now supported our claims with a significant amount of new experimental data included in Supplemental Materials and with revisions of the main text. These changes are described in detail in our response to the two other reviewers and include demonstrating comb tunability via pump tuning and dispersion engineering (resonator geometry). Novelty of the solitons is clear, since these are 3-equation solitons. The pump field and idler field, which drive the signal, are dynamically varying detuning sensitive fields even in their simplest monochromatic approximations (even more so when the pump and idler host significant comb spectra), therefore, solitons are clearly of a novel type and are not described by the Lugiato-Lefever model. Connecting to the latter is useful, of course, to make the next step in our knowledge about these systems clearer to readers.\n2. The authors omit a large body of work using parametric interactions to generate pulses (soliton, dark soliton, simultons, etc.). Surprisingly, multi-pumped parametric interaction with DKS is cited but not properly linked to the authors’ work. As noted, the authors’ system resembles a multi-pump system, with the main pump and CW OPO, where parametric interaction expands the frequency comb (ref [8-9]), generates bright-dark states by cross-phase modulation binding (ref 11), or nonlinear mixes with the soliton to create a parametric optical trapping (Moille et al PRL 134 (19), 193802, 2025). A recent complete theoretical description by Menyuk et al. 33 (10), 21824-21835 Optics Express 2025 models a multi-color system similarly to the authors’ method but is uncited, making the claim ”Since no present concept or theory explains our measurements,” too strong, though I agree the system lacks thorough theoretical study. Linking this state-of-the-art with the authors’ work would aid readers and clarify the pump and idler comb spectra, also described by XPM-induced effects in Stone et al., Phys. Rev. Applied 17, 024038, 2022. The authors should also cite single-pump work on comb generation and parametric interaction, as their claim ”no microresonator nondegenerate OPO that generates soliton microcombs has been demonstrated to date” is at best inaccurate. They omit Yang et al., Communications Physics 3, 27 (2020) on satellite frequency combs (not soliton microcombs) and Anderson et al., Phys. Rev. X 13, 011040, 2023 (especially figure 7), where cavity geometry modulation enables other degree of freedom for phase matching enabling parametric interaction with dark solitons for coherent satellite comb generation. Though these systems differ from the authors’, these omissions bias the manuscript and fail to represent the field’s state-of-the-art.\nResponse:\nWe thank the reviewer for bringing to our attention a list of further papers on solitons in microresonators published in reputable journals. This particular PRL concerns synchronization, a large topic in its own right that falls outside our current scope. The Optics Express paper by Menyuk, et al studies three equations with 2 pumps and, crucially, operates outside the hyperparametric regimes we uncovered, with most of the energy transferred into the free-running (i.e., no-pump) signal field. We have added a reference to this work (Ref [15]), as well as to the next one by Stone (Ref [45]) and to Yang’s work (Ref. 46). Anderson’s work on dispersion modulation is very valuable, but it does not align closely enough with the scope of our work.\n3. The experimental results do not convincingly show a hyperparametric soliton. Both the main pump and signal lie in the anomalous dispersion regime, which should allow either to generate a soliton, and it is rather unclear how the authors prove that the soliton is not a “main-pump soliton” which is then spectrally translated around the signal/idler through XPM.\nFrom their simulations, the hyperparametric soliton should appear in the main-pump blue-detuned regime. In their simulation, although not really described, Figure 4-d shows that further detuning the main pump reveals additional regimes at the main pump where, I believe, a main-pump soliton may appear while the pump-degenerated OPO may persists.\nExperimentally in Figure 3-c, they show the system passing through multiple nonlinear regimes, including modulation instability, before reaching a soliton step, which begs the question if the authors are still blue-detuned. More importantly, the simulation does not corroborate such MI state appearing before reaching the hyperparametric soliton. Since cross-phase modulation alone explains the creation and bonding of the secondary pumped wave (see ref 11 and Menyuk et al), how do the authors experimentally exclude generating a main-pump soliton simply shifted around the CW-OPO signal? Without detuning measurements or a clear theory comparison of the intermediate states, their experimental evidence does not prove the hyperparametric soliton.\nResponse:\nWe thank the reviewer for raising this point. Our manuscript presents a comprehensive set of supporting evidence that the signal component is the primary component of the hyperparametric soliton bearing the strongest frequency comb.\na) Using experimental data, we have calculated the total power of pump comb (180-198 THz, excluding its central line) as 0.1 mW, which is only 4% of the signal comb power (225-258 THz, excluding the centra signal line), see line 286: “Specifically, the total power of the pump comb accounts for only 4% of the signal comb power, excluding the central lines of both combs.”\nb) At 425 mW of pump, we experimentally observed the OPO soliton, which transitions directly from the OPO primary comb (see the video at 00:18) without any intermediate states. The evolution ends by the transition from OPO solitons to a few lines near the signal/pump/idler. The difference between the experiment and the simulation is seen in Fig. 4d only for larger detunings (the right most data interval). The buildup of intraresonator power becomes noticeable at these detunings, leading to strong thermal effects that are not compensated for in our experimental setup. If they were to be compensated, we indeed could see non-hyperparametric solitons, i.e., DKSs, discussed in many previous papers. In support of this, the lowest panel in Fig. S5 shows the onset of pure-pump Turing patterns (comb lines are separated by 28×FSRs), which, with thermal compensation techniques on, would evolve to Kerr-like pump solitons, see new text in line 446, “We also observed the generation of spectrally broad..”\nc) To further support the claim of hyperparametric, i.e., non-DKS, soliton regime, we have also performed a new set of repetition rate measurements by injecting the OPO single-soliton microcomb into the cascaded intensity-phase modulators, which gave us 199.58 GHz repletion rate. This rate is much closer to the simulated FSR of the signal (D =199.75 GHz) than that of the pump (D =198.99 GHz). The results are presented in Figs. 3e to 3f and the new section added to the main text starting in Line 288: “ Coherence of the soliton state was confirmed via…”\n4. Related to the claim of experimental demonstration of hyperparametric soliton, figure 5 is not convincing. The multi-DKS simulation shows clear comb spectral asymmetry around the pump mode, absent experimentally except in the top case. Also, the soliton crystal simulation shows dark teeth that are not very dark, possibly from the hyperparametric soliton’s background bump. Yet, the authors show perfect soliton crystals, (where interestingly Cnoidal waves/Turing rolls, and soliton crystals can occur in the blue-detuned regime see Qi et al, Optica 6, 9, 2019). These discrepancies cast doubt that the soliton forms at the signal rather than at the pump and then is translated through XPM around the signal. The authors could compare direct (multi)soliton generation at the main pump with their observed spectra to help confirm a hyperparametric soliton.\nResponse:\nHyperparametric solitons are distinctly different from DKS, see our point 3(a) above and the Concept section in the main manuscript text. The spectral asymmetry around the pump, seen in Fig. 5, is small, but it is clearly observed in both modelling and experiment for single- and multisoliton states. For solitons composed of three combs all propagating with different group velocities, the asymmetries of the individual combs are to be expected. Our experiment and modelling show clear spectra corresponding to soliton crystals. The spectral periodicity of the dominant lines is perfect. Extremely weak secondary spectra can sometimes be present or absent in the modelling and occur due to minor offsets of soliton locations from a perfect 2π/N angles along the ring circumference. This is a very natural, noise- and higher-order dispersion-sensitive feature that does not cast any doubt on the simulation results. These weak spectral backgrounds are always present in the experimental data shown in Fig. 5. As the number of solitons in a crystal grows, the central component of the signal spectra drops towards the comb level (our spectra should be judged on the signal, not the pump or idler). This effect is due to the background energy being taken by soliton cores. Stepping from the every-line soliton spectra to every 2nd, every 3rd and so on in the spectra is another clear signature of soliton crystals. Turing patterns usually pop up around their critical periodicity, which is often associated with much larger line-to-line spacings.\n5. The noise measurements are not thorough enough and lack a lot of detail to be able to claim low-noise operation. The authors do not report any detected power. If no light or light below the NEP is sent to the photodiode, only the noise floor appears. Therefore, it would help to compare their measurement with a noisy state to show the (multi)-soliton regime low-noise operation. The authors also fail to explain the beat in the noise measurement, which is puzzling. The system is not described as a breather, and no spectral overlap among the three comb components appears (which is not stated but should not be on the same frequency grid, but rather offset from one another). Hence, without a better explanation of the noise measurement, the soliton regime demonstration remains unconvincing.\nResponse:\nTo address this point in the revised manuscript, we directly measured the soliton coherence by performing a heterodyne beat measurement between a selected comb line and an independent tunable laser, see Fig. 3d. The new noise characterization setup can be found in Fig. 3a.\nMoreover, for single-soliton, other than the newly performed heterodyne measurement, we have also characterized the repetition rate measurement of the hyperparametric single-soliton by injecting it into the cascaded intensity-phase modulators, see new Figure S3 and new Section III in the supplementary document. For multi-solitons and soliton crystals, we have added the noise measurement results discussion in line 470 of the main text and Fig. 5b.\n6. The ring resonator’s wide width allows for multi-mode operation at the signal. Because the sidewalls are not straight, avoided mode crossings could occur. This has been shown to enable high-power multi-mode family OPO generation (see ref 21) with minimal XPM-mediated loss channels (see Stone et al., Phys. Rev. Applied 17, 024038, 2022). The spectrum in Figure 3b is notably clean, resembling the normal-dispersion regime main pump (see 24), despite the main pump being in the anomalous regime and at very high power. The authors do not measure the mode in the O-Band, where lasers are directly available. Their claim that they work with the same mode family as the main pump relies on their dispersion model, which, given Figure 2e and the high measurement noise, is not convincing. I believe it is a crucial point to clarify as this could lead to significant discrepancy between the theory and experiment.\nResponse:\nWe thank the reviewer for raising this important point. We carefully reviewed the transmission data and found that some resonance wavelengths were extracted incorrectly due to the strong interaction of the low-extinction ratio TM modes with the FP modes induced by the reflection between the waveguide facets. We have carefully checked the data and the corrected data are shown in Figs. 2d-2f. The FSR agreement between the simulation and measurement was greatly improved. Below is a zoom-in view of the one resonance case.\nAlso, to further confirm the measurements, we have acquired an O-band laser (1260-1360 nm) to measure the transmission. Corresponding data have been added to the Figs. 2d and 2e. The measurement and simulation also show a good overall agreement.\nAdditionally, we have pumped different modes and all of them can excite the OPO comb, as illustrated in Fig. S2b, which agrees with the simulation very well. Such OPO behavior are also demonstrated across several resonators with similar dimensions and all of them can support OPO operation with the same behavior which can be predicted by the simulation, as shown in Fig. S4. Modeling and measurements confirm that the signal and idler are the same mode family as the pump.\nMinor remarks:\n• The authors claim the pump is quasi-CW based on the azimuthal field profile. However, a comb clearly forms from XPM interaction with the hyperparametric soliton, altering the pump’s azimuthal profile. It is mostly CW-driven but not quasi-CW; otherwise, no comb would appear.\nResponse:\nYes sure, but we see no difference between the mostly-CW and quasi-CW cases.\n• It is unclear why the authors use a pulley configuration when the system’s width is limited. A straight waveguide could offer similar coupling performance. Unless system losses must balance perfectly, since the parametric interaction occurs on-chip, the signal’s Qc affects only extraction efficiency, not parametric interaction efficiency.\nResponse:\nSuch pulley coupling was used to enhance the extraction efficiency and suppress the Raman scattering [7, 48] through the engineering of the Q factors. In the previous demonstrations, we found that only main-pump combs can be generated when the pump is in the under-coupling state. We intended to use the pulley coupling to enhance the coupling rate for the pump. While later, we noted that the system losses must be balanced carefully for soliton generation and pulley coupling offers more flexibility.\n1. We thank Reviewer #1 for the positive evaluation of our revisions.\n2. We thank Reviewer #2 for the positive evaluation of our revisions and address further comments below.\n(1) Microresonators are most frequently characterised in terms of their repetition rates matching the Free Spectral Range at specific frequencies, which is a more natural and widely adopted description of the physics of microresonator frequency combs if compared to the group velocity language natural in the context of waveguides. Equation linking a group index, n_g, to the rep rate, D_1, is well known, D_1=(c/n_g) / (2*pi*R), where c is the vacuum light speed, c/n_g is the group velocity and R is the resonator radius. This, for example, gives 2.088 for the signal group index. Values of the repetition rates and their role in the observed effects are given and discussed in our manuscript; see, e.g., the text lines 247-249 and 676-678, and figure 2e. Furthermore, the experimental techniques we use measure the repetition rates directly, see section III in the supplemental materials. For these reasons, including values of the group indices and group velocities in the manuscript text would make the text appear disconnected from the standard methodology and techniques.\n(2) By fitting the measured power spectra of the signal part of the soliton with a sech² function, the corresponding spectral width is found to be 2.0 THz. Assuming transform-limited pulses, this width corresponds to a pulse duration of 158 fs (FWHM). We have added the corresponding sentence in the main text (see lines 304-308): “By fitting the signal spectrum with a sech-squared function, we found that the soliton has a 3-dB bandwidth of 2 THz, giving a transform-limited pulse duration of 158 fs at full width half maximum.”\n3. We thank Reviewer #3 for the positive evaluation of our revisions and address further comments below.\nThe reviewer further comments on their previous point that this is not a new soliton but “rather a new method for generating the driving field.” From the first version of our manuscript, we stated that our qualitative interpretation is in fact close to this point of view, see, in particular, see lines 667-675. However, this interpretation is not in contradiction and co-exists with the fact that the effective monochromatic pump field (entering the signal equation) is not a constant field as in the LLE model, but is a function of the true physical pump and idler fields satisfying two further equations. Therefore, even with this simplified interpretation, this is not an LLE system. We have added a new sentence to clarify this further in the Method section, lines 675-680: “It is important to note that, even in this approximation, the effective pump in the signal equation, Eq. (8b), is not an independent parameter like the pump term in the LLE model [4]. Instead, it is a variable whose stationary values and evolution are determined by the pump, a0(t), and idler amplitudes, a~-μ0(t), which satisfy additional equations.” On the further point, we consistently describe the idler and pump components of the solitons as quasi-CW fields.\nResponse to the XPM comment from Referee 3:\nComment: \"However, these XPM-induced additional comb teeth exist and cannot be dismissed from experimental observation.\"\nReply: This comment connects to the ones on the quasi-CW nature of the idler and pump fields. The XPM-induced effects the referee is referring to are reflected in our experimental and modelling data via additional comb teeth, and are explicitly mentioned next to Eqs. 11-13. The term \"quasi-CW,\" as used extensively in our discussions, explicitly indicates that the central CW component of a given group of modes, e.g., clustered around the idler, is dominant within this group, where XPM and other effects induce the weaker comb teeth."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have address my comments. I support the publication of the manuscript in Nature Communications.", "reviewer_2": "The manuscript has substantially improved. The rebuttal adds coherence and noise measurements, repetition-rate measurements, corrects the dispersion/FSR extraction, and reports absolute power and conversion efficiency.\nTwo focused items remain:\n(1) Group-velocity mismatch. The response is still vague. Please quantify GVM and report ( \\beta_{1} ) or ( n_g ) at the pump/signal/idler centers.\n(2) Pulse-width metric and experimental cross-check. State unambiguously whether you use intensity FWHM; if 1/e or 1/e(^2) widths appear, convert them to FWHM.", "reviewer_3": "The authors have significantly improved their manuscript and I commend them for providing new experimental data. The repetition rate measurements, the heterodyne beat measurements, and the additional data showing that the soliton forms at the signal frequency rather than being a pump-centered DKS spectrally translated via XPM and OPO. The noise measurements are also now convincing.\nHowever, as stated in my first report, the authors do not demonstrate a new soliton regime but rather a new method for generating the driving field. The signal equation takes the form of a generalized Lugiato-Lefever equation in which the four-wave mixing term acts as an effective pump. Since this driving term is dominated by CW components from the nondegenerate OPO, the resulting signal soliton behaves as a standard DKS. Furthermore, coupled-LLE systems describing multi-color soliton dynamics have been studied extensively in the literature, as cited in my first report. This is not the first theoretical framework of its kind, even if the authors dismissed most prior work by claiming a different regime despite employing the same equation. The novelty lies in the pumping mechanism, which enables DKS generation outside the spectral window of the main pump, not in the soliton physics itself.\nDoes this distinction diminish the importance of their findings? I think not. Their work will be of interest for the community.\nShould this disagreement prevent publication? Again, I think not. This represents an opinion disagreement that time and the community will resolve.\nA minor point on \"quasi-CW\": the experimental data show comb teeth around the pump and idler, indicating that these fields are not physically CW. Otherwise, no comb would form around them, and only a single tone would appear in the experimental spectra. These waves are indeed dominated by the CW tone, which justifies the approximation of the parametric driving field of the DKS. However, these XPM-induced additional comb teeth exist and cannot be dismissed from experimental observation.\nGiven the solid results and overall contribution, I recommend publication in Nature Communications with the reservation that the authors address my comments either as a response or modifications."}, "author_response": ""}]

Decision letter