Introduction

Parametric nonlinear processes in integrated microresonators have attracted increasing attention in the scientific community. Of particular interest is the generation of chip-scale optical frequency comb using optical microresonators, commonly known as microcombs1,2. The emergence of microcombs holds great potential for a wide range of applications spanning microwave photonics3,4, optical communications5,6,7, precision measurements8,9,10,11,12,13,14, optical computing15,16,17 and quantum optics18,19.

Dispersion engineering of microresonators is one of the most important elements for efficient nonlinear processes in general and broadband microcomb generation in particular that require specific phase matching conditions. Previous research on microcombs predominantly focused on bright solitons within the anomalous dispersion regime. Conventionally, dispersion engineering involves the careful design of the cross-section of the ring waveguide, which imparts the requisite anomalous group velocity dispersion (GVD) necessary for degenerate four-wave mixing (FWM)—a fundamental mechanism underlying microcomb generation.

In contrast to bright solitons, dark pulses (or ‘platicons’) generated in normal-dispersion microresonators20,21,22,23 offer several advantages, including high energy conversion efficiency, thermal management-free operation, deterministic generation, and ease of achieving normal dispersion21,24. However, in the normal dispersion regime, phase matching conditions are not naturally satisfied, necessitating the presence of local anomalous dispersion for microcomb initiation25. The attainment of local anomalous dispersion, which involves a red shift of the pump mode or a blue shift of the side mode, can be accomplished by controlling the mode splitting26,27,28,29. However, the method of pump shifting suffers limitations stemming from (1) deterioration of robustness because of the thermal drift of the blue-shifted splitting mode and (2) higher threshold power for microcomb initiation as part of the pump light is coupled to other modes30,31. Moreover, the blue shift of a single side mode introduces odd-order dispersion components, which are thought to be responsible for the asymmetry observed in the spectral power distribution of the comb27,28,32,33. How to restore the spectral symmetry in the platicon comb while ensuring robustness and performance is crucial for various applications such as quantum optics and optical communications, which are sensitive to power discrepancies between comb lines located symmetrically with respect to the pump mode (at \(\mu =0\)), with a mode spacing of \(\mu =\pm N\).

In this study, we present a approach to generate stable, power-efficient microcombs with high spectral symmetry in normal dispersion microresonators. We introduce two local dispersion alterations positioned symmetrically on both sides of the pump mode, resulting in a symmetrically engineered dispersion profile. This configuration facilitates the spontaneous formation of mode-locked dark pulses that are not only highly stable but also exhibit spectral symmetry across various pump power levels and pump-resonance detuning. Furthermore, by adjusting the interval between the positions of the dispersion alterations and the pump mode, we achieve deterministic generation of a symmetric perfect platicon crystal in normal-GVD regime. This demonstrates the capability of our method to control and fine-tune the spectral characteristics of the generated microcombs. Our proposed approach provides a reliable means to access efficient microcombs, as demonstrated in the context of silicon nitride microresonators, and has the potential to be applied to other integrated nonlinear platforms.

Results and discussion

Principle of symmetrical dispersion-engineering for microcomb generation

We consider a Kerr microresonator with pure second-order dispersion (SOD), where the dispersion profile follows a parabolic function. In this case, the formation of a symmetric localized structure is theoretically possible20, although experimentally challenging in the normal GVD regime. However, when the influence of third-order dispersion (TOD) becomes significant and cannot be neglected, it leads to distortions in the comb spectrum and temporal profile34. As a result, there is a power discrepancy between symmetrically located comb lines. It is worth noting that not only TOD but also all odd-order dispersion components contribute to the odd-function component in the dispersion profile, thereby breaking the spectral symmetry of the comb system. Consequently, the formation of symmetric platicon is prevented.

To investigate the relation between dispersion profile and comb spectrum symmetry, we transform the Lugiato-Lefever equation (LLE) to the frequency domain (see the Supplementary Note 1 for its derivation):

$$\frac{{d\widetilde{A}}_{\mu }}{{dt}}=-i{D}_{{{{{\mathrm{int}}}}}}(\mu ){\widetilde{A}}_{\mu }-\left(\frac{\kappa }{2}+i{\alpha }\right){\widetilde{A}}_{\mu }+{ig}\mathop{\sum }\limits_{{\mu }_{1}+{\mu }_{2}-{\mu }_{3}=\mu }{\widetilde{A}}_{{\mu }_{1}}{\widetilde{A}}_{{\mu }_{2}}{\widetilde{A}}_{{\mu }_{3}}^{* }+{\delta }_{\mu ,0}\sqrt{{\kappa }_{c}}{s}_{{in}}$$
(1)

where \(t\) is the slow time describing the evolution of the complex amplitude \({\widetilde{A}}_{\mu }\) in the frame rotating at the cavity FSR \({D}_{1}=\Delta {\omega }_{{FSR}}\). \(\alpha ={\omega }_{0}-{\omega }_{p}\) represents the pump-cavity detuning and \({\delta }_{\mu ,0}\) is the Kronecker delta. \(g\) is the nonlinear coupling coefficient. \({\kappa }_{c}\) denotes the coupling rate from bus waveguide to microresonator and \({|{s}_{{in}}|}^{2}\) represents the photon flux of the pump related to the pump power. The Total dispersion profile, \({D}_{{{{{\mathrm{int}}}}}}(\mu )={\omega }_{\mu }-{\omega }_{0}-\mu {D}_{1}=\frac{1}{2}{D}_{2}{\mu }^{2}+\frac{1}{6}{D}_{3}{\mu }^{3}+o({\mu }^{3})\), describes the cumulative effect of dispersion on the system and is typically expanded as a series. It can be decomposed into an even-function part, \({D}_{{{{{\mathrm{int}}}}},{even}}(\mu )=\frac{1}{2}{D}_{2}{\mu }^{2}+\frac{1}{4!}{D}_{4}{\mu }^{4}+\ldots +\frac{1}{2n!}{D}_{2n}{\mu }^{2n}\), and an odd-function part, \({D}_{{{{{\mathrm{int}}}}},{odd}}(\mu )=\frac{1}{6}{D}_{3}{\mu }^{3}+\frac{1}{5!}{D}_{5}{\mu }^{5}+\ldots +\frac{1}{\left(2n+1\right)!}{D}_{2n+1}{\mu }^{2n+1}\). For a stable frequency comb state with perfect spectral symmetry, \({\widetilde{A}}_{\mu }={\widetilde{A}}_{-\mu }\) and \(\frac{d{\widetilde{A}}_{\mu }}{{dt}}=\frac{d{\widetilde{A}}_{-\mu }}{{dt}}=0\) must be satisfied, which requires the odd-fuction part \({D}_{{{{{\mathrm{int}}}}},{odd}}\) to be zero (see the Supplementary Note 1 for details).

As shown in Fig. 1a, the conventional approach to local dispersion engineering involves introducing a sharp variation in the dispersion curve, which is otherwise slowly varying (considering only SOD here, for simplicity). This variation aims to create phase-matching conditions for the initiation of the frequency comb. However, the single discontinuity comprises multiple high-order odd dispersion terms, leading to a significant asymmetry in the comb spectrum. As alternative solution to mitigate this issue is to select the perturbed mode as the pump mode22,30. However, this configuration is handicapped by the fact that the pump light is strongly coupled to other modes at the split resonance mode, which increases the required input power to reach the comb initiation threshold. In addition, the use of the red-shifted splitting mode as pump mode may results in reduced robustness of the device, as large thermal drift of the blue-shifted splitting mode could render the red-shifted splitting mode inaccessible during the sweeping of the pump laser31,35. These challenges highlight the limitation and trade-offs associated with the conventional local dispersion engineering.

Fig. 1: Principle of symmetrical dispersion-engineering for microcomb generation.
figure 1

a Dispersion profile with single perturbation at mode \(\mu =-1\) is divided into even and odd components. The purple dashed lines indicate the additional phase shift induced by the second perturbation at mode \(\mu =+1\), which cancels out the odd component of the dispersion. b Microresonator with both inner and outer sidewall Bragg gratings enables the spontaneous formation of platicon microcomb in normal-dispersion regime. Inset: transmission spectrum where two selected mode splittings are independently controlled by the two sets of gratings with different numbers of periods. c Evolution of comb spectra with increased frequency detuning. The color gradient represents the optical power on a logarithmic scale. d Dispersion profiles and phase-matching condition for FWM. The cold- and hot-cavity resonances are marked by blue triangles and orange circles, respectively. The phase-matching conditions are marked by orange dashed lines (perfect) and the yellow thick lines (within a range of \(\pm \kappa /2\)). \({\delta }_{{SPM}}\) and \({\delta }_{{XPM}}\) are the phase shifts caused by Kerr nonlinearity with a relation of \({\delta }_{{XPM}}=2{\delta }_{{SPM}}\), while the local dispersion alterations have been included in both cold- and hot-cavity dispersion profiles. A: asymmetric dispersion engineering with a single dispersion alteration positioned at mode \(\mu =-1\). S: symmetric dispersion engineering with two dispersion alterations positioned at modes \(\mu =\pm 1\).

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In our study, we propose a solution to address this problem by independently controlling two dispersion alterations. We consider a pure-SOD cavity where a local dispersion alteration \({\delta }_{1}\) is introduced at mode \(\mu =-1\). Additionally, we position an additional perturbation, \({\delta }_{2}={\delta }_{1}\), at mode \(\mu =1\) to compensate for the dispersion profile’s asymmetry, effectively eliminating the odd components. Figure 1d dipicts the excitation of microcomb with asymmetric (left) and symmetric (right) local dispersion alterations. The phase-matching condition for FWM relies on the balance between nonlinearity and local anomalous dispersion, \(2{\delta }_{{XPM}}-2{\delta }_{{SPM}}={\delta }_{1}+{\delta }_{2}\), represented by the thick yellow line in Fig. 1d, where cross-phase modulation (XPM) is twice the self-phase modulation (SPM). Consequently, the frequency combs have the same initiation condition (i.e., the detuning point at a certain input power), while the sum of dispersion alterations \({\delta }_{1}+{\delta }_{2}\) remains fixed. It is important to note that in the single-dispersion-alteration scheme, the phase mismatch for modes \(\mu =\pm N(N\ge 2)\) is asymmetric. This asymmetry results in the generation of an asymmetric microcomb through cascaded FWM processes. However, with the introduction of symmetrical dispersion perturbations, as shown in the right panel in Fig. 1c, the simulated comb spectra exhibit stable microcomb initiation and platicon formation with a larger range of existence and improved spectral symmetry. The numerical simulations presented in our study are based on the LLE model (see Methods for details).

Microcomb generation with symmetrical dispersion engineering

In a microring resonator (MRR) with sidewall angular Bragg grating36, the coupling rate \({\kappa }_{{cw},{ccw}}\) between counterclockwise (CCW) and clockwise (CW) propagating modes is significantly enhanced at the Bragg reflection wavelength \({\lambda }_{B}=2\varLambda \cdot {n}_{{eff}}\), where \(\varLambda\) represents the pitch of Bragg grating and \({n}_{{eff}}\) is the effective refractive index in the ring waveguide. When the resonance wavelength \({\lambda }_{m}\) matches \({\lambda }_{B}\), the strong coupling between CW and CCW modes induced by the Bragg grating breaks their degeneracy, leading to a mode split. This mode split results in a blue-shifted resonance at \({\omega }_{m}^{+}={\omega }_{m}+{\kappa }_{{cw},{ccw}}\) and a red-shifted resonance at \({\omega }_{m}^{-}={\omega }_{m}-{\kappa }_{{cw},{ccw}}\), while the other resonance modes remain unaffected37. As a result, a local dispersion alteration is created at this specific point, altering the dispersion properties of the system. To achieve two local dispersion alterations, we employ two sets of gratings with different periods that are fabricated on opposite sides of the deuterated silicon nitride (SiNx:D) ring waveguide38,39, as shown in Fig. 1b. The inset transmission spectrum illustrates the presence of two mode splittings at the resonance modes \(\mu =\pm 1\). By adjusting the parameters of the two sets of gratings, we can independently control the strength and position of these two splittings (see Supplementary Note 2 for details). The controlling accuracy of the dispersion disruption is on the order of 100 MHz with the utilization of grating’s duty cycle as a new degree of freedom.

Figure 2 showcases one of our symmetrically dispersion-engineered devices and experimental evidence for platicon generation assisted by the two symmetrical mode splittings. In Fig. 2a, we present a fabricated SiNx MRR device with a radius of 80 μm and a waveguide cross-section of 2.25 μm × 0.6 μm (width × height). The inner sidewall of the microring features 1138 grating ‘teeth’ (i.e., first-order grating resonant with mode m = 569) with dimensions of 40 nm × 44 nm (lengths in the radial and angular dimensions), and a pitch of 446 nm. Similarly, the outer sidewall contains 1142 grating ‘teeth’ (first-order grating resonant with mode m = 571) with dimensions of 15 nm × 44 nm (radial and angular length), and a pitch of 446 nm. To achieve comparable coupling strengths, the radial length of the outer grating is smaller than that of the inner grating, as the intensity of the light field is higher in the outer sidewall. The measured transmission spectrum and dispersion profile exhibit a mode splitting of 0.83 GHz at 1539.6 nm and a mode splitting of 0.79 GHz at 1544.3 nm (see Fig. 2b, c). These mode splittings correspond to two local dispersion alterations symmetrically positioned on both sides of the pump mode (m = 570). The normal GVD of \({D}_{2}/2\pi =-8.3{{{{{\rm{MHz}}}}}}\) is obtained through parabolic fitting.

Fig. 2: Platicon frequency comb generation in microresonator enabled by symmetric frequency engineering.
figure 2

a Microscope photograph of one of the fabricated microresonators with inner and outer angular gratings. Inset: electron microscopy image of the coupling section of the microring. b Measured transmission spectrum around the pump wavelength 1542 nm. The arrows indicate two mode splittings (MS1 and MS2, red) occurring around the pump mode (orange). c Measured TE00 mode dispersion \({D}_{{{{{\mathrm{int}}}}}}\) of the device. \({D}_{2}/2{{{{{\rm{\pi }}}}}}=-8.3{{{{{\rm{MHz}}}}}}\) is obtained by parabolic fitting. d Existence map showing conversion efficiency of the platicon microcombs. e Simulated intensity profiles in the cavity at different input power and frequency detuning (normalized by resonance linewidth κ) indicated in (d). f Measured output optical spectrum at different input power. The gray dashed lines show the simulated output optical spectrum corresponding to different platicon states in (d, e).

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We conducted a numerical simulation using the LLE model. We set two identical dispersion alterations of 0.4 GHz at modes \(\mu =\pm 1\) with a normal GVD of \({D}_{2}/2{{{{{\rm{\pi }}}}}}=-8.3\,{{{{{\rm{MHz}}}}}}\). The device operated in the over-coupling regime to achieve high efficiency, and we used a coupling factor of \({\rm K}={\kappa }_{c}/{\kappa }_{i}=1.75\) in the simulation. The existence map presented in Fig. 2d illustrates the conversion efficiency CE of the dark pulse or platicon state. All the states obtained in our stability simulation are stable, and no chaos state has been observed. Figure 2e presents the simulated temporal waveforms in a fast time frame normalized by the roundtrip time \(\phi ={D}_{1}\tau\), which represents the angular location of the temporal structure around the resonator (in a frame rotating at the group velocity). As the detuning increases, the conversion efficiency also increases, and the simulated waveform exhibits additional low-intensity oscillation at the center of the dark-pulse structure. This behavior is consistent with previous observation in platicon microcombs30,32. It should be noted that the conversion efficiency of the platicon state primarily depends on the coupling factor \(K\). If the intrinsic quality factor \({Q}_{i}\) of the microresonator can be further improved (to >6 million), a stable platicon state with \({{{{{\rm{CE}}}}}} > 70 \%\) is achievable (see Supplementary Note 3 for more details). Such high \({Q}_{i}\) values have been successful demonstrated in LPCVD Si3N4 platform40,41.

In the experiment, we pumped the fabricated device with different EDFA output powers ranging from 600 mW to 2 W (equivalent to ~120 mW to 400 mW on-chip power), without filtering amplifier spontaneous emission (ASE) noise. We captured the stable states using an optical spectrum analyzer. The selected pump mode has a loaded \(Q\) of \(0.5\times {10}^{6}\) while a high \({Q}_{i}\) up to \(1.5\times {10}^{6}\) is maintained without observable degradation. Figure 2f illustrates the measured output optical spectra (blue solid line) and the simulated results (gray dashed line), which show good agreement. The transmission trace of the converted comb power (with pump light filtered out) in Supplementary Note 4 demonstrates a continuously increasing comb power, with the maximum measured value of CE reaching 27.6%. The intensity noise spectra were captured using an electrical spectrum analyzer, confirming the stable generation of mode-locked platicon microcomb without transitioning through a chaos state (see Supplementary Note 4). This is in contrast to the bright solitons observed in anomalous dispersion cases. Additionally, better symmetry in the comb spectra was observed at various pump power, consistent with our simulation.

To explore the range of existence for the stable platicon state at different dispersion perturbations, numerical simulations were performed with the same parameters, except for the values of symmetric local dispersion alteration \({\delta }_{1}={\delta }_{2}=\delta\). The simulation results indicate that a large \(\delta\) leads to a broader existence range for the platicon state during the sweeping of the pump laser. However, it is crucial to maintain the dispersion alterations within the stability limit to enable the platicon generation, since too large a dispersion disruption (\({\delta }_{s} > 1{{{{{\rm{GHz}}}}}}\) in this case) will results in a chaotic state and prevent stable platicon formation (see Supplementary Note 3). Based on the simulations, an alteration ranging from 0.4 GHz to 0.8 GHz is preferable.

In our study, we conducted both numerically and experimentally investigation to explore the spectrum symmetry by controlling the two dispersion alterations (\({\delta }_{1}\) for \(\mu =-1\) and \({\delta }_{2}\) for \(\mu =1\)). Figure 3a presents the localized dissipative structures and optical spectra in the \({\delta }_{1}/{\delta }_{2}\) plane. The microcomb spectrum exhibits distinct ‘wing’ and ‘horn’ features of a platicon state22,30. The positions of the two horns, which correspond to the oscillation frequency of the temporal switching wave at the rising and falling edges, serve as direct indicators of spectral symmetry. When the dispersion alteration \({\delta }_{1}={\delta }_{2}\), the generated microcomb exhibits perfect symmetry in its spectrum and temporal waveform. However, when \({\delta }_{1} \, \ne \, {\delta }_{2}\), asymmetry emerges, leading to several noteworthy phenomena. In the frequency domain, the comb horns shift to a lower (higher) frequency for \({\delta }_{1} \, < \, {\delta }_{2}\) (\({\delta }_{1} \, > \, {\delta }_{2}\)). This shift implies that there are more comb lines with significant power in the lower (higher) frequency region compared to the frequency of continuous wave pump laser. In the time domain, the non-identical dispersion alterations introduce odd derivative in fast time \(\phi\) in the LLE model (see Supplementary Note 1). This breaks the time reversibility of the temporal waveform, resulting in an asymmetric localized structure with respect to the pulse center. The oscillation frequencies of the two switching waves decrease or increase, causing the two horns to move closer or further away from the pump.

Fig. 3: Spectral symmetry control with two local dispersion alterations.
figure 3

a Localized dissipative structures and optical spectra in the two local dispersion alterations (\({\delta }_{1}/{\delta }_{2}\)) plane. The red region represents highly asymmetric comb states with a large difference \({{{{{\rm{|}}}}}}{\delta }_{1}-{\delta }_{2}{{{{{\rm{|}}}}}}\), while the blue region corresponds to symmetric comb states with a small difference \(\left|{\delta }_{1}-{\delta }_{2}\right|\approx 0\). b Experimental platicon comb states in three MRR devices with different value of \({\delta }_{1}-{\delta }_{2}\). The arrows indicate the position of the ‘horn’, which serves as direct evidence of spectral asymmetry. Top: \({\delta }_{1}-{\delta }_{2}=-0.19{{{{{\rm{GHz}}}}}}\). Middle: \({\delta }_{1}-{\delta }_{2}=0.02{{{{{\rm{GHz}}}}}}\). Bottom: \({\delta }_{1}-{\delta }_{2}=0.05{{{{{\rm{GHz}}}}}}\).

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The refs. 42,43, based on the nonlinear dispersion relation analysis, provide an intuitive understanding of the origin of this asymmetry. These two asymmetrical horns correspond to the crossing points between the platicon line and the dispersion parabola. When \({\delta }_{1} \, \ne \, {\delta }_{2}\), spontaneous pulse is excited by the CW pump, with a repetition rate smaller (\({\delta }_{1} \, > \, {\delta }_{2}\)) or larger (\({\delta }_{1} \, < \, {\delta }_{2}\)) than microresontator FSR. This asynchrony leads to a ‘tilted’ platicon line, shifting the crossing points to higher (\({\delta }_{1} \, > \, {\delta }_{2}\)) or lower (\({\delta }_{1} \, < \, {\delta }_{2}\)) frequency. Moreover, the repetition frequency of the asymmetric platicon varies with pump-cavity detuning (see Supplementary Note 5 for details), further influencing the spectrum symmetry. This tunable repetition rate, however, may lead to reduced stability of the microcomb under environmental changes in the absence of a feedback system.

In Fig. 3b, we present the experimental generation of microcomb in MRR devices with different values of \({\delta }_{1}\) and \({\delta }_{2}\). Firstly, we investigated MRR device 1, which had a single dispersion alteration \({\delta }_{2}=0.19{{{{{\rm{GHz}}}}}}\). The corresponding position of this device in the \({\delta }_{1}/{\delta }_{2}\) plane is indicated in Fig. 3a. The measured optical spectrum of the generated platicon microcomb exhibited two horns located at mode \(\mu =16\) and \(\mu =-31\), demonstrating significant spectral asymmetry. To improve the spectral symmetry, we applied two dispersion alterations with a small difference of \(0.03{{{{{\rm{GHz}}}}}}\) in MRR device 2, which was the same device characterized in Fig. 2f. The output optical spectrum of this device indicated a symmetric microcomb with two horns located at mode \(\mu =17\) and \(\mu =-15\), respectively. In MRR device 3, we set the second dispersion alteration \({\delta }_{1}\) to be significantly greater than the first one \({\delta }_{2}\). Specifically, we used \({\delta }_{1}=0.31{{{{{\rm{GHz}}}}}}\) and \({\delta }_{2}=0.26\,{{{{{\rm{GH}}}}}}{{{{{\rm{z}}}}}}\). As a result, the two horns of the generated microcomb moved to higher-frequency modes \(\mu =34\) and \(\mu =-23\), respectively. Moreover, they exhibited an opposite spectral skewness compared to MRR device 1. These experimental results were in excellent agreement with the simulated results, highlighting the decisive influence of (\({\delta }_{1}-{\delta }_{2}\)) on the spectral asymmetry. It’s worth mentioning that the green circles in Fig. 3b indicate discontinuity, which can be attributed to avoided-mode-crossing (AMX). Optimized coupling designs can help mitigate this effect and improve the overall performance of the system44,45.

Deterministic generation of perfect platicon crystals with high spectral symmetry

The perfect soliton crystal (PSC) is one of the soliton microcombs with \(N\times\)FSR comb line spacing corresponding to temporally ordered ensembles of \(N\) soliton pulses regularly arranged in the microresonator, where \(N\) is an integer46. In contrast to the symmetric spectrum profile of PSC with anomalous dispersion, most of the reported \(N\)-FSR-spaced microcombs in normal dispersion microresonators show large spectral asymmetry because of a single dispersion alteration (see Supplementary Note 6).

To address this issue and achieve symmetric perfect platicon crystal (PPC) generation, we introduced two symmetric dispersion alterations of ~0.4 GHz at modes \(\mu =\pm N\) in normal-dispersion microresonators with a FSR of 298 GHz. The locations of the two mode splittings corresponding to these alterations are clearly indicated in the transmission spectra in Fig. 4a. Experimental results in Fig. 4b demonstrate the successful generation of \(N\times {{{{{\rm{FSR}}}}}}\) platicon crystals with symmetric output optical spectra, which are in good agreement with the simulation results. The simulated intensity profiles within the microresonator reveal a series of orderly arranged temporal dark pulses, forming the platicon crystals. The number of platicons is determined by the spacing between the pump mode and the defect modes \(N\), which can be easily designed by adjusting the number of periods of the inner and outer gratings. Throughout the evolution, all the generated PPCs exhibit low intensity noise levels below our experimental sensitivity, further highlighting their stability and high-quality operation.

Fig. 4: Deterministic generation of perfect platicon crystals with high spectral symmetry.
figure 4

a Measured transmission spectra of MRR devices with different interval between the positions of the dispersion alterations and the pump mode. The red arrows indicate two mode splittings occurring around the pump mode (indicated by orange arrow). b Measured (blue) and simulated (orange) output optical spectra when the positions of two defected modes are set to be from \(\pm 1\times\) to \(\pm 5\times {{{{{\rm{FSR}}}}}}\) away from the pump mode, corresponding to panel (a). c The simulated intensity profiles in the cavity corresponding to panels (a, b).

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In the experimental investigation, we made an observation regarding the conversion efficiency of PPCs in the normal-dispersion regime. We found that the conversion efficiency of these multi-platicon states is nearly equal to that of the single-platicon case (\({{{{{\rm{CE}}}}}}=27.6 \%\)). This is in contrast to the \(N\times\) enhancement typically observed in bright solitons. This observation is further supported by numerical simulations based on the LLE model (\({{{{{\rm{CE}}}}}}=20 \% \sim 30 \%\) for different numbers of platicons, see Fig. 4b). It should be noted that in our experimental and numerical results (Fig. 4c) the PPCs appear to be limited to dark-pulse states due to the large GVD and insufficient pump power. As the number of platicons increases, there are fewer oscillating tails between two dark pulses, and the characteristic ‘wing’ feature is weakened. This can give the appearance of Turing patterns, which are typically observed in anomalous dispersion microresonators. However, with carefully designed GVD and dispersion alteration \(\delta\), it is possible to achieve a bright-pulse state with a large number of platicons (see Supplementary Note 7).

To investigate bright-pulse states of PPCs in a numerical study, we utilized typical parameters derived from a 50-GHz microresonator. Specifically, we employed a normal GVD of \({D}_{2}/2{{{{{\rm{\pi }}}}}}=0.5{{{{{\rm{MHz}}}}}}\) and introduced two symmetrical dispersion alterations of \({\delta }_{1}={\delta }_{2}=0.1{{{{{\rm{GHz}}}}}}\) at \(\mu =\pm N\). The remaining parameters were determined based on experimentally measured values. Perfect platicon crystals with \(2\times\) and \(4\times {{{{{\rm{FSR}}}}}}\) were numerically obtained in symmetrically engineered dispersion profiles by simply tuning the continuous wave pump laser into the resonance mode \(\mu =0\) (see Fig. 5a, b). In contrast to bright soliton in anomalous dispersion microresonators, both multi-platicon and single-platicon states showed similar conversion efficiency, ranging from 30% to 40%. However, our observation unveiled a counter-intuitive behavior of PPCs: the power of comb lines at the wing region demonstrated an enhancement of \(\sim {N}^{2}\) for \(N\)-FSR PPCs. Figure 5a indicates a 6-dB enhancement of power per comb line in the wing region when the spacing between two adjacent comb lines is double. This finding suggests that a greater proportion of comb power is distributed in the flattened wing region rather than the center-lope region. Such a characteristic renders PPCs more appealing for certain applications such as optical communications.

Fig. 5: Observation of perfect platicon crystals.
figure 5

a Simulated output optical spectrum of perfect platicon crystals with \(4\times\), \(2\times\) and \(1\times {{{{{\rm{FSR}}}}}}\). The power difference between 4-platicon and 2-platicon at comb line \(\mu =40\) is 6.73 dB while the power difference between 2-platicon and 1-platicon at comb line \(\mu =40\) is 6.03 dB. b Simulated cavity power profiles of the platicon crystals. c Chaos microcomb with pump-defect mode spacing of \(2\) (\({\delta }_{s}\approx 0.25\,{{{{{\rm{GHz}}}}}}\; {{{{{\rm{at}}}}}}\; {{{{{\rm{mode}}}}}}\) \(\mu =\pm 2\)). d–f Experimental observation of perfect platicon crystals with \(4\times {{{{{\rm{FSR}}}}}}\) (\({\delta }_{s}\approx 0.4{{{{{\rm{GHz}}}}}}\) at mode \(\mu =\pm 4\)), \(2\times {{{{{\rm{FSR}}}}}}\left(\right.{\delta }_{s}\approx 0.25{{{{{\rm{GHz}}}}}}\; {{{{{\rm{at}}}}}}\; {{{{{\rm{mode}}}}}}\) \(\mu =\pm 2\)) and the single platicon microcomb (generated in the device 3 in Fig. 3b). Inset: calculated intensity profiles in the cavity corresponding to the simulated optical spectrum in gray line.

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By using the symmetrically dispersion engineering method, we successfully observed the formation of perfect platicon crystals with \(4\times\) and \(2\times {{{{{\rm{FSR}}}}}}\) in 148-GHz MRRs. Figure 5d, e present the measured output optical spectrum of the perfect platicon crystal, which are in good agreement with our numerical results. The simulated temporal waveforms show 6 and 5 intensity dips between two platicons in the 4-platicon and 2-platicon states, respectively. The generated perfect platicon crystals have higher conversion efficiency than the single platicon (see Fig. 5f) and possess two smooth wing regions on both sides of the pump, thanks to their high spectral symmetry.

In our experiment, not all the symmetrically dispersion-engineered MRRs can enter a PPC state successfully, even with precisely defined GVD and mode splittings. A significant interference factor is the randomly occurring AMX in our multi-transverse mode microring. Figure 5c presents one of the symmetrically dispersion-engineered MRRs initially generated a 2-FSR microcomb with the help of local dispersion alterations set at \(\mu =\pm 2\). However, the random dispersion perturbation caused by AMX induced a competition effect and finally entered a chaos state. We believe that this problem can be resolved by carefully optimizing the coupling structure44,45 or adopting special design for filtering high-order modes47.

Conclusions

In this study, we successfully demonstrated the generation of symmetric platicon microcombs by introducing two defect modes in microresonators with inner and outer sidewall gratings. Our approach of pumping the non-splitting mode centered between the two defect modes offers a pathway to stably access low-noise, high-efficiency platicon states with highly symmetric spectra in normal dispersion microring resonators. By controlling the spacing between the pump mode and the splitting modes, we deterministically generated symmetric perfect platicon crystals with selectable comb line spacings.

The key principle underlying our approach is the restoration of the even-symmetry of the dispersion profile. This scheme provides a robust method for generating symmetric platicon microcombs that can be applied to other integrated nonlinear platforms. This opens up possibilities for various potential applications that rely on optical spectrum symmetry and conversion efficiency, such as quantum optics and optical communications.

Methods

Numerical simulation

The simulation results are based on the split-step Fourier method and LLE model. The simulation were performed with 512 modes. In the asymmetric (A) scheme in Fig. 1c, a detuning of \({\delta }_{1}=2\kappa\) is added to the splitting mode \(\mu =1\). In the symmetric (S) scheme, the detuning of \({\delta }_{1}={\delta }_{2}=\kappa\) is added to the two splitting modes \(\mu =\pm 1\). The remaining parameters are based on the measured parameters of our devices: \({{{{{\rm{FSR}}}}}}=148\,{{{{{\rm{GHz}}}}}}\), \({D}_{2}/2{{{{{\rm{\pi }}}}}}=-2.6\,{{{{{\rm{MHz}}}}}}\), \({Q}_{i}={Q}_{c}=1.5\times {10}^{6}\), \({\omega }_{0}/2\pi =193.4\,{{{{{\rm{THz}}}}}}\).

For the stability map simulations presented in Fig. 2d, the simulation parameters were based on the measured values: \({{{{{\rm{FSR}}}}}}=298\,{{{{{\rm{GHz}}}}}}\), \({D}_{2}/2{{{{{\rm{\pi }}}}}}=-8.3\,{{{{{\rm{MHz}}}}}}\), \({Q}_{i}=1.4\times {10}^{6}\), \({Q}_{c}=0.8\times {10}^{6}\), \({\omega }_{0}/2\pi =193.4\,{{{{{\rm{THz}}}}}}\),\({\delta }_{1}={\delta }_{2}=0.4\,{{{{{\rm{GHz}}}}}}\). The system was initialized with a single platicon solution at various pump-cavity detuning α and pump power \({P}_{{in}}\), and was propagating for about 50,000 roundtrips to reach the stable state.

Device fabrication

The SiNx:D film with a thickness of 600 nm is deposited on a silicon wafer with a 3 μm thick thermally-grown silica (SiO2) layer at a temperature of 270 °C, using deuterated silane (SiD4) and pure N2 as the source gases. The layer is deposited in one continuous run utilizing the inductance-coupled plasma chemical vapor deposition (ICP-CVD) technology without high-temperature annealing and CMP polishing. The refractive index of the SiNx:D film at 1550 nm is measured by ellipsometry as about 1.96, which is used for the simulation of mode profiles and dispersion. The thickness is chosen to satisfy the normal dispersion condition of platicons. Devices in this work are fabricated solely by subtractive processing. The pattern of MRR devices is defined by means of electron beam lithography (EBL) in AR-P 6200 resist with a thickness of 800 nm and then is transferred to the SiNx:D film by reactive ion etching (RIE) with the gases of CHF3 and O2. The etched waveguides, with the photoresist removed, are cladded with an ICP-CVD silica layer of 2.5 μm, which is deposited using silane (SiH4) as the source gas. The width of the MRR waveguide is designed to be 2.25 μm, aiming to minimize optical field overlap with the etched waveguide sidewalls while avoiding excessive higher-order modes. The width of the access bus waveguide is set to 1.6 μm, enabling selective excitation of TE00 mode in the MRR waveguide. Pulley-coupling is utilized in microresonators with a radius of 160 μm to suppress high-order modes. Inverse taper design is applied to enlarge mode size of light and reduce the chip-to-fiber coupling loss.

Device characterization

The transmission spectra of MRRs were measured using a Keysight 8164B Lightwave Measurement System. A narrow-linewidth tunable laser (Keysight 81606 A TLS) was used as the light source, and it was edge-coupled through a single-mode lensed fiber to the inverse-tapered SiNx:D waveguides with a tip width of 350 nm. The Loaded \(Q\) values obtained by fitting each resonance using a Lorentzian line shape model or a resonance doublet model48 with excellent goodness-of-fit (GOF) statistics throughout the whole wavelength range were used to extract \({Q}_{i}\) and \({Q}_{c}\).

To characterize the dispersion of SiNx microresonators, a fiber MZI consisting of two optical couplers with a splitting ratio of 50:50, and a zero-dispersion fiber with a length of 40 m was utilized to calibrate the frequency difference49 and the interferometric period is around 5 MHz. The probe light was divided into two channels, with one channel passing through the SiNx:D MRR and the other through the fiber MZI. Then, they were received by two photodetectors (Thorlabs, PDB482C-AC) and monitored by an oscilloscope (SIGLENT SSA3075X Plus) to record transmission traces.

OFC generation setup

A tunable continuous wave laser (Toptica DLC CTL 1550), amplified by an EDFA, was used as the pump light for the MRRs. The generated optical signal from the through port was divided into three branches for further characterization. The first branch, with the pump laser filtered out, was monitored by a photodetector and fed into a digital oscilloscope (SIGLENT SDS6204 H12 Pro) to track the comb power evolution. The second branch was connected to an optical spectrum analyzer (Anritsu MS9740A) to record the spectra. The third branch was used to measure the intensity noise and coherence of the Kerr comb using an electric spectrum analyzer (SIGLENT SSA3075X Plus).