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A scanning electron microscopy (SEM) image of the dispersion-managed microresonator is shown in Fig. 1a (1). Figure 1a (2) depicts a conceptual picture of the Si3N4 microresonator, which consists of a waveguide with varying widths to provide the oscillating group velocity dispersion (GVD) along the cavity length (a detailed design of the waveguide width is shown in Fig. 1a (3)). Figure 1a (4) shows the GVD and nonlinear coefficient varying along the cavity length. Since the effect of the varying nonlinear coefficient is negligible compared to the changing GVD, we only consider the impact of GVD management in this work. Such an oscillating GVD results in the periodic stretching and compression of dispersion-managed dissipative solitons (DM-DSs) and is more resistant to breather soliton instability, increasing the attainable pulse energy from a passive resonator34 (see supplementary Information Section Ⅱ). In our current design, the microresonator waveguide width first changes from 1 µm at the coupling region to 4 µm in the middle of the microcavity and then changes back to 1 µm in the second half of the microcavity, resulting in a GVD oscillation from -59 fs2 mm-1 to +58 fs2 mm-1 (see supplementary Information Section Ⅰ for details). We solve a system consisting of the cavity coupling equation and the nonlinear Schrödinger equation numerically to describe the DM-DS formation physics:
Fig. 1 Dispersion-managed dissipative Kerr soliton generation with an adiabatically tapered Si3N4 microring.
a (1) SEM image of the dispersion-managed microcavity. (2) Conceptual schematic of the tapered Si3N4 microring and the breathing pulse evolution along the cavity length. The varying widths of the cavity waveguide provide an oscillating group velocity dispersion (GVD) and varying nonlinear coefficient. (3) The waveguide width changes at different locations of the microcavity. (4) The GVD (blue curve) and nonlinear coefficient (red curve) at the pump wavelength (1598.5 nm) change at different locations of the microcavity. b Cold cavity transmission of the tapered Si3N4 microring, measured with a high-resolution coherent swept wavelength interferometer (SWI) (see Supplementary Section Ⅰ). The existence of higher-order transverse modes is not observed across the wavelength region of interest. The Q factors and wavelength-dependent free spectral range (FSR) are determined from the transmission measurement. c COMSOL-modeled GVD and third-order dispersion (TOD) of the Si3N4 waveguide with respect to the waveguide width, taking into consideration both the waveguide dimensions and the material dispersion. At the pump wavelength of 1598.5 nm, the path-averaged GVD and TOD are-2.6 fs2 mm-1 and-397 fs3 mm-1, respectively. The red dots are the measured GVD for waveguides with widths of 1.2 μm, 1.5 μm, and 1.6 μm, showing good agreement with the simulation results (see Supplementary Section Ⅰ). d Wavelength dependence of the FSR, determining the residual non-equidistance of the modes, D=-β2ωFSR2c/2πn, of 54 ± 3 kHz. The extracted GVD is anomalous at-6.4 ± 0.4 fs2 mm-1, in good agreement with the simulation result. e RF amplitude noise of the Kerr frequency microcomb in different states, showing the transition into the low-phase noise state with amplitude noise reaching the detector background. The 5 GHz scan range is more than 50 times the cavity linewidth. f The measured optical spectrum of the dispersion-managed dissipative soliton, which fits better to a Gaussian profile (red curve) than a sech2 profile (blue curve). The 3 dB bandwidth of the measured spectrum is 4.78 THz, and the corresponding FWHM of the transformed-limited pulse is 92 fs. Inset: simulated comb spectrum, also showing a better match with a Gaussian profile than a sech2 profile. g Pulse shape (black line) and temporal phase (blue line) retrieved from the FROG measurement. The FWHM pulse duration is measured to be 167 fs$$ \left\{ {\begin{array}{*{20}{l}} {{A^{m + 1}} = \sqrt {T{P_{in}}} + \sqrt {1 - T} {A^m}\exp \left( { - j\delta } \right)}\\ {\frac{\partial }{{\partial z}}{A^m} = - \frac{\alpha }{2}{\mathit{\boldsymbol{A}}^m} - \frac{j}{2}{\beta _2}\left( z \right)\frac{{{\partial ^2}}}{{\partial {t^2}}}{A^m} + j\gamma \left( z \right){{\left| {{A^m}} \right|}^2}{A^m}} \end{array}} \right. $$ where m, β2, α, T, γ, and δ are the number of roundtrips, GVD, cavity loss, coupling loss, Kerr nonlinear coefficient, and pump-resonance detuning, respectively. The cavity length is discretized into a total of 120 steps, and at each step, β2 and γ are re-evaluated based on the local waveguide geometry. Figure 2a plots the numerically solved DM-DS evolution within the microcavity. The asymmetry of the pulse width and peak power shown in Fig. 2a ii and iii is due to the chirp change (see the detailed simulation result and discussion in Supplementary Section Ⅱ). For each round trip, the DM-DS experiences a cycle of stretching and compression between 32.3 fs and 29.9 fs (Supplementary Fig. 6).
Fig. 2 Observing the evolution and transition dynamics with an ultrafast temporal magnifier.
a, i: NLSE-modeled dissipative Kerr soliton dynamics with an oscillating pulse width due to dispersion management. The pulse width changes along the cavity length length. ii: The variation in the FWHM of the dispersion-managed dissipative soliton along the cavity length. iii: The variation in the peak power of the dispersion-managed dissipative soliton along the cavity length. b Conceptual schematic of the ultrafast temporal magnifier (UTM), the time-domain counterpart of a high-speed digital microscope system. c Schematic setup of the UTM system. Along the fast time axis, the temporal structure of the intracavity field is magnified and captured by a real-time oscilloscope. The temporal magnification and time resolution of the UTM system are ×72 and 600 fs, respectively. Along the slow time axis, the evolution and transition dynamics are sampled optically with a stabilized femtosecond Mode-locked fiber laser. The frame rate of the UTM system is thus 250 MHz, limited by the repetition rate of the Mode-locked fiber laser. All of the electronics are commonly referenced to an Rb-disciplined crystal oscillator for accurate synchronization. Inset: two pulses separated by 30 ps, originally unresolved (black curve), are distinguishable via the UTM system. ECDL: external cavity diode laser. WDM: wavelength-division multiplexing. EDFA: erbium-doped fiber amplifier. AWG: arbitrary waveform generator. EOM: electro-optic modulator. PD: photodetector. D1, D2, and Df are the dispersions for the UTMs (see Supplementary Table Ⅰ). To increase the SNR and reduce the aberration, the measurement frame rate is reduced to 25 MHz by picking 1 pulse out of 10 with an EOM driven by a synchronized AWG (red dashed box). d Total power transmission as the pump frequency is scanned across a cavity resonance at a speed of 2.1 THz s-1 for an on-chip pump power of 30 dBm. The step signature is characteristic of the low-phase noise soliton state (state 2). e The dissipative Kerr soliton dynamics around these transmission steps are studied and portrayed with the UTM systemThe Si3N4 microresonator design not only provides an oscillating GVD along the cavity length but also suppresses higher-order mode families, which suppresses the mode-crossing-induced perturbation to dissipative soliton generation, leaving only the rare mode-crossing caused by TE-TM mode hybridization from the fabrication imperfection35. Both the bus waveguide and the cavity waveguide in the coupling region are designed to be strictly single-mode, thereby ensuring selective excitation of the fundamental mode and suppressing other transverse mode families, leading to near-ideal coupling37. The cavity transmission around the pump mode is plotted in Fig. 1b, with no observable higher-order transverse modes in the transmission spectrum. Near-critical coupling is attained, with a loaded Q of 1.9 × 106 and a cavity loading of 90% for the pump mode (see Supplementary Information Section Ⅰ for details). The microresonator free spectral range is measured to be ≈88 GHz. The GVD and TOD shown in Fig. 1c are calculated with a commercial full-vectorial finite-element-method solver (COMSOL Multiphysics), taking into consideration both the cavity geometry and the material dispersion. The path-averaged GVD is slightly anomalous at -2.6 fs2 mm-1, leading to intracavity pulse dynamics in the stretched-pulse regime. Operation in the stretched-pulse regime with near-zero GVD has been demonstrated to be beneficial in femtosecond Mode-locked fiber lasers for achieving a narrow linewidth, low phase noise, and attosecond timing jitter38, which are important merits for advancing microwave photonics39 and coherent pulse synthesis40. A similar timing jitter reduction from decreasing the net cavity dispersion is also theoretically predicted in Kerr-active resonators41. To verify the near-zero path-averaged GVD, high-resolution frequency-comb-assisted diode laser spectroscopy42 is employed to characterize the cold cavity properties of our designed Si3N4 microresonator. With active control of the on-chip temperature, passive shielding against acoustic noise and an absolute wavelength calibration with the hydrogen cyanide gas standard, the method provides a GVD accuracy of 0.4 fs2 mm-1, determined as the standard deviation calculated from 10 independent measurements (see Supplementary Information Section Ⅰ for details). The mean value of the net cavity GVD from the 10 measurements is -6.4 fs2 mm-1 (Fig. 1d).
Tuning the pump frequency into the cavity resonance from the blue side with a scan speed of 3.5 THz s-1, a stable DM-DS can be observed, with a typical optical spectrum shown in Fig. 1f. The 3 dB bandwidth of the measured spectrum is 4.78 THz, and the transform-limited FWHM pulse duration is 92 fs. Notably, the measured DM-DS spectrum fits well with a Gaussian profile (red line) rather than the squared hyperbolic shape (blue line), indicative of stretched-pulse operation35. The inset plots the numerically simulated comb spectrum, which also shows a better match with a Gaussian profile than a sech2 profile. Furthermore, the TOD effect is augmented due to the near-zero GVD, resulting in the observed asymmetry in the optical spectrum and carrier frequency shift (green dashed line) from the pump. A stable pulse train and low-noise operation is confirmed by an RF amplitude noise spectrum measurement (Fig. 1e), showing that the noise level approaches the detector background, and a frequency-resolved-optical-gating (FROG) measurement (Fig. 1g), showing consistently low retrieval errors below 10-2. A singlet DM-DS pulse with a negative chirp, in qualitative agreement with the numerical simulation, is retrieved from the FROG spectrogram. We note that a quantitative comparison cannot be made due to the limited bandwidth of the C-band Er-doped fiber amplifier used in our FROG measurement (see Methods and Supplementary Information Section Ⅲ for details).
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While the temporal structure of the intracavity field is detailed at the sub-picosecond time scale, the evolution and transition dynamics are portrayed at a much slower sub-microsecond time scale, which is associated with the cavity photon time of the microresonator. The orders-of-magnitude difference in the time scale between the two time dimensions poses an experimental challenge for capturing a comprehensive picture of the dynamics. Here, we demonstrate that UTMs are invaluable solutions that can fully characterize the evolution and transitional dynamics of dissipative Kerr solitons. A UTM is the time-domain counterpart of a high-speed digital microscope system, utilizing the space-time duality principle where diffraction in space and dispersion in time share the same mathematical expression29 (Fig. 2b). Incorporating suitable GDDs (D1 and D2) before and after the four-wave mixing stage in a highly nonlinear fiber (HNLF), a temporal magnification of ×72 and a time resolution of 600 fs are attained in our UTM, enabling a detailed depiction of the dissipative soliton's temporal structure. The evolution and transition dynamics, on the other hand, are sampled optically with a synchronized and stabilized femtosecond Mode-locked fiber laser (MenloSystems GmbH). The frame rate of the first UTM is 250 MHz, higher than the cavity resonance linewidth of 100 MHz, and is determined by the laser repetition rate stabilized to an Rb-disciplined crystal oscillator (Fig. 3c, see Methods and Supplementary Section Ⅳ). At a frame rate of 25 MHz for the second UTM and with a 190 ps single-shot record length, we can record a complete intracavity field of 16 roundtrips in real time, but we miss the information between each measurement. Then, multiple frames are stitched to reconstruct the soliton evolution through the whole transmission step. As long as the intracavity field pattern evolves much slower than 25 MHz, our technique could provide valuable information about the intracavity waveform evolution.
Fig. 3 UTM-enabled comparison of the stability zone and temporal dynamics between static and dispersion-managed dissipative Kerr solitons.
a Total power transmission (left panel) and the 2D evolution portrait (right panel) of static soliton formation in a homogenous microresonator with a measured GVD of-33.1 fs2 mm-1. A single soliton is only observed in the last transmission step and remains stable for the pump resonance detuning range of 30 kHz, where the cavity loading is reduced to 30%. The on-chip pump power is 30 dBm. b, c Total power transmission (left panel) and the 2D evolution portrait (right panel) of dispersion-managed soliton formation in the tapered microresonator, showing increased stability zones at a higher pump power than static solitons. In panel b with an on-chip pump power of 30 dBm, transitions from a chaotic oscillation to a multiple soliton state and eventually to a single soliton state are also observed. However, unlike the static soliton formation shown in a the transition from multiple to single soliton states does not require an ejection of excessive intracavity power. Instead, a monotonic increase in the cavity loading from 38% to 52% is measured along the gradual transition into a single soliton state. In panel c as the on-chip pump power increases to 32 dBm, a single soliton state can persist across the whole transmission step and remain stable for a pump resonance detuning of 400 kHz, while the cavity loading monotonically increases to 57%. At this pump power level, a low-noise stable soliton state is not observed in a homogeneous microresonator. Insets: measured pulse shapes at the pump resonance detuning denoted by the white dashed lines in the 2D mappingsFigure 2d shows the total transmission of the microresonator with respect to the pump resonance detuning when the pump frequency is scanned across the resonance at a speed of 2.1 THz s-1. The transmission deviates from a Lorentzian lineshape and follows a triangular profile defined by the combined effect of the thermal and nonlinear resonance shifts, resulting in optical bistability that eventually leads to dissipative Kerr soliton formation. At the edge of the resonance, multiple discrete transmission steps are observed, which have been identified as important attributes of dissipative Kerr soliton (Fig. 2e). As solitons are formed, excessive optical power is ejected from the cavity, resulting in a stepwise increase in the total transmission10. To compare the difference in the transition dynamics and stability zone between static and dispersion-managed dissipative Kerr solitons, we synchronize both the transmission and UTM measurements with pump frequency scanning and focus on the dynamics around these transmission steps. A representative result of static soliton formation in a homogenous microresonator is summarized in Fig. 3a. The Si3N4 microresonator consists of a 1.2 × 0.8 μm2 uniform waveguide, with a loaded Q of 1.5 × 106, a cavity loading of 86%, and a measured GVD of -33.1 fs2 mm-1 (Fig. 1c). At the first transmission step where the cavity loading decreases to 45% (corresponding to a transmission of 55%), a doublet soliton state is reached and remains stable for a 160 kHz change in the pump resonance detuning before the second transmission jump. A further reduction in the cavity loading to 30% results in the formation of a singlet soliton state, which nevertheless only exists in a small stability region of 30 kHz. An on-chip pump power of 30 dBm is used in this example, but similar behavior is noted for pump powers of 24 dBm and 27 dBm. However, no stable soliton states are observed from the UTM when the on-chip pump power exceeds 30 dBm.