Fig. 1. Illustration of the bichromatic pumping scheme. The ACWB consisting of N optical pulses is breathing in the nonlinear optical microresonator. The red solid curve represents a breathing maximum while the blue dashed curve represents a breathing minimum. The corresponding averaged optical spectrum features a quasi-triangle envelope (in logarithmic scale).

Fig. 2. Numerical simulations of a typical ACWB state with CLS=13×FSR under f12=13.5, f22=3, ζ0=−0.8, and Δζ0=−6.8. (a) Intracavity intensity patterns of the ACWB sample at successive breathing minimums (SP1 and SP2), breathing maximums (CP1 and CP2), and an approximate frequency-doubled moment (A) depicted in (b). (b) Evolution of the intracavity power over 10 power breathing periods. (c) Evolution of the intracavity pulse waveform. (d) Optical spectra of the ACWB were sampled at seven moments over a power breathing period. The averaged spectrum over one period features a quasi-triangle envelope (in logarithmic scale) with some enhanced comb lines.

Fig. 3. (a) Simulated averaged optical spectrum of the ACWB state with CLS=13×FSR. The red dashed lines indicate the triangle envelope of the spectrum, and the enhanced comb lines are marked by the red arrows. (b) Simulated power evolution of the comb lines around the center (Pcenter) and wings (Pwings) as depicted in (a) (Pcenter and Pwings are scaled to the same range). (c) Simulated power evolution of the comb lines with mode index μ=13 and μ=26, and the numbers in parentheses denote the corresponding breathing depth.

Fig. 4. (a) Schematic of the experimental setup. The inset shows the scanning electron microscopy image of a Si3N4 microresonator with the radius of 240 μm. CW Laser: continues-wave laser; DP-MZI: dual-parallel Mach-Zehnder interferometer; VCO: voltage-controlled oscillator; AWG: arbitrary waveform generator; EDFA: erbium-doped fiber amplifier; FPC: fiber polarization controller; EC: electric coupler; OC: optical coupler; OSA: optical spectrum analyzer; OSC: oscilloscope; ESA: electronic spectrum analyzer. (b) Measured transmission spectrum (blue circles) of the pumped resonance centered around 1564 nm. The Lorentz fitting curve (red solid line) indicates a loaded quality factor (Qload) of 1.2×106. (c) Measured (blue circles) and FDE simulated (red solid line) integrated dispersion (Dint=ωμ−ω0−μD1) of the fundamental quasi-TE mode as a function of relative mode number μ (μ=0 is at 1564 nm). Here, ω0 is the angular frequency of the pumped mode, ωμ is the angular frequency of the μth cavity mode relative to ω0, and D1 is the FSR measured at ω0. The inset shows the simulated mode profile of the fundamental TE00 mode.

Fig. 5. (a) Illustration of the two-step tuning method for ACWB generation. (b) Three optical spectra sampled during the two-step tuning process as depicted in (a). II: primary combs generated by the secondary pump. III: chaotic combs generated by the primary pump. IV: ACBW microcombs with CLS=13×FSR.

Fig. 6. (a) Optical spectrum of the ACWB microcomb with CLS=13×FSR. The red dashed lines indicate the triangle envelope of the spectrum, and the enhanced comb lines are marked by the red arrows. (b) The corresponding RF spectrum (blue trace) of the ACWB in (a) and the PD noise floor (red trace). The resolution bandwidth (RBW) of the measured traces is 200 Hz. (c) Recorded fast power evolution of the filtered comb mode μ=−13 and mode μ=−26.

Fig. 7. Optical spectra of the ACWB microcombs with different CLSs ranging from 10 to 17 times the FSR. The red dashed lines indicate the triangle envelope of the spectrum, and the enhanced comb lines are marked by the red arrows. The pumped mode is around 1564 nm for all microcombs. The parameters (total on-chip power, power difference, frequency difference) of the two pump lasers are set as (26.3 dBm, 5.6 dB, 530 MHz), (26.8 dBm, 5.6 dB, 535 MHz), (26.5 dBm, 6.5 dB, 520 MHz), (26.5 dBm, 6.5 dB, 520 MHz), (28.4 dBm, 6.4 dB, 544 MHz), (27 dBm, 6.5 dB, 495 MHz), (27.4 dBm, 6.5 dB, 500 MHz), and (29 dBm, 6.5 dB, 574 MHz), respectively. Due to the uncertainty of the edge coupling loss and the drift of the coupling state, the power parameter has some uncertainty from measurement to measurement.

Fig. 8. Optical spectra of the molecular crystal-like breathers with different CLSs ranging from 2 to 9 times the FSR. The parameters (total on-chip power, power difference, frequency difference) of the two pump lasers are set as (28 dBm, 5.6 dB, 484 MHz), (27 dBm, 6.2 dB, 493 MHz), (28 dBm, 6.5 dB, 531 MHz), (26 dBm, 6.3 dB, 500 MHz), (29 dBm, 7 dB, 630 MHz), (25 dBm, 5.7 dB, 430 MHz), (26.5 dBm, 6 dB, 478 MHz), and (27 dBm, 6.5 dB, 530 MHz), respectively.

Fig. 9. (a) Experimentally measured optical spectrum of a molecular crystal-like breather state with CLS=6×FSR. The total on-chip power, power difference, and frequency difference of the two pump lasers are set as 29 dBm, 7 dB, and 630 MHz, respectively. (b) Simulated averaged optical spectrum of a molecular crystal-like breather state with CLS=6×FSR. (c) Evolution of the simulated intracavity pulse waveform. (d) Simulated intracavity pulse waveform sampled at two breathing maxima depicted in (c) by the white dashed lines. The shaded regions in (d) indicate one period of the corresponding waveforms. For all the simulations, the parameters are set as f12=20, f22=40, ζ0=−2, and Δζ0=−10, respectively.

Fig. 10. High periodicity and irregular breathing phenomena. (a)–(c) Optical spectra of the microcombs with CLS equal to 9×FSR, 16×FSR, and 4×FSR, respectively. (d)–(f) RF spectra of the microcomb states in (a)–(c). The insets show the corresponding time-domain power evolution recorded by a fast photodetector. (d)–(f) show the period-2, period-4, and irregular breathing phenomena, respectively. The RBW of the measured traces is 100 kHz. The frequency difference is marked by the red arrow.

Fig. 11. (a) The spectrum of the single pumped breathing soliton. (b) The RF spectrum of the breathing soliton.

Fig. 12. (a) The spectrum of the single pumped Turing rolls. (b) The spectrum of the bichromatically pumped ACWB. (c) The spectrum of single pumped perfect soliton crystal (PSC).

Fig. 13. Probable parameters of the primary and secondary pump lasers illustrated in the stability chart of the monochromatic pumping LLE. The probable parameter regions for the primary pump and secondary pump are respectively shaded with blue and red. Note: the regions of the DKS breather, transient chaos, and spatiotemporal chaos are not displayed (see Ref. [28] for a detailed description).

Fig. 14. Simulated two-step tuning process. (a) Envelope of the intracavity power. The Roman numbers correspond to the four stages shown in Fig. 5(a) in the main text. The black dashed line separates the forward and backward tuning processes. (b) Evolution of the intracavity pulse waveform. The inset shows local details. (c) Evolution of the intracavity optical spectrum.

Fig. 15. Simulated high periodicity breathing and irregular breathing phenomena. (a) Period-2 breathing state for a 9-FSR CLS under f12=20, f22=3.5, ζ0=−2, and Δζ0=−10 (see Visualization 4). (b) Period-4 breathing state for a 16-FSR CLS under f12=14, f22=2.8, ζ0=−0.1, and Δζ0=−6.1 (see Visualization 5). (c) Irregular breathing state for a 4-FSR CLS under f12=21, f22=4, ζ0=−2.1, and Δζ0=−9.7 (see Visualization 6). In (a)–(c), top left: averaged optical spectrum; bottom left: RF spectrum of the corresponding microcomb over 200,000 roundtrips (the equivalent resolution bandwidth is 500 kHz); top right: evolution of the total intracavity power; bottom right: evolution of the intracavity waveform.

Fig. 16. Experimentally measured optical spectra of other molecular crystal-like breathers. The parameters (total on-chip power, power difference, frequency difference) of the two pump lasers are set as (27 dBm, 6 dB, 450 MHz), (29.3 dBm, 7 dB, 596 MHz), (26.6 dBm, 5.7 dB, 473 MHz), (27 dBm, 6 dB, 503 MHz), (28.4 dBm, 6.5 dB, 541 MHz), and (25.6 dBm, 6 dB, 453 MHz), respectively.

Fig. 17. (a) Optical spectrum of a period-2 breathing molecular crystal-like breathing state with CLS=2×FSR. (b) Evolution of the RF spectra during the tuning process. (c) RF spectra sampled at three different stages as depicted in (b). The resolution bandwidth of the measured traces is 100 kHz.

Fig. 18. (a) Optical spectrum of an ACWB state with CLS=14×FSR combined with a CoBrite DX laser. The inset shows the local details of the spectrum. (b) Top: RF spectrum of the ACWB. Bottom: RF spectrum of the heterodyne signal. The black arrows indicate the frequency difference (around 640 MHz) of the primary and secondary pumps. The red arrow indicates the heterodyne frequency (around 7.5 GHz).

Fig. 19. (a) Improved experiment setup. The VCOs are replaced by pure RF sources, and an optical filter is used to suppress the ASE noise from the EDFA. (b) Optical spectrum of the ACWB. (c) ESA recorded beatnote. (d) Phase noise of the beatnote when using VCOs and pure RF sources.