Fig. 5: Temporal observation of the breathing dynamics of soliton crystals (SCs) with a 66 GHz Si₃N₄ microresonator.

a Experimental setup for the simultaneous observation of the slow-time evolution (blue dashed rectangular) using oscilloscope directly and the fast-time evolution using an improved time-lens imaging system (purple dashed rectangular). Measurements with two different time scales are recorded separately. The SC microcombs are generated with a breathing angular frequency f_B. BPF1 and BPF2 have the different bandwidths. Optical buffer effectively extends the one-shot length of time-lens system to 2 ns. DUT device under test, EDFA erbium-doped fiber amplifier, ESA electronic spectrum analyzer, OSA optical spectrum analyzer, OSC oscilloscope, MLL mode-locked fiber laser, AWG arbitrary waveform generator. b Schematic representation of the phase-differential sampling based on the stroboscopic effect. The blue circle represents the time lens system and the orange region ϕ_F represents the one-shot angular length. The red circle represents the period of the SC breather. We can use the stroboscopic effect to stitch the dynamics of breathers in a full period. ϕ_res is the accumulated phase difference between the SC breather and time-lens sampling after 500 ns. c The simulated intensity distribution of 2-defect SC after BPF2 as a function of the fast time. One roundtrip is angularly normalized to be 2π. The two defects are set to be temporally separated by 15 solitons. d The simulated 2D evolution map of the intracavity intensity after a similar bandpass filter as BPF2. Solitons near two defects move in different phases, creating two sequent dips in the spatio-temporal evolution map. Two dips are named “dip 1” and “dip 2” in the evolution map. The temporal period, spatial spacing and breathing phase difference of two dips are presented via white, black and purple dashed line, respectively. e The 2D evolution map resampled from d to match the PARTI frame rate, where two dips are reconstructed via resampling and stitching different breathing periods. f The 2D evolution map of the 2-defect breather recorded by our improved time-lens system, showing that the two solitons move in different phases as highlighted by the white dashed parallelogram. Intracavity one roundtrip is angularly normalized to be 2π. The reconstructed temporal period, spatial spacing and breathing phase difference of two dips are presented via white, black and purple dashed line, respectively. g The 2D evolution map of 2-defect breather with a decreased oscillation period. We clearly observe the drift in breather frequency due to noise in addition to the different breathing phases. h The experimental radiofrequency (RF) spectra of the 2-defect breather. The occurrence of the second-harmonic peak near 140 MHz shows that the slow-time evolution is not a perfect sine wave. The broadening of the peak near 76 MHz explains the drift of breathing frequency in g. i Experimental panoramic-reconstruction temporal imaging (PARTI) trace of a chaotic SC with the inset showing the corresponding RF spectrum