Fig. 1. Experimental real-time measurements from undispersed events (i.e., not using TS-DFT) for the formation and evolution of a soliton, where the beating phenomenon cannot be discovered [Video 1, MOV, 3.34 MB (URL: https://doi.org/10.1117/1.AP.1.1.016003.1)].

Fig. 2. Experimental real-time observation for the formation and evolution of a soliton, including Q-ML, beating dynamics, and stable mode-locking [Video 2, MOV, 3.69 MB (URL: https://doi.org/10.1117/1.AP.1.1.016003.2)].

Fig. 3. Experimental real-time observation for the formation and evolution of a soliton, including Q-ML, beating dynamics, transient bound state, and stable mode-locking [Video 3, MOV, 6.99 MB (URL: https://doi.org/10.1117/1.AP.1.1.016003.3)].

Fig. 4. Experimental real-time display of the buildup dynamics of solitons in mode-locked laser, captured simultaneously by using (a) an undispersed and (b) a dispersed element, respectively. (a) Direct detection with a high-speed photodetector and a real-time oscilloscope. (A) The expanded view shows that the timing data represent the temporal information. (b) The buildup dynamics are obtained by dispersing the solitons in 5-km DCF prior to detection, as denotes the single-shot spectral information shown in (B). The timing data advance the real-time spectral data about 24.5μs that is delayed by 5-km DCF. (c) Close-up of the data from (b) for the fifth laser spike, revealing the quasi mode-locking (Q-ML) and beating dynamics with the duration of ∼30μs. (C) There are multiple subordinate pulses together with a dominant pulse in the laser cavity. The period for (A) to (C) is 38.03 ns, corresponding to the roundtrip time of the laser cavity. The buildup process of mode-locked lasers includes the raised relaxation oscillation, Q-ML stage, beating dynamics, and stable mode-locking.

Fig. 5. Formation of a soliton with beating dynamics. The recorded time series is segmented with respect to the roundtrip time and displays the buildup dynamics of a soliton. The intensity profile evolves along with time (vertical axis) and roundtrips (horizontal axis). (a) Experimental real-time observation during the formation of a soliton from the Q-ML and beating behavior to the stable mode-locking (see Video 2 for the full animation). The experimental data are from Fig. 4(b). TS-DFT maps the spectral information into the temporal domain. (b) Exemplary single-shot spectrum, corresponding to the last frame in (a). (c) Optical spectrum of soliton measured by an OSA. (d) Close-up of the data from (a), revealing the interference pattern for the beating dynamics. (e) Experimental real-time observation via direct measurement (see Video 1 for the full animation), without using TS-DFT technique. The beating dynamics are not revealed in the direct measurement. [RT: roundtrip (scale bar).]

Fig. 6. Experimental real-time display of the buildup dynamics of solitons with transient bound state. (a) Buildup dynamics via the TS-DFT technique. The entire buildup process includes the raised relaxation oscillation, transition region, and stable mode-locking. (b) Close-up of the data from (a) at the stable mode-locking state, as denotes the single-shot spectral information. (c) Experimental real-time interferogram during the formation of a soliton, accessed via the dynamics of Q-ML, beating dynamics, transient bound state, and stable mode-locking. For a full animation of the observations, see Video 3. For convenience of reference, the beginning time of stable mode-locking is set to zero roundtrips. (d) Dynamics of the solitons mapped in the interaction plane over 1000 roundtrips. The angle, α, represents the relative phase of two solitons. The radius, R, corresponds to the bound-state separation. Two solitons gradually depart from each other. (e) The Fourier transform of each single-shot spectrum corresponds to a field autocorrelation of the momentary bound state, tracing the separation between both solitons. (f) The relative phase between both solitons along with roundtrips.

Fig. 7. Interaction and evolution of two solitons in the transient bound state. The temporal solitons are extracted from the experimental data shown in Fig. 6 by using the Levenberg–Marquardt algorithm. The stronger soliton gradually evolves to the stationary state; however, the weaker one ultimately vanishes via the complex dynamics. The roundtrips are from about −2800 to −300, and the corresponding autocorrelation intensity of the momentary bound state is shown in Fig. 6(e).

Fig. 8. Experimental buildup process of mode-locked laser with Q-switched lasing. The laser experiences an unstable Q-switched lasing stage during the buildup process. (a) Hybrid saturable absorber based on the single-wall carbon nanotubes together with NPR technique. The duration of Q-switched lasing is about 3 ms with five lines. The separation between the neighboring Q-switched lasing lines is ∼700μs. Inset (left): close-up of the data from (a) for a Q-switched lasing. Inset (right): expanded view of a part from the left inset figure. (b) NPR-based saturable absorber together with weaker fluctuation of pumping strength. The duration of Q-switched lasing is >80ms with 76 lines. (c) NPR-based saturable absorber together with stronger fluctuation of pumping strength. The duration of Q-switched lasing is >160ms with 189 lines.

Fig. 9. Buildup process of mode-locked laser without Q-switched lasing. (a) Experimental results. (b) Numerical simulation, corresponding to (a). To numerically simulate the raised relaxation oscillation, we assume that the phase difference θ between any two neighboring modes is uniformly distributed from −4.8 to 4.8, from −4.2 to 4.2, from −3.65 to 3.65, and from −2.86 to 2.86 for the second to fifth laser spikes, respectively. Red curves indicate the evolution of pumping rate along with time. Blue curves show the evolution of raised relaxation oscillation. (c) Close-up of (b). Two insets in (c) are the expanded view of the fourth and fifth laser spikes. Note that a.u. denotes arbitrary unit. The raised relaxation oscillation at the beginning stage of the mode-locking can be precisely predicted with several key features, matching excellently with the experimental measurements.

Fig. 10. Numerical simulations based on the roundtrip circulating-pulse method. (a)–(c) Soliton buildup process with beating dynamics in one pathway. (d)–(f) Soliton buildup process with transient bound state in the other pathway. (a) and (d) Spectral evolution of soliton along with roundtrips. (c) and (f) Temporal evolution of soliton along with roundtrips. (b) and (e) Close-ups of the data from the white dotted-line boxes in (a) and (c), respectively. (g) Initial signals with the noise background. The intensity of initial signals is in the order of magnitude of 10−6W, which is eight orders of magnitude lower than the intensity of stable solitons (i.e., 102W). Inset: close-up of the data from the blue box. (h) Spectral and (i) temporal profiles at the finally stationary solution. Although the solitons experience through two different buildup pathways, they evolve to the same steady soliton solution with Kelly sidebands.

Fig. 11. Schematic diagram of the experimental setup for the mode-locked laser, generating the temporal solitons. The output can be characterized with real-time acquisition and time-averaged spectral acquisition (i.e., via optical spectrum analyzer). Real-time acquisition harnesses a high-speed real-time oscilloscope (20GSa/s sampling rate) together with a high-speed photodetector. Timing data are achieved with direct detection, and real-time spectral data are obtained by dispersing the solitons in dispersion-compensating fiber (DCF) prior to detection. EDF, erbium-doped fiber; PC, polarization controller; LD, laser diode; CNT-SA, carbon nanotube saturable absorber; WTI, hybrid combiner of wavelength division multiplexer, tap coupler, and isolator.

Fig. 12. Numerical results at the beginning stage (from 1 to 62 roundtrips) for the soliton buildup process with beating dynamics. (a) Spectral and (b) temporal evolutions of solitons along with roundtrips.

Fig. 13. Numerical results at the beginning stage (from 1 to 62 roundtrips) for the soliton buildup process with transient bound state. (a) Spectral and (b) temporal evolutions of solitons along with roundtrips.