State-of-the-art ultrafast mid-IR fiber lasers deliver optical solitons with durations of several hundred femtoseconds. The Er- or Ho-doped fluoride gain fibers generally used in these lasers have strong anomalous dispersion at ∼3 µm, which generally forces them to operate in the soliton regime. Here we report that a pulse-energy clamping effect, caused by the buildup of intracavity nonlinearities, limits the shortest obtainable pulse durations in these mid-infrared soliton fiber lasers. Excessive intra-cavity energy results in soliton instability, collapse and fragmentation into a variety of stable multi-pulse states, including phase-locked soliton molecules and harmonically mode-locked states. We report that the spectral evolution of the mid-IR laser pulses can be recorded between roundtrips through stretching their second-harmonic signal in a 25-km-length of single-mode fiber. Using a modified dispersive Fourier transform set-up, we were able to perform for the first time spectro-temporal measurements of mid-IR laser pulses both in the pulsed state and during pulse collapse and fragmentation. The results provide insight into the complex nonlinear dynamics of mid-IR soliton fiber lasers and open up new opportunities for obtaining a variety of stable multi-pulse mode-locked states at mid-IR wavelengths.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Mode-locked fiber lasers employing Er- or Ho-doped fluoride fibers as gain media, generating pulsed light at mid-IR wavelengths, have attracted considerable interest due to potential applications in many fields, from molecular spectroscopy to invasive skin surgery [1–6]. Compared to their bulk material counterparts [7–9], mid-IR fiber lasers offer several distinct advantages, including a simple configuration, low cost, high brightness, and an emission band broad enough to support ultrashort pulses. Several examples of such lasers, delivering high-quality ultrashort pulses at ∼3 µm with average powers of sub-W levels and pulse durations of several hundreds of femtoseconds, have recently been reported [4–6]. Compared to fiber lasers at near-IR wavelengths, mid-IR lasers are expected to deliver higher pulse energies [1–6], since the core diameters for single-mode operation are larger and the effective fiber nonlinearity is lower .
In practice, however, pulse energies directly generated by state-of-the-art mode-locked fiber lasers at ∼3 µm are always in the range of several nJ [4–6]—similar to the pulse energies of mode-locked near-IR fiber lasers [11–15]. This is because the soft-glass gain fiber used in these lasers generally has strong anomalous dispersion at ∼3 µm. Due to the absence of commercially-available dispersion compensating fibers or devices in this wavelength region, most of the ∼3 µm mode-locked fiber lasers reported so far have strong anomalous cavity dispersion, leading to the generation of optical solitons in the laser cavity [4–6,16]. The resulting short pulse durations, high peak powers and rapidly-accumulating intra-cavity nonlinearity make these lasers unstable at high pump power levels. Transient nonlinear processes [6, 16–20] and complex pulsed states [21–23] are usually observed, limiting both the achievable average power and the shortest pulse duration. A detailed study of soliton instability and dynamics at high pump power levels is therefore crucial if short pulse lasers at high average power are to be developed in the mid-IR. Such a study can also provide insight into the behavior of other nonlinear optical systems, such as passive fiber cavities  and microresonators . Along with the development of the dispersive Fourier transform (DFT) method in recent several years, measurements of the spectro-temporal evolution of laser pulses during a variety of transient nonlinear processes have attracted extensive interest in the fields of mode-locked lasers and nonlinear fiber optics [26–28]. However, spectro-temporal measurements of nonlinear soliton dynamics at mid-IR wavelengths using the conventional DFT method are generally regarded as challenging because of the lack of low-loss, km-long optical fiber at these wavelengths .
In this paper we report, first, that a strongly-pumped, Er-doped fluoride fiber laser emitting at 2.8 µm can be adjusted to stably operate in a variety of multi-soliton states, including phase-locked soliton molecules (soliton-pairs, soliton-triplets and soliton-quadruplets) and harmonically mode-locked states with two or three solitons equally-spaced in the laser cavity. This allows the total intra-cavity pulse energy to be increased from up to ∼10 nJ without leading to any instability. Second, we report that a modified DFT set-up based on frequency doubling the laser output , using a 25-km-length of single-mode (SMF28) fiber, can be used to measure the spectro-temporal evolution of the mid-IR laser pulses between round-trips. As a result we were able to follow the evolution of the mid-IR laser pulses through the phases of pulsation instability [21–23], soliton collapse and soliton pair buildup [26–29], unveiling the complex nonlinear dynamics of these transient processes.
2. Experimental set-up
A schematic of the ring laser cavity used in the experiments is shown in Fig. 1. The gain fiber was a 3.5 m length of double-clad Er-doped ZBLAN (ZrF4 − BaF2 − LaF3 − AlF3 − NaF) fiber with a core diameter of 15 µm and an inner cladding diameter of 250 µm. The pump laser was a multimode 975 nm laser diode (LD) with maximum power 25 W. Two dichroic mirrors with >90% transmission at 975 nm and 66% reflection at 2.8 µm were used to separate the pump and signal light, providing an energy extraction ratio of ∼22%. The launch efficiency of the pump light into the gain fiber is estimated to be ∼70%. Unidirectional operation was ensured using a polarization-dependent optical isolator which also worked as a polarizer. Three wave-plates, working together with the polarizer, acted as a fast saturable absorber through nonlinear polarization rotation (NPR), enabling self-start of mode-locking and background noise suppression during stable mode-locked operation of the laser . The free-space length in the cavity was ∼0.5 m which, combined with a 3.5-m-long fluoride gain fiber (effective refractive index ∼1.5), gave a cavity free spectral range (FSR) of 51.8 MHz. The total cavity loss was estimated to be ∼7 dB, including the loss induced by the output mirrors. In order to avoid water absorption, the entire laser cavity was enclosed in a chamber filled with dry nitrogen, as sketched in Fig. 1.
In the diagnostic set-up, the laser output pulses at ∼2.8 µm were monitored using a mid-IR photodiode (PD 1) with 200 MHz bandwidth (∼5 ns response time), connected to an oscilloscope with 33 GHz bandwidth. The optical spectrum was measured using a Fourier-transform infrared spectrometer (FTIR, continuously purged with dry nitrogen) with 0.1 nm resolution. A mid-IR autocorrelator (AC) was used to measure the pulse duration with 50 fs resolution. For high-temporal-resolution of the output pulse train, the laser beam was focused through a 1-mm-thick AgGaS2 (AGS) crystal. A second harmonic signal at ∼1.4 µm was generated in the crystal and monitored using a fast (25-GHz-bandwidth) near-IR photodiode (PD 2). The electrical signal from PD 2 was also recorded by the 33 GHz oscilloscope, yielding a temporal resolution of ∼40 ps. With these diagnostics we were able to resolve the laser output trace over a broad range of timescales. On the one hand, details of the laser pulses at fast timescales could be resolved using the autocorrelator with 50 fs resolution over a scanning span of 200 ps, while on the other hand the second harmonic signal, recorded by the fast PD2 and the oscilloscope, monitored the laser output at longer timescales (tens of ps to several ms).
3. Stable mode-locked states
3.1 Single-pulse mode-locked state
The threshold pump power for CW lasing was ∼250 mW. By increasing the pump power and adjusting the three wave-plates in the laser cavity, stable single-pulse mode-locking could be achieved at an in-fiber pump power of 1.75 W, estimated using a launch efficiency of ∼70% and 2.5 W pump laser power. At this pump level, the laser output power at 2.8 µm was 36 mW. Fundamental mode-locking (FSR 51.8 MHz) at 2.8 µm could be maintained by adjusting the three intra-cavity wave-plates when the pump power was further increased to ∼2.2 W. At a pump power of 2.1 W, the laser output power was 44 mW, corresponding to an average power of ∼200 mW at the output port of the gain fiber. At 51.8 MHz the maximum single-pulse energy inside the laser cavity was estimated to be ∼3.9 nJ. At this pump power the output pulse train, recorded using the mid-IR PD, is shown as the red dashed line in Fig. 2(a). The second harmonic signal at 1.4 µm, recorded using the fast near-IR PD, is also shown in Fig. 2(a) as the blue solid line. The fast Fourier transform (FFT) spectrum of the second harmonic signal is shown in Fig. 2(b). The many sharp spectral harmonics of the cavity FSR have almost equal intensity, indicating stable continuous-wave (CW) mode-locking. Note that, first in Fig. 2(b) some small sidebands are observed in addition to the main cavity harmonics, most probably corresponding to periodic fluctuations in the laser pulse energy. This is because that we measured the FFT spectrum at a pump power of 2.1 W, quite close to the ∼2.2 W threshold for pulsation instability. In practice, such residual sidebands (small periodic pulse-energy fluctuations) can be efficiently suppressed by slightly decreasing the pump power or properly adjusting the intra-cavity wave-plates. Second, the high noise floor in Fig. 2(b) is caused by calculating the FFT spectrum using the time-domain pulse trace recorded by the oscilloscope. The frequency resolution of this measurement is determined by the duration of the time-domain signal, which in practice is 4 µs (limited by the memory size of the oscilloscope), corresponding to a frequency resolution of 0.25 MHz. High-resolution laser radio-frequency (RF) spectra, with much higher signal-to-noise ratios, can be measured using an electrical spectrum analyzer.
The laser optical spectrum, measured by the FTIR, has a 3 dB bandwidth of 12 nm (blue curve, Fig. 2(c)). The Kelly sidebands in the optical spectrum indicate that the laser is operating in the soliton regime . The autocorrelation of the pulses shown in Fig. 2(d) (blue curve) has a width of 1.24 ps, corresponding to a full width half-maximum (FWHM) pulse duration of 805 fs. The time-bandwidth product was calculated to be ∼0.37, close to the transform limit for a hyperbolic secant pulse [10,16]. Single-pulse mode-locked operation could be stably maintained over 24 hours (the limit of the experiment) without any re-alignment; fluctuations in the laser output power were <5% during this measurement.
When the pump power was increased slightly above 2.2 W, soliton instabilities were always observed, i.e., the pulse energy and optical spectrum became unstable. At relatively high pump powers, by carefully adjusting the wave-plates we were able to observe a quasi-stable single-pulse state consisting of a soliton on a strong CW background. At a gain fiber length of 3.5 m, the energy of the intra-cavity solitons was always clamped below 4 nJ. This clamped pulse energy could be increased by shortening the length of gain fiber to 1.5 m, thus reducing the intra-cavity nonlinearity. As well as increasing the cavity FSR to 107 MHz, this increased the mode-locking threshold to ∼7 W. When using a 1.5-m-long gain fiber, stable fundamental mode-locking could be obtained at pump powers between ∼7 W and ∼9 W. At 9 W the laser output power was 124 mW, corresponding a pulse energy of ∼5.3 nJ at the output port of the gain fiber. The optical spectrum (red curve in Fig. 2(c)) has a 3 dB bandwidth of 21 nm, and the autocorrelation of the output pulses (red curve in Fig. 2(d)) has a FWHM width of 660 fs, corresponding to a pulse duration of 430 fs. The time-bandwidth product is ∼0.35.
3.2 Soliton-molecule mode-locked states
In the laser with the 3.5-m-long gain fiber, a variety of stable multi-pulse states  were observed at pump powers above 3 W. Phase-locked soliton pairs [33,34] with different pulse-to-pulse separations could be obtained at 3.6 W, when the laser output power was 80 mW, corresponding to an intra-cavity pulse energy of 3.5 nJ, clamped below 4 nJ. The recorded time-domain traces and their FFT spectra are the same as those shown in Figs. 2(a) and 2(b), while the laser optical spectra and autocorrelation functions are shown in Figs. 3(a)–3(d). The 3 dB spectral widths are ∼10 nm, slightly narrower than in the single-soliton case (Fig. 2(c)) due to the slightly lower pulse energies. The 5.4 ps and 12.4 ps pulse separations, directly retrieved from the autocorrelation functions, perfectly match the interferometric spectral fringes [27,33,34] with periods of 4.8 nm and 2.1 nm (see Figs. 3(a) and 3(c)), respectively. When the pump power was maintained at 3.6 W, stable soliton pairs with several discrete pulse separations ranging from several ps to several tens of ps were observed . Once formed, the pair separation was found to be quite robust against moderate variations of the pump power. In the experiments we observed that the soliton pair with a 5.4 ps separation was robust against pump power variations < ±5%, while the pair with a 12.4 ps separation was robust against variations < ±2%. We also observed that moderate changes in pump power could slightly alter the pulse separation without destroying the bound state.
Further increasing the pump power to 4.4 W produced phase-locked soliton-triplet states, a laser output power of ∼100 mW, and an intra-cavity pulse energy of ∼2.9 nJ. The optical spectrum and autocorrelation function are shown in Figs. 3(e) and 3(f), indicating identical pulse separations of 7.4 ps between the three solitons. Stable phased-locked soliton quadruplets were seen at a pump power of 5.3 W, a laser output power of ∼125 mW and an intra-cavity pulse energy of ∼2.7 nJ.
3.3 Harmonically mode-locked states
Besides phase-locked soliton molecules, we were also able to obtain stable harmonically mode-locked (HML) states of the laser at two and three times the fundamental round-trip frequency . At 3.6 W and 4.4 W (the pump powers needed to generate soliton pairs and triplets), stable pulse trains at 103.8 MHz and 155.4 MHz could be obtained by carefully adjusting the three wave-plates. The recorded time-domain traces and their FFT spectra are shown in Figs. 4(a)–4(d). We attribute the noisy spikes in the FFT spectra (see Figs. 4(b) and 4(d)) to relatively large amplitude fluctuations and timing jitter in the time-domain pulse trains (see Figs. 4(a) and 4(c)) . The optical spectra and autocorrelation functions of the two harmonically mode-locked states are very similar to those shown as blue curves in Figs. 2(c) and 2(d), with 3 dB spectral bandwidths of ∼10 nm and pulse durations of ∼1 ps.
4. Pulse instability and transient dynamics
4.1 SH-DFT set-up
To resolve transient dynamics of the laser pulses, we adapted the time-stretch dispersive Fourier transform (TS-DFT) method [26–29,38], constructing a second-harmonic set-up (Fig. 5(a)) to measure the pulse spectrum in each successive roundtrip (assuming that the second harmonic signal preserves the temporal profile and carrier phase information of the mid-IR pulses). We used the laser with 3.5-m-long gain fiber, generating its second-harmonic in an AGS crystal and then stretching the 1.4 µm signal using 25 km of SMF28 fiber [26–29]. The resulting SH-DFT signal was then detected by a fast (25-GHz-bandwidth) near-IR PD, and the electrical signal amplified and then recorded using a fast oscilloscope. A portion of the second harmonic signal was directly detected after the AGS crystal using a near-IR PD and an oscilloscope, to allow monitoring of the intensities of successive second-harmonic pulses. Simultaneously, the laser pulses at 2.8 µm were directly measured using a 200-MHz-bandwidth mid-IR PD (see experimental set-up shown in Fig. 5(a)) to record the pulse energies. SH-DFT measurements for a typical soliton-pair mode-locked state are shown in Figs. 5(b) and 5(c). For comparison, the optical spectrum of the soliton pair, measured using the FTIR, is shown in Fig. 5(d). Note that this SH-DFT technique has some limitations. For example, as shown in panel (II) of Fig. 5(c) and Fig. 5(d) it is difficult to use the SH-DFT technique to measure the Kelly sidebands  on the laser spectrum. This is because the Kelly sidebands correspond to quasi-CW dispersive waves  in the temporal domain, with much lower intensities and longer durations than the laser solitons. The relatively low intensities of these dispersive waves mean that they are only very weakly converted in the second harmonic crystal, making it difficult to detect the Kelly sidebands by SH-DFT. However, as demonstrated below, the SH-DFT set-up can provide a convenient means of recording the shot-by-shot variations of the pulse spectrum.
4.2 Quasi-stable pulsation state
For the laser with 3.5-m-long gain fiber, the single-pulse mode-locked state became unstable when the laser pump power was increased to slightly higher than 2.2 W. At a pump power of 2.25 W, a few quasi-stable pulsation states were observed with different oscillation periods ranging from several to tens of cavity roundtrips [21–23]. One example of these pulsation states is illustrated in Fig. 6. As shown in Fig. 6(a), when the laser operated at this quasi-stable pulsation state, both the pulse energy (see panel (I) of Fig. 6(a)) and spectral bandwidth (see panel (II) of Fig. 6(a)) oscillated with a constant period of 4 cavity roundtrips. The relative variation of pulse energy is ∼15%, estimated using the time-domain trace of mid-IR pulse train. The relative variation of spectral bandwidth is estimated to be >50% through fitting the SH-DFT signal at successive roundtrips (see panel (I) of Fig. 6(b)). The variations of pulse energy and spectral bandwidth as the functions of the round-trip number are plotted in panel (II) of Fig. 6(b), and the evolution trajectory of this pulsation state, plotted in the bandwidth-energy plane [21,22], is shown in Fig. 6(c). The 4-round-trip pulse train repeats over thousands of cavity roundtrips without obvious deviation, as shown in Fig. 6(c), i.e. the pulsation period exactly equals to 4 cavity roundtrips [21,22]. Note that the two traces shown in panel (I) and (II) of Fig. 6(a) have been synchronized in time, and that the mismatch between the pulse peaks in these two traces arises from the fact that while the 200 MHz mid-IR PD records only the pulse-energy variations at successive roundtrips (see panel (I) of Fig. 6(a)), the SH-DFT signal shown in panel (II) of Fig. 6(a) contains the spectral-width information [26–28] of the laser pulses. Laser pulses with higher pulse energies generally have broader spectral widths, leading higher stretching ratios in the SH-DFT set-up.
At pump powers slightly higher than 2.2 W, the laser can operate in a few pulsation states with different periods, all of which can be stably preserved over several hours. A more complex quasi-stable pulsation state, with a period of 10 cavity roundtrips, is illustrated in Fig. 7.
4.3 Pulse-collapse dynamics
When we further increased the pump power of the laser, the pulsation state became unstable, leading to the collapse of the laser pulse. After pulse collapse, the laser most probably operated in a quasi-stable Q-switched mode-locking state. Using SH-DFT we recorded the transient dynamics during the pulse-collapse process. In the experiment we first set the laser pump power to 2 W and adjusted the wave-plates so as to obtain a stable single-pulse mode-locked state. The pump power was then increased from 2 W to 2.4 W (an increase of ∼20%) by modulating the driving current of the pump laser diode using an electrical signal generator with <1 µs response time (the response time of the intra-cavity pulse energy is mainly determined by the ∼10 µs relaxation time of the gain fiber, corresponding to ∼500 cavity roundtrips).
As shown in Fig. 8(a), this abrupt increase of pump power led to an intra-cavity pulse energy slightly higher than the clamping value (∼4 nJ in this laser), resulting first in a transient pulsation stage and then in the collapse of the laser pulse. The dynamics of the entire process were recorded over 700 µs, corresponding to 35,000 consecutive cavity roundtrips (see Fig. 8(a)). Zoom-in plots of this process show that at a pump power of 2 W both the pulse energy and spectral bandwidth of the laser pulse remained stable (see Fig. 8(b)). Then the operation of the laser pulse evolved from stable single-pulse mode-locking to a transient pulsation stage with a period of 4 cavity roundtrips (see Fig. 8(c)), as a result of the increased pulse energy. This transient pulsation stage was unstable, lasted ∼30,000 cavity roundtrips (∼600 µs), finally collapsing after experiencing irregular oscillations in the pulse parameters over several hundreds of cavity roundtrips (some details are shown in Fig. 8(d)).
A bandwidth-energy plot during the transition from the single-pulse mode-locking to transient pulsation is shown in Fig. 8(e), where the yellow dots indicate the starting single-pulse stage, the blue lines the 4-roundtrip-period pulsation stage and the gray lines the transition between these two stages (also see Supplementary Visualization 1). This transient pulsation state at relatively-high pump power was observed to be unstable since, compared to solitons, pulsating solutions are generally weaker attractors in dissipative nonlinear systems [21,22] and are therefore more easily destroyed by laser noise perturbations. A bandwidth-energy plot during collapse of this transient pulsation state is shown in Fig. 8(f), and the dynamics are illustrated in Supplementary Visualization 2.
We performed several independent measurements of pulse collapse by increasing the laser pump power to different levels. The results showed essentially the same features: transient pulsation instabilities and sudden irregular oscillations leading to pulse collapse. We found that higher pulse energies cause shorter transient pulsation stages with longer periods [21,22]. As shown in Fig. 9, a ∼30% increase in laser pump power from 2 W to 2.6 W led to a shorter transient pulsation stage lasting ∼10,000 roundtrips (∼200 µs), with an oscillation period of 7 cavity roundtrips. We believe that this general behavior could also be observed in mode-locked soliton fiber lasers at other wavelengths.
4.4 Dynamics of soliton-pair buildup
We also recorded, using the SH-DFT technique, the dynamics during the laser evolution from a single-pulse mode-locked state (with a strong CW background) to stable soliton-pair mode-locked state. During these measurements the pump power was kept at 3.5 W and the wave-plates adjusted. At such high pump power the laser could operate in a quasi-single-pulse mode-locked state with the pulse energy clamped below 4 nJ, while the excessive intra-cavity energy appeared as a CW background. The optical spectrum of this quasi-single-pulse state, measured by the FTIR, is shown in panel (I) of Fig. 10(a). The narrow spike observed on top of the broad soliton spectrum indicates the existence of the CW background. In such a state, competition between the NPR effect (working as a saturable absorber that provides lower loss to high-intensity laser pulses ) and gain filtering (working as a saturable amplifier that provides higher loss to broad-spectrum laser pulses) results in the coexistence of a laser soliton with a CW background.
By slightly adjusting one of the intra-cavity wave-plates, we were able to alter the working point of the NPR effect , enhancing the saturable absorber effect. This could cause a transfer of intra-cavity energy from the CW background to the laser pulse. As shown in Fig. 10(b), a sudden increase in the pulse energy led to intense oscillations of both its intensity and optical spectrum (see details in Figs. 10(b) and 10(c)), which finally resulted in pulse fragmentation and buildup of a stable soliton pair . The pulse separation in this newly-generated soliton pair was estimated to be ∼2.6 ps, and the optical spectrum of the soliton pair was also measured using the FTIR (see panel (II) of Fig. 10(a)).
The FFT power spectrum of the SH-DFT signal, corresponding to the autocorrelation of the time-domain pulse trace [27,38], is calculated at each successive cavity roundtrip (see Fig. 10(d)). The pulse-to-pulse separation, after pulse fragmentation has happened, is estimated using the autocorrelation, and the carrier phase difference between the two pulses in a pair is estimated by calculating the position of the interferometric fringes relative to the pulse spectral envelope [27,38]. The results of both pulse-separation and phase-difference variations are plotted in Fig. 10(e), and a separation-phase plot is shown in Fig. 10(f) (also see Supplementary Visualization 3). As shown in Fig. 10(e), during soliton-pair buildup the pulse-to-pulse separation first oscillated over tens of cavity roundtrips and then stabilized at 2.6 ps, while the phase-difference between the solitons first experienced a fast drift over ∼8π before stabilizing at ∼0.4π . The entire process, including pulse fragmentation and soliton-pair stabilization, lasted only ∼100 cavity roundtrips (∼2 µs).
Pulse energy clamping and soliton instabilities occur in strongly-pumped, mid-IR soliton fiber lasers, as a result of the buildup of strong intra-cavity nonlinearities. At higher pump powers, both stable multi-pulse mode-locking of soliton-molecules and harmonically mode-locked states are seen. The SH-DFT method provides a convenient means of studying transient pulse-to-pulse dynamics during pulsation instability, pulse collapse and buildup of soliton pairs. The results will be useful for designing mid-IR mode-locked fiber lasers with higher pulse energies and shorter pulse durations, and may open up new opportunities for resolving complex nonlinear dynamics and developing high-repetition-rate light sources at mid-IR wavelengths.
1. D. D. Hudson, “Short pulse generation in mid-IR fiber lasers,” Opt. Fiber Technol. 20(6), 631–641 (2014). [CrossRef]
2. S. Jackson, “Towards high-power mid-infrared emission from a fibre laser,” Nat. Photonics 6(7), 423–431 (2012). [CrossRef]
3. P. Tang, Z. Qin, J. Liu, C. Zhao, G. Xie, S. Wen, and L. Qian, “Watt-level passively mode-locked Er3+-doped ZBLAN fiber laser at 2.8 µm,” Opt. Lett. 40(21), 4855–4858 (2015). [CrossRef]
4. T. Hu, S. Jackson, and D. Hudson, “Ultrafast pulses from a mid-infrared fiber laser,” Opt. Lett. 40(18), 4226–4228 (2015). [CrossRef]
5. S. Duval, M. Bernier, V. Fortin, J. Genest, M. Piché, and R. Vallée, “Femtosecond fiber lasers reach the mid-infrared,” Optica 2(7), 623–626 (2015). [CrossRef]
6. S. Antipov, D. Hudson, A. Fuerbach, and S. Jackson, “High-power mid-infrared femtosecond fiber laser in the water vapor transmission window,” Optica 3(12), 1373–1376 (2016). [CrossRef]
7. M. Cizmeciyan, H. Cankaya, A. Kurt, and A. Sennaroglu, “Kerr-lens mode-locked femtosecond Cr2+:ZnSe laser at 2420 nm,” Opt. Lett. 34(20), 3056–3058 (2009). [CrossRef]
8. Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012). [CrossRef]
9. U. Elu, M. Baudisch, H. Pires, F. Tani, M. Frosz, F. Köttig, A. Ermolov, P. S. J. Russell, and J. Biegert, “High average power and single-cycle pulses from a mid-IR optical parametric chirped pulse amplifier,” Optica 4(9), 1024–1029 (2017). [CrossRef]
10. R. W. Boyd, Nonlinear Optics (Academic, 2008).
11. M. E. Fermann and I. Hartl, “Ultrafast fibre lasers,” Nat. Photonics 7(11), 868–874 (2013). [CrossRef]
12. P. Grelu and N. Akhmediev, “Dissipative solitons for mode-locked lasers,” Nat. Photonics 6(2), 84–92 (2012). [CrossRef]
13. K. Tamura, E. Ippen, H. Haus, and L. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18(13), 1080–1082 (1993). [CrossRef]
14. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2(1-2), 58–73 (2008). [CrossRef]
15. F.Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-Similar Evolution of Parabolic Pulses in a Laser,” Phys. Rev. Lett. 92(21), 213902 (2004). [CrossRef]
16. I. Duling, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16(8), 539–541 (1991). [CrossRef]
17. M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and A. M. Weiner, “Passive mode locking in erbium fiber lasers with negative group delay,” Appl. Phys. Lett. 62(9), 910–912 (1993). [CrossRef]
18. K. Tamura, L. E. Nelson, H. A. Haus, and E. P. Ippen, “Soliton versus nonsoliton operation of fiber ring lasers,” Appl. Phys. Lett. 64(2), 149–151 (1994). [CrossRef]
19. D. Noske, N. Pandit, and J. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17(21), 1515–1517 (1992). [CrossRef]
20. P. Grelu and J. M. Soto-Crespo, “Multisoliton states and pulse fragmentation in a passively mode-locked fibre laser,” J. Opt. B: Quantum Semiclassical Opt. 6(5), S271–S278 (2004). [CrossRef]
21. J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 70(6), 066612 (2004). [CrossRef]
22. J. M. Soto-Crespo, N. Akhmediev, and A. Ankiewicz, “Pulsating, creeping, and erupting solitons in dissipative systems,” Phys. Rev. Lett. 85(14), 2937–2940 (2000). [CrossRef]
23. L. Lou, T. J. Tee, and P. L. Chu, “Chaotic behavior in erbium-doped fiber-ring lasers,” J. Opt. Soc. Am. B 15(3), 972–978 (1998). [CrossRef]
24. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express 21(7), 9180–9191 (2013). [CrossRef]
25. C. Bao, J. A. Jaramillo-Villegas, Y. Xuan, D. E. Leaird, M. Qi, and A. M. Weiner, “Observation of Fermi-Pasta-Ulam recurrence induced by breather solitons in an optical microresonator,” Phys. Rev. Lett. 117(16), 163901 (2016). [CrossRef]
26. G. Herink, B. Jalali, C. Ropers, and D. R. Solli, “Resolving the build-up of femtosecond mode-locking with single-shot spectroscopy at 90 MHz frame rate,” Nat. Photonics 10(5), 321–326 (2016). [CrossRef]
27. G. Herink, F. Kurtz, B. Jalali, D. R. Solli, and C. Ropers, “Real-time spectral interferometry probes the internal dynamics of femtosecond soliton molecules,” Science 356(6333), 50–54 (2017). [CrossRef]
28. P. Ryczkowski, M. Närhi, C. Billet, J. M. Merolla, G. Genty, and J. M. Dudley, “Real-time full-field characterization of transient dissipative soliton dynamics in a mode-locked laser,” Nat. Photonics 12(4), 221–227 (2018). [CrossRef]
29. S. Hamdi, A. Coillet, and P. Grelu, “Real-time characterization of optical soliton molecule dynamics in an ultrafast thulium fiber laser,” Opt. Lett. 43(20), 4965–4968 (2018). [CrossRef]
30. S. Tan, X. Wei, B. Li, Q. T. K. Lai, K. K. Tsia, and K. K. Y. Wong, “Ultrafast optical imaging at 2.0 µm through second-harmonic-generation-based time-stretch at 1.0 µm,” Opt. Lett. 43(16), 3822–3825 (2018). [CrossRef]
31. M. Hofer, M. Fermann, F. Haberl, M. Ober, and A. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16(7), 502–504 (1991). [CrossRef]
32. S. M. J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Electron. Lett. 28(8), 806–808 (1992). [CrossRef]
33. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton Solutions of the Complex Ginzburg-Landau Equation,” Phys. Rev. Lett. 79(21), 4047–4051 (1997). [CrossRef]
34. M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett. 95(14), 143902 (2005). [CrossRef]
35. J. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28(19), 1757–1759 (2003). [CrossRef]
36. A. Grudinin and S. Gray, “Passive harmonic mode locking in soliton fiber lasers,” J. Opt. Soc. Am. B 14(1), 144–154 (1997). [CrossRef]
37. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. 29(3), 983–996 (1993). [CrossRef]
38. X. Liu, X. Yao, and Y. Cui, “Real-Time Observation of the Buildup of Soliton Molecules,” Phys. Rev. Lett. 121(2), 023905 (2018). [CrossRef]