## 6 Other Multisoliton States and Dissipative Soliton Dynamics
Figure 24: Experimental observation of a vibrating soliton pair at \(\sim\)1950 nm: (**a**) optical spectrum; (**b**) optical autocorrelation trace (inset: magnified view of the central peak fitted by sech\({}^{2}\) profile); (**c**) pulse train on oscilloscope; (**d**) RF spectrum (inset: RF spectrum over a 1-GHz span). (Reproduced with permission from [23]. © The Optical Society of America (Washington, DC, USA), 2016). Soliton rains were first observed in an anomalous dispersion Er-doped fibre laser [88; 89; 90]. The name originates from the analogy to evaporative cycle of the water [89]. In a soliton rain, multiple closely spaced dissipative solitons form a condensed phase; new weak dissipative solitons (termed as drifting dissipative solitons afterwards) arise from the noisy background spontaneously and drift towards the condensed phase, collide and annihilate. Multiple solitons (several tens) generation and different operation centre frequencies between the condensed phase and the drifting solitons (related to relative drift) are two requirements of soliton rains [89]. These two requirements can be easily satisfied in the anomalous dispersion regime. Hence, it was first observed in anomalous dispersion fibre lasers. In contrast, it is more difficult to get multiple solitons operation in the normal dispersion regime under a moderate pump power, as pulse propagating in the normal dispersion tolerates larger nonlinear phase shift [91]. By using a dual-filter design, soliton rains were also observed in a normal dispersion fibre laser (Figure 25a) [92]. The dual-filter consists of a narrowband Gaussian filter and a broadband, multiple transmission peak birefringent filter. The narrowband filter enables the mode-locked pulse break into multiple pulses easily [92; 93]. The multiple transmission peak filter causes the emission of dissipative solitons in the condensed phase and the drifting dissipative solitons at different centre wavelengths. Thus, they can have relative move due to the difference in group velocity. Figure 25b is the measured temporal trace of the soliton rain state, showing a similar temporal dynamic to those in anomalous dispersion regime. The mode-locked spectrum is relatively narrow (Figure 25c) and the measured RF spectrum reflects the complex temporal dynamics. The measured autocorrelation trace shows the multiple pulses are relatively far spaced, different from a typical soliton-molecule state. Rogue waves (RWs) are extreme waves that exceed expectation based on long-term observation and Gaussian statistics. In particular, optical RWs go beyond the significant wave height by a factor of 2, which have been widely investigated in different nonlinear optical systems, such as mode-locked Ti:sapphire lasers [94], Raman fibre amplifiers [95; 96], parametric processes [97] and fibre lasers [98; 99; 100; 101; 102; 103; 104; 105; 106]. Figure 26 shows typical characterizations of dissipative RWs, which were obtained  
Figure 25: **(a)** Illustration of the mode-locked Yb-fibre laser with dual-filter for observing soliton rains. iso: isolator, WDM: wavelength division multiplexer, PBS: polarization beam splitter, QWP: quarter waveplate, HWP: half waveplate, BP: birefringent plate; **(b)** Temporal trace of soliton rains, showing that the drifting solitons move towards a condensed phase; **(c)** Optical spectrum of the state; **(d)** RF spectrum of the state; **(e)** Autocorrelation trace of the soliton rain state. (Reproduced with permission from [92]. © The Optical Society of America (Washington, DC, USA), 2013). in a Tm-doped fibre laser mode-locked by monolayer MoS\({}_{2}\)[107]. The net GVD of the cavity is approximately \(-3.82\) ps\({}^{2}\). Under enough pumping and appropriate PC settings, the pulses become aggregated as a bunch of intense pulses with internal structure fluctuating, namely noise-like pulses (NLPs). The features of NLPs can be significantly affected by the polarization state of the PCs.