## 5 Discussion on Generation and Stability of Various Soliton Molecules
### Interaction Mechanisms
have found the important role of the oscillatory tails of single solitons [77; 78], via which the discrete bound states are determined by quantized local minima of nonlinear losses [7]. Based on their theoretical frame, the effective interaction potential enables prediction of the existence of in-, opposite- and \(\pi/2\)-phase bound solitons. Among the specific interaction terms, nonlinear Kerr and Raman effects [79] dominate, concerning their response time on the order of tens of femtoseconds (for silica fibre, the response time of the Kerr nonlinearity is \(<\)10 fs and Raman 60-70 fs [62]). Those nonlinear effects lead to change of mutual frequency content of the partially overlapped pulses and dispersive effects lead to change of group velocity. Depending on the sign of the dispersion, the adjacent solitons can then repel or attract each other. Because dispersion distribution is usually inhomogeneous inside a fibre laser, particularly in the case of dispersion-managed cavities, so that solitons exhibit strong stretching and compression when travelling along the cavity. Therefore the nonlinear effects via pulse overlapping can play a role typically in the range of several to several tens of times pulse duration [19], or say, several picoseconds. Another short-range interaction is mediated by saturable absorption recovery time, i.e., the time-dependent loss on a timescale of femtoseconds to picoseconds, typical of most effective and natural saturable absorbers. Simultaneously, indirect interaction is very common via long-range media such as dispersive waves, continuous-wave emission [31] and acoustic waves [69; 80]. They influence interaction forces by again changing the refractive indices of the optical fibres, so that widely spaced pulses are able to feel each other and possess distinct group velocities. Such long-range media are typically responsible for the weak interaction on a scale of picoseconds to microseconds. The key role of dispersive waves in formation of bound states has attracted great attention. In anomalous dispersion regime, periodic soliton reshaping results in the radiation of dispersive waves. The phase-matched interaction at specific frequencies between soliton and radiation components produces resonant sidebands of significant magnitude, known as Kelly sidebands of fibre lasers [81; 82]. As an example to unravel the vital impact of dispersive waves, Figure 23 shows the optical spectra of several bound states in a 2 \(\upmu\)m fibre laser similar to Figure 9. Remarkably, the Kelly sidebands perfectly coincide with the interference fringes of the spectrum, yielding integer number of interference fringes between the two lowest-order Kelly sidebands (see Figure 23a,c). Induced by a slight perturbation  
Figure 22: Schematic of soliton molecules generated and stabilized via interactions mediated by different parameters. All factors marked by coloured arrows play important roles in the interactions, hampering a simple interpretation and modelling of the underlying physical mechanisms. However, a systematic analysis of the interaction forces at different length scales provides significant insights. on the pumping power, the little deviation from the obtained equilibrium point of the soliton pair, with the pulse separation of ~7.2 ps (Figure 23b), eventually evolves into another equilibrium point (Figure 23d). These experimental results provide clear evidence to the importance of dispersive waves on formation of bound states and confirm that the quantized separation is a universal property of the phase-locked solitons [18; 19]. The internal repulsion or attraction between two pulses of the soliton pair, which is induced by the process of soliton-continuum interaction, is a periodical function of the initial pulse separation with a set of equilibrium point, where the repulsion and attraction balance each other.