# Real-Time Intensity Domain Characterization of Fibre Lasers Using Spatio-Temporal Dynamics
## 3 Results
### Spatio-Temporal Dynamics of Partially Mode Locked Fibre Lasers
The panels give examples of regimes exhibiting (**a**) stationary characteristics over evolution co-ordinate T; (**b**) periodicity over evolution co-ordinate T; (**c**) interactions between the pulse and the background; and (**d**) highly chaotic, turbulent generation. As observed in Figure 5, the spatio-temporal regimes are composed of an exceeding diversity of features moving with different velocities. By choosing the right value of the round trip time \(\tau_{RT}\), one can then arrive at the rest frame of the particular feature of interest. A background-subtracted autocorrelation based analysis was proposed in [33] for the identification of the constituents of radiation and their respective rest frames. However, the exact temporal location of the underlying features along the recorded intensity dynamics remains inaccessible in such an approach. Alternatively, one can locate a region of interest in the spatio-temporal dynamics and analyze its content. For example, Figure 7 shows the application of a cross correlation methodology to arrive at a characteristic round trip time for the background features of Figure 5c. In this approach, an arbitrary time slice \(I(t,T)_{T=T_{1}}\) centred around a feature of interest is cross correlated with another time slice \(I(t,T)_{T=T_{2}}\). The existence of a positive correlation for \(T_{1}\
eq T_{2}\) then indicates the possible existence of coherent features, and the shift of the peak away from the zero-lag position, \(\Delta\tau_{peak}\), indicates a difference in its group velocity from that used to obtain the spatio-temporal dynamics I(t,T) itself. One such instance is shown in Figure 7b, where \(T_{1}=500\). The difference in the round trip time can then be simply calculated as  
\[\Delta\tau_{RT}=\frac{\Delta\tau_{peak}}{T_{1}-T_{2}} \tag{8}\]  
Figure 7c shows the cross correlation function carried out for a range of values of \(T_{2}\). The shift of the peak of the cross correlation function away from \(T_{2}=T_{1}\) indicates the existence of features which have a velocity different from that of the rest frame of Figure 7a. The spatio-temporal dynamics corresponding to the rest frame of these features is shown in Figure 7d. This reveals the existence  
Figure 7: Cross-correlation based analysis for revealing the constituents of radiation—(**a**) Spatio-temporal dynamics \(I(t,T)\) of the background region between the pulses for the regime of Figure 5c; (**b**) Cross correlation of intensity dynamics, \(\kappa(\tau,T_{2})_{T_{1}}=\langle I(t,T_{1})I(t+\tau,T_{2})\rangle\). Here \(T_{1}\) is chosen to be the 500 th round trip of Figure 7(**a**); (**c**) Close up of cross correlation matrix \(\kappa(\tau,T_{2})_{T_{1}}\). The peak progressively shifting with \(T_{2}\) indicates the possible existence of coherent features moving with a distinct velocity; and (**d**) the spatio-temporal dynamics of the background corresponding to its rest frame, revealing the existence of dark solitons. of dark features in the background, which move with their own characteristic velocity. Note that the existence of such dark coherent features have been earlier reported in numerical simulations of similar ultralong mode locked fibre lasers, and have been confirmed to be dark solitons [82]. Note that the difference in velocity between the pulse and these background dark solitons are of the order of \(\frac{\Delta\tau_{\mathrm{RF}}}{\tau_{\mathrm{RF}}}\sim 10^{-6}\). While the existence of with features differing in periodicity by this order of magnitude would be barely revealed by conventional RF analyzers, the spatio-temporal dynamics can confirm their existence, even if they are exceedingly intermittent, and even if they last over time scales of only a few round trips. Figure 8 shows how the dark solitons interact with the pulse. The dark solitons move faster than the pulse, as evidenced by their relative inclinations in Figure 8a.