# Real-Time Intensity Domain Characterization of Fibre Lasers Using Spatio-Temporal Dynamics
## 2 Experimental Section  
### Principle  
The availability of state of the art DSOs allow the continuous observation and record of laser intensity dynamics over time scales ranging from a few hundreds to thousands of round trips. Such long-term intensity variation records have proven to be very fruitful in the statistical studies of laser emission (see for example [13; 28; 29; 30; 31]). More importantly, the records would also reveal coherent features such as mode locked pulses, which would appear periodically at the output port of the laser after making round trip circulations inside the cavity. In this case, one can select temporal segments along the intensity dynamics centered about consecutive re-occurrences of these features, and stack them one atop another. The resulting two-dimensional representation then depicts the intensity evolution of the feature of interest as it makes successive round trips within the cavity. The temporal co-ordinate _along_ each segment takes the notion of a temporal co-ordinate in the rest frame of the feature. Successive segments describe the evolution of the feature as it travels over a physical length along the cavity. Hence the round trip number \(N\) takes the notion of a spatial co-ordinate. Indeed, it can be rescaled into appropriate distance units \(N\times L\) (or \(N\times 2L\), for a Fabry Perot configuration), where \(L\) is the physical length of the cavity. Thus, this temporally segmented and collated representation of one-dimensional intensity dynamics is called the _spatio-temporal evolution of laser intensity_, or simply, _spatio-temporal dynamics_[32; 33] (also see Supplementary, [33]). In a broader context, spatio-temporal representations of intensity dynamics have been employed in describing the dynamic behaviour of semiconductor lasers [34; 35] including vertical cavity surface emitting lasers [36; 37], transverse pattern formation in various lasers and parametric oscillators [38; 39], in the study of multimode instabilities [40; 41; 42], including those observed in high power fibre lasers [43; 44] and in the influence of stimulated Brillouin scattering [45; 46]. Spatio-temporal representations of intensity evolution have also been routinely used in the numerical modelling of various kinds of fibre-based light sources [47; 48; 49]. From an experimental perspective, intensity evolution has been represented in similar two dimensional forms as in the study of inter-pulse interactions in multi-soliton fibre lasers [50; 51], ultraweak soliton interaction via the transverse Brillouin effect [52],and in the study of phase bits in a driven semiconductor laser [24]. Here, we extend this concept to the study of quasi-continuous wave and partially mode locked fibre lasers. The methodology for arriving at the spatio-temporal dynamics is as follows. Say the laser radiation consists of features which move with a characteristic round trip time \(\tau_{RT}\) and survive over multiple round cavity passes (Figure 1a). These features would then appear periodically at the output of the laser at intervals of the time \(\tau_{RT}\). Such a train of pulses can be represented mathematically as  
\[ \begin{split} I(t)&=I_{0}\ S(t)\ \otimes\sum_{T=-\infty}^{\infty} \delta(t-\tau_{RT}T)\\ &=I_{0}\sum_{T=-\infty}^{\infty}S(t-\tau_{RT}T)\end{split}  \tag{1}\]  
where \(S(t)\) gives the shape of the pulse, \(I_{0}\) the peak intensity, \(\tau_{RT}\) the round trip periodicity of the pulse, \(T\) is an integer indicating the round trip number, and '\(\otimes\)' indicates a convolution operation. An estimate of the round trip time \(\tau_{RT}\) can be obtained by calculating the autocorrelation function (ACF) \(K(\tau)\) of \(I(t)\):  
\[K(\tau)=\int_{-\infty}^{\infty}I(\tau^{{}^{\prime}})I(\tau^{{}^{\prime}}+\tau) d\tau^{{}^{\prime}}, \tag{2}\]  
where \(\tau\) is defined as the delay coefficient, which defines the relative overlap of the function \(I(\tau^{{}^{\prime}})\) and its copy \(I(\tau^{{}^{\prime}}+\tau)\).