# Real-Time Intensity Domain Characterization of Fibre Lasers Using Spatio-Temporal Dynamics
## 2 Experimental Section
### Experimental Considerations  
Ideally when performing spatio-temporal measurements for fibre lasers, one would like to make real-time observations of the laser dynamics over time scales several times over the characteristic round trip time period \(\tau_{RT}\), while simultaneously maintaining a high temporal resolution along the instantaneous temporal co-ordinate \(t\). However, several technical aspects limit high bandwidth measurements of spatio-temporal dynamics over laboratory time scales. These can be summarized as follows.  
As the output characteristics are typically measured at one point along the cavity, the temporal resolution along the evolution co-ordinate is limited to the cavity round trip time. The resolution of small scale features in the spatio-temporal dynamics depends on the combined electrical bandwidth of the photodetector-oscilloscope (PD-DSO) combination. An estimate of the requisite resolution can be obtained using the value of the transform limited pulse width corresponding to the spectral width of emission. For example, for \(sech^{2}\)-like pulses corresponding to a spectral emission of width \(\Delta\lambda\), the transform limited pulse width \(\Delta t\) is given via the relation  
\[\Delta t=0.315\frac{\lambda^{2}}{c\Delta\lambda} \tag{4}\]  
where \(c\) is the velocity of light in vacuum. Thus, the electrical bandwidth must be at least \(1/\Delta t\) to resolve such features. The optimal temporal resolution can also be determined by performing a second harmonic generation based intensity autocorrelation measurement, which would give a statistical estimate of the scale of small features present in the radiation [53; 54; 55]. This bandwidth condition can be relaxed when one is more interested in the dynamics of interaction between the different components of radiation than their internal structures.  
The shortest laser that can be investigated using the spatio-temporal methodology is limited by the finite time domain impulse response function of the PD-DSO combination. If the rise time \((t_{R})_{PD}\) of the photodetector is known, the full-width at half maximum of the impulse response can be given as  
\[(t_{FWHM})_{PD}=0.915(t_{R})_{PD}. \tag{5}\]  
A similar expression can also be arrived at for the impulse response of the DSO, in which case the FWHM of the impulse response of the PD-DSO combination can be obtained to be  
\[t_{FWHM}=\sqrt{(t_{FWHM})_{PD}^{2}+(t_{FWHM})_{DSO}^{2}}. \tag{6}\]  
For example, if the photodetector has a cut-off frequency of 60 GHz, and the oscilloscope has a cut-off of 33 GHz, Equation (6) gives \(t_{FWHM}=10.75\) ps. For adequate resolution of the spatio-temporal features, the round trip time of the laser should be several times this width.  
The longest laser that can be investigated is limited by the memory of the oscilloscope. The total number of round trips that can be observed depends on the round trip time \(\tau_{RT}\) of the cavity, the sampling resolution \(\delta t\) of the oscilloscope, and the total number of data points \(M_{tot}\) that can be recorded in a single sweep. These values are related via the expression  
\[N\leq\frac{M_{tot}\delta t}{\tau_{RT}} \tag{7}\]  
For example, the spatio-temporal evolution in a 375 km long linear cavity based laser can be investigated over \(N=10\) round trips, using a 33 GHz oscilloscope with a memory of \(3\times 10^{9}\) points and performing a time sampling \(\delta t=12.5\) ps.