## 3 Results and Discussion
### Amplification of Shaped Bursts
A similar behavior is seen in Figure 5. Here, one observes an oscillation between too much and too little extraction. To compensate for the disturbances of the temporal burst shape, a mathematical model is required that describes the observed changes. For this purpose, we followed the calculation by [PERSON] _et al._[10] and adapted it for our system. To obtain a linear differential equation, we introduce the deviation of the actual excitation density from the equilibrium value using \(\Delta n(t)=n(t)-n_{\text{eq}}\). Combining Equations (3) and (4) and using the small signal gain for a multi-pass amplifier leads to:  
\[I_{\text{out}}=I_{\text{in}}\cdot e^{m\,\sigma\,n}. \tag{5}\]  
Here, \(m\) is the number of passes through the active medium and \(\sigma\) is the emission cross-section of the laser medium. Furthermore, one finds:  
\[\frac{\text{d}\Delta n(t)}{\text{d}t}=-\frac{\Delta n(t)}{\tau_{\text{f}}}- \frac{I_{\text{in}}}{h\
u}\cdot e^{m\,\sigma\,n_{\text{eq}}}\cdot\left(e^{m\, \sigma\,\Delta n(t)}-1\right). \tag{6}\]  
Using the approximation \(e^{m\,\sigma\,\Delta n(t)}\approx 1+m\,\sigma\,\Delta n(t)\) for small deviations of \(n(t)\) from \(n_{\text{eq}}\), the resulting differential equation:  
\[\frac{\text{d}\Delta n(t)}{\text{d}t}=-\Delta n(t)\cdot\left(\frac{1}{\tau_{ \text{f}}}+\frac{m\,I_{\text{in}}\,\sigma}{h\
u}\cdot e^{m\,\sigma\,n_{\text{ eq}}}\right) \tag{7}\]  
can be solved by an exponential approach:  
\[\Delta n(t)=\Delta n(t{=}0)\cdot e^{-\frac{t}{\tau_{\text{eff}}}} \tag{8}\]  
Here, \(\tau_{\text{eff}}\) is the characteristic time constant, defining the exponential approach of \(n(t)\) towards \(n_{\text{eq}}\). Defining the steady-state output intensity \(I_{\text{out/eq}}=I_{\text{in}}\cdot\exp(m\sigma n_{\text{eq}})\) and the saturation intensity \(I_{\text{sat}}=h\
u/(\sigma\tau_{\text{f}})\), \(\tau_{\text{eff}}\) can be calculated by:  
Figure 3:0  
For our system operating in a 20-pass configuration, \(I_{\text{out/eq}}\) was found to approximately vary between 7.5 kW/cm\({}^{2}\) and 9 kW/cm\({}^{2}\) for an input ranging from 20% to 100% of the available seed power. The values for \(I_{\text{out/eq}}\) were retrieved from measurements of the the output power for bursts with a uniform energy distribution at different seed levels in combination with the beam area (for details, see [15]). The comparably small variation of \(I_{\text{out/eq}}\) can be explained by the fact that the equilibrium output power in a burst mode amplifier is mainly influenced by the available pump power, while the gain can be widely tuned by only minor variations of \(n\). \(I_{\text{sat}}\) is calculated to be 21.6 kW/cm\({}^{2}\) (based on spectroscopic data from [17] at 100 K). This results in a range from 239 \(\upmu\)s to 204 \(\upmu\)s for \(\tau_{\text{eff}}\) depending on \(I_{\text{in}}\), as shown in Figure 3:1  
Using Equations (5) and (8), the gain coefficient \(g=\log\left(I_{\text{out}}/I_{\text{in}}\right)\) can be calculated as:  
Figure 3:2  
Therefore, an initial mismatch between \(I_{\text{in}}\) and the initial excitation density will decay exponentially with the time constant \(\tau_{\text{eff}}\). A second important parameter required to determine the output burst shape is \(m\,\sigma\,\Delta n(t{=}0)\) as a function of \(I_{\text{in}}\). We use Equation (5) for calculating the gain factor \(g_{1}\) under equilibrium conditions for a matched parameter set of \(I_{\text{in1}}\), \(I_{\text{out/eq1}}\) and \(n_{\text{eq1}}\):  
Figure 3:3  
Here, \(I_{\text{in2}}\) is an input intensity differing from \(I_{\text{in1}}\). For reasonable changes of \(I_{\text{in}}\) in the range between 20% and 100%, the according change of \(I_{\text{out/eq1}}\) is in the range of \(\pm\)10% of the amplifier output. In combination with a gain of more than 100, we can approximate \(\log(I_{\text{out/eq1}}/I_{\text{in2}})\approx\log(I_{\text{out/eq2}}/I_{ \text{in2}})\) with an error of less than 0.1%.