## 3 Results and Discussion
### Amplification of Shaped Bursts
This leads to:  
Figure 3:4  
If we now consider that our amplifier has a certain starting excitation density matching, e.g., \(I_{\text{in/max}}\), this allows us to calculate the gain difference \(m\,\sigma\,(n_{\text{eq}}-n_{\text{eq/max}})=m\,\sigma\,\Delta n\) to any other \(I_{\text{in}}\) by:  
Figure 3:5  
This means that as long as it is known, which \(I_{\text{in}}\) in the beginning of a burst will fulfill Equation (4), the gain coefficient required to satisfy the equilibrium condition of any other \(I_{\text{in}}\) can be directly calculated, as shown in Figure 3:6 In our case, this is solved by adjusting the delay between pump diodes to obtain a burst with a homogeneous energy distribution at maximum seed. Using \(I_{\text{out}}=I_{\text{in}}\cdot\exp(g)\) together with Equation (11), we are now able to predict the output shape of any input burst starting with a uniform input shape. For the final step towards arbitrarily-shaped bursts,we now subdivide the burst into subsections with uniform input energy distributions. Each subsection may be as small as one single pulse. Starting at the beginning of the burst, where we know our initial target \(I_{\text{in}}\), we can now calculate the change of the excitation _versus_ time using Equation (8). The new excitation density at the end of each subsection will be used as the initial one for the next subsection. Knowing this allows for the calculation of the gain, the output energy distribution and the change of excitation density in the next subsection, and so on. Hence, the final output burst shape of any input burst shape can be calculated with this iterative technique. The predicted forms from this theoretical modeling for the amplified bursts in Figures 4 and 5 are plotted together with the amplified shape. For the numerical calculation of the output pulse shapes, a value of 240 \(\upmu\)s for \(\tau_{\text{eff}}\) was found empirically. This is close to the predicted range from the theoretical calculation. The calculated curves are in good agreement with the measured output. The inversion of this recursive approach is more elaborate than the inversion of the Frantz-Nodvik equation, which was described in [18] for pulse shapes and extended in [19] for bursts in amplifier chains, because pumping and spontaneous fluorescence during the amplification process cannot be neglected. Therefore, we used the method of successive approximation, _i.e._, varying the input burst shape until an acceptable agreement with the calculated output shape was reached. This front-upcalculation renders a feedback loop in the amplifier system unnecessary if only predictable saturation effects have to be compensated. Results of the algorithm are shown in Figures 7 and 8. Here, we used the same energy distributions as before for the final target shapes. From them, we calculated the required shapes of seed pulse bursts that have to be further amplified. In the experiments, seed bursts were shaped accordingly, amplified afterwards and finally recorded. Figures 7 and 8 reveal the good agreement of the measurement with the initial target function of output burst shapes for these examples. This proves the practicability of the used methods to produce quasi-arbitrarily-shaped pulse bursts, within the given range of \(I_{\text{in}}\).