## 3 Results and Discussion
### Amplification of Shaped Bursts  
To generate shaped pulse bursts at a high power, the influence of the amplification process has to be taken into account. In an idealized burst amplifier, which produces bursts with a uniform temporal energy distribution, an equilibrium excitation has to be maintained by balancing the excitation of the active medium by the optical pump and the de-excitation by the laser extraction and spontaneous emission during the full duration of the burst. Taking into account a longitudinally-pumped amplifier, this balance can be expressed following the considerations given in [10]:  
\[\frac{\text{d}n(t)}{\text{d}t}=P_{\text{pump}}-\frac{n(t)}{\tau_{\text{f}}}- \frac{I_{\text{out}}(t)-I_{\text{in}}}{h\
u} \tag{3}\]  
Here, \(n(t)\) is the volume density of the excited ions integrated over the crystal length, \(P_{\text{pump}}\) is the excitation rate density integrated over the crystal length, \(\tau_{\text{f}}\) is the fluorescence lifetime and \(I_{\text{out}}\) and \(I_{\text{in}}\) are the output and input intensities of the amplifier, respectively. \(\
u\) is the frequency of the extracted laser radiation, and \(h\) is [PERSON]'s constant. In the following, we will assume that the change of \(n\) in the amplification medium is small. This implies that \(P_{\text{pump}}\) can be considered to be constant (for a constant pump power), as the pump  
Figure 3: **(Left)** Gaussian-shaped burst envelope; **(right)** house-shaped burst envelope. absorption does not undergo significant changes. As the extracted laser intensity \(I_{\text{out}}\) increases with increasing \(n\), a particular \(n\) allows one to balance Equation (3) according to \(P_{\text{pump}}\) to give \(\frac{\text{d}n}{\text{d}t}=0\). The quantities required for this equilibrium case will be noted as \(n_{\text{eq}}\) and \(I_{\text{out/eq}}\). If the initial density of the excited ions \(n_{0}\) is equal to \(n_{\text{eq}}\), the temporal energy distribution of the amplified burst will be homogeneous, as long as the input has a homogeneous energy distribution. In this case, Equation (3) becomes:  
\[0=P_{\text{pump}}-\frac{n_{\text{eq}}}{\tau_{\text{f}}}-\frac{I_{\text{out/eq }}-I_{\text{in}}}{h\
u} \tag{4}\]  
This mode of operation can be achieved in practice, e.g., by adjusting the pump duration prior to the arrival of the burst or by a temporal modulation of the pump pulse, as demonstrated in [9,10]. Using a pre-shaped burst as the seed for the amplifier, the input shape will no longer be constant. According to Equation (3), this means that the inversion and, hereby, the gain will no longer be constant throughout the burst. A lower \(I_{\text{in}}\) at the beginning of the burst will lead to lower \(I_{\text{out}}\), increasing \(n\) and, finally, increasing the gain over time. In contrast, a higher \(I_{\text{in}}\) will lead to a higher \(I_{\text{out}}\), decreasing \(n\) and decreasing the gain. In Figures 4 and 5, the input and output energy distributions for a step-shaped burst and a burst with an alternating level are shown, demonstrating this effect. For these measurements, the amplifier's initial inversion was adjusted to produce a uniform energy distribution at an input seed level of 100% by adjusting the pump duration prior to the burst. Figure 4: **(Left)** Stair-shaped input pulse train; **(right)** resulting output pulse train after amplification. Figure 5: **(Left)** Input pulse train with alternating amplitude; **(right)** resulting output pulse train after amplification. It can be seen from Figure 4 that at the beginning of the pulse, \(I_{\text{in}}\) is too low, resulting in an increase of the gain and \(n\) over time, accordingly. Reaching the second step of the input signal, \(n\) has increased sufficiently to balance the output for about 60% of the maximum \(I_{\text{in}}\). In the final steps, however, the input is too high, leading to an extraction that is too high, which reduces the gain and \(n\) over time in the following.