## 3 Results and Discussion  
Before laser operation we analyzed the dynamic polarization contrast of the BBO Pockels cell at high repetition rates (see Figure 2). While we obtained a contrast of more than 1:2000 at low repetition rates it was reduced by piezoelectric ringing of the BBO crystal with a resonance frequency of 465 kHz. Outside this spectral linewidth (in this case 25 kHz) the measured polarization contrast was suitable for laser operation (see inset Figure 2). The resonator quality including the Pockels cell (switched off) was checked in continuous-wave (cw) operation using the QWP and TFP for output coupling (see Figure 3a). A higher slope efficiency of 32% was measured for Yb:YAG compared to Yb:CaF\({}_{2}\) (17%). This can be explained by a different gain and loss balance considering the emission section, the degree of output coupling at the TFP and the transmission loss of around 2% for the Pockels cell. The quarter-wave plate was adjusted in order to optimize the output power at the maximum pump power separately for both gain media. In pulsed operation the average output power was set to 1 W while a slightly higher pump power of 5 W (Yb:YAG) and 8 W (Yb:CaF\({}_{2}\)) was required as compared to cw-mode. The output beam profile of the Yb:YAG  
Figure 3: (**a**) Laser output power _vs._ pump power at continuous-wave operation; (**b**) output beam profile with a size of 2.8 mm (D4\(\sigma\) x width) and 2.5 mm (D4\(\sigma\) y width). Figure 2: Dynamic polarization contrast of the BBO Pockels cell _vs._ repetition rate when supplied voltage = 0; Inset: Polarization contrast _vs._ time at two repetition rates. laser cavity is shown in Figure 3b similar to case of Yb:CaF\({}_{2}\). A beam quality factor of better than \(M^{2}<1.2\) can be given which was measured with a Shack-Hartmann sensor (Optocraft, Erlangen, Germany; SSHCam HR-150-GE) [18]. Figure 4: Pulse train measurement: (**a**) intra-cavity signal showing 40 round-trips in the case of Yb:CaF\({}_{2}\); (**b**) output signal at 1 MHz for Yb:CaF\({}_{2}\) and 2 MHz for Yb:YAG. Figure 5: Average output power and pulse energy _vs._ repetition rate at regenerative amplification for Yb:YAG (**a**) and Yb:CaF\({}_{2}\) (**b**). Figure 6: Simulation of the rate equations in the case of Yb:CaF\({}_{2}\) for a repetition rate of 1 MHz and continuous pumping at 8 W; output pulse energy (**a**) and stored energy within the gain medium (**b**) _vs._ time. The intra-cavity and extra-cavity pulse trains at regenerative amplification are shown in Figure 4. The number of round trips is set to 32 (Yb:YAG) and 40 (Yb:CaF\({}_{2}\)) to prevent instable pulse train operation such as period doubling or bifurcations at a repetition rate of 20 kHz. An optimum extraction of the output power is obtained at several 100 kHz which is shown in Figure 5. In general, the extractable pulse energy is proportional to the inverse of the repetition rate with impact on the small-signal gain. We have also observed an increasing insensitivity [19] of the output power against the variation of the pass number especially for very high repetition rates. This phenomenon can be explained by a numerical simulation (see Table 1) of the rate equations [20] while the results were normalized to a maximum average output power of 1 W at 1 MHz. In Figure 6 the results in steady-state are illustrated for different numbers of round trips in the case of Yb:CaF\({}_{2}\). At a pass number of 80 an instable pulse train operation becomes likely when operating close to gain depletion. A rising inversion level (stored energy) is obtained for a lower number of round trips while the total gain and therefore the cumulatively extracted power remains rather constant within a wider range. When modeling for repetition rates lower than 1 MHz we observe a drop of the average output power (see Figure 5) depending on the fluorescence lifetime of the gain media, while the power would remain constant for an infinite fluorescence lifetime.