## 2 Numerical Simulation of Ytterbium-Based Amplification
### Regenerative Amplifier: Free-Running Cavity  
In this section we will make a preliminary evaluation of the wavelength tunability for both media considering a free-running cavity, _i.e._, where amplified stimulated emission grows from noise rather than from a seed pulse. The amplified pulse is then extracted at the peak energy. The initial pulse traveling in the cavity has a rectangular spectral profile between 1020 and 1060 nm and an energy of 10\({}^{-6}\) nJ, corresponding to \(\sim\)1 photon for each wavelength slice used in the calculation. The pulse duration is 10 ns, which is similar to that obtained by cavity-dumping. We assume that there is no spectral filtering in the cavity (apart from the gain-induced modulation), so losses are equal for all wavelengths. Figures 2 (Yb:CaF\({}_{2}\)) and 3 (Yb:YAG) show the results in terms of output spectrum (left) and energy (right) performance _vs._ pump power. For two discrete ranges of pump powers, 46-66 W for Yb:CaF \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Yb:CaF\({}_{2}\)** & **Yb:YAG** \\ \hline Pump pulse & & \\ \hline Power (W) & 46–66 & 20–30 \\ Duration (ms) & 2.4 & 1 \\ Beam diameter (mm) & 0.7 & 0.7 \\ Wavelength (nm) & 938 & 938 \\ \hline Seed pulse & & \\ \hline Energy (pJ) & 100 & 100 \\ Duration (ns) & 1 & 1 \\ Beam diameter (mm) & 0.7 & 0.7 \\ FWHM Bandwidth (nm) & 12 & 12 \\ \hline Gain medium & & \\ \hline Length (mm) & 3.8 & 8 \\ Yb\({}^{3+}\) concentration (at \%) & 5 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used in the numerical simulations. Figure 1: Absorption (solid lines) and emission (dashed lines) for Yb:CaF\({}_{2}\) (green) and Yb:YAG (blue). Adapted from Ref. [13]. and 20-30 W for Yb:YAG, we have determined the maximum energy extractable and the number of roundtrips needed to achieve it. A common trend to both media is immediately apparent: for lower pump powers, there is a significant fraction of the amplified spectrum centered around 1048-1050 nm, corresponding to the case of a lower population inversion. As the pump power increases, gain shifting brings the spectral peak down to \(\sim\)1030-1035 nm, resulting from the higher population inversion. The spectral distribution is however different for each case. For Yb:CaF\({}_{2}\) at the lowest pump power we observe a broad peak centered at 1048 nm. As the pump power is raised, the output energy is transferred to a second peak which forms around 1034 nm. At higher powers (54-58 W) it is possible to obtain a balance in the energy distribution between the two peaks. By increasing the pump power even further the longer wavelength peak will eventually vanish and the second one increases and slightly shifts towards lower wavelengths. This behavior confirms experimental results reported for different pump conditions by [PERSON] _et al_. [PERSON] _et al._ (2013). For Yb:YAG, a similar evolution is numerically witnessed, although the transition is more abrupt and the parameters are much more restrictive. The pump power required to overcome the lasing threshold (\(\sim\)20 W) is very close to that when the output spectrum is shifted to 1032 nm (\(\sim\)26 W), leaving a narrow range of pump powers and cavity loss combinations that allow amplification at 1049 nm. As a consequence of this, more than a thousand passes become necessary to reach high energies at this wavelength, as can be seen in Figure 3 (right). In contrast, an Yb:CaF\({}_{2}\) amplifier operating at a similar wavelength would only require less than half the passes. This should be taken in consideration for the practical development of mJ-level amplifiers at 1050 nm based in Yb:YAG. The behavior described above can be understood recurring to a model for the population inversion at the two different wavelength regions. Taking the case of Yb:YAG, Figure 4 shows its energy level scheme, that can be considered a quasi-4 level. Both the ground level and the upper level are located within manifolds, separated by approximately 10,000 cm\({}^{-1}\).