## 2 Numerical Simulation of Ytterbium-Based Amplification  
### Introduction  
As described in our previous work, we have developed a one-dimensional numerical code for modeling energy, gain and spectral performance of an ytterbium-based regenerative amplifier, to which we have added routine for the spectral response of a generic Nd:glass amplifier following the first stage. The model for ytterbium-based amplification can be applied to different host materials and consists of two routines: the first one simulates the pump process and calculates the pump absorption and population inversion, and the second one simulates the overall amplification process calculating the pulse energy and spectrum after each pass through the gain medium. Both routines are ruled by the following discrete rate equations:  
\[\frac{dN_{{}_{\mathrm{ex}}}(z,t)}{dt}=-\frac{N_{{}_{\mathrm{ex}}}(z,t)}{\tau_ {{}_{\mathrm{ex}}}}+\left[\sigma_{{}_{\mathrm{ex}}}(\lambda)N_{{}_{\mathrm{ex} \mathrm{ex}}}(z,t)-\sigma_{{}_{\mathrm{ex}}}(\lambda)N_{{}_{\mathrm{ex}}}(z,t) \right]\frac{\lambda}{h_{C}}I(z,t) \tag{1}\]  
\[\frac{dI(z,t)}{dz}=\left[\sigma_{e}(\lambda)N_{{}_{\mathrm{ex}}}(z,t)-\sigma_ {{}_{\mathrm{a}}}(\lambda)N_{{}_{\mathrm{grad}}}(z,t)\right]\!I(z,t) \tag{2}\]  
where \(N_{\mathrm{ex}}\) is the excited state population, \(N_{\mathrm{grad}}\) is the ground state population, (\(z\),\(t\)) are the longitudinal position and time inside the gain medium, \(I\) is the wavelength-dependent pulse intensity, \(\lambda\) is the wavelength, \(\sigma_{\mathrm{a}}\) and \(\sigma_{\mathrm{e}}\) are respectively the absorption and emission cross-sections, \(\tau_{\mathrm{f}}\) is the fluorescence lifetime, \(h\) is [PERSON]'s constant and \(c\) is the speed of light in vacuum.  
Ytterbium-doped laser media are described by a quasi-three-level model where the relevant laser transitions take place between the sub-levels of two manifolds, the ground-state \({}^{2}F_{7/2}\) and the excited state \({}^{2}F_{5/2}\). At room temperature, the sub-levels of the ground-state manifold are thermally populated due to the proximity between them, and in consequence reabsorption losses can be generated at the laser wavelength. This property has the drawback of making ytterbium lasers inefficient for very low pump intensities, but it can also be used beneficially in order to explore different sub-level transitions. This effectively means that one is able to tune the laser amplifier for different wavelengths by changing the pump power.  
Two good candidates for efficient laser amplification near 1053 nm are Yb:CaF\({}_{2}\) and Yb:YAG, in the first case because of the relatively broad tunability [10], and in the second because of the presence of a peak in the emission cross-section near this wavelength. We have used spectroscopic data for both materials (Figure 1) [13] and evaluated their amplification performance under different pump powers.  
Table 1 lists the parameters that were used for the two materials. Where realistic input parameters for the pump source and the laser media were required, we have assumed typical values without loss of generality. The chosen pump pulse length matches the fluorescence lifetime of each gain media, _i.e._, 2.4 ms for Yb:CaF\({}_{2}\) and 1 ms for Yb:YAG. It is assumed that the gain medium is longitudinally pumped in a two-pass configuration, in order to reach a high population inversion in Yb:CaF\({}_{2}\). The focused pump spot FWHM diameter is 700 \(\upmu\)m. For Yb:CaF\({}_{2}\) we have considered 5 at% doping level and a length of 3.8 mm, while for Yb:YAG the doping level is 3 at% and the length is 8 mm. Furthermore no considerations about the damage threshold of the items in the cavity were taken into account.