## 2 Average Power Scaling of Modelocked Thin-Disk Lasers: Challenges and Milestones
### Harnessing Intracavity Nonlinearity  
In a typical cavity for a soliton modelocked TDL (Figure 2), the circulating pulse experiences SPM by propagating through nonlinear materials like the gain medium, a Brewster plate and the air atmosphere. Most commonly, the disk is used as a folding mirror in a linear cavity, resulting in 4 gain passes through the thin disk. This configuration is usually referred to as a \"single-pass\" configuration. A Brewster plate is most commonly used to obtain a linearly polarized output and for fine adjustment of the SPM by placing it at a position in the cavity where there is a focus. Negative dispersion is introduced with dispersive mirrors throughout the cavity. Most commonly, Gires Tournois Interferometer (GTI)-type mirrors are used because they can provide large amounts of negative dispersion [30, 31] required for stable soliton modelocking at the pulse durations typically obtained in state-of-the-art TDLs. The total nonlinear phase shift \(\varPhi_{\mathrm{nl}}\) accumulated by the pulses due to self-phase modulation (SPM) per cavity round trip is given by  
\[\phi_{\mathrm{nl}}=2\cdot\frac{2\pi}{\lambda}\int\limits_{\mathrm{cav}}n_{2}(z)I( z)\,dz \tag{1}\]  
Most commonly, the \(\gamma\)-factor (in mrad/MW) is used, which is independent of the pulse parameters but takes into account the different spot sizes of the beam throughout the cavity. \[\phi_{\mathrm{nl}}=4\cdot\frac{2\pi}{\lambda}P_{\mathrm{pk}}\int\limits_{ \mathrm{cav}}\frac{n_{2}(z)}{A(z)}\,dz=P_{\mathrm{pk}}\cdot\gamma_{\mathrm{cav}} \tag{2}\]  
Therefore, the \(\gamma\)-factor can be written as:  
\[\gamma_{\mathrm{cav}}=4\cdot\frac{2\pi}{\lambda}\int\limits_{\mathrm{cav}} \frac{n_{2}(z)}{A(z)}\,dz \tag{3}\]  
Different elements in the cavity contribute to the total \(\gamma\)-factor and the total nonlinear phase shift undergone by the pulses. In Table 1, the contribution of these different elements to the total nonlinear phase shift in state-of-the-art modelocked thin disk lasers is presented. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Parameter** & **Yb:YAG [32]** & **Yb:YAG [17]** & **Yb:YAG [18]** & **Yb:LuO [26]** \\ \hline Average output power & 44 W & 275 W & 145 W & 141 W \\ Repetition rate & 4 MHz & 16.3 MHz & 3.5 MHz & 60 MHz \\ Pulse duration & 791 fs & 583 fs & 1100 fs & 740 fs \\ Output pulse energy & 11.3 μJ & 16.9 μJ & 41 μJ & 2.35 μJ \\ Output peak power & 12.6 MW & 25.6 MW & 33 MW & 2.8 MW \\ Intracavity average power & 450 W & 2.5 kW & 0.2 kW & 1.5 kW \\ Intracavity pulse energy & 113 μJ & 154 μJ & 57 μJ & 25 μJ \\ Intracavity peak power & 127 MW & 236 MW & 46 MW & 30 MW \\ Nonlinear phase shift environment \(\varPhi_{\mathrm{env}}\) & helium 108 mrad & vacuum 8 mrad & air 1725 mrad & air 44 mrad \\ Nonlinear phase shift & 95 mrad & 17 mrad & - & - \\ Brewster plate \(\varPhi_{\mathrm{np}}\) & & & & \\ Nonlinear phase shift disk \(\varPhi_{\mathrm{disk}}\) & 19 mrad & 3 mrad & - & 4 mrad \\ Other nonlinear phase shifts \(\varPhi_{\mathrm{other}}\) & - & 45 mrad & - & - \\ Total nonlinear phase shift \(\varPhi_{\mathrm{cav}}\) & 222 mrad & 75 mrad & 1725 mrad & 48 mrad \\ GDD per roundtrip & –20,000 fs\({}^{2}\) & –8,100 fs\({}^{2}\) & –346,500 fs\({}^{2}\) & –9,900 fs\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of performance and contribution of different cavity elements to nonlinear phase shift due to self-phase modulation (SPM) in state-of-the-art high-power TDLs. Figure 2: Typical layout of a modelocked TDL cavity. HR: highly-reflective mirror, OC: output coupling mirror, DM: dispersive mirror. * Gain material: The SPM from the disk can typically be neglected as the laser mode size on the disk is large and the pulse passes only through very little material. * Brewster plate: The influence of the Brewster plate can be controlled by the choice of the thickness, the material and the mode size at the location in the cavity where it is placed.