# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 3 Influence of Thermal Lensing and Intrinsic Compensation  
In a bulk laser, the thermal load originating from the laser process generates a temperature profile in the laser medium due to thermal expansion and the temperature dependence of the refractive index. This effect is known as thermal lensing. Typically, the most prominent effect is a focusing or defocusing of the beam by the thermal lens leading to increased intensities on amplifier optics and finally to optical damages. This should therefore be considered in the setup. In the imaging multi-pass schemes presented in this work, the laser active medium is envisioned to be situated in the primary image plane. Hence, due to the imaging properties, the actual beam shape in this plane will be constant irrespective of a thermal lens. Nevertheless, the collimation state will be altered, adding up the focusing of the thermal lens with every round-trip. It is therefore essential to include a means of compensation within the system counteracting this effect. In the following, we will demonstrate that this is possible due to slight adaptions to the optics distances in systems of both types. The thermal lens will be represented by the matrix \(L_{t}\) as an ideal lens with focal length \(f_{t}\). In a Type I system, this lens is only passed once per half round trip represented by \(G_{2}\). Hence, the modified matrix \(G_{2}^{*}\) is calculated by:  
\[G_{2}^{*}=G_{2}\cdot L_{t} \tag{9}\]  
To achieve a compensation in \(G_{2}^{*}\), again, all entries have to be zero, except for the main axis. The corrected distances are:  
\[d_{1}=f_{1}+\frac{f_{1}}{2}\cdot\left(1-\sqrt{1+\frac{4f_{1}^{2} \cdot(d_{3}-f_{2})}{f_{t}f_{2}^{2}}}\right) \tag{10}\] \[d_{2}=\frac{f_{1}f_{2}+(f_{1}+f_{2})\cdot(f_{t}-d_{1})}{f_{t}+f_ {1}-d_{1}}\] (11) \[d_{3}=f_{2}+\frac{f_{2}^{2}\cdot(f_{1}-d_{1})}{f_{1}^{2}}\cdot \left(1+\frac{f_{1}-d_{1}}{f_{t}}\right) \tag{12}\]  
By deriving these compensating solution, all other solutions that do not converge to the original Type I solution for \(f_{t}\rightarrow\infty\) are omitted, as these would represent systems integrating the thermal lens as an additional imaging element. A more intuitive formulation of Equations (10)-(12) can be obtained for a standard layout with \(d_{1}=f_{1}\):  
\[d_{1}=f_{1} \tag{13}\] \[d_{2}=f_{1}+f_{2}-\frac{f_{1}^{2}}{f_{t}}\] (14) \[d_{3}=f_{2} \tag{15}\]  
According to this solution, the spherical part of a thermal lens in a Type I system can be completely compensated just by adjusting the distance between the imaging optics. In this case, it can further be deduced that the focal length of the thermal lens can only be compensated as long as:  
\[f_{t}\geq\frac{f_{1}^{2}}{f_{1}+f_{2}} \tag{16}\]  
otherwise, \(d_{2}\) will become negative. However, in real life, setups \(f_{t}\) should be much bigger than \(f_{1}\) in order to avoid intensity spikes on the imaging elements. The modified matrix for a Type II system, in which the thermal lens is passed in the beginning and at the end of each round-trip, is calculated using:  
\[G_{3}^{*}=L_{t}\cdot G_{3}\cdot L_{t} \tag{17}\]To compensate for a thermal lens, \(G_{3}^{*}\) again needs to be equal to \(U\), which leads to:  
\[d_{1}=f_{1}+\frac{f_{t}}{2}\cdot\left(1-\sqrt{1-\frac{2f_{1}^{2}}{ f_{2}f_{t}}}\right) \tag{18}\] \[d_{1}=f_{1}+\frac{f_{1}}{2}\cdot\left(1-\sqrt{1+\frac{4f_{1}^{2} \cdot(d_{3}-f_{2})}{f_{t}f_{2}^{2}}}\right) \tag{10}\]0  
As for the Type I system, all solutions that do not converge to the original Type II solution for \(f_{t}\rightarrow\infty\) are omitted; though the length of both arms is changed. The lens Equation (8) is also valid for the new solution. Hence, the major effect of the thermal lens can be seen as a virtual change of the focal length of \(L_{2}\). In Figure 3, the validity of the derived compensation formulas is demonstrated using the ray-tracing software FRED. The traces are performed for compensated, as well as for uncompensated systems for different values of \(f_{t}\) and for up to five round-trips.