# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 4 Astigmatism in Mirror-Based Layouts
As a second parameter, we use the ratio of the system's focal length and the radius of the output wave front \(\
icefrac{{f}}{{R_{o}}}\) in the respective plane, which allows one to access a changed collimation state, as well as the astigmatism of the beam. Assuming that the wavefront is orthogonal to each single ray, \(\
icefrac{{f}}{{R_{o}}}\) can be calculated within the small angle approximation using:  
\[\frac{f}{R_{o}}=\frac{\alpha f}{x}=\frac{f}{x_{0}}\cdot x_{0}\cdot\frac{ \alpha}{x} \tag{26}\]  
Using Equations (24) and (25) \(\
icefrac{{f}}{{R}}\) becomes independent of \(x_{0}\) and \(f\). Though the exact terms for further round-trips become more complex, so that we refrain from displaying them here, \(x_{0}\) and \(x_{0}\
icefrac{{f}}{{f}}\) can still be isolated in the respective terms as in Equations (24) and (25) in the same way as before. Hence, the chosen expressions can still be used for describing the behavior of the system. A Type II system can be treated in the same way obtaining the output ray after one round-trip:  
\[\vec{v}_{mII}=\left( \begin{array}{c}-x_{0}\cdot\frac{7\cos^{2}\theta-16\cos \theta+8}{\cos^{2}\theta}\\ \\ -\frac{x_{0}}{f}\cdot\frac{2\cos^{2}\theta-6\cos\theta+4}{\cos^{2}\theta} \end{array} \right) \tag{27}\]  
\[\vec{v}_{sII}=\left( \begin{array}{c}x_{0}\cdot\left(-8\cos^{2}\theta+16\cos \theta-7\right)\\ \\ -\frac{x_{0}}{f}\cdot\left(4\cos^{2}\theta-6\cos\theta+2\right)\end{array} \right) \tag{28}\]  
Again, the same expressions for describing the system's behavior as before can be used. The results for the deformation of the beam are given for up to five round-trips for both system types in Figure 5. Within the given range of angles, the deformation of the beam in the image plane is less than \(5\,\%\). The influence in both system types is approximately the same. Because the changes in the tangential and the sagittal direction are nearly identical, the beam shape itself will not be influenced; only the size of the beam is slightly shrunken. The absolute astigmatic deformation of the wavefront \(\
icefrac{{|f}}{{R}}\) in comparison for Type I and Type II is given in Figure 6. We used the absolute value here for a clearer presentation. The sagittal component results in a defocused beam, while the tangential direction results in a focused one. Again, we used the FRED software to benchmark the results from the matrix calculations. The only difference from the previous simulations is that the beam diameter is set to \(3\,\mathrm{mm}\) and the wave length to \(500\,\mathrm{nm}\) for Type I systems and \(200\,\mathrm{nm}\) for Type II. Both changes were made to minimize the propagation effects in the system, which might disturb the comparability of both approaches and to assure a stable wavefront reconstruction. The actual wavefront radius was determined from the quadratic part of the sixth Zernike term. The results show a reasonably good agreement of both calculation methods. The slight difference of the tangential and sagittal direction in the ABCD-matrix calculation further suggests that a slight spherical term is also added to the beam. As this is much smaller than the actual astigmatic terms, we refrained from further discussing it here. Comparing Type I and Type II systems, we find that the astigmatic error for the same folding angle in a Type I system is twice the error in a Type II system. This can be explained logically, as the sources for the astigmatism in both systems are the large spherical mirrors, which are hit twice as often per round-trip in a Type I system. Furthermore, it can be seen that the astigmatic error for both systems increases over-proportionally with the folding angle and rather linearly with the number of round-trips. Hence, the folding angle in such a system should be chosen as small as reasonable possible. Figure 5: Change in beam diameter in the tangential and sagittal plane as a function of the folding angle \(\theta\) for Type I and Type II systems for up to five round-trips.