# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 4 Astigmatism in Mirror-Based Layouts  
In the amplifier, the use of spherical mirrors instead of lenses as imaging elements has many advantages. Typically, the losses within a mirror-based layout are lower than for the corresponding lens design. Furthermore, due to the folding of the setup, a smaller footprint of the system can be realized. Another drawback for a lens-based design is the presence of ghost foci that are generated by residually reflected light from the transmitting surfaces. In high energy laser systems, such ghost foci must be considered very carefully to avoid damages. For the case of a mirror-based design, it is nearly impossible to generate any ghost foci, as light transmitted through the high reflective surfaces is de-focused and typically not re-coupled to the rest of the setup. Another advantage is that a mirror-based design will significantly reduce the collected nonlinear phase, the so-called B-integral. Nevertheless, the main drawback for a mirror-based design is that to separate the incoming and outgoing beams on a spherical mirror, an incident angle \(\theta\) has to be used as shown in Figures 1 and 2. This adds a certain amount of astigmatism to the beam, which will also increase during the round-trips in an amplifier. This effect is also problematic in spectroscopic multi-pass cells, as discussed in [3,10,11]. In the following, we derive a quantitative calculation of this effect using the ABCD-algorithm. In dependence of \(\theta\), the focal length in a plane tangential to the optic \(f_{m}\) and vertical (sagittal) to the optic \(f_{s}\) are defined as, e.g., in [12]:  
\[f_{m} = f\cdot\cos\theta \tag{20}\] \[f_{s} = \frac{f}{\cos\theta} \tag{21}\]  
In the following, we describe the relay imaging system in each of the planes by modified system matrices \(G_{m}\) and \(G_{s}\), which can be obtained for the corresponding planes by altering the focal lengths of the optics accordingly, while maintaining the original distances. For simplicity reasons, we will limit our considerations to the basic layouts of Types I and II. In case of Type I, this means that we assume \(f_{1}=f_{2}=f\) and \(d_{1}=d_{2}=f\). For Type II, we use \(f_{1}=2f_{2}=f\) and \(d_{1}=d_{2}=2f\) accordingly. In the following, we will refer to \(f\) as the system's focal length. It should be mentioned that the simplifications made are not a general limit of the matrix model, but rather used to reduce the complexity of the resulting expressions. Additionally, the assumptions made here will still meet most real layouts of such systems. We assume that the input beam into our system is a collimated beam. A parallel ray with distance \(x_{0}\) with respect to the optical axis is represented by:  
\[\vec{v}_{0}= \begin{pmatrix}x_{0}\\ 0\end{pmatrix}  \tag{22}\]  
The resulting output ray after one round-trip in the system in each plane \(\vec{v}_{s/m}\) is obtained by multiplying \(\vec{v}_{0}\) with the corresponding system matrix:  
\[\vec{v}_{s/m}=G_{s/m}\cdot\vec{v}_{0} \tag{23}\]  
In the case of a Type I system, one obtains for the output after one round-trip:  
\[\vec{v}_{mI}=\left( \begin{array}{c}x_{0}\cdot\frac{\cos^{4}\theta-16\cos^{3 }\theta+40\cos^{2}\theta-32\cos\theta+8}{\cos^{4}\theta}\\ \\ \frac{x_{0}}{f}\cdot\frac{-4\cos^{3}\theta+20\cos^{2}\theta-24\cos\theta+8}{ \cos^{4}\theta}\end{array} \right) \tag{24}\]\[\vec{v}_{sI}=\left( \begin{array}{c}x_{0}\cdot\left(1+8\cos^{4}\theta-32\cos^{3} \theta+40\cos^{2}\theta-16\cos\theta\right)\\ \frac{x_{0}}{f}\cdot\left(8\cos^{4}\theta-24\cos^{3}\theta+20\cos^{2}\theta-4 \cos\theta\right)\end{array} \right) \tag{25}\]  
For further evaluation, we want to use parameters that are independent of both the exact focal length \(f\) of the system and the input beam parameter \(x_{0}\). The position of a single ray in the image plain relative to the original position with \(\
icefrac{{x}}{{x_{0}}}\) in the respective plane fulfills these requirements and can be used to predict a deformation and a size change of the beam in the image plane.