# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 6 Intrinsic Astigmatism Compensation Methods  
Though astigmatism can be compensated in a straightforward way by using cylindrical optics for pre- and or post-compensation, it is often more convenient to have an intrinsic compensation. If this is achieved in each single pass of the system, it further reduces issues arising from the deformation of the beam profile on optics outside the image plane. The basic idea for such a compensation mechanism is to introduce astigmatism in two orthogonal directions with the same strength, which finally cancel each other. Such approaches were also discussed, e.g., in [3; 10; 11] for spectroscopic cells. We will  
Figure 8: Absolute astigmatic deformation of the wavefront as a function of the ratio between the diameter \(D\) of optic \(L_{1}\) and its focal width \(f\) for Type I and Type II systems. Lines are calculated by the matrix method, and single data points originate from FRED ray tracings. Figure 9: Evolution of the astigmatic deformation of the wavefront for 5 round-trips (**a**) and for 10 round-trips (**b**). Lines are only for illustration purposes. present two approaches to achieve this within the presented Type I and Type II systems in the basic configurations, as described in Section 4. In the case of the astigmatism within a Type I system caused by the folding angle of the arms, a compensating effect can be achieved when both arms are tilted in orthogonal planes by approximately the same amount (cf. Figure 10). Using this configuration in each plane, the beam will have one optic that is hit in the asagittal plane having a focal length of \(f_{s}\) and one in the tangential plane having a focal length of \(f_{m}\), where only the order is reversed for each beam axis. Hence, using Equation (4) results in:  
\[d_{2}=f_{m}+f_{s}=f\cdot\left(\cos(\theta)+\frac{1}{\cos(\theta)}\right) \tag{37}\]  
Furthermore, Equation (3) must be fulfilled for both directions to achieve a complete compensation. An exchange of \(f_{1}\) and \(f_{2}\) in Equation (3) must not alter \(d_{1}\) or \(d_{3}\). This results in:  
\[d_{1}=d_{3}=\frac{f_{m}f_{s}^{2}+f_{s}f_{m}^{2}}{f_{m}^{2}+f_{s}^{2}}=f\cdot \cos\theta\cdot\frac{1+\cos^{2}\theta}{1+\cos^{4}\theta} \tag{38}\]  
A system with the according setup will be fully compensated, at least according to the matrix model. To cross-check this, we modeled a Type I system according to the ones presented in Section 4 with a folding angle of \(\theta=10^{\circ}\) and five round-trips in the compensated arrangement. The resulting astigmatic deformation of the wavefront \(f/R\) was less then \(4.5\times 10^{-3}\) in all individual round-trips. Compared with the original value of \(f/R\approx 0.3\), this practically corresponds to a full compensation and proves the results from the matrix model. The residual deformation corresponds to the astigmatism caused by the round-trips, which is in the same range for the chosen configuration (D/f = 0.15; cf. Figure 8). In the case of a Type II system, a compensation according to the just described method in Type I systems is only possible if the system is extended to get a second arm that can be tilted. In the following, we will discuss another way to compensate for astigmatism, which is based on compensating the astigmatism due to the folding angle with the astigmatism originating from the round-trips. We will use a basic Type II system (cf. Section 4) with 10 round-trips, a diameter of \(L_{1}\) of 200 mm and a focal length of \(f=500\) mm, resulting in \(\
icefrac{{D}}{{f}}=0.3\). According to Section 5, the total astigmatic deformation due to the round-trips of the system \(f/R_{o}\) is calculated to be 0.0329 in the tangential plane and \(-0.0328\) in the sagittal plane. Now, we use the matrix method according to Section 4 to numerically find an angle that will produce the inverse  
Figure 10: Different views of a compensated Type I system with five round-trips and \(\theta=10^{\circ}\). amount of astigmatic deformation.