# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 5 Astigmatism due to Round-Trips  
Another source of astigmatism within the presented relay imaging setups is introduced by the angular separation of the consecutive passes. This leads to a non-normal incidence on optics \(L_{1}\) and \(L_{2}\) in Type I systems and on \(L_{2}\) in Type II systems. As before, we will use the basic system layouts for analyzing this effect. The astigmatism will be included in the calculation by the same means as before, but using a varying incidence angle \(\varphi(k)\). In the case of a Type II system, the sole source of astigmatism due to the round-trips is the mirror \(L_{2}\). As the mirror \(L_{1}\) is placed in the radius distance to the image plane, the incidence angle on \(L_{1}\) is always normal to its surface. Including the astigmatism terms for \(L_{2}\) according to Equations (20) and (21) for \(G_{3}^{m}\) and \(G_{3}^{s}\) for a single round-trip in the tangential and asgittal plane respectively results in:  
\[G_{3}^{m}= \begin{bmatrix}1&0\\ \frac{2\cos(\varphi(k))-2}{f\cos(\varphi(k))}&1\end{bmatrix} \qquad G_{3}^{s}= \begin{bmatrix}1&0\\ -\frac{2\cos(\varphi(k))-2}{f}&1\end{bmatrix}  \tag{29}\]  
Here, \(\varphi(k)\) is the incident angle, which varies for each round-trip. Given, that the system matrix represents an ideal lens in each plane, the inverse focal width \(\
icefrac{{1}}{{f}}\) will just add up over the round trips. Hence, \((\
icefrac{{1}}{{R_{o}}})_{m/s}\) after \(n\) round-trips can be expressed in each plane by a sum:  
\[\left(\frac{1}{R_{o}}\right)_{m}=-\frac{2}{f}\cdot\sum_{k=1}^{n} \frac{\cos(\varphi(k))-1}{\cos(\varphi(k))} \tag{30}\] \[\left(\frac{1}{R_{o}}\right)_{s}=\frac{2}{f}\cdot\sum_{k=1}^{n} \cos(\varphi(k))-1 \tag{31}\]  
The maximum incidence angle \(\varphi_{max}\) can be calculated using:  
\[\varphi_{max}=\arctan\left(\frac{a_{n}}{2d_{2}}\right)=\arctan \left(\frac{a_{n}}{4f}\right) \tag{32}\]  
Here, \(a_{n}\) is the maximum separation on the optic \(L_{1}\), and it depends on the total number of round-trips \(n\). \(a_{n}\) again can be calculated assuming an optimum usage of the optics aperture \(D\) (cf. Figure 7) using:  
\[a_{n}=D\cdot\frac{n-1}{n} \tag{33}\]  
Hence, the change in the incidence angle for each pass is given by:  
\[\Delta\varphi=\frac{2\varphi_{max}}{n-1} \tag{34}\]  
Now, \(\varphi(k)\) can be calculated by:  
\[\varphi(k)=\varphi_{max}-(k-1)\cdot\Delta\varphi \tag{35}\]  
The wavefront radius \(R_{o}\) of the sagittal and tangential plane of the astigmatic wavefront can be directly calculated using Equations (30) and (31). For the case of Type I systems, such a closed solution no longer exists as both \(L_{1}\) and \(L_{2}\) generate the astigmatism during the round-trips and are not situated in an image plane or in an equivalent one. Hence, propagation effects will also be present. The handling of this type is nevertheless identical to the procedure described in the previous section for the astigmatism due to the folding angle except fora varying incidence angle \(\varphi(k)\) for every pass. The computations for \(\varphi(k)\) are the same as for Type II systems except for \(\varphi_{max}\), which is half of the value in Type I systems (cf. Figure 7):  
\[\varphi_{max}=\frac{1}{2}\cdot\arctan\left(\frac{a_{n}}{2d_{1}}\right)=\frac{1} {2}\cdot\arctan\left(\frac{a_{n}}{2f}\right) \tag{36}\]  
For the following results, we used a collimated input beam according to the procedure in Section 4. In Figure 8, the absolute astigmatic deformation of the wavefront as a function of the ratio between the diameter of the optic \(L_{1}\) and its focal width is shown for both system types. Using \(\
icefrac{{D}}{{f}}\) for the x-axis and \(\
icefrac{{|f/k|}}{{R}}\) for the y-axis allows being independent from the exact focal width and the actual diameter of \(L_{1}\). Hence, the results are applicable to a wide range of specific systems. For the ray tracing simulation, systems identical to the ones used in Section 4 were used. To keep the influence of the folding angle negligible, this angle was fixed at 0.5\({}^{\circ}\).