# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 3 Influence of Thermal Lensing and Intrinsic Compensation
The exact parameters of the test systems are displayed in \[d_{1}=f_{1}+\frac{f_{1}}{2}\cdot\left(1-\sqrt{1+\frac{4f_{1}^{2} \cdot(d_{3}-f_{2})}{f_{t}f_{2}^{2}}}\right) \tag{10}\]1 The three-dimensional layouts are shown in \[d_{1}=f_{1}+\frac{f_{1}}{2}\cdot\left(1-\sqrt{1+\frac{4f_{1}^{2} \cdot(d_{3}-f_{2})}{f_{t}f_{2}^{2}}}\right) \tag{10}\]2 All test systems were mirror based with a folding angle \(2\theta=2^{\circ}\) to keep the influence of astigmatism negligible (cf. the next section). All traces were done with a collimated coherent Gaussian TEM00 beam with a full width at half maximum of 2 mm. The source area was 5 mm wide and situated at a 1-mm distance from the image plane, directly pointing towards it. The short distance was chosen to avoid deformation of the wavefront due to propagation effects. The beam was sampled with \(255\times 255\) points. The analysis surface for recording the wave fronts was placed in front of the image plane and was 10 mm wide with \(201\times 201\) sampling points. The thermal lens was implemented by replacing the plane mirror sitting in the image plane by a curved one. The radius of this mirror equals the focal width of the thermal lens, thus simulating the double pass of a laser medium in an active mirror scheme. \[d_{1}=f_{1}+\frac{f_{1}}{2}\cdot\left(1-\sqrt{1+\frac{4f_{1}^{2} \cdot(d_{3}-f_{2})}{f_{t}f_{2}^{2}}}\right) \tag{10}\]3 Inverse absolute value of the radius \(R_{o}\) as a function of the focal width of a thermal lens \(f_{t}\) for compensated and uncompensated relay imaging systems with 5 round-trips. Results were derived from ray-tracing simulation. Values small than \(10^{-3}\,\mathrm{1/m}\) can be considered as numerical noise. The incident wavefront was extracted before each pass of the image plane and saved for processing using the scripting functions included in FRED. The extracted wavefronts were then processed using MATLAB for fitting Zernike polynomials (we used the same numbering as [PERSON] et al. [9]). The radius of the wavefront \(R_{o}\) was finally calculated from the spherical term of the Zernike polynomials. From the results displayed in Figure 3, it can be seen that for both system types, a good compensation of the thermal lens can be achieved, which is equivalent two a low value of \(1/|R_{o}|\). Further, at least for the simulated range, \(1/|R_{o}|\) is widely independent of \(f_{t}\) for a compensated system. In uncompensated systems, the predicted add-up effect of the thermal lens aberrations with an increasing number of round-trips can also be clearly seen. Comparing the Type I and Type II results, it is further visible that the Type II systems achieve a higher compensation than the Type I systems where a lensing effect slightly increasing with the number of round trips can be seen. As this does not depend on \(f_{t}\), it can be attributed to a system-specific effect, which is very likely caused by spherical aberrations on the large spherical mirrors that are hit out of center. In Type II systems, this is not the case, since the beams are always coming from and are reflected into a common spot in the radius distance of \(L_{1}\). For this, a spherical optic is the optimum case. As the numerical accuracy of the evaluation algorithm can be estimated to be in the range of \(R_{o}=\)1 km, variations for values smaller than \(1/|R_{o}|=10^{-3}\,1/m\) should be attributed to negligible noise. Furthermore, residual defocus errors in this order of magnitude have no impact in real systems. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Type I** & **Type II** \\ \hline \(f_{1}\) & 0.5 m & 0.5 m \\ \(f_{2}\) & 0.5 m & 0.25 m \\ \(d_{1}\) & 0.5 m & 1 m \\ \(d_{2}\) & 1 m & 1 m \\ \(d_{3}\) & 0.5 m & - \\ diameter \(L_{1}\) & 75 mm & 75 mm \\ diameter \(L_{2}\) & 75 mm & 19 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Configuration parameters for the test systems used in the ray tracings. Figure 4: Three dimensional setup of the basic layouts for Type I (**a**) and Type II (**b**) systems in FRED.