# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 8 Conclusions  
In this work, we presented a matrix model for the categorization and evaluation of imaging setups for multi-pass laser amplifiers in which the imaging elements are repeatedly used to reduce the total number of optics needed. It was found that such systems can be classified into two basic types, which we referred to as Type I and Type II. Here, Type I systems resemble a double-passed telescope, with a total of four interactions with imaging elements. Type II systems resemble a three-element imaging with three interactions of imaging elements per round-trip, accordingly. With the matrix  
Figure 12: Analysis of ray-tracing results from the compensated Type I system with five round-trips and the compensated Type II system with 10 round-trips. The upper diagrams show the Zernike coefficients of the output wave front (numbering according to [PERSON] et al. [9]) for a wavelength corresponding to 1 μm. The lower images show the corresponding residual wavefront after subtracting all terms lower than seven. formalism, we directly derived laws for the distances between the optics in dependence of their individual focal lengths. Further we discussed the influence of aberrations in the setup of such systems. It was shown that both types of systems can be used to internally compensate a thermal lens within their image plane. Correction terms for the distances were again derived by the matrix formalism accordingly. The problem of astigmatism was analyzed and quantified as well in dependence of the inevitable folding angle of the arms in mirror-based systems as for the tilted incidence required to geometrically separate individual round-trips for in- and out-put coupling. It was found that comparing both system types, equally dimensioned Type II systems exhibit a lower astigmatism by a factor of about two. Further using the matrix formalism, we demonstrated different methods, where due to geometrical folding, an intrinsic compensation of the system's astigmatism could be achieved. Here, a Type I system was compensated for the folding angle of the arms, while in a Type II system, compensation of the arms' folding angle was achieved with the astigmatism generated by the round-trips. In both cases, a nearly complete compensation was demonstrated. All presented calculations with the matrix method were cross-checked by analyzing the Zernike polynomials of wavefronts calculated by coherent ray-tracing of representative systems in the commercial software FRED. Hereby, a very good agreement between both calculation methods was achieved, proving the validity of the matrix method. The ray-tracing software also allowed analyzing higher order Zernike terms. It was found that both systems exhibit practically negligible higher order distortions even for a high number of round-trips and comparably short focal length. From a practical point of view, the results of our calculations suggest that, in general, Type II systems show a superior performance when compared to Type I systems. This statement is supported by the lower amount of astigmatic aberrations generated under similar conditions, the higher degree of compensation of effects from a thermal lens and by the overall lower amount of residual aberrations in a compensated system. The latter point is even more pronounced, as the investigated Type II system allows twice the number of passes, and therefore, it is also more compact. Furthermore, from an engineer's point of view, Type II systems seems to be more convenient as the total number of optics is smaller, in particular for the large diameter optics, on which the round trips are separated. Until now, mostly Type I systems can be found in the literature, but we believe that the use of Type II systems can lead to more compact laser designs with less aberrations, where shorter focal lengths can be employed. This work was partly supported by the European Social Fund (ESF) through the Thuringian Ministry of Economy, Employment and Technology (Project Number 2011 FGR 0122).