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We present the results from a theoretical investigation of laser beam propagation in relay imaging multi-pass layouts, which recently found application in high-energy laser amplifiers. Using a method based on the well-known ABCD-matrix formalism and proven by ray tracing, it was possible to derive a categorization of such systems. Furthermore, basic rules for the setup of such systems and the compensation for low order aberrations are derived. Due to the introduced generalization and parametrization, the presented results can immediately be applied to any system of the investigated kinds for a wide range of parameters, such as number of round-trips, focal lengths and optics sizes. It is shown that appropriate setups allow a close-to-perfect compensation of defocus caused by a thermal lens and astigmatism caused by non-normal incidence on the imaging optics, as well. Both are important to avoid intensity spikes leading to damages of optics in multi-pass laser amplifiers

Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers