# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 1 Introduction  
In the last two decades, the output energy of diode pumped high energy lasers based on Ytterbium doped materials has scaled the achievable output energies into the range of tens of Joules for nanosecond pulses [1], as well as for femtosecond pulses within laser architectures based on the chirped pulse amplification technique [2]. The major advantage of these materials is the long upper state lifetime in comparison to neodymium-doped materials, which are widely used in flash-lamp pumped, high energy systems. This allows one to store a large amount of energy within the active medium with comparably low pump power. Thus, a lower power laser diode pump engine can be used, which reduces size and cost. Nevertheless, this advantage comes at the price of significantly lower cross-sections of the laser medium. Hence, extracting the energy and achieving a reasonable gain in the amplifier requires a higher number of material passes. A major task for the development of such laser systems is the design of multi-pass setups that maintain the typically top-hat-shaped beam profile throughout many amplification steps. As such, beam profiles cannot be propagated without significant diffraction effects, only imaging systems are suitable for this task. In these systems, aberrations should be as little as possible. These includes low order aberrations, as these may cause high intensities, leading to damages of optical elements, clipping of the beam profile due to the deformation of the beam or inhomogeneous depletion of the excitation and, therefore, less efficient amplification. Though this task is pretty similar to spectroscopic multi-pass cells, the readily available setups, such as Herriott or White cells [3], are no solution for laser amplifiers, since these cells either do not have an image plane that is passed multiple times or as there are non-planar optics within their imageplanes, which is not feasible in a laser amplifier, since the laser material should be placed in these image planes. In recent years, some compact imaging multi-pass layouts have been demonstrated. With these setups, a high beam quality and a high number of round-trips have been achieved [4; 5; 6; 7]. Furthermore, due to the repeated use of the imaging optics, the setups described in [5; 6; 7] allowed one to achieve a high number of material passes with a low number of optics. Such systems offer to scale the number of round-trips without increasing the number of optics, which allows one to achieve simple and compact designs for multi-pass imaging laser amplifiers. Similar layouts with integration into cavities of regenerative amplifiers to achieve multiple amplification passes per resonator round-trip were patented by [PERSON] et al. [8]. In this work, we will present a method to analyze such systems with multiple round-trips and repeated use of the imaging optics by employing the ABCD-matrix formalism. These imaging elements will be modeled as ideal lenses, while in the following ray tracing simulations, spherical optics will be used for practical reasons. Thus, starting with only a few basic assumptions, we are able to categorize these systems into two fundamental types. These two kinds of setups are investigated separately for the introduced low order aberrations. First, the defocus, caused by thermal lensing, is investigated. Secondly, astigmatism, caused by tilted incidence on the imaging elements, is analyzed. In both cases, the generalized design parameters and the number of round-trips are varied. We will also derive means for corrections to the system parameters that allow one to compensate for these unwanted effects. The presented results from the paraxial one-dimensional matrix model are cross-checked using the ray tracing software FRED (Version 9.110 Optimum, Photon Engineering LLC, Tucson, AZ, USA, 2010). Here, an automated routine was programmed to compute the wavefront produced by sample systems with varying parameters.