Ellipticity and astigmatism compensation in a double-pass AOM setup via the generalized ABCD matrix approach

High-spatial-quality laser beams are a crucial tool in the field of atomic, molecular, and optical physics. Although the preparation of a pure TEM₀₀ mode seems to be straightforward, it turns out to be demanding when the beams have to be controlled by means of free-space acousto-optic modulators (AOMs), due to their immanent property of introducing ellipticity, astigmatism, and higher-order aberrations to the diffracted beams. In this paper, we focus on the frequently used double-pass setup, in which the beam undergoes diffraction twice in the same modulator. First, we provide a theoretical description of the beam propagation through this system, in the framework of an extended ABCD formalism, allowing us to describe the nontrivial behavior of the AOM. Then, we apply it experimentally for two popular wavelengths (780 and 1560 nm) in the custom optical setup. The beam exiting the realized double-pass configuration, being nearly perfectly round and non-astigmatic, can be in particular coupled to the high-finesse optical cavities, being part of the optical flywheels in optical atomic clocks.