The Simons Observatory: Rotation Performance of Cryogenic Half-Wave Plate Polarization Modulators

We present the on-site rotation performance of the first three cryogenic continuously rotating half-wave plate (HWP) polarization modulators for the Simons Observatory small aperture telescopes (SATs). The SATs operate at an altitude of 5200 m in the Atacama Desert in northern Chile, and measure the degree-scale cosmic microwave background polarization to search for primordial $B$-mode polarization. To this end, the SATs employ a 505 mm diameter 50 K cryogenic HWP polarization modulator to suppress atmospheric $1/f$ noise and to mitigate systematic uncertainties. We present methods for reconstructing the rotation of our polarization modulators, enabling detailed evaluations of their rotation angle accuracy, stability, displacements, and vibrations. We achieve the required rotation angle accuracy in more than 99.9% of observations, with a median noise level of 0.16 $\mu$rad$\sqrt{\text{s}}$. We also achieve one-dimensional measurement of the rotor displacement with an accuracy of 0.06 $\mu\text{m}\sqrt{\text{s}}$. Our results demonstrate the on-site rotation performance, rotation angle reconstruction method, and robustness of the polarization angle modulators. This is a crucial step towards achieving the SAT science goals.