Apparatus for the measurement of birefringence maps of optical materials: the case of crystalline silicon for Einstein Telescope

Einstein Telescope (ET) is expected to achieve sensitivity improvements exceeding an order of magnitude compared to current gravitational-wave detectors. The rigorous characterization in optical birefringence of materials and coatings has become a critical task for next-generation detectors, especially since this birefringence is generally spatially non-uniform. A highly sensitive optical polarimeter has been developed at the Department of Physics and Earth Sciences of the University of Ferrara and INFN - Ferrara Section, Italy, aimed at performing two-dimensional birefringence mapping of substrates. In this paper we describe the design and working principle of the system and present results for crystalline silicon, a candidate material for substrates in the low-frequency (LF) interferometers of ET. We find that the birefringence is of order $10^{-7}$ for commercially available samples and is position dependent in the silicon (100)-oriented samples, with variations in both magnitude and axis orientation. We also measure the intrinsic birefringence of the (110) surface. Implications for the performance of gravitational-wave interferometers are discussed.