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

Abstract 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 interferometers of ET. We find that the birefringence is $$\lesssim 10^{-7}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>≲</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> 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: $$\Delta n^{(110)}=-(1.50\pm 0.15)\times 10^{-6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>110</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1.50</mml:mn> <mml:mo>±</mml:mo> <mml:mn>0.15</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> @ $$\lambda =1550$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1550</mml:mn> </mml:mrow> </mml:math> nm. Implications for the performance of gravitational-wave interferometers are discussed.