Theoretical analysis and simulation verification for measuring the geometric distances between the silicon spheres with the laser interferometer in G measurement

Abstract The Newtonian gravitational constant G is one of the most fundamental constants in nature. In G measurement with the angular acceleration feedback method, the largest error comes from the distances between the geometric centers of the source masses. In the on-going experiment, the silicon spheres with a more homogeneous density are used as the source masses. Here a scheme of measuring the geometric distances between the silicon spheres with the laser interferometer is proposed. The measurement principle is analyzed, and the error sources, such as the laser, the sphere, the alignment of optical path, and the environment are evaluated. With this method, the horizontal and vertical geometric distances can be measured with uncertainties of 11 nm and 9 nm, respectively. The simulation is performed to verify the theoretical model of measuring the distance, where the maximum deviation between the simulation result and the theoretical one is only −2.7 nm. When the sphericities of the four silicon spheres are at the level of 0.1 µ m, the uncertainty of each distance after considering the sphericity is about 0.1 µ m, corresponds to a combined uncertainty of 0.6 ppm for G measurement with the angular acceleration feedback method. This provides an effective method to reduce the measurement uncertainty of geometric distance between the silicon spheres, and makes it possible to measure G with a higher precision.