A Multiaxial Grating Interferometer System With Spatially Separated Beams in Joule Balance

The Joule balance method is a kilogram realization method based on Planck constant. Alignment error is one of the largest errors in Joule balance. It can be suppressed by the integrated mechanism of force measurement and guidance based on the reciprocity principle. However, a six axial displacement measurement system is required for force measurement and motion guidance. This article proposes a six-axial displacement measurement method based on grating interferometry. First, we design a grating interferometry with spatially separated beams. With the design, the optical mixing can be avoided, and nonlinear errors is suppressed. The analysis of the high tolerance angle design is also conducted. With the cube corner retroreflector (CCR) and grating, the tolerance angle can be significantly increased. A decoupling model for six-axial displacement measurement is established. High-precision displacement decoupling measurement is achieved by using spatial coordinates and vectors. Finally, an experimental setup is constructed. The nonlinear error of the proposed interferometer is measured, and the results show the amplitude of the nonlinear error is less than 0.5 nm. The measurement resolution for displacement is tested, the resolution of the vertical and lateral displacement are 0.19 and 0.20 nm, respectively. With decoupling model, the measurement resolution of the three-axial displacements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta x,\Delta y$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta z$ </tex-math></inline-formula> are 0.2, 0.2, and 0.19 nm, respectively. The measurement resolution of the three-axial rotation <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta \theta _{x},\Delta \theta _{y}$ </tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Delta \theta _{z}$ </tex-math></inline-formula> are 0.001, 0.001, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.004~\mu $ </tex-math></inline-formula>rad, respectively. The results indicate that the proposed grating interferometer system can be used for high-precision six-axial measurement.