Synthetical modeling and experimental study of fabrication and assembly errors of two-dimensional gratings

The geometrical errors resulting from imperfection of the fabrication and assembly of two-dimensional(2D) grating were investigated. Based on the Doppler frequency shift theory and the coordinate transformation method, the general error model versus non-orthogonal and misalignment angles of 2D gratings was established. Then the effects of these error angles were quantitatively studied, and the cosine and the cross-talk errors of both directions were simulated and analyzed. The results illustrated that the geometrical errors were independent from the diffraction orders, diffraction times and the optical subdivision multiple, just determined by the error angles and the 2D displacement. Meanwhile, the cosine errors could be induced by all the error angles, but the cross-talk errors were mainly influenced by the non-orthogonal and raw angles. Besides, the cross-talk errors were more serious than the cosine errors, which is the dominant component of the geometrical errors. A planar displacement measurement system with 2D cross-grating was constructed, and the theoretical analyses and the numerical simulation were verified with the 10 mm square displacement measurement.