Error analysis and realization of a phase-modulated diffraction grating used as a displacement sensor.

A grating-based interferometric cavity produces coherent diffraction light field in a compact size, serving as a promising candidate for displacement measurement by taking advantage of both high integration and high accuracy. Phase-modulated diffraction gratings (PMDGs) make use of a combination of diffractive optical elements, allowing for the diminishment of zeroth-order reflected beams and thus improving the energy utilization coefficient and sensitivity of grating-based displacement measurements. However, conventional PMDGs with submicron-scale features usually require demanding micromachining processes, posing a significant challenge to manufacturability. Involving a four-region PMDG, this paper establishes a hybrid error model including etching error and coating error, thus providing a quantitative analysis of the relation between the errors and optical responses. The hybrid error model and the designated process-tolerant grating are experimentally verified by micromachining and grating-based displacement measurements using an 850 nm laser, confirming the validity and effectiveness. It is found the PMDG achieves an energy utilization coefficient (the ratio of the peak-to-peak value of the ±1st order beams to the 0th-order beam) improvement of nearly 500% and a four-fold reduction in 0th-order beam intensity compared with the traditional amplitude grating. More importantly, this PMDG maintains very tolerant process requirements, and the etching error and coating error can be up to 0.5 µm and 0.6 µm, respectively. This offers attractive alternatives to the fabrication of PMDGs and grating-based devices with wide process compatibility. This work first systematically investigates the influence of fabrication errors and identifies the interplay between the errors and the optical response for PMDGs. The hybrid error model allows further avenues for the fabrication of diffraction elements with practical limitations of micromachining fabrication.