A study by the Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, and the University of Chinese Academy of Sciences, titled "Hole-pillar coupled structure: a universal design model for two-dimensional gratings fabricated by dual-beam interference lithography," published in the journal Photonics Research, reports a coupled design approach for optical structures. The fabricated gratings exhibit diffraction efficiencies above twenty percent under normal incidence at 670 nanometers.

High-precision displacement measurement equipment acts as a core component for modern industrial manufacturing, including advanced semiconductor lithography machines and precise instrumentation. Within these measurement systems, two-dimensional gratings function as the central optical elements that split, modulate, and shift the phase of incoming light beams, which proves advantageous for compact system designs. As high-end manufacturing progresses toward meter-scale travel ranges and sub-nanometer resolution, the optical components require stable diffraction efficiency across different light polarizations. Currently, dual-beam interference lithography serves as a primary fabrication technique to produce these large-area optical elements.

Despite the extensive application of dual-beam interference lithography, researchers encounter an ongoing challenge regarding structural fidelity. During the exposure and development stages, the interplay of light waves generates continuous intensity variations. These variations cause the microscopic material to evolve gradually from hole-like cavities into pillar-like protrusions. Traditional theoretical design frameworks often assume idealized shapes such as perfect prisms, which may not capture the actual transition from holes to pillars. This gap between theoretical assumptions and actual physical transitions can lead to deviations in diffraction efficiency from theoretical predictions.

To address these morphology-related issues, the research team introduces a hole-pillar coupled design model tailored for dual-beam interference lithography. This approach examines the actual distribution of light intensity generated during fabrication. The researchers use a shape factor parameter to quantitatively describe the gradual bottom morphology changes of the optical grating. By scanning this parameter, the model mathematically represents the structural variation from holes to pillars. Furthermore, the team establishes a unified definition for structural proportions, constructing a representative three-dimensional model of the surface morphology. Employing an inverse design method, the researchers optimize critical physical dimensions to achieve stable diffraction efficiency for the incident light beams.

The experimental implementations show that the fabricated two-dimensional gratings exhibit measured diffraction efficiencies above twenty percent for the four first orders under normal incidence. Characterizations show a correlation coefficient of 0.988 between the actual fabricated microscopic structures and the theoretical designs. The experimental measurements exhibit a deviation of approximately six percent in total diffraction efficiency from the theoretical value, a per-order deviation of one to two percent, and an efficiency fluctuation of less than three percent across different light polarizations. By providing a descriptive model of the microscopic structural evolution, the work expands the manufacturing tolerance for large-area optical elements. Ultimately, this specific design methodology offers a design reference for nanoscale measurement systems, contributing to the development of next-generation precision instrumentation.