We propose a simple and compact reading head with the Littrow configuration that will increase measurement range and reduce the complexity of a two-dimensional grating-based interferometer. The reading head contains only a beam splitter, two polarizing beam splitter modules, and two mirrors. The theoretical resolutions in two directions are 0.27 nm and 0.18 nm, respectively. In comparison with a dual-frequency laser interferometer, the proposed interferometer can measure displacement from 3 nm to 10 mm with high accuracy. The $3\sigma$ values in two directions for the difference are 1.67 nm and 1.35 nm for a displacement of 9 nm. Repeatability for a displacement of 1 μm is better than 2 nm.

We propose a new symmetrical heterodyne grating displacement measurement method, based on 2D grating and single diffraction quadruple subdivision method. Using a dual-frequency laser with a wavelength of 632.8 nm, output power of 2.2 mW, and a 1200 l/mm 2D grating, eight diffracted light beams interfere in pairs in the X and Y directions through a turning element. The detection system’s measurement accuracy was assessed experimentally. The system measurement resolution in the X and Y directions is better than 3 nm; the grating displacement measurement errors within a 10 mm range are better than ±30 nm and ±40 nm, and the repeatability error is better than ±25 nm. The method is not only applicable to nanoscale 2D displacement measurement technology but also can be used for ultra-precision positioning and ultra-precision processing, with the potential for picometer-level improvement.

We propose a 3D measurement method based on 2D grating dual-channel and Littrow equal-optical path incidence to detect the 3D displacement of a 2D grating in the X-, Y-, and Z-directions. The 2D grating is combined with the Littrow incidence method and a turning element to cause the Littrow diffracted light with frequency f₁ to interfere with the reference light at frequency f₂, and the displacement data in the X-, Y-, and Z-directions are obtained using the separation-dual-channel phase decoupling algorithm. A corresponding test experimental platform is constructed, and linear error evaluation and step error evaluation experiments are performed to determine the displacements in the X-, Y-, and Z-directions. The results obtained show that all linearity errors are within ±60 nm in the 10 mm measurement ranges in the X-, Y-, and Z-directions, and the test resolution is within ±5 nm. The proposed method can thus realize nanoscale synchronous measurement of X-, Y-, and Z-direction 3D displacements.

The grating interferometer in the Littrow configuration uses quarter wave plates (QWPs) to modulate the polarization in the measurement system to determine the autocollimation optical path. Fabrication errors and mounting errors of the QWPs lead to phase changes in the grating interferometer that generate measurement errors. As an alternative, we propose a grating interferometer that produces conical diffraction. Using the grating instead of QWPs to modulate the beam’s polarization bypasses this source of error. A 45 mm range experiment was performed that yielded a repeated measurement error of 40 nm. Experiments show that the system has a simple structure and good repeatability and is capable of high-precision displacement measurements.

Grating-based interferometers play important roles in precision displacement measurements. Gratings are the core components of grating-based interferometers, and grating surface errors and line errors seriously affect measurement accuracy, especially in systems with Littrow configurations. A fast, accurate method to calculate displacement errors caused by grating surface errors and line errors in a grating-based interferometer with a Littrow configuration is proposed. Displacement errors are calculated using the diffracted wavefronts at the $\pm 1$st orders of the grating. Experimental comparison of the displacements of a grating-based interferometer and a laser interferometer verifies the correctness of the proposed method. The differences between the calculated and measured displacement error results are within 40 nm. The method is accurate, fast, and low cost and is highly significant for system error compensation and improvement of the measurement accuracy of grating-based interferometers.

To fabricate plain holographic gratings with high wavefront quality and to obtain the wavefront required in varied line-space grating, an active control technology of a diffraction grating wavefront by modulating the phase distribution of the scanning-beam interference lithography system was proposed. Sinusoidal wavefront control is simulated, and the controlled wavefront being almost the same as the target wavefront. A photoresist grating was fabricated whose surface is uniform and the wavefront is ideally sinusoidal. The theoretical analysis and experimental results confirmed that the wavefront of the diffraction grating can be actively controlled by modulating the phase distribution of the scanning-beam interference lithography system.