Objective  Nano-displacement measurement technology is an important branch in the field of precision measurement, its development and improvement are important guarantee for realizing high-precision nano-manufacturing. With the rise of laser self-mixing interference technology, the precision displacement measurement method with simple structure, low manufacturing cost and measurement accuracy up to nanometer level has been vigorously developed. Laser self-mixing interference technology has been widely used in displacement measurement, absolute distance measurement, speed measurement, and vibration measurement, etc. With the advantages of single optical path structure and the comparable measurement accuracy as double beam interference, the self-mixing interference technology has better application prospect in the industrial area. Traditional laser self-mixing interference schemes mostly take mirrors or scattering surfaces as target mirrors, which take laser wavelengths as measurement benchmarks and are easily disturbed by environmental changes. In order to increase the robustness of the measurement benchmark, this paper studies a laser self-mixing nanometer displacement measurement method based on a planar reflective holographic grating. Different from traditional laser self-mixing interference, the displacement measurement value based on grating feedback is determined by the period of the grating.

  Methods  For the laser self-mixing displacement measurement method based on the plane reflective grating feedback, the vibration displacement value of the holographic grating is reconstructed in this paper. The displacement measurement value of this method is based on the grating period. The system setup is shown (Fig.1). The light emitted by the laser is incident on the grating surface at the Littrow angle, so the retro-reflect one-order diffraction light carry the Doppler phase shift caused by the displacement along the grating period direction. The self-mixing interference output laser is splitted by the structure composed of a half-wave plate and a polarized beam splitter, and the self-mixing signal is collected through a photodetector. In terms of signal processing, the grating self-mixing interference signal is firstly denoised by a low-pass filter and then normalized. Combining the threshold setting method to decide the inversion point of the displacement direction and the phase unwrapping algorithm of arccosine, the displacement of the grating is reconstructed. The grating used in this experiment is a plane diffraction grating with the period of 2400 lines/mm, which equals 416.67 nm. The constructed displacement is compared with the measurement result of a commercial laser interferometer.

  Results and Discussions  In the grating self-mixing interference experiment, the signal under the condition of weak feedback intensity was measured, and the normalized interference signal was shown (Fig.5). After signal processing based on the arccosine method, the corresponding nano-displacement reconstruction results were obtained (Fig.7). The result represents the linear displacement of reciprocating motion as shown in the experiment setting. By calculating the variance of the linear displacement, the entire system has a displacement noise of 5.82 nm, which is expected to be optimized by performing a finer filtering on the signal. From the displacement reconstruction results, the entire measurement result has a linear deviation coefficient of 1.1086 times the actual displacement. A commercial laser interferometer and a grating self-mixing interferometer were also used to compare the displacement measurement data. After the linear correction, the measurement results show that the displacement error does not exceed 0.241% (Tab.2).

  Conclusions  Laser self-mixing nano-displacement measurement method based on the feedback of a planar diffraction grating is studied in this article, and a calculation method using the arccosine method for wrapping phase is proposed. Experimental research was carried out under weak feedback conditions, and the experimental results were reconstructed based on the arccosine method. Compared with the measurement results of commercial laser interferometers, it was found that the laser self-mixing interferometry method based on planar diffraction grating feedback could be used as an effective scheme for nano-displacement measurement. In the future, the measurement accuracy and precision of the grating self-mixing interferometer can be further improved by optimizing the geometric alignment, adopting a more accurate grating, and performing more effective filtering on the signal.