Driven by rapid advancements in modern precision manufacturing and machining, the demand for high-precision measurements is continually increasing, with displacement measurement accuracy requirements reaching the nanometer scale. Owing to their high resolution and precision, laser interferometers serve as primary tools for nanometer-scale measurements. Unlike traditional Gaussian beams, vortex beams possess a helical phase factor exp(i?θ) in the complex amplitude of their optical field, thus carrying orbital angular momentum, where ? is the topological charge, and θ is the azimuthal angle. The helical phase center contains a phase singularity, resulting in a hollow ring-shaped intensity distribution. These characteristics provide new degrees of freedom for optical field manipulation and analysis. Among these, vortex interferometers are formed by employing optical vortex beams as the carrier within the laser interferometer. Vortex interferometers encode axial phase variations into the azimuthal rotation of the resulting interference fringes. Furthermore, as the azimuthal angle provides an inherent 2π metrological datum, this approach enables high-precision phase demodulation. Moreover, vortex interferometers permit the direct determination of phase or displacement changes from the azimuthal rotation angle within a single interference fringe pattern, thus inherently possessing dynamic phase demodulation capabilities. However, conventional vortex interferometry demodulation relies on area-array cameras for real-time fringe capture and frame-by-frame image processing. The finite pixel size limits the phase shift resolution, while camera frame rate constraints and the high data volume associated with image sequences impede the measurement of dynamic displacement velocities.

Consequently, this paper establishes a carrier vortex interferometer and proposes a novel demodulation method based on the Doppler effect. A vortex beam generated by a gradual-width Fermat spiral mask (GW-FSM) is used as the test beam of a Mach-Zehnder interferometer to produce vortex interference fringes. A rotating chopper, a focusing lens, and a photodetector are successively set at the exit of the Mach-Zehnder interferometer to convert the vortex interference field into a one-dimensional (1D) temporal modulated signal. When the measured surface is stationary, the carrier frequency is obtained by the Fourier transform of the 1D temporal signal. When the measured surface moves at a certain velocity, the vortex interference fringes rotate azimuthally, generating a Doppler frequency shift relative to the carrier frequency, and the surface displacement velocity and instantaneous displacement can be obtained.

The results of the experiment are as follows:

1) The intensity distributions of the vortex beam captured at different observation distances, as well as the vortex interference fringes recorded at different observation distances after introducing the vortex beam into a Mach-Zehnder interferometer, are compared with the simulation results. The experimental results show a high degree of agreement with the simulations (Fig. 6).

2) When the measured surface is stationary, the experimental results under different carrier frequencies show a high degree of agreement with the simulation results (Fig. 7). The carrier frequency error is mainly caused by the jitter of the vortex interference fringes induced by the rotational vibration of the hollow rotary motor (HRM), as well as the angular velocity error of the HRM.

3) Experiments show that the experimental values of the Doppler frequency shift of the measured surface at different displacement velocities under different carrier frequencies have errors in the range of [-0.13 Hz, 0.1 Hz] compared with the theoretical values. The errors of the measured values of the surface displacement velocity compared with the theoretical values are in the range of [-34 nm/s, 26.6 nm/s] (Fig. 9).

Finally, this paper discusses the angular deviation between the reference wavefront and the measurement wavefront in a vortex interferometer, which is induced by the inclination of the measured surface. It indicates that the carrier vortex interferometer can solve the difficulty encountered by traditional vortex interferometric image post-processing algorithms in handling distorted interference fringes and exhibits strong robustness in dynamic surface displacement measurements (Fig. 10). The upper and lower limits of the displacement velocity are discussed.

This paper proposes a carrier vortex interferometric system and method for dynamic displacement measurement. On the one hand, a Bessel-Gauss (BG) vortex beam generated by GW-FSM diffraction is used as the measurement carrier in a Mach-Zehnder interferometer, endowing the vortex interferometric system with the potential for compactness, robustness, and ease of integration. On the other hand, a rotating chopper and a single-point photodetector are used to convert the two-dimensional (2D) vortex interference field into a 1D temporal modulated signal. The conventional phase demodulation algorithm based on pixel-wise processing of interference image sequences is improved into a frequency-domain analysis of time-domain signals, enabling the extraction of dynamic displacement velocity information of the surface through carrier frequency and Doppler shift. This method demonstrates strong robustness against distorted vortex interference fringes caused by surface inclination. The current study only carries out experimental research on surface displacement under uniform motion. However, by applying time-frequency analysis methods for non-stationary signals, such as short-time Fourier transform and wavelet transform, this approach can be extended in the future to dynamic displacement measurements of surfaces under non-uniform motion, exhibiting application potential in the field of nanometer-scale dynamic displacement measurement.