Centroid identification and displacement calculation method for vortex grating interferometer or vortex light interferometer

Abstract

This invention discloses a centroid identification method and a matching displacement sensing system for vortex grating interferometers (OVGI) or vortex optical interferometers (OVI), belonging to the field of precision optical measurement technology. The centroid identification method includes color-weighted grayscale preprocessing of a color image; extraction of pixel coordinates and intensity to construct a three-dimensional feature vector; and the use of a two-stage segmented fuzzy C-means (FCM) clustering strategy. In the first stage, initial centroids are obtained by clustering single-frame images; in the second stage, stable centroids are obtained by secondary clustering of multiple initial centroids. Finally, the displacement is calculated based on the analytical relationship between the petal rotation angle and the displacement. The displacement sensing system includes a light source module, a vortex light generation module, a diffraction and displacement sensing module, an interference optical path construction module, a detection module, and an independent reference measurement module. The system converts the displacement of the sensing module into the azimuth angle rotation of the interference petals through vortex light interference, and the image is acquired by a CCD. This invention achieves nanometer-level precision, high environmental robustness, system self-calibration, and lightweight real-time processing.

Description

Technical Field

[0001] This invention belongs to the field of precision optical measurement technology, specifically relating to a displacement sensing system based on vortex optical interferometry and a centroid clustering identification method. Background Technology

[0002] High-precision displacement measurement is the cornerstone of advanced manufacturing processes such as photolithography, precision robotics, and CNC machining. Laser interferometry is widely used for displacement measurement due to its non-contact nature, high resolution, and wide dynamic range. However, it is sensitive to environmental disturbances such as temperature fluctuations and air turbulence, making it difficult to maintain stable accuracy in non-constant temperature industrial environments. In recent years, vortex-based interferometers (OVIs) have attracted considerable attention: they convert displacement into azimuth rotation of a vortex interferogram, using a 2π circle as a natural reference, theoretically achieving nanometer to picometer-level resolution. However, OVIs still use the laser wavelength as the reference when measuring displacement, remaining sensitive to environmental disturbances and thus requiring strict environmental control. In contrast, grating interferometers utilize the thermally and mechanically stable grating pitch as the measurement reference, which is less affected by environmental factors, enhancing resistance to environmental interference. Therefore, applying vortex light to grating interferometers can leverage the high-precision displacement measurement potential of OVIs while maintaining the high stability of grating measurements, showing great promise for applications in precision displacement measurement. However, this novel vortex grating interferometer (OVGI) currently lacks a systematic solution that can simultaneously ensure high accuracy and robustness. OVI and potential OVGI methods typically rely on identifying the centroids of the petals in the vortex interferogram for displacement calculation, using the centroid rotation angle to calculate the displacement. However, existing centroid identification methods have significant drawbacks that fail to meet the needs of practical industrial applications, specifically in the following six aspects: First, environmental disturbances and pattern distortions lead to large centroid identification errors, making it difficult to guarantee measurement accuracy. Traditional methods often directly calculate the geometric center of the petal pattern as the centroid, but interference patterns are highly susceptible to deformation caused by external random noise, uneven illumination, and wavefront distortion, resulting in a shift in the geometric center and a significant decrease in measurement accuracy. Improving the anti-interference capability of centroid identification algorithms to achieve high-precision centroid positioning in complex environments is a primary technical challenge. Second, the measurement system lacks an independent and traceable displacement reference and self-verification mechanism, resulting in insufficient reliability of measurement results. Existing solutions typically lack mechanisms for real-time, independent verification of measurement results, leading to insufficient reliability and traceability of the system's output displacement values, making them unsuitable for application in metrological scenarios requiring high reliability. Establishing an absolute displacement reference for the system and implementing self-verification to ensure the metrological reliability and long-term stability of measurement data is a critical issue that urgently needs to be addressed. Third, the interference optical path requires stringent assembly and adjustment precision, presenting significant engineering challenges and resulting in poor system stability. The high precision required for the assembly and adjustment of the interferometric optical path is crucial; even minor misalignments can lead to signal quality degradation, increasing the difficulty of system engineering implementation and maintenance. Optimizing the optical path design to ensure high-sensitivity displacement-angle conversion while reducing assembly and adjustment difficulty and improving system usability and long-term stability is a significant challenge in practical applications. Fourth, image preprocessing and feature extraction methods are often crude, failing to fully utilize and enhance the effective information in the interferogram, thus limiting the accuracy of subsequent processing. Designing more intelligent image preprocessing and feature representation methods to extract more accurate and robust petal feature information is a crucial step in improving the overall system performance. Fifth, the core algorithm relies on large amounts of data and complex models, resulting in high computational demands and limitations in real-time performance and generalization ability. To address pattern distortion, some studies have attempted to use deep learning-based neural networks for angle recognition. However, these methods require large amounts of diverse labeled data for training, and the complex models consume significant computational resources, making them difficult to deploy in embedded systems with high real-time requirements or limited computational resources. Their generalization ability is also limited. Developing a core algorithm that is computationally lightweight and highly generalizable, independent of big data, to reduce system complexity and application barriers is essential for the practical application of the technology. Sixth, low system integration and a lack of modular design make industrialization difficult. Existing solutions typically involve tightly coupled components, lacking clear module divisions and standardized interfaces, leading to difficulties in system integration, debugging, and maintenance, and hindering the development of standardized products suitable for mass production and widespread adoption.

[0003] Therefore, for the practical application of vortex grating interferometers (OVGI), this invention aims to comprehensively solve the above six core problems through systematic hardware and software innovation, and provide a complete displacement measurement solution that takes into account high precision, high robustness, self-calibration capability, ease of use, real-time performance and high industrial feasibility. Summary of the Invention

[0004] To address the problems existing in the prior art, this invention provides a method for centroid identification and displacement calculation applicable to vortex grating interferometers or vortex optical interferometers. The method first acquires a sequence of petal-shaped interference images formed by vortex light diffraction and interference through a grating using a CCD. Then, the original color images undergo color-weighted grayscale preprocessing to enhance the contrast and visibility of the petal structure. Next, pixel coordinates and normalized intensity values ​​are extracted from each frame to construct a three-dimensional feature vector, forming point cloud data. Centroid identification employs a two-stage fuzzy C-means (FCM) clustering strategy: In the first stage, FCM clustering is performed on the feature point cloud of a single frame image, setting the cluster number to the number of petals, and iteratively solving for the initial centroid of each petal; in the second stage, the initial centroid set obtained from multiple frames is used as input, and FCM clustering is performed again to obtain a stable and noise-resistant final centroid estimate. Finally, based on the analytical relationship between the petal rotation angle and the grating displacement, the rotation angle is calculated using the stable centroids obtained from clustering, thereby accurately calculating the displacement.

[0005] Corresponding to the above method, this invention also provides a grating interferometric displacement sensing system based on vortex light. The system includes a light source module, a vortex light generation module, a diffraction and displacement sensing module, an interference optical path construction module, a detection module, and an independent reference measurement module arranged sequentially along the optical path. The light source module provides a Gaussian beam, which is converted into a Laguerre-Gaussian beam (vortex light) by the vortex light generation module. The grating in the diffraction and displacement sensing module diffracts the vortex light into diffracted beams including +1st and -1st orders. The grating is mounted on a displacement stage, and its in-plane displacement modulates the phase of the diffracted beams. The interference optical path construction module guides and combines the two diffracted beams through a mirror and a beam splitter to form a coaxial interference beam. The detection module uses a charge-coupled device (CCD) to acquire interference patterns presenting a specific number of petals. The reference measurement module uses a laser interferometer to independently and with high precision measure the displacement of the displacement stage, serving as the reference for system displacement calculation. This system linearly and with high sensitivity converts the grating displacement into the azimuth rotation of the interference petals, achieving high-precision and robust displacement measurement in conjunction with an independent reference.

[0006] Compared with the prior art, the present invention has the following technical effects: 1. Achieving a balance between nanometer-level measurement accuracy and high environmental robustness This invention, through systematic hardware and software co-design, achieves nanometer-level displacement measurement accuracy while significantly enhancing the system's robustness in complex environments. Specifically: 1) The system uses the grating pitch of an optical grating as the displacement reference, replacing the laser wavelength reference which is sensitive to environmental disturbances. The excellent thermal and mechanical stability of the grating itself fundamentally reduces the impact of factors such as temperature fluctuations and air turbulence on the measurement reference. 2) The core algorithm employs a two-stage fuzzy C-means (FCM) clustering strategy: In the first stage, FCM clustering is performed on a single frame image, using a fuzzy membership mechanism to adaptively handle petal distortion and noise, forming the initial cluster centers for each frame image; in the second stage, secondary FCM clustering is performed on the initial cluster centers of multiple frames, effectively filtering out random noise in the time dimension and generating extremely stable final centroid estimates. Experiments show that this scheme can still stably control the displacement measurement error within ±0.2 micrometers under simulated interference conditions, with a typical step deviation of only tens of nanometers, successfully balancing high precision and strong anti-interference capability.

[0007] 2. Provide a traceable absolute displacement reference and implement system self-calibration function. This invention constructs a highly reliable measurement system with self-calibration capabilities through optical path design and system integration. In addition to the main vortex optical interferometer path, a high-precision laser interferometer is set up in parallel as an independent reference measurement module. The measurement optical path of this laser interferometer is independent, and its measurement target mirror is directly aligned with the displacement stage of the driving grating, thereby obtaining the absolute displacement truth value traceable to the laser wavelength, unaffected by the main optical path. This design brings two beneficial effects: First, this truth value provides an authoritative calibration and verification benchmark for the displacement calculation results of the vortex grating interferometer (OVGI), ensuring the metrological reliability of the measurement data. Second, it realizes the system's online self-calibration and performance monitoring functions, enabling real-time evaluation of the state stability and potential drift of the main measurement system, providing a guarantee for long-term reliable operation.

[0008] 3. High sensitivity and ease of use are achieved through displacement-angle conversion mechanism and coaxial interference design. This invention achieves high-sensitivity measurement and reduces engineering complexity through a unique optical conversion mechanism and optimized optical path layout. First, the system utilizes the strict linear correspondence between the azimuth angle of the interference petal pattern of vortex light after diffraction through a grating and the grating displacement, thus enabling the measurement of minute linear displacements from sub-nanometer to micrometer scales. Linear amplification to a rotation angle change that can be clearly detected by a charge-coupled device (CCD) This conversion mechanism significantly improves the sensitivity of displacement detection, while its natural circular reference with a 2π period avoids the complexity of phase demodulation. Secondly, the optical path employs a symmetrical layout to guide ±1st-order diffracted light, and a beam splitter achieves strict coaxial interference. This design ensures excellent interference fringe contrast and signal quality, while significantly reducing the stringent requirements for the assembly and adjustment precision of optical components, thus improving the system's assemblability, repeatability, and long-term stability.

[0009] 4. Utilize intelligent image preprocessing and feature extraction to enhance useful information and accurately characterize the measurement target. This invention employs an innovative method at the image processing front end, effectively improving the quality of the raw data and providing superior feature representations for subsequent calculations. For the acquired color interferograms, an innovative color-weighted grayscale preprocessing method (such as...) is used. Instead of traditional simple binarization or standard grayscale conversion, this method selectively highlights the intensity information of petal features in specific color channels (such as green and red channels) through weighted coefficients. This effectively suppresses background noise while preserving key photometric gradients and texture details, significantly enhancing the visibility and distinguishability of the petal structure. Based on this, the algorithm constructs a three-dimensional feature vector for each pixel, fusing spatial coordinates (x, y) and normalized intensity values ​​(z). This integrates the geometric shape of the petal with its light intensity distribution, forming three-dimensional point cloud data that better reflects its physical essence, laying a precise data foundation for high-precision centroid clustering.

[0010] 5. A lightweight adaptive clustering algorithm is adopted to achieve strong generalization ability with low computational cost. The centroid recognition algorithm proposed in this invention offers significant advantages such as lightweight design, low dependency, and strong generalization while maintaining high performance. The core of the algorithm is a fuzzy C-means (FCM) clustering framework based on optimization theory, requiring no large amount of training data. Compared to traditional deep learning methods that require massive amounts of labeled data for training, this algorithm only needs to perform mathematical iterative optimization on a limited feature point cloud, resulting in low computational resource consumption, strong real-time performance, and easy deployment on embedded platforms. Furthermore, its fuzzy clustering mechanism allows it to adaptively handle unique petal shapes caused by uneven illumination or slight distortion in each frame of the image, without needing to retrain the model for different interference conditions. This principle-based rather than data-driven design gives the algorithm excellent generalization adaptability to different topological charges, pattern shapes, and interference types, making it highly practical in engineering.

[0011] 6. It forms a complete modular technical solution with high potential for industrial transformation. This invention provides a complete and modular high-precision displacement measurement solution, encompassing physical principles, optical systems, and signal processing algorithms. In terms of system design, the boundaries and interfaces of each functional module (light source, vortex transformation, diffraction interferometry, detection, and reference calibration) are clearly defined. This modular architecture facilitates system integration, debugging, maintenance, and upgrades. The algorithm flow, from image acquisition, preprocessing, feature extraction, cluster analysis to displacement calculation, forms a standardized and automated processing chain. The decoupling and standardized design between hardware and software greatly enhances the portability of the technology. In summary, this invention not only achieves high-performance measurement but also provides a solid technical foundation for the direct industrialization of miniaturized, low-cost, and highly reliable commercial displacement sensor products, with broad application prospects. Attached Figure Description

[0012] Figure 1 Flowchart of vortex optical interferometry displacement sensing system and centroid clustering recognition algorithm; Figure 2 Original petal color weight grayscale processing image; Figure 3 Feature extraction map; Figure 4 3D secondary clustering results diagram; Figure 5 Displacement measurement results diagram; Figure 6 The effect diagram of two-stage centroid clustering; Figure 7 Error curve; Figure 8 Diagram of a vortex optical interferometer system. Detailed Implementation

[0013] Example 1 A method for centroid identification and displacement calculation applicable to vortex grating interferometers or vortex optical interferometers includes the following steps: S1, Image Acquisition An interference pattern formed by the diffraction of a vortex beam through a grating is captured using a charge-coupled device (CCD6). The vortex beam, after diffraction by the grating, produces ±1st order diffracted light, which, after reflection and beam combining, forms an interference pattern with... Interference patterns of lobes, among which Let be the topological charge of the vortex beam. The displacement of the grating. This will drive the interference petals to rotate rigidly, and the CCD6 will record this rotation process in real time, providing raw image data for subsequent displacement calculation.

[0014] S2, Image Processing The acquired raw color interferometric image Color-weighted grayscale conversion is performed to preserve luminance contrast, prioritizing the presentation of yellows before reds, rather than using traditional standard grayscale or binary thresholding methods. The core conversion formula is:

[0015] This processing method helps enhance the visibility and discriminative power of the petal structure in subsequent feature extraction. (See attached image) Figure 3 As shown, Figure (a) shows the overall processing effect, and Figure (b) is a magnified view of the lower petals, demonstrating the effect of preserving contrast before and after processing.

[0016] S3. Feature Extraction Pixel feature vectors are constructed from each processed image frame. ,in Image coordinates of pixels, This represents the normalized intensity value. By treating intensity as a third-dimensional coordinate, point cloud data encoding the 3D petal geometry can be generated. M pixel vectors are extracted from each image to construct an enhanced feature vector. ,in Scale the intensity values ​​to the [0,1] range. (See attached image) Figure 4 The three-dimensional point cloud structure formed after feature extraction is shown.

[0017] S4, Cluster Centroid Stable cluster centers are generated using a cross-frame two-stage fuzzy C-means clustering (FCM) algorithm. First, the pixel feature vectors in each frame are subjected to the first FCM clustering, with the cluster number set to c=4 (corresponding to four petals). The initial cluster center for each petal is obtained by minimizing the following objective function:

[0018] In the above formula, c is the number of cluster centers, N is the number of data points, xj is the j-th data point, ci is the ith cluster center, and m is the fuzziness index, a parameter controlling the influence of membership degree. Let the membership degree of the i-th data point to the j-th cluster center satisfy:

[0019] From the objective function, we can obtain the formula for calculating the membership matrix:

[0020] Calculate the cluster centers of each cluster:

[0021] The process iterates until the change in cluster centers between two adjacent frames is less than a set threshold ξ, obtaining the initial centroids for each frame. Then, using the initial centroids obtained from multiple frames as input, a second FCM clustering is performed to obtain the final stable centroid estimate, effectively suppressing the influence of scattering and environmental noise. (Appendix) Figure 5 The results of single-frame and multi-frame FCM clustering are shown. Figure (a) shows the single-frame clustering of the normal pattern, Figure (b) shows the multi-frame clustering results, and Figure (c) shows the centroid rotation trajectory of a single petal at the initial position and the step position.

[0022] S5, Solve for displacement In the polar coordinate system (r, θ), the intensity distribution of the interference pattern can be expressed as:

[0023] In the above formula, Radial field distribution, It is grating translation Phase shift caused by (grating spacing d, diffraction order n). Phase change. This will cause the petal pattern to undergo a rigid azimuth rotation. The relationship is as follows:

[0024] Combining the above relationships, we can obtain the expression for displacement sensitivity:

[0025] The petal rotation angle was calculated using the stable centroids obtained from clustering. Substituting these values ​​into the above formula allows for a high-precision calculation of the grating's displacement. This improves the system's robustness and measurement stability under interference environments.

[0026] Appendix Figure 6 The displacement measurement results are shown, with the initial and post-displacement centroids of petals 1 and 3 superimposed on the figure, along with the polar coordinate mapping used to calculate the rotation. (Attached) Figure 7 The effect of two-stage centroid clustering is demonstrated. Figures (a) and (c) show the original interferograms under normal and mismatched optical path conditions, respectively. Figures (b) and (d) show the polar coordinate distribution of centroid rotation under the corresponding conditions, indicating that stable centroid clustering can still be achieved even with distortion. (Appendix) Figure 8 The error curves show that under both normal and asymmetrical optical path conditions, the errors are controlled within ±0.2 micrometers, with a typical step deviation of only tens of nanometers, verifying the robustness and measurement accuracy of this method.

[0027] Example 2 A grating interferometric displacement sensing system based on vortex light is provided. The system includes a light source module, a vortex light generation module, a diffraction and displacement sensing module, an interferometric optical path construction module, a detection module, and an independent reference measurement module for providing a displacement reference, arranged sequentially along the optical path. The modules are optically connected to form a closed-loop interferometric measurement optical path.

[0028] Specifically, the light source module includes a laser 1 for providing a stable Gaussian beam output. The vortex beam generation module includes a spiral phase plate 2 disposed on the output optical path of the laser 1. The spiral phase plate 2 receives the Gaussian beam and converts it into a Laguerre-Gaussian beam, i.e., vortex beam. The diffraction and displacement sensing module includes a grating 3 disposed on the propagation optical path of the vortex beam. The grating 3 receives the vortex beam and diffracts it to generate a diffracted beam including at least +1 and -1 orders. The grating 3 is mounted on a displacement stage. The displacement to be measured drives the displacement stage to move the grating 3 in its plane, thereby introducing and displacing the diffracted beam. Related phase modulation The interference optical path construction module includes at least one beam splitter 5 and multiple reflectors 4, connected as follows: the +1st order diffraction beam and the -1st order diffraction beam of the grating 3 are guided and redirected by different reflectors among the multiple reflectors 4; the redirected +1st order diffraction beam and the -1st order diffraction beam are incident on the beam splitter 5, and combined by the beam splitter 5 to form a coaxially propagating interference beam; the spatial positions of the multiple reflectors 4 and the beam splitter 5 are configured to match the optical path lengths of the +1st order diffraction beam and the -1st order diffraction beam, ensuring interference conditions. The detection module includes a charge-coupled device (CCD) 6, which is disposed on the combined output optical path of the beam splitter 5, for receiving the interference beam and acquiring and presenting it. The interference pattern of each petal converts the optical signal into a digital image signal. The independent reference measurement module is a laser interferometer 7. The measurement optical path of the laser interferometer 7 is independent of the aforementioned vortex light interference optical path, and its measurement target mirror is aligned with the displacement stage or fixed to the grating 3. It is used to perform high-precision independent measurement of the displacement of the displacement stage, and the measurement result serves as the reference value for the system displacement calculation.

[0029] In the system, the laser 1, the spiral phase plate 2, the grating 3, the reflector 4, the beam splitter 5, and the CCD 6 are connected sequentially along the optical path to form the main measurement optical path of the vortex light interferometric displacement sensor; the laser interferometer 7 is set in parallel with the main measurement optical path to form the calibration and reference optical path for displacement measurement.

[0030] The central axis of the emitted beam of the laser 1, the center of the spiral phase plate 2, the center of the diffraction surface of the grating 3, and the center of the reflecting surface of the main mirror 4 for receiving the +1st and -1st order diffracted beams are located in the same horizontal reference plane. The mirrors for guiding the +1st order diffracted beam and the mirrors for guiding the -1st order diffracted beam are symmetrically arranged with respect to the normal of the grating 3. The beam splitter 5 is located at the convergence point of the optical paths of the two symmetrically arranged mirrors 4, and its splitting surface forms a 45-degree angle with the horizontal reference plane, allowing the two beams to be combined in a near-perpendicular incident manner. The photosensitive surface of the CCD 6 is set perpendicular to the propagation direction of the combined beam, and its center is aligned with the center of the beam. The line connecting the laser emitter head of the laser interferometer 7 and the mirror is parallel to the displacement direction of the grating 3, and its measurement point is located on the moving axis of the displacement stage.

[0031] The system will displace the grating 3 in the plane. Linear, high-sensitivity conversion of petal azimuth rotation in interferometric patterns acquired by CCD6 And through the laser interferometer 7, an absolute displacement reference is provided, thereby realizing high-precision and highly robust nanometer-level displacement measurement.

[0032] The technical solutions and technical details disclosed in the embodiments of this invention are merely illustrative of the inventive concept of this invention and do not constitute a limitation on the technical solutions of this invention. Any conventional changes, substitutions, or combinations made to the technical details disclosed in the embodiments of this invention have the same inventive concept as this invention and are within the protection scope of the claims of this invention.

Claims

1. A method for centroid identification and displacement calculation using a vortex grating interferometer or a vortex optical interferometer, characterized in that, Includes the following steps: S1. Image Acquisition: Acquire a sequence of petal-shaped interference images formed by the interference of vortex light through the displacement sensing system; S2. Image Processing: Perform color-weighted grayscale preprocessing on the acquired raw color interferometric image; S3. Feature Extraction: Extract pixel coordinates and normalized intensity values ​​from each processed image frame to construct a three-dimensional feature vector to form point cloud data; S4. Cluster centroid: A two-stage fuzzy C-means clustering (FCM) strategy is adopted. In the first stage, FCM clustering is performed on the feature point cloud of a single frame image to obtain the initial centroid of each petal. The second stage uses the initial centroid set obtained from multiple frames of images as input, and performs FCM clustering again to obtain stable final centroid estimates. S5. Calculate the displacement: Based on the analytical relationship between the petal rotation angle and the grating displacement, the rotation angle is calculated using the final centroid estimate, and then the displacement is calculated.

2. The centroid identification and displacement calculation method according to claim 1, characterized in that, In step S2, the original color interferogram is processed. After color-weighted grayscale conversion, the luminous contrast of the petal-shaped interference image is preserved, for example, yellow is presented first and then red.

3. The centroid identification and displacement calculation method according to claim 1, characterized in that, In step S3, each processed image contains M pixel vectors { x i , y i , z i } M i=1 Constructing enhanced feature vectors x j = ( x i , y i , α z i ), where α normalizes the intensity to the interval [0, 1]; Where ξ is the threshold for terminating the iteration, thus obtaining the initial centroid. Based on obtaining the initial centroid, a fuzzy C-means clustering (FCM) is performed on these initial centroids to obtain the final centroid.

4. The centroid identification and displacement calculation method as described in claim 1, characterized in that, The displacement sensing system described in step S1 is suitable for a vortex grating interferometer or a vortex optical interferometer. It linearly converts the in-plane displacement of the displacement sensing module into the petal azimuth rotation of the interference pattern acquired by the detection module, and provides an absolute displacement reference through the laser interferometer.

5. The centroid identification and displacement calculation method as described in claim 4, characterized in that, The displacement sensing system includes a light source module, a vortex light generation module, a diffraction and displacement sensing module, an interference optical path construction module, a detection module, and an independent reference measurement module arranged sequentially along the optical path. The light source module is used to provide a Gaussian beam. The vortex light generation module is used to convert the Gaussian beam into a Laguerre-Gaussian beam, i.e., vortex light.

6. The centroid identification and displacement calculation method as described in claim 5, characterized in that, The diffraction and displacement sensitive module includes a grating disposed on the propagation path of the vortex light, used to diffract the vortex light to generate a diffracted beam including at least +1 and -1 orders; the grating is mounted on a displacement stage, and its in-plane displacement is used to introduce phase modulation related to the displacement amount into the diffracted beam. The interference optical path construction module includes at least one beam splitter and multiple mirrors, used to guide and combine the +1st and -1st order diffracted beams to form a coaxial interference beam. The detection module includes a charge-coupled device (CCD) positioned in the propagation direction of the interference beam to acquire interference patterns that present a specific number of petals. The reference measurement module is a laser interferometer, whose measurement optical path is independent of the aforementioned vortex interferometer optical path. It is used to independently measure the displacement of the displacement stage, and the measurement result serves as the reference value for the system displacement calculation.

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