Micro-displacement measurement based on opposed dual-surface differential conjugate vortex beam interference
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摘要:
针对传统共轭涡旋光干涉微位移测量方法中干涉图样旋转角变化量小、亚纳米级微位移检测精度不足的问题，本文提出了一种对向双端面差分共轭涡旋光干涉微位移测量方法。该方法利用共轭涡旋光干涉图样旋转角与两臂相位差之间的线性关系，将待测微位移转换为花瓣状干涉图样的角向旋转。为克服相机像素分辨率和图样畸变对微小旋转角解算的限制，系统构建对向双端面差分光路，使两束共轭涡旋光分别作用于同一被测目标的相对表面，使相对光程差相比传统单端面反射结构翻倍，从而使相同位移下的干涉图样旋转角实现理论倍增。进一步地，采用基于圆谐波卷积的H-Nets模型对旋转图样进行等变特征表征，将连续角向变化映射为复数特征空间中的相位偏移，实现微小旋转角的稳定表征与位移反演。实验结果表明，在0-500 nm范围内最大绝对误差小于0.61 nm，平均绝对误差为0.37 nm。该方法通过差分光程增强与旋转等变解码的协同作用，有效提升了微位移测量的精度与稳定性。
Abstract:To address the limitations of conventional conjugate vortex beam interference-based micro-displacement measurement methods, including limited rotation-angle variations of interference patterns and insufficient capability for sub-nanometer displacement detection, a micro-displacement measurement method based on opposed dual-surface differential conjugate vortex beam interference is proposed. The proposed method exploits the linear relationship between the rotation angle of the conjugate vortex interference pattern and the phase difference between two interferometric paths, converting micro-displacement into the angular rotation of petal-shaped interference patterns. To overcome the constraints imposed by camera pixel resolution and pattern distortion on tiny rotation-angle extraction, an opposed dual-surface differential optical path is developed, where two conjugate vortex beams interact with opposite surfaces of the same target. Compared with the conventional single-surface reflection configuration, the proposed structure doubles the relative optical path difference, thereby theoretically doubling the interference-pattern rotation angle under identical displacement conditions. Furthermore, an H-Nets model based on circular harmonic convolution is introduced to extract rotation-equivariant features from the rotating interference patterns. Continuous angular variations are encoded as phase shifts in the complex-valued feature space, enabling stable characterization of tiny rotation angles and accurate displacement inversion. Experimental results demonstrate that, within a displacement range of 0–500 nm, the maximum absolute error is below 0.61 nm, with a mean absolute error of 0.37 nm. The proposed method achieves enhanced micro-displacement measurement accuracy and stability through the synergistic integration of differential optical-path amplification and rotation-equivariant feature decoding.
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图 2 干涉花瓣旋转效果对比图。(a)传统光路位移前(b)传统光路位移后(c)差分光路位移前(d)差分光路位移后
Figure 2. Comparison of Interference Petal Rotation Effects. (a) Conventional path, before displacement;(b) Conventional path, after displacement; (c) Differential path, before displacement;(d) Differential path, after displacement
图 3 花瓣旋转在圆谐波特征空间中的相位偏移表示
Figure 3. Phase Offset Expression of Petal Rotation in Circular Harmonic Feature Space
图 9 不同位移测量方法MAE比较
Figure 9. Comparison of MAE for Different Displacement Measurement Methods
表 1 H-Nets模型参数表
Table 1. Parameter Table of H-Nets Model
Layer Index Layer Name Number of Channel-s Ker-nel size Output Size 1 Input 1 - 640×640×1 2 Conv1+HNonlin 16 5×5 640×640×16 3 Conv2+BN+Mea-nPool 16 5×5 320×320×16 4 Conv3+HNonlin 32 5×5 320×320×32 5 Conv4+BN+Mea-nPool 32 5×5 160×160×32 6 Conv5+HNonlin 64 5×5 160×160×64 7 Conv6+BN 64 5×5 160×160×64 8 Conv7 1 5×5 160×160×1 9 Complex-to-Real Conversion 2 - 160×160×2 10 Global Average Pooling (Complex) 2 - 2 11 FC 1 Line-ar 1 12 Output 1 - 1 
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表 2 两种光路结构下花瓣干涉图样旋转角度
Table 2. Rotation Angles of Interference Petal Patterns under Two Optical Path Structures
Displacement(nm) CVBI Rotation(°) DSVBI Rotation(°) 100 11.1 22.3 200 22.1 45.4 300 33.3 67.1 400 44.5 90.2 500 55.2 112.3
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表 3 H-Nets消融实验
Table 3. Ablation Study of the H-Nets Model
Model Variant MAE(nm) MSE(nm2) R2 Full H-Nets 0.324 0.297 0.956 W/o Harmonic Filter 0.367 0.338 0.924 W/o Complex Feature 0.379 0.356 0.909 W/o Rotation Equivariance 0.413 0.401 0.891
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表 4 不同位移量评估结果
Table 4. Evaluation Results at Different Displacement Levels
Displacement(nm) MAE(nm) MSE(nm2) R2 10 0.366 0.297 0.956 50 0.327 0.317 0.924 100 0.359 0.324 0.914 300 0.369 0.321 0.909 500 0.348 0.356 0.901
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表 5 位移测量方法性能比较
Table 5. Performance Comparison of Micro-Displacement Measurement Methods
Method MAE/nm MSE/nm2 Centroid Identification enhanced CVBI 1.987 4.237 Centroid Identification enhanced DSVBI 1.643 3.967 CNN enhanced CVBI 0.579 0.496 H-Nets enhanced CVBI 0.457 0.384 CNN enhanced DSVBI 0.419 0.342 H-Nets enhanced DSVBI 0.370 0.299
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