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Filtering technology is the key to accurate phase reconstruction in off-axis digital holography. Due to the limitations of resolution of charge coupled device (CCD) and off-axis digital holography itself, the filtering process of the step-phase objects is often accompanied by spectral loss, spectral aliasing and spectral leakage when non-integer periods are intercepted. At present, much research has been done on adaptive filtering in the frequency domain, but the above problems have not been fundamentally solved. In this work, the influence of spatial filtering on the accuracy of step-phase reconstruction is first analyzed theoretically. The analysis shows that even if the size of the filter window is equal to the sampling frequency of the CCD, the reconstructed object cannot retain all the spectral information of the object due to the limitation of the resolution power of the CCD itself. In addition, in the off-axis holographic recording process, considering the interference of zero-order terms and conjugate terms, the actual filter width is usually only 1/24 of the sampling frequency of the CCD, at which the average absolute error of the step is about 10% of the height of the step, the oscillation is relatively severe, and after further smoothing filtering, the details of the object are lost, the edge is blurred, and the tiny structure cannot be resolved. Second, according to the definition of discrete Fourier transform, the one-dimensional Fourier transform of a two-dimensional function integrates only in one direction, while the other dimension remains unchanged. When performing one-dimensional Fourier transform along the direction perpendicular to the holographic interference fringes and performing one-dimensional full-spectrum filtering, the distribution of reconstructed object light waves in the direction parallel to the fringes follows the original distribution, is not affected by the filtering, and has high accuracy. Therefore, by combining the reconstructed light waves obtained from one-dimensional full-spectrum filtering of two orthogonal off-axis holograms, an accurate two-dimensional differential phase can be obtained, which provides a basic guarantee for the accurate phase unwinding of Poisson equation. On this basis, a spectral lossless phase reconstruction algorithm based on orthogonal holography and optical experiment method is proposed. In this paper, the ideal sample simulation, including irregular shapes such as gear, circle, V, diamond, drop, hexagon and pentagram, and the corresponding experiment based on USFA1951 standard plate and silicon wafer are carried out. The AFM-calibrated average step heights of the standard plate and the silicon wafer are 100 nm and 240 nm, respectively. The experimental results show that compared with the currently widely used adaptive filter phase reconstruction, the proposed method naturally avoids spectrum loss, spectrum aliasing and spectrum leakage caused by filtering, the reconstruction accuracy is high, and it is suitable for three-dimensional contour reconstruction of any shape step object, which provides a practical way for reconstructing the high-precision phase of off-axis holography.
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Keywords:
- digital holography /
- step phase /
- in-situ reconstruction
Erratum: In-situ reconstruction of step phase based on orthogonal holograms.
HAO Aihua, HUANG Jingyan, ZHANG Shiji, WANG Zhijun, WANG Xiaolong. Erratum: In-situ reconstruction of step phase based on orthogonal holograms. Acta Phys. Sin., doi: 10.7498/aps.74.079901
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图 1 不同占空比下的平均绝对误差随(a)旁瓣数和(b)空间频率的变化
Figure 1. Variation of average absolute error with (a) the number of side lobes and (b) the spatial frequency at different duty cycles.
图 2 正交全息记录光路(He-Ne为氦氖激光器, SF为空间滤波器, L1和L2为透镜, BS为5∶5分光棱镜, MO1和MO2为显微物镜, CCD为电荷耦合器件, 返回光束用蓝色表示)
Figure 2. Optical configuration for recording two orthogonal holograms. He-Ne represents helium-neon laser, SF represents spatial filter, L1 and L2 represent lens, BS represents 5∶5 beam splitter, MO1 and MO2 represent microscope objective, CCD represents charge-coupled device, the return light is shown in blue.
图 3 基于两个正交全息图的频谱无损相位重建算法, 其中右边一列的图形对应条纹平行于Y方向的全息图的分步运算结果, 最后一个图为二维相位解缠绕的结果
Figure 3. Phase reconstruction algorithm without spectral losses based on two orthogonal holograms. The images in the right column correspond to the results for the Y-hologram at each step and the last image shows the two-dimensional unwrapped phase.
图 4 不规则图形重建结果 (a) RRFM方法重建的相位; (b)图(a)中白色虚线位置处的剖面线; (c) SLPR方法重建的相位; (d), (e)图(c)中红线和黑线位置对应的轮廓线
Figure 4. Results of irregular image reconstruction: (a) The reconstructed phase image by FFRM; (b) the section line at white dashed line in panel (a); (c) the reconstructed phase image by SLPR; (d), (e) the profile lines at the position indicated by the red and black lines in panel (c).
图 5 硅片相位重建 (a) RRFM方法重建的相位; (b) SLPR方法重建的相位; (c)图(b)中值滤波后的结果; (d)图(c)中白色虚线位置的轮廓图
Figure 5. Results of wafer phase reconstruction: (a) The phase image reconstructed by RRFM; (b) the phase image reconstructed by SLPR; (c) the output of median filtering in panel (b); (d) the profile map at white dashed line in panel (c).
图 6 USAF1951标准板相位重建 (a), (b) SLPR和RRFM方法的重建相位图; (c)图(a), (b)中相同位置的轮廓线, 分别对应图中洋红和蓝色线条位置; (d)图(c)中第1个台阶的放大图
Figure 6. Phase reconstruction of USAF1951 target: (a), (b) The reconstructed phase images by SLPR and RRFM; (c) the profile lines at the position indicated by the magenta line in panel (a) and blue line in panel (b); (d) the enlarged view of the first step in panel (c)
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[1] Zhang T, Yamaguchi I 1998 Opt. Lett. 23 1221
Google Scholar
[2] Li E B, Yao J Q, Yu D Y, Xi J D, Chicharo J 2005 Opt. Lett. 30 189
Google Scholar
[3] Atlan M, Gross M, Absil E 2007 Opt. Lett. 32 1456
Google Scholar
[4] Takeda M 1990 Industrial Metrology 1 79
Google Scholar
[5] Debnath S K, Park Y 2011 Opt. Lett. 36 4677
Google Scholar
[6] Li J S, Wang Z, Gao J M, Liu Y, Huang J H 2014 Opt. Eng. 54 031103
Google Scholar
[7] He X, Nguyen C V, Pratap M, Zheng Y, Wang Y, Nisbet D R, Williams R J, Rug M, Maier A G, Lee W M 2016 Biomed. Opt. Express 7 3111
Google Scholar
[8] Weng J, Li H, Zhang Z, Zhong J 2014 Optik 125 2633
Google Scholar
[9] Xiong H, Zhang D 2023 Photonics 10 194
Google Scholar
[10] Ma Z, Long J L, Ding Y, Zhang J M, Xi J T, Li Y R, Peng Y Y 2025 Opt. Laser Technol. 181 111807
Google Scholar
[11] Matrecano M, Memmolo P, Miccio L, Persano A, Quaranta F, Siciliano P, Ferraro P 2015 Appl. Opt. 54 3428
Google Scholar
[12] Hong Y, Shi T, Wang X, Zhang Y, Chen K, Liao G 2017 Opt. Commun. 382 624
Google Scholar
[13] Yu H Q, Jia S H, Lin Z H, Gao L M, Zhou X 2023 J. Mod. Opt. 70 77
Google Scholar
[14] Wei J, Wu J, Wang C 2024 Sensors 24 5928
Google Scholar
[15] Lin Z H, Jia S H, Zhou X, Zhang H J, Wang L N, Li G J, Wang Z 2023 Opt. Lasers Eng. 166 107571
Google Scholar
[16] Lin Z H, Jia S H, Wen B, Zhang H J, Yang Z H, Zhou X, Wang L N, Wang Z, Li G J 2024 Opt. Laser Technol. 179 111366
Google Scholar
[17] Xiao W, Wang Q, Pan F, Cao R, Wu X, Sun L 2019 Biomed. Opt. Express 10 1613
Google Scholar
[18] Sun Q Y, Liu Y W, Chen H, Jiang Z Q 2022 Opt. Continuum 1 475
Google Scholar
[19] Kim H W, Cho M, Lee M C 2024 Sensors 24 1950
Google Scholar
[20] Cheng Y Y, Wyant J C 1984 Appl. Opt. 23 4539
Google Scholar
[21] 石炳川, 朱竹青, 王晓雷, 席思星, 贡丽萍 2014 63 244201
Google Scholar
Shi B C, Zhu Z Q, Wang X L, Xi S X, Gong L P 2014 Acta Phys. Sin. 63 244201
Google Scholar
[22] Ghiglia D C, Pritt M D 1998 Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (New York: Wiley-Interscience
[23] Girshovitz P, Shaked N T 2014 Opt. Lett. 39 2262
Google Scholar
[24] Girshovitz P, Shaked N T 2015 Opt. Express 23 8773
Google Scholar
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