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Dual-wavelength digital holography can expand the unambiguous measurement depth in phase unwrapping by using a differential synthetic wavelength which is longer than the single illumination wavelength. However, the phase noise is significantly amplified due to the magnification of the differential synthetic wavelength, resulting in a lower measurement accuracy. On the other hand, a lower noise level can be achieved by using additive synthetic-wavelength which is shorter than the single illumination wavelength. However, the corresponding unambiguous measurement depth is greatly reduced due to the phase ambiguity. In this case, combining the merits of the differential synthetic-wavelength and the additive synthetic-wavelength, different low noise phase unwrapping algorithms have been developed in recent years. However, these algorithms are complex and time consuming because they need to calculate multiple intermediate variables or search for the constrained boundary conditions in two-dimensional space. Therefore, in this paper, we develop a hierarchical phase unwrapping algorithm by using the two synthetic wavelengths for dual-wavelength digital holography to realize low noise and fast unambiguous measurement with large depth. In this algorithm, the unwrapped phase difference obtained by the differential synthetic wavelength is used to guide the wrapped phase of one single wavelength to realize phase unwrapping, and then the optical path difference obtained by the single-wavelength unwrapped phase is employed to guide the wrapped phase sum, and thus realizing phase unwrapping. As a result, the phase noise is attenuated and the depth sensitivity is preserved for dual-wavelength phase unwrapping. After theoretical analysis, a series of simulation experiments is carried out on the reconstructed quality, anti-noise characteristics and speed through comparing with state-of-the-art dual-wavelength phase unwrapping algorithms, including the conventional algorithm, the linear programming algorithm and the direct linear programming algorithm. In this case, a flipping dual-wavelength common-path digital holography with orthogonal carrier is built to acquire multiplexed off-axis hologram in one shot and illustrate the operation of the algorithm with circular step target, and stability test of the setup. Both the simulation and experimental results show that the proposed method can be simplified and deterministic, resulting in a lower noise phase unwrapping in a time of 20.5 ms for a phase map of one megapixel. We expect that the proposed method can have practical applications in measurement that requires high accuracy, fast speed, and large depth.
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图 1 分级解包裹流程图
Figure 1. Flowchart of Hierarchical phase unwrapping.
图 2 (a)待测物体高度图; (b)离轴全息图; (c) λ1和(d) λ2对应的包裹相位
Figure 2. (a) Original height map of the simulated sample; (b) simulated multiplexed off-axis hologram; and corresponding wrapped phase maps at (c) λ1 and (d) λ2.
图 3 (a)算法I, (b)算法Ⅱ, (c)算法Ⅲ和(d)算法Ⅳ的恢复结果; (e)算法I, (f)算法Ⅱ, (g)算法Ⅲ和(h)算法Ⅳ恢复结果的残差图
Figure 3. Reconstructed results by (a) Algorithm I, (b) Algorithm Ⅱ, (c) Algorithm Ⅲ, and (d) Algorithm Ⅳ; and the corresponding residue maps obtained by (e) Algorithm I, (f) Algorithm Ⅱ, (g) Algorithm Ⅲ, (h) Algorithm Ⅳ.
图 6 视场翻转双波长载波正交数字全息原理图
Figure 6. Flipping dual-wavelength common-path digital holography with orthogonal carrier.
图 7 (a)复合全息图; (b)频谱图; (c) λ1和(d) λ2对应的包裹相位
Figure 7. (a) Multiplexed hologram and its (b) power spectrum; and corresponding wrapped phase maps at (c) λ1 and (d) λ2.
图 8 圆柱恢复结果 (a)算法I; (b)算法Ⅱ; (c)算法Ⅲ; (d)算法Ⅳ
Figure 8. Reconstructed results of the step target by (a) Algorithm I; (b) Algorithm Ⅱ; (c) Algorithm Ⅲ; (d) Algorithm Ⅳ.
图 10 液滴酒精恢复结果 (a)算法I; (b)算法Ⅱ; (c)算法Ⅲ; (d)算法Ⅳ;
Figure 10. Reconstructed result of a drop of alcohol: (a) Algorithm I; (b) Algorithm Ⅱ; (c) Algorithm Ⅲ, ; (d) Algorithm Ⅳ..
图 12 稳定性实验
Figure 12. Stability test for the proposed method.
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[1] Gabor D 1948 Nature 161 777
Google Scholar
[2] Lu X X, Chen J P, Liu S D, Ma Z J, Zhang Z, Zhong L Y 2012 Opt. Las. Eng. 50 1431
Google Scholar
[3] Popescu P, Ikeda T, Dasari R R, Feld M S 2006 Opt. lett. 31 775
Google Scholar
[4] Girshovitz P Shaked N T 2013 Opt. Express 21 5701
Google Scholar
[5] Guo R L, Yao B L, Gao P, Min J W, Han J, Yu X, Lei M, Yan S H, Yang Y L, Dan D, Ye T 2013 Appl. Opt. 52 3484
Google Scholar
[6] Hao B G, Diao M, Shan M G, Zhang Y B, Zhong Z 2013 Opt. Express 21 2126
Google Scholar
[7] Frenklach I, Girshovitz P, Shaked N T 2014 Opt. Lett. 39 1525
Google Scholar
[8] Rawat S, Komatsu S, Markman A, Anand A, Javidi B 2017 Appl. Opt. 56 D127
Google Scholar
[9] Hu C F, Zhu S S, Gao L, Popescu G 2018 Opt. Lett. 43 3373
Google Scholar
[10] Picazo-Bueno J A, Trusiak M, MicóV 2019 Opt. Express 27 5655
Google Scholar
[11] Shaked N T, Micó V, Trusiak M, Kuś A, Mirsky S K 2020 Adv. Opt. Photon. 12 556
Google Scholar
[12] Polhemus C 1973 Appl. Opt. 12 2071
Google Scholar
[13] Gass J, Dakoff A, Kim M K 2003 Opt. Lett. 28 1141
Google Scholar
[14] Kühn J, Colomb T, Montfort F, Charrière F, Emery Y, Cuche E, Marquet P, Depeursinge C 2007 Opt. Express 15 7231
Google Scholar
[15] Min J W, Yao B L, Zhou M L, Guo R L, Lei M, Yang Y L, Dan D, Yan S H, Peng T 2014 J. Opt. 16 125409
Google Scholar
[16] 袁飞, 袁操今, 聂守平, 朱竹青, 马青玉, 李莹, 朱文艳, 冯少彤 2014 物理学报 63 166
Google Scholar
Yuan F, Yuan C J, Nie S P, Zhu Z Q, Ma Q Y, Li Y, Zhu W Y, Feng S T 2014 Acta Phys. Sin. 63 166
Google Scholar
[17] Khmaladze A, Matz R L, Zhang C, Wang T, Banaszak Holl M M, Chen Z 2011 Opt. Lett. 36 912
Google Scholar
[18] Wang Z M, Jiao J N, Qu W J, Yang F, Li H R, Tian A L, Asundi A 2017 Appl. Opt. 56 424
Google Scholar
[19] Di J L, Zhang J W, Xi T L, Ma C J, Zhao J L 2015 J. Micro/Nanolithography, MEMS, and MOEMS 14 041313
Google Scholar
[20] Xiong J X, Zhong L Y, Liu S D, Qiu X, Zhou Y F, Tian J D, Lu X X 2017 Opt. Express 25 7181
Google Scholar
[21] Shan M G, Liu L, Zhong Z, Liu B, Zhang Y B 2019 Opt. Las. Eng. 117 1
Google Scholar
[22] Liu Q, Li L L, Huang X J, Zhang H, Yue X B 2020 J. Opt. 22 045701
Google Scholar
[23] Shan M G, Liu L, Zhong Z, Liu B, Luan G Y, Zhang Y B 2017 Opt. Express 25 26253
Google Scholar
[24] Liu L, Shan M G, Zhong Z, Liu B, Luan G Y, Diao M, Zhang Y B 2017 Opt. Lett. 42 4331
Google Scholar
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