Abstract

Quantitatively measuring a dynamic liquid surface often presents a challenge due to high transparency, fluidity and specular reflection. Here, a novel Transmission-Lattice based Geometric Phase Analysis (TLGPA) method is introduced. In this method, a special lattice is placed underneath a liquid to be tested and, when viewed from above, the phase of the transmission-lattice image is modulated by the deformation of the liquid surface. Combining this with multi-directional Newton iteration algorithms, the dynamic deformation field of the liquid surface can be calculated from the phase variation of a series of transmission-lattice images captured at different moments. The developed method has the advantage of strong self-adaption ability to initial lattice rotational errors and this is discussed in detail. Dynamic 3D ripples formation and propagation was investigated and the results obtained demonstrated the feasibility of the method.

© 2014 Optical Society of America

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2013 (2)

C. W. Li, Z. W. Liu, H. M. Xie, D. Wu, “Novel 3D SEM Moiré method for micro height measurement,” Opt. Express 21(13), 15734–15746 (2013).
[CrossRef] [PubMed]

Z. W. Liu, X. F. Huang, H. M. Xie, “A novel orthogonal transmission-virtual grating method and its applications in measuring micro 3-D shape of deformed liquid surface,” Opt. Lasers Eng. 51(2), 167–171 (2013).
[CrossRef]

2012 (1)

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

2011 (3)

2010 (1)

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[CrossRef]

2009 (3)

2008 (2)

D. M. Liu, P. Z. Lin, “A numerical study of three-dimensional liquid sloshing in tanks,” J. Comput. Phys. 227(8), 3921–3939 (2008).
[CrossRef]

Y. Tang, X. Y. Su, Y. K. Liu, H. L. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16(19), 15090–15096 (2008).
[CrossRef] [PubMed]

2007 (2)

A. Skydan, M. J. Labor, D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Mater. Sci. Technol. 18, 106–114 (2007).

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

2005 (2)

Q. C. Zhang, X. Y. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005).
[CrossRef] [PubMed]

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

2004 (2)

T. Bothe, W. S. Li, C. V. Kopylow, W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

X. F. Gao, L. Jiang, “Biophysics: water-repellent legs of water striders,” Nature 432(7013), 36 (2004).
[CrossRef] [PubMed]

2003 (2)

D. L. Hu, B. Chan, J. W. M. Bush, “The hydrodynamics of water strider locomotion,” Nature 424(6949), 663–666 (2003).
[CrossRef] [PubMed]

M. J. Hÿtch, J. L. Putaux, J. M. Pénisson, “Measurement of the displacement field of dislocations to 0.03 A by electron microscopy,” Nature 423(6937), 270–273 (2003).
[CrossRef] [PubMed]

2001 (1)

R. Ibrahim, V. Pilipchuk, T. Ikeda, “Recent advances in liquid sloshing dynamics,” Appl. Mech. Rev. 54(2), 133–199 (2001).
[CrossRef]

2000 (1)

F. Chen, G. M. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

1989 (1)

L. D. Peterson, E. F. Crawley, R. J. Hansman, “Nonlinear fluid slosh coupled to the dynamics of a spacecraft,” AIAA 27(9), 1230–1240 (1989).
[CrossRef]

1985 (1)

1978 (1)

Asundi, A. K.

Bothe, T.

T. Bothe, W. S. Li, C. V. Kopylow, W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Brown, G. M.

F. Chen, G. M. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Burton, D. R.

A. Skydan, M. J. Labor, D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Mater. Sci. Technol. 18, 106–114 (2007).

Bush, J. W. M.

D. L. Hu, B. Chan, J. W. M. Bush, “The hydrodynamics of water strider locomotion,” Nature 424(6949), 663–666 (2003).
[CrossRef] [PubMed]

Chan, B.

D. L. Hu, B. Chan, J. W. M. Bush, “The hydrodynamics of water strider locomotion,” Nature 424(6949), 663–666 (2003).
[CrossRef] [PubMed]

Chen, F.

F. Chen, G. M. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Crawley, E. F.

L. D. Peterson, E. F. Crawley, R. J. Hansman, “Nonlinear fluid slosh coupled to the dynamics of a spacecraft,” AIAA 27(9), 1230–1240 (1989).
[CrossRef]

Dai, F. L.

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Dai, M. J.

Du, H.

Z. W. Liu, X. F. Huang, H. M. Xie, X. H. Lou, H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[CrossRef]

Eiju, T.

Fang, D. N.

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Feng, X. Q.

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

Gao, J. X.

Gao, X. F.

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

X. F. Gao, L. Jiang, “Biophysics: water-repellent legs of water striders,” Nature 432(7013), 36 (2004).
[CrossRef] [PubMed]

Gu, C. Z.

Z. W. Liu, H. M. Xie, C. Z. Gu, Y. G. Meng, “The digital geometric phase technique applied to the deformation evaluation of MEMS devices,” J. Micromech. Microeng. 19(1), 015012 (2009).
[CrossRef]

Hansman, R. J.

L. D. Peterson, E. F. Crawley, R. J. Hansman, “Nonlinear fluid slosh coupled to the dynamics of a spacecraft,” AIAA 27(9), 1230–1240 (1989).
[CrossRef]

He, G.

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

Hinsch, K. D.

Hu, D. L.

D. L. Hu, B. Chan, J. W. M. Bush, “The hydrodynamics of water strider locomotion,” Nature 424(6949), 663–666 (2003).
[CrossRef] [PubMed]

Huang, L.

Huang, X. F.

Z. W. Liu, X. F. Huang, H. M. Xie, “A novel orthogonal transmission-virtual grating method and its applications in measuring micro 3-D shape of deformed liquid surface,” Opt. Lasers Eng. 51(2), 167–171 (2013).
[CrossRef]

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

Z. W. Liu, X. F. Huang, H. M. Xie, X. H. Lou, H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[CrossRef]

Hÿtch, M. J.

M. J. Hÿtch, J. L. Putaux, J. M. Pénisson, “Measurement of the displacement field of dislocations to 0.03 A by electron microscopy,” Nature 423(6937), 270–273 (2003).
[CrossRef] [PubMed]

Ibrahim, R.

R. Ibrahim, V. Pilipchuk, T. Ikeda, “Recent advances in liquid sloshing dynamics,” Appl. Mech. Rev. 54(2), 133–199 (2001).
[CrossRef]

Ikeda, T.

R. Ibrahim, V. Pilipchuk, T. Ikeda, “Recent advances in liquid sloshing dynamics,” Appl. Mech. Rev. 54(2), 133–199 (2001).
[CrossRef]

Jia, J. F.

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Jiang, L.

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

X. F. Gao, L. Jiang, “Biophysics: water-repellent legs of water striders,” Nature 432(7013), 36 (2004).
[CrossRef] [PubMed]

Jing, H. L.

Jüptner, W.

T. Bothe, W. S. Li, C. V. Kopylow, W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Kopylow, C. V.

T. Bothe, W. S. Li, C. V. Kopylow, W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Labor, M. J.

A. Skydan, M. J. Labor, D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Mater. Sci. Technol. 18, 106–114 (2007).

Li, C. W.

Li, W. S.

T. Bothe, W. S. Li, C. V. Kopylow, W. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).
[CrossRef]

Lin, P. Z.

D. M. Liu, P. Z. Lin, “A numerical study of three-dimensional liquid sloshing in tanks,” J. Comput. Phys. 227(8), 3921–3939 (2008).
[CrossRef]

Liu, D. M.

D. M. Liu, P. Z. Lin, “A numerical study of three-dimensional liquid sloshing in tanks,” J. Comput. Phys. 227(8), 3921–3939 (2008).
[CrossRef]

Liu, H.

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Liu, Y. K.

Liu, Z. W.

Z. W. Liu, X. F. Huang, H. M. Xie, “A novel orthogonal transmission-virtual grating method and its applications in measuring micro 3-D shape of deformed liquid surface,” Opt. Lasers Eng. 51(2), 167–171 (2013).
[CrossRef]

C. W. Li, Z. W. Liu, H. M. Xie, D. Wu, “Novel 3D SEM Moiré method for micro height measurement,” Opt. Express 21(13), 15734–15746 (2013).
[CrossRef] [PubMed]

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

Z. W. Liu, J. X. Gao, “Deformation-pattern-based digital speckle correlation for coefficient of thermal expansion evaluation of film,” Opt. Express 19(18), 17469–17479 (2011).
[CrossRef] [PubMed]

Z. W. Liu, X. F. Huang, H. M. Xie, X. H. Lou, H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[CrossRef]

Z. W. Liu, H. M. Xie, C. Z. Gu, Y. G. Meng, “The digital geometric phase technique applied to the deformation evaluation of MEMS devices,” J. Micromech. Microeng. 19(1), 015012 (2009).
[CrossRef]

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Lou, X. H.

Z. W. Liu, X. F. Huang, H. M. Xie, X. H. Lou, H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[CrossRef]

Matsuda, K.

Meng, Y. G.

Z. W. Liu, H. M. Xie, C. Z. Gu, Y. G. Meng, “The digital geometric phase technique applied to the deformation evaluation of MEMS devices,” J. Micromech. Microeng. 19(1), 015012 (2009).
[CrossRef]

Ng, C. S.

Pénisson, J. M.

M. J. Hÿtch, J. L. Putaux, J. M. Pénisson, “Measurement of the displacement field of dislocations to 0.03 A by electron microscopy,” Nature 423(6937), 270–273 (2003).
[CrossRef] [PubMed]

Peterson, L. D.

L. D. Peterson, E. F. Crawley, R. J. Hansman, “Nonlinear fluid slosh coupled to the dynamics of a spacecraft,” AIAA 27(9), 1230–1240 (1989).
[CrossRef]

Pilipchuk, V.

R. Ibrahim, V. Pilipchuk, T. Ikeda, “Recent advances in liquid sloshing dynamics,” Appl. Mech. Rev. 54(2), 133–199 (2001).
[CrossRef]

Putaux, J. L.

M. J. Hÿtch, J. L. Putaux, J. M. Pénisson, “Measurement of the displacement field of dislocations to 0.03 A by electron microscopy,” Nature 423(6937), 270–273 (2003).
[CrossRef] [PubMed]

Skydan, A.

A. Skydan, M. J. Labor, D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Mater. Sci. Technol. 18, 106–114 (2007).

Song, M.

F. Chen, G. M. Brown, M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[CrossRef]

Su, X. Y.

Tang, Y.

Wang, Y.

Watanabe, S.

Wu, D.

Wu, F.

Wu, Z. N.

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

Xie, H. M.

C. W. Li, Z. W. Liu, H. M. Xie, D. Wu, “Novel 3D SEM Moiré method for micro height measurement,” Opt. Express 21(13), 15734–15746 (2013).
[CrossRef] [PubMed]

Z. W. Liu, X. F. Huang, H. M. Xie, “A novel orthogonal transmission-virtual grating method and its applications in measuring micro 3-D shape of deformed liquid surface,” Opt. Lasers Eng. 51(2), 167–171 (2013).
[CrossRef]

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

Z. W. Liu, X. F. Huang, H. M. Xie, X. H. Lou, H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[CrossRef]

Z. W. Liu, H. M. Xie, C. Z. Gu, Y. G. Meng, “The digital geometric phase technique applied to the deformation evaluation of MEMS devices,” J. Micromech. Microeng. 19(1), 015012 (2009).
[CrossRef]

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Xue, Q. K.

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Zhang, Q. C.

Zhang, S.

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[CrossRef]

Zhao, Y. C.

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

Zheng, Q. S.

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

AIAA (1)

L. D. Peterson, E. F. Crawley, R. J. Hansman, “Nonlinear fluid slosh coupled to the dynamics of a spacecraft,” AIAA 27(9), 1230–1240 (1989).
[CrossRef]

Appl. Mech. Rev. (1)

R. Ibrahim, V. Pilipchuk, T. Ikeda, “Recent advances in liquid sloshing dynamics,” Appl. Mech. Rev. 54(2), 133–199 (2001).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Z. W. Liu, H. M. Xie, D. N. Fang, F. L. Dai, Q. K. Xue, H. Liu, J. F. Jia, “Residual strain around a step edge of artificial Al/Si (111)-7×7 nanocluster,” Appl. Phys. Lett. 87(20), 201908 (2005).
[CrossRef]

Chin. J. Lasers (1)

Y. C. Zhao, X. F. Huang, Z. W. Liu, H. M. Xie, G. He, “Transmission-virtual grating method and its applications in measuring deformed liquid surface,” Chin. J. Lasers 39(9), 0908001 (2012).
[CrossRef]

J. Comput. Phys. (1)

D. M. Liu, P. Z. Lin, “A numerical study of three-dimensional liquid sloshing in tanks,” J. Comput. Phys. 227(8), 3921–3939 (2008).
[CrossRef]

J. Micromech. Microeng. (1)

Z. W. Liu, H. M. Xie, C. Z. Gu, Y. G. Meng, “The digital geometric phase technique applied to the deformation evaluation of MEMS devices,” J. Micromech. Microeng. 19(1), 015012 (2009).
[CrossRef]

Langmuir (1)

X. Q. Feng, X. F. Gao, Z. N. Wu, L. Jiang, Q. S. Zheng, “Superior Water Repellency of Water Strider Legs with Hierarchical Structures: Experiments and Analysis,” Langmuir 23(9), 4892–4896 (2007).
[CrossRef] [PubMed]

Mater. Sci. Technol. (1)

A. Skydan, M. J. Labor, D. R. Burton, “3D shape measurement of automotive glass by using a fringe reflection technique,” Mater. Sci. Technol. 18, 106–114 (2007).

Meas. Sci. Technol. (1)

Z. W. Liu, X. F. Huang, H. M. Xie, X. H. Lou, H. Du, “The artificial periodic lattice phase analysis method applied to deformation evaluation of TiNi shape memory alloy in micro scale,” Meas. Sci. Technol. 22(12), 125702 (2011).
[CrossRef]

Nature (3)

M. J. Hÿtch, J. L. Putaux, J. M. Pénisson, “Measurement of the displacement field of dislocations to 0.03 A by electron microscopy,” Nature 423(6937), 270–273 (2003).
[CrossRef] [PubMed]

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[CrossRef]

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Opt. Lett. (1)

Proc. SPIE (1)

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

An illustration showing schematic diagrams of mathematical relations for the measurement of 3D ripples on the water. (a) A water ripple in the established Cartesian coordinate system. (b) an arbitrary cross section (y = y0) of the ripple; A(xi - Si, y0, H) is a point in the physical lattice pattern and its virtual image in the observing plane moves to A′(xi, y0, H) when the water surface lowers down h(xi, y0) from the horizontal plane, i.e. mathematically, a distribution of h(x, y) aroused a corresponding distribution of S(x, y). (c) a magnified view of (b) showing a very small section of the curve about the ripple shape; refractions lead to the change of propagation directions of transmitted light.

Fig. 2
Fig. 2

A schematic diagram showing the experiment layout.

Fig. 3
Fig. 3

The calculation process for a liquid surface morphology using the TLGPA method. (a) A distorted lattice implying liquid surface deformation. (b) the diffraction spectrum after performing FFT of (a). (c) and (d) show the raw phase images in g1 and g2 directions after performing IFFT of (b). (e) and (f) show the U and V fields (i.e. displacement in the x and y directions) of (a). (g) the calculated surface morphology under an assumption of H = 15 mm and calculation step of 0.2 mm.

Fig. 4
Fig. 4

Eliminating the angle error using TLGPA. (a) A transmission-lattice image with an intersection angle of 15° against the x direction; a tensile strain of 5% is introduced in the x direction at the bottom half of the image. (b) and (c) are the raw phase images in the two selected principal directions. (d) the calculated displacement field in the x direction. (e) the calculated strain field in the x direction.

Fig. 5
Fig. 5

The test layout for measuring the 3D shape of a plano-concave lens. (a) experiment setups. (b) physical picture of the plano-concave lens. (c) a cross-section through the geometric center of the lens.

Fig. 6
Fig. 6

The calculation process for the validation test and the comparison of measurement value with real height. (a) Distorted lattice caused by a plano-concave lens and the diffraction spectrum after performing FFT to the calculation region (framed by the red square). (b) and (c) are the U and V fields of the selected region; (d) the measurement value (up) and comparing with the real height (down). (e) quantitative comparison between the real height of the spherical lens and the measurement value along three different cross sections. (f) the relative measurement error distribution in the calculation.

Fig. 7
Fig. 7

The photographs of a ripple when a droplet drops into still water showing the resultant displacement vectors and final surface morphology evolution measured at five moments; (a) t = 0.0 ms; (b) t = 29 ms; (c) t = 75 ms; (d) t = 100 ms; and (e) t = 110 ms.

Equations (10)

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{ Hh(x,y,t)= S(x,y,t) tan(ϕθ) S(x,y,t) | x=0 = θ(x,y,t) | x=0 =0
n a sinϕ= n w sinθ
{ h(x,y,t)= 0 x h(x,y,t) x dx = 0 x tanϕdx h(x,y,t) | x=0 = h(x,y,t) x | x=0 = ϕ(x,y,t) | x=0 =0
{ h(x,y,t)=HS(x,y,t)Ψ{ h(x,y,t) x } Ψ(u)= n w 2 +( n w 2 1) u 2 + u 2 u×{ n w 2 +( n w 2 1) u 2 1 } h(x,y,t) | x=0 = h(x,y,t) x | x=0 =0
I(r)= g I g (r) e 2πigr
I g (r)= A g (r) e i P g (r)
P g (r)=2πgS(r)
S(r)= 1 2π P g (r)a
h(x,y,t)=H S x (x,y,t)Ψ{ h(x,y,t) x } =H+ 1 2π ( a 1x P g1 + a 2x P g2 )Ψ{ h(x,y,t) x }
S x = 1 2 { S 1 /cos[ arctan( y Q y O x Q x O ) ]+ S 2 /cos[ arctan( y P y O x P x O ) ] }

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