Abstract

From a sequence of simultaneous multi-wavelength phase-shifting interferograms (SMWPSIs), a novel single-wavelength phase retrieval method based on the least-squares iterative algorithm is proposed and utilized in dual-wavelength interferometry. Firstly, only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, and then the accurate wrapped phases of each single-wavelength can be respectively retrieved from SMWPSIs by the least-squares iterative operation, so the phase of synthetic wavelength can be obtained by the subtraction easily. Using the proposed method, both the simulation and the experimental results demonstrate that the optical setup is simpler; the requirements for the displacement of the phase-shifting device and the number of the captured interferograms are smaller compared to the traditional phase-shifting multi-wavelength interferometry or off-axis multi-wavelength interferometry. Even in the case that the phase-shifts are unknown, the wrapped phases and the phase-shifts of each single-wavelength can be obtained by the proposed method.

© 2014 Optical Society of America

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References

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2014 (1)

2012 (1)

2011 (5)

2009 (1)

U. P. Kumar, B. Bhaduri, M. Kothiyal, and N. K. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

2008 (1)

2007 (2)

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15(12), 7231–7242 (2007).
[Crossref] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

2006 (1)

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

2005 (1)

2004 (2)

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

2003 (1)

1998 (1)

1991 (1)

1988 (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1987 (1)

1985 (1)

1984 (1)

Abdelsalam, D. G.

Asundi, A.

Barada, D.

Bhaduri, B.

U. P. Kumar, B. Bhaduri, M. Kothiyal, and N. K. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

Carazo, J. M.

Charrière, F.

Cheng, Y.-Y.

Colomb, T.

Coppola, G.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28(14), 1257–1259 (2003).
[Crossref] [PubMed]

Creath, K.

Cuche, E.

De Nicola, S.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28(14), 1257–1259 (2003).
[Crossref] [PubMed]

Depeursinge, C.

Emery, Y.

Estrada, J. C.

Fei, L.

Ferraro, P.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28(14), 1257–1259 (2003).
[Crossref] [PubMed]

Finizio, A.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

P. Ferraro, G. Coppola, S. De Nicola, A. Finizio, and G. Pierattini, “Digital holographic microscope with automatic focus tracking by detecting sample displacement in real time,” Opt. Lett. 28(14), 1257–1259 (2003).
[Crossref] [PubMed]

Grilli, S.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

Guo, Z.

Han, B.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Hariharan, P.

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

Iodice, M.

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

Ishii, Y.

Kawata, S.

Kemper, B.

Kiire, T.

Kim, D.

Kothiyal, M.

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red-Green-Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

U. P. Kumar, B. Bhaduri, M. Kothiyal, and N. K. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

Kühn, J.

Kumar, U. P.

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red-Green-Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

U. P. Kumar, B. Bhaduri, M. Kothiyal, and N. K. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

Magnusson, R.

Marquet, P.

Miao, J.

Mohan, N. K.

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red-Green-Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

U. P. Kumar, B. Bhaduri, M. Kothiyal, and N. K. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

Montfort, F.

Onodera, R.

Peng, X.

Pierattini, G.

Quiroga, J. A.

Schmit, J.

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

Sorzano, C. O.

Sugisaka, J.-i.

Vargas, J.

von Bally, G.

Wang, H.

Wang, Z.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Wangping Zhang, X. L.

Wyant, J. C.

Xu, L.

Yatagai, T.

Zhao, H.

Zhong, L.

Appl. Opt. (8)

J. Opt. (1)

U. P. Kumar, N. K. Mohan, and M. Kothiyal, “Red-Green-Blue wavelength interferometry and TV holography for surface metrology,” J. Opt. 40(4), 176–183 (2011).
[Crossref]

Meas. Sci. Technol. (1)

G. Coppola, P. Ferraro, M. Iodice, S. De Nicola, A. Finizio, and S. Grilli, “A digital holographic microscope for complete characterization of microelectromechanical systems,” Meas. Sci. Technol. 15(3), 529–539 (2004).
[Crossref]

Opt. Eng. (1)

J. Schmit and P. Hariharan, “Two-wavelength interferometric profilometry with a phase-step error-compensating algorithm,” Opt. Eng. 45(11), 115602 (2006).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (2)

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra-and inter-frame intensity variations,” Opt. Lasers Eng. 45(2), 274–280 (2007).
[Crossref]

U. P. Kumar, B. Bhaduri, M. Kothiyal, and N. K. Mohan, “Two-wavelength micro-interferometry for 3-D surface profiling,” Opt. Lasers Eng. 47(2), 223–229 (2009).
[Crossref]

Opt. Lett. (6)

Prog. Opt. (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Other (1)

N. Warnasooriya and M. Kim, “Phase-Shifting Interference Microscopy with Multi-Wavelength Optical Phase Unwrapping,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2007), p. DTuD2.

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Figures (13)

Fig. 1
Fig. 1 (a) One experimental single-wavelength phase-shifting interferogram at 532nm; (b-d) the retrieved wrapped phase, background and modulation of (a), respectively; (e) one experimental single-wavelength phase-shifting interferogram at 632.8nm; (f-h) the retrieved wrapped phase, background and modulation of (e), respectively.
Fig. 2
Fig. 2 A sequence of 5-frame simulated SPSDWIs
Fig. 3
Fig. 3 The wrapped phases of single-wavelength retrieved from Fig. 2. (a) at 532nm; (b) at 632.8nm; the difference between the reference and the retrieved wrapped phase (c) at 532nm; (d) at 632.8nm.
Fig. 4
Fig. 4 (a). The phase of synthetic wavelength by the proposed algorithm; (b) the reference phase of synthetic wavelength; (c) the difference between Figs. 4(a) and 4(b).
Fig. 5
Fig. 5 Experimental setup for recording SPSMWIs. ND, the netural density filter; BS1 and BS2, beam splitter; PZT, piezoelectric transducer; M, mirror; BE, beam expander; MO, microscope.
Fig. 6
Fig. 6 One of single-wavelength phase-shifting interferograms (a) at 532nm; (b) at 632.8nm; (c) One of SPSDWIs
Fig. 7
Fig. 7 The wrapped phases retrived from a squence of single-wavelength phase-shifting interfergrams. (a)at 532nm; (b) at 632.8nm; (c) the phase of synthetic wavelength obtained from Fig. 7(a) and 7(b), the wrapped phase retrieved from 16-frame SPSDWIs (d)at 532nm; (e) at 632.8nm; (f) the phase of synthetic wavelength obtained from Figs. 7(d) and 7(e).
Fig. 8
Fig. 8 (a) The height distribution of the 200th line in Fig. 7(c); (b) the magnification of the black dotted line area in Fig. 8(a); (c) the height distribution of the 200th line in Fig. 7(f); (d) the magnification of the black dotted line area in Fig. 8(c).
Fig. 9
Fig. 9 A sequence of 5-frame experimental SPSDWIs without the object
Fig. 10
Fig. 10 A sequence of 5-frame experimental SPSDWIs with the vortex phase plate
Fig. 11
Fig. 11 The additional wrapped phases (a) at 532nm; (b) at 632.8nm; The total wrapped phase of vortex phase plate including the additional phase (c) at 532nm, (d) at 632.8nm.
Fig. 12
Fig. 12 The actual wrapped phases of vortex phase plate (a) at 532nm, (b) at 632.8nm; (c) height map of the vortex phase plate. (d) height distribution of the 200th line in (c), in which a sequence of 5-frame SPSDWIs is used for the iterative operation.
Fig. 13
Fig. 13 The wrapped phases of vortex phase plate (a) at 532nm, (b) at 632.8nm; (c) height distribution map of vortex phase plate, (d) height distribution of 200th in (c), in which a sequence of 20-frame SPSDWIs is used for the iterative operation.

Tables (1)

Tables Icon

Table1 Phase-shifts extracted from 5-frame simulated SPSDWIs by the proposed algorithm.

Equations (22)

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i ' lmn = a lm + b lm cos( φ lm + δ ln )
i ' mn = l=1 L i ' lmn = l=1 L a lm + l=1 L b lm cos( φ lm + δ ln )
i ' mn = c 0m + l=1 L b lm (cos φ lm cos δ ln sin φ lm sin δ ln )= c 0m + l=1 L ( c 1lm cos δ ln + c 2lm sin δ ln )
i ' mn = c 0m + l=1 L b lm (cos φ lm cos δ ln sin φ lm sin δ ln )= c 0m + l=1 L (c ' 1ln cos φ lm +c ' 2ln sin φ lm )
E n = n=1 N (i ' mn i mn ) 2 = n=1 N [ c 0m + l=1 L ( c 1lm cos δ ln + c 2lm sin δ ln ) i mn ] 2
E m = m=1 M (i ' mn i mn ) 2 = m=1 M [ c 0m + l=1 L (c ' 1ln cos φ lm +c ' 2ln sin φ lm ) i mn ] 2
E n c 0m =0 , E n c 1l'm =0 , E n c 2l'm =0
E m c 0m =0 , E m c 1l'n =0 , E m c 2l'n =0
AC=I
A'C'=I'
A=[ N n=1 N cos δ 1n n=1 N cos δ Ln n=1 N sin δ 1n n=1 N sin δ Ln n=1 N cos δ 1n n=1 N cos 2 δ 1n n=1 N cos δ 1n cos δ Ln n=1 N cos δ 1n sin δ 1n n=1 N cos δ 1n sin δ Ln n=1 N cos δ Ln n=1 N cos δ 1n cos δ Ln n=1 N cos 2 δ Ln n=1 N cos δ Ln sin δ 1n n=1 N cos δ Ln sin δ Ln n=1 N sin δ 1n n=1 N cos δ 1n sin δ 1n n=1 N cos δ Ln sin δ 1n n=1 N sin 2 δ 1n n=1 N sin δ 1n sin δ Ln n=1 N sin δ Ln n=1 N cos δ 1n sin δ Ln n=1 N cos δ Ln sin δ Ln n=1 N sin δ 1n sin δ Ln n=1 N sin 2 δ Ln ]
C= [ c 0m c 11m c 1Lm c 21m c 2Lm ] T
I= [ n=1 N i mn n=1 N cos δ 1n i mn n=1 N cos δ Ln i mn n=1 N sin δ 1n i mn n=1 N sin δ Ln i mn ] T .
A'=[ M m=1 M cos φ 1m m=1 M cos φ Lm m=1 M sin φ 1m m=1 M sin φ Lm m=1 M cos φ 1m m=1 M cos 2 φ 1m m=1 M cos φ 1m cos φ Lm m=1 M cos φ 1m sin φ 1m m=1 M cos φ 1m sin φ Lm m=1 M cos φ Lm m=1 M cos φ 1m cos φ Lm m=1 M cos 2 φ Lm m=1 M cos φ Lm sin φ 1m m=1 M cos φ Lm sin φ Lm m=1 M sin φ 1m m=1 M cos φ 1m sin φ 1m m=1 M cos φ Lm sin φ 1m m=1 M sin 2 φ 1m m=1 M sin φ 1m sin φ Lm m=1 M sin φ Lm m=1 M cos φ 1m sin φ Lm m=1 M cos φ Lm sin φ Lm m=1 M sin φ 1m sin δ Lm n=1 M sin 2 φ Lm ]
C= [ c 0m c ' 11n c ' 1Ln c ' 21n c ' 2Ln ] T
I'= [ m=1 M i mn m=1 M cos φ 1m i mn m=1 M cos φ Lm i mn m=1 M sin φ 1m i mn m=1 M sin φ Lm i mn ] T
C= A 1 I
φ lm =arctan( c 2lm c 1lm )
C'=A ' 1 I'
δ ln =arctan( c ' 2ln c ' 1ln ).
Th=max{|[ δ ln (k) δ l1 (k)][ δ ln (k1) δ l1 (k1)]|}<ε
h=ΛΦ/2π = λ 1 λ 2 | λ 2 λ 1 | ( φ λ 1 φ λ 2 )/2π

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