Abstract

A novel implementation of lensless multiwavelength digital holography with autocalibration of temporal phase shifts and artificial wavelength is presented. The algorithm we used to calculate the phase shifts was previously proposed [Opt. Lett. 29 183 (2004)] and, to our knowledge, is now used for the first time in lensless holography. Because precise knowledge of the generated artificial wavelength is crucial for absolute measurement accuracy, a simple and efficient method to determine the artificial wavelength directly is presented. The calibration method is based on a simple modification of the experimental setup and needs just one additional image acquisition per wavelength. The results of shape measurement of a metallic test object with a rough surface and steep edges are shown and the measurement accuracy is discussed.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008

2007

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

2006

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

D. Parshall and M. Kim, “Digital holographic microscopy with dual-wavelength phase unwrapping,” Appl. Opt. 45, 451-459(2006).
[CrossRef]

2005

2004

2000

M.-A. Beeck and W. Hentschel, “Laser metrology--a diagnostic tool in automotive development processes,” Opt. Lasers Eng. 34, 101-120 (2000).

C. Furlong and R. J. Pryputniewicz, “Absolute shape measurements using high-resolution optoelectronic holography methods,” Opt. Eng. 39, 216-223 (2000).

C. Wagner, W. Osten, and S. Seebacher, “Automatic processing of direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 3979-85(2000).

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994-7001 (1999).
[CrossRef]

1997

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-245(1997).

S. Kuwamura and I. Yamaguchi, “Wavelength scanning profilometry for real-time surface shape measurement,” Appl. Opt. 36, 4473-4482 (1997).
[CrossRef]

1995

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

1994

1976

R. Jones, “The design and application of a speckle pattern interferometer for total plane strain field measurement,” Opt. Laser Technol. 8215-219(1976).
[CrossRef]

Adams, M.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-245(1997).

Barbosa, E. A.

Beeck, M.-A.

M.-A. Beeck and W. Hentschel, “Laser metrology--a diagnostic tool in automotive development processes,” Opt. Lasers Eng. 34, 101-120 (2000).

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Blu, T.

Brandenburg, A.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

Bredebusch, I.

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

Butusov, M. M.

Y. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography, Springer Series in Optical Sciences (Springer, 1980).

Cai, L. Z.

Carl, D.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high resolution living-cell analysis,” Appl. Opt. 43, 6536-6544(2004).
[CrossRef]

D. Carl, M. Fratz, D. Strohmeier, D. M. Giel, and H. Höfler, “Digital holography with arbitrary temporal phase-shifts and multiple wavelengths for shape measurement of rough sSurfaces,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2008), paper DMC7.

Colomb, T.

Cuche, E.

Curcio, B. G.

Depeursinge, C.

Domschke, W.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

Emery, Y.

Esslinger, T.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Filho, A. A. V.

Fratz, M.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

D. Carl, M. Fratz, D. Strohmeier, D. M. Giel, and H. Höfler, “Digital holography with arbitrary temporal phase-shifts and multiple wavelengths for shape measurement of rough sSurfaces,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2008), paper DMC7.

Furlong, C.

C. Furlong and R. J. Pryputniewicz, “Absolute shape measurements using high-resolution optoelectronic holography methods,” Opt. Eng. 39, 216-223 (2000).

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Gesualdi, M. R. R.

Ghighlia, D. C.

D. C. Ghighlia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

Giel, D. M.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

D. Carl, M. Fratz, D. Strohmeier, D. M. Giel, and H. Höfler, “Digital holography with arbitrary temporal phase-shifts and multiple wavelengths for shape measurement of rough sSurfaces,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2008), paper DMC7.

Hänsch, T. W.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Hemmerich, A.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Hentschel, W.

M.-A. Beeck and W. Hentschel, “Laser metrology--a diagnostic tool in automotive development processes,” Opt. Lasers Eng. 34, 101-120 (2000).

Hoffmann, C.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

Höfler, H.

D. Carl, M. Fratz, D. Strohmeier, D. M. Giel, and H. Höfler, “Digital holography with arbitrary temporal phase-shifts and multiple wavelengths for shape measurement of rough sSurfaces,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2008), paper DMC7.

Höink, A.

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

Ishii, Y.

Ito, T.

Jones, R.

R. Jones, “The design and application of a speckle pattern interferometer for total plane strain field measurement,” Opt. Laser Technol. 8215-219(1976).
[CrossRef]

Jueptner, W. P. O.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-245(1997).

Jüptner, W.

Kato, M.

Kemmler, M.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

Kemper, B.

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high resolution living-cell analysis,” Appl. Opt. 43, 6536-6544(2004).
[CrossRef]

Kim, M.

König, W.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Kreis, T.

T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Publishing1996).

Kreis, T. M.

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-245(1997).

Kuwamura, S.

Liebling, M.

Liu, Q.

Lo, C.-M.

Magistretti, P.

Magistretti, P. J.

Mann, C.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Marquet, P.

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Miura, J.

Muramatsu, M.

Osten, W.

C. Wagner, W. Osten, and S. Seebacher, “Automatic processing of direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 3979-85(2000).

Ostrovskaya, G. V.

Y. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography, Springer Series in Optical Sciences (Springer, 1980).

Ostrovsky, Y. I.

Y. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography, Springer Series in Optical Sciences (Springer, 1980).

Y. I. Ostrovsky, V. V. Shchepinov, and V. V. Yakovlev, Holographic Interferometry in Experimental Mechanics, Springer Series in Optical Sciences (Springer, 1991).

Parshall, D.

Pisarev, V. S.

V. P. Shchepinov and V. S. Pisarev, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Pritt, M. D.

D. C. Ghighlia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

Pryputniewicz, R. J.

C. Furlong and R. J. Pryputniewicz, “Absolute shape measurements using high-resolution optoelectronic holography methods,” Opt. Eng. 39, 216-223 (2000).

Rappaz, B.

Ricci, L.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Sato, Y.

Saum, N.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

Schäfer, M.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

Schnars, U.

Schnekenburger, J.

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, “Automatic processing of direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 3979-85(2000).

Shchepinov, V. P.

V. P. Shchepinov and V. S. Pisarev, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

Shchepinov, V. V.

Y. I. Ostrovsky, V. V. Shchepinov, and V. V. Yakovlev, Holographic Interferometry in Experimental Mechanics, Springer Series in Optical Sciences (Springer, 1991).

Shimobaba, T.

Soga, D.

Strohmeier, D.

D. Carl, M. Fratz, D. Strohmeier, D. M. Giel, and H. Höfler, “Digital holography with arbitrary temporal phase-shifts and multiple wavelengths for shape measurement of rough sSurfaces,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2008), paper DMC7.

Takeda, M.

Takenouchi, M.

Unser, M.

von Bally, G.

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, “Parameter-optimized digital holographic microscope for high resolution living-cell analysis,” Appl. Opt. 43, 6536-6544(2004).
[CrossRef]

Vuletic, V.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Wada, A.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, “Automatic processing of direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 3979-85(2000).

Weidemüller, M.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Wernicke, G.

Yakovlev, V. V.

Y. I. Ostrovsky, V. V. Shchepinov, and V. V. Yakovlev, Holographic Interferometry in Experimental Mechanics, Springer Series in Optical Sciences (Springer, 1991).

Yamaguchi, I.

Yamamoto, H.

Yang, X. L.

Yu, L.

Zimmermann, C.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

Appl. Opt.

J. Biomed. Opt.

M. Kemmler, M. Fratz, D. M. Giel, N. Saum, A. Brandenburg, and C. Hoffmann, “Noninvasive time-dependent cytometry monitoring by digital holography,” J. Biomed. Opt. 12, 064002(2007).

B. Kemper, D. Carl, J. Schnekenburger, I. Bredebusch, M. Schäfer, W. Domschke, and G. von Bally, “Investigation of living pancreas tumor cells by digital holographic microscopy,” J. Biomed. Opt. 11, 034005 (2006).

J. Opt. Soc. Am. A

Opt. Commun.

L. Ricci, M. Weidemüller, T. Esslinger, A. Hemmerich, C. Zimmermann, V. Vuletic, W. König, and T. W. Hänsch, “A compact grating-stabilized diode laser system for atomic physics,” Opt. Commun. 117, 541-549 (1995).
[CrossRef]

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233-245 (1999).
[CrossRef]

Opt. Eng.

C. Furlong and R. J. Pryputniewicz, “Absolute shape measurements using high-resolution optoelectronic holography methods,” Opt. Eng. 39, 216-223 (2000).

C. Wagner, W. Osten, and S. Seebacher, “Automatic processing of direct shape measurement by digital wavefront reconstruction and multiwavelength contouring,” Opt. Eng. 3979-85(2000).

Opt. Express

Opt. Laser Technol.

R. Jones, “The design and application of a speckle pattern interferometer for total plane strain field measurement,” Opt. Laser Technol. 8215-219(1976).
[CrossRef]

Opt. Lasers Eng.

M.-A. Beeck and W. Hentschel, “Laser metrology--a diagnostic tool in automotive development processes,” Opt. Lasers Eng. 34, 101-120 (2000).

Opt. Lett.

Proc. SPIE

B. Kemper, D. Carl, A. Höink, G. von Bally, I. Bredebusch, and J. Schnekenburger, “Modular digital holographic microscopy system for marker-free quantitative phase contrast imaging of living cells,” Proc. SPIE 6191, 61910T(2006).

T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-245(1997).

Other

D. C. Ghighlia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley-Interscience, 1998).

V. P. Shchepinov and V. S. Pisarev, Strain and Stress Analysis by Holographic and Speckle Interferometry (Wiley, 1996).

T. Kreis, Holographic Interferometry: Principles and Methods (Akademie Publishing1996).

Y. I. Ostrovsky, M. M. Butusov, and G. V. Ostrovskaya, Interferometry by Holography, Springer Series in Optical Sciences (Springer, 1980).

Y. I. Ostrovsky, V. V. Shchepinov, and V. V. Yakovlev, Holographic Interferometry in Experimental Mechanics, Springer Series in Optical Sciences (Springer, 1991).

D. Carl, M. Fratz, D. Strohmeier, D. M. Giel, and H. Höfler, “Digital holography with arbitrary temporal phase-shifts and multiple wavelengths for shape measurement of rough sSurfaces,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2008), paper DMC7.

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for temporal phase-shifting lensless multiwavelength digital holography with a movable reference object (BK7 glass plate, d = 3.0 mm ).

Fig. 2
Fig. 2

(a) Three temporal phase-shifted speckle interferograms of a single object state and (b) additionally captured interferogram with the phase object ( 3.0 mm BK7 glass plate) moved partly into the reference path of the interferometer.

Fig. 3
Fig. 3

(a) Reconstructed phase map φ λ 1 ( x , y ) in the hologram plane without a reference object; (b) reconstructed phase map ϕ λ 1 ( x , y ) with a reference object of the same wavelength λ 1 ; (c),(d) corresponding phase maps φ λ 2 ( x , y ) and ϕ λ 2 ( x , y ) of the second wavelength λ 2 ; (e) difference phase map of wavelength λ 1 ; (f) difference phase map of wavelength λ 2 ; (g) difference phase map of generated artificial wavelength λ a , ROI1 without and ROI2 with a reference object, respectively.

Fig. 4
Fig. 4

Single wavelength reconstructions of amplitude and phase at (a), (b) λ = 781.591 nm and (c), (d) λ = 781.531 nm .

Fig. 5
Fig. 5

(a) Reconstructed amplitude of the test object (gray values scaled logarithmically); (b) corresponding phase map of a generated artificial wavelength of 10.14 mm with cross section; (c) pseudo-3D representation of the phase map with the amplitude as texture; (d) digital photograph of the test object.

Fig. 6
Fig. 6

(a) Height z versus lateral position x along the cross section marked in Fig. 5b; (b) enlarged section of (a) with the 3 σ range highlighted.

Tables (1)

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Table 1 Comparison of Measured Artificial Wavelengths with a Wavemeter and the Reference Object Method

Equations (10)

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p = 1 M x , y | I 1 ( x , y ) I 2 ( x , y ) | , q = 1 M x , y | I 2 ( x , y ) I 3 ( x , y ) | , r = 1 M x , y | I 1 ( x , y ) I 3 ( x , y ) | , c = 2 p q r [ 2 ( p 2 q 2 + p 2 r 2 + q 2 r 2 ) ( p 4 + q 4 + r 4 ) ] 1 / 2 .
φ 1 = 2 arcsin ( p / c ) , φ 2 = 2 arcsin ( q / c ) .
C λ ( x , y ) = exp ( i φ 1 / 2 ) sin [ ( φ 1 φ 2 ) / 2 ) ] ( I 1 ( x , y ) I 3 ( x , y ) ) exp [ i ( φ 1 φ 2 ) / 2 ) ] sin ( φ 1 / 2 ) ( I 1 ( x , y ) I 2 ( x , y ) ) .
Ψ z , λ ( x , y ) FFT 1 { FFT [ C λ ( x , y ) ] exp [ i π λ z ( x 2 Δ x 2 N x 2 + y 2 Δ y 2 N y 2 ) ] } .
δ λ , i ( x , y ) = arctan ( Im { Ψ z , λ , i ( x , y ) } Re { Ψ z , λ , i ( x , y ) } ) ,
δ d ( x , y ) = mod ( δ λ , i ( x , y ) δ λ , j ( x , y ) , 2 π ) .
λ a = λ 1 λ 2 | λ 1 λ 2 | = λ 1 λ 2 Δ λ .
h ( x , y ) = λ a 2 δ d ( x , y ) 2 π .
λ a = 2 π Δ φ ref d ( n BK 7 n air ) .
Δ φ ref = ( φ λ 1 ϕ λ 1 ) ( φ λ 2 ϕ λ 2 ) .

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