Abstract

In temporal phase-shifting-based digital holographic microscopy, high-resolution phase contrast imaging requires optimized conditions for hologram recording and phase retrieval. To optimize the phase resolution, for the example of a variable three-step algorithm, a theoretical analysis on statistical errors, digitalization errors, uncorrelated errors, and errors due to a misaligned temporal phase shift is carried out. In a second step the theoretically predicted results are compared to the measured phase noise obtained from comparative experimental investigations with several coherent and partially coherent light sources. Finally, the applicability for noise reduction is demonstrated by quantitative phase contrast imaging of pancreas tumor cells.

© 2009 Optical Society of America

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2008 (2)

B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52-A61 (2008).
[CrossRef] [PubMed]

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

2007 (1)

S. Knoche, B. Kemper, G. Wernicke, and G. von Bally, “Modulation analysis in spatial phase shifting electronic speckle pattern interferometry and application for automated data selection on biological specimens,” Opt. Commun. 270, 68-78 (2007).
[CrossRef]

2006 (1)

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).
[CrossRef]

2005 (2)

2004 (1)

2002 (2)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

A. Dubois, L. Vabre, A. C. Boccara, and E. Beaurepaire, “High-resolution full-field optical coherence tomography with a Linnik microscope,” Appl. Opt. 41, 805-812 (2002).
[CrossRef] [PubMed]

2001 (1)

1999 (3)

1998 (1)

1997 (2)

1992 (1)

H. P. Elsässer, U. Lehr, B. Agricola, and H. F. Kern, “Establishment and characterization of two cell lines with different grade of differentiation derived from one primary human pancreatic adenocarcinoma,” Virchows Arch. B 61, 295-306 (1992).
[CrossRef] [PubMed]

1990 (1)

Aebischer, H. A.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Agricola, B.

H. P. Elsässer, U. Lehr, B. Agricola, and H. F. Kern, “Establishment and characterization of two cell lines with different grade of differentiation derived from one primary human pancreatic adenocarcinoma,” Virchows Arch. B 61, 295-306 (1992).
[CrossRef] [PubMed]

Beaurepaire, E.

Beuthan, J.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Bevilacqua, F.

Boccara, A. C.

Bothe, T.

Bredebusch, I.

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).
[CrossRef]

Brophy, C. P.

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing (Wiley, 1992), pp. 501-598.

Buchstaller, A.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Burke, J.

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).
[CrossRef]

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]

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Charrière, F.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Colomb, T.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Creath, K.

K. Creath, “Temporal phase measurement methods,” in Interferogramm Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 94-140.

Cuche, E.

Depeursinge, C.

Dietrich, C.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

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).
[CrossRef]

Dubois, A.

Dubois, F.

Elsässer, H. P.

H. P. Elsässer, U. Lehr, B. Agricola, and H. F. Kern, “Establishment and characterization of two cell lines with different grade of differentiation derived from one primary human pancreatic adenocarcinoma,” Virchows Arch. B 61, 295-306 (1992).
[CrossRef] [PubMed]

Emery, Y.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength accuracy,” Opt. Lett. 30, 468-470 (2005).
[CrossRef] [PubMed]

Gersonde, I.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing (Wiley, 1992), pp. 501-598.

Gutmann, B.

Helmers, H.

Irion, K.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Johannes, L.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Kato, J.

Kempe, M.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Kemper, B.

B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52-A61 (2008).
[CrossRef] [PubMed]

S. Knoche, B. Kemper, G. Wernicke, and G. von Bally, “Modulation analysis in spatial phase shifting electronic speckle pattern interferometry and application for automated data selection on biological specimens,” Opt. Commun. 270, 68-78 (2007).
[CrossRef]

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).
[CrossRef]

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]

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Kern, H. F.

H. P. Elsässer, U. Lehr, B. Agricola, and H. F. Kern, “Establishment and characterization of two cell lines with different grade of differentiation derived from one primary human pancreatic adenocarcinoma,” Virchows Arch. B 61, 295-306 (1992).
[CrossRef] [PubMed]

Kim, K.

Knoche, S.

S. Knoche, B. Kemper, G. Wernicke, and G. von Bally, “Modulation analysis in spatial phase shifting electronic speckle pattern interferometry and application for automated data selection on biological specimens,” Opt. Commun. 270, 68-78 (2007).
[CrossRef]

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry--Optical and Digital Methods (Wiley-VCH, 2005).

Kreis, T. M.

T. M. Kreis, Holographic Interferometry: Principles and Methods (Academie-Verlag, 1996).

Kühn, J.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Legros, J. C.

Lehr, U.

H. P. Elsässer, U. Lehr, B. Agricola, and H. F. Kern, “Establishment and characterization of two cell lines with different grade of differentiation derived from one primary human pancreatic adenocarcinoma,” Virchows Arch. B 61, 295-306 (1992).
[CrossRef] [PubMed]

Lo, C.-M.

Magistretti, P. J.

Mann, C. J.

Marquet, P.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength accuracy,” Opt. Lett. 30, 468-470 (2005).
[CrossRef] [PubMed]

Mizuno, J.

Monfort, F.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Ohta, S.

Rappaz, B.

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogramm Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 194-229.

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).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

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).
[CrossRef]

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Schütze, K.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Stich, M.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Stutz, M.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Vabre, L.

von Bally, G.

B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52-A61 (2008).
[CrossRef] [PubMed]

S. Knoche, B. Kemper, G. Wernicke, and G. von Bally, “Modulation analysis in spatial phase shifting electronic speckle pattern interferometry and application for automated data selection on biological specimens,” Opt. Commun. 270, 68-78 (2007).
[CrossRef]

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).
[CrossRef]

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]

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Waldner, S.

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Weber, H.

Wernicke, G.

S. Knoche, B. Kemper, G. Wernicke, and G. von Bally, “Modulation analysis in spatial phase shifting electronic speckle pattern interferometry and application for automated data selection on biological specimens,” Opt. Commun. 270, 68-78 (2007).
[CrossRef]

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]

Wolleschensky, R.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
[CrossRef]

Yamaguchi, I.

Yu, L.

Zhang, T.

Appl. Opt. (7)

J. Biomed. Opt. (1)

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).
[CrossRef]

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (2)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Monfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Opt. Commun. (2)

S. Knoche, B. Kemper, G. Wernicke, and G. von Bally, “Modulation analysis in spatial phase shifting electronic speckle pattern interferometry and application for automated data selection on biological specimens,” Opt. Commun. 270, 68-78 (2007).
[CrossRef]

H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Virchows Arch. B (1)

H. P. Elsässer, U. Lehr, B. Agricola, and H. F. Kern, “Establishment and characterization of two cell lines with different grade of differentiation derived from one primary human pancreatic adenocarcinoma,” Virchows Arch. B 61, 295-306 (1992).
[CrossRef] [PubMed]

Other (7)

D. W. Robinson, “Phase unwrapping methods,” in Interferogramm Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 194-229.

User's guide, Sony XCD X700 (2000).

K. Creath, “Temporal phase measurement methods,” in Interferogramm Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, 1993), pp. 94-140.

T. M. Kreis, Holographic Interferometry: Principles and Methods (Academie-Verlag, 1996).

T. Kreis, Handbook of Holographic Interferometry--Optical and Digital Methods (Wiley-VCH, 2005).

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing (Wiley, 1992), pp. 501-598.

G. von Bally, B. Kemper, D. Carl, S. Knoche, M. Kempe, C. Dietrich, M. Stutz, R. Wolleschensky, K. Schütze, M. Stich, A. Buchstaller, K. Irion, J. Beuthan, I. Gersonde, and J. Schnekenburger, “New methods for marker-free live cell and tumour analysis,” in Biophotonics: Visions for Better Health Care (VCH-Wiley, 2006), pp. 301-359.
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Figures (9)

Fig. 1
Fig. 1

Reference wave shift β for a minimized error D in the reconstructed object phase in dependence of the phase-shifting error Δ β .

Fig. 2
Fig. 2

Average phase error D ¯ versus shift error Δ β for commonly used shift values β. The solid curve represents the phase error minimum and is achieved by using the shift β N = β min 1 or β N = β min 2 that is given by Eqs. (23).

Fig. 3
Fig. 3

(a) Modified Michelson interferometer setup for temporal phase-shifting holography; (b): Linnik-type interferometer setup for temporal phase-shifting digital holographic microscopy [LS, light source; SF, spatial filter; L, lens; BS, beam splitter; M, mirror; MO, microscope lens; P, piezo actuator; CCD, digital image recording device (CCD camera)].

Fig. 4
Fig. 4

Characterization of the phase noise quantification algorithm. Phase noise σ obtained from simulated phase distributions in dependence of the expected noise J.

Fig. 5
Fig. 5

Phase noise σ versus the utilized fraction h of the camera dynamic range. The measurement points correspond to the exposure times τ = 1 / 128 , 1 / 64 , 1 / 32 , 1 / 16 , and 1 / 8 s .

Fig. 6
Fig. 6

Phase noise σ versus phase shift β for holograms recorded by application of (a) LEDs with different spectral emission distributions and of (b) a He–Ne laser, a laser diode, and a superluminescent diode (SLD).

Fig. 7
Fig. 7

Phase noise σ versus fringe contrast V.

Fig. 8
Fig. 8

Normalized fringe contrast V * versus the phase-shift velocity v, respectively camera frame rate F.

Fig. 9
Fig. 9

Digital holographic investigation on pancreas tumor cells; illumination with laser light (upper row) and LED light (bottom row). (a), (f) Holograms (exemplarily of one of three recorded temporal phase-shifted holograms); (b), (g) amplitude distribution; (c), (h) phase distribution modulo 2 π ; (d), (i) unwrapped phase distribution; (e), (j) gray-level-coded pseudo-three-dimensional representation of the phase maps in (d) and (i).

Equations (30)

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I k ( n , m ) = [ I O ( n , m ) + I R ( n , m ) ] [ 1 + γ ( n , m ) cos ( ϕ O ( n , m ) ϕ R k ( n , m ) ) ] .
γ ( n , m ) = 2 I O ( n , m ) I R ( n , m ) I ¯ ( m , n ) ,
I ¯ = 2 cos β I 2 I 1 I 3 2 cos β 2 ,
γ = [ ( 1 cos β ) ( I 1 I 3 ) ] 2 + [ sin β ( 2 I 2 I 1 I 3 ) ] 2 ( I 1 + I 3 2 I 2 cos β ) sin β ,
ϕ O = arctan ( 1 cos β sin β I 1 I 3 2 I 2 I 1 I 3 ) .
1 2 I ¯ ( n , m ) γ ( n , m ) = I O ( n , m ) I R | E O ( n , m ) | .
E O ( n , m ) I ¯ ( n , m ) γ ( n , m ) exp ( i ϕ O ( n , m ) ) .
σ = k = 1 3 l = 1 3 ( ϕ O I k ) ( ϕ O I l ) Δ I k Δ I l ,
ϕ O = arctan k A k I k l B l I l ,
σ ϕ 2 = 1 ( C ˜ γ I ¯ ) 2 k l ( A k cos ϕ O B k sin ϕ O ) · ( A l cos ϕ O B l sin ϕ O ) Δ I k Δ I l ,
C ˜ 2 ( k A k I k ) 2 + ( l B l I l ) 2 ( γ I ¯ ) 2 .
Δ I k Δ I k = 1 12 ,
Δ I k Δ I k + 1 = 0 ,
Δ I k Δ I k + 2 = 1 12 .
σ = 1 3 1 h Q .
σ 2 = Δ I 2 ( C γ I ¯ ) 2 k ( A k 2 cos 2 ϕ O + B k 2 sin ϕ O 2 A k B k sin ϕ O cos ϕ O ) .
σ = Δ I 2 2 γ I ¯ 1 sin 2 β + 3 ( 1 cos β ) 2 .
β min = 2 π / 3 ( = ^ 120 ° ) .
tan ϕ O = 1 cos β sin β I 1 + I 3 2 I 2 I 1 I 3 ,
tan ϕ O * = 1 cos ( β + Δ β ) sin ( β + Δ β ) I 1 + I 3 2 I 2 I 1 I 3 .
D ( β , Δ β , ϕ O ) = ϕ O * ϕ O arctan ( tan ϕ O * tan ϕ O tan ϕ O ) ϕ O = arctan ( 1 cos β sin β sin ( β + Δ β ) 1 cos ( β + Δ β ) tan ϕ O ) ϕ O .
D β = sin ϕ O ( sin ( β + Δ β ) ) + sin β ( 1 + cos ( β + Δ β ) ) ( cos β + 1 ) cos ϕ O ( 1 + ( 1 cos 2 β ) sin 2 ( β + Δ β ) tan 2 ϕ O sin 2 β ( 1 cos ( β + Δ β ) ) 2 ) = ! 0 ,
β min 1 = arctan 2 ( 1 2 2 + 2 cos Δ β , 1 2 2 + 2 cos Δ β ( cos Δ β 1 ) sin Δ β ) , β min 2 = arctan 2 ( 1 2 2 + 2 cos Δ β , 1 2 2 + 2 cos Δ β ( cos Δ β 1 ) sin Δ β ) .
Δ ϕ O ( n , m ) = { | ϕ O ¯ ( n , m ) ϕ O ( n , m ) |     | ϕ O ¯ ( n , m ) ϕ O ( n , m ) | π 2 π | ϕ O ¯ ( n , m ) ϕ O ( n , m ) |     | ϕ O ¯ ( n , m ) ϕ O ( n , m ) | > π .
σ = 1 N M n m Δ ϕ O ( n , m ) 2 .
σ 2 = 1 2 J J J ( ϕ O * + ϵ ϕ O ϕ O * ) 2 d ϵ = 1 2 J J J ϵ 2 d ϵ = 1 3 J 2 ,
σ = J / 3 .
γ = γ 0 sinc ( Θ / 2 ) .
Θ = v τ 2 π λ ,
V * ( v ) = V ( v ) V ( v 0 ) ,

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