Abstract

We present experimental validation of a new reconstruction method for off-axis digital holographic microscopy (DHM). This method effectively suppresses the object autocorrelation, namely, the zero-order term, from holographic data, thereby improving the reconstruction bandwidth of complex wavefronts. The algorithm is based on nonlinear filtering and can be applied to standard DHM setups with realistic recording conditions. We study the robustness of the technique under different experimental configurations, and quantitatively demonstrate its enhancement capabilities on phase signals.

© 2009 Optical Society of America

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  1. C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2009), pp. 201-204.
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    [CrossRef]
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    [CrossRef]
  4. O. Coquoz, R. Conde, F. Taleblou, and C. Depeursinge, “Performances of endoscopic holography with a multicore optical fiber,” Appl. Opt. 34, 7186-7193 (1995).
    [CrossRef]
  5. J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
    [CrossRef]
  6. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468-470 (2005).
    [CrossRef]
  7. B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52-A61 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
    [CrossRef]
  31. F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944-2953 (2006).
    [CrossRef]
  32. E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59-69(2000).
    [CrossRef]
  33. F. Charrière, T. Colomb, F. Montfort, E. Cuche, P. Marquet, and C. Depeursinge, “Shot-noise influence on the reconstructed phase image signal-to-noise ratio in digital holographic microscopy,” Appl. Opt. 45, 7667-7673 (2006).
    [CrossRef]

2009 (1)

2008 (5)

2007 (1)

2006 (3)

2005 (1)

2004 (4)

2003 (2)

2002 (2)

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

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

2000 (2)

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59-69(2000).
[CrossRef]

1999 (3)

1997 (2)

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
[CrossRef]

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357-2360 (1997).
[CrossRef]

1995 (1)

1994 (1)

1982 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

1962 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef]

Aspert, N.

Bevilacqua, F.

Blu, T.

Bo, F.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Bourquin, S.

Cai, L.

Chang, C.-C.

Charrière, F.

Chen, G.-L.

Cheng, X.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Colomb, T.

Conde, R.

Coquoz, O.

Cuche, E.

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

F. Charrière, T. Colomb, F. Montfort, E. Cuche, P. Marquet, and C. Depeursinge, “Shot-noise influence on the reconstructed phase image signal-to-noise ratio in digital holographic microscopy,” Appl. Opt. 45, 7667-7673 (2006).
[CrossRef]

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177-3190(2006).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944-2953 (2006).
[CrossRef]

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

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59-69(2000).
[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]

E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291-293 (1999).
[CrossRef]

Demoli, N.

Depeursinge, C.

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

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177-3190(2006).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944-2953 (2006).
[CrossRef]

F. Charrière, T. Colomb, F. Montfort, E. Cuche, P. Marquet, and C. Depeursinge, “Shot-noise influence on the reconstructed phase image signal-to-noise ratio in digital holographic microscopy,” Appl. Opt. 45, 7667-7673 (2006).
[CrossRef]

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

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59-69(2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[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]

E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291-293 (1999).
[CrossRef]

O. Coquoz, R. Conde, F. Taleblou, and C. Depeursinge, “Performances of endoscopic holography with a multicore optical fiber,” Appl. Opt. 34, 7186-7193 (1995).
[CrossRef]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2009), pp. 201-204.

Devaney, A.

Emery, Y.

J. Kühn, F. Charrière, T. Colomb, E. Cuche, F. Montfort, 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, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468-470 (2005).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef]

Garbusi, E.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Guo, P.

Han, B.

Hu, C.

Ina, H.

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]

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357-2360 (1997).
[CrossRef]

U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179-181 (1994).
[CrossRef]

Kawai, H.

Kemper, B.

Kobayashi, S.

Kreis, T. M.

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357-2360 (1997).
[CrossRef]

Kühn, J.

Kuo, M.-K.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Leith, E. N.

Leval, J.

Li, Y.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Liebling, M.

Lin, C.-Y.

Liu, C.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Liu, J.-P.

Liu, Q.

Liu, Z.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Magistretti, P. J.

Marian, A.

Marquet, P.

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

T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23, 3177-3190(2006).
[CrossRef]

F. Montfort, F. Charrière, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position,” J. Opt. Soc. Am. A 23, 2944-2953 (2006).
[CrossRef]

F. Charrière, T. Colomb, F. Montfort, E. Cuche, P. Marquet, and C. Depeursinge, “Shot-noise influence on the reconstructed phase image signal-to-noise ratio in digital holographic microscopy,” Appl. Opt. 45, 7667-7673 (2006).
[CrossRef]

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

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59-69(2000).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[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]

Mestrovic, J.

Montfort, F.

Ohzu, H.

Osten, W.

Pavillon, N.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2009), pp. 201-204.

Pedrini, G.

Picart, P.

Poon, T.-C.

Pruss, C.

Rappaz, B.

Schnars, U.

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

U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179-181 (1994).
[CrossRef]

Seelamantula, C. S.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2009), pp. 201-204.

Sovic, I.

Takaki, Y.

Takeda, M.

Taleblou, F.

Tiziani, H.

Unser, M.

M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21, 367-377 (2004).
[CrossRef]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2009), pp. 201-204.

Upatnieks, J.

von Bally, G.

Wang, Z.

Weng, J.

Yamaguchi, I.

Yang, X.

Zhang, T.

Zhang, Y.

Zhong, J.

Zhu, J.

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Appl. Opt. (9)

U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179-181 (1994).
[CrossRef]

O. Coquoz, R. Conde, F. Taleblou, and C. Depeursinge, “Performances of endoscopic holography with a multicore optical fiber,” Appl. Opt. 34, 7186-7193 (1995).
[CrossRef]

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

Y. Takaki, H. Kawai, and H. Ohzu, “Hybrid holographic microscopy free of conjugate and zero-order images,” Appl. Opt. 38, 4990-4996 (1999).
[CrossRef]

E. Garbusi, C. Pruss, and W. Osten, “Single frame interferogram evaluation,” Appl. Opt. 47, 2046-2052 (2008).
[CrossRef]

N. Demoli, J. Mestrović, and I. Sović, “Subtraction digital holography,” Appl. Opt. 42, 798-804 (2003).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

F. Charrière, T. Colomb, F. Montfort, E. Cuche, P. Marquet, and C. Depeursinge, “Shot-noise influence on the reconstructed phase image signal-to-noise ratio in digital holographic microscopy,” Appl. Opt. 45, 7667-7673 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

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. Montfort, Y. Emery, P. Marquet, and C. Depeursinge, “Axial sub-nanometer accuracy in digital holographic microscopy,” Meas. Sci. Technol. 19, 074007 (2008).
[CrossRef]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777-778 (1948).
[CrossRef]

Opt. Commun. (1)

E. Cuche, P. Marquet, and C. Depeursinge, “Aperture apodization using cubic spline interpolation: application in digital holographic microscopy,” Opt. Commun. 182, 59-69(2000).
[CrossRef]

Opt. Eng. (2)

T. M. Kreis and W. P. O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357-2360 (1997).
[CrossRef]

C. Liu, Y. Li, X. Cheng, Z. Liu, F. Bo, and J. Zhu, “Elimination of zero-order diffraction in digital holography,” Opt. Eng. 41, 2434-2437 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

Other (1)

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2009), pp. 201-204.

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Figures (11)

Fig. 1
Fig. 1

Frequency response of the high-pass filters corresponding to different kernel sizes as defined in Eq. (8). The position of the interference terms is shown, corresponding to the experimental results in Subsection 3C.

Fig. 2
Fig. 2

Optimal spectral configuration of the Fourier transform of a hologram in the case of (a) Fourier technique (FT) and (b) nonlinear (NL) reconstruction with object autocorrelation suppressed, improving the usable bandwidth for the imaging order.

Fig. 3
Fig. 3

MO normalized discrete spatial bandwidth a MO / N in comparison with the maximal available bandwidth for standard ( a max FT / N ) and nonlinear ( a max NL / N ) reconstruction techniques.

Fig. 4
Fig. 4

Optical sketch of the (a) transmission and (b) reflection holographic microscopes. (P)BS, (polarizing) beam splitter; λ / 2 , half-wave plate; M, mirror; BE, beam expander; C, condenser lens; MO, microscope objective; FL, field lens; IM, image plane; CL, curvature lens.

Fig. 5
Fig. 5

Solution of yew pollens reconstructed with (a) the linear Fourier technique with the full zero-order term, (b) the nonlinear technique with the reference intensity term, and (c) the nonlinear technique with subtraction of the experimental reference and mean suppressed, where the zero-order term is suppressed (contrast in each image is scaled independently on full dynamic range)

Fig. 6
Fig. 6

Relevant quadrant of the yew pollen cells hologram spectrum after (a) linear Fourier filtering and (b) the nonlinear method.

Fig. 7
Fig. 7

Specimen solution of yew pollen cells. Reconstruction with (a),(b) the linear Fourier technique and (c),(d) the nonlinear method with (a),(c) comparison in phase and (b),(d) amplitude. Insets show a part of the field of view (contrast scaled independently on full dynamic range).

Fig. 8
Fig. 8

Magnitude of a quadrant of the Fourier transform of the hologram of the mirror scratch. The dashed circle shows the region of the spectrum that is spatially filtered.

Fig. 9
Fig. 9

Specimen mirror with a scratch (a) amplitude and (b) phase reconstructed with the standard linear technique. The black dashed square in (a) shows the artifacts induced by the zero-order term in amplitude. The white dashed square in (b) shows the region in the phase image where the standard deviation is computed.

Fig. 10
Fig. 10

Phase standard deviation on the measured zone of Fig. 9b as a function of the mean intensity ratio between the reference and the object waves.

Fig. 11
Fig. 11

Phase standard deviation on the measured zone of Fig. 9b in comparison with the nonlinear technique with high-pass filters with different kernels as defined in Eq. (8).

Tables (1)

Tables Icon

Table 1 Specifications of the MOs Used for the Calculations of Fig. 3

Equations (14)

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i ( x , y ) = | r ( x , y ) + o ( x , y ) | 2 = | o ( x , y ) | 2 + | r ( x , y ) | 2 + o ( x , y ) * r ( x , y ) + o ( x , y ) r ( x , y ) * ,
o FT ( x , y ) = r D ( x , y ) i F ( x , y ) , i F ( x , y ) = F 1 { F { i ( x , y ) } W ( ω x , ω y ) } ,
Ψ ( x , y ) = A exp ( 2 π i d / λ ) i λ d F 1 { F { r D ( x , y ) i F ( x , y ) } G ( ω x , ω y ) } , G ( ω x , ω y ) = exp { i π λ d [ ω x 2 + ω y 2 ] } ,
i | r | 2 = ( 1 + o r ) ( 1 + ( o r ) * ) .
ln ( i | r | 2 ) = ln ( 1 + o r ) + ln ( 1 + ( o r ) * ) ,
i F ( x , y ) = ln ( 1 + o r ) = F 1 { F { ln ( i | r | 2 ) } × 1 [ 0 , ) × [ 0 , ) } ,
o NL ( x , y ) = r D ( x , y ) ( exp ( i F ( x , y ) ) 1 ) ,
i HP = i i * h k , h k ( m , n ) = 1 k 2 ,     | m , n | < k 2 ,
a max FT = N 2 + 3 2 .
a max NL = N 4 ,
b = ( 2 1 ) a NL .
p MO = sin ( α ) λ = NA n i λ ,
a MO = NA n i λ N Δ x M ,
μ = I ¯ r ( x , y ) I ¯ o ( x , y ) ,

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