Abstract

This paper presents an experimental investigation and an analytical modeling of the nonlinear pixel saturation effect in digital off-axis holography. The theoretical analysis is based on a semiempirical modeling and supported by the experimental analysis. Taking into account the nonlinearity of the phenomenon, an exponential law for the high-order harmonic amplitude is proposed and validated by the experimental results. The conclusion of this analysis is that the saturation effect can be described by the use of a linear operator that involves autoconvolution of the initial object wave, even though the saturation phenomenon is nonlinear.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  2. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. 23, 1221–1223(1998).
    [CrossRef]
  3. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  4. B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25, 610–612 (2000).
    [CrossRef]
  5. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28–30 (2000).
    [CrossRef]
  6. T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481–483 (2007).
    [CrossRef] [PubMed]
  7. I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
    [CrossRef]
  8. S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
    [CrossRef]
  9. P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
    [CrossRef]
  10. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
    [CrossRef]
  11. P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263–8274(2007).
    [CrossRef] [PubMed]
  12. J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763–5772 (2005).
    [CrossRef] [PubMed]
  13. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
    [CrossRef] [PubMed]
  14. P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882–8892 (2005).
    [CrossRef] [PubMed]
  15. I. Yamaguchi, T. Matsumura, and J. Kato, “Phase shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
    [CrossRef]
  16. P. Picart, D. Mounier, and J. M. Desse, “High resolution digital two-color holographic metrology,” Opt. Lett. 33, 276–278 (2008).
    [CrossRef] [PubMed]
  17. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef] [PubMed]
  18. Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
    [CrossRef]
  19. J. C. Li, P. Tankam, Z. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. 34, 572–574(2009).
    [CrossRef] [PubMed]
  20. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820(1999).
    [CrossRef]
  21. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008).
    [CrossRef]
  22. A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. 25, 1630–1632(2000).
    [CrossRef]
  23. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974–976(2001).
    [CrossRef]
  24. X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232–237 (2008).
    [CrossRef]
  25. T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077–6085 (2006).
    [CrossRef] [PubMed]
  26. G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216–1225(2005).
    [CrossRef] [PubMed]
  27. 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] [PubMed]
  28. F. Charriere, B. Rappaz, J. Kuhn, T. Colomb, P. Marquet, and C. Depeursinge, “Influence of shot noise on phase measurement accuracy in digital holographic microscopy,” Opt. Express 15, 8818–8831 (2007).
    [CrossRef] [PubMed]
  29. M. Gross and M. Atlan, “Digital holography with ultimate sensitivity,” Opt. Lett. 32, 909–911 (2007).
    [CrossRef] [PubMed]
  30. M. Gross, M. Atlan, and E. Absil, “Noise and aliases in off-axis and phase-shifting holography,” Appl. Opt. 47, 1757–1766(2008).
    [CrossRef] [PubMed]
  31. Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
    [CrossRef]
  32. Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
    [CrossRef]
  33. C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
    [CrossRef]

2009 (1)

2008 (4)

2007 (4)

2006 (2)

2005 (3)

2004 (1)

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
[CrossRef]

2003 (2)

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
[CrossRef] [PubMed]

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

2002 (3)

Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
[CrossRef]

I. Yamaguchi, T. Matsumura, and J. Kato, “Phase shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
[CrossRef]

2001 (3)

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974–976(2001).
[CrossRef]

2000 (3)

1999 (2)

1998 (1)

1997 (1)

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

1995 (1)

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

1994 (1)

1993 (1)

Absil, E.

Adams, M.

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Atlan, M.

Baumbach, T.

T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077–6085 (2006).
[CrossRef] [PubMed]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

Berthelot, J.-M.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
[CrossRef]

Bevilacqua, F.

Boileau, J.-P.

Breteau, J.-M.

Cai, X.

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232–237 (2008).
[CrossRef]

Charriere, F.

Charrière, F.

Colomb, T.

Cuche, E.

Dalmont, J.-P.

De Nicola, S.

Depeursinge, C.

Desse, J. M.

Diouf, B.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
[CrossRef]

Ferraro, P.

Finizio, A.

Gautier, B.

Gillet, S.

Gougeon, S.

Grill, M.

Gross, M.

Guimezanes, Th.

Guo, C. S.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Javidi, B.

Juptner, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

Jüptner, W.

Kato, J.

I. Yamaguchi, T. Matsumura, and J. Kato, “Phase shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
[CrossRef]

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

Kebbel, V.

Kolenovic, E.

Kreis, Th.

Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
[CrossRef]

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Kuhn, J.

Leval, J.

Li, J. C.

Lolive, E.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
[CrossRef]

Marquet, P.

Massig, J. H.

Matsumura, T.

Mills, G. A.

Montfort, F.

Mounier, D.

Murata, S.

Nitanai, E.

Nomura, T.

Numata, T.

Ohta, S.

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

Onural, L.

Osten, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820(1999).
[CrossRef]

Pascal, J. C.

Pascal, J.-C.

Pedrini, G.

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

Peng, Z.

Picart, P.

J. C. Li, P. Tankam, Z. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. 34, 572–574(2009).
[CrossRef] [PubMed]

P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008).
[CrossRef]

P. Picart, D. Mounier, and J. M. Desse, “High resolution digital two-color holographic metrology,” Opt. Lett. 33, 276–278 (2008).
[CrossRef] [PubMed]

P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263–8274(2007).
[CrossRef] [PubMed]

P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882–8892 (2005).
[CrossRef] [PubMed]

J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763–5772 (2005).
[CrossRef] [PubMed]

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
[CrossRef]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
[CrossRef] [PubMed]

Pierattin, G.

Piquet, F.

Rappaz, B.

Rong, Z. Y.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Schnars, U.

Seebacher, S.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820(1999).
[CrossRef]

Stadelmaier, A.

Tajahuerce, E.

Tankam, P.

Tiziani, H. J.

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

Wagner, C.

Wand, H.

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232–237 (2008).
[CrossRef]

Wang, H. T.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Yamaguchi, I.

Zhang, L.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

Zhang, T.

Appl. Opt. (7)

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

Measurement (1)

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995).
[CrossRef]

Opt. Commun. (1)

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232–237 (2008).
[CrossRef]

Opt. Eng. (4)

Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. 41, 771–778 (2002).
[CrossRef]

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002).
[CrossRef]

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. 42, 2768–2772 (2003).
[CrossRef]

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. 43, 1169–1176(2004).
[CrossRef]

Opt. Express (3)

Opt. Lasers Eng. (1)

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of micro components using digital holography,” Opt. Lasers Eng. 36, 103–126 (2001).
[CrossRef]

Opt. Lett. (13)

T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. 23, 1221–1223(1998).
[CrossRef]

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

B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. 25, 610–612 (2000).
[CrossRef]

B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28–30 (2000).
[CrossRef]

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481–483 (2007).
[CrossRef] [PubMed]

I. Yamaguchi, T. Matsumura, and J. Kato, “Phase shifting color digital holography,” Opt. Lett. 27, 1108–1110 (2002).
[CrossRef]

P. Picart, D. Mounier, and J. M. Desse, “High resolution digital two-color holographic metrology,” Opt. Lett. 33, 276–278 (2008).
[CrossRef] [PubMed]

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef] [PubMed]

A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. 25, 1630–1632(2000).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974–976(2001).
[CrossRef]

M. Gross and M. Atlan, “Digital holography with ultimate sensitivity,” Opt. Lett. 32, 909–911 (2007).
[CrossRef] [PubMed]

J. C. Li, P. Tankam, Z. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett. 34, 572–574(2009).
[CrossRef] [PubMed]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
[CrossRef] [PubMed]

Opt. Rev. (1)

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85–89 (2001).
[CrossRef]

Proc. SPIE (1)

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224–233 (1997).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (32)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

Evolution of saturation ratio versus optical power.

Fig. 3
Fig. 3

Evolution of mean value of hologram versus optical power.

Fig. 4
Fig. 4

Effect of saturation on image quality.

Fig. 5
Fig. 5

Modeling for saturated hologram.

Fig. 6
Fig. 6

Amplitude of 0 order harmonics versus saturation ratio.

Fig. 7
Fig. 7

Amplitude of + 1 order harmonics, in percent of amplitude of 0 order, versus saturation ratio.

Fig. 8
Fig. 8

Amplitude of + 2 order harmonics, in percent of amplitude of 0 order, versus saturation ratio.

Fig. 9
Fig. 9

Amplitude of + 3 order harmonics, in percent of amplitude of 0 order, versus saturation ratio.

Fig. 10
Fig. 10

Amplitude of + 4 order harmonics, in percent of amplitude of 0 order, versus saturation ratio.

Fig. 11
Fig. 11

Representation of 0 and 1 orders.

Fig. 12
Fig. 12

Spatial aliasing of orders.

Fig. 13
Fig. 13

Periodicity of the reconstructed field.

Fig. 14
Fig. 14

Representation of 0, 1, and 2 orders.

Fig. 15
Fig. 15

Representation of 0, 1, and 3 orders.

Fig. 16
Fig. 16

Representation of 0, 1, and 4 orders.

Fig. 17
Fig. 17

Influence of object amplitude on the reconstructed hologram.

Fig. 18
Fig. 18

Harmonics 0 to 4 of exact and approximated modeling for τ = 5.7 % .

Fig. 19
Fig. 19

Harmonics 0 to 4 of exact and approximated modeling for τ = 26.1 % .

Fig. 20
Fig. 20

Harmonics 0 to 4 of exact and approximated modeling for τ = 57.7 % .

Fig. 21
Fig. 21

Harmonics 0 to 4 of exact and approximated modeling for τ = 79.1 % .

Fig. 22
Fig. 22

Reconstructed holograms with τ = 5.7 % : (a) experimental saturation, (b) numerical saturation, (c) exact modeling, and (d)  approximated modeling.

Fig. 23
Fig. 23

Reconstructed holograms with τ = 18.8 % : (a) experimental saturation, (b) numerical saturation, (c) exact modeling, and (d)  approximated modeling.

Fig. 24
Fig. 24

Reconstructed holograms with τ = 26.1 % : (a) experimental saturation, (b) numerical saturation, (c) exact modeling, and (d)  approximated modeling.

Fig. 25
Fig. 25

Reconstructed holograms with τ = 57.7 % : (a) experimental saturation, (b) numerical saturation, (c) exact modeling, and (d)  approximated modeling.

Fig. 26
Fig. 26

Reconstructed holograms with τ = 79.1 % : (a) experimental saturation, (b) numerical saturation, (c) exact modeling, and (d)  approximated modeling.

Fig. 27
Fig. 27

Reconstructed holograms with τ = 96.1 % : (a) experimental saturation, (b) numerical saturation, (c) exact modeling, and (d)  approximated modeling.

Fig. 28
Fig. 28

Evolution of the mean value of the hologram versus order α.

Fig. 29
Fig. 29

Evolution of the mean value of the hologram versus order α for coefficient equal to 1.

Fig. 30
Fig. 30

Evolution of the mean value of the hologram versus order α for coefficient equal to 2.

Fig. 31
Fig. 31

Evolution of the mean value of the hologram versus order α for coefficient equal to 4.

Fig. 32
Fig. 32

Evolution of the mean value of the hologram versus order α for coefficient equal to 10.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

U r ( X , Y ) = a r exp ( j φ r ( X , Y ) ) ,
φ r ( X , Y ) = 2 π ( u r X + v r Y ) .
H = t 1 t 1 + Δ t | U r + U 0 | 2 d t = Δ t [ | U r | 2 + | U 0 | 2 + U r * U 0 + U r U 0 * ] ,
A r ( x , y ) = j exp ( 2 j π d r / λ ) λ d r exp [ j π λ d r ( x 2 + y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 H ( k p x , l p y ) exp [ j π λ d r ( k 2 p x 2 + l 2 p y 2 ) ] × exp [ 2 j π λ d r ( k x p x + l y p y ) ] ,
A r + 1 ( x , y ) = λ 2 d 0 2 exp [ j π λ d 0 ( x 2 + y 2 ) ] { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * W ˜ ( x , y , d 0 ) * δ ( x λ d 0 u r , y λ d 0 v r ) ,
H ( x , y ) = a ( x , y ) + b ( x , y ) cos ( φ 0 ( x , y ) + 2 π ( u r x + v r y ) ) ,
H ( x ) = { a ( x ) + b ( x ) cos ( φ 0 ( x ) + 2 π u r x ) if     a + b < H sat H sat if     a + b H sat .
H sat = a + b cos ( φ 0 + 2 π u r x sat ) .
x sat = 1 2 π u r cos 1 ( H sat a b ) .
τ = x sat + x sat H sat d x 1 / 2 u r + 1 / 2 u r H sat d x = 2 x sat H sat H sat / u r = 2 u r x sat = 1 π cos 1 ( H sat a b ) .
H ( x ) = n = + c n exp ( 2 j π n u r x + n j φ 0 ) = k = 0 + H k ( x ) ,
c n = u r 1 / 2 u r 1 / 2 u r H ( x ) exp ( 2 j π n u r x n j φ 0 ) d x .
c 0 = u r 1 / 2 u r x sat ( a + b cos ( 2 π u r x ) ) d x + u r x sat x sat H sat d x + u r x sat 1 / 2 u r ( a + b cos ( 2 π u r x ) ) d x .
c 0 = a ( 1 τ ) + τ H sat b π sin ( π τ ) .
c 1 = b 2 ( 1 τ ) + ( H sat a ) π sin ( π τ ) b 4 π sin ( 2 π τ ) .
c n = ( H sat a ) n π sin ( n π τ ) + b 2 π ( n 1 ) sin ( π ( 1 n ) τ ) b 2 π ( 1 + n ) sin ( π ( 1 + n ) τ )     n > 1 .
H ( τ ) = ( a 0 2 + a r 2 ) ( 1 τ ) + τ H sat 2 a r a 0 π sin ( π τ ) + [ a r a 0 ( 1 τ ) + H sat a π sin ( π τ ) a r a 0 2 π sin ( 2 π τ ) ] [ exp ( j φ r ) exp ( j φ 0 ) + cc ] + n = 2 [ H sat a n π sin ( n π τ ) a r a 0 ( n 1 ) π sin ( ( n 1 ) π τ ) a r a 0 ( n + 1 ) π sin ( ( n + 1 ) π τ ) ] [ exp ( n j φ r ) exp ( n j φ 0 ) + cc ] ,
H 0 = ( a r 2 + a 0 2 ) ( 1 τ ) + τ H sat 2 a r a 0 sin ( π τ ) π .
H 1 = [ ( 1 τ ) sin ( 2 π τ ) 2 π ] [ a r exp ( j φ r ) a 0 exp ( j φ 0 ) + cc ] + H sat a r 2 π sin ( π τ ) [ exp ( j φ r ) exp ( j φ 0 ) + cc ] sin ( π τ ) π [ exp ( j φ r ) a 0 2 exp ( j φ 0 ) + cc ] .
H n ( τ ) = 1 π [ sin ( ( n 1 ) π τ ) n 1 + sin ( ( n + 1 ) π τ ) n + 1 ] [ a r exp ( j n φ r ) n a 0 1 / n exp ( j φ 0 ) + cc ] + H sat a r 2 n π sin ( n π τ ) [ exp ( j n φ r ) n exp ( j φ 0 ) + cc ] sin ( n π τ ) n π [ exp ( j n φ r ) n a 0 2 / n exp ( j φ 0 ) + cc ] .
β 11 = log [ | A r + 1 ( x , y ) | | A r , 0 + 1 ( x , y ) | ] ,
β 12 = log [ | A r , 2 + 1 ( x , y ) | | A r + 1 ( x , y ) | ] .
A r , α + 1 ( x , y ) exp ( β 11 ( α 1 ) ) A r + 1 ( x , y ) for all     0 α 1 ,
A r , α + 1 ( x , y ) exp ( β 12 ( α 1 ) ) A r + 1 ( x , y ) for all     1 α 2 .
H 1 = [ ( 1 τ ) sin ( 2 π τ ) 2 π ] [ a r exp ( j φ r ) a 0 exp ( j φ 0 ) + cc ] + H sat a r 2 π sin ( π τ ) exp ( β 11 ) [ exp ( j φ r ) a 0 exp ( j φ 0 ) + cc ] sin ( π τ ) π exp ( β 12 ) [ exp ( j φ r ) a 0 exp ( j φ 0 ) + cc ] .
β n = log [ | A r , 1 + n ( x , y ) | | A r , 0 + n ( x , y ) | ] .
β α = exp ( 0 , 25 ) exp ( ( α 1 / 2 ) 2 ) for     0 α < 1.
A r , α + n ( x , y ) β α exp ( β n ( α 1 ) ) A r , 1 + n ( x , y ) for     0 α 1 and 2 n 4.
H n ( τ ) = 1 π [ sin ( ( n 1 ) π τ ) n 1 + sin ( ( n + 1 ) π τ ) n + 1 ] β 1 / n exp [ β n ( 1 n 1 ) ] [ a r exp ( j n φ r ) n a 0 exp ( j φ 0 ) + cc ] + H sat a r 2 n π sin ( n π τ ) β 0 exp ( β n ) [ exp ( j n φ r ) n a 0 exp ( j φ 0 ) + cc ] sin ( n π τ ) n π β 2 / n exp [ β n ( 2 n 1 ) ] [ exp ( j n φ r ) n a 0 exp ( j φ 0 ) + cc ] .
A r + 1 ( x , y ) λ 2 d 0 2 exp [ j π λ d 0 ( u r 2 + v r 2 ) ] exp [ 2 j π ( u r x + v r y ) ] × [ ( 1 τ ) a r sin ( 2 π τ ) 2 π a r + ( H sat a r 2 ) sin ( π τ ) π exp ( β 11 ) sin ( π τ ) π exp ( β 12 ) ] × A ( x , y ) * δ ( x λ u r d 0 , y λ v r d 0 ) * W ˜ NM ( x , y , d 0 ) .
A r + 3 ( x , y ) = λ 5 d 0 5 exp [ j π λ d 0 ( x 2 + y 2 ) ] × [ a r 4 π [ 2 sin ( 2 π τ ) + sin ( 4 π τ ) ] β 1 / 3 exp ( β 3 / 3 ) + ( H sat a r 2 ) sin ( 3 π τ ) 3 π β 0 exp ( β 3 ) sin ( 3 π τ ) 3 π β 2 / 3 exp ( β 3 / 3 ) ] × { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * W ˜ ( x , y , d 0 ) * W ˜ 2 ( x , y ) * δ ( x 3 λ d 0 u r + λ d 0 p x , y 3 λ d 0 v r + λ d 0 p y ) .
A r + 4 ( x , y ) = λ 6 d 0 6 exp [ j π λ d 0 ( x 2 + y 2 ) ] × [ a r 3 π [ sin ( 3 π τ ) + 3 5 sin ( 5 π τ ) ] β 1 / 4 exp ( 3 β 4 / 4 ) + ( H sat a r 2 ) sin ( 4 π τ ) 4 π β 0 exp ( β 4 ) sin ( 4 π τ ) 4 π β 1 / 2 exp ( β 4 / 2 ) ] × { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * { A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * W ˜ ( x , y , d 0 ) * W ˜ 3 ( x , y ) * δ ( x 4 λ d 0 u r + λ d 0 p x , y 4 λ d 0 v r + λ d 0 p y ) .
A r 2 ( x , y ) = λ 3 d 0 3 exp [ j π λ d 0 ( x 2 + y 2 ) ] × [ a r 3 π [ 3 sin ( π τ ) + sin ( 3 π τ ) ] β 1 / 2 exp ( β 2 / 2 ) + ( H sat a r 2 ) sin ( 2 π τ ) 2 π β 0 exp ( β 2 ) sin ( 2 π τ ) 2 π ] × { A * ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * { A * ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] } * W ˜ ( x , y , d 0 ) * W ˜ 3 ( x , y ) * δ ( x + 2 λ d 0 u r λ d 0 p x , y + 2 λ d 0 v r λ d 0 p y ) .
A r , sat + 1 = A r + 1 + A r 2 + A r + 3 + A r + 4 .

Metrics