## General theoretical formulation of image formation in digital Fresnel holography

JOSA A, Vol. 25, Issue 7, pp. 1744-1761 (2008)

http://dx.doi.org/10.1364/JOSAA.25.001744

Acrobat PDF (1005 KB)

### Abstract

We present a detailed analysis of image formation in digital Fresnel holography. The mathematical modeling is developed on the basis of Fourier optics, making possible the understanding of the different influences of each of the physical effects invoked in digital holography. Particularly, it is demonstrated that spatial resolution in the reconstructed plane can be written as a convolution product of functions that describe these influences. The analysis leads to a thorough investigation of the effect of the width of the sensor, the surface of pixels, the numerical focusing, and the aberrations of the reference wave, as well as to an explicit formulation of the Shannon theorem for digital holography. Experimental illustrations confirm the proposed theoretical analysis.

© 2008 Optical Society of America

## 1. INTRODUCTION

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

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

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

5. P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. **31**, 1405–1407 (2006). [CrossRef] [PubMed]

6. G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. **41**, 4489–4496 (2002). [CrossRef] [PubMed]

7. C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express **13**, 8693–8698 (2005). [CrossRef] [PubMed]

8. K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express **15**, 3047–3052 (2007). [CrossRef] [PubMed]

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

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

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

12. Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. **26**, 1478–1480 (2001). [CrossRef]

13. B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. **30**, 236–238 (2005). [CrossRef] [PubMed]

14. T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. **32**, 2146–2148 (2007). [CrossRef] [PubMed]

15. 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]

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

17. I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. **45**, 7610–7616 (2006). [CrossRef] [PubMed]

18. M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. **45**, 684–689 (2007). [CrossRef]

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

20. Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. **45**, 65–70 (2005). [CrossRef]

21. P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. **42**, 1947–1957 (2003). [CrossRef] [PubMed]

22. 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. (Bellingham) **43**, 1169–1176 (2004). [CrossRef]

23. T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. **45**, 925–934 (2006). [CrossRef] [PubMed]

24. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement **18**, 251–260 (1995). [CrossRef]

25. Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. **46**, 5719–5727 (2007). [CrossRef] [PubMed]

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

28. N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express **13**, 4812–4817 (2005). [CrossRef] [PubMed]

29. A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. **45**, 2391–2395 (2006). [CrossRef] [PubMed]

30. 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]

31. 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]

32. T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. **45**, 4534–4539 (2006). [CrossRef]

33. 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]

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

35. 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]

36. L. Onural and M. T. Ozgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A **9**, 252–260 (1992). [CrossRef]

37. S. Coetmellec, D. Lebrun, and C. Oskul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A **19**, 1537–1546 (2002). [CrossRef]

39. F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “‘Suppression of the Moiré effect in sub-picosecond digital in-line holography,” Opt. Express **15**, 887–895 (2007). [CrossRef] [PubMed]

40. M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express **12**, 2270–2279 (2004). [CrossRef] [PubMed]

41. L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. **45**, 944–952 (2006). [CrossRef] [PubMed]

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

43. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang–Gu algorithm,” Appl. Opt. **42**, 6452–6457 (2003). [CrossRef] [PubMed]

44. Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. **29**, 1787–1789 (2004). [CrossRef] [PubMed]

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

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

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

48. 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]

## 2. GENERAL MODELING OF THE OBJECT–IMAGE RELATION

### 2A. Fundamental Basics

*z*axis is perpendicular to the surface and is considered to be the propagation direction of a diffracted light beam. The object surface illuminated by a coherent beam of wavelength

*λ*produces an object wavefront noted

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

48. 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]

*a priori*focusing error cannot occur. Now, in the interference plane, the hologram

*H*is written asNote that in the case where

58. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. **44**, 337–343 (2005). [CrossRef] [PubMed]

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

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

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

### 2B. Different Contributions to Image Formation

### 2C. General Linear Relation

*κ*includes irrelevant constants and phase terms of Eq. (20). This relation can also be written in the general form of Eq. (1) by introducing the impulse response of the process In what follows, function

## 3. INTRINSIC RESOLUTION

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

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

48. 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]

58. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. **44**, 337–343 (2005). [CrossRef] [PubMed]

*n*and

*m*vary from

## 4. INFLUENCE OF THE ACTIVE SURFACE OF PIXELS

### 4A. Resolution Function

*x*direction. Numerical values are

*x*profile. The numerical values are

### 4B. Criterion for Influence

*x*coordinate, for which we haveand

*N*. The numerical values are

## 5. INFLUENCE OF FOCUS

### 5A. Analytical Formulation

### 5B. Comparison with a Digital Holographic Simulation

**3098**, 224–233 (1997). [CrossRef]

*x*profile for perfect focusing and the

*z*profile along the

*z*axis. It can be seen that the

*x*profiles overlap, whereas the

*z*profiles are slightly different but have equivalent width according to the Rayleigh criterion. No explications were found for understanding this slight difference. Globally, it can be admitted that the modeling presented in this section is a good approximation of the spatial resolution in digital Fresnel holography in the presence of focusing error.

## 6. INFLUENCE OF REFERENCE WAVE ABERRATION

### 6A. Modeling for Aberration

### 6B. Aberration in the Recording Plane

*ξ*into

*η*into

### 6C. Resolution in the Presence of Aberrations

### 6D. Numerical Simulation

21. P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. **42**, 1947–1957 (2003). [CrossRef] [PubMed]

22. 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. (Bellingham) **43**, 1169–1176 (2004). [CrossRef]

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

30. 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]

31. 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]

33. 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]

58. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. **44**, 337–343 (2005). [CrossRef] [PubMed]

## 7. SHANNON THEOREM FOR DIGITAL HOLOGRAPHY

### 7A. Structure of the Diffracted Field

### 7B. Optimization of Recording

*a*, the full width of the reconstructed field (

*x*and

*y*directions) must be equal to four times the width of the object (i.e.,

*a*, one gets [Fig. 18b]and the optimal spatial frequencies are given byNote that for

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

**38**, 4812–4820 (1999). [CrossRef]

56. L. Xu, X. Peng, Z. Guo, J. Mia, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express **13**, 2444–2452 (2005). [CrossRef] [PubMed]

## 8. CONCLUSION

## Appendix A

## Appendix B

1. | M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. |

2. | U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

3. | T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase shifting digital holography,” Opt. Lett. |

4. | E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase contrast imaging,” Opt. Lett. |

5. | P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. |

6. | G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. |

7. | C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express |

8. | K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express |

9. | J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express |

10. | B. Javidi and E. Tajahuerce, “Three-dimensional object recognition by use of digital holography,” Opt. Lett. |

11. | B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. |

12. | Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. |

13. | B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. |

14. | T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. |

15. | 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. |

16. | I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. |

17. | I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. |

18. | M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. |

19. | S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. |

20. | Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. |

21. | P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. |

22. | 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. (Bellingham) |

23. | T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. |

24. | G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement |

25. | Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. |

26. | G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. |

27. | P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. |

28. | N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express |

29. | A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. |

30. | J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. |

31. | 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 |

32. | T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. |

33. | 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 |

34. | A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. |

35. | 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. |

36. | L. Onural and M. T. Ozgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A |

37. | S. Coetmellec, D. Lebrun, and C. Oskul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A |

38. | F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A |

39. | F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “‘Suppression of the Moiré effect in sub-picosecond digital in-line holography,” Opt. Express |

40. | M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express |

41. | L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. |

42. | L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. |

43. | Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang–Gu algorithm,” Appl. Opt. |

44. | Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. |

45. | Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE |

46. | Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. (Bellingham) |

47. | Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. (Bellingham) |

48. | C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. |

49. | M. Liebling, “On Fresnelets, interferences fringes, and digital holography Ph.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004). |

50. | I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase shifting digital holography and application to microscopy,” Appl. Opt. |

51. | T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. |

52. | 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. |

53. | R. Binet, J. Colineau, and J. C. Lehureau, “‘Short-range synthetic aperture imaging at |

54. | J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. |

55. | G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. |

56. | L. Xu, X. Peng, Z. Guo, J. Mia, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express |

57. | J. W. Goodman, |

58. | P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. |

59. | 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. (Bellingham) |

60. | M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. |

61. | F. Le Clerc, L. Collot, and M. Gross, “Numerical heterodyne holography with two-dimensional photo detector arrays,” Opt. Lett. |

62. | Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) |

63. | E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. |

64. | I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. |

**OCIS Codes**

(090.0090) Holography : Holography

(090.2870) Holography : Holographic display

(090.2880) Holography : Holographic interferometry

(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.4630) Instrumentation, measurement, and metrology : Optical inspection

(090.1995) Holography : Digital holography

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Holography

**History**

Original Manuscript: December 19, 2007

Revised Manuscript: April 8, 2008

Manuscript Accepted: May 4, 2008

Published: June 25, 2008

**Citation**

Pascal Picart and Julien Leval, "General theoretical formulation of image formation in digital Fresnel holography," J. Opt. Soc. Am. A **25**, 1744-1761 (2008)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-25-7-1744

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### References

- M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333-334 (1972).
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