## Digital Fresnel holography beyond the Shannon limits |

Optics Express, Vol. 20, Issue 16, pp. 18303-18312 (2012)

http://dx.doi.org/10.1364/OE.20.018303

Acrobat PDF (3050 KB)

### Abstract

This paper presents a detailed analysis of the influence of the pixel dimension in digitally-recorded holograms. The investigation is based on both theoretical and experimental viewpoints for recordings beyond the Shannon limits. After discussing the pixel paradox, the sinc amplitude modulation is experimentally demonstration. The experimental analysis is well correlated to the theoretical basics; in addition, the filling factor of the sensor can be estimated. The analysis of the phase changes of the object show that they can be obtained with a very good contrast and that they are only limited by the decorrelation noise, as when the Shannon conditions are fulfilled.

© 2012 OSA

## 1. Introduction

## 2. Theoretical basics

### 2.1 Digital hologram recording and reconstruction

*R*(

*x*,

*y*) =

*a*exp[2

_{R}*i*π(

*u*

_{0}

*x*+

*v*

_{0}

*y*)] is the reference wave with spatial frequencies {

*u*

_{0},

*v*

_{0}}, and

*O*is the wave diffracted in the recording plane by the object located at distance

*d*

_{0}from this plane. In the Fresnel approximations, we have [23]:

*A*(

*X*,

*Y*) =

*A*

_{0}(

*X*,

*Y*)exp[

*iψ*

_{0}(

*X*,

*Y*)] is the object wave front. Due to the spatial extension of the pixel, the detected holograms, which will be transferred to the digital acquisition board, are composed of elementary spatially-integrated zones of the real ones. At any pixel coordinate (

*lp*,

_{x}*kp*) at which the digital hologram is recorded (

_{y}*l*,

*k*: integers;

*p*,

_{x}*p*: pixel pitches), we have:

_{y}12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A **25**(7), 1744–1761 (2008). [CrossRef] [PubMed]

*d*

_{0}from the recording plane is given by the discrete Fresnel transform [3

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

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

### 2.2 The pixel paradox

9. C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. **38**(22), 4812–4820 (1999). [CrossRef] [PubMed]

21. L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express **13**(7), 2444–2452 (2005). [CrossRef] [PubMed]

*x*) = sin(

*x*)/

*x*. According to Eq. (7), the useful + 1 order is amplitude-modulated by the sinc functions; therefore, the reconstructed object will also be amplitude-modulated by the sinc functions [21

21. L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express **13**(7), 2444–2452 (2005). [CrossRef] [PubMed]

*pixel paradox*. Indeed, the first approach states that the pixel function is a convolution function, whereas the latter states that it is a multiplicative sinc function. One may wonder what magical reason explains why the same element provides different influences. Furthermore, note that the Fourier transform of the recorded hologram provides the spatial frequency spectrum of the object. Thus, it is multiplied by the sinc function according to Eq. (5). Nevertheless, the Fourier transform of the hologram is also the discrete Fresnel transform for a reconstruction distance tending toward infinity, leading to an out-of-focus object. There is no logical reason for the out-of-focus object to be not convoluted by the pixel function in the same manner as the focused object will be. But it is quite common sense that the Fourier transform of the hologram is multiplied by the sinc function.

### 2.3 The Shannon limits

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A **25**(7), 1744–1761 (2008). [CrossRef] [PubMed]

16. N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt. **48**(34), H186–H195 (2009). [CrossRef] [PubMed]

25. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**(16), 1268–1270 (1997). [CrossRef] [PubMed]

12. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A **25**(7), 1744–1761 (2008). [CrossRef] [PubMed]

22. N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express **17**(18), 15842–15852 (2009). [CrossRef] [PubMed]

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

*X*, the mean spatial frequency of the hologram becomes

*u*

_{0}= Δ

*X*/

*λd*

_{0}. When laterally moving the object in the field of view, the reconstructed object will be localized at the corresponding spatial shift until spatial aliasing occurs. Indeed, the field of view reconstructed by the discrete Fresnel transform is sized

*λd*

_{0}/

*p*×

_{x}*λd*

_{0}/

*p*. Because of the sampling of recording and reconstruction spaces, the field of view is virtually periodic. This means that the reconstructed field is repeated infinitely, with a period equal to its width. However, the numerical computation can only give the data of the main window corresponding to the field of view. Figure 1 illustrates the virtual periodicity of the reconstructed field. The central window (not translucent) is that reconstructed by the FFT algorithms, whereas the 8 other ones (translucent) are not reachable by FFT calculation. However, these 8 neighborhood zones indicate that, when the object is shifted from the central window to one of the adjacent ones, its reconstructed image comes from the symmetric zone. Figure 1(left) shows the aliasing that occurs when exceeding Shannon limits: the under-sampled part of the object is shifted out of the main window and re-appears in the opposite corner. Thus, for a physical object shifted along the

_{y}*x*-axis with Δ

*X*∈{(2

*k*−1)

*λd*

_{0}/2

*p*,(2

_{x}*k*+ 1)

*λd*

_{0}/2

*p*},

_{x}*k*integer, the position of the numerically-reconstructed object is given by Δ

*X*= Δ

_{r}*X*−

*kλd*

_{0}/

*p*. Beyond the Shannon limits, the reconstructed object appears in the field of view at cyclic positions and with an amplitude attenuated by the sinc(

_{x}*π*Δ

*X*Δ

*/*

_{x}*λd*

_{0}) function.

*=*

_{x}*p*).

_{x}## 3. Experimental set-up and methodology

*L*×

*K*= 1024 × 1360 pixels sized

*p*=

_{x}*p*= 4.65μm. The laser is a continuous frequency doubled NdYAG laser (

_{y}*λ*= 532nm,

*P*

_{0}= 150mW) and the object is a 50 cents euro coin, 24mm in diameter. It is illuminated with a circular spot. The object is mounted on a translation stage (Fig. 3); the mean horizontal spatial frequencies of the digital hologram, {

*u*

_{0}}, may thus vary continuously. The amplitude modulation due to the pixel is then formulated with sinc(πΔ

*Δ*

_{x}*X*/

*λd*

_{0}). In order to measure the amplitude modulation of the object, a power meter estimates the injected laser power into the interferometer. A rotation stage (mirror M2) ensures that the object is correctly illuminated when translating in the

*x*-axis direction. Note that change in the illuminating angle, due to the object translation, requires the measurement of the bidirectional reflectance distribution function (BRDF) of the object for angles similar to those provided by the illumination.

*d*

_{0}= 900mm during the experiment. The object can be slightly tilted on its mount. This facility provides the possibility to study the effect of the pixel dimension on phase differences extracted between two object states, especially the decorrelation noise and the coherence degree of the two speckle phases [24

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

*μ*| being a quality marker extracted from experimental data.

## 4 Experimental results

*X*-axis direction from 0 to 280mm. The Shannon limits for the set-up is 51.48mm for the lateral position. The BRDF for the experimental conditions was measured and was transposed as a lateral shift variation. It was evidenced that its contribution to the amplitude modulation is a multiplication by exp(−Δ

*X*/

*b*). This means that the amplitude modulation on the reconstructed object is a

*f*(Δ

*X*) = |sinc(

*π*Δ

*X*/

*a*)exp(−Δ

*X*/

*b*)| function rather than |sinc(

*π*Δ

*X*/

*a*)|. Figure 4 shows the measured amplitude modulation from 0 to 280mm. The experimental curve is compared to the analytical model

*f*(Δ

*X*), giving

*a*= 119.15mm and

*b*= 168.35mm (99.824% correlation). The |sinc(

*π*Δ

*X*/

*a*)| curve is also plotted. Fit

*f*(Δ

*X*) is in excellent agreement with the experimental curve. This confirms a part of the pixel paradox discussed in this paper: the pixel dimension has a sinc influence on the reconstructed object amplitude. Furthermore, this validates the modeling for the BRDF of the object, that must be taken into account to measure pixel influence.

22. N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express **17**(18), 15842–15852 (2009). [CrossRef] [PubMed]

29. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. **162**(4-6), 205–210 (1999). [CrossRef]

24. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]

26. P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, “A simple method for measuring the random variation of an interferometer,” Meas. Sci. Technol. **12**(8), 1311–1317 (2001). [CrossRef]

*μ*| of Eq. (8) according to the object lateral shift. The Shannon limit and the red points of Fig. 4 are also shown.

*a*= 119.15mm. Thus, the pixel width is estimated to Δ

*=*

_{x}*λd*

_{0}/

*a*= 4.01μm, leading to a filling factor of Δ

*x*/

*p*= 86.4%.

_{x}## 5. Conclusion

## Acknowledgments

## References and Links

1. | J. W. Goodman and R. W. Laurence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. |

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

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

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

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

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

7. | Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. |

8. | Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. |

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

10. | M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A |

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

12. | P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A |

13. | D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE |

14. | D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. |

15. | D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS rapid publication |

16. | N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt. |

17. | P. Picart, P. Tankam, and Q. Song, “Experimental and theoretical investigation of the pixel saturation effect in digital holography,” J. Opt. Soc. Am. A |

18. | N. Pandey and B. M. Hennelly, “Quantization noise and its reduction in lensless Fourier digital holography,” Appl. Opt. |

19. | 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. |

20. | A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. |

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

22. | N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express |

23. | J. W. Goodman, |

24. | M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef] |

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

26. | P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, “A simple method for measuring the random variation of an interferometer,” Meas. Sci. Technol. |

27. | D. Middleton, |

28. | W. B. Davenport and W. L. Root, |

29. | H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. |

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.3010) Image processing : Image reconstruction techniques

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: June 1, 2012

Revised Manuscript: July 5, 2012

Manuscript Accepted: July 16, 2012

Published: July 25, 2012

**Citation**

Mathieu Leclercq and Pascal Picart, "Digital Fresnel holography beyond the Shannon limits," Opt. Express **20**, 18303-18312 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18303

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### References

- J. W. Goodman and R. W. Laurence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967). [CrossRef]
- 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).
- U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt.33(2), 179–181 (1994). [CrossRef] [PubMed]
- L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett.18(11), 846–848 (1993). [CrossRef] [PubMed]
- Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang-Gu algorithm,” Appl. Opt.42(32), 6452–6457 (2003). [CrossRef] [PubMed]
- Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997). [CrossRef]
- Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng.41(4), 771–778 (2002). [CrossRef]
- Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng.41(8), 1829–1839 (2002). [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(22), 4812–4820 (1999). [CrossRef] [PubMed]
- M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004). [CrossRef] [PubMed]
- I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt.40(34), 6177–6186 (2001). [CrossRef] [PubMed]
- P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A25(7), 1744–1761 (2008). [CrossRef] [PubMed]
- D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE7072, 707215 (2008). [CrossRef]
- D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009). [CrossRef]
- D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS rapid publication6, 11034 (2011). [CrossRef]
- N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt.48(34), H186–H195 (2009). [CrossRef] [PubMed]
- P. Picart, P. Tankam, and Q. Song, “Experimental and theoretical investigation of the pixel saturation effect in digital holography,” J. Opt. Soc. Am. A28(6), 1262–1275 (2011). [CrossRef] [PubMed]
- N. Pandey and B. M. Hennelly, “Quantization noise and its reduction in lensless Fourier digital holography,” Appl. Opt.50(7), B58–B70 (2011). [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(9), 2768–2772 (2003). [CrossRef]
- A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng.43(1), 239–250 (2004). [CrossRef]
- L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express13(7), 2444–2452 (2005). [CrossRef] [PubMed]
- N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express17(18), 15842–15852 (2009). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (Second Edition, McGraw-Hill Editions, 1996).
- M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. (2012), doi:. [CrossRef]
- I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997). [CrossRef] [PubMed]
- P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, “A simple method for measuring the random variation of an interferometer,” Meas. Sci. Technol.12(8), 1311–1317 (2001). [CrossRef]
- D. Middleton, Introduction to Statistical Communication Theory (McGraw Hill, 1960).
- W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw Hill, 1958).
- H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun.162(4-6), 205–210 (1999). [CrossRef]

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