## Long time exposure digital in-line holography for 3-D particle trajectography |

Optics Express, Vol. 21, Issue 20, pp. 23522-23530 (2013)

http://dx.doi.org/10.1364/OE.21.023522

Acrobat PDF (1297 KB)

### Abstract

One advantage of digital in-line holography is the ability for a user to know the 3-D location of a moving particle recorded at a given time. When the time exposure is much larger than the time required for grabbing the particle image at a given location, the diffraction pattern is spread along the trajectory of this particle. This can be seen as a convolution between the diffraction pattern and a blurring function resulting from the motion of the particle during the camera exposure. This article shows that the reconstruction of holograms recorded under such conditions exhibit traces that could be processed for extracting 3D trajectories.

© 2013 OSA

## 1. Introduction

## 2. Hologram recording of moving particle

### 2.1 Digital recording of the hologram

*z*from a quadratic sensor, as shown in Fig. 1. Note that the case of a plane wave is used for the sake of clarity, but as shown in [20

_{e}20. D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. **50**(34), H1–H9 (2011). [CrossRef] [PubMed]

21. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. **18**(11), 846–848 (1993). [CrossRef] [PubMed]

11. C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng. **33**(6), 409–421 (2000). [CrossRef]

### 2.2 Influence of the displacement during the exposure

19. L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express **19**(5), 4393–4398 (2011), doi:. [CrossRef] [PubMed]

**. For this reasoning,**

*V***is supposed to be a constant and parallel with respect to the camera sensor during the exposure time**

*V**τ*(i.e.,

*V**= (V*. The particular case of zero velocity in

_{x},V_{y},0))*z*direction is first considered for the sake of clarity. Displacement in

*z*direction will be introduced later in subsection 2.3. The angle between

**and the**

*V**x-*axis is noted as

*τ*is given bywhere

*t*. Considering a new axis system

*(x’,y’,z)*obtained from a rotation of the

*(x,y,z)*system around the z axis by an angle

*E*can be rewritten as:It is not difficult to show that Eq. (5) can also be expressed as a convolution operation of the exposure of a fixed pattern

_{V}22. D. Lebrun, C. E. Touil, and C. Özkul, “Methods for the deconvolution of defocused-image pairs recorded separately on two CCD cameras: application to particle sizing,” Appl. Opt. **35**(32), 6375–6381 (1996). [CrossRef] [PubMed]

*x’*). Thus, the expression of

*Δ*in the

*(x,y)*coordinate system is:As a result, the influence of the 3-D displacement during the exposure time on the digitized diffraction pattern can be described by: By introducing Eq. (3) into Eq. (9), Eq. (8) gives: Finally, the hologram produced by a moving particle can be expressed as the hologram

*Δ(x,y)*represents the displacement of this particle during the exposure time.

### 2.3 Hologram reconstruction

*z*from the camera, can be calculated by a convolution operation:By using the relation of Eq. (9), Eq. (12) can be rewritten as:where

_{r}*R(x,y)*is no more than the noiseless reconstructed image for a fixed particle whose expression can be found in [20

20. D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. **50**(34), H1–H9 (2011). [CrossRef] [PubMed]

*Δ(x,y)*. The recorded distance

*z*may also vary during the recording, especially if a long exposure time is considered. In that case, the 3-D trajectory of particles should be considered by interrogating several planes. Afterwards, a tilted plane adapted to the main direction of the particle should be defined in order to visualize the entire trajectory of the object. This situation is not different from the case where fiber holograms are recorded under arbitrary angles [23

_{e}23. D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express **11**(3), 224–229 (2003). [CrossRef] [PubMed]

5. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. **45**(5), 836–850 (2006). [CrossRef] [PubMed]

*x,y*and

*z*direction. Another important consideration concerns the Signal to noise ratio (SNR) of holographic images. Even if the noise term is reduced by a time average during the exposure, Eq. (13) shows also that the image contrast is reduced by the convolution operation with

*Δ(x,y)*. These contradictory effects are studied in the following section.

## 3. Simulations

15. N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol. **19**(7), 074001 (2008). [CrossRef]

*60-μm*particle located at a distance

*z*from the CCD sensor and moving with a constant velocity

_{e}= 258 mm*V*during an exposure time of

_{x}= 0.025 m/s, V_{y}= 0.5 m/s24. G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. **29**(2), 610–611 (1958). [CrossRef]

*0.1–3 m.s*].

^{−1}*Δ(x,y)*as introduced in section 2. Figure 4 show the variations of the SNR of the reconstructed images with respect to the object velocity and for several exposure durations, i.e.,

*τ = 200, 500*and

*1000 µs*. The SNR is defined here by the ratio of the image intensity to the standard deviation of the background intensity. As shown on this figure, the SNR remains unchanged for object velocities

*0.05 m.s*. This result is not surprising because under this limit, the particle displacement

^{−1}*d = 60 µm*). Therefore, the smoothing effect due to the convolution operation with the function

*0.1 m.s*a decrease of

^{−1}*-10 dB/decade*is observed. This is in good accordance with Eq. (13), where the convolution operation leads to a reduction by a factor

25. H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. **15**(4), 673–685 (2004). [CrossRef]

*2 mm*for a particle diameter

*d = 60 µm*.

## 4. Experimental results

*λ = 635 nm*. This light source is controlled by a synchronization device that enables the generation of a TTL signal. Although a diverging beam is used, the equivalent plane wave scheme of Fig. 1 can be used because that the far field conditions are assumed [20

20. D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. **50**(34), H1–H9 (2011). [CrossRef] [PubMed]

*τ*can be adjusted continuously from 5 to 1000 μs. The moving objects are bubbles produced in a cavitation tunnel and observed through a transparent pipe. More details can be found in Ref [20

**50**(34), H1–H9 (2011). [CrossRef] [PubMed]

*.*Hence, as shown by Slimani

*et al.*[26

26. F. Slimani, G. Gréhan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt. **23**(22), 4140–4148 (1984). [CrossRef] [PubMed]

*1280 × 1024*CMOS camera. Figure 6 shows a set of experimental diffraction patterns recorded by the probe for different pulse durations τ in the interval [10-1000 µs]. In order to compare the experimental results with the model, we have chosen images of bubbles with diameters around 60 µm. The mean velocities are roughly evaluated by calculating the ratio between the estimated length of the traces and the time exposure.

*z*play an important role in the image contrast [27

_{e}27. S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids **39**(1), 1–9 (2005). [CrossRef]

*τ*. The graph of Fig. 7 shows that the −5dB behavior of the

*SNR(τ)*graph is rather well confirmed by the experimental points.

15. N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol. **19**(7), 074001 (2008). [CrossRef]

## 5. Conclusion

## Acknowledgments

## References and links

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

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

3. | P. Picart and J. Li, |

4. | G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. |

5. | J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. |

6. | A. El Mallahi, C. Minetti, and F. Dubois, “Automated three-dimensional detection and classification of living organisms using digital holographic microscopy with partial spatial coherent source: application to the monitoring of drinking water resources,” Appl. Opt. |

7. | F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in Particle Digital Holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A |

8. | F. Lamadie, L. Bruel, and M. Himbert, “Digital holographic measurements of liquid-liquid two-phase flows,” Opt. Lasers Eng. |

9. | S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” Appl. Opt. |

10. | W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. |

11. | C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng. |

12. | M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. |

13. | F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital In-line holography with a sub-picosecond laser beam,” Opt. Commun. |

14. | M. Brunel, H. Shen, S. Coëtmellec, and D. Lebrun, “Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond Digital In-line Holography with anamorphic optical systems,” Appl. Opt. |

15. | N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol. |

16. | M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter |

17. | J. F. Restrepo and J. Garcia-Sucerquia, “Automatic three-dimensional tracking of particles with high-numerical-aperture digital lensless holographic microscopy,” Opt. Lett. |

18. | F. Dubois, N. Callens, C. Yourassowsky, M. Hoyos, P. Kurowski, and O. Monnom, “Digital holographic microscopy with reduced spatial coherence for three-dimensional particle flow analysis,” Appl. Opt. |

19. | L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express |

20. | D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. |

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

22. | D. Lebrun, C. E. Touil, and C. Özkul, “Methods for the deconvolution of defocused-image pairs recorded separately on two CCD cameras: application to particle sizing,” Appl. Opt. |

23. | D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express |

24. | G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. |

25. | H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol. |

26. | F. Slimani, G. Gréhan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt. |

27. | S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids |

**OCIS Codes**

(090.0090) Holography : Holography

(100.3010) Image processing : Image reconstruction techniques

(100.6890) Image processing : Three-dimensional image processing

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: June 27, 2013

Revised Manuscript: August 17, 2013

Manuscript Accepted: September 4, 2013

Published: September 26, 2013

**Citation**

D. Lebrun, L. Méès, D. Fréchou, S. Coëtmellec, M. Brunel, and D. Allano, "Long time exposure digital in-line holography for 3-D particle trajectography," Opt. Express **21**, 23522-23530 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-23522

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### References

- J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett.11(3), 77–79 (1967). [CrossRef]
- 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]
- P. Picart and J. Li, Digital holography (John Wiley & Sons Ed. 2012)
- G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt.42(5), 827–833 (2003). [CrossRef] [PubMed]
- J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt.45(5), 836–850 (2006). [CrossRef] [PubMed]
- A. El Mallahi, C. Minetti, and F. Dubois, “Automated three-dimensional detection and classification of living organisms using digital holographic microscopy with partial spatial coherent source: application to the monitoring of drinking water resources,” Appl. Opt.52(1), A68–A80 (2013). [CrossRef] [PubMed]
- F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in Particle Digital Holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A24(12), 3708–3716 (2007). [CrossRef] [PubMed]
- F. Lamadie, L. Bruel, and M. Himbert, “Digital holographic measurements of liquid-liquid two-phase flows,” Opt. Lasers Eng.50(12), 1716–1725 (2012). [CrossRef]
- S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” Appl. Opt.41(2), 312–319 (2002). [CrossRef] [PubMed]
- W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett.28(3), 164–166 (2003). [CrossRef] [PubMed]
- C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three dimensional location of particles,” Opt. Lasers Eng.33(6), 409–421 (2000). [CrossRef]
- M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol.15(4), 699–705 (2004). [CrossRef]
- F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital In-line holography with a sub-picosecond laser beam,” Opt. Commun.268(1), 27–33 (2006). [CrossRef]
- M. Brunel, H. Shen, S. Coëtmellec, and D. Lebrun, “Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond Digital In-line Holography with anamorphic optical systems,” Appl. Opt.51(8), 1137–1148 (2012). [CrossRef] [PubMed]
- N. Salah, G. Godard, D. Lebrun, P. Paranthoen, D. Allano, and S. Coetmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard-von Karman vortex flow,” Meas. Sci. Technol.19(7), 074001 (2008). [CrossRef]
- M. Heydt, P. Divós, M. Grunze, and A. Rosenhahn, “Analysis of holographic microscopy data to quantitatively investigate three-dimensional settlement dynamics of algal zoospores in the vicinity of surfaces,” Eur Phys J E Soft Matter30(2), 141–148 (2009). [CrossRef] [PubMed]
- J. F. Restrepo and J. Garcia-Sucerquia, “Automatic three-dimensional tracking of particles with high-numerical-aperture digital lensless holographic microscopy,” Opt. Lett.37(4), 752–754 (2012). [CrossRef] [PubMed]
- F. Dubois, N. Callens, C. Yourassowsky, M. Hoyos, P. Kurowski, and O. Monnom, “Digital holographic microscopy with reduced spatial coherence for three-dimensional particle flow analysis,” Appl. Opt.45(5), 864–871 (2006). [CrossRef] [PubMed]
- L. Dixon, F. C. Cheong, and D. G. Grier, “Holographic particle-streak velocimetry,” Opt. Express19(5), 4393–4398 (2011), doi:. [CrossRef] [PubMed]
- D. Lebrun, D. Allano, L. Méès, F. Walle, F. Corbin, R. Boucheron, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt.50(34), H1–H9 (2011). [CrossRef] [PubMed]
- L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett.18(11), 846–848 (1993). [CrossRef] [PubMed]
- D. Lebrun, C. E. Touil, and C. Özkul, “Methods for the deconvolution of defocused-image pairs recorded separately on two CCD cameras: application to particle sizing,” Appl. Opt.35(32), 6375–6381 (1996). [CrossRef] [PubMed]
- D. Lebrun, A. M. Benkouider, S. Coëtmellec, and M. Malek, “Particle field digital holographic reconstruction in arbitrary tilted planes,” Opt. Express11(3), 224–229 (2003). [CrossRef] [PubMed]
- G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat.29(2), 610–611 (1958). [CrossRef]
- H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: from film to digital recording,” Meas. Sci. Technol.15(4), 673–685 (2004). [CrossRef]
- F. Slimani, G. Gréhan, G. Gouesbet, and D. Allano, “Near-field Lorenz-Mie theory and its application to microholography,” Appl. Opt.23(22), 4140–4148 (1984). [CrossRef] [PubMed]
- S. Pu, D. Lebrun, D. Allano, B. Patte-Rouland, M. Malek, and C. Cen, “Particle field characterization by digital in-line holography: 3D location and sizing,” Exp. Fluids39(1), 1–9 (2005). [CrossRef]

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