## Particle field digital holographic reconstruction in arbitrary tilted planes

Optics Express, Vol. 11, Issue 3, pp. 224-229 (2003)

http://dx.doi.org/10.1364/OE.11.000224

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### Abstract

Digital holography is applied to the reconstruction of small particles in a plane whose orientation is arbitrary as specified by the user. The diffraction pattern produced by the particles is directly recorded by a conventional CCD camera. The digital recorded image enables the recovery of particle-images in several parallel planes of the probe volume. Afterwards, an interrogation slice corresponding to a thin layer around a theoretical arbitrary tilted plane is fixed. The pixels whose 3D coordinates belong to this slice are selected and juxtaposed to rebuild the particle images. The feasibility is demonstrated on a fiber tilted with respect to the camera plane. A second example is given on an experimental particle field. These results let us predict future applications such as the characterization of particle fields in planes other than those parallel with the camera plane.

© 2003 Optical Society of America

## 1. Introduction

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

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

5. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. **36**, 1947–1951 (1997) [CrossRef]

6. C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Optics and Lasers in Eng. **33**, 09–421 (2000) [CrossRef]

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

8. S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform,” App. Optics , **41**, 312–319, (2002) [CrossRef]

9. S. Coëtmellec, C. Buraga-Lefebvre, D. Lebrun, and C. Özkul, “Application of in-line digital holography to multiple plane velocimetry,” Meas. Sci and Tech. **12**, 1392–1397 (2001) [CrossRef]

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

*et al.*[11

11. L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express **10**, 1250–1257, (2002) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1250 [CrossRef] [PubMed]

## 2. Digital reconstruction of in-line holograms

*1*if (ξ,,η) belongs to the object

*0*elsewhere

_{0}from the object can be expressed as the following convolution product:

_{0}>> π

*d*

^{2}/λ where d is the object diameter, this term can be treated as a constant and can be neglected [5

5. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. **36**, 1947–1951 (1997) [CrossRef]

*a*by:

_{r}can be seen as the WT of the intensity distribution recorded by the photosensitive plane :

*WT*. When

_{Iz0}(a_{r},x, y)*a*=

_{r}*a*(

_{0}*i.e.*the interrogation plane located at a distance

*z*corresponds to the object plane located at a distance

_{r}*z*), it can be shown that by dropping a multiplicative constant:

_{0}*2*from the reconstruction plane. More details can be found in Ref. [5

_{z0}5. S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. **36**, 1947–1951 (1997) [CrossRef]

## 3. Digital reconstruction in a tilted plane

*a*in Eq. (5).

_{r}_{0}with respect to the η direction (see Fig. 2).

_{0}from the CCD camera. If we assume that the diffraction pattern produced by the fiber is the contribution given by each fiber element, the intensity distribution in the diffraction pattern I(x,y) is given by Eq. (1) where

*z*=

_{0}*D*+

_{0}*ytanθ*. D

_{0}_{0}is the distance between the fiber and the camera for y=

_{0}.

*I(x,y)*produced by a fiber of diameter

*d=60 µm*which is illuminated by a plane wave of wavelength : λ=0.6328 µm and where

*θ*=

_{0}*84,2°*. This figure consists of a 512×512 array of 256 gray level pixels. The parameters

*d*and θ

_{0}have been chosen in order to highlight the variations of the diffraction pattern along

*y*axis. On this example, z

_{0}lies in the interval [50, 95 mm]. As we can see here, the width of the central fringe grows as the distance

*z*increases. Figure 3b shows the image obtained by applying to the diffraction pattern

_{0}*I(x,y)*the reconstruction formulae given by Eq. (5) (

*WT*) and for

_{I}(a_{r}, x,y)*z*=

_{r}*D*. Here, except for y=0, the condition

_{0}*z*=

_{r}*z*is not satisfied. Consequently, the reconstruction process seen as a wavelet transform is only valid for the pixels located on the line y=0. As we can see on Fig. 3b,

_{0}*WT*leads to a satisfactory reconstruction only in the central region of the figure. This focusing process reveals itself by a narrowing of the central fringe. Observe that the image is in focus along 1 mm in the

_{I}(a_{r}, x,y)*y*direction. Consequently, the reconstructed images presented in this paper represent a slice rather than a plane. The thickness of a given slice depends on the depth of field given by the numerical reconstruction (see Ref. [9

9. S. Coëtmellec, C. Buraga-Lefebvre, D. Lebrun, and C. Özkul, “Application of in-line digital holography to multiple plane velocimetry,” Meas. Sci and Tech. **12**, 1392–1397 (2001) [CrossRef]

_{0}=D,

*WT*should be computed with a scale parameter in accordance with the reconstruction in another plane. Thus, by combining all the reconstructed images, the totality of the fiber portion corresponding to the field seen by the camera could be analyzed. These operations can be synthesized in the following two-step process : 1 - Reconstruction plane by plane of the whole volume for different values of z

_{I}(a_{r}, x, y)_{r}, 2 - Rebuilding of the intensity in a tilted direction by selecting within a given z

_{r}-plane the pixels belonging to this plane (i.e we hold in a given z

_{r}-plane only the pixels whose coordinates verify : z

_{r}=D

_{0}+ ytanθ

_{0}). An example of reconstruction is given in Fig. 3c. Observe that the lateral fringes due to the twin image move away as z

_{r}increases.

## 4. Experimental results

_{0}=67° with respect to the y direction. The central fiber element is located at a distance D

_{0}=40 mm from the CCD sensor. Figure 4b is obtained by applying the digital reconstruction formula given by the Eq. (5) for z

_{r}=50 mm. Note that, as is the case for the numerical simulations, the image of the fiber is only focused in a small region of the image (top of the figure). The image in Fig. 4c has been calculated by applying a numerical reconstruction process to the whole volume contained between z

_{r}=30 mm and z

_{r}=50 mm. Afterwards, we selected only the pixels that verify z

_{r}=D

_{0}+ ytanθ

_{0}. The fiber is well reconstructed within the whole field of view but the contrast at the center of the image decreases for the smallest values of z

_{r}. This can be explained by the windowing function applied on the daughter wavelet

*ψ*for the reconstruction. Indeed, for an object located near the CCD camera,

_{ar}(x, y)*a*takes smaller values, so the windowing function must be narrowed enough to verify the spatial sampling condition (see Ref. 5

_{r}**36**, 1947–1951 (1997) [CrossRef]

_{r}=120 mm. We can see that one of the particle images indicated by dashed arrows is in focus (upper left corner) whereas the other image at the bottom of the figure is defocused. The image shown in Fig. 6c has been obtained by adjusting the two parameters (D

_{r},θ

_{r}) such that these particle images are both focused. We found D

_{r}=100 mm and θ

_{r}=76°. Note that this example exhibits a singular case where nearly all the droplet images belong to the same slice.

## 5. Conclusion

## References and links

1. | T.M. Kreis and W.P.O. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. |

2. | M.K. Kim, “Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography,” Opt. Express |

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

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

5. | S. Belaïd, D. Lebrun, and C. Özkul, “Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. |

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

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

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

9. | S. Coëtmellec, C. Buraga-Lefebvre, D. Lebrun, and C. Özkul, “Application of in-line digital holography to multiple plane velocimetry,” Meas. Sci and Tech. |

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

11. | L. Yu, Y. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express |

**OCIS Codes**

(090.0090) Holography : Holography

(090.1760) Holography : Computer holography

(100.3010) Image processing : Image reconstruction techniques

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 6, 2003

Revised Manuscript: January 31, 2003

Published: February 10, 2003

**Citation**

Denis Lebrun, A. Benkouider, S. Coëtmellec, and M. Malek, "Particle field digital holographic reconstruction in arbitrary tilted planes," Opt. Express **11**, 224-229 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-3-224

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### References

- T.M. Kreis, W.P.O. Jüptner, "Suppression of the dc term in digital holography," Opt. Eng. 36, 2357-2360 (1997). [CrossRef]
- M.K. Kim, "Tomographic three-dimensional imaging of a biological specimen using wavelength-scanning digital interference holography," Opt. Express 7, 305-310, (2000) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-9-305</a>. [CrossRef] [PubMed]
- O. Schnars and W. Juptner, �??Direct recording of holograms by a CCD target and numerical reconstruction,�?? Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
- Yamaguchi and T. Zhang, �??Phase-shifting digital holography,�?? Opt. Lett. 22, 1268-1270, (1997) [CrossRef] [PubMed]
- S. Belaïd, D. Lebrun and C. �?zkul, "Application of two dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame," Opt. Eng. 36, 1947-1951 (1997). [CrossRef]
- C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun and C. �?zkul, "Application of wavelet transform to hologram analysis : three-dimensional location of particles," Opt. Lasers Eng. 33, 409-421 (2000). [CrossRef]
- L. Onural, "Diffraction from a wavelet point of view," Opt. Lett. 18, 846-848, (1993). [CrossRef] [PubMed]
- S. Coëtmellec, D. Lebrun and C. �?zkul, "Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier Transform," Appl. Opt. 41, 312-319 (2002). [CrossRef]
- S. Coëtmellec, C. Buraga-Lefebvre, D. Lebrun and C. �?zkul, "Application of in-line digital holography to multiple plane velocimetry," Meas. Sci. Tech. 12, 1392-1397 (2001). [CrossRef]
- S. Coëtmellec, D. Lebrun and C. �?zkul, "Application of two-dimensional fractional-order Fourier transformation to particle field digital holography," J. Opt. Soc. Am. A. 19, 1537-1546 (2002). [CrossRef]
- L. Yu, Y. An and L. Cai, �??Numerical reconstruction of digital holograms with variable viewing angles,�?? Opt. Express 10, 1250-1257, (2002) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1250">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1250</a>. [CrossRef] [PubMed]

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