## Modified convolution method to reconstruct particle hologram with an elliptical Gaussian beam illumination |

Optics Express, Vol. 21, Issue 10, pp. 12803-12814 (2013)

http://dx.doi.org/10.1364/OE.21.012803

Acrobat PDF (2802 KB)

### Abstract

Application of the modified convolution method to reconstruct digital inline holography of particle illuminated by an elliptical Gaussian beam is investigated. Based on the analysis on the formation of particle hologram using the Collins formula, the convolution method is modified to compensate the astigmatism by adding two scaling factors. Both simulated and experimental holograms of transparent droplets and opaque particles are used to test the algorithm, and the reconstructed images are compared with that using FRFT reconstruction. Results show that the modified convolution method can accurately reconstruct the particle image. This method has an advantage that the reconstructed images in different depth positions have the same size and resolution with the hologram. This work shows that digital inline holography has great potential in particle diagnostics in curvature containers.

© 2013 OSA

## 1. Introduction

1. T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Digital in-line holography in particle measurement,” in Interferometry ’99: Techniques and Technologies, SPIE **3744**, 54–64 (1999) [CrossRef] .

2. J. Lu, R. A. Shaw, and W. Yang, “Improved particle size estimation in digital holography via sign matched filtering,” Opt. Express **20**, 12666–12674 (2012) [CrossRef] [PubMed] .

1. T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Digital in-line holography in particle measurement,” in Interferometry ’99: Techniques and Technologies, SPIE **3744**, 54–64 (1999) [CrossRef] .

3. L. Wilson and R. Zhang, “3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation,” Opt. Express **20**, 16735–16744 (2012) [CrossRef] .

4. G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng. **43**, 1039–1055 (2005) [CrossRef] .

5. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009) [CrossRef] [PubMed] .

6. J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. **42**, 531–555 (2010) [CrossRef] .

7. Y. Yang and B. Kang, “Measurements of the characteristics of spray droplets using in-line digital particle holography,” J. Mech. Sci. Technol. **23**, 1670–1679 (2009) [CrossRef] .

8. Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. **50**, H22–H29 (2011) [CrossRef] [PubMed] .

9. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express **14**, 5895–5908 (2006) [CrossRef] [PubMed] .

10. L. Tian, N. Loomis, J. A. Domíanguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt. **49**, 1549–1554 (2010) [CrossRef] [PubMed] .

5. F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express **17**, 13071–13079 (2009) [CrossRef] [PubMed] .

11. E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, T. J. Naughton, and A. K. Asundi, “Microparticle characterization using digital holography,” Chem. Eng. Sci. **65**, 1037–1044 (2010) [CrossRef] .

13. M. DaneshPanah and B. Javidi, “Tracking biological microorganisms in sequence of 3D holographic microscopy images,” Opt. Express **15**, 10761–10766 (2007) [CrossRef] [PubMed] .

1. T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Digital in-line holography in particle measurement,” in Interferometry ’99: Techniques and Technologies, SPIE **3744**, 54–64 (1999) [CrossRef] .

14. S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express **9**, 294–302 (2001) [CrossRef] [PubMed] .

15. N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. **47**, 4147–4157 (2008) [CrossRef] [PubMed] .

16. N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express **18**, 7807–7819 (2010) [CrossRef] [PubMed] .

15. N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. **47**, 4147–4157 (2008) [CrossRef] [PubMed] .

17. J. Crane, P. Dunn, B. J. Thompson, J. Knapp, and J. Zeiss, “Far-field holography of ampule contaminants,” Appl. Opt. **21**, 2548–2553 (1982) [CrossRef] [PubMed] .

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

15. N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. **47**, 4147–4157 (2008) [CrossRef] [PubMed] .

16. N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express **18**, 7807–7819 (2010) [CrossRef] [PubMed] .

20. 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**, 1137–1148 (2012) [CrossRef] [PubMed] .

**3744**, 54–64 (1999) [CrossRef] .

24. D. Lebrun, S. Belad, and C. zkul, “Hologram Reconstruction by use of Optical Wavelet Transform,” Appl. Opt. **38**, 3730–3734 (1999) [CrossRef] .

17. J. Crane, P. Dunn, B. J. Thompson, J. Knapp, and J. Zeiss, “Far-field holography of ampule contaminants,” Appl. Opt. **21**, 2548–2553 (1982) [CrossRef] [PubMed] .

18. C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng. **25**, 251189 (1986) [CrossRef] .

14. S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express **9**, 294–302 (2001) [CrossRef] [PubMed] .

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

21. F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A **22**, 2569–2577 (2005) [CrossRef] .

25. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express **13**, 9935–9940 (2005) [CrossRef] [PubMed] .

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

21. F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A **22**, 2569–2577 (2005) [CrossRef] .

**47**, 4147–4157 (2008) [CrossRef] [PubMed] .

16. N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express **18**, 7807–7819 (2010) [CrossRef] [PubMed] .

22. N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A **25**, 1459–1466 (2008) [CrossRef] .

26. M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, “Digital phase contrast with the fractional Fourier transform,” Appl. Opt. **48**, 579–583 (2009) [CrossRef] [PubMed] .

## 2. Hologram formation

29. G. Gouesbet and G. Gréhan, *Generalized Lorenz-Mie Theories* (Springer, 2011) [CrossRef] .

29. G. Gouesbet and G. Gréhan, *Generalized Lorenz-Mie Theories* (Springer, 2011) [CrossRef] .

30. J. Collins and A. Stuart, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A **60**, 1168–1177 (1970) [CrossRef] .

*u*,

*v*) is where

*ω*

_{x1,y1},

*R*

_{x1,y1}, being the beam radius and wave curvature in the

*x*and

*y*directions respectively.

*A*,

_{x,y}*B*,

_{x,y}*D*are the elements of the ABCD transform matrix

_{x,y}*x*and

*y*directions.

31. J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. **83**, 1752 (1988) [CrossRef] .

*R*is

*R*is still an elliptical Gaussian beam, and the objective wave

*O*is

*C*=

_{obj}*π*.

*R · R̄*and

*O · Ō*are the directly transmitted light and cannot be used to reconstruct the object. The

*O·R̄*and

*R·Ō*determine the interference pattern and can be used to reconstruct the virtual and real image of the object respectively.

## 3. Modified convolution reconstruction

*O · R̄*and

*R · Ō*

*z*

_{(x,y),eq}=

*B*

_{(x,y)}·

*R*

_{(x,y)}can be treated as the equivalent propagation distance from the particle to the CCD, which determines the frequency of the fringes in

*x*and

*y*directions in the hologram. For a plane wave propagating through the flat surface,

*ω*

_{x1,y1}→ ∞,

*R*

_{x1,y1}→ ∞, the

*R*= 1, and

_{x,y}*z*

_{(x,y),eq}equal to the wave propagation distance

*z*. The hologram can be reconstructed with the classical methods. For a circular Gaussian beam,

*R*equals to

_{x}*R*but not 1, the

_{y}*z*=

_{x,eq}*z*with the same magnifications in both

_{y,eq}*x*and

*y*directions. The fringe patterns in the hologram of a spherical particle are still composed of concentric rings. The reconstructed optical field using the classical methods should be rescaled to take the magnifications into account. For an elliptical Gaussian beam, the

*R*≠

_{x}*R*, and the

_{y}*B*≠

_{x}*B*; it can be interpreted that the beam has propagated different equivalent distances from the particle to the CCD. This results in the different magnifications and the spatial frequency of the fringes in the transverse directions. Thus, the hologram cannot be simultaneously focused in both

_{y}*x*and

*y*directions at a certain depth position with the classical methods for reconstruction. Figure 2 shows a simulated hologram of a spherical particle illuminated by an elliptical Gaussian beam, and the image slices reconstructed using the classical convolution method. The object is focused at

*z*= 7.2 cm in the

*x*direction (in Fig. 2(b)) and at

*z*= 4.0 cm in the

*y*direction (in Fig. 2(c)). However, no clear particle image can be observed in the two reconstructed images.

*G*(

_{z}*u*,

*v*) =

*F*(

*g*) can be obtained by direct calculation of the analytic expansion [28

28. T. Kreis, “Digital Recording and Numerical Reconstruction of Wave Fields,” in *Handbook of Holographic Interferometry* (Wiley-VCH Verlag GmbH & Co. KGaA, 2005), pp. 81–183 [CrossRef] .

*I*(

_{re}*x*,

*y*) =

*F*

^{−1}[

*F*(

*I*) ·

_{holo}*G*], saving one Fourier transform.

## 4. Simulated holograms

29. G. Gouesbet and G. Gréhan, *Generalized Lorenz-Mie Theories* (Springer, 2011) [CrossRef] .

*ω*

_{x0}= 2

*μ*m and

*ω*

_{y0}= 3 mm in the

*x*and

*y*directions respectively. The particle with a diameter of 140

*μ*m located at

*z*= 5.0 cm. The wave radii of the beam at the depth position of the particle were 4.2 mm and 3.0 mm, with the wave curvature of 5 cm and 5.65 × 10

^{4}m in the

*x*and

*y*directions respectively. Detailed parameters of the simulated hologram are given in Table 1. The hologram is characterized with elliptical fringes. The hologram was reconstructed through using the modified convolution method, with the

*z*= 4.0 cm, as shown in Fig. 3(b). The hologram was also reconstructed by using fractional Fourier transform (FRFT) for comparison. The particle was focused at optional fractional orders

*α*= 0.743,

_{x}*α*= 0.582 in FRFT. The scaling factors

_{y}*S*and optimal fractional orders

_{x,y}*α*satisfy the relation in Eq. (11). By rescaling the reconstructed image to make its resolution equaling to the recorded hologram, Fig. 3(c) shows the selected ROI region of the reconstructed image. Comparisons between the reconstructed images in Fig. 3(b) and Fig. 3(c) show that they are the same.

_{x,y}*z*= −2.3 cm, and the detailed simulation parameters are listed in Table 1. Hyperbolic fringes can be observed in the hologram. The in-focus image of the particle was reconstructed at

*z*= 7.3 cm using the modified convolution method with

*α*= −0.878,

_{x}*α*= 0.746 using fractional Fourier transform, as shown in Fig. 4(b) and Fig. 4(c) respectively.

_{y}20. 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**, 1137–1148 (2012) [CrossRef] [PubMed] .

22. N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A **25**, 1459–1466 (2008) [CrossRef] .

*x*and

*y*directions. For a spherical particle, the cross-section of the reconstructed in-focus image might be elliptical, as shown in Fig. 2–7. Table 1 compares the ratio of simulated particle size with its reconstructed particle size, and results agree well with Eq. (12). In the far field, both the beam radius and wave curvature increase linearly with the distance, and the wave can be approximately considered as a spherical wave centered at the beam waist. The transverse shift can be evaluated according to geometric optics [22

22. N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A **25**, 1459–1466 (2008) [CrossRef] .

## 5. Experiments

*x*and

*y*direction respectively. Holograms of particles were recorded by the CCD (LaVision ImagePro) with a squared pixel size of 7.4

*μ*m. Two particle fields were used: one was the sprayed water droplets, and the other was coal particles.

## 6. Parallel fringes

*x*) direction, while not focused in the vertical (

*y*) direction. Supposing a particle locating at or very near to the beam waist of the elliptical Gaussian beam, From Eq. (5), in the

*x*direction, the

*ω*

_{x1}and the particle diameter

*d*are about at the same size order of magnitude, and the radius of curvature

*R*

_{x1}→ ∞. Thus the

*R*is much larger than 1 (up to 10

_{x}^{3}or larger), and the equivalent propagation distance

*z*

_{(x),eq}far greater than the actual propagation distance

*z*. In the

*y*direction, while

*ω*

_{y1}≫

*d*, and the radius of curvature

*R*

_{y1}→ ∞. Thus the

*R*≃ 1, and

_{y}*z*

_{(y),eq}=

*z*. From the quadratic phase term in Eq. (5), the frequency of the fringes in the

*x*direction is not only much smaller than that in the

*y*direction, but also so small that the CCD could not even sample a period. Therefore, the recorded hologram fringes of a particle are composed of parallel fringes. Figure 10(a) and 10(b) shows the simulated and experimental holograms with parallel fringes.

*S*= 1, as shown in Fig. 10(c) and 10(d). The focused particle image is distorted to a line, due to the huge magnification in the

_{x,y}*x*direction.

## 7. Conclusion

*S*and

_{x}*S*are introduced to compensate the different spatial frequencies of the fringe patterns in the

_{y}*x*and

*y*directions caused by the astigmatism. This algorithm is validated by both simulated and experimental holograms of transparent droplets and opaque particles. The reconstructed images with the modified convolution method are compared with those using FRFT reconstruction, and results show that the modified convolution method can accurately reconstruct the particle image. Transverse shift and magnification of the hologram by an elliptical Gaussian beam are also discussed. This method has the advantage that both the size and resolution of the reconstructed image equal to the recorded hologram, and thus the reconstructed images can be easily extended for post-processing.

## Acknowledgments

## References and links

1. | T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Digital in-line holography in particle measurement,” in Interferometry ’99: Techniques and Technologies, SPIE |

2. | J. Lu, R. A. Shaw, and W. Yang, “Improved particle size estimation in digital holography via sign matched filtering,” Opt. Express |

3. | L. Wilson and R. Zhang, “3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation,” Opt. Express |

4. | G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng. |

5. | F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express |

6. | J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. |

7. | Y. Yang and B. Kang, “Measurements of the characteristics of spray droplets using in-line digital particle holography,” J. Mech. Sci. Technol. |

8. | Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. |

9. | F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express |

10. | L. Tian, N. Loomis, J. A. Domíanguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt. |

11. | E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, T. J. Naughton, and A. K. Asundi, “Microparticle characterization using digital holography,” Chem. Eng. Sci. |

12. | L. L. Taixé, M. Heydt, A. Rosenhahn, and B. Rosenhahn, “ |

13. | M. DaneshPanah and B. Javidi, “Tracking biological microorganisms in sequence of 3D holographic microscopy images,” Opt. Express |

14. | S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express |

15. | N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt. |

16. | N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express |

17. | J. Crane, P. Dunn, B. J. Thompson, J. Knapp, and J. Zeiss, “Far-field holography of ampule contaminants,” Appl. Opt. |

18. | C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng. |

19. | S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. |

20. | 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. |

21. | F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A |

22. | N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A |

23. | Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “ |

24. | D. Lebrun, S. Belad, and C. zkul, “Hologram Reconstruction by use of Optical Wavelet Transform,” Appl. Opt. |

25. | S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express |

26. | M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, “Digital phase contrast with the fractional Fourier transform,” Appl. Opt. |

27. | U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. |

28. | T. Kreis, “Digital Recording and Numerical Reconstruction of Wave Fields,” in |

29. | G. Gouesbet and G. Gréhan, |

30. | J. Collins and A. Stuart, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A |

31. | J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. |

**OCIS Codes**

(090.1000) Holography : Aberration compensation

(090.1995) Holography : Digital holography

**ToC Category:**

Holography

**History**

Original Manuscript: March 6, 2013

Revised Manuscript: April 18, 2013

Manuscript Accepted: April 26, 2013

Published: May 17, 2013

**Citation**

Xuecheng Wu, Yingchun Wu, Jing Yang, Zhihua Wang, Binwu Zhou, Gérard Gréhan, and Kefa Cen, "Modified convolution method to reconstruct particle hologram with an elliptical Gaussian beam illumination," Opt. Express **21**, 12803-12814 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12803

Sort: Year | Journal | Reset

### References

- T. M. Kreis, M. Adams, and W. P. O. Jueptner, “Digital in-line holography in particle measurement,” in Interferometry ’99: Techniques and Technologies, SPIE3744, 54–64 (1999). [CrossRef]
- J. Lu, R. A. Shaw, and W. Yang, “Improved particle size estimation in digital holography via sign matched filtering,” Opt. Express20, 12666–12674 (2012). [CrossRef] [PubMed]
- L. Wilson and R. Zhang, “3D Localization of weak scatterers in digital holographic microscopy using Rayleigh-Sommerfeld back-propagation,” Opt. Express20, 16735–16744 (2012). [CrossRef]
- G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng.43, 1039–1055 (2005). [CrossRef]
- F. C. Cheong, B. Sun, R. Dreyfus, J. Amato-Grill, K. Xiao, L. Dixon, and D. G. Grier, “Flow visualization and flow cytometry with holographic video microscopy,” Opt. Express17, 13071–13079 (2009). [CrossRef] [PubMed]
- J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech.42, 531–555 (2010). [CrossRef]
- Y. Yang and B. Kang, “Measurements of the characteristics of spray droplets using in-line digital particle holography,” J. Mech. Sci. Technol.23, 1670–1679 (2009). [CrossRef]
- Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt.50, H22–H29 (2011). [CrossRef] [PubMed]
- F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express14, 5895–5908 (2006). [CrossRef] [PubMed]
- L. Tian, N. Loomis, J. A. Domíanguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt.49, 1549–1554 (2010). [CrossRef] [PubMed]
- E. Darakis, T. Khanam, A. Rajendran, V. Kariwala, T. J. Naughton, and A. K. Asundi, “Microparticle characterization using digital holography,” Chem. Eng. Sci.65, 1037–1044 (2010). [CrossRef]
- L. L. Taixé, M. Heydt, A. Rosenhahn, and B. Rosenhahn, “Automatic tracking of swimming microorganisms in 4D digital in-line holography data,” in (IEEE, 2009), pp. 1–8.
- M. DaneshPanah and B. Javidi, “Tracking biological microorganisms in sequence of 3D holographic microscopy images,” Opt. Express15, 10761–10766 (2007). [CrossRef] [PubMed]
- S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express9, 294–302 (2001). [CrossRef] [PubMed]
- N. Verrier, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography in thick optical systems: application to visualization in pipes,” Appl. Opt.47, 4147–4157 (2008). [CrossRef] [PubMed]
- N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express18, 7807–7819 (2010). [CrossRef] [PubMed]
- J. Crane, P. Dunn, B. J. Thompson, J. Knapp, and J. Zeiss, “Far-field holography of ampule contaminants,” Appl. Opt.21, 2548–2553 (1982). [CrossRef] [PubMed]
- C. Vikram and M. Billet, “Fraunhofer holography in cylindrical tunnels: neutralizing window curvature effects,” Opt. Eng.25, 251189 (1986). [CrossRef]
- S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett.26, 974–976 (2001). [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, 1137–1148 (2012). [CrossRef] [PubMed]
- F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A22, 2569–2577 (2005). [CrossRef]
- N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A25, 1459–1466 (2008). [CrossRef]
- Y. Yuan, K. Ren, S. Coëtmellec, and D. Lebrun, “Rigorous description of holograms of particles illuminated by an astigmatic elliptical Gaussian beam,” in (IOP Publishing, 2009), 012052.
- D. Lebrun, S. Belad, and C. zkul, “Hologram Reconstruction by use of Optical Wavelet Transform,” Appl. Opt.38, 3730–3734 (1999). [CrossRef]
- S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express13, 9935–9940 (2005). [CrossRef] [PubMed]
- M. Brunel, S. Coëtmellec, D. Lebrun, and K. A. Ameur, “Digital phase contrast with the fractional Fourier transform,” Appl. Opt.48, 579–583 (2009). [CrossRef] [PubMed]
- U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13, R85–R101 (2002). [CrossRef]
- T. Kreis, “Digital Recording and Numerical Reconstruction of Wave Fields,” in Handbook of Holographic Interferometry (Wiley-VCH Verlag GmbH & Co. KGaA, 2005), pp. 81–183. [CrossRef]
- G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories (Springer, 2011). [CrossRef]
- J. Collins and A. Stuart, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A60, 1168–1177 (1970). [CrossRef]
- J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am.83, 1752 (1988). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.