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

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References

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    [CrossRef] [PubMed]
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    [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. Express17, 13071–13079 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13, R85–R101 (2002).
    [CrossRef]
  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]
  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. A60, 1168–1177 (1970).
    [CrossRef]
  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]

2012

2011

2010

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]

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

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]

2009

2008

2007

2006

2005

2002

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13, R85–R101 (2002).
[CrossRef]

2001

1999

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

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]

1988

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

1986

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

1982

1970

J. Collins and A. Stuart, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A60, 1168–1177 (1970).
[CrossRef]

Adams, M.

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]

Alfieri, D.

Allano, D.

Amato-Grill, J.

Ameur, K. A.

Asundi, A. K.

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]

Barbastathis, G.

Belad, S.

Billet, M.

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

Breazeale, M.

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

Brunel, M.

Callens, N.

Cen, K.

Chen, L.

Cheong, F. C.

Coëtmellec, S.

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]

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]

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]

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]

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]

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]

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.

Collins, J.

J. Collins and A. Stuart, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A60, 1168–1177 (1970).
[CrossRef]

Crane, J.

DaneshPanah, M.

Darakis, E.

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]

De Nicola, S.

Dixon, L.

Domíanguez-Caballero, J. A.

Dreyfus, R.

Dubois, F.

Dunn, P.

Ferraro, P.

Finizio, A.

Gouesbet, G.

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

Gréhan, G.

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

Grier, D. G.

Grilli, S.

Heydt, M.

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.

Janssen, A. J.

Janssen, A. J. E. M.

Javidi, B.

Jueptner, W. P. O.

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]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13, R85–R101 (2002).
[CrossRef]

Kang, B.

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]

Kariwala, V.

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]

Katz, J.

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

Khanam, T.

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]

Knapp, J.

Kreis, T.

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]

Kreis, T. M.

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]

Lebrun, D.

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]

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]

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]

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]

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]

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]

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

Loomis, N.

Lu, J.

Meucci, R.

Naughton, T. J.

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]

Nicolas, F.

Pierattini, G.

Rajendran, A.

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]

Remacha, C.

Ren, K.

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.

Rosenhahn, A.

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.

Rosenhahn, B.

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.

Schnars, U.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13, R85–R101 (2002).
[CrossRef]

Schockaert, C.

Shaw, R. A.

Shen, G.

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]

Shen, H.

Sheng, J.

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

Stuart, A.

J. Collins and A. Stuart, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A60, 1168–1177 (1970).
[CrossRef]

Sun, B.

Taixé, L. L.

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.

Thompson, B. J.

Tian, L.

Verrier, N.

Vikram, C.

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

Wang, Z.

Wei, R.

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]

Wen, J.

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

Wilson, L.

Wu, X.

Wu, Y.

Xiao, K.

Yang, W.

Yang, Y.

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]

Yourassowsky, C.

Yuan, Y.

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.

Zeiss, J.

Zhang, R.

zkul, C.

Annu. Rev. Fluid Mech.

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

Appl. Opt.

Chem. Eng. Sci.

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]

Interferometry ’99: Techniques and Technologies, SPIE

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. Acoust. Soc. Am.

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

J. Mech. Sci. Technol.

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]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

U. Schnars and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol.13, R85–R101 (2002).
[CrossRef]

Opt. Eng.

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

Opt. Express

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]

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]

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]

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]

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]

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]

Opt. Lasers Eng.

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]

Opt. Lett.

Other

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.

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]

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.

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Figures (10)

Fig. 1
Fig. 1

Schematics of digital particle holography with elliptical Gaussian beam illumination.

Fig. 2
Fig. 2

Hologram of a particle with elliptical Gaussian beam illumination and its reconstructed images with the classical convolution method. (a. simulated hologram, b. reconstructed image at z = 7.2cm, c. reconstructed image at z = 4.0cm.)

Fig. 3
Fig. 3

Particle hologram and its reconstructed image using modified convolution and FRFT. (a. simulated hologram, b. convolution reconstruction with S x 2 = 0.556, S y 2 = 1, c. FRFT reconstruction with optimal fractional orders αx = 0.743, αy = 0.582).)

Fig. 4
Fig. 4

Particle hologram and its reconstructed image using modified convolution and FRFT. (a. simulated hologram, b. convolution reconstruction with S x 2 = 0.460, S y 2 = 1, c. FRFT reconstruction with optimal fractional orders αx = −0.878, αy = 0.746).)

Fig. 5
Fig. 5

Hologram of droplet and its reconstructed image using modified convolution and FRFT. (a. simulated hologram, b. convolution reconstruction with S x 2 = 0.540, S y 2 = 1, c. FRFT reconstruction with optimal fractional orders αx = 0.601, αy = 0.408).)

Fig. 6
Fig. 6

Hologram of droplet and its reconstructed image using modified convolution and FRFT. (a. simulated hologram, b. convolution reconstruction with S x 2 = 0.540, S y 2 = 1, c. FRFT reconstruction with optimal fractional orders αx = −0.865, αy = 0.758).)

Fig. 7
Fig. 7

Effect of magnification and position shift. (a. hologram of a cloud of particles with elliptical Gaussian beam illumination, b. comparison of the 3D position and cross-section between the simulated and reconstructed particles.)

Fig. 8
Fig. 8

Experimental holograms with typical elliptical fringes, hyperbolic fringes and parallel line fringes. (Water droplet located before (a) and after (b) the beam waist, Coal particle located before (c) and after (d) the beam waist.)

Fig. 9
Fig. 9

Reconstructed image holograms in Fig. 8 using modified convolution and fractional Fourier transform (a. modified convolution reconstruction of Fig. 8(a) with S x 2 = 0.625, S y 2 = 1, z=51.85mm, b. FRFT reconstruction of Fig. 8(a) with optimal fractional orders αx = 0.478, αy = 0.336, c. modified convolution reconstruction of Fig. 8(b) with S x 2 = 0.693, S y 2 = 1, z=231.2mm, d. FRFT reconstruction of Fig. 8(b) with optimal fractional orders αx = −0.834, αy = 0.736. e. modified convolution reconstruction of Fig. 8(c) with S x 2 = 0.392, S y 2 = 1, z=96.6mm, f. FRFT reconstruction of Fig. 8(c) with optimal fractional orders αx = −0.785, αy = 0.535, h. modified convolution reconstruction of Fig. 8(d) with S x 2 = 0.324, S y 2 = 1, z=206mm, i. FRFT reconstruction of Fig. 8(d) with optimal fractional orders αx = −0.915, αy = 0.750)

Fig. 10
Fig. 10

Holograms with parallel fringes. (a. simulated hologram with parallel line fringes, with ωx0 = 16μm, ωy0 = 3mm, particle diameter d = 10μm, particle located at the beam waist center, CCD position zCCD = 5.0cm, b. experimental hologram of water droplet with parallel line fringes, c. reconstructed image of Fig. 10(a) at z = 5.0cm, d. reconstructed image of Fig. 10(b) at z = 134.0mm.)

Tables (1)

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Table 1 Detailed parameters of simulated holograms, ωx0=2μm, ωy0=3×10−3mm in x and y directions, CCD pixel=4μm. (particle diameter: d /μm, refractive index: n, CCD position: zCCD /cm, wave curvature: Rx1 /cm and Ry1/m, beam radius at the particle: ωx1 /mm and ωy1 /mm, beam radius at the CCD: ωx,CCD /mm and ωy,CCD /mm, reconstructed particle diameter: dx and dy /μm.)

Equations (12)

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U ( u , v ) = G 1 ( x , y ) [ 1 T ( x , y ) ] exp [ i π λ B x ( A x u 2 2 u x + D x x 2 ) ] exp [ i π λ B y ( A y v 2 2 v y + D y y 2 ) ] d x d y .
R ( u , v ) = G 1 ( x , y ) exp [ i π λ B x ( A x u 2 2 u x + D x x 2 ) ] exp [ i π λ B y ( A y v 2 2 v y + D y y 2 ) ] d x d y = C ref exp ( i π D x x 2 λ B x + i π D y y 2 λ B y ) exp [ π x 2 λ B x ( λ B x π ω x 1 2 + i B x R x 1 i A x ) + π y 2 λ B y ( λ B y π ω y 1 2 + i B y R y 1 i A y ) ]
O ( u , v ) = G 1 ( x , y ) T ( x , y ) exp [ i π λ B x ( A x u 2 2 u x + D x x 2 ) ] exp [ i π λ B y ( A y v 2 2 v y + D y y 2 ) ] d x d y = C obj exp ( i π D x x 2 λ B x + i π D y y 2 λ B y ) k = 1 10 { A k exp [ π x 2 λ B x ( λ B x π ω x 1 2 + i B x R x 1 i A x λ B x B k π r 2 ) + π y 2 λ B y ( λ B y π ω y 1 2 + i B y R y 1 i A y λ B y B k π r 2 ) ] ( λ B x π ω x 1 2 + i B x R x 1 i A x λ B x B k π r 2 ) ( λ B y π ω y 1 2 + i B y R y 1 i A y λ B y B k π r 2 )
I holo ( u , v ) = U ( u , v ) U ( u , v ) ¯ = R R ¯ + O O ¯ + R O ¯ + O R ¯
arg ( O R ¯ + R O ¯ ) = [ π x 2 λ B x ( λ B x π ω x 1 2 + i B x R x 1 i A x ) + π y 2 λ B y ( λ B y π ω y 1 2 + i B y R y 1 i A y ) ] = π x 2 λ B x R x + π y 2 λ B y R y = π x 2 λ z B x R x z + π y 2 λ z B y R y z = π x 2 λ z x , e q + π y 2 λ z y , eq
g ( x , y , u , v ) = i exp ( i k S x 2 ( x u ) 2 + S y 2 ( y v ) 2 + z 2 ) λ S x 2 ( x u ) 2 + S y 2 ( y v ) 2 + z 2
S x 2 = z B x R x , S y 2 = z B y R y .
I re ( x , y ) = I holo ( x , y ) g ( x , y , u , v ) = F 1 [ F ( I holo ) F ( g ) ]
G z ( u , v ) = { exp ( i 2 π z ) 1 λ 2 u 2 S x 2 v 2 S y 2 ) , 1 λ 2 > u 2 S x 2 v 2 S y 2 0 , otherwise
α x , y = 2 π arctan ( λ B x , y N x , y δ x , y 2 ( M x , y D x , y ) )
S x , y 2 tan π α x , y 2 = λ z ( x , y ) , eq N x , y δ x , y 2 = constant
M a g ( x , y ) = ω ( x , y ) , C C D ω x 1 , y 1

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