Abstract

The field of view of digital in-line holography for flow field diagnostics is restricted to a small volume due to the finite size and the low spatial resolution of the available CCD. Expansion of the measurement cross section of digital holographic particle image velocimetry was investigated with a lens-based holography configuration. By sampling the chirp signal in the center lobe completely and undersampling the chirp signal in the second- and higher-order lobes by a magnified virtual recording plane produced by an imag ing camera lens, the field of view is expanded. Simulation results show that the three-dimensional (3D) location and size of the relatively large particle can be reconstructed with good accuracy. A digital holographic particle image velocimetry system was established for coal particle flow field diagnostics. Compared with the lensless configuration, the field of view of the digital holography system was enlarged 1.9 times, up to 2.78cm×2.78cm×3cm. The 3D location, size distribution, and the 3D vector field of coal powder were obtained. The results show that the application of digital in-line holography to measure large particle flow field is feasible.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
    [CrossRef]
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    [CrossRef]
  5. Y.-J. Choo and B.-S. Kang, “Measurements of three-dimensional velocities of spray droplets using the holographic velocimetry system,” J. Mech. Sci. Tech. 17, 1095–1103 (2003).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. M. Raffel, C. E. Willert, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer Verlag, 1998).

2010

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

2009

2008

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002(2008).
[CrossRef]

L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47, 4501–4508(2008).
[CrossRef] [PubMed]

Y. Yamamoto and T. Uemura, “3D particle measurements by single beam two-views magnified digital in-line holography,” Exp. Fluids 45, 813–821 (2008).
[CrossRef]

2006

2005

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]

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

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

2004

J. Coupland, “Holographic particle image velocimetry: signal recovery from under-sampled CCD data,” Meas. Sci. Technol. 15, 711–717 (2004).
[CrossRef]

J. Müller, V. Kebbel, and W. Jüptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15, 706–710(2004).
[CrossRef]

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

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

2003

Y.-J. Choo and B.-S. Kang, “Measurements of three-dimensional velocities of spray droplets using the holographic velocimetry system,” J. Mech. Sci. Tech. 17, 1095–1103 (2003).

2002

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

R. Konrath, W. Schröder, and W. Limberg, “Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine,” Exp. Fluids 33, 781–793 (2002).
[CrossRef]

2000

Y. Pu and H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197(2000).
[CrossRef]

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

1997

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

1976

G. Tyler and B. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23, 685–700 (1976).
[CrossRef]

Adams, M.

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Allano, D.

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

Amato-Grill, J.

ariri, K.

Atkinson, C.

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002(2008).
[CrossRef]

Buraga-Lefebvre, C.

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

Cao, L.

Cen, K. F.

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

Cheong, F.

Choi, Y. S.

Choo, Y.-J.

Y.-J. Choo and B.-S. Kang, “Measurements of three-dimensional velocities of spray droplets using the holographic velocimetry system,” J. Mech. Sci. Tech. 17, 1095–1103 (2003).

Coetmellec, S.

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

Coupland, J.

J. Coupland, “Holographic particle image velocimetry: signal recovery from under-sampled CCD data,” Meas. Sci. Technol. 15, 711–717 (2004).
[CrossRef]

de Jong, J.

De Souza, R.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Demoli, N.

Dixon, L.

Dreyfus, R.

Goncalves, E.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Grier, D.

Halaq, H.

Jin, H.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

Jueptner, W. P. O.

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[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]

Jüptner, W.

J. Müller, V. Kebbel, and W. Jüptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15, 706–710(2004).
[CrossRef]

Kang, B.-S.

Y.-J. Choo and B.-S. Kang, “Measurements of three-dimensional velocities of spray droplets using the holographic velocimetry system,” J. Mech. Sci. Tech. 17, 1095–1103 (2003).

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]

J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45, 3893–3901 (2006).
[CrossRef] [PubMed]

Kebbel, V.

J. Müller, V. Kebbel, and W. Jüptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15, 706–710(2004).
[CrossRef]

Kompenhans, J.

M. Raffel, C. E. Willert, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer Verlag, 1998).

Konrath, R.

R. Konrath, W. Schröder, and W. Limberg, “Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine,” Exp. Fluids 33, 781–793 (2002).
[CrossRef]

Kreis, T. M.

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Lebrun, D.

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

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

Lee, S. J.

Li, Y.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

Limberg, W.

R. Konrath, W. Schröder, and W. Limberg, “Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine,” Exp. Fluids 33, 781–793 (2002).
[CrossRef]

Ma, L.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

Malek, M.

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

Malkiel, E.

Meng, H.

L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47, 4501–4508(2008).
[CrossRef] [PubMed]

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

Y. Pu and H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197(2000).
[CrossRef]

Müller, J.

J. Müller, V. Kebbel, and W. Jüptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15, 706–710(2004).
[CrossRef]

Ozkul, C.

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

Palacios, D.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Palacios, F.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Pan, G.

L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47, 4501–4508(2008).
[CrossRef] [PubMed]

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

Patte-Rouland, B.

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

Pu, S. L.

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

Pu, Y.

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

Y. Pu and H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197(2000).
[CrossRef]

Raffel, M.

M. Raffel, C. E. Willert, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer Verlag, 1998).

Ricardo, J.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Schnars, U.

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

Schröder, W.

R. Konrath, W. Schröder, and W. Limberg, “Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine,” Exp. Fluids 33, 781–793 (2002).
[CrossRef]

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]

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]

J. Sheng, E. Malkiel, and J. Katz, “Digital holographic microscope for measuring three-dimensional particle distributions and motions,” Appl. Opt. 45, 3893–3901 (2006).
[CrossRef] [PubMed]

Soria, J.

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002(2008).
[CrossRef]

Sun, B.

Thompson, B.

G. Tyler and B. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23, 685–700 (1976).
[CrossRef]

Torzynski, M.

Tyler, G.

G. Tyler and B. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23, 685–700 (1976).
[CrossRef]

Uemura, T.

Y. Yamamoto and T. Uemura, “3D particle measurements by single beam two-views magnified digital in-line holography,” Exp. Fluids 45, 813–821 (2008).
[CrossRef]

Valin, J. L.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Vukicevic, D.

Wang, H.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

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]

Willert, C. E.

M. Raffel, C. E. Willert, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer Verlag, 1998).

Woodward, S.

L. Cao, G. Pan, J. de Jong, S. Woodward, and H. Meng, “Hybrid digital holographic imaging system for three-dimensional dense particle field measurement,” Appl. Opt. 47, 4501–4508(2008).
[CrossRef] [PubMed]

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

Xiao, K.

Yamamoto, Y.

Y. Yamamoto and T. Uemura, “3D particle measurements by single beam two-views magnified digital in-line holography,” Exp. Fluids 45, 813–821 (2008).
[CrossRef]

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.

Exp. Fluids

Y. Yamamoto and T. Uemura, “3D particle measurements by single beam two-views magnified digital in-line holography,” Exp. Fluids 45, 813–821 (2008).
[CrossRef]

R. Konrath, W. Schröder, and W. Limberg, “Holographic particle image velocimetry applied to the flow within the cylinder of a four-valve internal combustion engine,” Exp. Fluids 33, 781–793 (2002).
[CrossRef]

Y. Pu and H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197(2000).
[CrossRef]

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

J. Mech. Sci. Tech.

Y.-J. Choo and B.-S. Kang, “Measurements of three-dimensional velocities of spray droplets using the holographic velocimetry system,” J. Mech. Sci. Tech. 17, 1095–1103 (2003).

J. Mod. Opt.

G. Tyler and B. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23, 685–700 (1976).
[CrossRef]

J. Opt. A

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A 6, 396–400 (2004).
[CrossRef]

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]

J. Müller, V. Kebbel, and W. Jüptner, “Characterization of spatial particle distributions in a spray-forming process using digital holography,” Meas. Sci. Technol. 15, 706–710(2004).
[CrossRef]

J. Coupland, “Holographic particle image velocimetry: signal recovery from under-sampled CCD data,” Meas. Sci. Technol. 15, 711–717 (2004).
[CrossRef]

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

J. Soria and C. Atkinson, “Towards 3C-3D digital holographic fluid velocity vector field measurement-tomographic digital holographic PIV (Tomo-HPIV),” Meas. Sci. Technol. 19, 074002(2008).
[CrossRef]

Opt. Commun.

F. Palacios, J. Ricardo, D. Palacios, E. Goncalves, J. L. Valin, and R. De Souza, “3D image reconstruction of transparent microscopic objects using digital holography,” Opt. Commun. 248, 41–50 (2005).
[CrossRef]

Opt. Express

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]

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

Proc. SPIE

M. Adams, T. M. Kreis, and W. P. O. Jueptner, “Particle size and position measurement with digital holography,” Proc. SPIE 3098, 234–240 (1997).
[CrossRef]

Other

M. Raffel, C. E. Willert, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer Verlag, 1998).

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Figures (14)

Fig. 1
Fig. 1

One-dimensional transmittance of a spherical object in a hologram.

Fig. 2
Fig. 2

Schematic of lens-based holography.

Fig. 3
Fig. 3

Normalized gray-level variance along depth direction.

Fig. 4
Fig. 4

Normalized intensity profile along the x and y direction.

Fig. 5
Fig. 5

Flow chart of the simulation process.

Fig. 6
Fig. 6

Simulated hologram with noise.

Fig. 7
Fig. 7

Superimposed 3D position of reconstructed particles of the 20 simulated holograms.

Fig. 8
Fig. 8

Comparison of the size distribution of the reconstructed and simulated particles.

Fig. 9
Fig. 9

Relation between the depth position error and the particle size error.

Fig. 10
Fig. 10

Schematic of experiment setup.

Fig. 11
Fig. 11

Pair of raw holograms of the coal particle flow.

Fig. 12
Fig. 12

Reconstructed plane of the hologram of the coal particle flow.

Fig. 13
Fig. 13

Size distribution of the reconstructed coal particles.

Fig. 14
Fig. 14

Superimposed velocity field of seven pairs of holograms. (a) 3D visualization; (b) at the X Y plane.

Equations (7)

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I ( r ) = 1 2 π d 2 λ z sin ( π r 2 λ z ) [ 2 J 1 ( 2 π d r / ( λ z ) ) 2 π d r / ( λ z ) ] + π 2 d 4 λ 2 z 2 [ 2 J 1 ( 2 π d r / ( λ z ) ) 2 π d r / ( λ z ) ] 2 ,
f chirp = d ( π r 2 / ( λ z ) ) d r | r = 3.83 λ z / ( 2 π d ) = 3.83 d .
Δ x < d 7.66 ,
Γ z ( x , y ) = 1 I ( x , y ) ψ z ( x , y ) = 1 I ( x , y ) { π λ z [ sin ( π x 2 + y 2 λ z ) M ψ ] × exp ( π x 2 + y 2 λ z σ 2 ) } ,
M ψ = σ 2 1 + σ 4 .
Var ( z ) = 1 N h N v i = 1 N h j = 1 N v [ Γ z ( i , j ) Γ ¯ z ] ,
v = 2 g h ,

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