Abstract

Holography of small particles is a newly revived topic because of its importance in holographic particle image velocimetry (HPIV). However, the property of particle images formed through holography remains largely unexplored. This fact undermines the measurement reliability of HPIV techniques and has become one of the obstacles in the full deployment of HPIV. We study the intrinsic aberrations in the holographic particle image introduced by particle light scattering and investigate how accurately holography can deliver information about the particles that are being imaged. Consistent with our experimental observations, simulations based on Mie scattering theory show that even with a perfect hologram the reconstructed particle images demonstrate complex three-dimensional morphologies and bodily shifts. These characteristics, manifested as image aberrations, result from uneven scattering amplitude and phase distributions across the finite aperture of the hologram. Such aberrations degrade the signal-to-noise ratio in the reconstructed image as well as introducing systematic errors in detected particle image positions. We examine the effect of these aberrations on HPIV measurements.

© 2003 Optical Society of America

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References

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  1. R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
    [CrossRef]
  2. M. P. Arroyo, C. A. Greated, “Stereoscopic particle velocimetry,” Meas. Sci. Technol. 2, 1181–1186 (1991).
    [CrossRef]
  3. K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
    [CrossRef]
  4. H. Meng, F. Hussain, “In-line recording and off-axis viewing technique for holographic particle velocimetry,” Appl. Opt. 34, 1827–1840 (1995).
    [CrossRef] [PubMed]
  5. J. D. Trolinger, R. B. Lal, D. McIntosh, W. K. Witherow, “Holographic particle-image velocimetry in the first International Microgravity Laboratory aboard the Space Shuttle Discovery,” Appl. Opt. 35, 681–689 (1996).
    [CrossRef] [PubMed]
  6. J. O. Scherer, L. P. Bernal, “In-line holographic particle image velocimetry for turbulent flows,” Appl. Opt. 36, 9309–9318 (1997).
    [CrossRef]
  7. D. H. Barnhart, R. J. Adrian, C. D. Meinhart, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle image velocimetry,” Appl. Opt. 33, 7159–7169 (1994).
    [CrossRef] [PubMed]
  8. Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
    [CrossRef]
  9. J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
    [CrossRef]
  10. J. D. Trolinger, R. A. Belz, W. M. Farmer, “Holographic techniques for the study of dynamic particle fields,” Appl. Opt. 8, 957–961 (1969).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. M. S. Marshall, R. E. Benner, “Sizing opaque spherical particles using classical matched filters and holographic ring detectors,” Opt. Eng. 31, 947–955 (1992).
    [CrossRef]
  14. P. R. Hobson, “Precision coordinate measurements using holographic recording,” J. Phys. E 21, 139–145 (1988).
    [CrossRef]
  15. H. Meng, W. L. Anderson, F. Hussain, D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
    [CrossRef]
  16. H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).
  17. S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
    [CrossRef]
  18. A. Lozano, J. Kostas, J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
    [CrossRef]
  19. K. T. Chan, Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
    [CrossRef]
  20. D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Holographic particle image velocimetry: analysis using a conjugate reconstruction geometry,” Opt. Laser Technol. 32, 527–533 (2000).
    [CrossRef]
  21. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  22. K. T. Chan, T. P. Leung, “Holographic imaging of side-scattering particles,” Opt. Laser Technol. 28, 565–571 (1996).
    [CrossRef]
  23. B. Ovryn, S. H. Izen, “Imaging of transparent spheres through a planar interface using a high-numerical-aperture optical microscope,” J. Opt. Soc. Am. A 17, 1202–1213 (2000).
    [CrossRef]
  24. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.
  25. Y. Pu, X. Song, H. Meng, “Off-axis holographic particle image velocimetry for diagnosing particulate flows,” Exp. Fluids 29, S117–S128 (2000).
    [CrossRef]
  26. N. Nikolaidis, I. Pitas, 3-D Image Processing Algorithms (Wiley, New York, 2001), p. 82.

2000 (5)

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

S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
[CrossRef]

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Holographic particle image velocimetry: analysis using a conjugate reconstruction geometry,” Opt. Laser Technol. 32, 527–533 (2000).
[CrossRef]

Y. Pu, X. Song, H. Meng, “Off-axis holographic particle image velocimetry for diagnosing particulate flows,” Exp. Fluids 29, S117–S128 (2000).
[CrossRef]

B. Ovryn, S. H. Izen, “Imaging of transparent spheres through a planar interface using a high-numerical-aperture optical microscope,” J. Opt. Soc. Am. A 17, 1202–1213 (2000).
[CrossRef]

1999 (1)

A. Lozano, J. Kostas, J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
[CrossRef]

1998 (2)

K. T. Chan, Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[CrossRef]

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

1997 (2)

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
[CrossRef]

J. O. Scherer, L. P. Bernal, “In-line holographic particle image velocimetry for turbulent flows,” Appl. Opt. 36, 9309–9318 (1997).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

1993 (2)

H. Meng, W. L. Anderson, F. Hussain, D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
[CrossRef]

K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

1992 (1)

M. S. Marshall, R. E. Benner, “Sizing opaque spherical particles using classical matched filters and holographic ring detectors,” Opt. Eng. 31, 947–955 (1992).
[CrossRef]

1991 (2)

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

M. P. Arroyo, C. A. Greated, “Stereoscopic particle velocimetry,” Meas. Sci. Technol. 2, 1181–1186 (1991).
[CrossRef]

1988 (1)

P. R. Hobson, “Precision coordinate measurements using holographic recording,” J. Phys. E 21, 139–145 (1988).
[CrossRef]

1980 (1)

1974 (1)

B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
[CrossRef]

1969 (1)

Adrian, R. J.

D. H. Barnhart, R. J. Adrian, C. D. Meinhart, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle image velocimetry,” Appl. Opt. 33, 7159–7169 (1994).
[CrossRef] [PubMed]

K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Anderson, W. L.

Arroyo, M. P.

M. P. Arroyo, C. A. Greated, “Stereoscopic particle velocimetry,” Meas. Sci. Technol. 2, 1181–1186 (1991).
[CrossRef]

Barnhart, D. H.

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Holographic particle image velocimetry: analysis using a conjugate reconstruction geometry,” Opt. Laser Technol. 32, 527–533 (2000).
[CrossRef]

D. H. Barnhart, R. J. Adrian, C. D. Meinhart, G. C. Papen, “Phase-conjugate holographic system for high-resolution particle image velocimetry,” Appl. Opt. 33, 7159–7169 (1994).
[CrossRef] [PubMed]

Belz, R. A.

Benner, R. E.

M. S. Marshall, R. E. Benner, “Sizing opaque spherical particles using classical matched filters and holographic ring detectors,” Opt. Eng. 31, 947–955 (1992).
[CrossRef]

Bernal, L. P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.

Chan, K. T.

K. T. Chan, Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[CrossRef]

K. T. Chan, T. P. Leung, “Holographic imaging of side-scattering particles,” Opt. Laser Technol. 28, 565–571 (1996).
[CrossRef]

Coupland, J. M.

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Holographic particle image velocimetry: analysis using a conjugate reconstruction geometry,” Opt. Laser Technol. 32, 527–533 (2000).
[CrossRef]

Estevadeordal, J.

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

Ewan, B. C. R.

Farmer, W. M.

Gogeneni, S.

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

Goss, L.

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

Greated, C. A.

M. P. Arroyo, C. A. Greated, “Stereoscopic particle velocimetry,” Meas. Sci. Technol. 2, 1181–1186 (1991).
[CrossRef]

Halliwell, N. A.

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Holographic particle image velocimetry: analysis using a conjugate reconstruction geometry,” Opt. Laser Technol. 32, 527–533 (2000).
[CrossRef]

Herrmann, S.

S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
[CrossRef]

Hinrichs, H.

S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
[CrossRef]

Hinsch, K. D.

S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
[CrossRef]

Hobson, P. R.

P. R. Hobson, “Precision coordinate measurements using holographic recording,” J. Phys. E 21, 139–145 (1988).
[CrossRef]

Hussain, F.

Izen, S. H.

Katz, J.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
[CrossRef]

Kostas, J.

A. Lozano, J. Kostas, J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
[CrossRef]

Lal, R. B.

Leung, T. P.

K. T. Chan, T. P. Leung, “Holographic imaging of side-scattering particles,” Opt. Laser Technol. 28, 565–571 (1996).
[CrossRef]

Li, Y. J.

K. T. Chan, Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[CrossRef]

Liu, D.

Lozano, A.

A. Lozano, J. Kostas, J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
[CrossRef]

Marshall, M. S.

M. S. Marshall, R. E. Benner, “Sizing opaque spherical particles using classical matched filters and holographic ring detectors,” Opt. Eng. 31, 947–955 (1992).
[CrossRef]

McIntosh, D.

Meinhart, C. D.

Meng, H.

Y. Pu, X. Song, H. Meng, “Off-axis holographic particle image velocimetry for diagnosing particulate flows,” Exp. Fluids 29, S117–S128 (2000).
[CrossRef]

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

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

H. Meng, F. Hussain, “In-line recording and off-axis viewing technique for holographic particle velocimetry,” Appl. Opt. 34, 1827–1840 (1995).
[CrossRef] [PubMed]

H. Meng, W. L. Anderson, F. Hussain, D. Liu, “Intrinsic speckle noise in in-line particle holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
[CrossRef]

Nikolaidis, N.

N. Nikolaidis, I. Pitas, 3-D Image Processing Algorithms (Wiley, New York, 2001), p. 82.

Ovryn, B.

Papen, G. C.

Pitas, I.

N. Nikolaidis, I. Pitas, 3-D Image Processing Algorithms (Wiley, New York, 2001), p. 82.

Prasad, K.

K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

Pu, Y.

Y. Pu, X. Song, H. Meng, “Off-axis holographic particle image velocimetry for diagnosing particulate flows,” Exp. Fluids 29, S117–S128 (2000).
[CrossRef]

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

Roquemore, W. M.

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

Scherer, J. O.

Song, X.

Y. Pu, X. Song, H. Meng, “Off-axis holographic particle image velocimetry for diagnosing particulate flows,” Exp. Fluids 29, S117–S128 (2000).
[CrossRef]

Soria, J.

A. Lozano, J. Kostas, J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
[CrossRef]

Surmann, C.

S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
[CrossRef]

Tao, B.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
[CrossRef]

Thompson, B. J.

B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
[CrossRef]

Trolinger, J. D.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Witherow, W. K.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.

Zhang, J.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
[CrossRef]

Annu. Rev. Fluid Mech. (1)

R. J. Adrian, “Particle-imaging techniques for experimental fluid mechanics,” Annu. Rev. Fluid Mech. 23, 261–304 (1991).
[CrossRef]

Appl. Opt. (6)

Exp. Fluids (6)

Y. Pu, X. Song, H. Meng, “Off-axis holographic particle image velocimetry for diagnosing particulate flows,” Exp. Fluids 29, S117–S128 (2000).
[CrossRef]

K. Prasad, R. J. Adrian, “Stereoscopic particle image velocimetry applied to liquid flows,” Exp. Fluids 15, 49–60 (1993).
[CrossRef]

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

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
[CrossRef]

S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
[CrossRef]

A. Lozano, J. Kostas, J. Soria, “Use of holography in particle image velocimetry measurements of a swirling flow,” Exp. Fluids 27, 251–261 (1999).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. E (2)

P. R. Hobson, “Precision coordinate measurements using holographic recording,” J. Phys. E 21, 139–145 (1988).
[CrossRef]

B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
[CrossRef]

J. Vision (1)

H. Meng, J. Estevadeordal, S. Gogeneni, L. Goss, W. M. Roquemore, “Holographic flow visualization as a tool for studying 3D coherent structures and instabilities,” J. Vision 1, 51–63 (1998).

Meas. Sci. Technol. (1)

M. P. Arroyo, C. A. Greated, “Stereoscopic particle velocimetry,” Meas. Sci. Technol. 2, 1181–1186 (1991).
[CrossRef]

Opt. Eng. (1)

M. S. Marshall, R. E. Benner, “Sizing opaque spherical particles using classical matched filters and holographic ring detectors,” Opt. Eng. 31, 947–955 (1992).
[CrossRef]

Opt. Laser Technol. (3)

K. T. Chan, Y. J. Li, “Pipe flow measurement by using a side-scattering holographic particle imaging technique,” Opt. Laser Technol. 30, 7–14 (1998).
[CrossRef]

D. H. Barnhart, N. A. Halliwell, J. M. Coupland, “Holographic particle image velocimetry: analysis using a conjugate reconstruction geometry,” Opt. Laser Technol. 32, 527–533 (2000).
[CrossRef]

K. T. Chan, T. P. Leung, “Holographic imaging of side-scattering particles,” Opt. Laser Technol. 28, 565–571 (1996).
[CrossRef]

Other (3)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 464.

N. Nikolaidis, I. Pitas, 3-D Image Processing Algorithms (Wiley, New York, 2001), p. 82.

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Figures (13)

Fig. 1
Fig. 1

Recording and reconstruction in particle holography. (a) Recording. (b) Vector projections. Eo is decomposed into three mutually orthogonal components: Eo, in parallel with Er, and Eo(1) and Eo(2), perpendicular to Er. (c) Reconstruction.

Fig. 2
Fig. 2

Generalized geometry of off-axis particle holography. (a) Recording. The scattering sphere is located at the origin OP. The normal vector nH of hologram H is pointing toward OP. The hologram has an aperture AH (with a radius aH). The scattered field at an arbitrary receiving point P in the hologram plane H consists of components Eθ and Eφ. (b) Reconstruction. The particle image is ideally reconstructed at the principal location OP. The image is interrogated in an observing plane S, which is parallel to H. The observing distance between S and Π (observing plane containing OP, or the principal plane) is δs.

Fig. 3
Fig. 3

Phase deviations of the reconstructed wave front from the sphere centered at the principal image. Since this sphere is the Gaussian reference sphere of the principal image, such deviations are a direct measure of aberrations in the reconstructed image.

Fig. 4
Fig. 4

Phase deviation of the reconstructed wave front immediately behind the hologram plane H. (a) θH=0° (forward scattering). The major aberration component is spherical aberration. (b) θH=30°. Major aberration components are coma and astigmatism. (c) θH=90°. The major aberration component is astigmatism. In the figure, coordinates xH and yH are normalized with the aperture radius aH. In all cases, particle size is α=60, m=1.15 and the angular aperture is 2Ω=30°.

Fig. 5
Fig. 5

Focal image of a particle (α=60) at various scattering angles. (a) θH=0°. (b) θH=15°. (c) θH=30°. (d) θH=90°. (e) θH=165°. (f) θH=180°. All pictures are sized 15λ×15λ.

Fig. 6
Fig. 6

Reconstructed particle image projected on a selected plane. In all cases α=60 except in (f), where α=75. (a) Principal image at θH=0°. (b) Focal image at θH=0°, δS=-31λ. (c) Principal image at θH=180°. (d) Focal image at θH=180°, δS=-56λ. (e) Focal image at θH=90°. Note that at θH=90° the focal image is identical to the principal image. (f) Focal image of a particle with a different size at θH=90°. Intensities are normalized with focal intensity If.

Fig. 7
Fig. 7

Examples of axial intensity distributions and focal images in an extreme case at θH=0° and α=58. (a) Axial intensity distribution of the image center due to complete Ey*. (b) Axial intensity distribution due to the central scattering lobe of Ey*. (c) Focal image at δS=26λ produced by complete Ey*. (d) Focal image at δS=0 produced by the central scattering lobe of Ey*.

Fig. 8
Fig. 8

Three-dimensional morphology of reconstructed particle images. (a) Simulated image at θH=0°. (b) Simulated image at θH=90°. (c) A typical image acquired from a hologram at θH=90°. The threshold is chosen to best demonstrate the 3D exterior surface. All length is nondimensionalized with the wavelength.

Fig. 9
Fig. 9

Image integrity γ as a function of size α for various θH. Steep changes seen at most angles are due to the multiple, competing foci. Not shown in this figure: γ at θH=15° is similar to θH=0°, and γ at θH=150° and θH=165° are similar to θH=180°.

Fig. 10
Fig. 10

Deviations (erors) of image centroids from the principal location OP as a function of particle size α. εx and εz are transverse and axial errors, respectively. (a) θH=0°. (b) θH=15°. (c) θH=30°. (d) θH=90°. (e) θH=150°. (f) θH=165°. (g) θH=180°. Locations of the particle image is calculated by an intensity-weighted mean (the first momentum) based on pixels with intensities above 50% If. Deviation is normalized with particle diameter dp.

Fig. 11
Fig. 11

Geometry of Mie scattering theory. The shaded plane is the scattering plane. The scattering sphere is located at the origin OP. Ei is the incident wave propagating along ki. Eθ and Eφ are scattered field components at an arbitrary point P.

Fig. 12
Fig. 12

Geometry for the analysis of error due to the Fresnel approximation.

Fig. 13
Fig. 13

Comparison of the intensity distribution according to Eqs. (B2) and (B3). (a) Transverse cross section. (b) Axial cross section.

Tables (1)

Tables Icon

Table 1 Imaging Characteristics of Particle Holography

Equations (33)

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IH=(Er+Eo)  (Er+Eo)*,
IH=(Er+Eo)(Er+Eo)*+|Eo(1)|2+|Eo(2)|2.
E=CIHEr*=C(Er2Er*+Er2Eo*+Er*Er*Eo),
Eθ=-ikrexp(ikr)cos φS2(θ),
Eφ=-ikrexp(ikr)sin φS1(θ),
Eo=Exuˆx+Eyuˆy+Ezuˆz,
Ex=Eθ[cos(φ-φH)cos θ cos θH+sin θ sin θH]+Eφ sin(φ-φH)cos θH,Ey=Eθ sin(φ-φH)cos θ-Eφ cos(φ-φH), Ez=Eθ[-cos(φ-φH)cos θ sin θH+sin θ cos θH]-Eφ sin(φ-φH)sin θH.
Ey*=ikrexp(-ikr)H(θ, φ),
H(θ, φ)=S2*(θ)cos2 φ cos θ+S1*(θ)sin2 φ.
E(s)=AHEy*(r) 1iλexp[ikd(s, r)]d(s, r)cos(s-r, nH)dA,
E(ξ, η)=exp(iφ1)iλ(rH+δS)exp(iφ2)AU(x, y)×exp-ik(xξ+yη)2(rH+δS)dxdy,
U(x, y)=Ey*(x, y)expik(x2+y2)2(rH+δS).
Ey*(x, y)=ikrHexp(-ikrH)exp-ik(x2+y2)2rHH(θ, φ),
U(x, y)=ikrHexp(-ikrH)exp-ikδs(x2+y2)2rH2H(θ, φ).
γ=If/I0,
I0=πaλtan Ω2A2.
xc=VTI(x)  xdxdydzIs,
Is=VTI(x)dxdydz
Eθ=-ikrexp(ikr)cos φS2(θ),
Eφ=ikrexp(ikr)sin φS1(θ),
k=2π/λ,
S1=n=12n+1n(n+1)[anπn(cos θ)+bnτn(cos θ)],
S2=n=12n+1n(n+1)[bnπn(cos θ)+anτn(cos θ)],
πn(cos θ)=1sin θPn1(cos θ),
τn(cos θ)=ddθPn1(cos θ),
an=ψn(mα)ψn(α)-ψn(mα)ψn(α)ψn(mα)ζn(α)-ψn(mα)ζn(α),
bn=mψn(mα)ψn(α)-ψn(mα)ψn(α)mψn(mα)ζn(α)-ψn(mα)ζn(α),
ψn(z)=πz2 Jn+1/2(z),
ζn(z)=πz2 Hn+1/2(2)(z)
E0=Ar0exp(-ikr0),
E(ξ, η, Δz)=-iA2λΣ1r0exp(-ikr0) 1rexp(ikr)cos χdxdy,
E=-iA exp(ikΔz)2λz0(z0+Δz)expξ2+η22(z0+Δz)×Σ exp-ik x2+y22z0expik x2+y22(z0+Δz)×exp-ik ξx+ηyz0+Δzdxdy.
[(x-ξ)2+(y-η)2]max2(z0+Δz)min4λ.

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