Abstract

In holographic imaging of particle fields, the interference among coherent wave fronts associated with particle scattering gives rise to intrinsic speckle noise, which sets a fundamental limit on the amount of information that particle holography can deliver. It has been established that the intrinsic speckle noise is especially severe in in-line holography because of superposition of virtual image waves, the direct transmitted wave, and the real image. However, at sufficiently high particle number densities, such as those typical in holographic particle image velocimetry (HPIV) applications, intrinsic speckle noise also arises in off-axis particle holography from self-interference among wave fronts that form the real image of particles. To overcome the latter problem we have constructed a mathematical model that relates the first- and second-order statistical properties of the intrinsic speckle noise to relevant holographic system parameters. Consistent with our experimental data, the model provides a direct estimate of the information capacity of particle holography. We show that the noise-limited information capacity can be expressed as the product of particle number density and the extent of the particle field along the optical axis. A large angular aperture of the hologram contributes directly to achievement of high information capacity. We also show that filtering in either digital or optical form is generally ineffective in removing the intrinsic speckle noise from the particle image as a result of the similar spectral properties of the two. These findings emphasize the importance of angular aperture in designing holographic particle imaging systems.

© 2004 Optical Society of America

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References

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  1. 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]
  2. B. J. Thompson, “Holographic particle sizing techniques,” J. Phys. E 7, 781–788 (1974).
    [CrossRef]
  3. B. C. R. Ewan, “Fraunhofer plane analysis of particle field holograms,” Appl. Opt. 19, 1368–1372 (1980).
    [CrossRef] [PubMed]
  4. M. S. Marshall, R. E. Benner, “Sizing opaque spherical particles using classical matched filters and holographic ring detector,” Opt. Eng. 31, 947–955 (1992).
    [CrossRef]
  5. A. R. Jones, M. Sarjeant, C. R. Davis, R. O. Denham, “Application of in-line holography to drop size measurement in dense fuel sprays,” Appl. Opt. 17, 328–330 (1978).
    [CrossRef]
  6. P. R. Hobson, “Precision coordinate measurements using holographic recording,” J. Phys. E 21, 139–145 (1988).
    [CrossRef]
  7. H. Meng, F. Hussain, “In-line recording and off-axis viewing (IROV) technique for holographic particle velocimetry,” Appl. Opt. 34, 1827–1840 (1995).
    [CrossRef] [PubMed]
  8. 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]
  9. J. O. Scherer, L. P. Bernal, “In-line holographic particle image velocimetry for turbulent flows,” Appl. Opt. 36, 9309–9318 (1997).
    [CrossRef]
  10. L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” Publ. 85-0526 (American Institute of Aeronautics and Astronautics, New York, 1985).
  11. H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
    [CrossRef]
  12. 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]
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    [CrossRef] [PubMed]
  14. Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
    [CrossRef]
  15. J. Zhang, B. Tao, J. Katz, “Turbulent flow measurement in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–381 (1997).
    [CrossRef]
  16. S. Herrmann, H. Hinrichs, K. D. Hinsch, C. Surmann, “Coherence concepts in holographic particle image velocimetry,” Exp. Fluids 29, S108–S116 (2000).
    [CrossRef]
  17. 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]
  18. 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]
  19. R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).
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    [CrossRef]
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    [CrossRef] [PubMed]
  23. N. Andres, P. Arroyo, M. Quintanilla, “Velocity measurements in a convective flow by holographic interferometry,” Appl. Opt. 36, 6997–7007 (1997).
    [CrossRef]
  24. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  25. D. J. Bone, “Fourier fringe analysis—the 2-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  26. N. A. Ochoa, F. M. Santoyo, A. J. Moore, C. P. Lopez, “Contrast enhancement of electronic speckle pattern interferometry addition fringes,” Appl. Opt. 36, 2783–2787 (1997).
    [CrossRef] [PubMed]
  27. R. Kumar, S. K. Singh, C. Shakher, “Wavelet filtering applied to time-average digital speckle pattern interferometry fringes,” Opt. Laser Technol. 33, 567–571 (2001).
    [CrossRef]
  28. U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  32. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  33. H. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
    [CrossRef]

2003 (1)

2001 (1)

R. Kumar, S. K. Singh, C. Shakher, “Wavelet filtering applied to time-average digital speckle pattern interferometry fringes,” Opt. Laser Technol. 33, 567–571 (2001).
[CrossRef]

2000 (2)

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]

1999 (2)

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]

N. Andres, P. Arroyo, “Digital speckle-pattern interferometry as a full-field fluid-velocimetry technique,” Opt. Lett. 24, 575–577 (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. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
[CrossRef]

1997 (4)

1996 (1)

1995 (1)

1994 (2)

1993 (1)

1992 (1)

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

1991 (2)

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

D. J. Bone, “Fourier fringe analysis—the 2-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

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

1983 (1)

1980 (1)

1978 (1)

1974 (1)

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

1969 (1)

1967 (1)

Adrian, R. J.

Anderson, W. L.

Andres, N.

Arroyo, P.

Barnhart, D. H.

Beeler, G. B.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” Publ. 85-0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

Belz, R. A.

Benner, R. E.

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

Bernal, L. P.

Böhmer, M.

H. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
[CrossRef]

Bone, D. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York1980), 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]

Davis, C. R.

Denham, R. O.

Ennos, A. E.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., (Springer-Verlag, Berlin, 1984), pp. 207–210.

Ewan, B. C. R.

Farmer, W. M.

Goodman, J. W.

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]

H. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
[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]

H. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
[CrossRef]

Hobson, P. R.

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

Huntley, J. M.

Hussain, F.

Jones, A. R.

Juptner, W.

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]

Kickstein, J.

H. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
[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]

Kumar, R.

R. Kumar, S. K. Singh, C. Shakher, “Wavelet filtering applied to time-average digital speckle pattern interferometry fringes,” Opt. Laser Technol. 33, 567–571 (2001).
[CrossRef]

Lal, R. B.

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]

Linderman, A. M.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” Publ. 85-0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

Liu, D.

Lopez, C. P.

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 detector,” Opt. Eng. 31, 947–955 (1992).
[CrossRef]

McIntosh, D.

Meinhart, C. D.

Meng, H.

Meynart, R.

Moore, A. J.

Ochoa, N. A.

Papen, G. C.

Pu, Y.

Y. Pu, H. Meng, “Intrinsic aberrations due to Mie scattering in particle holography,” J. Opt. Soc. Am. A 20, 1920–1932 (2003).
[CrossRef]

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

Quintanilla, M.

Santoyo, F. M.

Sarjeant, M.

Scherer, J. O.

Schnars, U.

Shakher, C.

R. Kumar, S. K. Singh, C. Shakher, “Wavelet filtering applied to time-average digital speckle pattern interferometry fringes,” Opt. Laser Technol. 33, 567–571 (2001).
[CrossRef]

Singh, S. K.

R. Kumar, S. K. Singh, C. Shakher, “Wavelet filtering applied to time-average digital speckle pattern interferometry fringes,” Opt. Laser Technol. 33, 567–571 (2001).
[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.

Weinstein, L. W.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” Publ. 85-0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

Witherow, W. K.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York1980), 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]

Appl. Opt. (13)

D. J. Bone, “Fourier fringe analysis—the 2-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

U. Schnars, W. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
[CrossRef] [PubMed]

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]

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

N. Andres, P. Arroyo, M. Quintanilla, “Velocity measurements in a convective flow by holographic interferometry,” Appl. Opt. 36, 6997–7007 (1997).
[CrossRef]

N. A. Ochoa, F. M. Santoyo, A. J. Moore, C. P. Lopez, “Contrast enhancement of electronic speckle pattern interferometry addition fringes,” Appl. Opt. 36, 2783–2787 (1997).
[CrossRef] [PubMed]

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

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]

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]

A. R. Jones, M. Sarjeant, C. R. Davis, R. O. Denham, “Application of in-line holography to drop size measurement in dense fuel sprays,” Appl. Opt. 17, 328–330 (1978).
[CrossRef]

B. C. R. Ewan, “Fraunhofer plane analysis of particle field holograms,” Appl. Opt. 19, 1368–1372 (1980).
[CrossRef] [PubMed]

R. Meynart, “Instantaneous velocity field measurements in unsteady gas flow by speckle velocimetry,” Appl. Opt. 22, 535–540 (1983).
[CrossRef] [PubMed]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

Exp. Fluids (5)

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]

H. Hinrichs, K. D. Hinsch, J. Kickstein, M. Böhmer, “Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modeling,” Exp. Fluids 24, 333–339 (1998).
[CrossRef]

Fluid Dyn. Res. (1)

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. E (2)

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

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

Opt. Eng. (1)

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

Opt. Laser Technol. (2)

R. Kumar, S. K. Singh, C. Shakher, “Wavelet filtering applied to time-average digital speckle pattern interferometry fringes,” Opt. Laser Technol. 33, 567–571 (2001).
[CrossRef]

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]

Opt. Lett. (1)

Other (5)

R. K. Erf, ed., Speckle Metrology (Academic, New York, 1978).

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., (Springer-Verlag, Berlin, 1984), pp. 207–210.

L. W. Weinstein, G. B. Beeler, A. M. Linderman, “High-speed holocine-matographic velocimeter for studying turbulent flow control physics,” Publ. 85-0526 (American Institute of Aeronautics and Astronautics, New York, 1985).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York1980), p. 464.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (8)

Fig. 1
Fig. 1

Diagram for holographic reconstruction of the real particle image field. H, hologram plane; AH, hologram aperture; aH, hologram aperture radius; rH, hologram recording distance; Ω, hologram angular aperture.

Fig. 2
Fig. 2

Diagrams for evaluation of mean speckle intensity at point P. (a) Mean noise intensity at point P with distance z0 away from the image field. Only images enclosed in the solid angle 2Ω (shaded region) contribute noise to point P. (b) Evaluation of mean noise intensity on an image plane inside the particle field. Particle images within the depth of focus δf are considered signal images.

Fig. 3
Fig. 3

Diagram and results for experimental validation of the mathematical model for the mean noise intensity. (a) Optical setup for the experiment. (b) IN as a function of ns; L=25 mm, f2.8, Ω=10°. (c) IN as a function of L; ns=14 mm-3, f2.8, Ω=10°. (d) IN as a function of Ω; ns=18 mm-3, L=25 mm. Dark background intensity of 59.6 has been subtracted from the measurement.

Fig. 4
Fig. 4

Diagram of derivation of 3D form of van Cittert–Zernike theorem. Qi(ξi, ηi, ζi) is the center of the ith phasors contributing to both P1 and P2. ρi(P1) and ρi(P2) are the distance from Qi to P1 and P2, respectively. V is the volume that contains phasors contributing light to both P1 and P2.

Fig. 5
Fig. 5

Representation of analysis of transverse mutual intensity JXY. (a) P1 and P2 are on a plane perpendicular to z axis (the optical axis) with distance z0 away from the image field. The angular aperture is 2Ω. Only particle images in the shaded region contribute to JXY. (b) Evaluation of JXY on an image plane inside the particle image field.

Fig. 6
Fig. 6

Mutual intensity of the speckle. (a) Transverse mutual intensity. (b) Axial mutual intensity. Note that the position of the first zero serves as an estimation of the mean speckle size along the respective directions. Dotted curves are from experimental data. Deviation between theory and experimental data is due to insufficient spatial and intensity resolution.

Fig. 7
Fig. 7

Schematic diagram of analysis of axial mutual intensity JZ. (a) P1 and P2 are on the optical axis. P1 is z0 away from the image field, and P2 is Δz away from P1. Only particle images in the shaded region contribute to JZ. (b) Evaluation of JZ on an image plane that is inside the image field.

Fig. 8
Fig. 8

Power spectral density of intrinsic speckle noise and particle images. Sampling pixel size is 5 μm and speckle size is ≈2.1 μm. Note the striking similarity among the three spectra. The slow varying curvature is due to the aperture of the lens that is used to obtain these images.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

IT=|R+O|2=|R|2+RO*+R*O+|O|2,
Et=R*τ=C|R|2R*+CR*|O|2+CR*R*O+C|R|2O*,
ISσN=I0/IN1+2I0/IN,
IS=I0=πaHλtan Ω2A2rH2=π2λ-2A2tan4 Ω,
Ei(x, y, z)=Aρiexp(ikρi)exp(iϕi),
IN=iEi(x, y, z)jEj*(x, y, z)=ijEi(x, y, z)Ej*(x, y, z).
IN=i|Ei(x, y, z)|2=VnsA2(z0-z)2dxdydz=πnsA2L tan2 Ω.
I0IN=π tan2 Ωλ2nsL.
J12=iE(Qi;P1)jE*(Qj;P2),
J12=iE(Qi;P1)E*(Qi;P2)=iA2ρi(P1)ρi(P2)exp(iψi),
J12(P1;P2)=VnsA2×exp(-iψ)λ2(z1-ζ)(z2-ζ)×exp-i πλξ12+η12z1-ζ-ξ22+η22z2-ζ×expi πλx1ξ+y1ηz1-ζ-x2ξ+y2ηz2-ζdξdηdζ,
ψ=πλ(x12+y12)(z1-ζ)-(x22+y22)(z2-ζ),
JXY(x1, y1; x2, y2)
=-L0nsA2exp(-iψ)λ2(z0-ζ)2×Sexpi πλ(x1-x2)ξ+(y1-y2)ηz0-ζdξdηdξ,
ψ=πλΔr2(z0-ζ),Δr=(x1-x2)2+(y1-y2)2
JXY(Δr)=2πnsA2tan2 Ωg(Δr) J1(2πΔr tan Ω/λ)2πΔr tan Ω/λ,
|μXY(Δr)|2=JXYIN2=4 J12(2πΔr tan Ω/λ)(2πΔr tan Ω/λ)2.
JZ(z0, Δz)=-L0nsA2λ2(z0-ζ)(z0+Δz-ζ)×Sexp-i πλξ2+η2z0-ζ-ξ2+η2z0+Δz-ζdξdηdζ.
JZ(z0, Δz)=-L0nsA2λ2(z0-ζ)2×Sexp-i πλ Δz ξ2+η2(z0-ζ)2dξdηdζ,
|μZ(Δz)|2=JZIN2=sin2πΔz tan2 Ω2λπΔz tan2 Ω2λ2.
I^N=MIN,
nsL|max=π tan2 Ωλ2(I0/IN)min.
nsL|max=πγ tan2 ΩMλ2(I0/IN)min,

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