Abstract

The process of in-line Fraunhofer holography of particles is studied by an analysis of the four terms in the reconstructed field. Specific results are discussed for objects with circular cross section. The image shape and contrast are described in detail. The study is particularly useful when the recording is performed at a few far fields. It has been found that the recording at a higher far field is better due to a higher image-to-background irradiance ratio at the edges of the reconstructed image.

© 1984 Optical Society of America

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References

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  1. B. J. Thompson, P. Dunn, “Recent Advances in Holography,” Proc. Soc. Photo-Opt. Instrum, Eng. 215, 102 (1980).
  2. S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle Sizing Using Far-Field Holography: New Developments,” Opt. Eng. 19, 727 (1980).
    [CrossRef]
  3. C. S. Vikram, M. L. Billet, “Gaussian Beam Effects in Far-Field In-line Holography,” Appl. Opt. 22, 2830 (1983).
    [CrossRef] [PubMed]
  4. K. Murata, H. Fujiwara, T. Asakura, in Proceedings, Symposium on Engineering Uses of Holography (Strathclyde University, Glasgow1970), p. 289.
  5. J. T. Bartlett, R. J. Adams, “Development of a Holographic Technique for Sampling Particles in Moving Aerosols,” Microscope 20, 375 (1972).
  6. J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
    [CrossRef]
  7. P. Dunn, J. M. Walls, “Improved Microimages from In-line Absorption Holograms,” Appl. Opt. 18, 263 (1979).
    [CrossRef] [PubMed]
  8. P. Dunn, J. M. Walls, “Absorption and Phase In-line Holograms: a Comparison,” Appl. Opt. 18, 2171 (1979).
    [CrossRef] [PubMed]
  9. J. B. DeVelis, G. B. Parrent, B. J. Thompson, “Image Reconstruction with Fraunhofer Holograms,” J. Opt. Soc. Am. 56, 423 (1966).
    [CrossRef]
  10. G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
    [CrossRef]
  11. B. J. Thompson, J. H. Ward, W. R. Zinky, “Application of Hologram Techniques for Particle Size Analysis,” Appl. Opt. 6, 519 (1967).
    [CrossRef] [PubMed]
  12. D. M. Robinson, “A Calculation of Edge Smear in Far-Field Holography Using a Short-cut Edge Trace Technique,” Appl. Opt. 9, 496 (1970).
    [CrossRef] [PubMed]
  13. R. A. Belz, in “Holography and Optical Filtering,” NASA Report SP-299 (1973), p. 193.
  14. B. J. Thompson, “Holographic Particle Sizing Techniques,” J. Phys. E 7, 781 (1974).
    [CrossRef]
  15. See, for example, J. D. Trolinger, R. A. Belz, W. M. Farmer, “Holographic Techniques for the Study of Dynamic Particle Fields,” Appl. Opt. 8, 957 (1969).
    [CrossRef] [PubMed]
  16. G. B. Parrent, B. J. Thompson, “On the Fraunhofer (far field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
    [CrossRef]
  17. See, for example, R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983), p. 180.
  18. R. Bexon, M. G. Dalzell, M. C. Stainer, “In-Line Holography and the Assessment of Aerosols,” Opt. Laser Technol. 8, 161 (1976).
    [CrossRef]
  19. G. Haussmann, W. Lauterborn, “Determination of Size and Position of Fast Moving Gas Bubbles in Liquids by Digital 3-D Image Processing of Hologram Reconstructions,” Appl. Opt. 19, 3529 (1980).
    [CrossRef] [PubMed]
  20. P. R. Payne, K. L. Carder, R. G. Steward, “Image Analysis Techniques for Holograms of Dynamic Oceanic Particles,” Appl. Opt. 23, 204 (1984).
    [CrossRef] [PubMed]

1984 (1)

1983 (1)

1980 (3)

B. J. Thompson, P. Dunn, “Recent Advances in Holography,” Proc. Soc. Photo-Opt. Instrum, Eng. 215, 102 (1980).

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle Sizing Using Far-Field Holography: New Developments,” Opt. Eng. 19, 727 (1980).
[CrossRef]

G. Haussmann, W. Lauterborn, “Determination of Size and Position of Fast Moving Gas Bubbles in Liquids by Digital 3-D Image Processing of Hologram Reconstructions,” Appl. Opt. 19, 3529 (1980).
[CrossRef] [PubMed]

1979 (2)

1976 (2)

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[CrossRef]

R. Bexon, M. G. Dalzell, M. C. Stainer, “In-Line Holography and the Assessment of Aerosols,” Opt. Laser Technol. 8, 161 (1976).
[CrossRef]

1975 (1)

J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
[CrossRef]

1974 (1)

B. J. Thompson, “Holographic Particle Sizing Techniques,” J. Phys. E 7, 781 (1974).
[CrossRef]

1972 (1)

J. T. Bartlett, R. J. Adams, “Development of a Holographic Technique for Sampling Particles in Moving Aerosols,” Microscope 20, 375 (1972).

1970 (1)

1969 (1)

1967 (1)

1966 (1)

1964 (1)

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (far field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[CrossRef]

Adams, R. J.

J. T. Bartlett, R. J. Adams, “Development of a Holographic Technique for Sampling Particles in Moving Aerosols,” Microscope 20, 375 (1972).

Asakura, T.

K. Murata, H. Fujiwara, T. Asakura, in Proceedings, Symposium on Engineering Uses of Holography (Strathclyde University, Glasgow1970), p. 289.

Bartlett, J. T.

J. T. Bartlett, R. J. Adams, “Development of a Holographic Technique for Sampling Particles in Moving Aerosols,” Microscope 20, 375 (1972).

Belz, R. A.

Bexon, R.

R. Bexon, M. G. Dalzell, M. C. Stainer, “In-Line Holography and the Assessment of Aerosols,” Opt. Laser Technol. 8, 161 (1976).
[CrossRef]

Billet, M. L.

Carder, K. L.

Cartwright, S. L.

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle Sizing Using Far-Field Holography: New Developments,” Opt. Eng. 19, 727 (1980).
[CrossRef]

Dalzell, M. G.

R. Bexon, M. G. Dalzell, M. C. Stainer, “In-Line Holography and the Assessment of Aerosols,” Opt. Laser Technol. 8, 161 (1976).
[CrossRef]

DeVelis, J. B.

Dunn, P.

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle Sizing Using Far-Field Holography: New Developments,” Opt. Eng. 19, 727 (1980).
[CrossRef]

B. J. Thompson, P. Dunn, “Recent Advances in Holography,” Proc. Soc. Photo-Opt. Instrum, Eng. 215, 102 (1980).

P. Dunn, J. M. Walls, “Improved Microimages from In-line Absorption Holograms,” Appl. Opt. 18, 263 (1979).
[CrossRef] [PubMed]

P. Dunn, J. M. Walls, “Absorption and Phase In-line Holograms: a Comparison,” Appl. Opt. 18, 2171 (1979).
[CrossRef] [PubMed]

Farmer, W. M.

Fujiwara, H.

K. Murata, H. Fujiwara, T. Asakura, in Proceedings, Symposium on Engineering Uses of Holography (Strathclyde University, Glasgow1970), p. 289.

Haussmann, G.

Jones, R.

See, for example, R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983), p. 180.

Lauterborn, W.

Murata, K.

K. Murata, H. Fujiwara, T. Asakura, in Proceedings, Symposium on Engineering Uses of Holography (Strathclyde University, Glasgow1970), p. 289.

Parrent, G. B.

J. B. DeVelis, G. B. Parrent, B. J. Thompson, “Image Reconstruction with Fraunhofer Holograms,” J. Opt. Soc. Am. 56, 423 (1966).
[CrossRef]

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (far field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[CrossRef]

Payne, P. R.

Robinson, D. M.

Stainer, M. C.

R. Bexon, M. G. Dalzell, M. C. Stainer, “In-Line Holography and the Assessment of Aerosols,” Opt. Laser Technol. 8, 161 (1976).
[CrossRef]

Steward, R. G.

Thompson, B. J.

B. J. Thompson, P. Dunn, “Recent Advances in Holography,” Proc. Soc. Photo-Opt. Instrum, Eng. 215, 102 (1980).

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle Sizing Using Far-Field Holography: New Developments,” Opt. Eng. 19, 727 (1980).
[CrossRef]

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[CrossRef]

B. J. Thompson, “Holographic Particle Sizing Techniques,” J. Phys. E 7, 781 (1974).
[CrossRef]

B. J. Thompson, J. H. Ward, W. R. Zinky, “Application of Hologram Techniques for Particle Size Analysis,” Appl. Opt. 6, 519 (1967).
[CrossRef] [PubMed]

J. B. DeVelis, G. B. Parrent, B. J. Thompson, “Image Reconstruction with Fraunhofer Holograms,” J. Opt. Soc. Am. 56, 423 (1966).
[CrossRef]

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (far field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[CrossRef]

Trolinger, J. D.

Tyler, G. A.

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[CrossRef]

Vikram, C. S.

Walls, J. M.

Ward, J. H.

Wykes, C.

See, for example, R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983), p. 180.

Zinky, W. R.

Appl. Opt. (8)

J. Opt. Soc. Am. (1)

J. Phys. E (1)

B. J. Thompson, “Holographic Particle Sizing Techniques,” J. Phys. E 7, 781 (1974).
[CrossRef]

Microscope (1)

J. T. Bartlett, R. J. Adams, “Development of a Holographic Technique for Sampling Particles in Moving Aerosols,” Microscope 20, 375 (1972).

Opt. Acta (2)

G. B. Parrent, B. J. Thompson, “On the Fraunhofer (far field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background,” Opt. Acta 11, 183 (1964).
[CrossRef]

G. A. Tyler, B. J. Thompson, “Fraunhofer Holography Applied to Particle Size Analysis: A Reassessment,” Opt. Acta 23, 685 (1976).
[CrossRef]

Opt. Eng. (2)

S. L. Cartwright, P. Dunn, B. J. Thompson, “Particle Sizing Using Far-Field Holography: New Developments,” Opt. Eng. 19, 727 (1980).
[CrossRef]

J. D. Trolinger, “Particle Field Holography,” Opt. Eng. 14, 383 (1975).
[CrossRef]

Opt. Laser Technol. (1)

R. Bexon, M. G. Dalzell, M. C. Stainer, “In-Line Holography and the Assessment of Aerosols,” Opt. Laser Technol. 8, 161 (1976).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum, Eng. (1)

B. J. Thompson, P. Dunn, “Recent Advances in Holography,” Proc. Soc. Photo-Opt. Instrum, Eng. 215, 102 (1980).

Other (3)

K. Murata, H. Fujiwara, T. Asakura, in Proceedings, Symposium on Engineering Uses of Holography (Strathclyde University, Glasgow1970), p. 289.

See, for example, R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U.P., London, 1983), p. 180.

R. A. Belz, in “Holography and Optical Filtering,” NASA Report SP-299 (1973), p. 193.

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Figures (4)

Fig. 1
Fig. 1

Reconstructed image intensity variation for N = 3 and p = 0.1. Dashed curve corresponds to the optimum case when N is very large.

Fig. 2
Fig. 2

Reconstructed image intensity variation for N = 3 and p = 0.2. Dashed curve corresponds to the optimum case when N is very large.

Fig. 3
Fig. 3

Reconstructed image intensity variation for N = 3 and p = 0.3. Dashed curve corresponds to the optimum case when N is very large.

Fig. 4
Fig. 4

Variation of image-to-background intensity ratio α at the edges against p for a few values of N. The dashed curve represents the optimum case when N is very large.

Equations (18)

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τ ( x , y ) = τ b - K I ( x , y ) ,
I ( μ , ν ) = I 1 + I 2 + I 3 + I 4 2 ,
I 1 = p ,
I 2 = A * ( μ , ν ) ,
I 3 = - ( 2 λ z ) - 1 exp [ i ( π R 2 / 2 λ z + π / 2 ) ] A ˜ ( μ / 2 λ z , ν / 2 λ z ) ,
I 4 = - ( λ z ) - 2 A ˜ ( μ / λ z , ν / λ z ) 2 .
p = τ 0 / ( τ b - τ 0 ) ,
I ( μ , ν ) = [ p + A ( μ , ν ) ] 2 + ( λ z ) - 1 [ p + A ( μ , ν ) ] sin ( π R 2 / 2 λ z ) A ˜ ( μ / 2 λ z , ν / 2 λ z ) - ( λ z ) - 3 sin ( π R 2 / 2 λ z ) A ˜ ( μ / 2 λ z , ν / 2 λ z ) [ A ˜ ( μ / λ z , ν / λ z ) ] 2 - 2 ( λ z ) - 2 [ p + A ( μ , ν ) ] [ A ˜ ( μ / λ z , ν / λ z ) ] 2 + ( 2 λ z ) - 2 [ A ˜ ( μ / 2 λ z , ν / 2 λ z ) ] 2 + ( λ z ) - 4 [ A ˜ ( μ / λ z , ν / λ z ) ] 4 ,
R 2 = μ 2 + ν 2 .
I ( μ , ν ) [ p + A ( μ , ν ) ] 2 .
A ( ξ , η ) = 1 for ( ξ 2 + η 2 ) 1 / 2 a = 0 for ( ξ 2 + η 2 ) 1 / 2 > a ,
A ˜ ( r / λ z ) = 2 π a 2 Λ 1 ( 2 π a r / λ z ) ,
I ( R ) = [ p + circ ( R / a ) ] 2 + ( π / 2 N ) [ p + circ ( R / a ) ] sin ( π R 2 / 8 a 2 N ) Λ 1 ( π R / 4 N a ) - ( π / 2 N ) 3 sin ( π R 2 / 8 a 2 N ) Λ 1 ( π R / 4 N a ) Λ 1 2 ( π R / 2 N a ) - 2 ( π / 2 N ) 2 [ p + circ ( R / a ) ] Λ 1 2 ( π R / 2 N a ) + ( π / 4 N ) 2 Λ 1 2 ( π R / 4 N a ) + ( π / 2 N ) 4 Λ 1 4 ( π R / 2 N a ) ,
circ ( R / a ) = 1 for R a = 0 for R > a ,
N = λ z ( 2 a ) 2 .
I ( a ) background = p 2 + 2 p sin ( π / 8 N ) J 1 ( π / 4 N ) - 2 sin ( π / 8 N ) J 1 ( π / 4 N ) J 1 2 ( π / 2 N ) - 2 p J 1 2 ( π / 2 N ) + J 1 2 ( π / 4 N ) + J 1 4 ( π = 2 N )
I ( a ) image = I ( a ) background + 2 p + 1 + 2 sin ( π / 8 N ) J 1 ( π / 4 N ) - 2 J 1 2 ( π / 2 N )
α = I ( a ) image / I ( a ) background

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