Abstract

Wavelet analysis provides an efficient tool in numerous signal processing problems and has been implemented in optical processing techniques, such as in-line holography. This paper proposes an improvement of this tool for the case of an elliptical, astigmatic Gaussian (AEG) beam. We show that this mathematical operator allows reconstructing an image of a spherical particle without compression of the reconstructed image, which increases the accuracy of the 3D location of particles and of their size measurement. To validate the performance of this operator we have studied the diffraction pattern produced by a particle illuminated by an AEG beam. This study used mutual intensity propagation, and the particle is defined as a chirped Gaussian sum. The proposed technique was applied and the experimental results are presented.

© 2013 Optical Society of America

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  1. C. Vikram, Particle Field Holography (Cambridge University, 1992).
  2. G. Hausmann and W. Lauterborn, “Determination of size and position of fast moving glass bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
    [CrossRef]
  3. D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
    [CrossRef]
  4. D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
    [CrossRef]
  5. W. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249–255 (1995).
    [CrossRef]
  6. C. Buraga-Lefebvre, S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
    [CrossRef]
  7. L. Onural, “Diffraction from wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef]
  8. T. M. Kreis and W. Jüeptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
    [CrossRef]
  9. M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
    [CrossRef]
  10. F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
    [CrossRef]
  11. N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
    [CrossRef]
  12. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  13. J. Ricklin and F. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802(2002).
    [CrossRef]
  14. C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
    [CrossRef]
  15. J. Goodman, Statistical Optics (Wiley, 2000).
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  17. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Courier Dover, 1970).
  18. S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
    [CrossRef]
  19. J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  20. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
    [CrossRef]
  21. B. Ge, Q. Lu, and Y. Zhang, “Particle digital in-line holography with spherical wave recording,” Chin. Opt. Lett. 01, 517 (2003).
  22. R. Owen and A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
    [CrossRef]
  23. S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
    [CrossRef]
  24. M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
    [CrossRef]
  25. N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
    [CrossRef]
  26. S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
    [CrossRef]

2013 (2)

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
[CrossRef]

2011 (1)

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

2010 (2)

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
[CrossRef]

2008 (1)

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

2006 (1)

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

2005 (1)

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

2004 (2)

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[CrossRef]

2003 (1)

B. Ge, Q. Lu, and Y. Zhang, “Particle digital in-line holography with spherical wave recording,” Chin. Opt. Lett. 01, 517 (2003).

2002 (2)

S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

J. Ricklin and F. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802(2002).
[CrossRef]

2000 (2)

R. Owen and A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

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

1999 (1)

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

1997 (1)

T. M. Kreis and W. Jüeptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

1995 (1)

W. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249–255 (1995).
[CrossRef]

1993 (1)

L. Onural, “Diffraction from wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef]

1988 (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1980 (1)

G. Hausmann and W. Lauterborn, “Determination of size and position of fast moving glass bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Courier Dover, 1970).

Allano, D.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[CrossRef]

Anderson, W.

W. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249–255 (1995).
[CrossRef]

Belaïd, S.

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

Bengtsson, B.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Brunel, M.

C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
[CrossRef]

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
[CrossRef]

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

Buraga-Lefebvre, C.

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

Callens, N.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

Coëtmellec, S.

C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
[CrossRef]

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
[CrossRef]

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

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

Corbin, F.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

Davidson, F.

J. Ricklin and F. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802(2002).
[CrossRef]

Diao, H.

W. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249–255 (1995).
[CrossRef]

Dubois, F.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

Egelberg, P.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Foucaut, J.-M.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

Ge, B.

B. Ge, Q. Lu, and Y. Zhang, “Particle digital in-line holography with spherical wave recording,” Chin. Opt. Lett. 01, 517 (2003).

Godard, G.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

Goodman, J.

J. Goodman, Statistical Optics (Wiley, 2000).

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Gustafsson, M.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Hausmann, G.

G. Hausmann and W. Lauterborn, “Determination of size and position of fast moving glass bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
[CrossRef]

Janssen, A.

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

Jüeptner, W.

T. M. Kreis and W. Jüeptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Kreis, T. M.

T. M. Kreis and W. Jüeptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Lauterborn, W.

G. Hausmann and W. Lauterborn, “Determination of size and position of fast moving glass bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
[CrossRef]

Lebrun, D.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
[CrossRef]

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
[CrossRef]

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

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

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

Lecordier, B.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

Lenart, T.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Lu, Q.

B. Ge, Q. Lu, and Y. Zhang, “Particle digital in-line holography with spherical wave recording,” Chin. Opt. Lett. 01, 517 (2003).

Malek, M.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[CrossRef]

Nicolas, F.

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

Onural, L.

L. Onural, “Diffraction from wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef]

Owen, R.

R. Owen and A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Özkul, C.

S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

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

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

Paranthoën, P.

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

Pettersson, S.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Remacha, C.

C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
[CrossRef]

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

Ricklin, J.

J. Ricklin and F. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802(2002).
[CrossRef]

Salah, N.

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

Schockaert, C.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

Sebesta, M.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Courier Dover, 1970).

Verrier, N.

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
[CrossRef]

Vikram, C.

C. Vikram, Particle Field Holography (Cambridge University, 1992).

Walle, F.

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yourassowsky, C.

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

Zhang, Y.

B. Ge, Q. Lu, and Y. Zhang, “Particle digital in-line holography with spherical wave recording,” Chin. Opt. Lett. 01, 517 (2003).

Zozulya, A.

R. Owen and A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Appl. Opt. (5)

G. Hausmann and W. Lauterborn, “Determination of size and position of fast moving glass bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
[CrossRef]

D. Allano, M. Malek, F. Walle, F. Corbin, G. Godard, S. Coëtmellec, B. Lecordier, J.-M. Foucaut, and D. Lebrun, “Three-dimensional velocity near-wall measurements by digital in-line holography: calibration and results,” Appl. Opt. 52, A9–A17 (2013).
[CrossRef]

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730–3734 (1999).
[CrossRef]

W. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249–255 (1995).
[CrossRef]

C. Remacha, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a rectangular complex coherence factor,” Appl. Opt. 52, A147–A160 (2013).
[CrossRef]

Chin. Opt. Lett. (1)

B. Ge, Q. Lu, and Y. Zhang, “Particle digital in-line holography with spherical wave recording,” Chin. Opt. Lett. 01, 517 (2003).

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (2)

S. Coëtmellec, N. Verrier, M. Brunel, and D. Lebrun, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Publ. 5, 10027 (2010).
[CrossRef]

S. Coëtmellec, C. Remacha, M. Brunel, D. Lebrun, and A. Janssen, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Publ. 6, 11060 (2011).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Ricklin and F. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802(2002).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569–2577 (2005).
[CrossRef]

Meas. Sci. Technol. (1)

N. Salah, G. Godard, D. Lebrun, P. Paranthoën, D. Allano, and S. Coëtmellec, “Application of multiple exposure digital in-line holography to particle tracking in a Benard–von Karman vortex flow,” Meas. Sci. Technol. 19, 074001 (2008).
[CrossRef]

Opt. Eng. (2)

R. Owen and A. Zozulya, “In-line digital holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

T. M. Kreis and W. Jüeptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997).
[CrossRef]

Opt. Express (3)

N. Verrier, C. Remacha, M. Brunel, D. Lebrun, and S. Coëtmellec, “Micropipe flow visualization using digital in-line holographic microscopy,” Opt. Express 18, 7807–7819 (2010).
[CrossRef]

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, “Focus plane detection criteria in digital holography microscopy by amplitude analysis,” Opt. Express 14, 5895–5908 (2006).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, and D. Lebrun, “Digital in-line holography: influence of the shadow density on particle field extraction,” Opt. Express 12, 2270–2279 (2004).
[CrossRef]

Opt. Lasers Eng. (2)

M. Gustafsson, M. Sebesta, B. Bengtsson, S. Pettersson, P. Egelberg, and T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

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

Opt. Lett. (1)

L. Onural, “Diffraction from wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
[CrossRef]

Other (5)

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. Goodman, Statistical Optics (Wiley, 2000).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Courier Dover, 1970).

C. Vikram, Particle Field Holography (Cambridge University, 1992).

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Figures (7)

Fig. 1.
Fig. 1.

Theoretical setup.

Fig. 2.
Fig. 2.

Diffraction pattern with D=150μm, ze=154mm, λ=632.8nm, ωξ=7mm, ων=2.27mm, Rξ=4.38·105m, and Rν=65mm. (a) Figure 4 of [22] and (b) simulated from Eq. (13).

Fig. 3.
Fig. 3.

Profiles along the (a) x axis and (b)  y axis of Fig. 2(a) (dotted curve) and of Fig. 2(b) (solid curve).

Fig. 4.
Fig. 4.

Application of the EWT. (a) Diffraction pattern from Eq. (14) with D=150μm, ze=154mm, λ=632.8nm, ωξ=7mm, ων=2.27mm, Rξ=4.38·105m, and Rν=65mm. (b) Reconstruction by EWT with ax=3.22·107m2 and ay=2.35·107m2. (c) Application of a factor of F=ax/ay=1.4 along the x axis.

Fig. 5.
Fig. 5.

Application of the Fresnel transformation on the hologram in Fig. 4(a) with z=146mm.

Fig. 6.
Fig. 6.

Reconstruction processing: comparison between FRFT and EWT. (a) Hologram of a latex ball (5 μm) in micropipe (100 μm) with Fig. 5 of [11]. (b)–(c) (zoom): FRFT of the intensity distribution with orders along x:0.89 and along y:0.96. (d)–(e) (resizing): EWT of the intensity distribution with ax=4.13·107m2 and ay=1.37·107m2.

Fig. 7.
Fig. 7.

Reconstruction processing: comparison between FRFT and EWT. (a) Hologram of a latex ball (5 μm) in micropipe (100 μm). (b)–(c) (zoom): FRFT of the intensity distribution with orders along x:0.96 and along y:0.985. (d)–(e) (resizing): EWT of the intensity distribution with ax=1.4·107m2 and ay=6.9·106m2.

Equations (57)

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Is(x,y)=1(λze)2R4Jt(ξ1,ξ2,ν1,ν2)exp{iπλze[ξ22ξ122x(ξ2ξ1)]}exp{iπλze[ν22ν122y(ν2ν1)]}dξ1dξ2dν1dν2,
Ji(ξ1,ξ2,ν1,ν2)=A(ξ1,ν1)A*(ξ2,ν2).
A(ξ,ν)=exp[(ξ2ωξ2+ν2ων2)]exp[iπλ(ξ2Rξ+ν2Rν)].
Ji(ξ1,ξ2,ν1,ν2)=exp[(Cξ*ξ12+Cν*ν12)]exp[(Cξξ22+Cνν22)],
Cj=1ωj2iπλRj,
Jt(ξ1,ξ2,ν1,ν2)=P(ξ1,ν1)P*(ξ2,ν2)Ji(ξ1,ξ2,ν1,ν2)
P(ξ,ν)=1n=1NAnexp[BnR2(ξ2+ν2)]
Is(x,y)=1(λze)2[I1(x,y)I2(x,y)I3(x,y)+I4(x,y)].
I1(x,y)=C1exp[Dξ(Cξ+Cξ*)x2]exp[Dν(Cν+Cν*)y2],
I2(x,y)=π2n=1N{An(Cξ+iBz)1/2(Cξ*iBz+BnR2)1/2(Cν+iBz)1/2×(Cν*iBz+BnR2)1/2exp(Bz2x2Cξ+iBz)exp(Bz2x2Cξ*iBz+BnR2)exp(Bz2y2Cν+iBz)exp(Bz2y2Cν*iBz+BnR2)}.
I2(x,y)=I1(x,y)(AξAν)1/4FHyp1/2exp(iPxx2+iPyy2)n=1N{AnR2Bnexp[Bz2R2Bn(x2+y2)]}
I2(x,y)+I3(x,y)=I1(x,y)(AξAν)1/4×F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y){2ahcos[Pxx2+Pyy2]2bhsin[Pxx2+Pyy2]},
Is(x,y)=1(λze)2[I1(x,y)I2(x,y)I3(x,y)].
IH(x,y)=1(AξAν)1/4F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y){2ahcos[Pxx2+Pyy2]2bhsin[Pxx2+Pyy2]}.
Cb(xa,ya)={IH(x,y)**(|axay|)1/2(arcos[axx2+ayy2]+brsin[axx2+ayy2])},
Cb(xa,ya)=(|axay|)1/2R2IH(x,y)×{arcos[ax(xax)2+ay(yay)2]+brsin[ax(xax)2+ay(yay)2]}dxdy.
ajopt=Pj=Dl(Bz+ClCl*2i),
Cb(xa,ya)=π(ah+bh)π(ah2+bh2)circ[(xa2Rx2+ya2Ry2)1/2]cos(Pxxa2+Pyya2)Ju1(xa,ya),
Ju1(xa,ya)=π2(λze)2(ah2bh2)R2FP(x,y)cos[Px(2x22xxa+xa2)+Py(2y22yya+ya2)]dxdy,
Fp(x,y)=(AξAν)1/4F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y).
Jt(ξ1,ξ2,ν1,ν2)=Ji(ξ1,ξ2,ν1,ν2)(1n=1N{Anexp[BnR2(ξ12+ν12)]}n=1N{An*exp[Bn*R2(ξ22+ν22)]}+n=1N{Anexp[BnR2(ξ12+ν12)]}n=1N{An*exp[Bn*R2(ξ22+ν22)]}).
I1(x,y)=R4exp[i2Bz(xξ2xξ1)]exp[iBz(ξ22ξ12)]exp(Cξξ22Cξ*ξ12)exp[i2Bz(xν2xν1)]exp[iBz(ν22ν12)]exp(Cνξ22Cν*ν12)dξ1dξ2dν1dν2.
I1=C1exp[Dξ(Cξ+Cξ*)x2]exp[Dν(Cν+Cν*)y2].
I2(x,y)=R4exp[i2Bz(xξ2xξ1)]exp[iBz(ξ22ξ12)]exp(Cξξ22Cξ*ξ12)×exp[i2Bz(yν2yν1)]exp[iBz(ν22ν12)]exp(Cνν22Cν*ν12)×n=1NAnexp[Bgn(ξ12+ν12)]dξ1dξ2dν1dν2.
I2(x,y)=π2n=1NAn(Cξ+iBz)1/2(Cξ*iBz+Bgn)1/2(Cν+iBz)1/2(Cν*iBz+Bgn)1/2×exp(Bz2x2Cξ+iBz)exp(Bz2x2Cξ*iBz+Bgn)×exp(Bz2y2Cν+iBz)exp(Bz2y2Cν*iBz+Bgn).
I3(x,y)=I2(x,y)*.
I4(x,y)=R4exp[i2Bz(xξ2xξ1)]exp[iBz(ξ22ξ12)]exp(Cξξ22Cξ*ξ12)exp[i2Bz(yν2yν1)]exp[iBz(ν22ν12)]exp(Cνν22Cν*ν12)n=1N{Anexp[Bgn(ξ12+ν12)]}n=1N{An*exp[Bgn*(ξ22+ν22)]}dξ1dξ2dν1dν2.
I4(x,y)=π2n=1N[An*(Cξ+iBz+Bgn*)1/2(Cν+iBz+Bgn*)1/2×exp(Bz2y2Cν+iB+Bgn*)exp(Bz2x2Cξ+iBz+Bgn*)]×n=1N[An(Cξ*iBz+Bgn)1/2(Cν*iBz+Bgn)1/2×exp(Bz2x2Cξ*iBz+Bgn)exp(Bz2y2Cν*iBz+Bgn)].
πCξ+iBzπCξ*iBz+Bgn=πAξ1/4Aξ1/21Bgn1/2FxHyp1/2,
FxHyp1/2={Aξ1/2Bgn(Cξ+iBz)[(Cξ*iBz)+Bgn]}1/2.
|πλRξπλze||IBgnR2|,
1ωξ2R(BnR2).
FxHyp1/2=(Aξ1/2Cξ+iBz)1/2.
FHyp=(Cξ*iBz)(Cν*iBz)AξAν.
|1ωj||πλze+πλRj|,
FHyp=sgn[I(Cξ*)Bz]sgn[I(Cν*)Bz].
Bz2Cξ+iBzBz2Cξ*iBz+Bgn=Bz2Aξ(Cξ+Cξ*)+Bz2Aξ(iBz+CξCξ*2Bz2Bgn)+Δaeg,
Δaeg=Bz2Cξ*iBzBgn(Cξ*iBz+Bgn).
|R(Δaeg)||R[Bz2Aξ(Cξ+Cξ*)+Bz2Aξ(iBz+CξCξ*2)Bz2Bgn]|,
|I(Δaeg)||I[Bz2Aξ(Cξ+Cξ*)+Bz2Aξ(iBz+CξCξ*2)Bz2Bgn]|,
I2(x,y)=C1(AξAν)1/4FHyp1/2exp[Dξ(Cξ+Cξ*)x2]exp[Dν(Cν+Cν*)y2]exp(iPxx2+iPyy2)n=1N{AnBgnexp[Bz2Bgn(x2+y2)]},
I3(x,y)=C1(AξAν)1/4FHyp*1/2exp[Dξ(Cξ+Cξ*)x2]exp[Dν(Cν+Cν*)y2]exp(iPxx2Pyy2)n=1N{An*Bgn*exp[Bz2Bgn*(x2+y2)]}.
n=1NAn*Bgn*exp[Bz2Bgn*(x2+y2)]=n=1NAnBgnexp[Bz2Bgn(x2+y2)]=F1{(λze)2πcirc[(xa2+ya2)1/2R1]},
I2(x,y)+I3(x,y)=C1(AξAν)1/4exp[Dξ(Cξ+Cξ*)x2]exp[Dν(Cν+Cν*)y2]×F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)(2ah2{exp[iPxx2+iPyy2]+exp[iPxx2iPyy2]}2bh2i{exp[iPxx2+iPyy2]exp[iPxx2iPyy2]}).
I2(x,y)+I3(x,y)=C1(AξAν)1/4exp[Dξ(Cξ+Cξ*)x2]exp[Dν(Cν+Cν*)y2]×F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y){2ahcos[Pxx2+Pyy2]2bhsin[Pxx2+Pyy2]}.
Cb(xa,ya)=π(|PxPy|)1/2R2(AξAν)1/4F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)×{2ahcos[P1]2bhsin[P1]}{ahcos[P2]+bhsin[P2]}dxdy,
Cb(xa,ya)=π(|PxPy|)1/2(AξAν)1/4R2F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)×{2ah2cos[P1]cos[P2]+2ahbhcos[P1]sin[P2]2bhahsin[P1]cos[P2]2bh2sin[P1]sin[P2]}dxdy.
Cb(xa,ya)=π(PxPy)1/2(AξAν)1/4[T1(xa,ya)+T2(xa,ya)],
T1(xa,ya)=R2F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)×(ah22{exp[iPxx2+iPyy2]+exp[iPxx2iPyy2]}×{exp[iPx(x22xxa+xa2)+iPy(y22yya+ya2)]+exp[iPx(x22xxa+xa2)iPy(y22yya+ya2)]})dxdy.
T1(xa,ya)=R2F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)×[ah22(exp[2i(xxaPx+yyaPy)]exp[i(Pxx2iPyy2)]+exp[i(Pxxa2+Pyya2)]exp[i(Pxxa2+Pyy2)]+exp{i[Px(2x22xxa+xa2)+Py(2y22yya+ya2)]}+exp{i[Px(2x22xxa+xa2)+Py(2y22yya+ya2)]})]dxdy.
R2F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)exp[2i(xxaPx+yyaPy)]dxdy=R2F1{(λze)2πcirc[(xa2+ya2)1/2R1]}(x,y)exp[2i(xxaPx+yyaPy)]dxdy=(λze)2πcirc[(xa2Rx2+ya2Ry2)1/2],
T1(xa,ya)=ah2(λze)2πcirc[(xa2Rx2+ya2Ry2)1/2]cos(Pxxa2+Pyya2)+ah2R2cos[Px(2x22xxa+xa2)+Py(2y22yya+ya2)]dxdy.
T2(xa,ya)=bh2(λze)2πcirc[(xa2Rx2+ya2Ry2)1/2]cos(Pxxa2+Pyya2)bh2R2cos[Px(2x22xxa+xa2)+Py(2y22yya+ya2)]dxdy.
Cb(xa,ya)=π(ah2+bh2)Faegcirc[(xa2Rx2+ya2Ry2)1/2]cos(Pxxa2+Pyya2)Ju1(xa,ya),
Faeg=(λze)2π(AξAν)1/4(|PxPy|)1/2=(λze)2π(AξAν)1/4(|Bz4AξAν(BzI(Cξ))(BzI(Cν))|)1/2.
Faeg=π(AξAν)1/4(|(BzI(Cξ))(BzI(Cν))|)1/2,
Faeg=π.

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