Abstract

We propose in this paper the study of a particular spatially partially coherent source applied to digital in-line holography of dense particle flow. A source with a rectangular complex coherence factor is implemented. The effects of such a source on the intensity distribution of the diffraction pattern are described. In particular, we show that this type of source allows us to eliminate the diffraction pattern along one axis while all the information about the dimension of the particle is kept along the other perpendicular axis. So particle images can be well reconstructed along one direction and the speckle can be largely limited.

© 2012 Optical Society of America

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References

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    [CrossRef]
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2011 (1)

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

2010 (3)

2009 (3)

J. Lee, B. Miller, and A. Sallam, “Demonstration of digital holographic diagnostics for breakup of liquid jets using a commercial-grade CCD sensor,” Atomization Sprays 19, 445–456 (2009).
[CrossRef]

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17, 20343–20348 (2009).
[CrossRef]

M. A. Alonso, “Diffraction of paraxial partially coherent fields by planar obstacles in the Wigner representation,” J. Opt. Soc. Am. A 26, 1588–1597 (2009).
[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]

2007 (1)

2006 (2)

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006).
[CrossRef]

2004 (1)

2003 (1)

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

2002 (2)

2000 (1)

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

1999 (1)

1996 (1)

1993 (1)

1988 (1)

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

1984 (1)

1971 (1)

1968 (1)

Abramowitz, M.

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

Allano, D.

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. Slimani, G. Grehan, G. Gouesbet, and D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
[CrossRef]

Alonso, M. A.

Anderson, W. L.

Bishara, W.

Born, M.

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

Breazeale, M.

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

Brunel, M.

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

S. Coëtmellec, N. Verrier, D. Lebrun, and M. Brunel, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Pub. 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, and D. Lebrun, “Digital in-line holography with sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[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. Laser Eng. 33, 409–421 (2000).
[CrossRef]

Coëtmellec, S.

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

S. Coëtmellec, N. Verrier, D. Lebrun, and M. Brunel, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Pub. 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, and D. Lebrun, “Digital in-line holography with sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[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. Laser Eng. 33, 409–421 (2000).
[CrossRef]

C. Remacha, S. Coëtmellec, D. Lebrun, J.-M. Dorey, and F. David, “Inhomogeous dense flow: impact of the spatial coherence on digital holography,” presented at the International Symposium on Multiphase Flow and Transport Phenomena, Agadir, Morocco, 22–25 April 2012.

Coskun, A. F.

David, F.

C. Remacha, S. Coëtmellec, D. Lebrun, J.-M. Dorey, and F. David, “Inhomogeous dense flow: impact of the spatial coherence on digital holography,” presented at the International Symposium on Multiphase Flow and Transport Phenomena, Agadir, Morocco, 22–25 April 2012.

Davidson, F. M.

Denis, L.

Di, J.

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17, 20343–20348 (2009).
[CrossRef]

Dorey, J.-M.

C. Remacha, S. Coëtmellec, D. Lebrun, J.-M. Dorey, and F. David, “Inhomogeous dense flow: impact of the spatial coherence on digital holography,” presented at the International Symposium on Multiphase Flow and Transport Phenomena, Agadir, Morocco, 22–25 April 2012.

Dubois, F.

Fatih Erden, M.

Fournier, C.

Garcia-Sucerquia, J.

Godard, G.

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]

Goepfert, C.

Goodman, J. W.

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

Gouesbet, G.

Grehan, G.

Hernandez, C.

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Hussain, F.

Illueca, C.

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Jericho, M. H.

Jericho, S. K.

Joannes, L.

Klages, P.

Kreuzer, H. J.

Lebrun, D.

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

S. Coëtmellec, N. Verrier, D. Lebrun, and M. Brunel, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Pub. 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, and D. Lebrun, “Digital in-line holography with sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[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. Laser Eng. 33, 409–421 (2000).
[CrossRef]

C. Remacha, S. Coëtmellec, D. Lebrun, J.-M. Dorey, and F. David, “Inhomogeous dense flow: impact of the spatial coherence on digital holography,” presented at the International Symposium on Multiphase Flow and Transport Phenomena, Agadir, Morocco, 22–25 April 2012.

Lee, J.

J. Lee, B. Miller, and A. Sallam, “Demonstration of digital holographic diagnostics for breakup of liquid jets using a commercial-grade CCD sensor,” Atomization Sprays 19, 445–456 (2009).
[CrossRef]

Legros, J.-C.

Liu, D. D.

Lurie, M.

Lutomirski, R. F.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mas, D.

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Mendlovic, D.

Meng, H.

Miller, B.

J. Lee, B. Miller, and A. Sallam, “Demonstration of digital holographic diagnostics for breakup of liquid jets using a commercial-grade CCD sensor,” Atomization Sprays 19, 445–456 (2009).
[CrossRef]

Miret, J. J.

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Nicolas, F.

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

Ozaktas, H. M.

Ozcan, A.

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. Laser Eng. 33, 409–421 (2000).
[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]

Pérez, J.

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Piano, E.

Pontiggia, C.

Remacha, C.

S. Coëtmellec, C. Remacha, M. Brunel, and D. Lebrun, “Digital in-line holography with a spatially partially coherent beam,” J. Eur. Opt. Soc. Rapid Pub. 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]

C. Remacha, S. Coëtmellec, D. Lebrun, J.-M. Dorey, and F. David, “Inhomogeous dense flow: impact of the spatial coherence on digital holography,” presented at the International Symposium on Multiphase Flow and Transport Phenomena, Agadir, Morocco, 22–25 April 2012.

Repetto, L.

Ricklin, J. C.

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]

Sallam, A.

J. Lee, B. Miller, and A. Sallam, “Demonstration of digital holographic diagnostics for breakup of liquid jets using a commercial-grade CCD sensor,” Atomization Sprays 19, 445–456 (2009).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Slimani, F.

Soulez, F.

Stegun, I. A.

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

Su, T.-W.

Sun, W.

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17, 20343–20348 (2009).
[CrossRef]

Thiébaut, E.

Vazquez, C.

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Verrier, N.

S. Coëtmellec, N. Verrier, D. Lebrun, and M. Brunel, “General formulation of digital in-line holography from correlation with a chirplet function,” J. Eur. Opt. Soc. Rapid Pub. 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]

Wang, L.

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17, 20343–20348 (2009).
[CrossRef]

Wang, Q.

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17, 20343–20348 (2009).
[CrossRef]

Wen, J.

J. Wen and M. Breazeale, “A diffraction beam field expressed as 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).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Xu, W.

Yura, H. T.

Zhao, J.

W. Sun, J. Zhao, J. Di, Q. Wang, and L. Wang, “Real-time visualization of Karman vortex street in water flow field by using digital holography,” Opt. Express 17, 20343–20348 (2009).
[CrossRef]

Appl. Opt. (4)

Atomization Sprays (1)

J. Lee, B. Miller, and A. Sallam, “Demonstration of digital holographic diagnostics for breakup of liquid jets using a commercial-grade CCD sensor,” Atomization Sprays 19, 445–456 (2009).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

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

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

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

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. Commun. (2)

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with sub-picosecond laser beam,” Opt. Commun. 268, 27–33 (2006).
[CrossRef]

D. Mas, J. Pérez, C. Hernandez, C. Vazquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245–258 (2003).
[CrossRef]

Opt. Express (3)

Opt. Laser Eng. (1)

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

Opt. Lett. (1)

Other (7)

C. Remacha, S. Coëtmellec, D. Lebrun, J.-M. Dorey, and F. David, “Inhomogeous dense flow: impact of the spatial coherence on digital holography,” presented at the International Symposium on Multiphase Flow and Transport Phenomena, Agadir, Morocco, 22–25 April 2012.

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

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

A. E. Siegman, Lasers (University Science, 1986).

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

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

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, 1986).

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Figures (10)

Fig. 1.
Fig. 1.

Theoretical setup.

Fig. 2.
Fig. 2.

Illustration of the influence of the spatial filtering of an incoherent source on the coherence factor. At the top is the filter function, and at the bottom is the the modulus of the complex coherence factor ( | μ ( ξ , ν , Δ ξ , Δ ν ) | ) from the center point in the particle plane ( ζ s = 175 mm ) ( white = 1 , black = 0 ). (a)  ω β = ω α = 10 μm , (b)  ω β = ω α = 200 μm , and (c)  ω α = 10 μm , ω β = 200 μm .

Fig. 3.
Fig. 3.

Experimental setup. For examples presented Figs. 4 and 5 the parameters are filter- L 2 ( f = 50 mm ) = 50 mm , L 2 - Disk = 125 mm , Disk ( R = 35 μm ) - CDD = 115 mm .

Fig. 4.
Fig. 4.

Comparison between the analytic (a) and dotted curves of (c) and the experimental (b) and solid curves of (c), with ω β = 200 μm , ω α = 20 μm , ζ e = 115 mm , ζ s = 175 mm , R = 35 μm , and λ = 532 nm .

Fig. 5.
Fig. 5.

Comparison between the analytic (a) and dotted curves of (c) and the experimental (b) and the solid curves of (c), with ω α = 50 μm , ω β = 200 μm , ζ e = 115 mm , ζ s = 175 mm , R = 35 μm , and λ = 532 nm .

Fig. 6.
Fig. 6.

Holograms of oil drops and their profiles ( ζ s = 200 mm , ζ e = 440 mm , and R 150 μm ) illuminated by (a) a coherent source, (b) a spatially partially coherent source with a circular complex coherence factor ( ω α = ω β = 100 μm ), and (c) a spatially partially coherent source with a rectangular complex coherence factor ( ω α = 10 μm , ω β = 200 μm ).

Fig. 7.
Fig. 7.

Reconstruction at ζ e = 115 mm of Fig. 4 and its equivalent with a coherent source. The ellipticity is due to the source, here R ell = 1 .

Fig. 8.
Fig. 8.

Profiles along x a -axis of Fig. 7(a) (dotted curve) and Fig. 7(b) (solid curve).

Fig. 9.
Fig. 9.

Reconstruction at ζ e = 115 mm of the experimental holograms of Figs. 4(b) [panel (a)] and 5(b) [panel (b)].

Fig. 10.
Fig. 10.

Comparison between the calculations without (a) and the solid curve of (c) and with (a) and the dotted curve of (c) simplification. ω = 80 μm , ζ s = 100 mm ; ζ e = 100 mm ; R = 40 μm .

Equations (65)

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J i ( ξ , ν , Δ ξ , Δ ν ) = exp [ i 2 π ( ξ Δ ξ + ν Δ ν ) λ ζ s ] π ζ s 2 R 2 I f ( α , β ) exp [ i 2 π λ ζ s ( α Δ ξ + β Δ ν ) ] d β d α ,
I s ( x , y ) = 1 ( λ ζ e ) 2 R 4 J t ( ξ , ν , Δ ξ , Δ ν ) exp [ i 2 π λ ζ e ( ξ Δ ξ + ν Δ ν x Δ ξ y Δ ν ) ] d ξ d ν d Δ ξ d Δ ν .
J t ( ξ , ν , Δ ξ , Δ ν ) = P ( ξ Δ ξ 2 , ν Δ ν 2 ) P * ( ξ + Δ ξ 2 , ν + Δ ν 2 ) J i ( ξ , ν , Δ ξ , Δ ν ) ,
P ( ξ , ν ) = 1 n = 1 N A n exp [ B n R 2 ( ξ 2 + R ell 2 ν 2 ) ] ,
I f ( α , β ) = I 0 exp [ ( α 2 ω α 2 + β 2 ω β 2 ) ] ,
J i ( ξ , ν , Δ ξ , Δ ν ) = C 1 exp [ i 2 π ( ξ Δ ξ + ν Δ ν ) λ ζ s ] exp [ π 2 ω α 2 ( λ ζ s ) 2 Δ ξ 2 ] exp [ π 2 ω β 2 ( λ ζ s ) 2 Δ ν 2 ] ,
| μ ( ξ , ν , Δ ξ , Δ ν ) | = exp [ π 2 ω α 2 ( λ ζ s ) 2 Δ ξ 2 ] exp [ π 2 ω β 2 ( λ ζ s ) 2 Δ ν 2 ] ,
I s ( x , y ) = C 2 R 4 P ( ξ Δ ξ 2 , ν Δ ν 2 ) P * ( ξ + Δ ξ 2 , ν + Δ ν 2 ) × exp [ π 2 ω α 2 ( λ ζ s ) 2 Δ ξ 2 + { i 2 π λ ζ e ( ξ [ 1 + ζ e ζ s ] x ) } Δ ξ ] × exp [ π 2 ω β 2 ( λ ζ s ) 2 Δ ν 2 + { i 2 π λ ζ e ( ν [ 1 + ζ e ζ s ] y ) } Δ ν ] d ξ d ν d Δ ξ d Δ ν ,
I s ( x , y ) = C 2 { I 1 ( x , y ) [ I 2 ( x , y ) + I 3 ( x , y ) ] + I 4 ( x , y ) } .
I 1 ( x , y ) = ( λ ζ e ) 2 ( 1 + ζ e ζ s ) 2 .
I 2 ( x , y ) = n = 1 N A n π 2 R 2 λ ζ e R ell B n ( K α n ) 1 / 2 ( K β n ) 1 / 2 exp [ π 2 λ ζ e x 2 K α n ] exp [ π 2 λ ζ e y 2 K β n ] ,
K α n = π 2 ζ e ω α 2 λ ζ s 2 [ 1 + R 2 B n * ζ s 2 ( 1 + ζ e ζ s ) 2 ω α 2 B n B n * ζ e 2 + i λ ζ s 2 ( 1 + ζ e ζ s ) π ω α 2 ζ e ] , K β n = π 2 ζ e ω β 2 λ ζ s 2 [ 1 + R 2 B n * ζ s 2 ( 1 + ζ e ζ s ) 2 R ell 2 ω β 2 B n B n * ζ e 2 + i λ ζ s 2 ( 1 + ζ e ζ s ) π ω β 2 ζ e ] .
I 2 ( x , y ) + I 3 ( x , y ) = 2 λ ζ e ( 1 + ζ e ζ s ) sin [ π ( x 2 + y 2 ) λ ζ e ( 1 + ζ e ζ s ) ] exp [ π 2 ( ω α 2 x 2 + ω β 2 y 2 ) ( λ ζ s ) 2 ( 1 + ζ e ζ s ) 2 ] × n = 1 N A n π R 2 R ell B n exp [ R 2 π 2 ( λ ζ e ) 2 B n ( x 2 + y 2 R ell 2 ) ] .
I 2 ( x , y ) + I 3 ( x , y ) = 2 λ ζ e ( 1 + ζ e ζ s )    sin [ π ( x 2 + y 2 ) λ ζ e ( 1 + ζ e ζ s ) ] exp [ π 2 ( ω α 2 x 2 + ω β 2 y 2 ) ( λ ζ s ) 2 ( 1 + ζ e ζ s ) 2 ] × F 1 { ( λ ζ e ) 2 circ [ ( x a 2 + R ell 2 y a 2 ) 1 / 2 R 1 ] } ( x , y ) ,
I s ( x , y ) C 2 · ( λ ζ e ) 2 2 C 2 · λ ζ e sin [ π ( x 2 + y 2 ) λ ζ e ] exp [ π 2 ( ω α 2 x 2 + ω β 2 y 2 ) ( λ ζ s ) 2 ] × F 1 { ( λ ζ e ) 2 circ [ ( x a 2 + R ell 2 y a 2 ) 1 / 2 R 1 ] } ( x , y ) .
C a ( x a , y a ) = 1 a 2 R 2 I s ( x , y ) sin [ ( x a x ) 2 + ( y a y ) 2 a 2 ] d x d y .
C a ( x a , y a ) = π C 2 · ( ζ e λ ) 2 C 2 · 2 λ ζ e a 2 R 2 F O ( x , y ) sin [ π ( x 2 + y 2 ) λ ζ e ] × sin [ x 2 2 x x a + x a 2 + y 2 2 y y a + y a 2 a 2 ] d x d y ,
F O ( x , y ) = exp [ π 2 ( ω α 2 x 2 + ω β 2 y 2 ) ( λ ζ s ) 2 ] F 1 { ( λ ζ e ) 2 circ [ ( x a 2 + R ell 2 y a 2 ) 1 / 2 R 1 ] } ( x , y ) .
a opt = ( λ ζ e π ) 1 / 2 .
C a ( x a , y a ) = π C 2 · ( λ ζ e ) 2 π C 2 · F [ F O ( x , y ) ] ( x a π a opt 2 , y a π a opt 2 ) · cos [ π ( x a 2 + y a 2 ) λ ζ e ] + π C 2 · R 2 F O ( x , y ) cos [ 2 ( x 2 + y 2 ) 2 ( x a x + y a y ) + ( y a 2 + x a 2 ) λ ζ e ] d x d y .
F [ F O ( x , y ) ] ( x a λ ζ e , y a λ ζ e ) = ( λ ζ e ) 2 circ [ ( x a 2 + R ell 2 y a 2 ) 1 / 2 R ] * * ζ s 2 ζ e 2 1 π ω β ω α exp ( x a 2 ζ s 2 ζ e 2 ω α 2 y a 2 ζ s 2 ζ e 2 ω β 2 ) ,
lim ω α 0 ω β 0 ζ s 2 ζ e 2 1 π ω β ω α exp ( x a 2 ζ s 2 ζ e 2 ω α 2 ) = δ ( x a , y a ) ,
F [ F O ( x , y ) ] ( x a λ ζ e , y a λ ζ e ) = ( λ ζ e ) 2 circ [ ( x a 2 + R ell 2 y a 2 ) 1 / 2 R ] * ζ s ζ e 1 ω β π 1 / 2 exp ( y a 2 ζ s 2 ζ e 2 ω β 2 ) .
P ( ξ Δ ξ 2 , ν Δ ν 2 ) P * ( ξ + Δ ξ 2 , ν + Δ ν 2 ) = 1 n = 1 N A n exp [ B n R 2 ( ξ 2 ξ Δ ξ + Δ ξ 2 4 + R ell 2 ν 2 R ell 2 ν Δ ν + R ell 2 Δ ν 2 4 ) ] n = 1 N A n * exp [ B n * R 2 ( ξ 2 + ξ Δ ξ + Δ ξ 2 4 + R ell 2 ν 2 + R ell 2 ν Δ ν + R ell 2 Δ ν 2 4 ) ] + n = 1 N 2 E n exp [ C n R 2 ( ξ 2 + R ell 2 ν 2 ) + D n R 2 ( ξ Δ ξ + R ell 2 ν Δ ν ) C n 4 R 2 ( Δ ξ 2 + R ell 2 Δ ν 2 ) ] ,
I s ( x , y ) = C 2 { I 1 ( x , y ) [ I 2 ( x , y ) + I 3 ( x , y ) ] + I 4 ( x , y ) } .
I 1 ( x , y ) = [ ( λ ζ s ) 2 π ω α 2 ] 1 / 2 [ ( λ ζ s ) 2 ω β 2 π ] 1 / 2 [ π ζ e 2 ω α 2 ( ζ e + ζ s ) 2 ] 1 / 2 [ π ζ e 2 ω β 2 ( ζ e + ζ s ) 2 ] 1 / 2 = ( λ ζ e ) 2 ( 1 + ζ e ζ s ) 2 .
I 2 ( x , y ) = R 4 n = 1 N A n exp { ξ 2 ( B n R 2 ) + ξ [ Δ ξ ( B n R 2 i 2 π λ ζ e { 1 + ζ e ζ s } ) ] } × exp { ν 2 ( R ell 2 B n R 2 ) + ν [ Δ ν ( R ell 2 B n R 2 i 2 π λ ζ e { 1 + ζ e ζ s } ) ] } × exp [ Δ ξ 2 ( π 2 ω α 2 ( λ ζ s ) 2 + B n 4 R 2 ) ] exp [ i Δ ξ 2 π λ ζ e x ] × exp [ Δ ν 2 ( π 2 ω β 2 ( λ ζ s ) 2 + R ell 2 B n 4 R 2 ) ] exp [ i Δ ν 2 π λ ζ e y ] d ξ d ν d Δ ξ d Δ ν .
I 2 ( x , y ) = R 2 n = 1 N A n ( π R 2 R ell B n ) exp [ Δ ξ 2 K α n λ ζ e + Δ ξ ( j 2 π λ ζ e x ) ] × exp [ Δ ν 2 K β n λ ζ e + Δ ν ( j 2 π λ ζ e y ) ] d Δ ξ d Δ ν ,
K α n = π 2 ζ e ω α 2 λ ζ s 2 [ 1 + R 2 B n * ζ s 2 ( 1 + ζ e ζ s ) 2 ω α 2 B n B n * ζ e 2 + i λ ζ s 2 ( 1 + ζ e ζ s ) π ω α 2 ζ e ] , K β n = π 2 ζ e ω β 2 λ ζ s 2 [ 1 + R 2 B n * ζ s 2 ( 1 + ζ e ζ s ) 2 R ell 2 ω β 2 B n B n * ζ e 2 + i λ ζ s 2 ( 1 + ζ e ζ s ) π ω β 2 ζ e ] .
I 2 ( x , y ) = n = 1 N A n π 2 R 2 λ ζ e R ell B n ( K α n ) 1 / 2 ( K β n ) 1 / 2 exp [ π 2 λ ζ e x 2 K α n ] exp [ π 2 λ ζ e y 2 K β n ] .
I 3 ( x , y ) = I 2 ( x , y ) * .
I 4 ( x , y ) = R 2 n = 1 N 2 E n R 2 π R ell C n exp ( K v i x n Δ ξ 2 ) exp [ i Δ ξ 2 π x λ ζ e ] × exp ( K v i y n Δ ν 2 ) exp [ i Δ ν 2 π y λ ζ e ] d Δ ξ d Δ ν ,
K v i x n = C n * 4 C n C n * R 2 [ D n 2 + i 4 π D n R 2 ( 1 + ζ e ζ s ) λ ζ e + 4 π 2 R 4 ( 1 + ζ e ζ s ) 2 ( λ ζ e ) 2 ] π 2 ω α 2 ( λ ζ s ) 2 C n 4 R 2
K v i y n = C n * 4 R 2 R ell 4 C n C n * [ R ell 2 D n 2 + i 4 R ell 2 R 2 D n π ( 1 + ζ e ζ s ) λ ζ e + 4 π 2 R 4 ( 1 + ζ e ζ s ) 2 ( λ ζ e ) 2 ] π 2 ω β 2 ( λ ζ s ) 2 R ell 2 C n 4 R 2 .
I 4 ( x , y ) = n = 1 N 2 E n π R 2 R ell C n ( π K v i x n ) 1 / 2 ( π K v i y n ) 1 / 2 exp [ π 2 ( λ ζ e ) 2 x 2 K v i x n ] exp [ π 2 ( λ ζ e ) 2 y 2 K v i y n ] .
| I [ R 2 π 2 B n * ( 1 + ζ e ζ s ) 2 R ell 2 B n B n * ζ e 2 ] | π ( 1 + ζ e ζ s ) λ ζ e .
| ( B n * B n B n * ) | < 0.15 ,
0.15 R 2 π 2 B n * ( 1 + ζ e ζ s ) 2 R ell 2 B n B n * ( ζ e ) 2 < 0.1 π ( 1 + ζ e ζ s ) λ ζ e .
R R ell < [ λ ζ e 1.5 π ( 1 + ζ e ζ s ) ] ,
R R ell [ λ ζ e 0.25 π ( 1 + ζ e ζ s ) ] 1 / 2 ,
| R ( B n * B n B n * ) | < 0.25 ,
ω β 2 π 2 ( λ ζ s ) 2 π ( 1 + ζ e ζ s ) λ ζ e ,
ω β < [ 0.1 λ ζ s 2 ( 1 + ζ e ζ s ) π ζ e ] 1 / 2 .
| K in | = π ( 1 + ζ e ζ s ) ,
( a + i b ) 1 / 2 = ± { [ ( a 2 + b 2 ) 1 / 2 + a 2 ] 1 / 2 + isign ( b ) [ ( a 2 + b 2 ) 1 / 2 a 2 ] 1 / 2 } .
π λ ζ e ( K α n ) 1 / 2 ( K β n ) 1 / 2 = i λ ζ e ( 1 + ζ e ζ s ) .
A n B n = ( a + i b ) ,
exp { y 2 [ R 2 π i ( B n ) R ell 2 B n * B n ( λ ζ e ) 2 ] x 2 [ R 2 π i ( B n ) B n * B n ( λ ζ e ) 2 ] } = ( c + i d ) ,
i λ ζ e ( 1 + ζ e ζ s ) = i p ,
exp { y 2 [ i π λ ζ e ( 1 + ζ e ζ s ) ] + x 2 [ i π λ ζ e ( 1 + ζ e ζ s ) ] } = ( e + i f ) ,
C 3 = π 2 R 2 ζ e R ell exp [ y 2 R 2 π R ( B n ) R ell B n * B n ( λ ζ e ) 2 x 2 R 2 π R ( B n ) B n * B n ( λ ζ e ) 2 ] exp [ π 2 ( ω α 2 x 2 + ω β 2 y 2 ) ( λ ζ s ) ( 1 + ζ e ζ s ) ] ,
I 2 = n = 0 N C 3 i p ( a + i b ) ( c + i d ) ( e + i f ) .
I 2 = n = 0 N C 3 i p ( e + i f ) ( a + i b ) ( c + i d ) = n = 0 N C 3 ( e + i f ) [ p ( b c + d a ) + p i ( c a b d ) ] n = 0 N C 3 ( e + i f ) [ p ( b c d a ) + p i ( c a + b d ) ] ,
c d > 500 = > tan { y 2 [ R 2 π 2 ( B n ) R ell 2 B n * B n ( λ ζ e ) 2 ] + x 2 [ R 2 π 2 ( B n ) B n * B n ( λ ζ e ) 2 ] } < 1 500 ,
y 2 [ R 2 π 2 ( B n ) R ell 2 B n * B n ( λ ζ e ) 2 ] + x 2 [ R 2 π 2 ( B n ) B n * B n ( λ ζ e ) 2 ] 1 ,
R 2 R ell 2 < 0.07 ( λ ζ e ) 2 x max 2 .
I 2 ( x , y ) + I 3 ( x , y ) = i p ( e + i f ) F 1 ( n = 0 N T n ) + i p ( e i f ) F 1 ( n = 0 N T n * ) .
n = 0 N i b n = 0 = n = 0 N i b n = > n = 0 N T n = n = 0 N T n * = > F 1 ( n = 0 N T n ) = F 1 ( n = 0 N T n * ) .
I 2 ( x , y ) + I 3 ( x , y ) = 2 λ ζ e ( 1 + ζ e ζ s ) sin [ π ( x 2 + y 2 ) λ ζ e ( 1 + ζ e ζ s ) ] exp [ π 2 ( ω α 2 x 2 + ω β 2 y 2 ) ( λ ζ s ) 2 ( 1 + ζ e ζ s ) 2 ] × n = 1 N A n π R 2 R ell B n exp [ R 2 π 2 ( λ ζ e ) 2 B n ( x 2 + y 2 R ell 2 ) ] .
A = n = 1 N A n π R 2 R ell B n exp [ R 2 π 2 B n ( λ ζ e ) 2 ( x 2 + y 2 R ell 2 ) ] ,
F [ A ] ( x a , y a ) = R 2 n = 1 N A n π R 2 R ell B n exp [ R 2 π 2 B n ( λ ζ e ) 2 ( x 2 + y 2 R ell 2 ) ] × exp ( 2 π i x a x ) exp ( 2 π i y a y ) d x d y .
F [ A ] ( x a , y a ) = n = 1 N A n π R 2 R ell B n [ π B n ( λ ζ e ) 2 R 2 π 2 ] 1 / 2 [ R ell 2 π B n ( λ ζ e ) 2 R 2 π 2 ] 1 / 2 exp { x a 2 [ ( λ ζ e ) 2 R 2 ] B n } exp { y a 2 [ ( λ ζ e ) 2 R 2 ] B n R ell 2 } ,
TF [ A ] ( x a , y a ) = ( λ ζ e ) 2 n = 1 N A n exp [ B n R 1 2 ( x a 2 + R ell 2 y a 2 ) ] .
B = ( λ ζ e ) 2 n = 1 N A n exp [ B n R 1 2 ( x a 2 + R ell 2 y a 2 ) ] = ( λ ζ e ) 2 circ [ ( x a 2 + y a 2 R ell 2 ) 1 / 2 R 1 ] ,
A = F 1 [ B ] ( x , y ) .

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