Abstract

We investigate the use of a digital holographic microscope working in partially coherent illumination to study in three dimensions a micrometer-size particle flow. The phenomenon under investigation rapidly varies in such a way that it is necessary to record, for every camera frame, the complete holographic information for further processing. For this purpose, we implement the Fourier-transform method for optical amplitude extraction. The suspension of particles is flowing in a split-flow lateral-transport thin separation cell that is usually used to separate the species by their sizes. Details of the optical implementation are provided. Examples of reconstructed images of different particle sizes are shown, and a particle-velocity measurement technique that is based on the blurred holographic image is exploited.

© 2006 Optical Society of America

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  1. T.-C. Poon and M. Motamedi, "Optical/digital incoherent image processing for extended depth of field," Appl. Opt. 26, 4612-4615 (1987).
  2. E. R. Dowski, Jr., and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
  3. U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
  4. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
  5. T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).
  6. M. Sebesta and M. Gustafsson, "Object characterization with refractometric digital Fourier holography," Opt. Lett. 30, 471-473 (2005).
    [CrossRef]
  7. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase contrast imaging," Opt. Lett. 24, 291-293 (1999).
  8. F. Dubois, L. Joannes, and J.-C. Legros, "Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source," Appl. Opt. 38, 7085-7094 (1999).
  9. G. Indebetouw and P. Klysubun, "Spatiotemporal digital microholography," J. Opt. Soc. Am. A 18, 319-325 (2001).
  10. I. Yamaguchi, J.-I. Kato, S. Otha, and J. Mizuno, "Image formation in phase-shifting digital holography and applications to microscopy," Appl. Opt. 40, 6177-6186 (2001).
  11. D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
    [CrossRef]
  12. F. Dubois, C. Yourassowsky, and O. Monnom, "Microscopic en holographie digitale avec une source partiellement cohérente," in Imagerie et Photonique pour les Sciences du Vivant et la Médecine, M.Faupel, P.Smigielski, and R.Grzymala, eds. (Fontis Media, 2004), pp. 287-302.
  13. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
    [CrossRef]
  14. D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
    [CrossRef]
  15. B. Skarman, K. Wozniac, and J. Becker, "Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer," Flow Meas. Instrum. 7, 1-6 (1996).
    [CrossRef]
  16. P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
    [CrossRef]
  17. T.-C. Poon and T. Kim, "Optical image recognition of three-dimensional objects," Appl. Opt. 38, 370-381 (1999).
  18. B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000).
  19. F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, and J.-C. Legros, "Pattern recognition with digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).
  20. E. Cuche, P. Marquet, and C. Despeuringe, "Aperture apodization using cubic spline interpolation: application in digital holography microscopy," Opt. Commun. 182, 59-69 (2000).
    [CrossRef]
  21. F. Dubois, O. Monnom, C. Yourassowsky, and J.-C. Legros, "Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies," Appl. Opt. 41, 2621-2626 (2002).
  22. S.-G. Kim, B. Lee, and E.-S. Kim, "Removal of bias and the conjugate image in incoherent on-axis triangular holography and real-time reconstruction of the complex hologram," Appl. Opt. 36, 4784-4791 (1997).
  23. T.-C. Poon, T. Kim, G. Indebetouw, M. H. Wu, K. Shinoda, and Y. Suzuki, "Twin-image elimination experiments for three-dimensional images in optical scanning holography," Opt. Lett. 25, 215-217 (2000).
  24. P. Klysubun and G. Indebetouw, "A posteriori processing of spatiotemporal digital microholograms," J. Opt. Soc. Am. A 18, 326-331 (2001).
  25. T. Kim, T.-C. Poon, and G. Indebetouw, "Depth detection and image recovery in remote sensing by optical scanning holography," Opt. Eng. 41, 1331-1338 (2002).
    [CrossRef]
  26. C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992).
  27. W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography of microspheres," Appl. Opt. 41, 5367-5375 (2002).
  28. W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003).
  29. S. Coëtmellec, D. Lebrun, and C. Özkul, "Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform," Appl. Opt. 41, 312-319 (2002).
  30. L. Repetto, E. Piano, and C. Pontiggia, "Lensless digital holographic microscope with light-emitting diode illumination," Opt. Lett. 29, 1132-1134 (2004).
    [CrossRef]
  31. F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
    [CrossRef]
  32. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
  33. T. Kreis, "Digital holographic interference-phase measurement using the Fourier-transform method," J. Opt. Soc. Am. A 3, 847-855 (1986).
  34. J. C. Giddings, "A system based on split-flow lateral-transport thin (SPLITT) for rapid and continuous particle fractionation," Sep. Sci. Technol. 20, 749-768 (1985).
  35. P. S. Williams, "Particle trajectories in field-flow fractionation and SPLITT fractionation channels," Sep. Sci. Technol. 29, 11-45 (1994).
  36. D. Leighton and A. Acrivos, "Viscous resuspension," Chem. Eng. Sci. 41, 1377-1384 (1986).
    [CrossRef]

2005 (2)

2004 (4)

2003 (1)

2002 (5)

2001 (4)

2000 (3)

1999 (3)

1998 (1)

1997 (2)

1996 (1)

B. Skarman, K. Wozniac, and J. Becker, "Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer," Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

1995 (1)

1994 (2)

U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).

P. S. Williams, "Particle trajectories in field-flow fractionation and SPLITT fractionation channels," Sep. Sci. Technol. 29, 11-45 (1994).

1987 (1)

1986 (2)

1985 (1)

J. C. Giddings, "A system based on split-flow lateral-transport thin (SPLITT) for rapid and continuous particle fractionation," Sep. Sci. Technol. 20, 749-768 (1985).

1982 (1)

Acrivos, A.

D. Leighton and A. Acrivos, "Viscous resuspension," Chem. Eng. Sci. 41, 1377-1384 (1986).
[CrossRef]

Alfieri, D.

Becker, J.

B. Skarman, K. Wozniac, and J. Becker, "Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer," Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

Bevilacqua, F.

Carl, D.

Cathey, W. T.

Coëtmellec, S.

Colomb, T.

Coppola, G.

Cuche, E.

De Nicola, S.

Deiwick, M.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Delerlea, H.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Depeursinge, C.

Despeuringe, C.

E. Cuche, P. Marquet, and C. Despeuringe, "Aperture apodization using cubic spline interpolation: application in digital holography microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

Dirksena, D.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Dowski, E. R.

Drostea, H.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Dubois, F.

Emery, Y.

Ferraro, P.

Finizio, A.

Giddings, J. C.

J. C. Giddings, "A system based on split-flow lateral-transport thin (SPLITT) for rapid and continuous particle fractionation," Sep. Sci. Technol. 20, 749-768 (1985).

Gustafsson, M.

Ina, H.

Indebetouw, G.

Istasse, E.

Javidi, B.

Jericho, M. H.

Joannes, L.

Jüptner, W.

Kato, J.-I.

Kemper, B.

Kempera, B.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Kim, E.-S.

Kim, S.-G.

Kim, T.

Klysubun, P.

Kobayashi, S.

Kreis, T.

Kreuzer, H. J.

Lebrun, D.

Lee, B.

Legros, J.-C.

Leighton, D.

D. Leighton and A. Acrivos, "Viscous resuspension," Chem. Eng. Sci. 41, 1377-1384 (1986).
[CrossRef]

Magistretti, P. J.

Marquet, P.

Meinertzhagen, I. A.

Minetti, C.

Mizuno, J.

Monnom, O.

Motamedi, M.

Novella Requena, M.-L.

Otha, S.

Özkul, C.

Piano, E.

Pierattini, G.

Pontiggia, C.

Poon, T.-C.

Rappaz, B.

Repetto, L.

Scheld, H. H.

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Schnars, U.

Sebesta, M.

Shinoda, K.

Skarman, B.

B. Skarman, K. Wozniac, and J. Becker, "Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer," Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

Suzuki, Y.

Tajahuerce, E.

Takeda, M.

Vikram, C. S.

C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992).

von Bally, G.

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Wernicke, G.

Williams, P. S.

P. S. Williams, "Particle trajectories in field-flow fractionation and SPLITT fractionation channels," Sep. Sci. Technol. 29, 11-45 (1994).

Wozniac, K.

B. Skarman, K. Wozniac, and J. Becker, "Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer," Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

Wu, M. H.

Xu, W.

Yamaguchi, I.

Yourassowsky, C.

Zhang, T.

Appl. Opt. (13)

T.-C. Poon and M. Motamedi, "Optical/digital incoherent image processing for extended depth of field," Appl. Opt. 26, 4612-4615 (1987).

E. R. Dowski, Jr., and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).

U. Schnars and W. Jüptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).

F. Dubois, L. Joannes, and J.-C. Legros, "Improved three-dimensional imaging with digital holography microscope using a partial spatial coherent source," Appl. Opt. 38, 7085-7094 (1999).

I. Yamaguchi, J.-I. Kato, S. Otha, and J. Mizuno, "Image formation in phase-shifting digital holography and applications to microscopy," Appl. Opt. 40, 6177-6186 (2001).

D. Carl, B. Kemper, G. Wernicke, and G. von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

T.-C. Poon and T. Kim, "Optical image recognition of three-dimensional objects," Appl. Opt. 38, 370-381 (1999).

F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, and J.-C. Legros, "Pattern recognition with digital holographic microscope working in partially coherent illumination," Appl. Opt. 41, 4108-4119 (2002).

F. Dubois, O. Monnom, C. Yourassowsky, and J.-C. Legros, "Border processing in digital holography by extension of the digital hologram and reduction of the higher spatial frequencies," Appl. Opt. 41, 2621-2626 (2002).

S.-G. Kim, B. Lee, and E.-S. Kim, "Removal of bias and the conjugate image in incoherent on-axis triangular holography and real-time reconstruction of the complex hologram," Appl. Opt. 36, 4784-4791 (1997).

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography of microspheres," Appl. Opt. 41, 5367-5375 (2002).

F. Dubois, M.-L. Novella Requena, C. Minetti, O. Monnom, and E. Istasse, "Partial spatial coherence effects in digital holographic microscopy with a laser source," Appl. Opt. 43, 1131-1139 (2004).
[CrossRef]

S. Coëtmellec, D. Lebrun, and C. Özkul, "Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform," Appl. Opt. 41, 312-319 (2002).

Chem. Eng. Sci. (1)

D. Leighton and A. Acrivos, "Viscous resuspension," Chem. Eng. Sci. 41, 1377-1384 (1986).
[CrossRef]

Flow Meas. Instrum. (1)

B. Skarman, K. Wozniac, and J. Becker, "Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer," Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

E. Cuche, P. Marquet, and C. Despeuringe, "Aperture apodization using cubic spline interpolation: application in digital holography microscopy," Opt. Commun. 182, 59-69 (2000).
[CrossRef]

Opt. Eng. (1)

T. Kim, T.-C. Poon, and G. Indebetouw, "Depth detection and image recovery in remote sensing by optical scanning holography," Opt. Eng. 41, 1331-1338 (2002).
[CrossRef]

Opt. Lasers Eng. (1)

D. Dirksena, H. Drostea, B. Kempera, H. Delerlea, M. Deiwick, H. H. Scheld, and G. von Bally, "Lensless Fourier holography for digital holographic interferometry on biological samples," Opt. Lasers Eng. 36, 241-249 (2001).
[CrossRef]

Opt. Lett. (10)

I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).

T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23, 1221-1223 (1998).

M. Sebesta and M. Gustafsson, "Object characterization with refractometric digital Fourier holography," Opt. Lett. 30, 471-473 (2005).
[CrossRef]

E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase contrast imaging," Opt. Lett. 24, 291-293 (1999).

T.-C. Poon, T. Kim, G. Indebetouw, M. H. Wu, K. Shinoda, and Y. Suzuki, "Twin-image elimination experiments for three-dimensional images in optical scanning holography," Opt. Lett. 25, 215-217 (2000).

P. Ferraro, S. De Nicola, G. Coppola, A. Finizio, D. Alfieri, and G. Pierattini, "Controlling image size as a function of distance and wavelength in Fresnel-transform reconstruction of digital holograms," Opt. Lett. 29, 854-856 (2004).
[CrossRef]

B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000).

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, "Tracking particles in four dimensions with in-line holographic microscopy," Opt. Lett. 28, 164-166 (2003).

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, "Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy," Opt. Lett. 30, 468-470 (2005).
[CrossRef]

L. Repetto, E. Piano, and C. Pontiggia, "Lensless digital holographic microscope with light-emitting diode illumination," Opt. Lett. 29, 1132-1134 (2004).
[CrossRef]

Sep. Sci. Technol. (2)

J. C. Giddings, "A system based on split-flow lateral-transport thin (SPLITT) for rapid and continuous particle fractionation," Sep. Sci. Technol. 20, 749-768 (1985).

P. S. Williams, "Particle trajectories in field-flow fractionation and SPLITT fractionation channels," Sep. Sci. Technol. 29, 11-45 (1994).

Other (2)

C. S. Vikram, Particle Field Holography (Cambridge U. Press, 1992).

F. Dubois, C. Yourassowsky, and O. Monnom, "Microscopic en holographie digitale avec une source partiellement cohérente," in Imagerie et Photonique pour les Sciences du Vivant et la Médecine, M.Faupel, P.Smigielski, and R.Grzymala, eds. (Fontis Media, 2004), pp. 287-302.

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Figures (4)

Fig. 1
Fig. 1

Optical setup of the digital holographic microscope. L1, focusing lens; RGG, rotating ground glass for spatial coherence reduction; L2, collimating lens; L3, L4, identical microscope lenses (×20); L5, refocusing lens; CCD, charge-coupled device camera with the sensor placed in the back focal plane of L5; M1–M3, mirrors; BS1, BS2, beam splitters. The optical compensation in the reference arm is not indicated in the drawing.

Fig. 2
Fig. 2

(a) Digital hologram of size-calibrated 4   μm latex beads flowing at a mean velocity of v = 5.34 cm / s in a SPLITT channel. The particles are unfocused. The field of view is 300   μm × 300   μm . (b) Numerical refocusing of size-calibrated 4   μm latex beads flowing at a mean velocity of v = 5.34 cm / s in the SPLITT channel. It is the reconstruction of the digital hologram in (a). With respect to the hologram in (a), there is windowing to keep a field of view of 300   μm × 300   μm . The particles have a mean size of 4.2 ± 0.6   μm and a refocus distance of 23.4 ± 2.6   μm . The reconstructed image is processed by a border-processing algorithm.[21]

Fig. 3
Fig. 3

(a) Latex bead ( 4   μm ) , flowing at a mean velocity of v = 3.56 cm / s in the SPLITT channel, has a focus distance of 25 ± 2.6   μm . (b) Latex bead ( 7   μm ) , flowing at a mean velocity of v = 3.56 cm / s in the SPLITT channel, has a focus distance of 50 ± 2.6   μm .

Fig. 4
Fig. 4

(a) Blurred digital hologram of 5   μm Lichrospher particles flowing at a mean velocity of v = 12.46 cm / s in the SPLITT channel. (b) Reconstructed image from the digital hologram of (a) with a refocus distance of 40   μm . We observe a sharp trace of some 5   μm Lichrospher particles. The reconstructed image is processed by a border-processing algorithm.[21]

Tables (1)

Tables Icon

Table 1 Comparison between the Experimental and Theoretical Values of the 5 μm Lichrospher Particle Velocities inside the SPLITT Channel

Equations (21)

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

o o ( x , y , t ) = s ( x , y ) o i ( x , y , t ) .
a ( x , y , t ) = o o ( x , y , t ) + r ( x , y , t ) .
Γ ( x 1 , y 1 , x 2 , y 2 ) = a * ( x 2 , y 2 , t ) a ( x 1 , y 1 , t ) t ,
Γ ( x 1 , y 1 , x 2 , y 2 ) = o o * ( x 2 , y 2 , t ) o o ( x 1 , y 1 , t ) t + r * ( x 2 , y 2 , t ) r ( x 1 , y 1 , t ) t + o o * ( x 2 , y 2 , t ) r ( x 1 , y 1 , t ) t + r * ( x 2 , y 2 , t ) o ( x 1 , y 1 , t ) t .
Q ( x , y , t ) = P ( x , y ) N ( x , y , t ) ,
N * ( x 2 , y 2 , t ) N ( x 1 , y 1 , t ) t = δ ( x 1 x 2 , y 1 y 2 ) ,
o i ( x , y , t ) = exp { j k 2 f 2 } j λ f 2 ( F + N P ) ( x λ f 2 , y λ f 2 ) ,
( F ± W ) ( α , β ) = d x d y exp { 2 π j ( x α + y β ) } W ( x , y ) .
o i * ( x 2 , y 2 , t ) o i ( x 1 , y 1 , t ) t = A ( p p ) × ( x 1 x 2 λ f 2 , y 1 y 2 λ f 2 ) ,
Q ( x , y , t ) = P ( x + Δ x , y + Δ y ) N ( x + Δ x , y + Δ y , t ) .
r * ( x 2 , y 2 , t ) r ( x 1 , y 1 , t ) t = A ( p p ) ( x 1 x 2 λ f 2 , y 1 y 2 λ f 2 ) .
r * ( x 2 , y 2 , t ) o i ( x 1 , y 1 , t ) t = A exp { 2 π j ( Δ x x 2 + Δ y y 2 λ f 2 ) }
× ( p p ) ( x 1 x 2 λ f 2 , y 1 y 2 λ f 2 ) .
Γ ( x 1 , y 1 , x 2 , y 2 ) = A ( p p ) ( x 1 x 2 λ f 2 , y 1 y 2 λ f 2 )
× { s * ( x 2 , y 2 ) s ( x 1 , y 1 ) + 1 + s * ( x 2 , y 2 ) × exp { 2 π j ( Δ x x 1 + Δ y y 1 λ f 2 ) } + s ( x 1 , y 1 ) × exp { 2 π j ( Δ x x 2 + Δ y y 2 λ f 2 ) } } .
s ( x , y ) = s m ( x , y ) exp { j ϕ ( x , y ) } ,
i ( x , y ) = g { | s ( x , y ) | 2 + 1 + 2 s m ( x , y ) × cos { ϕ ( x , y ) 2 π ( Δ x x + Δ y y λ f 2 ) } } ,
I ( u , v ) = g { B ( u , v ) + δ ( u K 2 π ) S * ( u , v ) + δ ( u + K 2 π ) S ( u , v ) } ,
s d ( m , n ) = s ( m Δ , n Δ ) ,
s d ( m , n ) = exp { j k d } [ F m , n - 1 exp { j k λ 2 d 2 N 2 Δ 2 ( U 2 + V 2 ) } [ F U , V                 + 1 s d ( m , n ) ] ] ,
[ F m , n               ± 1 g ( m , n ) ] = 1 N k , l = 0 N 1 exp { 2 π j N × ( m k + n l ) } g ( k ,   l ) ,

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