Abstract

We present a detailed analysis of image formation in digital Fresnel holography. The mathematical modeling is developed on the basis of Fourier optics, making possible the understanding of the different influences of each of the physical effects invoked in digital holography. Particularly, it is demonstrated that spatial resolution in the reconstructed plane can be written as a convolution product of functions that describe these influences. The analysis leads to a thorough investigation of the effect of the width of the sensor, the surface of pixels, the numerical focusing, and the aberrations of the reference wave, as well as to an explicit formulation of the Shannon theorem for digital holography. Experimental illustrations confirm the proposed theoretical analysis.

© 2008 Optical Society of America

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2008 (1)

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

2007 (8)

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481-483 (2007).
[CrossRef] [PubMed]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “'Suppression of the Moiré effect in sub-picosecond digital in-line holography,” Opt. Express 15, 887-895 (2007).
[CrossRef] [PubMed]

K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047-3052 (2007).
[CrossRef] [PubMed]

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231-7242 (2007).
[CrossRef] [PubMed]

P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263-8274 (2007).
[CrossRef] [PubMed]

T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146-2148 (2007).
[CrossRef] [PubMed]

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719-5727 (2007).
[CrossRef] [PubMed]

2006 (8)

T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. 45, 944-952 (2006).
[CrossRef] [PubMed]

A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391-2395 (2006).
[CrossRef] [PubMed]

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
[CrossRef] [PubMed]

T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. 45, 4534-4539 (2006).
[CrossRef]

T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077-6085 (2006).
[CrossRef] [PubMed]

I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
[CrossRef] [PubMed]

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

2005 (9)

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337-343 (2005).
[CrossRef] [PubMed]

B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. 30, 236-238 (2005).
[CrossRef] [PubMed]

G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216-1225 (2005).
[CrossRef] [PubMed]

L. Xu, X. Peng, Z. Guo, J. Mia, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13, 2444-2452 (2005).
[CrossRef] [PubMed]

N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express 13, 4812-4817 (2005).
[CrossRef] [PubMed]

J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763-5772 (2005).
[CrossRef] [PubMed]

C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693-8698 (2005).
[CrossRef] [PubMed]

P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882-8892 (2005).
[CrossRef] [PubMed]

2004 (4)

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] [PubMed]

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787-1789 (2004).
[CrossRef] [PubMed]

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

M. Liebling, “On Fresnelets, interferences fringes, and digital holographyPh.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).

2003 (4)

2002 (7)

2001 (6)

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974-976 (2001).
[CrossRef]

Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478-1480 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase shifting digital holography and application to microscopy,” Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

2000 (5)

1999 (2)

1998 (1)

1997 (3)

I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
[CrossRef] [PubMed]

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-233 (1997).
[CrossRef]

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

1996 (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

1995 (1)

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251-260 (1995).
[CrossRef]

1994 (1)

1993 (1)

1992 (1)

1972 (1)

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Adams, M.

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-233 (1997).
[CrossRef]

Alferi, D.

Allano, D.

Asundi, A.

Baumbach, T.

Berthelot, J.-M.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

Bevilacqua, F.

Binet, R.

Boileau, J.-P.

Breteau, J.-M.

Brown, W.

Brunel, M.

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “'Suppression of the Moiré effect in sub-picosecond digital in-line holography,” Opt. Express 15, 887-895 (2007).
[CrossRef] [PubMed]

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

Cai, X.

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

Chalut, K.

Charrière, F.

Chun-Min, L.

Coetmellec, S.

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

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

Coëtmellec, S.

Colineau, J.

Collot, L.

Colomb, T.

Cuche, E.

Dalmont, J.-P.

De la Torre Ibarra, M.

De Nicola, S.

De Petrocellis, L.

Demoli, I.

Demoli, N.

Denis, L.

Depeursinge, C.

Diouf, B.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

Ducottet, C.

Emery, Y.

Ferraro, P.

Finizio, A.

Fjigaki, M.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Fournel, T.

Fournier, C.

Frauel, Y.

Fu, Y.

Gautier, B.

Gillet, S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gougeon, S.

Grill, M.

Gross, M.

Guimezanes, Th.

Guo, C. S.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Guo, Z.

Ida, T.

Jacquot, M.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Javidi, B.

Jeulin, D.

Juptner, W.

T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

Jüptner, W.

Kato, J.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase shifting digital holography and application to microscopy,” Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

Kebbel, V.

Kim, D.

Kim, M.

Kolenovic, E.

Kreis, Th.

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. (Bellingham) 41, 1829-1839 (2002).
[CrossRef]

Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. (Bellingham) 41, 771-778 (2002).
[CrossRef]

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-233 (1997).
[CrossRef]

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

Kronrod, M. A.

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Kühn, J.

Le Clerc, F.

Lebrun, D.

Lehureau, J. C.

Leval, J.

Liebling, M.

M. Liebling, “On Fresnelets, interferences fringes, and digital holographyPh.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).

Lolive, E.

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

Malek, M.

Mann, C.

Marquet, P.

Massig, J. H.

Merzlyakov, N. S.

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Mia, J.

Mills, G. A.

Mizuno, J.

Moisson, E.

Montfort, F.

Morimoto, Y.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Mosarraf, M.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Mounier, D.

Murata, S.

Nicolas, F.

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “'Suppression of the Moiré effect in sub-picosecond digital in-line holography,” Opt. Express 15, 887-895 (2007).
[CrossRef] [PubMed]

F. Nicolas, S. Coetmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” J. Opt. Soc. Am. A 268, 27-33 (2006).

Nitanai, E.

Nomura, T.

Numata, T.

Ohta, S.

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase shifting digital holography and application to microscopy,” Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

Onural, L.

Oskul, C.

Osten, W.

Ozgen, M. T.

Pascal, J. C.

Pascal, J.-C.

Pedrini, G.

Peng, X.

Picart, P.

Pierattin, G.

Pierattini, G.

Piquet, F.

Rong, Z. Y.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Sandoz, P.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

Santoyo, F. M.

Saucedo, T.

Schedin, S.

G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117-129 (2002).

Schnars, U.

Seebacher, S.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

Shakher, C.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Sheoran, G.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Singh, D.

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Singh, V. R.

Stadelmaier, A.

Tajahuerce, E.

Takahashi, I.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Tiziani, H. J.

Tribillon, G.

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

von Kopylow, C.

Wagner, C.

Wand, H.

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

Wang, H. T.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Wax, A.

Xu, L.

Yamaguchi, I.

Yamashita, K.

Yaroslavskii, L. P.

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Yokota, M.

Yoneyama, S.

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

Yu, L.

Zhang, L.

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Zhang, T.

Zhang, Y.

Appl. Opt. (18)

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

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070-4075 (2000).
[CrossRef]

I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase shifting digital holography and application to microscopy,” Appl. Opt. 40, 6177-6186 (2001).
[CrossRef]

G. Pedrini and H. J. Tiziani, “Short-coherence digital microscopy by use of a lensless holographic imaging system,” Appl. Opt. 41, 4489-4496 (2002).
[CrossRef] [PubMed]

R. Binet, J. Colineau, and J. C. Lehureau, “'Short-range synthetic aperture imaging at 633nm by digital holography,” Appl. Opt. 41, 4775-4782 (2002).
[CrossRef] [PubMed]

P. Picart, E. Moisson, and D. Mounier, “Twin sensitivity measurement by spatial multiplexing of digitally recorded holograms,” Appl. Opt. 42, 1947-1957 (2003).
[CrossRef] [PubMed]

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Image reconstruction for in-line holography with the Yang-Gu algorithm,” Appl. Opt. 42, 6452-6457 (2003).
[CrossRef] [PubMed]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337-343 (2005).
[CrossRef] [PubMed]

G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. 44, 1216-1225 (2005).
[CrossRef] [PubMed]

T. Baumbach, W. Osten, C. von Kopylow, and W. Juptner, “Remote metrology by comparative digital holography,” Appl. Opt. 45, 925-934 (2006).
[CrossRef] [PubMed]

L. Denis, C. Fournier, T. Fournel, C. Ducottet, and D. Jeulin, “Direct extraction of the mean particle size from a digital hologram,” Appl. Opt. 45, 944-952 (2006).
[CrossRef] [PubMed]

A. Asundi and V. R. Singh, “Time-averaged in-line digital holographic interferometry for vibration analysis,” Appl. Opt. 45, 2391-2395 (2006).
[CrossRef] [PubMed]

T. Saucedo, F. M. Santoyo, M. De la Torre Ibarra, G. Pedrini, and W. Osten, “Simultaneous two-dimensional endoscopic pulsed digital holography for evaluation of dynamic displacements,” Appl. Opt. 45, 4534-4539 (2006).
[CrossRef]

T. Baumbach, E. Kolenovic, V. Kebbel, and W. Jüptner, “Improvement of accuracy in digital holography by use of multiple holograms,” Appl. Opt. 45, 6077-6085 (2006).
[CrossRef] [PubMed]

I. Yamaguchi, T. Ida, M. Yokota, and K. Yamashita, “Surface shape measurement by phase shifting digital holography with a wavelength shift,” Appl. Opt. 45, 7610-7616 (2006).
[CrossRef] [PubMed]

J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763-5772 (2005).
[CrossRef] [PubMed]

Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. 46, 5719-5727 (2007).
[CrossRef] [PubMed]

Exp. Mech. (1)

Y. Morimoto, T. Nomura, M. Fjigaki, S. Yoneyama, and I. Takahashi, “Deformation measurement by phase shifting digital holography,” Exp. Mech. 45, 65-70 (2005).
[CrossRef]

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

Measurement (1)

G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 18, 251-260 (1995).
[CrossRef]

Opt. Commun. (2)

M. Jacquot, P. Sandoz, and G. Tribillon, “High resolution digital holography,” Opt. Commun. 190, 87-94 (2001).
[CrossRef]

X. Cai and H. Wand, “The influence of hologram aperture on speckle noise in the reconstructed image of digital holography and its reduction,” Opt. Commun. 281, 232-237 (2008).
[CrossRef]

Opt. Eng. (Bellingham) (5)

Th. Kreis, “Frequency analysis of digital holography,” Opt. Eng. (Bellingham) 41, 771-778 (2002).
[CrossRef]

Th. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. (Bellingham) 41, 1829-1839 (2002).
[CrossRef]

Th. Kreis and W. Juptner, “Suppression of the DC term in digital holography,” Opt. Eng. (Bellingham) 36, 2357-2360 (1997).
[CrossRef]

P. Picart, B. Diouf, E. Lolive, and J.-M. Berthelot, “Investigation of fracture mechanisms in resin concrete using spatially multiplexed digital Fresnel holograms,” Opt. Eng. (Bellingham) 43, 1169-1176 (2004).
[CrossRef]

C. S. Guo, L. Zhang, Z. Y. Rong, and H. T. Wang, “Effect of the fill factor of CCD pixels on digital holograms: comment on the paper,” Opt. Eng. (Bellingham) 42, 2768-2772 (2003).
[CrossRef]

Opt. Express (9)

C. Mann, L. Yu, L. Chun-Min, and M. Kim, “High-resolution quantitative phase-contrast microscopy by digital holography,” Opt. Express 13, 8693-8698 (2005).
[CrossRef] [PubMed]

P. Picart, J. Leval, M. Grill, J.-P. Boileau, J. C. Pascal, J.-M. Breteau, B. Gautier, and S. Gillet, “2D full field vibration analysis with multiplexed digital holograms,” Opt. Express 13, 8882-8892 (2005).
[CrossRef] [PubMed]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “'Suppression of the Moiré effect in sub-picosecond digital in-line holography,” Opt. Express 15, 887-895 (2007).
[CrossRef] [PubMed]

K. Chalut, W. Brown, and A. Wax, “Quantitative phase microscopy with asynchronous digital holography,” Opt. Express 15, 3047-3052 (2007).
[CrossRef] [PubMed]

J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231-7242 (2007).
[CrossRef] [PubMed]

P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263-8274 (2007).
[CrossRef] [PubMed]

L. Xu, X. Peng, Z. Guo, J. Mia, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express 13, 2444-2452 (2005).
[CrossRef] [PubMed]

N. Demoli and I. Demoli, “Dynamic modal characterization of musical instruments using digital holography,” Opt. Express 13, 4812-4817 (2005).
[CrossRef] [PubMed]

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] [PubMed]

Opt. Lasers Eng. (3)

G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117-129 (2002).

M. Mosarraf, G. Sheoran, D. Singh, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126 (2001).
[CrossRef]

Opt. Lett. (17)

B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28-30 (2000).
[CrossRef]

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

F. Le Clerc, L. Collot, and M. Gross, “Numerical heterodyne holography with two-dimensional photo detector arrays,” Opt. Lett. 25, 716-718 (2000).
[CrossRef]

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

I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
[CrossRef] [PubMed]

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

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

A. Stadelmaier and J. H. Massig, “Compensation of lens aberration in digital holography,” Opt. Lett. 25, 1630-1632 (2000).
[CrossRef]

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattin, “Correct-image reconstruction in the presence of severe anamorphism by means of digital holography,” Opt. Lett. 26, 974-976 (2001).
[CrossRef]

Y. Frauel and B. Javidi, “Neural network for three-dimensional object recognition based on digital holography,” Opt. Lett. 26, 1478-1480 (2001).
[CrossRef]

J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27, 2179-2181 (2002).
[CrossRef]

Y. Zhang, G. Pedrini, W. Osten, and H. J. Tiziani, “Reconstruction of in-line digital holograms from two intensity measurements,” Opt. Lett. 29, 1787-1789 (2004).
[CrossRef] [PubMed]

P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900-1902 (2003).
[CrossRef] [PubMed]

B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. 30, 236-238 (2005).
[CrossRef] [PubMed]

T. Nomura, B. Javidi, S. Murata, E. Nitanai, and T. Numata, “Polarization imaging of a 3D object by use of on-axis phase-shifting digital holography,” Opt. Lett. 32, 481-483 (2007).
[CrossRef] [PubMed]

P. Ferraro, D. Alferi, S. De Nicola, L. De Petrocellis, A. Finizio, and G. Pierattini, “Quantitative phase-contrast microscopy by a lateral shear approach to digital holographic image reconstruction,” Opt. Lett. 31, 1405-1407 (2006).
[CrossRef] [PubMed]

T. Nomura and B. Javidi, “Object recognition by use of polarimetric phase-shifting digital holography,” Opt. Lett. 32, 2146-2148 (2007).
[CrossRef] [PubMed]

Opt. Rev. (1)

I. Yamaguchi, J. Kato, and S. Ohta, “Surface shape measurement by phase shifting digital holography,” Opt. Rev. 8, 85-89 (2001).
[CrossRef]

Proc. SPIE (1)

Th. Kreis, M. Adams, and W. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE 3098, 224-233 (1997).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

M. A. Kronrod, N. S. Merzlyakov, and L. P. Yaroslavskii, “Reconstruction of a hologram with a computer,” Sov. Phys. Tech. Phys. 17, 333-334 (1972).

Other (2)

M. Liebling, “On Fresnelets, interferences fringes, and digital holographyPh.D. thesis (Ecole Polytechnique Fédérale de Lausanne, 2004).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (21)

Fig. 1
Fig. 1

Schematic diagram of image formation in digital holography.

Fig. 2
Fig. 2

Basics of recording a digital hologram.

Fig. 3
Fig. 3

Intrinsic resolution function.

Fig. 4
Fig. 4

Illustration of zero padding.

Fig. 5
Fig. 5

Profiles of the resolution function with nonextended and extended pixels.

Fig. 6
Fig. 6

Influence of the active surface of pixels.

Fig. 7
Fig. 7

Plot of criterion quantifying the influence of the active surface of pixels [circles, Eq. (34); curves, Eq. (30)].

Fig. 8
Fig. 8

Evolution of the resolution function versus the reconstruction distances.

Fig. 9
Fig. 9

Profiles of the resolution function for different reconstruction distances.

Fig. 10
Fig. 10

Comparison between the analytical formulation and the numerical simulation versus the reconstruction distance.

Fig. 11
Fig. 11

Profiles along x for the analytical formulation and numerical simulation at the best focus.

Fig. 12
Fig. 12

Profiles along z for the analytical formulation and full numerical for x = y = 0 .

Fig. 13
Fig. 13

Different schemes for generating a reference wave.

Fig. 14
Fig. 14

Zone of the wavefront “seen” by the sensor.

Fig. 15
Fig. 15

Resolution for ( δ x , δ y ) = ( 0 , 0 ) .

Fig. 16
Fig. 16

Resolution for ( δ x , δ y ) = ( 7.64 mm , 7.64 mm ) .

Fig. 17
Fig. 17

Resolution for ( δ x , δ y ) = ( 15.29 mm , 15.29 mm ) .

Fig. 18
Fig. 18

Schematic structure of the reconstructed field.

Fig. 19
Fig. 19

Focus on the + 1 order.

Fig. 20
Fig. 20

Focus on the 1 order.

Fig. 21
Fig. 21

Defocused image.

Tables (2)

Tables Icon

Table 1 Numerical Values for the Resolution Function Computed from Theory

Tables Icon

Table 2 Coefficients of the Polynomial Expansion of Aberration in the Recording Plane

Equations (71)

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

A R ( x , y ) = κ A ( x , y ) * R x y ( x , y ) ,
O ( x , y , d 0 ) = j exp ( 2 j π d 0 λ ) λ d 0 exp [ j π λ d 0 ( x 2 + y 2 ) ] × + + A ( x , y ) exp [ j π λ d 0 ( ( x x 0 ) 2 + ( y y 0 ) 2 ) ] exp [ 2 j π λ d 0 ( ( x x 0 ) x + ( y y 0 ) y ) ] d x d y .
r ( x , y ) = a R exp [ 2 j π ( u R x + v R y ) + j Δ Ψ a b ( x , y ) ] .
H ( x , y , d 0 ) = O ( x , y , d 0 ) 2 + r ( x , y ) 2 + r * ( x , y ) O ( x , y , d 0 ) + r ( x , y ) O * ( x , y , d 0 ) .
r * ( x , y ) O ( x , y , d 0 ) = j exp ( 2 j π d 0 λ ) λ d 0 exp [ j π λ d 0 ( x 0 2 + y 0 2 ) ] exp [ j π λ d 0 ( x 2 + y 2 ) ] a R exp [ 2 j π ( u R x 0 λ d 0 ) x 2 j π ( v R y 0 λ d 0 ) y j Δ Ψ a b ( x , y ) ] × + + A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] exp [ 2 j π λ d 0 ( x ( x + x 0 ) + y ( y + y 0 ) ) ] d x d y .
H P I X ( k p x , l p y , d 0 ) = [ H ( x , y , d 0 ) * Π Δ x , Δ y ( x , y ) ] ( k p x , l p y ) ,
Π Δ x , Δ y ( x , y ) = { 1 Δ x × 1 Δ y if x Δ x 2 and y Δ y 2 0 if not .
A R + 1 ( X , Y , d R ) = j exp ( 2 j π d R λ ) λ d R exp [ j π λ d R ( X 2 + Y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 H P I X + 1 ( k p x , l p y ) exp [ j π λ d R ( k 2 p x 2 + l 2 p y 2 ) ] × exp [ 2 j π λ d R ( k X p x + l Y p y ) ] .
H P I X + 1 ( k p x , l p y ) = { [ ( r * ( x , y ) O ( x , y , d 0 ) ] * Π Δ x , Δ y ( x , y ) } ( k p x , l p y ) ) .
A R 1 ( X , Y , d R ) = A R + 1 ( X , Y , d R ) * .
A R + 1 ( X , Y , d 0 ) = λ 2 d 0 2 exp [ j π λ d 0 ( u R 2 + v R 2 ) ] r * ( X , Y ) × A ( X , Y ) * Π Δ x , Δ y ( X , Y ) * W ̃ N M ( X , Y , d 0 ) * δ ( X λ u R d 0 , Y λ v R d 0 ) ,
W ̃ N M ( x , y , d 0 ) = exp [ j π ( N 1 ) x p x λ d 0 j π ( M 1 ) y p y λ d 0 ] × sin ( π N x p x λ d 0 ) sin ( π x p x λ d 0 ) sin ( π M y p y λ d 0 ) sin ( π y p y λ d 0 ) .
A R + 1 ( X , Y , d 0 ) = a R λ 2 d 0 2 exp [ j π λ d 0 ( X 2 + Y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 exp [ j Δ Ψ a b ( k p x , l p y ) ] exp [ 2 j π ( k p x u R + l p y v R ) ] × F ( k p x λ d 0 , l p y λ d 0 ) exp [ 2 j π λ d 0 ( X k p x + Y l p y ) ] ,
F ( x λ d 0 , y λ d 0 ) = + + F ̃ ( x , y ) exp [ 2 j π λ d 0 ( x x + y y ) ] d x d y = F T [ F ̃ ( x , y ) ] ( x λ d 0 , y λ d 0 ) ,
A R + 1 ( X , Y , d 0 ) = λ 2 d 0 2 exp [ j π λ d 0 ( u R 2 + v R 2 ) ] r * ( X , Y ) × A ( X , Y ) * W ̃ a b ( X , Y ) * W ̃ N M ( X , Y , d 0 ) * δ ( X λ u R d 0 , Y λ v R d 0 ) .
W ̃ a b ( x , y ) = exp [ j Δ Ψ a b ( x , y ) ] exp [ 2 j π λ d 0 ( x x + y y ) ] d x d y .
A R + 1 ( X , Y , d R ) = exp [ 2 j π ( d R + d 0 ) λ ] λ 2 d 0 d R exp [ j π λ d R ( X 2 + Y 2 ) ] × k = 0 k = K 1 l = 0 l = L 1 r * ( k p x , l p y ) F ( k p x λ d 0 , l p y λ d 0 ) × exp [ j π λ ( 1 d R + 1 d 0 ) ( k 2 p x 2 + l 2 p y 2 ) ] exp [ 2 j π λ d R ( X k p x + Y l p y ) ] .
F T 1 [ F T [ F ̃ ( x , y ) ] ( x λ d 0 , y λ d 0 ) ] ( X λ d R , Y λ d R ) = λ 2 d 0 2 [ F ̃ ( λ d 0 u , λ d 0 v ) ] ( X λ d R , Y λ d R ) = λ 2 d 0 2 F ̃ ( d 0 d R X , d 0 d R Y ) ,
F T 1 [ a R exp [ 2 j π ( u R x + v R y ) ] ] ( X λ d R , Y λ d R ) = λ 2 d R 2 a R δ ( X + u R λ d R , Y + v R λ d R ) .
A R + 1 ( X , Y , d R ) = λ 2 d 0 d R exp [ j π λ d 0 ( u R 2 + v R 2 ) ] exp [ j π λ d 0 ( x 0 2 + y 0 2 ) ] r ( X d 0 d R , Y d 0 d R ) exp [ 2 j π ( d 0 + d R ) λ ] exp [ + 2 j π ( X x 0 + Y y 0 ) ] A ( X d 0 d R , Y d 0 d R ) * W ̃ d R ( X , Y ) * W ̃ N M * ( X , Y , d R ) * δ ( X + λ u R d R x 0 d R d 0 , Y + λ v R d R y 0 d R d 0 ) .
W ̃ d R ( x , y ) = exp [ j π λ ( 1 d 0 + 1 d R ) ( x 2 + y 2 ) ] exp [ 2 j π λ d R ( x x + y y ) ] d x d y .
A R + 1 ( X , Y , d 0 ) = λ 2 d 0 2 exp [ j π λ d 0 ( u R 2 + v R 2 ) ] r * ( X , Y ) × A ( X , Y ) * W ̃ N M ( X , Y , d 0 ) * δ ( X λ u R d 0 , Y λ v R d 0 ) ,
A R + 1 ( X , Y , d R ) = κ × A ( X d 0 d R , Y d 0 d R ) * W ̃ a b ( X , Y ) * W ̃ d R ( X , Y ) * Π Δ x , Δ y ( X , Y ) * W ̃ N M * ( X , Y , d R ) * δ ( X + λ u R d R x 0 d R d 0 , Y + λ v R d R y 0 d R d 0 ) ,
R x y ( x , y ) = W ̃ a b ( x , y ) * W ̃ d R ( x , y ) * Π Δ x , Δ y ( x , y ) * W ̃ N M * ( x , y , d R ) * δ ( x + λ u R d R x 0 d R d 0 , y + λ v R d R y 0 d R d 0 ) .
R x y ( x , y ) = W ̃ N M ( x , y , d 0 ) * δ ( x λ u R d 0 , y λ v R d 0 ) .
ρ x = λ d 0 N p x and ρ y = λ d 0 M p y ,
Δ ξ = λ d 0 K p x and Δ η = λ d 0 L p y .
R x y ( x , y ) = Π Δ x , Δ y ( x , y ) * W ̃ N M ( x , y , d 0 ) .
C = + + R x y ( x , y ) 2 d x d y + + R x y Δ x = Δ y = 0 ( x , y ) 2 d x d y ,
C x = + R x ( x ) 2 d x + R x Δ x = 0 ( x ) 2 d x
W ̃ N ( x ) = R x Δ x = 0 ( x ) N e j ( N 1 ) x p x λ d 0 sinc ( π N p x λ d 0 x ) ,
1 N 2 + R x ( x ) 2 d x = λ d 0 N p x k = 0 k = ( 2 π N p x Δ x λ d 0 ) 2 k 2 × ( 1 ) k ( 2 k + 1 ) ( 2 k + 2 ) ! ;
C x = 2 k = 0 k = ( 2 π N p x Δ x λ d 0 ) 2 k ( 1 ) k ( 2 k + 1 ) ( 2 k + 2 ) ! ,
C x = 1 4 π 2 N 2 p x 2 Δ x 2 36 λ 2 d 0 2 + 16 π 4 N 4 p x 4 Δ x 4 1800 λ 4 d 0 4 64 π 6 N 6 p x 6 Δ x 6 141120 λ 6 d 0 6 + .
W ̃ d R ( x , y ) = N p x 2 + N p x 2 M p y 2 + M p y 2 exp [ j π λ ( 1 d 0 + 1 d R ) ( x 2 + y 2 ) ] exp [ 2 j π ( x λ d R x + y λ d R y ) ] d x d y .
u i = x λ ( 1 d 0 + 1 d R ) and v i = y λ ( 1 d 0 + 1 d R ) .
u i max = N p x 2 λ ( 1 d 0 + 1 d R ) and v i max = M p y 2 λ ( 1 d 0 + 1 d R ) .
Δ u = 2 u i max = N p x λ ( 1 d 0 + 1 d R ) .
ρ x d R = N p x ( 1 + d R d 0 ) .
Δ Ψ a b ( ξ , η ) = 2 π λ [ C S ( ξ 2 + η 2 ) 2 + C C ξ ( ξ 2 + η 2 ) ] ,
Δ Ψ a b ( x , y ) = 2 π λ [ a x 4 x 4 + a y 4 y 4 + a x 3 x 3 + a y 3 y 3 + a x 2 x 2 + a y 2 y 2 + a x 1 x + a y 1 y + a x 2 y 2 x 2 y 2 + a x 2 y 1 x 2 y + a x 1 y 2 x y 2 + a x 1 y 1 x y + c ] .
{ u i = 1 λ ( 4 a x 4 x 3 + 3 a x 3 x 2 + 2 a x 2 x + a x 1 + 2 a x 2 y 2 x y 2 + 2 a x 2 y 1 x y + a x 1 y 2 y 2 + a x 1 y 1 y ) v i = 1 λ ( 4 a y 4 y 3 + 3 a y 3 y 2 + 2 a y 2 y + a y 1 + 2 a x 2 y 2 x 2 y + a x 2 y 1 x 2 + 2 a x 1 y 2 x y + a x 1 y 1 x ) .
{ ρ x a b = λ d 0 Δ u 2 = d 0 ( a x 4 N 3 p x 3 2 + a x 2 N p x + a x 2 y 2 N p x M 2 p y 2 4 + a x 1 y 1 M p y 2 ) ρ y a b = λ d 0 Δ v 2 = d 0 ( a y 4 M 3 p y 3 2 + a y 2 M p y + a x 2 y 2 N 2 p x 2 M p y 4 + a x 1 y 1 N p x 2 ) .
{ δ X = d 0 a x 1 = 4 d 0 C S ( δ x 3 + δ x δ y 2 ) + d 0 C C ( 3 δ x 2 + δ y 2 ) δ Y = d 0 a y 1 = 4 d 0 C S ( δ x 2 δ y + δ y 3 ) + 2 d 0 C C δ x δ y .
{ ρ x a b = d 0 ( C S N 3 p x 3 2 + C S N p x ( 6 δ x 2 + 2 δ y 2 ) + 3 C C δ y N p x + C S N p x M 2 p y 2 2 + 4 C S δ x δ y M p y + C C δ y M p y ) ρ y a b = d 0 ( C S M 3 p y 3 2 + C S M p y ( 2 δ x 2 + 6 δ y 2 ) + C C δ x M p y + C S N 2 p x 2 M p y 2 + 4 C S δ x δ y N p x + C C δ y N p x ) .
{ ρ x a b = d 0 ( C S N 3 p x 3 2 + C S N p x ( 6 δ x 2 + 2 δ y 2 ) + C S N p x M 2 p y 2 2 + 4 C S δ x δ y M p y ) ρ y a b = d 0 ( C S M 3 p y 3 2 + C S M p y ( 2 δ x 2 + 6 δ y 2 ) + C S N 2 p x 2 M p y 2 + 4 C S δ x δ y N p x ) .
θ max 2 arcsin ( λ 4 max ( p x , p y ) ) .
O ( x , y , d 0 ) 2 = 1 λ 2 d 0 2 + + A ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] exp [ 2 j π λ d 0 ( x x + y y ) ] d x d y × + + A * ( x , y ) exp [ j π λ d 0 ( x 2 + y 2 ) ] exp [ + 2 j π λ d 0 ( x x + y y ) ] d x d y .
O ̃ 0 ( u , v ) = F T [ O ] ( u , v ) * F T [ O * ] ( u , v ) .
O ̃ 0 ( u , v ) = λ 4 d 0 4 exp [ j π λ d 0 ( u 2 + v 2 ) ] + + A ( λ d 0 u 1 , λ d 0 v 1 ) A * ( λ d 0 u λ d 0 u 1 , λ d 0 v λ d 0 u 1 ) exp [ 2 j π λ d 0 ( u 1 2 + v 1 2 ) ] d u 1 d v 1 .
O ̃ 0 ( u , v ) = λ 4 d 0 4 + + A ( λ d 0 u 1 , λ d 0 v 1 ) A * ( λ d 0 u λ d 0 u 1 , λ d 0 v λ d 0 u 1 ) exp [ 2 j π λ d 0 ( u 1 2 + v 1 2 ) ] d u 1 d v 1 ,
O ̃ 0 ( u , v ) λ 4 d 0 4 + + A 0 ( λ d 0 u 1 , λ d 0 v 1 ) A 0 ( λ d 0 u λ d 0 u 1 , λ d 0 v λ d 0 u 1 ) d u 1 d v 1 ,
O ̃ 0 ( u , v ) λ 4 d 0 4 C A 0 A 0 ( λ d 0 u , λ d 0 v ) .
A R 0 ( X , Y ) λ 4 d 0 4 C A 0 A 0 ( X , Y ) .
d 0 = ( 2 + 3 2 ) max ( p x , p y ) λ a ,
{ u R , v R } { ± 1 p x 3 6 + 2 2 + x 0 λ d 0 , ± 1 p y 3 6 + 2 2 + y 0 λ d 0 } .
d 0 = 4 max ( p x , p y ) λ a ,
( u R , v R ) = ( ± 3 8 p x + x 0 λ d 0 , ± 3 8 p y + y 0 λ d 0 ) .
d 0 = 2 max ( p x , p y ) λ a .
d 0 = max ( p x , p y ) λ a .
+ R x Δ x = 0 ( x ) 2 d x = N 2 + sinc 2 ( π N p x λ d 0 x ) d x = N λ d 0 p x = N 2 ( λ d 0 N p x ) .
+ R x ( x ) 2 d x = + R ̃ x ( k ) 2 d k ,
R ̃ x ( u ) = F T [ W ̃ N ( x ) ] ( u ) × F T [ Π Δ x ( x ) ] ( u ) ,
F T [ W ̃ N ( x ) ] ( u ) = λ d 0 p x Π N p x λ d 0 ( u ( N 1 ) p x 2 λ d 0 ) λ d 0 p x Π N p x λ d 0 ( u N p x 2 λ d 0 ) ,
F T [ Π Δ x ( x ) ] ( u ) = sinc ( π Δ x , u ) .
R ̃ x ( u ) = sinc ( π Δ x u ) λ d 0 p x Π N p x λ d 0 ( u ) ,
+ R x ( x ) 2 d x = λ 2 d 0 2 p x 2 + sinc ( π Δ x u ) Π N p x λ d 0 ( u N p x 2 λ d 0 ) 2 d u ,
+ R x ( x ) 2 d x = λ 2 d 0 2 p x 2 0 + N p x λ d 0 sinc 2 ( π Δ x u ) d u .
sin 2 ( π Δ x u ) ( π Δ x u ) = k = 0 k = ( 1 ) k 2 2 k + 1 ( π Δ x u ) 2 k ( 2 k + 2 ) ! ,
sin 2 ( π Δ x u ) ( π Δ x u ) 2 d u = k = 0 k = ( 1 ) k ( π Δ x ) 2 k 2 2 k + 1 u 2 k + 1 ( 2 k + 1 ) ( 2 k + 2 ) ! .
+ R x ( x ) 2 d x = 2 N 2 λ d 0 N p x k = 0 k = ( 2 π N p x Δ x λ d 0 ) 2 k ( 1 ) k ( 2 k + 1 ) ( 2 k + 2 ) ! .

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