Abstract

Advantages and disadvantages of the non-approximated numerical implementation of the Rayleigh–Sommerfeld diffraction integral (RSD) are revisited. In this work, it is shown that as trade-off for its large computation load, the non-approximated RSD removes any limitation on the propagation range and does not introduce any artifact in the computed wave field. A non-approximated GPU implementation of the RSD is contrasted with the angular spectrum, the Fresnel transform, and a fast Fourier transform implementation of the RSD. The forecasted phase shift introduced in the propagated wave fields as light is diffracted on complementary apertures and utilized as a metric to quantify the performance of the tested methods. An application to numerical reconstructions with arbitrary shape and size of digital recorded holograms from digital lensless holographic microscopy is presented.

© 2019 Optical Society of America

Full Article  |  PDF Article
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References

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2015 (2)

R. Castañeda, W. Toro, and J. Garcia-Sucerquia, “Evaluation of the limits of application for numerical diffraction methods based on basic optics concepts,” Optik 126, 5963–5970 (2015).
[Crossref]

D. Claus and J. M. Rodenburg, “Pixel size adjustment in coherent diffractive imaging within the Rayleigh–Sommerfeld regime,” Appl. Opt. 54, 1936–1944 (2015).
[Crossref]

2014 (3)

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

D. P. Kelly, “Numerical calculation of the Fresnel transform,” J. Opt. Soc. Am. A 31, 755–764 (2014).
[Crossref]

M. Hillenbrand, D. P. Kelly, and S. Sinzinger, “Numerical solution of nonparaxial scalar diffraction integrals for focused fields,” J. Opt. Soc. Am. A 31, 1832–1841 (2014).
[Crossref]

2013 (4)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

P. Clemente, V. Durán, E. Tajahuerce, P. Andrés, V. Climent, and J. Lancis, “Compressive holography with a single-pixel detector,” Opt. Lett. 38, 2524–2527 (2013).
[Crossref]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15, 075405 (2013).
[Crossref]

M. Leclercq and P. Picart, “Method for chromatic error compensation in digital color holographic imaging,” Opt. Express 21, 26456–26467 (2013).
[Crossref]

2012 (3)

2011 (1)

2010 (1)

2009 (2)

2006 (2)

2005 (1)

2004 (1)

2003 (1)

M. Sypek, C. Prokopowicz, and M. Gorecki, “Image multiplying and high-frequency oscillations effects in the Fresnel region light propagation simulation,” Opt. Eng. 42, 3158–3164 (2003).
[Crossref]

2002 (1)

U. Schnars and W. Juptner, “Digital recording and numerical,” Inst. Phys. Publ. 13, 17 (2002).

2001 (1)

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U. S. A. 98, 11301–11305 (2001).
[Crossref]

2000 (3)

1999 (2)

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

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[Crossref]

1997 (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[Crossref]

1994 (1)

1949 (1)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London A 197, 454–487 (1949).
[Crossref]

Andrés, P.

Beckers, M.

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[Crossref]

Bevilacqua, F.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 2005).

Castañeda, R.

R. Castañeda, W. Toro, and J. Garcia-Sucerquia, “Evaluation of the limits of application for numerical diffraction methods based on basic optics concepts,” Optik 126, 5963–5970 (2015).
[Crossref]

Choi, Y. S.

Claus, D.

Clemente, P.

Climent, V.

Cuche, E.

Dallas, W.

W. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, 1980), Vol. 41, pp. 291–366.

Depeursinge, C.

Durán, V.

Endo, Y.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

Ersoy, O. K.

O. K. Ersoy, Diffraction, Fourier Optics and Imaging (Wiley, 2006).

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[Crossref]

Frigo, M.

M. Frigo and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in Proceedings of the 1998 IEEE International Conference On Acoustics, Speech and Signal Processing (1998), Vol. 3, pp. 1381–1384.

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London A 197, 454–487 (1949).
[Crossref]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[Crossref]

Garcia-Sucerquia, J.

Giewekemeyer, K.

Goodman, J. W.

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

Gorecki, M.

M. Sypek, C. Prokopowicz, and M. Gorecki, “Image multiplying and high-frequency oscillations effects in the Fresnel region light propagation simulation,” Opt. Eng. 42, 3158–3164 (2003).
[Crossref]

Gorniak, T.

Grunze, M.

Hillenbrand, M.

Hirayama, R.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

Hiyama, D.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

Horstmeyer, R.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

Ito, T.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15, 075405 (2013).
[Crossref]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Javidi, B.

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39, 6595–6601 (2000).
[Crossref]

T. Nomura and B. Javidi, “Securing information using a digital holographic technique,” Opt. Comput. 4089, 619–624 (2000).
[Crossref]

Jericho, M. H.

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]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U. S. A. 98, 11301–11305 (2001).
[Crossref]

Jericho, S. K.

Johnson, S. G.

M. Frigo and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in Proceedings of the 1998 IEEE International Conference On Acoustics, Speech and Signal Processing (1998), Vol. 3, pp. 1381–1384.

Juan-chang, L.

P. Picart and L. Juan-chang, Digital Holography (Wiley, 2012).

Juptner, W.

U. Schnars and W. Juptner, “Digital recording and numerical,” Inst. Phys. Publ. 13, 17 (2002).

Kakue, T.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15, 075405 (2013).
[Crossref]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref]

Kelly, D. P.

Kim, M. K.

M. K. Kim, Digital Holographic Microscopy. Principles, Techniques, and Aplications (Springer, 2011).

Klages, P.

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[Crossref]

Kreuzer, H. J.

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]

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U. S. A. 98, 11301–11305 (2001).
[Crossref]

Lancis, J.

Leclercq, M.

Lee, S. J.

Li, J.

J. Li and P. Picart, “Calculating diffraction by fast Fourier transform,” in Digital Holography (Wiley, 2012), pp. 77–114.

Logofatu, P. C.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[Crossref]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[Crossref]

Masuda, N.

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Matsushima, K.

Meinertzhagen, I. A.

W. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U. S. A. 98, 11301–11305 (2001).
[Crossref]

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[Crossref]

Nagahama, Y.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

Nascov, V.

Nishitsuji, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Nomura, T.

T. Nomura and B. Javidi, “Securing information using a digital holographic technique,” Opt. Comput. 4089, 619–624 (2000).
[Crossref]

Oikawa, M.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15, 075405 (2013).
[Crossref]

Okada, N.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15, 075405 (2013).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Onural, L.

Picart, P.

M. Leclercq and P. Picart, “Method for chromatic error compensation in digital color holographic imaging,” Opt. Express 21, 26456–26467 (2013).
[Crossref]

J. Li and P. Picart, “Calculating diffraction by fast Fourier transform,” in Digital Holography (Wiley, 2012), pp. 77–114.

P. Picart and L. Juan-chang, Digital Holography (Wiley, 2012).

Prokopowicz, C.

M. Sypek, C. Prokopowicz, and M. Gorecki, “Image multiplying and high-frequency oscillations effects in the Fresnel region light propagation simulation,” Opt. Eng. 42, 3158–3164 (2003).
[Crossref]

Restrepo, J. F.

Rodenburg, J. M.

Roggemann, M. C.

Rosenhahn, A.

Rusch, J. J.

Sakurai, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Salditt, T.

Schnars, U.

Seo, E. S.

Seo, K. W.

Shen, F.

Shimobaba, T.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt. 15, 075405 (2013).
[Crossref]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Shiraki, A.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Sinzinger, S.

Sypek, M.

M. Sypek, C. Prokopowicz, and M. Gorecki, “Image multiplying and high-frequency oscillations effects in the Fresnel region light propagation simulation,” Opt. Eng. 42, 3158–3164 (2003).
[Crossref]

Tajahuerce, E.

Takada, N.

T. Shimobaba, Y. Nagahama, T. Kakue, N. Takada, N. Okada, Y. Endo, R. Hirayama, D. Hiyama, and T. Ito, “Calculation reduction method for color digital holography and computer-generated hologram using color space conversion,” Opt. Eng. 53, 24108 (2014).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Toro, W.

R. Castañeda, W. Toro, and J. Garcia-Sucerquia, “Evaluation of the limits of application for numerical diffraction methods based on basic optics concepts,” Optik 126, 5963–5970 (2015).
[Crossref]

Urbach, H. P.

Veerman, J. A. C.

Voelz, D. G.

Wang, A.

Weng, J.

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Figures (4)

Fig. 1.
Fig. 1. Babinet’s principle of complementary apertures using a circle of 1 mm as free space. The apertures in the left-most upper panels were illuminated with plane waves of 405 nm and propagated to 100 mm using Fresnel–Fraunhofer diffraction, such that the circle subtended an even number of Fresnel’s zones. The sum of the obstacle’s wave field U 1 and the aperture’s wave field U 2 appropriately matches the free space propagation U 0 and present the expected dark spot in the optical axis shown in the right-most upper panel.
Fig. 2.
Fig. 2. Normalized phase difference of the propagated wave fields of the aperture and its complement as the propagation distance changes, with Z c = M Δ x 0 2 / λ . The highlighted distance ranges emphasize the methods with negligible disturbances. The RSD shows an even behavior all over the propagation distance.
Fig. 3.
Fig. 3. Recovering of a phase object at a given distance. (a) Initial phase distribution. Recovered phase after propagating near to Z c = M Δ x 0 2 / λ and back using (b) angular spectrum, (c) Fresnel–Fraunhofer transform, (d) convolution Rayleigh–Sommerfeld, (e) Rayleigh–Sommerfeld diffraction integral. The insets in each panel plot the region bounded by the yellow square.
Fig. 4.
Fig. 4. Reconstruction of a DLHM hologram of a self-assembled monolayer of micrometer-sized polystyrene spheres. (a) Full-size intensity reconstruction of 1024 × 1024 hologram. The region outlined in yellow, originally bounded by 141 × 141 pixels, was independently reconstructed into a 512 × 512 image in (b) intensity and (c) phase. The same full-size DLHM hologram has been reconstructed via the AS in panel (d) and by CRSD in panel (e).

Equations (17)

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( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) U ( x , y , z ) = 0 ,
A ( f x , f y , 0 ) = U ( x 0 , y 0 , 0 ) exp ( i 2 π ( f x x 0 + f y y 0 ) ) d x 0 d y 0 ,
A ( f x , f y , z ) = A ( f x , f y , 0 ) exp ( i k z 1 λ 2 ( f x 2 + f y 2 ) ) .
U ( x , y , z ) = A ( f x , f y , z ) exp ( 2 π ( f x x + f y y ) ) d x d y ,
U ( x , y , z ) = F 1 { F { U ( x 0 , y 0 , 0 ) } exp ( i k z 1 λ 2 ( f x 2 + f y 2 ) ) } .
U ( x , y , z ) = 1 2 π U ( x 0 , y 0 , 0 ) ( i k + 1 r ) exp ( i k r ) r cos χ d x 0 d y 0 ,
U ( x , y , z ) = 1 2 π F 1 { F { U ( x 0 , y 0 , 0 ) } F { ( i k + 1 r ) exp ( i k r ) r z r } } ,
U ( x , y , z ) = i exp ( i k z ) λ z exp ( i k 2 z ( x 2 + y 2 ) ) U ( x 0 , y 0 , 0 ) exp ( i k 2 z ( x 0 2 + y 0 2 ) ) exp ( i k z ( x 0 x + y 0 y ) ) d x 0 d y 0 ,
U ( x λ z , y λ z , z ) = i exp ( i k z ) λ z exp i k 2 z ( x 2 + y 2 ) F { U ( x 0 , y 0 , 0 ) exp i k 2 z ( x 0 2 + y 0 2 ) } .
U ( p Δ x , q Δ y , z ) = r = 1 R s = 1 Q A ( r Δ f x , s Δ f y , 0 ) exp ( i μ z ) × exp i 2 π ( p Δ x r Δ f x + q Δ y s Δ f y ) Δ f x Δ f y ,
A ( r Δ f x , s Δ f y , 0 ) = m = 1 M n = 1 N U ( m Δ x 0 , n Δ y 0 , 0 ) exp i 2 π ( m Δ x 0 r Δ f x + n Δ y 0 s Δ f y ) Δ x 0 Δ y 0 ,
U ( p Δ x , q Δ y , z ) = 1 M R DFT 1 [ DFT [ U ( m Δ x 0 , n Δ y 0 , 0 ) ] × exp ( i z 2 π λ 1 λ 2 ( ( r M Δ x 0 ) 2 + ( s N Δ y 0 ) 2 ) ) ] .
U ( p Δ x , q Δ y , z ) = 1 2 π m = 1 M n = 1 N U ( m Δ x 0 , n Δ y 0 , 0 ) ( i k + 1 r ) exp ( i k r ) r z r Δ x 0 Δ y 0 ,
U ( p Δ x , q Δ y , z ) = 1 2 π M R DFT 1 [ DFT [ U ( m Δ x 0 , n Δ y 0 , 0 ) ] × DFT [ ( i k + 1 r ) exp ( i k r ) r z r ] ] .
U ( p Δ x λ z , q Δ y λ z , z ) = i exp ( i k z ) λ z exp i k 2 z ( ( p λ z M Δ x 0 ) 2 + ( q λ z N Δ y 0 ) 2 ) × DFT [ U ( m Δ x 0 , n Δ y 0 , 0 ) exp i k 2 z [ ( m Δ x 0 ) 2 + ( n Δ y 0 ) 2 ] Δ x 0 Δ y 0 ] .
λ 2 Z c [ 1 λ 2 ( 1 2 Δ x 0 ) 2 1 λ 2 ( M / 2 1 M Δ x 0 ) 2 ] ,
Z c = M Δ x 0 2 λ .

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