Abstract

This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg–Saxton (GS) algorithm applied in digital holography. In addition, a novel GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover a complex-valued wavefront using wavefront back-and-forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

© 2019 Optical Society of America

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Corrections

Tatiana Latychevskaia, "Iterative phase retrieval for digital holography: tutorial: publisher’s note," J. Opt. Soc. Am. A 37, 45-45 (2020)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-37-1-45

22 November 2019: A correction was made to the title. Typographical corrections were made in the abstract and in Section 4.A.

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2019 (2)

N. S. Balbekin, M. S. Kulya, A. V. Belashov, A. Gorodetsky, and N. V. Petrov, “Increasing the resolution of the reconstructed image in terahertz pulse time-domain holography,” Sci. Rep. 9, 180 (2019).
[Crossref]

T. Latychevskaia, “Reconstruction of missing information in diffraction patterns and holograms by iterative phase retrieval,” Opt. Commun. 452, 56–67 (2019).
[Crossref]

2018 (1)

C. Guo, C. Shen, Q. Li, J. B. Tan, S. T. Liu, X. C. Kan, and Z. J. Liu, “Fast-converging iterative method based on weighted feedback for multi-distance phase retrieval,” Sci. Rep. 8, 6436 (2018).
[Crossref]

2016 (1)

2015 (4)

T. Latychevskaia and H.-W. Fink, “Practical algorithms for simulation and reconstruction of digital in-line holograms,” Appl. Opt. 54, 2424–2434 (2015).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Reconstruction of purely absorbing, absorbing and phase-shifting, and strong phase-shifting objects from their single-shot in-line holograms,” Appl. Opt. 54, 3925–3932 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

L. Rong, T. Latychevskaia, C. Chen, D. Wang, Z. Yu, X. Zhou, Z. Li, H. Huang, Y. Wang, and Z. Zhou, “Terahertz in-line digital holography of human hepatocellular carcinoma tissue,” Sci. Rep. 5, 8445 (2015).
[Crossref]

2014 (1)

2013 (1)

2010 (3)

2009 (1)

2007 (1)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[Crossref]

2003 (1)

2000 (1)

J. Miao and D. Sayre, “On possible extensions of x-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sec. A 56, 596–605 (2000).
[Crossref]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1998 (1)

1993 (1)

1991 (1)

1987 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[Crossref]

1963 (1)

1957 (1)

G. Mollenstedt and M. Keller, “Elektroneninterfero metrisehe Messung des inneren Potentials,” Z. Phys. 148, 34–37 (1957).
[Crossref]

1956 (2)

G. Mollenstedt and H. Duker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[Crossref]

J. A. Ratcliffe, “Some aspects of diffraction theory and their application to the ionosphere,” Reports on progress in physics 19, 188–267 (1956).
[Crossref]

1952 (1)

1949 (1)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London, Ser. A 1051, 454–487 (1949).
[Crossref]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref]

1936 (1)

O. Scherzer, “Ueber einige Fehler von Elektronenlinsen,” Z. Phys. 101, 593–603 (1936).
[Crossref]

Allen, L. J.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[Crossref]

Almoro, P.

Arfire, C.

Balbekin, N. S.

N. S. Balbekin, M. S. Kulya, A. V. Belashov, A. Gorodetsky, and N. V. Petrov, “Increasing the resolution of the reconstructed image in terahertz pulse time-domain holography,” Sci. Rep. 9, 180 (2019).
[Crossref]

Belashov, A. V.

N. S. Balbekin, M. S. Kulya, A. V. Belashov, A. Gorodetsky, and N. V. Petrov, “Increasing the resolution of the reconstructed image in terahertz pulse time-domain holography,” Sci. Rep. 9, 180 (2019).
[Crossref]

Bergoend, I.

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[Crossref]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Chen, C.

L. Rong, T. Latychevskaia, C. Chen, D. Wang, Z. Yu, X. Zhou, Z. Li, H. Huang, Y. Wang, and Z. Zhou, “Terahertz in-line digital holography of human hepatocellular carcinoma tissue,” Sci. Rep. 5, 8445 (2015).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Depeursinge, C.

Duker, H.

G. Mollenstedt and H. Duker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Fienup, J. R.

Fink, H.-W.

Formanek, P.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: experimental comparison,” Ultramicroscopy 110, 472–482 (2010).
[Crossref]

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London, Ser. A 1051, 454–487 (1949).
[Crossref]

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref]

D. Gabor, “Improvements in and relating to microscopy,” PatentGB685286 (December17, 1947).

D. Gabor, Nobel Lecture (1971).

Gehri, F.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[Crossref]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2004).

Gorodetsky, A.

N. S. Balbekin, M. S. Kulya, A. V. Belashov, A. Gorodetsky, and N. V. Petrov, “Increasing the resolution of the reconstructed image in terahertz pulse time-domain holography,” Sci. Rep. 9, 180 (2019).
[Crossref]

Guo, C.

C. Guo, C. Shen, Q. Li, J. B. Tan, S. T. Liu, X. C. Kan, and Z. J. Liu, “Fast-converging iterative method based on weighted feedback for multi-distance phase retrieval,” Sci. Rep. 8, 6436 (2018).
[Crossref]

Haine, M. E.

Huang, H.

L. Rong, T. Latychevskaia, C. Chen, D. Wang, Z. Yu, X. Zhou, Z. Li, H. Huang, Y. Wang, and Z. Zhou, “Terahertz in-line digital holography of human hepatocellular carcinoma tissue,” Sci. Rep. 5, 8445 (2015).
[Crossref]

L. Rong, T. Latychevskaia, D. Wang, X. Zhou, H. Huang, Z. Li, and Y. Wang, “Terahertz in-line digital holography of dragonfly hindwing: amplitude and phase reconstruction at enhanced resolution by extrapolation,” Opt. Express 22, 17236–17245 (2014).
[Crossref]

Joyeux, D.

Jueptner, W.

U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).

Kan, X. C.

C. Guo, C. Shen, Q. Li, J. B. Tan, S. T. Liu, X. C. Kan, and Z. J. Liu, “Fast-converging iterative method based on weighted feedback for multi-distance phase retrieval,” Sci. Rep. 8, 6436 (2018).
[Crossref]

Keller, M.

G. Mollenstedt and M. Keller, “Elektroneninterfero metrisehe Messung des inneren Potentials,” Z. Phys. 148, 34–37 (1957).
[Crossref]

Kirkland, E. J.

E. J. Kirkland, Advanced Computing in Electron Microscopy (2010).

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Koch, C. T.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: experimental comparison,” Ultramicroscopy 110, 472–482 (2010).
[Crossref]

Koren, G.

Kulya, M. S.

N. S. Balbekin, M. S. Kulya, A. V. Belashov, A. Gorodetsky, and N. V. Petrov, “Increasing the resolution of the reconstructed image in terahertz pulse time-domain holography,” Sci. Rep. 9, 180 (2019).
[Crossref]

Latychevskaia, T.

T. Latychevskaia, “Reconstruction of missing information in diffraction patterns and holograms by iterative phase retrieval,” Opt. Commun. 452, 56–67 (2019).
[Crossref]

L. Rong, T. Latychevskaia, C. Chen, D. Wang, Z. Yu, X. Zhou, Z. Li, H. Huang, Y. Wang, and Z. Zhou, “Terahertz in-line digital holography of human hepatocellular carcinoma tissue,” Sci. Rep. 5, 8445 (2015).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Reconstruction of purely absorbing, absorbing and phase-shifting, and strong phase-shifting objects from their single-shot in-line holograms,” Appl. Opt. 54, 3925–3932 (2015).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Practical algorithms for simulation and reconstruction of digital in-line holograms,” Appl. Opt. 54, 2424–2434 (2015).
[Crossref]

L. Rong, T. Latychevskaia, D. Wang, X. Zhou, H. Huang, Z. Li, and Y. Wang, “Terahertz in-line digital holography of dragonfly hindwing: amplitude and phase reconstruction at enhanced resolution by extrapolation,” Opt. Express 22, 17236–17245 (2014).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Resolution enhancement in digital holography by self-extrapolation of holograms,” Opt. Express 21, 7726–7733 (2013).
[Crossref]

T. Latychevskaia, F. Gehri, and H.-W. Fink, “Depth-resolved holographic reconstructions by three-dimensional deconvolution,” Opt. Express 18, 22527–22544 (2010).
[Crossref]

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: experimental comparison,” Ultramicroscopy 110, 472–482 (2010).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Simultaneous reconstruction of phase and amplitude contrast from a single holographic record,” Opt. Express 17, 10697–10705 (2009).
[Crossref]

T. Latychevskaia and H.-W. Fink, “Solution to the twin image in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[Crossref]

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[Crossref]

Leith, E. N.

Li, G. B.

Li, J. C.

J. C. Li and P. Picart, Digital Holography (Wiley, 2012).

Li, L.

Li, Q.

C. Guo, C. Shen, Q. Li, J. B. Tan, S. T. Liu, X. C. Kan, and Z. J. Liu, “Fast-converging iterative method based on weighted feedback for multi-distance phase retrieval,” Sci. Rep. 8, 6436 (2018).
[Crossref]

Li, Z.

L. Rong, T. Latychevskaia, C. Chen, D. Wang, Z. Yu, X. Zhou, Z. Li, H. Huang, Y. Wang, and Z. Zhou, “Terahertz in-line digital holography of human hepatocellular carcinoma tissue,” Sci. Rep. 5, 8445 (2015).
[Crossref]

L. Rong, T. Latychevskaia, D. Wang, X. Zhou, H. Huang, Z. Li, and Y. Wang, “Terahertz in-line digital holography of dragonfly hindwing: amplitude and phase reconstruction at enhanced resolution by extrapolation,” Opt. Express 22, 17236–17245 (2014).
[Crossref]

Li, Z. Y.

Liu, G.

Liu, S. T.

C. Guo, C. Shen, Q. Li, J. B. Tan, S. T. Liu, X. C. Kan, and Z. J. Liu, “Fast-converging iterative method based on weighted feedback for multi-distance phase retrieval,” Sci. Rep. 8, 6436 (2018).
[Crossref]

Liu, Z. J.

C. Guo, C. Shen, Q. Li, J. B. Tan, S. T. Liu, X. C. Kan, and Z. J. Liu, “Fast-converging iterative method based on weighted feedback for multi-distance phase retrieval,” Sci. Rep. 8, 6436 (2018).
[Crossref]

Lubk, A.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: experimental comparison,” Ultramicroscopy 110, 472–482 (2010).
[Crossref]

McBride, W.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[Crossref]

Miao, J.

J. Miao and D. Sayre, “On possible extensions of x-ray crystallography through diffraction-pattern oversampling,” Acta Crystallogr. Sec. A 56, 596–605 (2000).
[Crossref]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998).
[Crossref]

Miao, J. W.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. W. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Mollenstedt, G.

G. Mollenstedt and M. Keller, “Elektroneninterfero metrisehe Messung des inneren Potentials,” Z. Phys. 148, 34–37 (1957).
[Crossref]

G. Mollenstedt and H. Duker, “Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen,” Z. Phys. 145, 377–397 (1956).
[Crossref]

Mulvey, T.

O’Leary, N. L.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[Crossref]

Osten, W.

Oxley, M. P.

L. J. Allen, W. McBride, N. L. O’Leary, and M. P. Oxley, “Exit wave reconstruction at atomic resolution,” Ultramicroscopy 100, 91–104 (2004).
[Crossref]

Pavillon, N.

Pedrini, G.

Petrov, N. V.

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Figures (8)

Fig. 1.
Fig. 1. Position of the object and its twin image during recording and reconstruction with (a), (b) spherical waves and (c), (d) plane waves.

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Fig. 2.
Fig. 2. Schematic drawing of Gerchberg–Saxton iterative phase retrieval algorithm from two intensity measurements, adapted from [10]. Measured intensities: $ |{u_0}|^2 $ is the intensity in the object plane, and $ |{U_0}|^2 $ is the intensity in the diffraction plane. From these two measured intensities, the algorithm recovers the complex-valued distributions in the object plane ($ {u_0}\exp (i{\varphi _0}) $) and in the diffraction plane ($ {U_0}\exp (i{\Phi _0}) $). (a) The algorithm starts in the object plane, where the initial complex-valued distribution is created by combining the measured amplitude distribution with the random phase distribution. (b) The Fourier transform of the object distribution gives the complex-valued distribution in the diffraction plane (c). The amplitude distribution in the diffraction plane is replaced with the measured amplitude distribution, creating an updated distribution of the complex-valued wavefront in the diffraction plane (d). (e) An inverse Fourier transform gives the complex-valued distribution in the object plane (f). The amplitude distribution in the object plane is replaced with the measured amplitude distribution, creating an updated object distribution for the next iteration starting at (a).

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Fig. 3.
Fig. 3. General scheme of iterative phase retrieval from a single-shot intensity measurement (hologram), adapted from [11]. (a) The algorithm starts in the hologram plane, where the initial complex-valued distribution is created by combining the measured amplitude distribution with the phase of the reference wave. (b) The wavefront propagates from the hologram plane to the sample plane, where it gives the distribution complex-valued transmission function $ t(x,y) $. (c) Constraints in the sample plane are applied, and the updated transmission function $ t^\prime (x,y) $ is obtained (d). (e) The wavefront is propagated from the sample plane to the detector plane (f). The amplitude of the wavefront distribution in the hologram plane is replaced with the measured amplitude. The complex-valued wavefront distribution in the detector plane is updated for the next iteration starting at (a).

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Fig. 4.
Fig. 4. 200 keV in-line hologram of latex sphere and its reconstruction. (a) In-line hologram of the sphere recorded at the defocus 180 µm. The blue lines mark the area outside which the transmission was set to 1 during the iterative reconstruction. (b) Retrieved amplitude distribution of the object wave. (c) Retrieved phase distribution of the object wave. Adapted from [27].

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Fig. 5.
Fig. 5. Object with absorbing and phase-shifting properties. (a) Distributions of transmittance (top) and phase (bottom) of the object. (b) Simulated hologram (top) and phase distributions at the detector plane (bottom). (c) Reconstructed amplitude (top) and phase (bottom) distributions of the transmission function. (d) Iteratively reconstructed amplitude (top) and phase (bottom) distributions in the object plane. The blue curves are the line scans through the centers of the corresponding images. Adapted from [12].

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Fig. 6.
Fig. 6. Reconstruction of 3D objects from two or more intensity measurements. (a) Experimental arrangement. The 3D sample is represented by a set of planes at different $ z $ positions. Here, the sample is sampled with four planes. Two holograms are acquired at different distances from the sample, $ {H_1} $ and $ {H_2} $. (b) Reconstructed amplitude distributions at the four planes within the sample distribution. Parameters of the simulations: wavelength is 532 nm, sample size is $ 1000 \;{\unicode{x00B5}{\rm m}} \times 1000\;{\unicode{x00B5}{\rm m}} $, sampled with $ 1000 \times 1000 $ pixels, distances between the planes within the sample are $ 50\;\unicode{x00B5}{\rm m} $, and $ {H_1} $ and $ {H_2} $ are acquired at distances 200 µm and 300 µm from the sample, respectively. In (b), only the central parts of the reconstructed distributions, $ 150\;{\unicode{x00B5}{\rm m}} \times 150\;{\unicode{x00B5}{\rm m}} $, sampled with $ 150 \times 150 $ pixels, are shown.

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Fig. 7.
Fig. 7. Reconstruction of 3D phase objects from two or more intensity measurements. (a) Experimental arrangement. The 3D sample is represented by a set of planes at different $z$ positions. Here, the sample is sampled with four planes. Two holograms are acquired at different distances from the sample, H1 and H2. (b) Reconstructed amplitude distributions at the four planes within the sample distribution. Parameters of the simulations are the same as in Fig. 6; the diameter of the spherical objects is 10 µm.

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Fig. 8.
Fig. 8. Resolution enhancement in digital holography by self-extrapolation of hologram. (a) Scanning electron microscope image of the sample. (b) $ 1000 \times 1000 $ pixels experimental optical hologram of the sample and its reconstruction (bottom). (c) Self-extrapolation of a piece of the hologram. The selected $ 500 \times 500 $ pixels part of the hologram is padded with zeros up to $ 1000 \times 1000 $ pixels and the corresponding reconstruction (bottom). (d) $ 1000 \times 1000 $ pixels self-extrapolated hologram from (c) after 300 iterations and its reconstruction (bottom). The details of the experiment and reconstruction procedure are available in Ref. [38].

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Equations (9)

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H ( X , Y ) = | R ( X , Y ) + O ( X , Y ) | 2 = | R ( X , Y ) | 2 + | O ( X , Y ) | 2 + R ∗ ( X , Y ) O ( X , Y ) | + R ( X , Y ) O ( X , Y ) ∗ .
R ( X , Y ) H ( X , Y ) ∝ | R ( X , Y ) | 2 O ( X , Y ) | + R 2 ( X , Y ) O ( X , Y ) ∗ .
t ( x , y ) = exp ⁡ [ − a ( x , y ) ] exp ⁡ [ i φ ( x , y ) ] ,
u ( x , y ) = u 0 ( x , y ) t ( x , y ) .
t ( x , y ) = 1 + o ( x , y ) ,
A u 0 ( x , y ) t ( x , y ) = A u 0 ( x , y ) [ 1 + o ( x , y ) ] → A [ R ( X , Y ) + O ( X , Y ) ] .
H ( X , Y ) = | A | 2 | R ( X , Y ) + O ( X , Y ) | 2 .
u z 2 = F T − 1 { F T ( u z 1 ) exp ⁡ ( 2 π i Δ z λ 1 − α 2 − β 2 ) } ,
F T ( u ) = ∬ u ( x , y ) exp ⁡ [ − 2 π i z ( α λ x + β λ y ) ] d x d y .