Abstract

A recently proposed technique introduced a time-resolved option of fast transient non-repetitive events to ptychographic microscopy. This technique, termed time-resolved imaging by multiplexed ptychography (TIMP), is based on algorithmic reconstruction of multiple frames from data recorded in a single camera acquisition of a single-shot ptychographic microscope. We demonstrate TIMP experimentally, reconstructing thirty-six frames of a dynamical complex-valued object from ptychographic data recorded in a single camera snapshot.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. Rodenburg, “Ptychography and related diffractive imaging methods,” in Advances in Imaging and Electron Physics, vol. 150 Hawkes, ed. (Elsevier, 2008), pp. 87–184.
    [Crossref]
  2. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
    [Crossref] [PubMed]
  3. J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
    [Crossref]
  4. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
    [Crossref] [PubMed]
  5. X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103, 171105 (2013).
    [Crossref]
  6. P. Sidorenko and O. Cohen, “Single-shot ptychography,” Optica 3, 9–14 (2016).
    [Crossref]
  7. P. Sidorenko, O. Lahav, and O. Cohen, “Ptychographic ultrahigh-speed imaging,” Opt. Express 25, 10997–11008 (2017).
    [Crossref] [PubMed]
  8. X. He, C. Liu, and J. Zhu, “Single-shot fourier ptychography based on diffractive beam splitting,” Opt. Lett. 43, 214–217 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  11. P. Li, T. Edo, D. Batey, J. Rodenburg, and A. Maiden, “Breaking ambiguities in mixed state ptychography,” Opt. Express 24, 9038–9052 (2016).
    [Crossref] [PubMed]
  12. J. Liang and L. V. Wang, “Single-shot ultrafast optical imaging,” Optica 5, 1113–1127 (2018).
    [Crossref]
  13. B. K. Chen, P. Sidorenko, O. Lahav, O. Peleg, and O. Cohen, “Multiplexed single-shot ptychography,” Opt. Lett. 43, 5379–5382 (2018).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  19. M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
    [Crossref]
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2018 (4)

2017 (1)

2016 (3)

2015 (1)

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

2013 (3)

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103, 171105 (2013).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68 (2013).
[Crossref] [PubMed]

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

2011 (1)

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

2009 (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

2004 (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

2001 (1)

Arnold, B.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Batey, D.

Brady, D. J.

D. J. Brady, Optical imaging and spectroscopy (John Wiley & Sons, 2009).
[Crossref]

Chen, B. K.

Cohen, O.

Edo, T.

Evans-Lutterodt, K.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Fainman, Y.

Faulkner, H. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Fienup, J. R.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Galtier, E. C.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Guizar-Sicairos, M.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Haham, G. I.

G. I. Haham, O. Peleg, P. Sidorenko, and O. Cohen, “High-resolution (diffraction limit) single-shot ptychography for ultrahigh-speed microscopy,” in Imaging and Applied Optics 2018 (DH), (Optical Society of America, 2018), p. JTh3A.4.

Hastings, J. B.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

He, X.

Hertz, H. M.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Hoppe, R.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Hore, A.

A. Hore and D. Ziou, “Image quality metrics: Psnr vs. ssim,” in 2010 20th International Conference on Pattern Recognition, (IEEE, 2010), pp. 2366–2369.
[Crossref]

Ishikawa, T.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical electrodynamics (John Wiley & Sons, 2007).

Lahav, O.

Lee, H. J.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Li, P.

Li, T.

Liang, J.

Liu, C.

Luo, Y.

Maiden, A.

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Meier, V.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Menzel, A.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68 (2013).
[Crossref] [PubMed]

Metzler, M.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Miao, J.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Murnane, M. M.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Nagler, B.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Narayanan, S.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Nilsson, D.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Pan, X.

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103, 171105 (2013).
[Crossref]

Patommel, J.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Peleg, O.

B. K. Chen, P. Sidorenko, O. Lahav, O. Peleg, and O. Cohen, “Multiplexed single-shot ptychography,” Opt. Lett. 43, 5379–5382 (2018).
[Crossref] [PubMed]

G. I. Haham, O. Peleg, P. Sidorenko, and O. Cohen, “High-resolution (diffraction limit) single-shot ptychography for ultrahigh-speed microscopy,” in Imaging and Applied Optics 2018 (DH), (Optical Society of America, 2018), p. JTh3A.4.

Robinson, I. K.

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Rodenburg, J.

P. Li, T. Edo, D. Batey, J. Rodenburg, and A. Maiden, “Breaking ambiguities in mixed state ptychography,” Opt. Express 24, 9038–9052 (2016).
[Crossref] [PubMed]

J. Rodenburg, “Ptychography and related diffractive imaging methods,” in Advances in Imaging and Electron Physics, vol. 150 Hawkes, ed. (Elsevier, 2008), pp. 87–184.
[Crossref]

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Rokitski, R.

Sandy, A. R.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Schroer, C. G.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Schropp, A.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Seiboth, F.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Shi, Y.

Sidorenko, P.

Stein, A.

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Sun, P.-C.

Thibault, P.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68 (2013).
[Crossref] [PubMed]

Uhlén, F.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Vogt, U.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Wang, L. V.

Xu, H.

Xu, W.

Zastrau, U.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Zhu, J.

Ziou, D.

A. Hore and D. Ziou, “Image quality metrics: Psnr vs. ssim,” in 2010 20th International Conference on Pattern Recognition, (IEEE, 2010), pp. 2366–2369.
[Crossref]

Appl. Phys. Lett. (3)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

X. Pan, C. Liu, and J. Zhu, “Single shot ptychographical iterative engine based on multi-beam illumination,” Appl. Phys. Lett. 103, 171105 (2013).
[Crossref]

M. Guizar-Sicairos, S. Narayanan, A. Stein, M. Metzler, A. R. Sandy, J. R. Fienup, and K. Evans-Lutterodt, “Measurement of hard x-ray lens wavefront aberrations using phase retrieval,” Appl. Phys. Lett. 98, 111108 (2011).
[Crossref]

Nature (1)

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68 (2013).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (3)

Optica (2)

Sci. Reports (1)

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. B. Hastings, D. Nilsson, F. Uhlén, U. Vogt, H. M. Hertz, and C. G. Schroer, “Full spatial characterization of a nanofocused x-ray free-electron laser beam by ptychographic imaging,” Sci. Reports 3, 1633 (2013).
[Crossref]

Science (1)

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348, 530–535 (2015).
[Crossref] [PubMed]

Ultramicroscopy (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Other (5)

J. Rodenburg, “Ptychography and related diffractive imaging methods,” in Advances in Imaging and Electron Physics, vol. 150 Hawkes, ed. (Elsevier, 2008), pp. 87–184.
[Crossref]

D. J. Brady, Optical imaging and spectroscopy (John Wiley & Sons, 2009).
[Crossref]

J. D. Jackson, Classical electrodynamics (John Wiley & Sons, 2007).

A. Hore and D. Ziou, “Image quality metrics: Psnr vs. ssim,” in 2010 20th International Conference on Pattern Recognition, (IEEE, 2010), pp. 2366–2369.
[Crossref]

G. I. Haham, O. Peleg, P. Sidorenko, and O. Cohen, “High-resolution (diffraction limit) single-shot ptychography for ultrahigh-speed microscopy,” in Imaging and Applied Optics 2018 (DH), (Optical Society of America, 2018), p. JTh3A.4.

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic diagram of 4fSSP microscope with ray tracing. (b) Schematic diagram of our TIMP microscope which is based on 4fSSP. A phase-only reflective SLM (SLMP) replaces the static pinhole array in (a). The dynamic object is produced by using a reflective amplitude-only SLM (SLMO). (c) The fixed component phase structure, which is induced by SLMP, that mimics a micro-lens array (MLA).
Fig. 2
Fig. 2 Reconstruction of 9 complex-valued objects and probes using OAME. (a) The intensity pattern recorded in a single camera snapshot. (b) Reconstructed frames – complex-valued objects and probes. Each frame is divided to 4 quarters (as marked on the first frame): top-left is object amplitude, top-right is object phase, bottom left is probe amplitude and bottom-right is probe phase. The amplitudes are normalized, and the phases are in the [−π, π] range.
Fig. 3
Fig. 3 Reconstruction of 9 complex-valued objects and probes using PGE. (a) The intensity pattern recorded in a single camera snapshot. (b) Reconstructed frames – complex-valued objects and probes. Each frame is divided to 4 quarters (as marked on the first frame): top-left is object amplitude, top-right is object phase, bottom left is probe amplitude and bottom-right is probe phase. The amplitudes are normalized, and the phases are in the [−π, π] range.
Fig. 4
Fig. 4 Reconstruction of 36 complex-valued objects and probes using OAM and PG encodings. (a) The intensity pattern recorded in a single camera snapshot. (b) Reconstructed frames – complex-valued objects and probes. Each frame is divided to 4 quarters (as marked on the first frame): top-left is object amplitude, top-right is object phase, bottom left is probe amplitude and bottom-right is probe phase. The amplitudes are normalized, and the phases are in the [−π, π] range.
Fig. 5
Fig. 5 The amplitude images and radially averaged spatial spectra that were used for comparison in Table 1. (a) the original (ground truth) image, (b) SSP reconstruction, (c) OAME TIMP reconstruction, (d) PGE TIMP reconstruction, (e) OAME+PGE (combined) TIMP reconstruction. (f) Radial average of the spatial spectra of (a)–(e). The dashed red line marks half of the maximum for FWHM comparison.
Fig. 6
Fig. 6 Reconstruction of 9 complex-valued spatial profiles of pulses in a burst from measured data shown in Fig. 2(a). Each frame is divided to 2 (as marked on the first frame): amplitude on the left and phase on the right. The amplitudes are normalized, and the phases are in the [−π, π] range.

Tables (1)

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Table 1 Reconstructions Comparison

Equations (6)

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I m ( ν ) = k = 1 K [ P k ( r R m ) O k ( r ) ] 2
Φ l ( r , θ ) = exp ( i π r 2 λ f MLA ) exp ( i l θ )
Φ k ( r ) = exp ( i π r 2 λ f MLA ) exp ( i k r )
b / 2 b / 2 b / 2 b / 2 Φ k ( r R m ) Φ k ˜ * ( r R m ) d x d y = 4 e i Δ k R m sin ( b 2 Δ k x ) Δ k x sin ( b 2 Δ k y ) Δ k y
Δ k { 2 π b ( i , j ) | ( i , j ) 2 , ( i , j ) ( 0 , 0 ) }
Δ k max < ν max Δ k obj = ( 1 n ) b 2 λ f 1 i max < ( 1 n ) b 2 4 π λ f 1

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