Abstract

Coherent diffraction imaging (CDI) is a method for reconstructing the complex-valued image of an object from diffraction intensities by using iterative phasing methods. X-ray ptychography is a scanning type of CDI using X-rays, allowing us to visualize the complex transmission function of an extended specimen. We here propose the use of the Kramers–Kronig relation (KKR) as an additional constraint in phase retrieval algorithms for multiple-energy X-ray ptychography using the absorption edge of a specific element. A numerical simulation showed that the speed of convergence was increased by using the improved algorithm with the KKR. We successfully demonstrated its usefulness in a proof-of-principle experiment at SPring-8. The present algorithm is particularly useful for imaging X-ray absorption fine structures of a specific element buried within thick samples by hard X-ray spectro-ptychography.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  34. Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
    [Crossref]
  35. M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
    [Crossref]

2017 (1)

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

2015 (2)

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

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]

2014 (1)

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

2013 (6)

A. M. Maiden, G. R. Morrison, B. Kaulich, A. Gianoncelli, and J. M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick-Baez focusing,” Acta Crystallogr. A 69, 490–497 (2013).
[Crossref]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref] [PubMed]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Faalkenberg, and C. G. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
[Crossref]

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

2012 (1)

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref] [PubMed]

2011 (3)

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011).
[Crossref]

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, K. Yamauchi, and T. Ishikawa, “Multiscale element mapping of buried structures by ptychographic x-ray diffraction microscopy using anomalous scattering,” Appl. Phys. Lett. 99, 131905 (2011).
[Crossref]

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
[Crossref]

2010 (1)

H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4, 833–839 (2010).
[Crossref]

2009 (2)

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

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

2008 (1)

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographic iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

2007 (2)

Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 98, 034801 (2007).

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

2005 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005).
[Crossref] [PubMed]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Prob. 21, 37–50 (2005).
[Crossref]

2004 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[Crossref] [PubMed]

W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
[Crossref] [PubMed]

2003 (4)

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100, 110–112 (2003).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spense, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from over-sampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003).
[Crossref]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[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]

1987 (1)

1982 (1)

1978 (1)

1972 (1)

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

Allen, L. J.

W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
[Crossref] [PubMed]

Als-Nielsen, J.

J. Als-Nielsen, Elements of Modern X-ray Physics (John Wiley & Sons, Copenhagen, 2002).

Bazylinski, D. A.

X. Zhu, A. P. Hitchcock, D. A. Bazylinski, P. Denes, J. Joseph, U. Lins, S. Marchesini, H.-W. Shiu, T. Tyliszczak, and D. A. Shapiro, “Measuring spectroscopy and magnetism of extracted and intracellular magnetosomes using soft X-ray ptychography,” Proc. Natl. Acad. Sci. U.S.A. early edition (2016).
[Crossref]

Bean, R.

Beckers, M.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011).
[Crossref]

Berenguer, F.

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographic iterative engine,” Ultramicroscopy 108, 481–487 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Cabana, J.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Celestre, R.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Chao, W.

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Chapman, H. N.

H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4, 833–839 (2010).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spense, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[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, B.

Chen, C.-C.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Damsgaard, C. D.

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Faalkenberg, and C. G. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
[Crossref]

David, C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
[Crossref] [PubMed]

Denes, P.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

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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).
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Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 98, 034801 (2007).

Morrison, G. R.

A. M. Maiden, G. R. Morrison, B. Kaulich, A. Gianoncelli, and J. M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
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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).
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Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 98, 034801 (2007).

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100, 110–112 (2003).
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Y. Nishino, J. Miao, and T. Ishikawa, “Image reconstruction of nanostructured nonperiodic objects only from over-sampled hard x-ray diffraction intensities,” Phys. Rev. B 68, 220101 (2003).
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S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spense, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
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Padmore, H.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
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Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
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Padmore, H. A.

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
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R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Faalkenberg, and C. G. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
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Perciano, T.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
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Peterson, I.

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
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O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographic iterative engine,” Ultramicroscopy 108, 481–487 (2008).
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J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
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Qian, D.

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
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Reese, M.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011).
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Reinhardt, J.

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Faalkenberg, and C. G. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
[Crossref]

Robinson, I. K.

Rodenburg, J. M.

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21, 13592–13606 (2013).
[Crossref] [PubMed]

A. M. Maiden, G. R. Morrison, B. Kaulich, A. Gianoncelli, and J. M. Rodenburg, “Soft X-ray spectromicroscopy using ptychography with randomly phased illumination,” Nat. Commun. 4, 1669 (2013).
[Crossref] [PubMed]

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
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A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
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J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-X-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98, 034801 (2007).
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H. M. L. Faulkner and J. M. Rodenburg, “Error tolerance of an iterative phase retrieval algorithm for moveable illumination microscopy,” Ultramicroscopy 103, 153–164 (2005).
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H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
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Rodriguez, J. A.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

Rosenhahn, A.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011).
[Crossref]

Salditt, T.

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick-Baez focusing,” Acta Crystallogr. A 69, 490–497 (2013).
[Crossref]

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011).
[Crossref]

Sarahan, M. C.

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
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R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Sayre, D.

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]

Schroer, C. G.

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Faalkenberg, and C. G. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
[Crossref]

Senba, Y.

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
[Crossref]

Senkbeil, T.

M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S.-C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys. Rev. Lett. 107, 208202 (2011).
[Crossref]

Shapiro, D. A.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

X. Zhu, A. P. Hitchcock, D. A. Bazylinski, P. Denes, J. Joseph, U. Lins, S. Marchesini, H.-W. Shiu, T. Tyliszczak, and D. A. Shapiro, “Measuring spectroscopy and magnetism of extracted and intracellular magnetosomes using soft X-ray ptychography,” Proc. Natl. Acad. Sci. U.S.A. early edition (2016).
[Crossref]

Shi, X.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

Shiu, H.-W.

X. Zhu, A. P. Hitchcock, D. A. Bazylinski, P. Denes, J. Joseph, U. Lins, S. Marchesini, H.-W. Shiu, T. Tyliszczak, and D. A. Shapiro, “Measuring spectroscopy and magnetism of extracted and intracellular magnetosomes using soft X-ray ptychography,” Proc. Natl. Acad. Sci. U.S.A. early edition (2016).
[Crossref]

Spense, J. C. H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spense, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Strobridge, F. C.

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

Suzuki, A.

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, K. Yamauchi, and T. Ishikawa, “Multiscale element mapping of buried structures by ptychographic x-ray diffraction microscopy using anomalous scattering,” Appl. Phys. Lett. 99, 131905 (2011).
[Crossref]

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
[Crossref]

Takahashi, Y.

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
[Crossref]

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, K. Yamauchi, and T. Ishikawa, “Multiscale element mapping of buried structures by ptychographic x-ray diffraction microscopy using anomalous scattering,” Appl. Phys. Lett. 99, 131905 (2011).
[Crossref]

Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 98, 034801 (2007).

Thibault, P.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494, 68–71 (2013).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

Tsutsumi, R.

Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 98, 034801 (2007).

Tyliszczak, T.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

X. Zhu, A. P. Hitchcock, D. A. Bazylinski, P. Denes, J. Joseph, U. Lins, S. Marchesini, H.-W. Shiu, T. Tyliszczak, and D. A. Shapiro, “Measuring spectroscopy and magnetism of extracted and intracellular magnetosomes using soft X-ray ptychography,” Proc. Natl. Acad. Sci. U.S.A. early edition (2016).
[Crossref]

Vassholz, M.

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick-Baez focusing,” Acta Crystallogr. A 69, 490–497 (2013).
[Crossref]

Vila-Comamala, J.

Warwick, T.

M. Farmand, R. Celestre, P. Denes, A. L. D. Kilcoyne, S. Marchesini, H. Padmore, T. Tyliszczak, T. Warwick, X. Shi, J. Lee, Y.-S. Yu, J. Cabana, J. Joseph, H. Krishnan, T. Perciano, F. R. N. C. Maia, and D. A. Shapiro, “Near-edge X-ray refraction fine structure microscopy,” Appl. Phys. Lett. 110, 063101 (2017).
[Crossref]

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Weierstall, U.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spense, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Wellenreuther, G.

R. Hoppe, J. Reinhardt, G. Hofmann, J. Patommel, J.-D. Grunwaldt, C. D. Damsgaard, G. Wellenreuther, G. Faalkenberg, and C. G. Schroer, “High-resolution chemical imaging of gold nanoparticles using hard x-ray ptychography,” Appl. Phys. Lett. 102, 203104 (2013).
[Crossref]

Wilke, R. N.

R. N. Wilke, M. Vassholz, and T. Salditt, “Semi-transparent central stop in high-resolution X-ray ptychography using Kirkpatrick-Baez focusing,” Acta Crystallogr. A 69, 490–497 (2013).
[Crossref]

Xu, R.

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

Yamauchi, K.

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
[Crossref]

Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, K. Yamauchi, and T. Ishikawa, “Multiscale element mapping of buried structures by ptychographic x-ray diffraction microscopy using anomalous scattering,” Appl. Phys. Lett. 99, 131905 (2011).
[Crossref]

Y. Takahashi, Y. Nishino, R. Tsutsumi, H. Kubo, H. Furukawa, H. Mimura, S. Matsuyama, N. Zettsu, E. Matsubara, T. Ishikawa, and K. Yamauchi, “High-resolution diffraction microscopy using the plane-wave field of a nearly diffraction limited focused x-ray beam,” Phys. Rev. B 98, 034801 (2007).

Yang, L. L.

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
[Crossref]

Yu, Y.

Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
[Crossref] [PubMed]

D. A. Shapiro, Y. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft X-ray microscopy,” Nat. Photonics 8, 765–769 (2014).
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Yu, Y.-S.

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X. Zhu, A. P. Hitchcock, D. A. Bazylinski, P. Denes, J. Joseph, U. Lins, S. Marchesini, H.-W. Shiu, T. Tyliszczak, and D. A. Shapiro, “Measuring spectroscopy and magnetism of extracted and intracellular magnetosomes using soft X-ray ptychography,” Proc. Natl. Acad. Sci. U.S.A. early edition (2016).
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Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, K. Yamauchi, and T. Ishikawa, “Multiscale element mapping of buried structures by ptychographic x-ray diffraction microscopy using anomalous scattering,” Appl. Phys. Lett. 99, 131905 (2011).
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Y. Yu, C. Kim, D. A. Shapiro, M. Farmand, D. Qian, T. Tyliszczak, A. L. D. Kilcoyne, R. Celestre, S. Marchesini, J. Joseph, P. Denes, T. Warwick, F. C. Strobridge, C. P. Grey, H. Padmore, Y. S. Meng, R. Kostecki, and J. Cabana, “Dependence on crystal size of the nanoscale chemical phase distribution and fracture in Lix FePO4,” Nano Lett. 15, 4282–4288 (2015).
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Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011).
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Figures (5)

Fig. 1
Fig. 1

Flowchart of the KKR constraint at the nth iteration in ptychographical iterative phase retrieval algorithms.

Fig. 2
Fig. 2

(a) Reconstructed amplitude image without (left) and with (right) the KKR constraint. (b) Reconstructed phase image without (left) and with (right) the KKR constraint. The scale bar is 1 μm and X-ray energy is 6.557keV. (c) Iteration number dependence of the RMS error of the object image with and without using the KKR constraint. (d) Original (top) and reconstructed near-edge XAFS spectra without (middle) and with (bottom) using the KKR constraint. The spectra were extracted from the 40 × 40 nm2 region (2 × 2 pixels) indicated by the arrows in (a) and (b).

Fig. 3
Fig. 3

(a) FE-SEM image of the test sample. The scale bar is 1 μm. (b) Experimental setup at BL29XUL in SPring-8. (c) Coherent diffraction pattern of the sample at 6.554 keV.

Fig. 4
Fig. 4

(a-b) Amplitude (left) and phase (right) images reconstructed from the ptychographic diffraction patterns at (a) 6.542 keV and (b) 6.554 keV. The scale bar is 1 μm. (c) PRTF curve of the image reconstructed at 6.564 keV. (d) Energy dependence of the full period spatial resolution determined using PRTF.

Fig. 5
Fig. 5

(a) Amplitude image reconstructed at 6.554 keV. The scale bar is 1 μm. (b) Near-edge XAFS spectrum of the sample measured in transmittance geometry using a focused X-ray beam (top). Near-edge XAFS spectra reconstructed by X-ray ptychography without (left) and with (right) using the KKR constraint. The spectra were extracted from the 40 × 40 nm2 regions (2 × 2 pixels) indicated by the arrows in (a). (c) Near-edge phase spectra at the same positions without (left) and with (right) using the KKR constraint.

Equations (9)

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| T ( r , E ) | = exp ( 2 π λ β ( r , E ) t ( r ) ) ,
arg ( T ( r , E ) ) = 2 π λ δ ( r , E ) t ( r ) .
δ = r e 2 π λ 2 i n i ( Z i + f i ) ,
β = r e 2 π λ 2 i n i f i ,
f ( ω ) = 2 π P 0 ω f ( ω ) ω 2 ω 2 d ω ,
2 π P 0 E { ln | T ( r , E ) | / λ } E 2 E 2 d E arg ( T ( r , E ) ) λ = r e t ( r ) i n i ( r ) Z i = c o n s t .
I j ( q , E k ) = | ψ j ( r , E k ) | 2 ,
ψ j ( r , E k ) = T ( r r j , E k ) × P ( r , E k ) ,
E e r r o r ( n ) = 1 K k = 1 k = K Σ r | T ( r , E k ) γ ( E k ) T n ( r , E k ) | 2 Σ r | T n ( r , E k ) | 2 ,