Hard X-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction

R. N. Wilke; M. Priebe; M. Bartels; K. Giewekemeyer; A. Diaz; P. Karvinen; T. Salditt

doi:10.1364/OE.20.019232

Hard X-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction

R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt

Optics Express
Vol. 20,
Issue 17,
pp. 19232-19254
(2012)
•https://doi.org/10.1364/OE.20.019232

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R. N. Wilke, M. Priebe, M. Bartels, K. Giewekemeyer, A. Diaz, P. Karvinen, and T. Salditt, "Hard X-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction," Opt. Express 20, 19232-19254 (2012)

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Related Topics
Table of Contents Category
- Image Processing
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Phase retrieval (100.5070)
- Tomographic image processing (100.6950)
- Cell analysis (170.1530)
- X-ray imaging (340.7440)

About this Article
History
- Original Manuscript: May 10, 2012
- Revised Manuscript: June 8, 2012
- Manuscript Accepted: June 13, 2012
- Published: August 8, 2012
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 7, Iss. 10

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Abstract

Ptychographic coherent X-ray diffractive imaging (PCDI) has been combined with nano-focus X-ray diffraction to study the structure and density distribution of unstained and unsliced bacterial cells, using a hard X-ray beam of 6.2keV photon energy, focused to about 90nm by a Fresnel zone plate lens. While PCDI provides images of the bacteria with quantitative contrast in real space with a resolution well below the beam size at the sample, spatially resolved small angle X-ray scattering using the same Fresnel zone plate (cellular nano-diffraction) provides structural information at highest resolution in reciprocal space up to 2nm⁻¹. We show how the real and reciprocal space approach can be used synergistically on the same sample and with the same setup. In addition, we present 3D hard X-ray imaging of unstained bacterial cells by a combination of ptychography and tomography.

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[Crossref]

2011 (4)

T. H. Jensen, M. Bech, O. Bunk, M. Thomsen, A. Menzel, A. Bouchet, G. L. Duc, R. Feidenhans’l, and F. Pfeiffer, “Brain tumor imaging using small-angle x-ray scattering tomography,” Phys. Med. Biol. 56, 1717–1726 (2011).
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[Crossref] [PubMed]

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]

M. Guizar-Sicairos, A. Diaz, M. Holler, M. S. Lucas, A. Menzel, R. A. Wepf, and O. Bunk, “Phase tomography from x-ray coherent diffractive imaging projections,” Opt. Express 19, 21345–21357 (2011).
[Crossref] [PubMed]

2010 (9)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467, 436–439 (2010).
[Crossref] [PubMed]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. von Knig, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” New J. Phys. 12, 035017 (2010).
[Crossref]

A. Schropp, P. Boye, J. M. Feldkamp, R. Hoppe, J. Patommel, D. Samberg, S. Stephan, K. Giewekemeyer, R. N. Wilke, T. Salditt, J. Gulden, A. P. Mancuso, I. A. Vartanyants, E. Weckert, S. Schoder, M. Burghammer, and C. G. Schroer, “Hard x-ray nanobeam characterization by coherent diffraction microscopy,” Appl. Phys. Lett. 96, 091102 (2010).
[Crossref]

C. M. Kewish, P. Thibault, M. Dierolf, O. Bunk, A. Menzel, J. Vila-Comamala, K. Jefimovs, and F. Pfeiffer, “Ptychographic characterization of the wavefield in the focus of reflective hard x-ray optics,” Ultramicroscopy 110, 325–329 (2010).
[Crossref] [PubMed]

S. Gorelick, V. A. Guzenko, J. Vila-Comamala, and C. David, “Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating,” Nanotechnology 21, 295303 (2010).
[Crossref] [PubMed]

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 107, 529–534 (2010).
[Crossref]

K. Giewekemeyer, H. Neubauer, S. Kalbfleisch, S. P. Krüger, and T. Salditt, “Holographic and diffractive x-ray imaging using waveguides as quasi-point sources,” New J. Phys. 12, 035008 (2010).
[Crossref]

J. Nelson, X. Huang, J. Steinbrener, D. Shapiro, J. Kirz, S. Marchesini, A. M. Neiman, J. J. Turner, and C. Jacobsen, “High-resolution x-ray diffraction microscopy of specifically labeled yeast cells,” Proc. Natl. Acad. Sci. USA 107, 7235–7239 (2010).
[Crossref] [PubMed]

2009 (7)

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

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]

P. Kraft, A. Bergamaschi, C. Broennimann, R. Dinapoli, E. F. Eikenberry, B. Henrich, I. Johnson, A. Mozzanica, C. M. Schleputz, P. R. Willmott, and B. Schmitt, “Performance of single-photon- counting pilatus detector modules,” J. Synchrotron Radiat. 16, 368–375 (2009).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty, A. G. Peele, and K. A. Nugent, “Ptychographic fresnel coherent diffractive imaging,” Phys. Rev. A 80, 063823 (2009).
[Crossref]

M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. 170, 4–12 (2009).
[Crossref]

2008 (4)

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

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

2007 (2)

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]

A. Gourrier, W. Wagermaier, M. Burghammer, D. Lammie, H. S. Gupta, P. Fratzl, C. Riekel, T. J. Wess, and O. Paris, “Scanning x-ray imaging with small-angle scattering contrast,” J. Appl. Crystallogr. 40, 78–82 (2007).
[Crossref]

2006 (5)

M. Eltsov and J. Dubochet, “Rebuttal: Ring-like nucleoids and dna repair in deinococcus radio-durans,” J. Bacteriol. 188, 6052 (2006).
[Crossref]

M. Eltsov and J. Dubochet, “Study of the deinococcus radiodurans nucleoid by cryoelectron microscopy of vitreous sections: supplementary comments,” J. Bacteriol. 188, 6053–6058 (2006).
[Crossref] [PubMed]

A. Minsky, E. Shimoni, and J. Englander, “Rebuttal: study of the deinococcus radiodurans nucleoid,” J. Bacteriol. 188, 6059 (2006).
[Crossref]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506 (2006).
[Crossref] [PubMed]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

2005 (3)

M. Eltsov and J. Dubochet, “Fine structure of the deinococcus radiodurans nucleoid revealed by cryoelectron microscopy of vitreous sections,” J. Bacteriol. 187, 8047–8054 (2005).
[Crossref] [PubMed]

J. Zimmerman and J. Battista, “A ring-like nucleoid is not necessary for radioresistance in the deinococcaceae,” BMC Microbiol. 5, 17 (2005).
[Crossref] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. USA 102, 15343–15346 (2005).
[Crossref] [PubMed]

2004 (1)

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

2003 (2)

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003).
[Crossref]

S. Levin-Zaidman, J. Englander, E. Shimoni, A. K. Sharma, K. W. Minton, and A. Minsky, “Ringlike structure of the deinococcus radiodurans genome: a key to radioresistance?,” Science 299, 254–256 (2003).
[Crossref] [PubMed]

2000 (1)

O. Paris, D. Loidl, H. Peterlik, M. Muller, H. Lichtenegger, and P. Fratzl, “The internal structure of single carbon fibers determined by simultaneous small- and wide-angle scattering,” J. Appl. Crystallogr. 33, 695–699 (2000).
[Crossref]

1999 (2)

S. Rinnerthaler, P. Roschger, H. F. Jakob, A. Nader, K. Klaushofer, and P. Fratzl, “Scanning small angle x-ray scattering analysis of human bone sections,” Calcified Tissue Int. 64, 422–429 (1999).
[Crossref]

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

1998 (1)

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]

1997 (1)

P. Fratzl, H. F. Jakob, S. Rinnerthaler, P. Roschger, and K. Klaushofer, “Position-resolved small-angle x-ray scattering of complex biological materials,” J. Appl. Crystallogr. 30, 765–769 (1997).
[Crossref]

1988 (1)

J. Dubochet, M. Adrian, J.-J. Chang, J.-C. Homo, J. Lepault, A.W. McDowall, and P. Schultz, “Cryo-Electron microscopy of vitrified specimens,” Q. Rev. Biophys. 21, 129–228 (1988).
[Crossref] [PubMed]

1985 (1)

Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985).
[Crossref] [PubMed]

1982 (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

Abbey, B.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Adrian, M.

Barrett, R.

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003).
[Crossref]

Barty, A.

Battista, J.

J. Zimmerman and J. Battista, “A ring-like nucleoid is not necessary for radioresistance in the deinococcaceae,” BMC Microbiol. 5, 17 (2005).
[Crossref] [PubMed]

Bech, M.

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

Beerlink, A.

Beetz, T.

Bergamaschi, A.

Bilger, Hans R.

Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985).
[Crossref] [PubMed]

Binderup, T.

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

Bouchet, A.

Boye, P.

Broennimann, C.

Bunk, O.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

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. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Burghammer, M.

Chang, J.-J.

Chapman, H.

Chapman, H. N.

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.

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003).
[Crossref]

Clark, J. N.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Cui, C.

Cullis, A. G.

David, C.

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

de Jonge, M.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

de Jonge, M. D.

Dhal, B. B.

Diaz, A.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

Dierolf, M.

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

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]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Dinapoli, R.

Dobson, B. R.

Dubochet, J.

M. Eltsov and J. Dubochet, “Rebuttal: Ring-like nucleoids and dna repair in deinococcus radio-durans,” J. Bacteriol. 188, 6052 (2006).
[Crossref]

Duc, G. L.

Eaton, W.

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003).
[Crossref]

Eikenberry, E. F.

Elser, V.

Eltsov, M.

M. Eltsov and J. Dubochet, “Rebuttal: Ring-like nucleoids and dna repair in deinococcus radio-durans,” J. Bacteriol. 188, 6052 (2006).
[Crossref]

Englander, J.

A. Minsky, E. Shimoni, and J. Englander, “Rebuttal: study of the deinococcus radiodurans nucleoid,” J. Bacteriol. 188, 6059 (2006).
[Crossref]

Faulkner, H. M. L.

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

Feidenhans’l, R.

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

Feldkamp, J. M.

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

Fienup, J.R.

M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

Fournet, G.

A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, 1955).

Fratzl, P.

Giewekemeyer, K.

K. Giewekemeyer, “A study on new approaches in coherent x-ray microscopy of biological specimens,” Ph.D. Thesis, Universität Göttingen (2011).

Glatter, O.

O. Glatter and O. Kratky, Small Angle X-Ray Scattering (Acad. Pr., 1982).

Gorelick, S.

Gourrier, A.

Guinier, A.

A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, 1955).

Guizar-Sicairos, M.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

M. Guizar-Sicairos, “Methods for coherent lensless imaging and x-ray wavefront measurement,” Ph.D. Thesis, University of Rochester (2010).

Gulden, J.

Gupta, H. S.

Guzenko, V. A.

Habib, Taufiq

Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985).
[Crossref] [PubMed]

Hau-Riege, S. P.

He, H.

Henrich, B.

Holler, M.

Holton, J.

Homo, J.-C.

Hoppe, R.

Howells, M.

Howells, M. R.

Huang, X.

Hurst, A. C.

Ishikawa, T.

Jacobsen, C.

Jakob, H. F.

Jefimovs, K.

Jensen, T. H.

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

Johnson, I.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Kalbfleisch, S.

Kewish, C. M.

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

Kirz, J.

Klaushofer, K.

Kohmura, Y.

Kraft, P.

Kratky, O.

O. Glatter and O. Kratky, Small Angle X-Ray Scattering (Acad. Pr., 1982).

Krüger, S. P.

Kynde, S.

Lammie, D.

Lepault, J.

Levin-Zaidman, S.

Lichtenegger, H.

Lima, E.

Loidl, D.

Lucas, M. S.

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]

Mancuso, A. P.

Marchesini, S.

Marti, O.

McDowall, A.W.

McNulty, I.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Menzel, A.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

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]

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Miao, H.

Miao, J.

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]

Minsky, A.

A. Minsky, E. Shimoni, and J. Englander, “Rebuttal: study of the deinococcus radiodurans nucleoid,” J. Bacteriol. 188, 6059 (2006).
[Crossref]

Minton, K. W.

Morrison, G.

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003).
[Crossref]

G. Morrison and B. Niemann, “Differential phase contrast x-ray microscopy,” in X-Ray Microscopy and Spectromicroscopy, Würzburg, 1996, J. Thieme, ed. (Springer, 1998), 85–94.

Mozzanica, A.

Muller, M.

Nader, A.

Neiman, A. M.

Nelson, J.

Neubauer, H.

Niemann, B.

G. Morrison and B. Niemann, “Differential phase contrast x-ray microscopy,” in X-Ray Microscopy and Spectromicroscopy, Würzburg, 1996, J. Thieme, ed. (Springer, 1998), 85–94.

Noy, A.

Nugent, K. A.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Nygård, K.

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

Ohashi, H.

Paris, O.

Paterson, D.

Patommel, J.

Peele, A. G.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Perret, E.

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

Peterlik, H.

Pfeifer, M. A.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Pfeiffer, F.

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

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]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Quiney, H. M.

Riekel, C.

Rinnerthaler, S.

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

Roschger, P.

Rosen, R.

Salditt, T.

Salome, M.

Samberg, D.

Sayre, D.

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]

Schleputz, C. M.

Schlichting, I.

Schmitt, B.

Schneider, P.

Schoder, S.

Schroer, C. G.

Schropp, A.

Schultz, P.

Senba, Y.

Shapiro, D.

Sharma, A. K.

Shimoni, E.

A. Minsky, E. Shimoni, and J. Englander, “Rebuttal: study of the deinococcus radiodurans nucleoid,” J. Bacteriol. 188, 6059 (2006).
[Crossref]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Spence, J.

Spence, J. C. H.

Starodub, D.

Steinbrener, J.

Stephan, S.

Suzuki, A.

Takahashi, Y.

Thibault, P.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

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]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Thomsen, M.

Thurman, S.T.

M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

Tran, C. Q.

Trtik, P.

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

Turner, J. J.

van der Veen, J.F.

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

Vartanyants, I. A.

Vila-Comamala, J.

Vine, D. J.

von Knig, K.

Wagermaier, W.

Weckert, E.

Weierstall, U.

Weitkamp, T.

T. Weitkamp, “Imaging and tomography with high resolution using coherent hard synchroton radiation,” Ph.D. Thesis, University of Hamburg (2002).

Wepf, R.

Wepf, R. A.

Wess, T. J.

Wilke, R. N.

Williams, G. J.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Willmott, P. R.

Yamauchi, K.

Zettsu, N.

Zimmerman, J.

J. Zimmerman and J. Battista, “A ring-like nucleoid is not necessary for radioresistance in the deinococcaceae,” BMC Microbiol. 5, 17 (2005).
[Crossref] [PubMed]

Appl. Opt. (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref] [PubMed]

Hans R. Bilger and Taufiq Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

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

BMC Microbiol. (1)

J. Zimmerman and J. Battista, “A ring-like nucleoid is not necessary for radioresistance in the deinococcaceae,” BMC Microbiol. 5, 17 (2005).
[Crossref] [PubMed]

Calcified Tissue Int. (1)

J. Appl. Cryst. (1)

K. Nygård, O. Bunk, E. Perret, C. David, and J.F. van der Veen, “Diffraction gratings as small-angle X-ray scattering calibration standards,” J. Appl. Cryst. 43, 350–351 (2010).
[Crossref]

J. Appl. Crystallogr. (3)

J. Bacteriol. (4)

M. Eltsov and J. Dubochet, “Rebuttal: Ring-like nucleoids and dna repair in deinococcus radio-durans,” J. Bacteriol. 188, 6052 (2006).
[Crossref]

A. Minsky, E. Shimoni, and J. Englander, “Rebuttal: study of the deinococcus radiodurans nucleoid,” J. Bacteriol. 188, 6059 (2006).
[Crossref]

J. Electron Spectrosc. (1)

J. Opt. Soc. Am. A (2)

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]

J. Phys. IV France (1)

G. Morrison, W. Eaton, R. Barrett, and P. Charalambous, “STXM imaging with a configured detector,” J. Phys. IV France 104, 547–550 (2003).
[Crossref]

J. Synchrotron Radiat. (2)

Nanotechnology (1)

Nat. Phys. (1)

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[Crossref]

Nature (2)

New J. Phys. (3)

O. Bunk, M. Bech, T. H. Jensen, R. Feidenhans’l, T. Binderup, A. Menzel, and F. Pfeiffer, “Multimodal x-ray scatter imaging,” New J. Phys. 11, 123016 (2009).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

M. Guizar-Sicairos, S.T. Thurman, and J.R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

Phys. Med. Biol. (1)

Phys. Rev. A (2)

P. Thibault, M. Dierolf, C. M. Kewish, A. Menzel, O. Bunk, and F. Pfeiffer, “Contrast mechanisms in scanning transmission x-ray microscopy,” Phys. Rev. A 80, 043813 (2009).
[Crossref]

Phys. Rev. B (2)

A. Diaz, P. Trtik, M. Guizar-Sicairos, A. Menzel, P. Thibault, and O. Bunk, “Quantitative x-ray phase nanotomography,” Phys. Rev. B 85, 020104 (2012).
[Crossref]

Phys. Rev. Lett. (2)

Proc. Natl. Acad. Sci. USA (3)

Q. Rev. Biophys. (1)

Science (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

Ultramicroscopy (4)

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]

Other (7)

O. Glatter and O. Kratky, Small Angle X-Ray Scattering (Acad. Pr., 1982).

A. Guinier and G. Fournet, Small-Angle Scattering of X-Rays (Wiley, 1955).

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Supplementary Material (2)

» Media 1: AVI (7973 KB)

» Media 2: MPG (22799 KB)

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Fig. 1 Schematic of the experimental setup used for X-ray cellular nano-diffraction and PCDI accentuating the important parameters: 1. gap of the source slit S0, which controls the flux and coherence properties of the beam, 2. the distance Z₀ between the zone plate optics (FZP) and the sample, which defines the size of the beam at the cellular specimen, and 3. the detector distance Z₁₂ with respect to the sample, which determines sampling of the recorded diffraction pattern and attainable diffraction angle. For different choices of these three parameters the biological specimen can then be translated in the XY – directions and additionally be rotated with respect to ϕ. In detail, the beamline setup consists of the X-ray undulator (period 19mm) source (S), horizontal and vertical slits (S1–S4), Si(111) monochromator (LN₂-cooled), fast shutter (FS), filter (Fi), central stop (CS), order sorting aperture (OSA) and focus (F) with bacterial specimen. Flight tube and beamstops are not shown.

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Fig. 2 (a) Phase from ‘ePIE’ reconstruction of the tantalum test structure. Scan points are indicated as magenta points. A fit of the error function to one of the edges of the line scans yields a line-spread function with FWHM-values of 16nm in the vertical direction and 17nm in the horizontal direction. The inner part of (a) is shown in (c). (b) Complex representation of reconstructed probe at the plane of the sample (top) and horizontal and vertical line scans through the centre of the back-propagated probe (bottom) (cf. Media 1). The white dashed lines highlight the position of the different focal positions. (d) (top) Gaussian fits (solid red lines) to the peaks of the line scans of the probe intensity are depicted below together with the data from the line scans (black dots). We obtain FWHM-values of 87nm in the vertical direction, and 93nm in the horizontal direction. (d) A typical diffraction pattern of the PCDI dataset (log-scale) is shown in the lower part.

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Fig. 3 The PRTFs of the ptychographic reconstruction of the resolution test chart and of a single projection of cells (Fig. 4(b)) are depicted in (a), green curve and blue curve respectively. In addition, the PRTF of the test sample according to an average over a single scan point is shown as red curve (cf. Eq. (2)). The high frequency ranges of the azimuthally averaged power spectral densities (PSD) of the reconstructed phase distributions are displayed in (b). Real space half-periods of 50nm, 25nm and 15nm are indicated as dashed, solid and dashed-dotted black lines, respectively. Legend of (b) is legend in (a).

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Fig. 4 (a)–(c) Phase reconstructions of D. radiodurans (‘DM’ - algorithm) showing dark areas, which we attribute to DNA rich regions. (d) 3D visualisation of the tomographic reconstruction of the set of projection images such as seen in (b) (cf. Media 2). The tomographic reconstruction is shown at three different angles. A magenta label in the tomogram (front-view) depicts voxel-regions which have been used to estimate the mass density ρ_c in the high-density regions of the bacterial specimen (red). We obtained <ρ_c> ≈ 1.6g/cm³. (a), (b) same colorbar as in (c).

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Fig. 5 Sample region A - coherent setting: differential phase contrast (a), (c) and darkfield (b), (d) of cells are shown for the mesh scans A₁ (top) and A₂ (bottom) - (cf. Table 2). The colorbar in (c) denotes the movement of the first moment whereas in (d) the grey scale denotes the ratio of scattered to unscattered photons, n_s and n₀ respectively. The intensity of the beam from the ptychographic reconstruction of the Siemens star (cf. Fig. 2) after back propagation of 1mm into its focus is visualized on top of the darkfield (d). Note, that the propagation of 1mm corresponds to the lateral displacement of the sample between the two different measurements (±100μm due to an uncertainty of the position on the sample holder with respect to the curvature of the polyimide foil). The centre of the reconstructed beam is chosen to be at the cell-position of A₄ (cf. Fig. 6). Sample region B - incoherent setting: a section of the scan B₁ is shown as darkfield (f) and differential phase contrast (e). The position of the cell diffraction of B₃ is indicated on top of the incoherent darkfield as red circle (f). Sample region B - coherent setting: darkfield (h) and phase contrast (g) of a corresponding coherent mesh scan B₂ are shown. Numerical simulations C: synthetic cell sample (top) and calculated differential phase contrast images (below) according to Eq. (1) using the back propagated probe (cf. Fig. 2) at different positions on the optical axis. The distance of propagation is displayed and the corresponding intensity of the probe is visualised as an overlay using the same colorbar as in (d).

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Fig. 6 Sample region A - coherent setting: (top) Diffraction data of A₂ and A₃ (cf. Table 2). The sets of diffraction patterns were segmented into areas with Σ₁ and without cell signal Σ₂ using the differential phase contrast image of A₂ (see inset). The difference between the two classes of diffraction sets averaged according to their set size is depicted next to the legend in the top right corner for A₂. Diffraction signals were obtained by azimuthally averaging each diffraction pattern of a scan and thereafter averaging according to their segmentation. The segmented diffraction curves of A₂ are shown and the difference signals are presented for both scans in combination with their power-law fits. Exponents of ν = −3.2 and ν = −3.7 were obtained for A₂ and A₃, respectively. (Bottom) presents the results of A₄ (cf. Fig. 5(d)). The diffraction signals obtained after azimuthally averaging are shown in the graph. The difference between the diffraction data of the cell material and of the sample holder is depicted in the lower left corner. A slightly modified power-law yields an exponent of ν = −3.6. In both graphs the points q = q_† ≡ (c/a)^1/ν are marked.

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Fig. 7 Sample region B - incoherent setting: incoherent diffraction curves were obtained from data B₁ and B₃ (cf. Table 2). The image segmentation of B₁ according to areas with cell signal Σ₁ and without cell signal Σ₂ is shown as an inset. Power-law fits yield exponents of ν = −3.5 and ν = −3.8 for B₁ and B₃, respectively. The points q = q_† ≡ (c/a)^1/ν are marked for both curves.

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Fig. 8 Photon surface densities ρ_γ of diffraction datasets A₂ (a), A₃ (b), B₁ (c) and ptychographic dataset (Fig. 4(a)) (d). The location of bacterial cells is indicated by black lines. (a)–(c) same colorbar as in (d).

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Fig. 9 (a) 2D PSD of ‘ePIE’ reconstruction of Siemens star test pattern. Note that the Siemens star structure can be clearly seen in the radial direction. (b) 2D PSD of ptychographic (‘DM’) reconstruction of D. radiodurans in Fig. 4(b). Scale bars correspond to spatial frequencies.

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Fig. 10 (a) Intensity of a model-probe exhibiting a dumbbell-shape. The calculated knife-edge scan along the horizontal direction is shown in (b). The white, dashed line in (a) indicates the region of highest focusing which is shown in (c) (solid, black line). The resulting Gaussian shape as estimated from the knife-edge scan is depicted as solid, red curve. (d) shows the intensity of the reconstructed probe at Δz = −0.686mm behind the plane of the sample. A white, dashed line indicates the part which has been used for estimating the beam width. The corresponding result of the calculated knife-edge scan is shown in (e). The FWHM values from calculated knife-edge scans are shown as function of distance to the sample plane in (f). The red and the blue curve denote horizontal and vertical beam-width, respectively. In addition, ϒ(Δz) is shown as solid, black curve which has been scaled arbitrarily for reasons of visualisation.

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Tables (2)

Table 1 Overview of PCDI data sets according to dwell time ΔT, grid spacing between adjacent rings Δ_r of a scan of one projection, fluence F of a single projection and dose estimations D of the whole reconstruction.

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Table 2 Overview of diffraction data sets according to dwell time ΔT, step size Δ_xy and number of scan points N_x × N_y within the rectangular mesh. In addition, Si filter thickness (BS denotes usage of beamstops instead of Si filter) and nominal gap of slits S0 are shown as well as the results from diffraction analysis for the exponent of the diffraction curves ν, highest scattering vector q_† and dose estimations D.

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Equations (13)

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I_{j} = {| ℱ {Ψ_{j} (\vec{x})} |}^{2} = {| ℱ {P (\vec{x} - {\vec{x}}_{j}) S (\vec{x})} |}^{2},

PRTF (| \vec{q} |) = \frac{{〈 {〈 \sqrt{I_{j}^{PCDI}} 〉}_{j} 〉}_{φ}}{{〈 {〈 \sqrt{I_{j}^{meas}} 〉}_{j} 〉}_{φ}} = \frac{{〈 {〈 | ℱ {{〈 Ψ_{j} 〉}_{it}} | 〉}_{j} 〉}_{φ}}{{〈 {〈 \sqrt{I_{j}^{meas}} 〉}_{j} 〉}_{φ}} .

𝒫 (O (\vec{x})) : = {\begin{array}{l} | O (\vec{x}) | \cdot exp (i min (Arg (O (\vec{x})), 0)) & \forall \vec{x} \in χ \\ | O (\vec{x}) | \cdot exp (i γ \cdot Arg (O (\vec{x}))) & \forall \vec{x} \notin χ \end{array},

D = \frac{μ E}{ρ_{c}} \cdot \frac{\int_{ℝ^{2}} d S \cdot χ (\vec{x}) \cdot ρ_{γ} (\vec{x})}{\int_{ℝ^{2}} d S \cdot χ (\vec{x})} .

Δ I (| \vec{q} |) = {〈 {〈 I_{j} (\vec{q}) 〉}_{ϕ} 〉}_{j \in Σ_{1}} - {〈 {〈 I_{j} (\vec{q}) 〉}_{ϕ} 〉}_{j \in Σ_{2}}

ϒ (Δ z) : = \int_{ℝ^{2}} d S {| 𝔉 {P} (\vec{x}, Δ z) |}^{4},

ρ_{c} (\vec{r}) \approx - \frac{2 u}{λ r_{0}} ϕ (\vec{r}),

q_{†} : = {(c / a)}^{1 / ν} .

w (x) = {\begin{matrix} I_{0} (β \sqrt{1 - {(2 x / L)}^{2}}) / I_{0} (β) & , - L / 2 \leq x \leq L / 2 \\ 0 & , | x | > L / 2 \end{matrix}

K (x, Δ z) = \int_{- \infty}^{x} d x^{'} \int_{- \infty}^{\infty} d y {| 𝔉 {P} (x^{'}, y, Δ z) |}^{2},

\partial_{x} K (x) = \int_{- \infty}^{\infty} {| 𝔉 {P} (x, y, Δ z) |}^{2} d y \equiv 𝔓_{α = 0} (x) .

{| 𝔉 {P} (x^{'}, y, Δ z) |}^{2} = exp (- {(x - μ)}^{2} / (2 σ^{2})) / (σ / \sqrt{2 π}) \cdot f (y)

\Rightarrow K (x, Δ z) = \frac{C_{0}}{2} (1 + erf (\frac{x - μ}{2 σ})),

reconstruction	ΔT [s]	Δ_r[μm]	F [photons/μm²]	D [Gy]
Fig. 4(a)	0.2	0.28	2.3 · 10⁹	4.9 · 10⁶
Fig. 4(b)	0.4	0.28	4.7 · 10⁹	9.9 · 10⁶
Fig. 4(c)	0.4	0.25	5.9 · 10⁹	1.3 · 10⁷
Fig. 4(d)	0.2	0.35	1.5 · 10⁹	2.4 · 10⁸

reconstruction	ΔT [s]	Δ_r[μm]	F [photons/μm²]	D [Gy]
Fig. 4(a)	0.2	0.28	2.3 · 10⁹	4.9 · 10⁶
Fig. 4(b)	0.4	0.28	4.7 · 10⁹	9.9 · 10⁶
Fig. 4(c)	0.4	0.25	5.9 · 10⁹	1.3 · 10⁷
Fig. 4(d)	0.2	0.35	1.5 · 10⁹	2.4 · 10⁸

region A	ΔT [s]	Δ_xy[μm]	N_x × N_y	Si[μm]	S0[μm]	ν	D [Gy]	q_†[ $\frac{1}{nm}$ ]

A₁	0.1	0.1	161 × 191	50	10	-	-	-
A₂	1	0.2	41 × 41	BS	10	−3.2	3.6 · 10⁷	0.75
A₃	10	0.8	11 × 11	BS	10	−3.7	2.2 · 10⁷	0.95
A₄	100	200	2 × 1	BS	10	−3.6	1.3 · 10⁸	1.88

region B

B₁	0.1	0.2	61 × 61	BS	420	−3.5	5.5 · 10⁷	2.30
B₂	0.1	0.2	61 × 61	50	10	-	-	-
B₃	100	200	2 × 1	BS	420	−3.8	2 · 10⁹	1.89