Abstract

We have implemented a deterministic method for solving the phase problem in hard x-ray in-line holography which overcomes the twin image problem. The phase distribution in the detector plane is retrieved by using two images with slightly different Fresnel numbers. We then use measured intensities and reconstructed phases in the detection plane to compute the exit wave in the sample plane. No further a priori information like a limited support or the assumption of pure phase objects is necessary so that it can be used for a wide range of complex samples. Using a nano-focused hard x-ray beam half period resolutions better than 30 nm are achieved.

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2012

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012).
[CrossRef] [PubMed]

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. Express20, 19232–19254 (2012).
[CrossRef] [PubMed]

2011

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography,” Opt. Express19, 10359–10376 (2011).
[CrossRef] [PubMed]

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011).
[CrossRef] [PubMed]

2010

2009

D. G. Voelz and M. C. Roggemann, “Digital simulation of scalar optical diffraction: revisiting chirp function sampling criteria and consequences,” Appl. Opt.48, 6132–6142 (2009).
[CrossRef] [PubMed]

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

2007

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

2006

2004

2003

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun.220, 49 – 58 (2003).
[CrossRef]

2002

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002).
[CrossRef] [PubMed]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
[CrossRef] [PubMed]

2000

D. P. J.B Tiller, A Barty, and K. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Comm.183, 7–14 (2000).
[CrossRef]

1999

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

1998

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
[CrossRef]

1997

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum.68, 2774–2782 (1997).
[CrossRef]

1996

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

1993

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1–92,” Atomic Data and Nuclear Data Tables54, 181–342 (1993).
[CrossRef]

1983

1982

1973

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: Ii. sources of error,” J. Phys. D: Appl. Phys6, 2217 (1973).
[CrossRef]

1972

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

1948

D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948).
[CrossRef] [PubMed]

1841

T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).

Abela, R.

Arhatari, B. D.

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Bartels, M.

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012).
[CrossRef] [PubMed]

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. Express20, 19232–19254 (2012).
[CrossRef] [PubMed]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
[CrossRef] [PubMed]

Barty, A

D. P. J.B Tiller, A Barty, and K. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Comm.183, 7–14 (2000).
[CrossRef]

Baruchel, J.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Baumbach, T.

Beerlink, A.

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012).
[CrossRef] [PubMed]

Beetz, T.

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]

Beta, C.

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

Bourgeois, L.

A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011).
[CrossRef] [PubMed]

Bronnikov, A.

Burvall, A.

Chapman, H.

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]

Cloetens, P.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
[CrossRef] [PubMed]

Cui, C.

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]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1–92,” Atomic Data and Nuclear Data Tables54, 181–342 (1993).
[CrossRef]

Davis, T.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
[CrossRef] [PubMed]

Davis, T. J.

Dhal, B.

Diaz, A.

Dyck, D. V.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Fienup, J. R.

Fink, H.-W.

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

Fuhse, C.

T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948).
[CrossRef] [PubMed]

Gao, D.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
[CrossRef] [PubMed]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum.68, 2774–2782 (1997).
[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

Gerchberg, R. W.

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

Giewekemeyer, K.

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

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. Express20, 19232–19254 (2012).
[CrossRef] [PubMed]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

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K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
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T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt.43, 2418–2430 (2004).
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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

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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).
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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).
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M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
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Kirz, J.

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).
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M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

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T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).

Krüger, S. P.

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
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A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011).
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P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
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Mai, D. D.

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

Mancuso, A.

Marchesini, S.

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]

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D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002).
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S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
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Miao, H.

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]

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D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002).
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S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
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Nakashima, P.

A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011).
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S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
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[CrossRef] [PubMed]

Olendrowitz, C.

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012).
[CrossRef] [PubMed]

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S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

Paganin, D.

A. Liu, D. Paganin, L. Bourgeois, and P. Nakashima, “Projected thickness reconstruction from a single defocused transmission electron microscope image of an amorphous object,” Ultramicroscopy111, 959–968 (2011).
[CrossRef] [PubMed]

D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002).
[CrossRef] [PubMed]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
[CrossRef] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80, 2586–2589 (1998).
[CrossRef]

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard x-rays,” Phys. Rev. Lett.77, 2961–2964 (1996).
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D. M. Paganin, Coherent X-ray Optics (New York: Oxford University Press, 2006).
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S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
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Peele, A.

Peele, A. G.

Pogany, A.

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt.43, 2418–2430 (2004).
[CrossRef] [PubMed]

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
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[CrossRef]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384, 335–338 (1996).
[CrossRef]

Priebe, M.

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

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. Express20, 19232–19254 (2012).
[CrossRef] [PubMed]

Roggemann, M. C.

Salditt, T.

C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012).
[CrossRef] [PubMed]

S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

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. Express20, 19232–19254 (2012).
[CrossRef] [PubMed]

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
[CrossRef] [PubMed]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
[CrossRef] [PubMed]

T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).

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Sayre, D.

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]

Schlenker, M.

P. Cloetens, W. Ludwig, J. Baruchel, D. V. Dyck, J. V. Landuyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays,” Appl. Phys. Lett.75, 2912–2914 (1999).
[CrossRef]

Scholten, R.

Shapiro, D.

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]

Spence, J.

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).
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S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
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M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
[CrossRef]

S. Kalbfleisch, H. Neubauer, S. P. Krüger, M. Bartels, M. Osterhoff, D. D. Mai, K. Giewekemeyer, B. Hartmann, M. Sprung, and T. Salditt, “The Göttingen holography endstation of beamline P10 at PETRA III/DESY,” AIPConf.Proc.1365, 96–99 (2011).

T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
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S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
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[CrossRef]

Wilke, R. N.

Wilkins, S.

S. Mayo, P. Miller, S. Wilkins, T. Davis, D. Gao, T. Gureyev, D. Paganin, D. Parry, A. Pogany, and A. Stevenson, “Quantitative x-ray projection microscopy: phase-contrast and multi-spectral imaging.” J. Microsc.207, 79–96 (2002).
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D. Paganin, S. Mayo, T. Gureyev, P. Miller, and S. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206, 33–40 (2002).
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S. P. Krüger, H. Neubauer, M. Bartels, S. Kalbfleisch, K. Giewekemeyer, P. J. Wilbrandt, M. Sprung, and T. Salditt, “Sub-10 nm beam confinement by X-ray waveguides: design, fabrication and characterization of optical properties,” J. Synchrotron Radiat.19, 227–236 (2012).
[CrossRef] [PubMed]

J.Electron Spectrosc.

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).
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[CrossRef]

Opt. Comm.

D. P. J.B Tiller, A Barty, and K. Nugent, “The holographic twin image problem: a deterministic phase solution,” Opt. Comm.183, 7–14 (2000).
[CrossRef]

Opt. Commun.

T. E. Gureyev, “Composite techniques for phase retrieval in the Fresnel region,” Opt. Commun.220, 49 – 58 (2003).
[CrossRef]

Opt. Express

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, “Sub-15 nm beam confinement by two crossed x-ray waveguides,” Opt. Express18, 13492–13501 (2010).
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B. D. Arhatari and A. G. Peele, “Optimisation of phase imaging geometry,” Opt. Express18, 23727–23739 (2010).
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T. Salditt, S. Kalbfleisch, M. Osterhoff, S. P. Krüger, M. Bartels, K. Giewekemeyer, H. Neubauer, and M. Sprung, “Partially coherent nano-focused x-ray radiation characterized by Talbot interferometry,” Opt. Express19, 9656–9675 (2011).
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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. Express20, 19232–19254 (2012).
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M. Bartels, M. Priebe, R. Wilke, S. Kruger, K. Giewekemeyer, S. Kalbfleisch, C. Olendrowitz, M. Sprung, and T. Salditt, “Low-dose three-dimensional hard x-ray imaging of bacterial cells,” Opt. Nanoscopy1, 10 (2012).
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C. Olendrowitz, M. Bartels, M. Krenkel, A. Beerlink, R. Mokso, M. Sprung, and T. Salditt, “Phase-contrast x-ray imaging and tomography of the nematode Caenorhabditis elegans,” Phys. Med. Biol.57, 5309–5323 (2012).
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Phys. Rev. A

K. Giewekemeyer, S. P. Krüger, S. Kalbfleisch, M. Bartels, C. Beta, and T. Salditt, “X-ray propagation microscopy of biological cells using waveguides as a quasipoint source,” Phys. Rev. A83, 023804 (2011).
[CrossRef]

Phys. Rev. B

T. Salditt, K. Giewekemeyer, C. Fuhse, S. P. Kruger, R. Tucoulou, and P. Cloetens, “Projection phase contrast microscopy with a hard x-ray nanofocused beam: Defocus and contrast transfer,” Phys. Rev. B79, 184112(2009).

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Figures (8)

Fig. 1
Fig. 1

General scheme of the Holo-TIE phase reconstruction: The wavefront ψobj in the object plane, i.e. in the exit plane of the sample, is modified by the complex valued transmission function of the object. The intensity distribution of the field propagating along the optical axis, and measured in the two planes z and z + Δz with Δzz, enables the reconstruction of an approximated phase ϕ̃, for example in the first of the two planes located at a distance z from the object plane. Along with the measured intensities in the ‘detector plane’ the reconstructed phase ϕ̃ forms a completely specified wave ψ1 which can be propagated back to the object plane in a straightforward manner.

Fig. 2
Fig. 2

Simulation results of the Holo-TIE reconstruction: (a), (b) Two intensity distributions of an object illuminated by a plane wave recorded in neighboring planes, in the holographic regime. (c) The difference of the images used to reconstruct the (d) phase in the detector plane. (e) The phase of the classical holographic reconstruction (Eq. (2)) shows contamination by the twin image. Together with the intensity shown in (a) and the reconstructed phase shown in (d), the correct wave function can be propagated back to the object plane (Eq. (8)). (f) The resulting phase in the object plane. Scale bars denote 5 μm.

Fig. 3
Fig. 3

Experimental setup used at the beamline P10 at PETRA III: (a) Sketch of the optical path. The beam is focused with a KB mirror system to some 100 nm. A pinhole or a waveguide is positioned in the focus to low pass filter the beam. The waveguide further confines the beam to less than 60 nm. (b) The resulting far field after using a pinhole with 1.4 μm diameter. (c) A far field created by the waveguide. The dashed rectangles denote the areas used in the KB- and WG-setup, respectively. Scale bars denote 5 mm on the detector. The divergent beam created by the focusing can be used to image samples with (d) a magnification setup which can be described in the (e) parallel beam geometry after a simple variable transformation.

Fig. 4
Fig. 4

(a) The general recording scheme. (b) A typical empty beam image recorded in the ‘KB-setup’ experiment. Due to the high flux of the KB-beam, it is possible to use a high resolution detector. Compared to other available detectors, this detector based on optical microscopy of a scintillation foil requires smaller geometric magnification for desired object pixel size, and hence enables larger defocus distances z. This combination is well suited for the KB beam with relatively small divergence. (c) The corresponding raw image of a test structure. In the waveguide setup (‘WG-setup’), a more efficient detector can be used (fiber coupled sCMOS camera) with larger pixels and larger active area, well matched to the larger divergence. (d) A typical empty beam image, with comparatively less artifacts from high frequency wave front distortions, which compromise image quality. (e) A corresponding raw image of the test structure. The scale bars denote 4 μm.

Fig. 5
Fig. 5

Experimental results of the Holo-TIE reconstruction in the KB experiment: (a), (b) Two empty beam corrected intensity distributions recorded with small defocus difference Δz. (c) Difference image, computed from (a) and (b) after compensating for lateral shift and different magnifications. (d) Phase reconstruction in the detector plane, computed from (a) and (b), using a regularization parameter compensating for non-uniformity in the illumination. (e) Conventional holographic reconstruction (Eq. (2)), showing artifacts due to the twin image. (f) The regularized Holo-TIE reconstruction (Eq. (8)). The zeros in the contrast-transfer-function are partially compensated in the Holo-TIE reconstruction. The inset shows a zoom of the central region (contrast adjusted). The 200 nm lines and spaces of the test structure are resolved. The dashed line in the inset shows approximately the region imaged in the WG-setup (Fig. 6). Scale bars denote 10 μm.

Fig. 6
Fig. 6

Experimental results of the Holo-TIE reconstruction in the waveguide experiment (‘WG-setup’ setting), presented in the same sequence as in Fig. 5: (a), (b) The two empty beam corrected images, and (c) the corresponding difference image after resizing and alignment. (d) The reconstructed phase distribution in the detector plane, and (f) the Holo-TIE reconstruction (Eq. (8)), which is found to exhibit less artifacts than (e) the holographic reconstruction (Eq. (2)). Compared to the reconstruction in Fig. 5 the Holo-TIE shows much less artifacts which is a result of the smaller regularization parameter. The central region in (f) is magnified in the inset. The inner structure size is 50 nm lines and spaces, which are clearly resolved in all directions. All scale bars denote 2 μm.

Fig. 7
Fig. 7

Experimental results of the hybrid approach. (a) The Holo-TIE reconstruction (Eq. (8)) of the short exposure data set with larger FOV. The cloudy background is attributed to shot noise in the difference image. (b) The phase distribution obtained by running a modified GS algorithm for 5 iterations after priming the algorithm with the holographic reconstruction, and (c) the same for the Holo-TIE initial guess, showing a much more uniform structure. The cloudy noise of the Holo-TIE reconstruction is significantly reduced by application of only 5 iterations. (d) A profile along the line shown in (c). Fitting an error function to a step in the central region yields a FWHM of 59 nm. Line profiles as shown here were used to estimate the phase difference between bright and dark regions of the structure shown in the table. The scale bars denote 2 μm.

Fig. 8
Fig. 8

(a) Power spectral densities of the hybrid approach with the short exposure (2 × 5 seconds) data set (Fig. 7(c)) and (b) of the Holo-TIE reconstruction of the long exposure (2 × 100 seconds) data set with smaller pixel size (Fig. 6(f)). For the long exposure no iterative refinement was necessary due to the better signal-to-noise ratio. The short exposure data set shows an isotropic resolution up to 70 nm half period length. In horizontal direction structures up to 50 nm resolution are visible. For the long exposure data set the Siemens star signal is isotropically resolved up to 40 nm. In the vertical direction the PSD shows spatial frequencies beyond 30 nm resolution.

Tables (1)

Tables Icon

Table 1 Experimental details used for the Holo-TIE reconstruction and parameters used for the simulation. E is the energy corresponding to the wavelength λ, dxeff denotes the (effective) pixel size, zeff the effective propagation distance and α is the regularization parameter. The Fresnel number F is calculated for a typical structure size a of 10 Pixels which allows a better comparison of the imaging regimes than taking the whole image or a fixed size due to the different numbers and sizes of pixel.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

ψ ( r , z ) = D z ( ψ ( r , 0 ) ) = e i k z 1 [ exp ( i z ( k x 2 + k y 2 ) 2 k ) ( ψ ( r , 0 ) ) ] ,
ψ obj = D z ( I 1 ( r ) ) .
( 2 + 2 i k z ) ψ ( r ) = 0 ,
( I ( r ) ϕ ( r ) ) = k I ( r ) z ,
ϕ ( r ) = k 2 ( { 1 I ( r ) [ 2 ( I ( r ) z ) ] } ) .
2 ( ) = 1 ( 1 k x 2 + k y 2 + α ( ) ) .
ϕ ˜ ( r ) = k Δ z 2 ( { 1 I 1 ( r ) [ 2 ( I 1 ( r ) I 2 ( r ) ) ] } ) .
ψ obj = D z ( ψ 1 = I 1 exp ( i ϕ ˜ ) ) .
ψ ( x , y ) exp ( k β τ ( x , y ) ) × exp ( i k δ τ ( x , y ) ) ,
abs ( ψ ) = e γ angle ( ψ ) .
D = μ I 0 h ν ρ ,

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