Abstract

It is generally believed that the resolution in digital holography is limited by the size of the captured holographic record. Here, we present a method to circumvent this limit by self-extrapolating experimental holograms beyond the area that is actually captured. This is done by first padding the surroundings of the hologram and then conducting an iterative reconstruction procedure. The wavefront beyond the experimentally detected area is thus retrieved and the hologram reconstruction shows enhanced resolution. To demonstrate the power of this concept, we apply it to simulated as well as experimental holograms.

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References

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  1. D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
    [CrossRef] [PubMed]
  2. D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. A 197, 454–487 (1949).
    [CrossRef]
  3. U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).
  4. T. Latychevskaia, J.-N. Longchamp, and H.-W. Fink, “When holography meets coherent diffraction imaging,” Opt. Express20(27), 28871–28892 (2012).
    [CrossRef] [PubMed]
  5. W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
    [CrossRef] [PubMed]
  6. J. Garcia-Sucerquia, W. B. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt.45(5), 836–850 (2006).
    [CrossRef] [PubMed]
  7. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for determination of phase from image and diffraction plane pictures,” Optik (Stuttg.)35, 237–246 (1972).
  8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982).
    [CrossRef] [PubMed]
  9. T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett.98(23), 233901 (2007).
    [CrossRef] [PubMed]
  10. T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
    [CrossRef]
  11. T. Latychevskaia and H.-W. Fink, “Simultaneous reconstruction of phase and amplitude contrast from a single holographic record,” Opt. Express17(13), 10697–10705 (2009).
    [CrossRef] [PubMed]

2012 (1)

2010 (1)

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
[CrossRef]

2009 (1)

2007 (1)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett.98(23), 233901 (2007).
[CrossRef] [PubMed]

2006 (1)

2001 (1)

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

1982 (1)

1972 (1)

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

1948 (1)

D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Fienup, J. R.

Fink, H.-W.

Formanek, P.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
[CrossRef]

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Garcia-Sucerquia, J.

Gerchberg, R. W.

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

Jericho, M. H.

J. Garcia-Sucerquia, W. B. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt.45(5), 836–850 (2006).
[CrossRef] [PubMed]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Jericho, S. K.

Klages, P.

Koch, C. T.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
[CrossRef]

Kreuzer, H. J.

J. Garcia-Sucerquia, W. B. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt.45(5), 836–850 (2006).
[CrossRef] [PubMed]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Latychevskaia, T.

T. Latychevskaia, J.-N. Longchamp, and H.-W. Fink, “When holography meets coherent diffraction imaging,” Opt. Express20(27), 28871–28892 (2012).
[CrossRef] [PubMed]

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
[CrossRef]

T. Latychevskaia and H.-W. Fink, “Simultaneous reconstruction of phase and amplitude contrast from a single holographic record,” Opt. Express17(13), 10697–10705 (2009).
[CrossRef] [PubMed]

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett.98(23), 233901 (2007).
[CrossRef] [PubMed]

Longchamp, J.-N.

Lubk, A.

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
[CrossRef]

Meinertzhagen, I. A.

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Saxton, W. O.

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

Xu, W. B.

J. Garcia-Sucerquia, W. B. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt.45(5), 836–850 (2006).
[CrossRef] [PubMed]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Appl. Opt. (2)

Nature (1)

D. Gabor, “A new microscopic principle,” Nature161(4098), 777–778 (1948).
[CrossRef] [PubMed]

Opt. Express (2)

Optik (Stuttg.) (1)

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

Phys. Rev. Lett. (1)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett.98(23), 233901 (2007).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, “Digital in-line holography for biological applications,” Proc. Natl. Acad. Sci. U.S.A.98(20), 11301–11305 (2001).
[CrossRef] [PubMed]

Ultramicroscopy (1)

T. Latychevskaia, P. Formanek, C. T. Koch, and A. Lubk, “Off-axis and inline electron holography: Experimental comparison,” Ultramicroscopy110(5), 472–482 (2010).
[CrossRef]

Other (2)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. A 197, 454–487 (1949).
[CrossRef]

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

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Figures (6)

Fig. 1
Fig. 1

Illustration describing the iterative self-extrapolation of a hologram of an object in the form of the word “ReSoLuTiOn”. The iterative loop includes the steps (i)-(iv) as described in the main text. The reconstructed object amplitude distribution in (ii) and (iii) are shown in inverted intensity scale.

Fig. 2
Fig. 2

Simulated example. (a) Synthetic object, consisting of two point scatterers, separated by six pixels; the central 50 × 50 pixels region is shown. (b) Simulated hologram, 2000 × 2000 pixels in size. (c) Hologram reconstruction, showing the central 50 × 50 pixels region. (d) Selected 500 × 500 pixels central region of the original hologram, as marked in (b) by a red square, and (e) its reconstruction. (f) Result of iterative reconstruction of (d) after 300 iterations. (g) The selected 500 × 500 pixels region is padded to 1000 × 1000 pixels with a constant background. (h) Self-extrapolated hologram after 300 iterations. (i) Reconstruction of the self-extrapolated hologram; the central 50 × 50 pixels region is shown. The blue curves in (c), (e), (f), and (i) show the intensity profiles of the reconstructions.

Fig. 3
Fig. 3

Profiles of the intensity in the central region of the holograms: original hologram (blue dots), 500 × 500 pixels truncated hologram after 300 iterations (red line), and 1000 × 1000 self-extrapolated hologram after 300 iterations (green line).

Fig. 4
Fig. 4

Experimental scheme for recording optical inline holograms. The screen is made up of a translucent tracing paper.

Fig. 5
Fig. 5

Experimental verification of the method. (a) Scanning electron microscope image of the sample. (b) 1000 × 1000 pixels experimental optical hologram of the sample and (c) its reconstruction; the 500 × 500 pixels central part is shown. (d) Selected 500 × 500 pixels central region of the experimental hologram, and (e) its reconstruction. (f) The 500 × 500 pixels hologram padded to 1000 × 1000 pixels. (g) The 1000 × 1000 pixels self-extrapolated hologram after 100 iterations. (h) Reconstruction of the self-extrapolated hologram; the 500 × 500 pixels central part is shown. The blue curves in (c), (e) and (h) show the intensity profiles of the reconstructions.

Fig. 6
Fig. 6

Self-extrapolation of a piece of the hologram. (a) The selected 500 × 500 pixels part of the hologram is padded up to 1000 × 1000 pixels; and (b) its reconstruction; the 500 × 500 pixels central part is shown. (c) 1000 × 1000 pixels self-extrapolated hologram after 300 iterations and (d) its reconstruction; the 500 × 500 pixels central part is shown.

Equations (4)

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R= λz N Δ S ,
t( r O )= i λ r O exp( ik r O ) U( r S ) exp(ik| r S r O |) | r S r O | d σ S ,
U( r S )= i λ t( r O ) exp( ik r O ) r O exp(ik| r S r O |) | r S r O | d σ O ,
K=( 1 1 1 1 4 1 1 1 1 )

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