Abstract

An efficient technique to achieve isotropic edge enhancement in optics involves applying a radial Hilbert transform on the object spectrum. Here we demonstrate a simple setup for isotropic edge-enhancement in soft-x-ray microscopy, using a single diffractive Laguerre-Gaussian zone plate (LGZP) for radial Hilbert transform. Since the LGZP acts as a beam-splitter, diffraction efficiency problems usually associated with x-ray microscopy optics are not present in this system. As numerically demonstrated, the setup can detect optical path differences as small as λ/50 with high contrast.

© 2009 Optical Society of America

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  1. A. Sakdinawat and Y. Liu, "Soft-x-ray microscopy using spiral zone plates," Opt. Lett. 32, 2635-2637 (2007).
    [CrossRef] [PubMed]
  2. G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
    [CrossRef]
  3. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
    [CrossRef]
  4. B. Kaulich, T. Wilhein, E. Di Fabrizio, F. Romanato, M. Altissimo, S. Cabrini, B. Fayard, and J. Susini, "Differential interference contrast x-ray microscopy with twin zone plates," J. Opt. Soc. Am. A 19, 797-806 (2002).
    [CrossRef]
  5. J. A. Davis, D. E. McNamara, D. M. Cottrell, and J. Campos, "Image processing with the radial Hilbert transform: theory and experiments," Opt. Lett. 25, 99-101 (2000).
    [CrossRef]
  6. S. Fürhaupter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral phase contrast imaging in microscopy," Opt. Express 13, 689-694 (2005).
    [CrossRef]
  7. I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
    [CrossRef] [PubMed]
  8. W. Leitenberger and A. Snigirev, "Microscopic imaging with high energy x-rays by Fourier transform holography," J. Appl. Phys. 90, 538-544 (2001).
    [CrossRef]
  9. C-S. Guo, Y-J. Han, J-B. Xu, and J. Ding, "Radial Hilbert transform with Laguerre-Gaussian spatial filters," Opt. Lett. 31, 1394-1396 (2006).
    [CrossRef] [PubMed]
  10. A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
    [CrossRef]
  11. http://www.ntt-at.com/products_e/x-ray_FZP/index.html
  12. N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, "Investigation of polarization effects for high-numerical-aperture first-order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead," Opt. Express 13, 10440-10447 (2005).
    [CrossRef] [PubMed]
  13. K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
    [CrossRef]
  14. http://www.princetoninstruments.com/products/xraycam/

2007 (2)

A. Sakdinawat and Y. Liu, "Soft-x-ray microscopy using spiral zone plates," Opt. Lett. 32, 2635-2637 (2007).
[CrossRef] [PubMed]

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

2006 (1)

2005 (2)

2003 (1)

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

2002 (2)

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

B. Kaulich, T. Wilhein, E. Di Fabrizio, F. Romanato, M. Altissimo, S. Cabrini, B. Fayard, and J. Susini, "Differential interference contrast x-ray microscopy with twin zone plates," J. Opt. Soc. Am. A 19, 797-806 (2002).
[CrossRef]

2001 (1)

W. Leitenberger and A. Snigirev, "Microscopic imaging with high energy x-rays by Fourier transform holography," J. Appl. Phys. 90, 538-544 (2001).
[CrossRef]

2000 (1)

1995 (1)

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

1992 (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Altissimo, M.

Anderson, E. H.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Bernet, S.

Bogart, G.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Bokor, N.

Cabrini, S.

Campos, J.

Choi, B. J.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Cottrell, D. M.

David, C.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

Davis, J. A.

Di Fabrizio, E.

Ding, J.

Domondon, A.

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

Evans-Lutterodt, K.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Fayard, B.

Fujii, M.

Fürhaupter, S.

Guo, C-S.

Guttmann, P.

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Han, Y-J.

Howells, M. R.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Iketaki, Y.

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, "Investigation of polarization effects for high-numerical-aperture first-order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead," Opt. Express 13, 10440-10447 (2005).
[CrossRef] [PubMed]

Jacobsen, C.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Jesacher, A.

Kaulich, B.

Kern, D. P.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Kirz, J.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Klemens, F.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Leitenberger, W.

W. Leitenberger and A. Snigirev, "Microscopic imaging with high energy x-rays by Fourier transform holography," J. Appl. Phys. 90, 538-544 (2001).
[CrossRef]

Liu, Y.

McNamara, D. E.

McNulty, I.

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Nagai, K.

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

Niemann, B.

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Nöhammer, B.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

Ocola, L. E.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Ritsch-Marte, M.

Romanato, F.

Rudolph, D.

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Sakdinawat, A.

Sato, Y.

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

Schmal, G.

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Schneider, G.

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Snigirev, A.

W. Leitenberger and A. Snigirev, "Microscopic imaging with high energy x-rays by Fourier transform holography," J. Appl. Phys. 90, 538-544 (2001).
[CrossRef]

Solak, H. H.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

Sreenivasan, S. V.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Stein, A.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Susini, J.

Tennant, D. M.

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Thieme, J.

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Watanabe, T.

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, "Investigation of polarization effects for high-numerical-aperture first-order Laguerre-Gaussian beams by 2D scanning with a single fluorescent microbead," Opt. Express 13, 10440-10447 (2005).
[CrossRef] [PubMed]

Wilhein, T.

Xu, J-B.

Ziegler, E.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287-3289 (2002).
[CrossRef]

IEEJ Trans. EIS (1)

K. Nagai, Y. Iketaki, A. Domondon, Y. Sato, and T. Watanabe, "A graphite-based multiplayer mirror for microscopy in the XANES region of biomolecules," IEEJ Trans. EIS 127, 1304-1307 (2007).
[CrossRef]

J. Appl. Phys. (1)

W. Leitenberger and A. Snigirev, "Microscopic imaging with high energy x-rays by Fourier transform holography," J. Appl. Phys. 90, 538-544 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Vac. Sci. Technol. B (1)

A. Stein, C. Jacobsen, K. Evans-Lutterodt, D. M. Tennant, G. Bogart, F. Klemens, L. E. Ocola, B. J. Choi, and S. V. Sreenivasan, "Diffractive x-ray optics using production fabrication methods," J. Vac. Sci. Technol. B 21, 214-219 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Rev. Sci. Instrum. (1)

G. Schmal, D. Rudolph, P. Guttmann, G. Schneider, J. Thieme, and B. Niemann, "Phase contrast studies of biological specimens with the x-ray microscope at BESSY," Rev. Sci. Instrum. 66, 1282-1286 (1995).
[CrossRef]

Science (1)

I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P. Kern, "High-resolution imaging by Fourier transform x-ray holography," Science 256, 1009-1012 (1992).
[CrossRef] [PubMed]

Other (2)

http://www.ntt-at.com/products_e/x-ray_FZP/index.html

http://www.princetoninstruments.com/products/xraycam/

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic view of a binary Laguerre-Gaussian zone plate (LGZP). (b) The proposed edge-enhancement Fourier hologram x-ray microscopy setup: the object (O, grey shaded area) is placed in a pinhole at the back focal plane of the LGZP illuminated by a collimated x-ray beam; a CCD is placed at a distance z from the object, where z is large enough to satisfy the condition for Fraunhofer diffraction. The CCD image is inverse Fourier transformed to obtain the edge-enhanced image of O.

Fig. 2.
Fig. 2.

(a) Amplitude object (axes are labeled in μm); calculated reconstructed image using (b) the Laguerre-Gaussian Hilbert filter setup of Fig. 1(b), and (c) a conventional Fourier transform x-ray microscope setup; (d), (e) and (f) show intensity cross-sections of (a), (b) and (c), respectively, along the vertical direction (denoted by the broken vertical lines) at the center of the field.

Fig. 3.
Fig. 3.

(a) Phase object with two areas having a difference Φ in phase shift (axes are labeled in μm); calculated edge-enhanced reconstructed images using the setup of Fig. 1(b), for (b) Φ = 2π/2, (c) Φ = 2π/10, and (d) Φ = 2π/50. Figures 3(e), (f) and (g) show intensity cross-sections of (b), (c) and (d), respectively, along the vertical direction denoted by the broken vertical lines.

Equations (3)

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

S r φ = circ ( r R ) exp ( j π λf r 2 )
LG r φ = r w exp [ ( r w ) 2 ] circ ( r R ) exp ( ) ,
T r φ S r φ + LG r φ 2 .

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