Abstract

We present an experimental and theoretical evaluation of an x-ray energy filter based on the chromatic properties of a prism-array lens (PAL). It is intended for small-scale applications such as medical imaging. The PAL approximates a Fresnel lens and allows for high efficiency compared to filters based on ordinary refractive lenses, however at the cost of a lower energy resolution. Geometrical optics was found to provide a good approximation for the performance of a flawless lens, but a field-propagation model was used for quantitative predictions. The model predicted a 0.29 ΔE/E energy resolution and an intensity gain of 6.5 for a silicon PAL at 23.5 keV. Measurements with an x-ray tube showed good agreement with the model in energy resolution and peak energy, but a blurred focal line contributed to a 29% gain reduction. We believe the blurring to be caused mainly by lens imperfections, in particular at the periphery of the lens.

© 2009 Optical Society of America

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  1. P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
    [CrossRef]
  2. R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).
  3. F. Sugiro, D. Li, and C. MacDonald, "Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging," Med. Phys. 31, 3288-3297 (2004).
    [CrossRef]
  4. J. Motz and M. Danos, "Image information content and patient exposure," Med. Phys. 5, 8-22 (1978).
    [CrossRef] [PubMed]
  5. G. Pfahler, "A roentgen filter and a universal diaphragm and protecting screen," Trans. Am. Roentgen Ray Soc., pp. 217-224 (1906).
  6. E. Fredenberg, B. Cederström, M. Åslund, C. Ribbing, and M. Danielsson, "A Tunable Energy Filter for Medical X-Ray Imaging," X-Ray Optics and Instrumentation 2008, Article ID 635024, 8 pages (2008), http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/635024.
  7. W. Jark, "A simple monochromator based on an alligator lens," X-Ray Spectrom. 33, 455-461 (2004).
    [CrossRef]
  8. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
    [CrossRef]
  9. B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).
  10. B. Cederström, C. Ribbing, and M. Lundqvist, "Generalized prism-array lenses for hard x-rays," J. Synchrotron Rad. 12, 340-344 (2005).
    [CrossRef]
  11. W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
    [CrossRef]
  12. L. D. Caro and W. Jark, "Diffraction theory applied to X-ray imaging with clessidra prism array lenses," J. Synchrotron Rad. 15, 176-184 (2008).
    [CrossRef]
  13. C. Fuhse and T. Salditt, "Finite-difference field calculations for two-dimensionally confined x-ray waveguides," Appl. Opt. 45, 4603-4608 (2006).
    [CrossRef] [PubMed]
  14. Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to X-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
    [CrossRef]
  15. V. Kohn, I. Snigireva, and A. Snigirev, "Diffraction theory of imaging with X-ray compound refractive lens," Opt. Commun. 216, 247-260 (2003).
    [CrossRef]
  16. D. Attwood, Soft x-rays and extreme ultraviolet radiation (Cambridge University Press, 1999), Ch. 9.
  17. B. Cederström, A multi-prism lens for hard x-rays, Ph.D. thesis (Royal Institute of Technology (KTH), Stockholm, 2002), Ch. 5.
  18. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005), Ch. 3.
  19. S. Panknin, A. K. Hartmann, and T. Salditt, "X-ray propagation in tapered waveguides: Simulation and optimization," Opt. Commun. 281, 2779-2783 (2008).
    [CrossRef]
  20. D. R. Lynch, Numerical Partial Differential Equations for Environmental Scientists and Engineers (Springer, 2005), Ch. 5.
  21. M. Berger, J. Hubbell, S. Seltzer, J.S., Coursey, and D. Zucker, XCOM: Photon Cross Section Database, (National Institute of Standards and Technology, Gaithersburg, MD, 2005), http://physics.nist.gov/xcom.
  22. B. Henke, E. Gullikson, and J. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," Atomic Data and Nuclear Data Tables 54, 181-342 (1993).
    [CrossRef]
  23. F. Laermer, A. Schilp, K. Funk, and M. Offenberg, "Bosch deep silicon etching: improving uniformity and etch rate for advanced MEMS applications," in Technical Digest MEMS’99, 211-216 (IEEE Robotics and Automation Society, 1999).

2008 (2)

L. D. Caro and W. Jark, "Diffraction theory applied to X-ray imaging with clessidra prism array lenses," J. Synchrotron Rad. 15, 176-184 (2008).
[CrossRef]

S. Panknin, A. K. Hartmann, and T. Salditt, "X-ray propagation in tapered waveguides: Simulation and optimization," Opt. Commun. 281, 2779-2783 (2008).
[CrossRef]

2006 (1)

2005 (3)

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).

B. Cederström, C. Ribbing, and M. Lundqvist, "Generalized prism-array lenses for hard x-rays," J. Synchrotron Rad. 12, 340-344 (2005).
[CrossRef]

2004 (3)

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

W. Jark, "A simple monochromator based on an alligator lens," X-Ray Spectrom. 33, 455-461 (2004).
[CrossRef]

F. Sugiro, D. Li, and C. MacDonald, "Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging," Med. Phys. 31, 3288-3297 (2004).
[CrossRef]

2003 (1)

V. Kohn, I. Snigireva, and A. Snigirev, "Diffraction theory of imaging with X-ray compound refractive lens," Opt. Commun. 216, 247-260 (2003).
[CrossRef]

2000 (1)

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

1996 (1)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
[CrossRef]

1995 (1)

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to X-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
[CrossRef]

1993 (1)

B. Henke, E. Gullikson, and J. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," Atomic Data and Nuclear Data Tables 54, 181-342 (1993).
[CrossRef]

1978 (1)

J. Motz and M. Danos, "Image information content and patient exposure," Med. Phys. 5, 8-22 (1978).
[CrossRef] [PubMed]

Arkadiev, V.

R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).

Baldelli, P.

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

Bohic, S.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Cahn, R.

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

Caro, L. D.

L. D. Caro and W. Jark, "Diffraction theory applied to X-ray imaging with clessidra prism array lenses," J. Synchrotron Rad. 15, 176-184 (2008).
[CrossRef]

Cederström, B.

B. Cederström, C. Ribbing, and M. Lundqvist, "Generalized prism-array lenses for hard x-rays," J. Synchrotron Rad. 12, 340-344 (2005).
[CrossRef]

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

Danielsson, M.

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

Danos, M.

J. Motz and M. Danos, "Image information content and patient exposure," Med. Phys. 5, 8-22 (1978).
[CrossRef] [PubMed]

Davis, J.

B. Henke, E. Gullikson, and J. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," Atomic Data and Nuclear Data Tables 54, 181-342 (1993).
[CrossRef]

Diekmann, F.

R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).

Fuhse, C.

Gambaccini, M.

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

Gilardoni, M.

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

Gullikson, E.

B. Henke, E. Gullikson, and J. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," Atomic Data and Nuclear Data Tables 54, 181-342 (1993).
[CrossRef]

Hartmann, A. K.

S. Panknin, A. K. Hartmann, and T. Salditt, "X-ray propagation in tapered waveguides: Simulation and optimization," Opt. Commun. 281, 2779-2783 (2008).
[CrossRef]

Henke, B.

B. Henke, E. Gullikson, and J. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," Atomic Data and Nuclear Data Tables 54, 181-342 (1993).
[CrossRef]

Jark, W.

L. D. Caro and W. Jark, "Diffraction theory applied to X-ray imaging with clessidra prism array lenses," J. Synchrotron Rad. 15, 176-184 (2008).
[CrossRef]

W. Jark, "A simple monochromator based on an alligator lens," X-Ray Spectrom. 33, 455-461 (2004).
[CrossRef]

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Kohn, V.

V. Kohn, I. Snigireva, and A. Snigirev, "Diffraction theory of imaging with X-ray compound refractive lens," Opt. Commun. 216, 247-260 (2003).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
[CrossRef]

Kopylov, Y. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to X-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
[CrossRef]

Krumrey, M.

R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).

Lawaczeck, R.

R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).

Lengeler, B.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
[CrossRef]

Li, D.

F. Sugiro, D. Li, and C. MacDonald, "Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging," Med. Phys. 31, 3288-3297 (2004).
[CrossRef]

Lundqvist, M.

B. Cederström, C. Ribbing, and M. Lundqvist, "Generalized prism-array lenses for hard x-rays," J. Synchrotron Rad. 12, 340-344 (2005).
[CrossRef]

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

MacDonald, C.

F. Sugiro, D. Li, and C. MacDonald, "Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging," Med. Phys. 31, 3288-3297 (2004).
[CrossRef]

Mancini, L.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Matteucci, M.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Montanari, L.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Motz, J.

J. Motz and M. Danos, "Image information content and patient exposure," Med. Phys. 5, 8-22 (1978).
[CrossRef] [PubMed]

Nygren, D.

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

Panknin, S.

S. Panknin, A. K. Hartmann, and T. Salditt, "X-ray propagation in tapered waveguides: Simulation and optimization," Opt. Commun. 281, 2779-2783 (2008).
[CrossRef]

Pérennès, F.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Popov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to X-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
[CrossRef]

Ribbing, C.

B. Cederström, C. Ribbing, and M. Lundqvist, "Generalized prism-array lenses for hard x-rays," J. Synchrotron Rad. 12, 340-344 (2005).
[CrossRef]

Rigon, L.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Salditt, T.

S. Panknin, A. K. Hartmann, and T. Salditt, "X-ray propagation in tapered waveguides: Simulation and optimization," Opt. Commun. 281, 2779-2783 (2008).
[CrossRef]

C. Fuhse and T. Salditt, "Finite-difference field calculations for two-dimensionally confined x-ray waveguides," Appl. Opt. 45, 4603-4608 (2006).
[CrossRef] [PubMed]

Snigirev, A.

V. Kohn, I. Snigireva, and A. Snigirev, "Diffraction theory of imaging with X-ray compound refractive lens," Opt. Commun. 216, 247-260 (2003).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
[CrossRef]

Snigireva, I.

V. Kohn, I. Snigireva, and A. Snigirev, "Diffraction theory of imaging with X-ray compound refractive lens," Opt. Commun. 216, 247-260 (2003).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
[CrossRef]

Somogyi, A.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Sugiro, F.

F. Sugiro, D. Li, and C. MacDonald, "Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging," Med. Phys. 31, 3288-3297 (2004).
[CrossRef]

Taibi, A.

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

Tromba, G.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Tucoulou, R.

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

Tuffanelli, A.

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

Vinogradov, A. V.

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to X-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
[CrossRef]

Appl. Opt. (1)

Atomic Data and Nuclear Data Tables (1)

B. Henke, E. Gullikson, and J. Davis, "X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50-30000 eV, Z=1-92," Atomic Data and Nuclear Data Tables 54, 181-342 (1993).
[CrossRef]

Invest. Radiol. (1)

R. Lawaczeck, V. Arkadiev, F. Diekmann, and M. Krumrey, "Monochromatic x-rays in digital mammography," Invest. Radiol. 40, 33-39 (2005).

J. Synchrotron Rad. (3)

B. Cederström, C. Ribbing, and M. Lundqvist, "Generalized prism-array lenses for hard x-rays," J. Synchrotron Rad. 12, 340-344 (2005).
[CrossRef]

W. Jark, F. Pérennès, M. Matteucci, L. Mancini, L. Montanari, L. Rigon, G. Tromba, A. Somogyi, R. Tucoulou, and S. Bohic, "Focusing X-rays with simple arrays of prism-like structures," J. Synchrotron Rad. 11, 248-253 (2004).
[CrossRef]

L. D. Caro and W. Jark, "Diffraction theory applied to X-ray imaging with clessidra prism array lenses," J. Synchrotron Rad. 15, 176-184 (2008).
[CrossRef]

Med. Phys. (2)

F. Sugiro, D. Li, and C. MacDonald, "Beam collimation with polycapillary x-ray optics for high contrast high resolution monochromatic imaging," Med. Phys. 31, 3288-3297 (2004).
[CrossRef]

J. Motz and M. Danos, "Image information content and patient exposure," Med. Phys. 5, 8-22 (1978).
[CrossRef] [PubMed]

Nature (2)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, "A compound refractive lens for focusing high-energy Xrays," Nature 384, 49-51 (1996).
[CrossRef]

B. Cederström, R. Cahn, M. Danielsson, M. Lundqvist, and D. Nygren, "Focusing hard X-rays with old LP’s," Nature 404, 951 (2000).

Opt. Commun. (3)

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, "Application of the parabolic wave equation to X-ray diffraction optics," Opt. Commun. 118, 619-636 (1995).
[CrossRef]

V. Kohn, I. Snigireva, and A. Snigirev, "Diffraction theory of imaging with X-ray compound refractive lens," Opt. Commun. 216, 247-260 (2003).
[CrossRef]

S. Panknin, A. K. Hartmann, and T. Salditt, "X-ray propagation in tapered waveguides: Simulation and optimization," Opt. Commun. 281, 2779-2783 (2008).
[CrossRef]

Phys. Med. Biol. (1)

P. Baldelli, A. Taibi, A. Tuffanelli, M. Gilardoni, and M. Gambaccini, "A prototype of a quasi-monochromatic system for mammography applications," Phys. Med. Biol. 50, 225-240 (2005).
[CrossRef]

X-Ray Spectrom. (1)

W. Jark, "A simple monochromator based on an alligator lens," X-Ray Spectrom. 33, 455-461 (2004).
[CrossRef]

Other (8)

G. Pfahler, "A roentgen filter and a universal diaphragm and protecting screen," Trans. Am. Roentgen Ray Soc., pp. 217-224 (1906).

E. Fredenberg, B. Cederström, M. Åslund, C. Ribbing, and M. Danielsson, "A Tunable Energy Filter for Medical X-Ray Imaging," X-Ray Optics and Instrumentation 2008, Article ID 635024, 8 pages (2008), http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/635024.

D. Attwood, Soft x-rays and extreme ultraviolet radiation (Cambridge University Press, 1999), Ch. 9.

B. Cederström, A multi-prism lens for hard x-rays, Ph.D. thesis (Royal Institute of Technology (KTH), Stockholm, 2002), Ch. 5.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005), Ch. 3.

D. R. Lynch, Numerical Partial Differential Equations for Environmental Scientists and Engineers (Springer, 2005), Ch. 5.

M. Berger, J. Hubbell, S. Seltzer, J.S., Coursey, and D. Zucker, XCOM: Photon Cross Section Database, (National Institute of Standards and Technology, Gaithersburg, MD, 2005), http://physics.nist.gov/xcom.

F. Laermer, A. Schilp, K. Funk, and M. Offenberg, "Bosch deep silicon etching: improving uniformity and etch rate for advanced MEMS applications," in Technical Digest MEMS’99, 211-216 (IEEE Robotics and Automation Society, 1999).

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Figures (5)

Fig. 1.
Fig. 1.

The transition from MPL to PAL. (Left) The MPL approximates a parabolic profile by straight line segments. (Right) In a PAL, each prism in the MPL is exchanged for a column of smaller prisms, resulting in a shorter lens with lower absorption. The focal lengths of the two lenses are the same if the prism angle (θ) and successive displacement (d) are equal, and the prism base (b) of the PAL corresponds to an x-ray phase shift of 2π.

Fig. 2.
Fig. 2.

Energy filtering with the PAL. Vertical and horizontal axes are not to scale.

Fig. 3.
Fig. 3.

(a) SEM image of the PAL used for the experiment. The optical axis is indicated by a dashed line. (b) A schematic of the lens with indicated parameters. The spacing between columns is not illustrated. (c) Schematic of the experimental bremsstrahlung setup. The two 50 µm collimator slits in the z-direction are not shown, but only the 200 µm slit in y.

Fig. 4.
Fig. 4.

The synchrotron setup. (a) The measured intensity distribution in the focal plane is shown on top, with the focal line at the depth corresponding to the maximum intensity below, measurements and predictions by the field-propagation (FP) model. (b) The focal line width (di) and the peak intensity gain over 14 µm (G◇) as a function of an increasing collimator slit in the y-direction.

Fig. 5.
Fig. 5.

The bremsstrahlung (BS) setup. (a) The measured PAL filtered and unfiltered spectra. (b) The measured intensity gain (G ), which is the fraction of the curves in (a), compared to the field-propagation (FP) and geometrical (GM) models.

Tables (1)

Tables Icon

Table 1. Summary of measurements and predictions by the field-propagation (FP) and geometrical (GM) models. LSF indicates that the line spread function of the CCD camera is included. The image distance (s i), image size (d i), and peak gain (G ) are tabulated for both the synchrotron (SR) and bremsstrahlung (BS) setups. E corresponds to the design energy in the SR setup and to the peak energy in the BS setup. ΔE/E is the energy resolution in the BS setup.

Equations (12)

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Fdiff=dh λ .
G(E)=so+si*dsso Dp ηi (y1,E)ηi(y1,E)dy1,
ηi=dsfi(y2,y2,E)dy2=dsfi*[y2+y1k(E)]dy2,
G*=4×0.76× sod0 1h δ*µ* ,
ΔEE*=2.6G* ,
G*GMPL=0.86δμ1λγandΔEE*=1.5 [ΔEE]MPL ,
2ikn0ux+2uy2+k2(n2n02)u=0 ,
U=U0exp[i2kn0υy2x]U0H,
u=12 [uqp+1+uqp1] ,
ux=12Δx[uqp+1uqp1] + O (Δx2) , and
2uy2=12Δy2[uq1p2uqp+uq+1p+uq+1p+12uqp+1+uq+1p+1] + O (Δy2) ,
uqp=exp (ik(nqpn0)xq) q {0,Q+1} ,

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