Abstract

We develop an analytical approach to refractive, blazed diffractive, and achromatic x-ray lenses of scalable dimensions for energies from 1 to 20 keV. Based on the parabolic wave equation, their wideband imaging properties are compared and optimized for a given spectral range. Low-Z lens materials for massive cores and rugged alternatives, such as polycarbonate or Si for flat Fresnel components, are investigated with respect to their suitability for diffraction-limited high-energy astronomy. Properly designed “hybrid” combinations can serve as an approach to x-ray telescopes with an enhanced efficiency throughout the whole considered band, nearly regardless of their inherent absorption.

© 2012 Optical Society of America

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  1. W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
    [CrossRef]
  2. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” Nature 384, 49–51 (1996).
    [CrossRef]
  3. B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
    [CrossRef]
  4. D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 1999).
  5. C. G. Schroer and B. Lengeler, “Focusing hard x-rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94, 054802 (2005).
    [CrossRef]
  6. S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
    [CrossRef]
  7. G. K. Skinner, “Diffractive-refractive optics for high energy astronomy—II. Variations on the theme,” Astron. Astrophys. 383, 352–359 (2002).
    [CrossRef]
  8. G. K. Skinner, “Design and imaging performance of achromatic diffractive-refractive x-ray and gamma-ray Fresnel lenses,” Appl. Opt. 43, 4845–4852 (2004).
    [CrossRef]
  9. C. Braig and P. Predehl, “Large-scale diffractive x-ray telescopes,” Exp. Astron. 21, 101–123 (2006).
    [CrossRef]
  10. C. Braig and P. Predehl, “Efficient Fresnel x-ray optics made simple,” Appl. Opt. 46, 2586–2599 (2007).
    [CrossRef]
  11. G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
    [CrossRef]
  12. G. K. Skinner, “Diffractive x-ray telescopes,” X-Ray Opt. Instrum. 2010, 743485 (2010).
    [CrossRef]
  13. P. Gorenstein, “Focusing x-ray optics for astronomy,” X-Ray Opt. Instrum. 2010, 109740 (2010).
    [CrossRef]
  14. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E=50–30000  eV, Z=1–92,” Atomic Data Nucl. Data Tables 54, 181–342 (1993).
    [CrossRef]
  15. B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).
    [CrossRef]
  16. C. Braig, P. Predehl, and E.-B. Kley, “Efficient extreme ultraviolet transmission gratings for plasma diagnostics,” Opt. Eng. 50, 066501 (2011).
    [CrossRef]
  17. 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]
  18. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  19. C. Braig, and P. Predehl, “Advanced Fresnel x-ray telescopes for spectroscopic imaging,” Exp. Astron. 27, 131–155(2009).
    [CrossRef]
  20. D. T. Grubb, “Radiation damage and electron microscopy of organic polymers,” J. Mater. Sci. 9, 1715–1736 (1974).
    [CrossRef]
  21. F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
    [CrossRef]
  22. B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
    [CrossRef]
  23. J. Krizmanic, G. Skinner, and N. Gehrels, “Formation flying for a Fresnel lens observatory mission,” Exp. Astron. 20, 497–503 (2005).
    [CrossRef]
  24. B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
    [CrossRef]
  25. J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
    [CrossRef]

2011 (1)

C. Braig, P. Predehl, and E.-B. Kley, “Efficient extreme ultraviolet transmission gratings for plasma diagnostics,” Opt. Eng. 50, 066501 (2011).
[CrossRef]

2010 (2)

G. K. Skinner, “Diffractive x-ray telescopes,” X-Ray Opt. Instrum. 2010, 743485 (2010).
[CrossRef]

P. Gorenstein, “Focusing x-ray optics for astronomy,” X-Ray Opt. Instrum. 2010, 109740 (2010).
[CrossRef]

2009 (3)

C. Braig, and P. Predehl, “Advanced Fresnel x-ray telescopes for spectroscopic imaging,” Exp. Astron. 27, 131–155(2009).
[CrossRef]

F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
[CrossRef]

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

2008 (1)

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

2007 (1)

2006 (1)

C. Braig and P. Predehl, “Large-scale diffractive x-ray telescopes,” Exp. Astron. 21, 101–123 (2006).
[CrossRef]

2005 (5)

J. Krizmanic, G. Skinner, and N. Gehrels, “Formation flying for a Fresnel lens observatory mission,” Exp. Astron. 20, 497–503 (2005).
[CrossRef]

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

C. G. Schroer and B. Lengeler, “Focusing hard x-rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

2004 (1)

2002 (1)

G. K. Skinner, “Diffractive-refractive optics for high energy astronomy—II. Variations on the theme,” Astron. Astrophys. 383, 352–359 (2002).
[CrossRef]

1999 (1)

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

1998 (1)

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[CrossRef]

1996 (1)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” 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 (2)

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

B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).
[CrossRef]

1974 (1)

D. T. Grubb, “Radiation damage and electron microscopy of organic polymers,” J. Mater. Sci. 9, 1715–1736 (1974).
[CrossRef]

Anderson, E. H.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Arzoumanian, Z.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Attwood, D.

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 1999).

Attwood, D. T.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Barkusky, F.

F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
[CrossRef]

Bayer, A.

F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
[CrossRef]

Benner, B.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Braig, C.

C. Braig, P. Predehl, and E.-B. Kley, “Efficient extreme ultraviolet transmission gratings for plasma diagnostics,” Opt. Eng. 50, 066501 (2011).
[CrossRef]

C. Braig, and P. Predehl, “Advanced Fresnel x-ray telescopes for spectroscopic imaging,” Exp. Astron. 27, 131–155(2009).
[CrossRef]

C. Braig and P. Predehl, “Efficient Fresnel x-ray optics made simple,” Appl. Opt. 46, 2586–2599 (2007).
[CrossRef]

C. Braig and P. Predehl, “Large-scale diffractive x-ray telescopes,” Exp. Astron. 21, 101–123 (2006).
[CrossRef]

Cash, W. C.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Chao, W.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Davis, J. C.

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

Drakopoulos, M.

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

Gehrels, N.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

J. Krizmanic, G. Skinner, and N. Gehrels, “Formation flying for a Fresnel lens observatory mission,” Exp. Astron. 20, 497–503 (2005).
[CrossRef]

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Gendreau, K.

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Gendreau, K. C.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Ghodssi, R.

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Gorenstein, P.

P. Gorenstein, “Focusing x-ray optics for astronomy,” X-Ray Opt. Instrum. 2010, 109740 (2010).
[CrossRef]

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Grubb, D. T.

D. T. Grubb, “Radiation damage and electron microscopy of organic polymers,” J. Mater. Sci. 9, 1715–1736 (1974).
[CrossRef]

Gullikson, E. M.

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

Günzler, T. F.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

Guttmann, P.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Heim, S.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

Henke, B. L.

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

Kley, E.-B.

C. Braig, P. Predehl, and E.-B. Kley, “Efficient extreme ultraviolet transmission gratings for plasma diagnostics,” Opt. Eng. 50, 066501 (2011).
[CrossRef]

Kohn, V.

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy x-rays,” 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]

Krizmanic, J.

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

J. Krizmanic, G. Skinner, and N. Gehrels, “Formation flying for a Fresnel lens observatory mission,” Exp. Astron. 20, 497–503 (2005).
[CrossRef]

Krizmanic, J. F.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Kuhlmann, M.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

Kurapova, O.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

Lengeler, B.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

C. G. Schroer and B. Lengeler, “Focusing hard x-rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[CrossRef]

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

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

Mann, K.

F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
[CrossRef]

Miller, M. C.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Morgan, B.

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Peth, C.

F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
[CrossRef]

Phillips, J. D.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[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]

Predehl, P.

C. Braig, P. Predehl, and E.-B. Kley, “Efficient extreme ultraviolet transmission gratings for plasma diagnostics,” Opt. Eng. 50, 066501 (2011).
[CrossRef]

C. Braig, and P. Predehl, “Advanced Fresnel x-ray telescopes for spectroscopic imaging,” Exp. Astron. 27, 131–155(2009).
[CrossRef]

C. Braig and P. Predehl, “Efficient Fresnel x-ray optics made simple,” Appl. Opt. 46, 2586–2599 (2007).
[CrossRef]

C. Braig and P. Predehl, “Large-scale diffractive x-ray telescopes,” Exp. Astron. 21, 101–123 (2006).
[CrossRef]

Raven, C.

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[CrossRef]

Reasenberg, R. D.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Rehbein, S.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

Reynolds, C. S.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Richwin, M.

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

Sambruna, R. M.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Schneider, G.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

Schroer, C.

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

Schroer, C. G.

C. G. Schroer and B. Lengeler, “Focusing hard x-rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

Skinner, G.

J. Krizmanic, G. Skinner, and N. Gehrels, “Formation flying for a Fresnel lens observatory mission,” Exp. Astron. 20, 497–503 (2005).
[CrossRef]

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Skinner, G. K.

G. K. Skinner, “Diffractive x-ray telescopes,” X-Ray Opt. Instrum. 2010, 743485 (2010).
[CrossRef]

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

G. K. Skinner, “Design and imaging performance of achromatic diffractive-refractive x-ray and gamma-ray Fresnel lenses,” Appl. Opt. 43, 4845–4852 (2004).
[CrossRef]

G. K. Skinner, “Diffractive-refractive optics for high energy astronomy—II. Variations on the theme,” Astron. Astrophys. 383, 352–359 (2002).
[CrossRef]

Snigirev, A.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[CrossRef]

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

Snigireva, I.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[CrossRef]

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

Streitmatter, R.

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

Streitmatter, R. E.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Tümmler, J.

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[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]

Werner, S.

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

Windt, D. L.

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yang, B. X.

B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).
[CrossRef]

Zontone, F.

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (1)

G. K. Skinner, “Diffractive-refractive optics for high energy astronomy—II. Variations on the theme,” Astron. Astrophys. 383, 352–359 (2002).
[CrossRef]

Atomic Data Nucl. Data Tables (1)

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

Exp. Astron. (4)

C. Braig and P. Predehl, “Large-scale diffractive x-ray telescopes,” Exp. Astron. 21, 101–123 (2006).
[CrossRef]

J. Krizmanic, B. Morgan, R. Streitmatter, N. Gehrels, K. Gendreau, Z. Arzoumanian, R. Ghodssi, and G. Skinner, “Development of ground-testable phase Fresnel lenses in silicon,” Exp. Astron. 20, 299–306 (2005).
[CrossRef]

J. Krizmanic, G. Skinner, and N. Gehrels, “Formation flying for a Fresnel lens observatory mission,” Exp. Astron. 20, 497–503 (2005).
[CrossRef]

C. Braig, and P. Predehl, “Advanced Fresnel x-ray telescopes for spectroscopic imaging,” Exp. Astron. 27, 131–155(2009).
[CrossRef]

J. Appl. Phys. (2)

F. Barkusky, A. Bayer, C. Peth, and K. Mann, “Direct photoetching of polymers using radiation of high energy density from a table-top extreme ultraviolet plasma source,” J. Appl. Phys. 105, 014906 (2009).
[CrossRef]

B. Lengeler, J. Tümmler, A. Snigirev, I. Snigireva, and C. Raven, “Transmission and gain of singly and doubly focusing refractive x-ray lenses,” J. Appl. Phys. 84, 5855–5861 (1998).
[CrossRef]

J. Mater. Sci. (1)

D. T. Grubb, “Radiation damage and electron microscopy of organic polymers,” J. Mater. Sci. 9, 1715–1736 (1974).
[CrossRef]

J. Phys. D (1)

B. Lengeler, C. G. Schroer, M. Kuhlmann, B. Benner, T. F. Günzler, O. Kurapova, F. Zontone, A. Snigirev, and I. Snigireva, “Refractive x-ray lenses,” J. Phys. D 38, A218–A222(2005).
[CrossRef]

J. Synchrotron Rad. (1)

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Rad. 6, 1153–1167 (1999).
[CrossRef]

Nature (2)

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft x-ray microscopy at a resolution better than 15 nm,” Nature 435, 1210–1213 (2005).
[CrossRef]

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

Nucl. Instrum. Methods Phys. Res. A (1)

B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).
[CrossRef]

Opt. Commun. (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]

Opt. Eng. (1)

C. Braig, P. Predehl, and E.-B. Kley, “Efficient extreme ultraviolet transmission gratings for plasma diagnostics,” Opt. Eng. 50, 066501 (2011).
[CrossRef]

Phys. Rev. Lett. (2)

C. G. Schroer and B. Lengeler, “Focusing hard x-rays to nanometer dimensions by adiabatically focusing lenses,” Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

S. Rehbein, S. Heim, P. Guttmann, S. Werner, and G. Schneider, “Ultrahigh-resolution soft x-ray microscopy with zone plates in high orders of diffraction,” Phys. Rev. Lett. 103, 110801 (2009).
[CrossRef]

Proc. SPIE (1)

G. K. Skinner, Z. Arzoumanian, W. C. Cash, N. Gehrels, K. C. Gendreau, P. Gorenstein, J. F. Krizmanic, M. C. Miller, J. D. Phillips, R. D. Reasenberg, C. S. Reynolds, R. M. Sambruna, R. E. Streitmatter, and D. L. Windt, “The Milli-Arc-Second Structure Imager, MASSIM: a new concept for a high angular resolution x-ray telescope,” Proc. SPIE 7011, 70110T (2008).
[CrossRef]

X-Ray Opt. Instrum. (2)

G. K. Skinner, “Diffractive x-ray telescopes,” X-Ray Opt. Instrum. 2010, 743485 (2010).
[CrossRef]

P. Gorenstein, “Focusing x-ray optics for astronomy,” X-Ray Opt. Instrum. 2010, 109740 (2010).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications (Cambridge University, 1999).

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Figures (27)

Fig. 1.
Fig. 1.

Plots of the real (black) and imaginary (red) contributions δ ( E ) and β ( E ) to the complex refractive index n = 1 δ i β for the low-Z elements with 3 Z 8 and energies 1 keV E 20 keV . The evaluated dataset for the density [ g cm 3 ] in the upper left corner is based on [14].

Fig. 2.
Fig. 2.

Critical zone number N 0 for atomic order numbers of low-Z elements with 1 Z 20 and photon energies up to 20 keV. Solid lines describe the observed proportionality in Z 3 . The evaluated dataset is based on [14].

Fig. 3.
Fig. 3.

Accuracy of the N 0 E 2 approximation within 1.0 keV E 10 keV . The thick solid line describes the mean percentaged error with respect to the zero level (perfect fit) for 3 Z 8 and its standard deviation (error bars).

Fig. 4.
Fig. 4.

Comparison of refractive (black/gray curves) and blazed diffractive (red/blue sawtooth lines) x-ray lenses. The radius and thickness are normalized by means of the geometrical lens radius σ and the radius of curvature ξ F δ .

Fig. 5.
Fig. 5.

Normalized focal spot size (HEW) of refractive lenses for s ratios 0.5 N / N 0 100 . Toward N / N 0 0 , the resolution of an absorption-free lens is approached. The solid line represents a semianalytical fit to calculated values (dots).

Fig. 6.
Fig. 6.

Spatial resolution of optimized refractive x-ray lenses made of Li, Be, B, and C in the intermediate Gaussian-diffractive regime as identified in Fig. 5. The dashed lines follow approximately the power law ( ρ HEW / R ) opt E c 2 .

Fig. 7.
Fig. 7.

Luminous power of elementary x-ray lenses. The refractive range is shown for the low- and high-absorption limit (hatched lines). Examples from Table 2 for Li and Be are indicated for various energies. For comparison, the performance of lossless blazed Fresnel lenses is also sketched.

Fig. 8.
Fig. 8.

Third-order aberrations of forward-oriented parabolic refractive profiles for δ = 10 4 . Spherical contributions are sketched in red. All point spread functions are shown for 10 ϕ 20 and κ 1 in steps of Δ κ = 0.1 .

Fig. 9.
Fig. 9.

Upper limits to the focal length of refractive lenses made of Li (dashed) and Be (solid) for E 6 keV due to spherical aberration as a function of the relative radius σ / ξ .

Fig. 10.
Fig. 10.

Successive reduction of a massive parabolic profile with N = 12 Fresnel zones to partially ( 1 < m < N / 2 ) and fully ( m = 1 ) reduced Fresnel lenses. The notations “Fresnel ring” and “groove” are considered as synonyms in this work.

Fig. 11.
Fig. 11.

Axial intensity of an absorbing GFL with N = 10 3 and N 0 = 10 for decreasing values of m N / 4 . The red solid lines represent the corresponding massive analog ( m = N / 2 ). If normalized to the GFL, the gray dashed curves form the envelopes to the interference peaks.

Fig. 12.
Fig. 12.

Linear normalized angular resolution Δ ϵ of Fresnel lenses for an increasing spectral bandwidth Δ E E of the incident radiation in units of N 1 2.5 and various groove numbers g . Absorption is neglected for the sake of clarity ( N 0 ).

Fig. 13.
Fig. 13.

Discrete spectrum of a GFL with m = 4 . The black bars are scaled to the spectral bandwidth. Absorption is represented by the gray-scaled background.

Fig. 14.
Fig. 14.

Luminous power of a Fresnel lens with R = 2 m and m = 4 , made of C 16 H 14 O 3 , Si and Ti in multiband usage (black dots). Dashed curves are fitted as guides to the eye. A lossless Fresnel lens marks the upper bound (red).

Fig. 15.
Fig. 15.

Peak intensity of a Fresnel lens with E c = 10 keV and m = 4 , made of C 16 H 14 O 3 , Si and Ti in multiband usage (black dots). Dashed curves are fitted as guides to the eye. A lossless Fresnel lens marks the upper bound (red).

Fig. 16.
Fig. 16.

Achromatic x-ray lens, made of a diffractive-refractive doublet with focal lengths F Z and F L . The detector in the focal plane contains typically up to 10 3 resolution elements in diameter, where the spot size is given by PSF . With slight modifications, the figure is adopted from [19].

Fig. 17.
Fig. 17.

Axial intensity as a function of the energy ψ for zone numbers 10 2 N 10 4 without absorption. The red solid line describes the parabolic envelope ψ 2 .

Fig. 18.
Fig. 18.

Energy-dependent response of hybrid lenses in the vicinity of the focus, neglecting absorption. Within the 80% Strehl ratio, the “classical” (a) diffraction-limited bandwidth can be enlarged by about 40% (b). The dots and diamonds measure this bandwidth for detector positions (a) and (b), respectively.

Fig. 19.
Fig. 19.

Normalized achromatic response under absorption for ratios 0 N / N 0 10 in the “enhanced-bandwidth mode” (b) from Fig. 18. The gray bar indicates the diffraction-limited band ψ + ψ from Eq. (60).

Fig. 20.
Fig. 20.

Usability of low-Z hybrid lenses. The black solid and dashed lines refer to the Q ( N ) criterion and the aspect ratio A , respectively. The red lines describe the f ratio with N / N 0 = 2.51 for the achromatic gain (see text).

Fig. 21.
Fig. 21.

Achromatic gain of low-Z elements as a function of the photon energy and the zone number N . The solid line represents the optimum according to N 2.51 N 0 .

Fig. 22.
Fig. 22.

Achromatic gain as a function of N for different values of N 0 . The PSF-corrected definition according to Eq. (63) is represented by the dashed “diamond” curves, and the standard gain without correction is drawn in black solid lines. Its optimization N = 2.51 N 0 is sketched in red.

Fig. 23.
Fig. 23.

Hybrid lenses and their performance with the same PSF , designed using the achromatic gain and the constant spot size criterion with s 2.5 and s = 5.0 , respectively. The aperture diameters are less than the maximum size that is obtained for s 6.25 (not shown).

Fig. 24.
Fig. 24.

Aberration-minimized “sandwich” hybrid lens with radius R whose refractive component ( L ) is split in two shells, embedding the Fresnel device. Their individual focal lengths are related to the diffractive one ( Z ) according to F L = 4 F Z . With slight modifications, the figure is adopted from [10].

Fig. 25.
Fig. 25.

Monomer of polycarbonate ( C 16 H 14 O 3 ) with two included benzene rings (red).

Fig. 26.
Fig. 26.

Minimized angular resolution of an achromatic sample lens made of PC for the Fe K α line with N 0 = 51 and other critical zone numbers 40 N 0 60 . The dashed red line and the center-of-mass symbols indicate the optimal zone ratio s = 5.58 . See text for details.

Fig. 27.
Fig. 27.

Fabrication errors of elementary refractive (left) and diffractive components (right). The rough thickness profile and the slightly irregular groove widths are indicated by the red solid lines, respectively (exaggerated drawing). Note that the refractive lens is limited by R 0 .

Tables (9)

Tables Icon

Table 1. Test of the Energy Dependence δ ( E ) = α E 2

Tables Icon

Table 2. Optimized Magnification for Li and Be

Tables Icon

Table 3. Transmission of Blazed Fresnel Lenses

Tables Icon

Table 4. Angular Resolution of Fresnel Lenses

Tables Icon

Table 5. Numerical Bandwidth Precision Test

Tables Icon

Table 6. Extended Bandwidth Precision Test

Tables Icon

Table 7. Focal Spot Size of Compact Hybrid Lenses

Tables Icon

Table 8. On-Axis Performance of Polycarbonate Sample Lenses

Tables Icon

Table 1 Global Symbols

Equations (101)

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

n ( E ) = 1 r e 2 π ( h c E ) 2 n a ( f 1 0 ( E ) + i f 2 0 ( E ) )
δ Z ( E ) = W ( E ) f 1 0 ( E ) , β Z ( E ) = W ( E ) f 2 0 ( E ) ,
δ ( E ) / δ ( E c ) = ( λ / λ c ) 2 = ( E c / E ) 2
N 0 δ 2 π β = Δ t abs Δ t π with N 0 > 0 ,
sin ( Δ ϑ + Δ θ ) = ( 1 δ ( E ) ) 1 sin ( Δ ϑ ) ,
E ( r⃗ ) u ( r⃗ ) e i k z + u * ( r⃗ ) e + i k z .
( Δ x , y 2 i k z ) u ( r⃗ ) = 0 with Δ x , y x 2 + y 2 .
u ( r⃗ ) = R 2 u ˜ 0 ( σ⃗ ) G ( r⃗ σ⃗ ) d 2 σ ,
t sph ( σ ) = ξ ξ 2 σ 2 = σ 2 2 ξ + σ 4 8 ξ 3 + O ( σ 6 ) ,
t ( σ ) = σ 2 2 ξ + ( 1 + b ) σ 4 8 ξ 3 + O ( σ 6 )
( 1 δ ) t + σ 2 + ( F t ) 2 = F = const ,
t par ( σ ) 1 2 σ 2 ξ + δ 4 σ 4 ξ 3 + O ( σ 6 ) ,
T par ( σ ) = e 4 π λ β t par ( σ ) = e 1 N 0 N ( σ ) ,
u ( r⃗ ) = C ( r⃗ ) 0 R u ˜ 0 ( σ ) e i k 2 z σ 2 J 0 ( k z ρ σ ) σ d σ ,
u ˜ 0 ( σ ) = e i k 2 q λ σ 2 with 1 q λ 1 F ( 1 + i 2 π N 0 )
u ( r⃗ ) = C ( r⃗ ) 0 R e i k 2 ( 1 q λ + 1 z ) σ 2 J 0 ( k z ρ σ ) σ d σ ,
I s ( υ ) = | 2 0 1 e s 2 τ 2 J 0 ( 2 π υ τ ) τ d τ | 2 .
2 υ HEW | s 1 = π 1 ( s ln 2 ) 1 2 s .
N ( E ) = N ( E c ) E 1 E c = N ( E c ) ( 1 + ε ) 1 .
ρ HEW R ς 2 f ( N ( E ) N 0 ( E c ) ( 1 + ε ) 2 ) 1 N ( E ) ,
3 s f ( s ) = f ( s ) s opt 5.38 .
I ( ψ , ζ ) = 1 2 e N 2 N 0 1 ψ cos ( G N ) + e N N 0 1 ψ ( N π ψ ) 2 [ ( ψ ζ ψ ) 2 + ( 1 2 π N 0 ζ ψ ) 2 ] ,
lim N N 0 Δ z 0 = F 2 π N 0 and lim N N 0 Δ z 0 = F N .
F ( E c ± Δ E 2 ) E c 2 ± E c Δ E + ( Δ E 2 ) 2 .
lim N N 0 Δ E E = 1 4 π N 0 and lim N N 0 Δ E E = 1 2 N .
1 4 h c F A eff × Δ E π 2 h c F
Δ ϵ⃗ = n = 1 3 M n σ 4 n ϕ n 1 e⃗ θ with e⃗ θ = ( cos θ sin θ )
M 1 = B [ 1 0 0 1 ] with B ( δ 2 ) δ 2 2 ξ 3 ,
M 2 = F [ 3 cos θ sin θ 2 sin θ 0 ] with F ( 1 δ ) δ 2 ξ 2 .
M 3 = 2 C [ 1 0 0 0 ] + D [ 1 0 0 1 ] with C δ 2 ξ
Δ ϵ x δ 2 κ 3 cos θ δ 2 ϕ κ 2 ( 1 + 2 cos 2 θ ) + 2 δ ϕ 2 κ cos θ , Δ ϵ y δ 2 κ 3 sin θ δ ϕ κ 2 sin θ cos θ + δ ϕ 2 κ sin θ .
Δ ϵ⃗ θ = 1 2 π 0 2 π | Δ ϵ⃗ ( θ ) | 2 d θ δ 2 κ 3 W κ ( δ , ϕ ) ,
F 0 ( 1 + δ κ 2 ) F 0 + ( 2 N ) 1 F 0 ,
F 0 1 2 λ δ 3 κ 4 .
T min = e 2 m N 0 and T tot = N 0 2 m ( 1 e 2 m N 0 ) ,
u ( r⃗ ) = p = 1 N 2 m F p ( m ) ( r⃗ ) e i k m N q λ R 2 ( p 1 ) ,
F p ( m ) ( r⃗ ) = C ( r⃗ ) σ ˜ p 1 σ ˜ p u ˜ 0 ( σ ) e i k 2 z σ 2 J 0 ( k z ρ σ ) σ d σ
I m ( ψ , ζ ) S m ( ψ , ζ ) = 1 2 e m N 0 1 ψ cos ( G m ) + e 2 m N 0 1 ψ ( N π ψ ) 2 [ ( ψ ζ ψ ) 2 + ( 1 2 π N 0 ζ ψ ) 2 ] ,
S m ( ψ , ζ ) = [ sin ( π N 2 ψ ζ ) / sin ( π m ψ ζ ) ] 2 ,
lim N 0 I m ( υ , ψ ) = | 2 ψ p = 1 N 2 m H p ( m ) ( υ , ψ ) e 2 π i m 1 ψ p | 2 ,
H p ( m ) ( υ , ψ ) = τ ˜ p 1 τ ˜ p e i π N ( ψ 1 ψ ) τ 2 J 0 ( 2 π ψ υ τ ) τ d τ .
Δ E = k 2 N 1 m E c ,
F m ( ψ ) = N 0 m ψ ( 1 e m N 0 1 ψ ) ,
A eff × Δ E = π h c N 0 2 k 4 ( 1 e k N 0 ) 2 F ( m E c ) .
I m ( ψ k , ζ k ) = [ N 0 ( m k E c ) ( 1 e k N 0 ( m k E c ) ) ] 2 .
N 0 ( m k E c ) N 0 ( m k E c ) = F ( m k E c ) 1 k N 0 ( m k E c ) k m E c 1 ,
t p ( L ) ( σ ) Δ t 2 π = N 4 [ 1 ( σ / R ) 2 ] ,
t p ( Z ) ( σ ) Δ t 2 π = 1 p + N 2 ( σ / R ) 2 .
t p ( A ) ( σ ) Δ t 2 π = 1 p + N 4 ( 1 + ( σ / R ) 2 ) ,
u ( r⃗ ) = p = 1 N / 2 F p ( 1 ) ( r⃗ ) e i k N q λ R 2 ( p 1 ) ,
F p ( 1 ) ( r⃗ ) = C ( r⃗ ) σ ˜ p 1 σ ˜ p u ˜ 0 ( σ ) e i k 2 z σ 2 J 0 ( k z ρ σ ) σ d σ
I ( ψ , ζ ) = G N 0 ( ψ , ζ ) × H N 0 ( ψ , ζ ) ,
G N 0 ( ψ , ζ ) = 1 2 e N 4 N 0 1 ψ cos ( π N 2 F + ( ψ , ζ ) ) + e N 2 N 0 1 ψ ( N π ψ ) 2 [ ( ψ ζ ψ ) 2 + ( 1 2 π N 0 ζ ψ ) 2 ] , H N 0 ( ψ , ζ ) = 1 2 e 1 2 N 0 1 ψ cos ( π F ( ψ , ζ ) ) + e 1 N 0 1 ψ 1 2 e + 1 2 N 0 1 ψ cos ( π F + ( ψ , ζ ) ) + e + 1 N 0 1 ψ ,
lim N 0 I ( ψ ) = ( sin ( π 2 G ( ψ ) ) N π 2 ( 1 ψ 2 ) sin ( N 2 π 2 G + ( ψ ) ) sin ( π 2 G + ( ψ ) ) ) 2 ,
4 × lim N 0 I ( ψ ) = 1 + 2 ( ψ 1 ) + O ( ψ 1 ) 2
F + ( ψ , ζ ) const. ζ max ( ψ ) = ( 2 ψ 1 ψ 2 ) 1 .
ψ + ψ = 2 2 / N = 2 Δ ψ 0 ,
| u ˜ 0 ( L ) ( σ ) | = exp [ ( s / 4 ) ψ 1 ( 1 ( σ / R ) 2 ) ] ,
u ^ ( ν⃗ ) L | u ˜ 0 ( L ) ( σ ) | e i ν⃗ σ⃗ d 2 σ .
u ^ s ( υ ) 2 π = 0 1 exp [ s 4 ( 1 τ 2 ) ] J 0 ( 2 π υ τ ) τ d τ .
A eff × Δ E G N ( E ) × F
G N ( E ) = 4 N 0 ( E ) N ( 1 e N 2 N 0 ( E ) )
Δ ϵ = ς R 1 λ Q ( s ) PSF = 2 ς N 1 R Q ( s ) ,
A eff × Δ E = π R 2 T ( s ) Δ E ( PSF ) 2 Q 2 ( s ) ,
G ( N , N 0 ) 2 N T ( s ) Q 2 ( s ) < G N ( E ) ,
Δ ϵ⃗ θ a = 1 8 f 1 ϕ 2 0.35 f 1 ϕ 2 ,
Δ ϵ⃗ θ z = 1 16 f 3 W Z ( f , ϕ ) 1 4 5 f 1 ϕ 2 ,
Δ t 2 π = 3.07 × 10 5 m for ρ = 1.2 g cm 3 ,
Δ t abs = 1 2 Δ t 2 π N 0 = 7.62 × 10 4 m ,
F = 5.08 × 10 6 m and Δ ϵ = 0.07 mas .
A eff × Δ E = 1635 cm 2 keV with Δ E = 1.137 keV .
Δ ϵ Z = ς λ c R = ς ( λ c N F c ) 1 / 2 = V λ × 1 N ,
Δ ϵ L ς λ c R 0 = ς ( λ c N 0 F c ) 1 / 2 = V λ × 1 N 0 ,
Δ ϵ A = ς λ c R Q ( s ) = [ ] = V λ × 1 N Q ( N / N 0 ) .
δ x h c 4 δ ( E c ) E 1 for E c = 1 keV .
δ σ ^ N / 2 R / 4 N = δ σ ^ p for 1 p N / 2 ,
u ( r⃗ ) = C ( r⃗ ) 0 R e i k 2 ( 1 q λ + 1 z ) σ 2 J 0 ( k z ρ σ ) σ d σ .
u ( ρ ) = C ( ρ ) 0 R e σ 2 2 λ F N 0 J 0 ( 2 π N R 2 ρ σ ) σ d σ ,
u ( υ ) = C ( υ ) 0 1 e s 2 τ 2 J 0 ( 2 π υ τ ) τ d τ ,
I s ( υ ) = | u ( υ ) N π | 2 = | 2 0 1 e s 2 τ 2 J 0 ( 2 π υ τ ) τ d τ | 2 .
u ( z ) = i 2 π λ z 0 R exp [ i π λ P N 0 ( z ) σ 2 ] σ d σ ,
u ( ψ , ζ ) = 2 π i ψ ζ N R 2 0 R exp [ i Q N 0 ( ψ , ζ ) σ 2 ] σ d σ ,
u ( ψ , ζ ) = 2 π i ψ ζ N 0 1 exp [ i Q N 0 ( ψ , ζ ) τ 2 ] τ d τ ,
u ( ψ , ζ ) = 2 π i ψ ζ N exp [ i Q N 0 ( ψ , ζ ) ] 1 2 i Q N 0 ( ψ , ζ ) .
I ( ψ , ζ ) = 1 2 e N 2 N 0 1 ψ cos ( G N ) + e N N 0 1 ψ ( N π ψ ) 2 [ ( ψ ζ ψ ) 2 + ( 1 2 π N 0 ζ ψ ) 2 ] .
u ( z ) = p = 1 N 2 m F p ( m ) ( z ) e i k m N q λ R 2 ( p 1 ) ,
F p ( m ) ( ψ , ζ ) = 2 π i ψ ζ N τ ˜ p 1 τ ˜ p exp [ i Q N 0 ( ψ , ζ ) τ 2 ] τ d τ .
F p ( m ) ( ψ , ζ ) K p ( m ) ( ψ , ζ ) = 2 π i ψ ζ m sin ( m N Q N 0 ( ψ , ζ ) ) m N Q N 0 ( ψ , ζ ) ,
u ( ψ , ζ ) = p = 1 N 2 m F p ( m ) ( ψ , ζ ) exp [ ( p 1 ) R N 0 ( m ) ( ψ ) ] ,
I m ( ψ , ζ ) S m ( ψ , ζ ) = 1 2 e m N 0 1 ψ cos ( G m ) + e 2 m N 0 1 ψ ( N π ψ ) 2 [ ( ψ ζ ψ ) 2 + ( 1 2 π N 0 ζ ψ ) 2 ] ,
u ( ρ ) = p = 1 N 2 m F p ( m ) ( ρ ) e i k m N q λ R 2 ( p 1 ) .
F p ( m ) ( ρ ) N π C ( ρ ) = σ ˜ p 1 σ ˜ p e i k 2 ( 1 q λ + 1 F c ) σ 2 J 0 ( k F c ρ σ ) σ d σ ,
F p ( m ) ( ρ ) N π R 2 C ( ρ ) = τ ˜ p 1 τ ˜ p e i π N J ( λ ) τ 2 J 0 ( 2 π λ F c ρ σ ) τ d τ ,
H p ( m ) ( υ , ψ ) = τ ˜ p 1 τ ˜ p e i π N ( ψ 1 ψ ) τ 2 J 0 ( 2 π ψ υ τ ) τ d τ .
lim N 0 I m ( υ , ψ ) = | 2 ψ p = 1 N 2 m H p ( m ) ( υ , ψ ) e 2 π i m 1 ψ p | 2 .
u ( z ) = p = 1 N / 2 F p ( 1 ) ( z ) e i k N q λ R 2 ( p 1 ) .
F p ( 1 ) ( z ) = i k z σ ˜ p 1 σ ˜ p u ˜ 0 ( σ ) e i k 2 z σ 2 σ d σ ,
u ˜ 0 ( σ ) = exp [ 2 π i 1 ψ ( 1 + i 2 π N 0 ) t p ( A ) ( σ ) Δ t 2 π ] ,
F p ( 1 ) ( ψ , ζ ) = i N π ψ ζ τ ˜ p 1 τ ˜ p u ˜ 0 ( τ ) e i N 2 π ψ ζ τ 2 τ d τ .
u ( ψ , ζ ) = p = 1 N / 2 F p ( 1 ) ( ψ , ζ ) e 2 π i 1 ψ ( 1 + i 2 π N 0 ) ( p 1 ) .
I ( ψ , ζ ) = G N 0 ( ψ , ζ ) × H N 0 ( ψ , ζ )

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