Abstract

We have measured the intensity profile and transmission of x rays focused by a series of biconcave spherical unit lenses fabricated from beryllium. The use of beryllium extends the range of operation of compound refractive lenses, improving transmission, aperture size, and gain. The compound refractive lens was composed of 160 biconcave unit lenses, each with a radius of curvature of 1.9 mm. Two-dimensional focusing with a gain of 1.5 was obtained at 6.5 keV with a focal length of 93 cm. The effective aperture of the compound refractive lens was measured as 600 µm, with 9% peak transmission.

© 2002 Optical Society of America

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References

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  1. T. Tomie, “X-ray lens,” Japanese patent6-045288 (February18, 1994); U.S. patents5,594,773 (January14, 1997) and 5,684,852 (November4, 1997).
  2. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
    [CrossRef]
  3. B. X. Yang, Nucl. Instrum. Methods Phys. Res. A 328, 578 (1993).
    [CrossRef]
  4. R. K. Smither, A. M. Khounsary, and S. Xu, Proc. SPIE 3151, 150 (1997).
    [CrossRef]
  5. A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
    [CrossRef]
  6. B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
    [CrossRef]
  7. B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
    [CrossRef]
  8. J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
    [CrossRef]
  9. H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).
  10. H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
    [CrossRef]
  11. R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
    [CrossRef]
  12. M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
    [CrossRef]
  13. M. A. Piestrup, R. H. Pantell, J. T. Cremer, and H. R. Beguiristain, “Compound refractive lens for x-rays,” U.S. patent6,269,145 (July31, 2001).
  14. F. L. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 60.
  15. H. Tompkins, SSRL User Research Administration, 2575 Sand Hill Road, MS 99, Menlo Park, Calif., 94025. (personal communication, June6, 1999) and the SSRL web page, http://www-ssrl.slac.stanford.edu/beamlines/bl_characteristics.html .
  16. B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
    [CrossRef]

2001

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

2000

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

1999

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

1998

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

1997

R. K. Smither, A. M. Khounsary, and S. Xu, Proc. SPIE 3151, 150 (1997).
[CrossRef]

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

1996

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
[CrossRef]

1993

B. X. Yang, Nucl. Instrum. Methods Phys. Res. A 328, 578 (1993).
[CrossRef]

Beguiristain, H. R.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

M. A. Piestrup, R. H. Pantell, J. T. Cremer, and H. R. Beguiristain, “Compound refractive lens for x-rays,” U.S. patent6,269,145 (July31, 2001).

H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).

Cahn, R. N.

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

Cederström, B.

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

Cremer, J. T.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

M. A. Piestrup, R. H. Pantell, J. T. Cremer, and H. R. Beguiristain, “Compound refractive lens for x-rays,” U.S. patent6,269,145 (July31, 2001).

H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).

Danielsson, M.

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

Drakopolulos, M.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

Elleaume, P.

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

Feinstein, J.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

Filseth, B.

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

Gary, C. K.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).

Khounsary, A. M.

R. K. Smither, A. M. Khounsary, and S. Xu, Proc. SPIE 3151, 150 (1997).
[CrossRef]

Klocke, Th.

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

Kohn, V.

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
[CrossRef]

Lengeler, B.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
[CrossRef]

Lundqvist, M.

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

Nygren, D. R.

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

Pantell, R. H.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

M. A. Piestrup, R. H. Pantell, J. T. Cremer, and H. R. Beguiristain, “Compound refractive lens for x-rays,” U.S. patent6,269,145 (July31, 2001).

H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).

Pedrotti, F. L.

F. L. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 60.

Pedrotti, L.

F. L. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 60.

Piestrup, M. A.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).

M. A. Piestrup, R. H. Pantell, J. T. Cremer, and H. R. Beguiristain, “Compound refractive lens for x-rays,” U.S. patent6,269,145 (July31, 2001).

Raven, C.

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

Richwin, M.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

Schroer, C. G.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

Smither, R. K.

R. K. Smither, A. M. Khounsary, and S. Xu, Proc. SPIE 3151, 150 (1997).
[CrossRef]

Snigirev, A.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
[CrossRef]

Snigireva, I.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
[CrossRef]

Souvorov, A.

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

Tatchyn, R.

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

Timmeler, J.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

Tomie, T.

T. Tomie, “X-ray lens,” Japanese patent6-045288 (February18, 1994); U.S. patents5,594,773 (January14, 1997) and 5,684,852 (November4, 1997).

Tompkins, H.

H. Tompkins, SSRL User Research Administration, 2575 Sand Hill Road, MS 99, Menlo Park, Calif., 94025. (personal communication, June6, 1999) and the SSRL web page, http://www-ssrl.slac.stanford.edu/beamlines/bl_characteristics.html .

Tuemmler, J.

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

Xu, S.

R. K. Smither, A. M. Khounsary, and S. Xu, Proc. SPIE 3151, 150 (1997).
[CrossRef]

Yang, B. X.

B. X. Yang, Nucl. Instrum. Methods Phys. Res. A 328, 578 (1993).
[CrossRef]

Appl. Phys. Lett.

B. Lengeler, C. G. Schroer, M. Richwin, J. Timmeler, M. Drakopolulos, A. Snigirev, and I. Snigireva, Appl. Phys. Lett. 74, 3924 (1999).
[CrossRef]

J. Appl. Phys.

B. Lengeler, J. Tuemmler, A. Snigirev, I. Snigireva, and C. Raven, J. Appl. Phys. 84, 5855 (1998).
[CrossRef]

Nature

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, Nature 384, 49 (1996).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A

B. X. Yang, Nucl. Instrum. Methods Phys. Res. A 328, 578 (1993).
[CrossRef]

Proc. SPIE

R. K. Smither, A. M. Khounsary, and S. Xu, Proc. SPIE 3151, 150 (1997).
[CrossRef]

A. Snigirev, B. Filseth, P. Elleaume, Th. Klocke, V. Kohn, B. Lengeler, I. Snigireva, A. Souvorov, and J. Tuemmler, Proc. SPIE 3151, 164 (1997).
[CrossRef]

H. R. Beguiristain, J. T. Cremer, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. Feinstein, Proc. SPIE 4144, 155 (2000).
[CrossRef]

B. Cederström, R. N. Cahn, M. Danielsson, M. Lundqvist, and D. R. Nygren, Proc. SPIE 3659, 407 (1999).
[CrossRef]

Rev. Sci. Instrum.

R. H. Pantell, J. Feinstein, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, and J. T. Cremer, Rev. Sci. Instrum. 72, 48 (2001).
[CrossRef]

M. A. Piestrup, H. R. Beguiristain, C. K. Gary, J. T. Cremer, and R. H. Pantell, Rev. Sci. Instrum. 71, 4375 (2000).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, and R. Tatchyn, Rev. Sci. Instrum. 70, 3545 (1999).
[CrossRef]

Other

H. R. Beguiristain, M. A. Piestrup, R. H. Pantell, C. K. Gary, and J. T. Cremer, Proceedings of the 11th Synchroton Radiation Instrumentation Conference, AIP Conf. Proc.521, 258 (2000).

T. Tomie, “X-ray lens,” Japanese patent6-045288 (February18, 1994); U.S. patents5,594,773 (January14, 1997) and 5,684,852 (November4, 1997).

M. A. Piestrup, R. H. Pantell, J. T. Cremer, and H. R. Beguiristain, “Compound refractive lens for x-rays,” U.S. patent6,269,145 (July31, 2001).

F. L. Pedrotti and L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3, p. 60.

H. Tompkins, SSRL User Research Administration, 2575 Sand Hill Road, MS 99, Menlo Park, Calif., 94025. (personal communication, June6, 1999) and the SSRL web page, http://www-ssrl.slac.stanford.edu/beamlines/bl_characteristics.html .

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Figures (6)

Fig. 1
Fig. 1

Series of unit lenses that form a CRL to reduce the overall focal length.

Fig. 2
Fig. 2

CRL attenuation apertures as a function of photon energy for lenses with focal length of 1 m made from Li, Be, Kapton (polyimide film), C, and Al. The figure establishes that x-ray CRLs assembled with Be unit lenses have approximately a factor-of-2 larger apertures at the softer x-ray energy range of the figure (roughly less than 15 keV).

Fig. 3
Fig. 3

Experimental setup for testing the prototype Be CRL.

Fig. 4
Fig. 4

Measured vertical profiles of 6.5-keV x rays focusing along the propagation direction. The minimum spot 42 µ m -wide FWHM was found at 93   cm from the lens.

Fig. 5
Fig. 5

Measured vertical profile of the 6.5-keV x-ray spot, indicating the FWHM produced by the spherical Be CRL at the image plane.

Fig. 6
Fig. 6

Measured transmission through the CRL as a function of the radius of the lens. The figure shows the attenuation aperture of the lens as well as the FWHM of the lens aperture.

Equations (5)

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

f = R / 2 N δ ,
n = 1 - δ - i β .
r a = 4 δ f / µ 1 / 2 ,
r p = 4 R 2 λ r i 1 / 4 2 R 3 λ δ N 1 / 4 .
G = r o 2 4 σ h σ v + fk 2 δ μ - 1 r o k 2 σ h σ v δ μ 2 × exp - µ Nd A A s M 2 T ,

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