Abstract

One of the main hurdles for nanometer focusing by a bending mirror lies in the theoretical surface errors by its approximations used for the traditional theory. The impacts of approximations and analytical corrections have been discussed, and the elliptically bent mirror theory has been described during exact mathematical analysis without any approximations. These approximations are harmful for the focusing system with bigger grazing angle, bigger mirror length, and bigger numerical aperture. The properties of equal-moment and single-moment bent mirrors have been described and discussed. Because of its obvious advantages, a single-moment bending mirror has high potential ability for nanometer focusing.

© 2011 Optical Society of America

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  1. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
    [CrossRef]
  2. C. G. Schroer, “Focusing hard x rays to nanometer dimensions using Fresnel zone plates,” Phys. Rev. B 74, 033405 (2006).
    [CrossRef]
  3. 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]
  4. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
    [CrossRef] [PubMed]
  5. A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
    [CrossRef] [PubMed]
  6. K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
    [CrossRef] [PubMed]
  7. L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
    [CrossRef]
  8. M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
    [CrossRef]
  9. F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analytical techniques: platform for nano materials science,” Spectrochim. Acta B 60, 13–26 (2005).
    [CrossRef]
  10. P. Kirkpatrick and A. V. Baez, “Formation of optical images by x-rays,” J. Opt. Soc. Am. 38, 766–773 (1948).
    [CrossRef] [PubMed]
  11. B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
    [CrossRef]
  12. A. C. Ugural and S. K. Fenster, Advanced Strength and Applied Elasticity (Prentice-Hall, 1995).
  13. A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer-Verlag, 2001).
  14. C. Mao and X. Yu, “A study on the grazing angle variability of bent elliptical microfocusing mirror,” AIP Conf. Proc. 879, 686–689 (2007).
    [CrossRef]
  15. S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
    [CrossRef]

2010 (2)

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

2007 (2)

C. Mao and X. Yu, “A study on the grazing angle variability of bent elliptical microfocusing mirror,” AIP Conf. Proc. 879, 686–689 (2007).
[CrossRef]

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

2006 (2)

C. G. Schroer, “Focusing hard x rays to nanometer dimensions using Fresnel zone plates,” Phys. Rev. B 74, 033405 (2006).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

2005 (2)

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analytical techniques: platform for nano materials science,” Spectrochim. Acta B 60, 13–26 (2005).
[CrossRef]

2001 (1)

A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer-Verlag, 2001).

2000 (1)

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

1998 (1)

L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (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 (2)

B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
[CrossRef]

A. C. Ugural and S. K. Fenster, Advanced Strength and Applied Elasticity (Prentice-Hall, 1995).

1948 (1)

Ablett, J. M.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Adams, F.

F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analytical techniques: platform for nano materials science,” Spectrochim. Acta B 60, 13–26 (2005).
[CrossRef]

Baez, A. V.

Barrett, R.

F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analytical techniques: platform for nano materials science,” Spectrochim. Acta B 60, 13–26 (2005).
[CrossRef]

Bozovic, N.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Cambie, D.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Celestre, R. S.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Church, M.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Conley, R.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

Duarte, R. M.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Eng, P. J.

B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
[CrossRef]

Evans-Lutterodt, K.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Fenster, S. K.

A. C. Ugural and S. K. Fenster, Advanced Strength and Applied Elasticity (Prentice-Hall, 1995).

Freund, A.

L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
[CrossRef]

Fuhse, C.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Goldberg, K. A.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Handa, S.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Hignette, O.

L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
[CrossRef]

Howells, M. R.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Hustache, R.

L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
[CrossRef]

Inagaki, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Irick, S.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Ishikawa, T.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Jarre, A.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Kang, H. C.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

Kimura, T.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Kirkpatrick, P.

Kirschman, J.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[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]

Lengeler, B.

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]

Liu, C.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

MacDowell, A. A.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Macrander, A. T.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

Mao, C.

C. Mao and X. Yu, “A study on the grazing angle variability of bent elliptical microfocusing mirror,” AIP Conf. Proc. 879, 686–689 (2007).
[CrossRef]

Maser, J.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

Matsuyama, S.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

McKinney, W. R.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Mimura, H.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Morrison, G.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Nishino, Y.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Noll, T.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Ollinger, C.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Padmore, H. A.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Pressley, A.

A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer-Verlag, 2001).

Rah, S.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Renner, T. R.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Rivers, M.

B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
[CrossRef]

Salditt, T.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Sandler, R.

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Sano, Y.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Schildkamp, W.

B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
[CrossRef]

Schroer, C. G.

C. G. Schroer, “Focusing hard x rays to nanometer dimensions using Fresnel zone plates,” Phys. Rev. B 74, 033405 (2006).
[CrossRef]

Seeger, J.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Snigirev, A.

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.

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]

Stein, A.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Stephenson, G. B.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

Tamasaku, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Taylor, A.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Tennant, D. M.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Tucoulou, R.

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
[CrossRef] [PubMed]

Ugural, A. C.

A. C. Ugural and S. K. Fenster, Advanced Strength and Applied Elasticity (Prentice-Hall, 1995).

Van Vaeck, L.

F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analytical techniques: platform for nano materials science,” Spectrochim. Acta B 60, 13–26 (2005).
[CrossRef]

Vogt, S.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

Warwick, T.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Yabashi, M.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Yamakawa, D.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Yamamura, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Yamauchi, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Yang, B. X.

B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
[CrossRef]

Yashchuk, V. V.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Yokoyama, H.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

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C. Mao and X. Yu, “A study on the grazing angle variability of bent elliptical microfocusing mirror,” AIP Conf. Proc. 879, 686–689 (2007).
[CrossRef]

Yuan, S.

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

Yumoto, H.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

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L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
[CrossRef]

Ziegler, E.

L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
[CrossRef]

AIP Conf. Proc. (1)

C. Mao and X. Yu, “A study on the grazing angle variability of bent elliptical microfocusing mirror,” AIP Conf. Proc. 879, 686–689 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Synchrotron Radiat. (1)

L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund, “Design optimization of a flexural hinge-based bender for x-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).
[CrossRef]

Nat. Phys. (1)

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[CrossRef]

Nature (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]

Opt. Eng. (1)

M. R. Howells, D. Cambie, R. M. Duarte, S. Irick, A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, and R. Sandler, “Theory and practice of elliptically bent x-ray mirrors,” Opt. Eng. 39, 2748–2762 (2000).
[CrossRef]

Phys. Rev. B (1)

C. G. Schroer, “Focusing hard x rays to nanometer dimensions using Fresnel zone plates,” Phys. Rev. B 74, 033405 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, “Nanometer linear focusing of hard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, and T. Salditt, “Two-dimensional hard x-ray beam compression by combined focusing and waveguide optics,” Phys. Rev. Lett. 94, 074801 (2005).
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K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99, 134801 (2007).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng, “GeoCARS microfocusing Kirkpatrick–Baez mirror bender development,” Rev. Sci. Instrum. 66, 2278–2280 (1995).
[CrossRef]

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F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analytical techniques: platform for nano materials science,” Spectrochim. Acta B 60, 13–26 (2005).
[CrossRef]

X-Ray Opt. Instrum. (1)

S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg, R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison, T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bent x-ray mirrors with active temperature stabilization,” X-Ray Opt. Instrum. 2010, 784732 (2010).
[CrossRef]

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A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer-Verlag, 2001).

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Figures (6)

Fig. 1
Fig. 1

Working condition and coordinate definition of the elliptical focusing mirror.

Fig. 2
Fig. 2

Schematic of a bent mirror with two different couples and definitions of coordinates, mirror parameters, etc.

Fig. 3
Fig. 3

Slope-error distributions: (a) approximation of coordinates, p = 5 m , q = 15 cm , and θ = 5 mrad , 10 mrad , 20 mrad ; (b) approximation of thickness, p = 5 m , q = 15 cm , and θ = 5 mrad , 10 mrad , and t = 1 cm , 2 cm ; (c) approximation of curvature, p = 5 m , q = 15 cm , and θ = 10 mrad .

Fig. 4
Fig. 4

Relative width distributions for different models of moment distribution with parameter configuration p = 5 m , q = 15 cm , θ = 5 mrad .

Fig. 5
Fig. 5

Width-variable mirror: (a) bent by single moment, k = 0 ; (b) bent by equal moments, k = 1 .

Fig. 6
Fig. 6

Relative width distributions of equal end-width mirrors with parameter configuration p = 5 m , q = 15 cm , θ = 5 mrad .

Equations (22)

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X 2 a 2 + Z 2 b 2 = 1.
z ( x ) = sin θ ( p + q ) 4 p q + ( p q ) 2 cos 2 θ × { 2 p q 2 [ ( p q ) 2 p q x 2 x p q ( p q ) cos θ ] 1 / 2 x cos θ ( p q ) } ,
z ( x ) = sin θ ( p + q ) 2 p q × [ 1 x 2 p q x ( p q ) cos θ p q ] 3 / 2 ,
z ( x ) = i = 2 a i x i .
a 2 = sin θ ( p + q ) 4 p q , a 3 = a 2 cos θ ( p q ) 2 p q , a 4 = a 2 [ 4 p q + 5 ( p q ) 2 cos 2 θ ] 16 ( p q ) 2 ,
E I ρ c ( s ) = M ( s ) ,
s ( x ) = 0 x [ z ( n ) 2 + 1 ] 1 / 2 d n ,
R c ( x ) = R ( x ) + t / 2 or ρ c ( s ) = ρ ( s ) + t / 2.
1 / R a ( x ) z ( x ) .
z c ( x ) = z ( x ) 1 + 1 / 2 t z ( x ) .
Δ T = z z c = t 2 z · z c d x .
1 / R = ( z 2 + 1 ) 3 / 2 z .
δ = R R a .
Δ a = z [ ( 1 + z a 2 ) 3 / 2 1 ] d x .
b ( s ) = b 0 ρ c ( s ) M ( s ) E I 0 ,
R c ( x ) = ( z 2 + 1 ) 3 / 2 z + t 2 .
M 1 = k M 2 .
M 2 = 4 a 2 E I 0 ( 1 + k ) ( 1 + a 2 t ) .
y ( s ) = K s + R c 1 ( l / 2 ) K s ( l / 2 ) ,
K = [ R c 1 ( l / 2 ) R c 1 ( l / 2 ) ] / L .
M ( s ) = 2 a 2 E I 0 R c 1 ( l / 2 ) K s ( l / 2 ) × y ( s ) .
M total = M + M g + M t + M b +

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