Abstract

We have derived a wave-optical model of curved nanofocusing x-ray multilayer mirrors used at synchrotron radiation sources, using a Takagi–Taupin-like approach. In a first approximation, the individual layers are assumed to be confocal elliptical. This assumption leads to a convenient spatial description in elliptical coordinates. As a first optimization, we study a rotation-like modification and compare numerical simulations to established results for planar multilayers.

© 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, Nature 435, 1210 (2005).
    [CrossRef]
  2. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
    [CrossRef]
  3. H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
    [CrossRef]
  4. S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, Opt. Express 18, 13492(2010).
    [CrossRef]
  5. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
    [CrossRef]
  6. Ch. Morawe and M. Osterhoff, X-Ray Opt. Instrum. 2010, 479631 (2010).
    [CrossRef]
  7. Ch. Morawe and M. Osterhoff, Nucl. Instrum. Methods A 616, 98 (2010).
    [CrossRef]
  8. S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, J. Opt. Soc. Am. A 29, 216 (2012).
    [CrossRef]
  9. C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003).
    [CrossRef]
  10. 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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
    [CrossRef]
  11. C. G. Schroer and B. Lengeler, Phys. Rev. Lett. 94, 054802 (2005).
    [CrossRef]
  12. H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
    [CrossRef]
  13. S. Takagi, Acta Cryst. 15, 1311 (1962).
    [CrossRef]
  14. D. Taupin, Acta Cryst. 23, 25 (1967).
    [CrossRef]
  15. J. Härtwig, J. Phys. D 34, 10A (2001).
    [CrossRef]
  16. A. Authier, Dynamical Theory of X-Ray Diffraction(Oxford University, 2001).
  17. L. G. Parratt, Phys. Rev. 95, 359 (1954).
    [CrossRef]

2012

2010

S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, Opt. Express 18, 13492(2010).
[CrossRef]

Ch. Morawe and M. Osterhoff, X-Ray Opt. Instrum. 2010, 479631 (2010).
[CrossRef]

Ch. Morawe and M. Osterhoff, Nucl. Instrum. Methods A 616, 98 (2010).
[CrossRef]

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

2008

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

2007

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

2006

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

2005

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

C. G. Schroer and B. Lengeler, Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

2003

C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef]

2001

J. Härtwig, J. Phys. D 34, 10A (2001).
[CrossRef]

1967

D. Taupin, Acta Cryst. 23, 25 (1967).
[CrossRef]

1962

S. Takagi, Acta Cryst. 15, 1311 (1962).
[CrossRef]

1954

L. G. Parratt, Phys. Rev. 95, 359 (1954).
[CrossRef]

Anderson, E. H.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

Aquila, A.

Attwood, D. T.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

Authier, A.

A. Authier, Dynamical Theory of X-Ray Diffraction(Oxford University, 2001).

Bajt, S.

Bartels, M.

Bergemann, C.

C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef]

Boye, P.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

Chao, W.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

Chapman, H. N.

Conley, R.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

Feldkamp, J.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

Giewekemeyer, K.

Gullikson, E.

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

Härtwig, J.

J. Härtwig, J. Phys. D 34, 10A (2001).
[CrossRef]

Holt, M. V.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Kalbfleisch, S.

Kang, H. C.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

Keymeulen, H.

C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef]

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Krüger, S. P.

Kurapova, O.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

Lengeler, B.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

C. G. Schroer and B. Lengeler, Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

Liu, C.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

Macrander, A.

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

Macrander, A. T.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

Maser, J.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Morawe, Ch.

Ch. Morawe and M. Osterhoff, X-Ray Opt. Instrum. 2010, 479631 (2010).
[CrossRef]

Ch. Morawe and M. Osterhoff, Nucl. Instrum. Methods A 616, 98 (2010).
[CrossRef]

Neubauer, H.

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Osterhoff, M.

Ch. Morawe and M. Osterhoff, X-Ray Opt. Instrum. 2010, 479631 (2010).
[CrossRef]

Ch. Morawe and M. Osterhoff, Nucl. Instrum. Methods A 616, 98 (2010).
[CrossRef]

Parratt, L. G.

L. G. Parratt, Phys. Rev. 95, 359 (1954).
[CrossRef]

Patommel, J.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

Salditt, T.

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Schroer, C. G.

C. G. Schroer and B. Lengeler, Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

Shen, Q.

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

Stephenson, G. B.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

Takagi, S.

S. Takagi, Acta Cryst. 15, 1311 (1962).
[CrossRef]

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Taupin, D.

D. Taupin, Acta Cryst. 23, 25 (1967).
[CrossRef]

van der Veen, J. F.

C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef]

Vogt, S.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

Winarski, R. P.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[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 Kazuto Yamauchi, Nat. Phys. 6, 122 (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 Kazuto Yamauchi, Nat. Phys. 6, 122 (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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Yamauchi, Kazuto

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Yan, H.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[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 Kazuto Yamauchi, Nat. Phys. 6, 122 (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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Acta Cryst.

S. Takagi, Acta Cryst. 15, 1311 (1962).
[CrossRef]

D. Taupin, Acta Cryst. 23, 25 (1967).
[CrossRef]

Appl. Phys. Lett.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, Appl. Phys. Lett. 92, 221114 (2008).
[CrossRef]

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, and B. Lengeler, Appl. Phys. Lett. 87, 124103 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

J. Härtwig, J. Phys. D 34, 10A (2001).
[CrossRef]

Nat. Phys.

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 Kazuto Yamauchi, Nat. Phys. 6, 122 (2010).
[CrossRef]

Nature

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Nature 435, 1210 (2005).
[CrossRef]

Nucl. Instrum. Methods A

Ch. Morawe and M. Osterhoff, Nucl. Instrum. Methods A 616, 98 (2010).
[CrossRef]

Opt. Express

Phys. Rev.

L. G. Parratt, Phys. Rev. 95, 359 (1954).
[CrossRef]

Phys. Rev. B

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, H. C. Kang, Phys. Rev. B 76, 115438 (2007).
[CrossRef]

Phys. Rev. Lett.

C. G. Schroer and B. Lengeler, Phys. Rev. Lett. 94, 054802 (2005).
[CrossRef]

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef]

C. Bergemann, H. Keymeulen, and J. F. van der Veen, Phys. Rev. Lett. 91, 204801 (2003).
[CrossRef]

X-Ray Opt. Instrum.

Ch. Morawe and M. Osterhoff, X-Ray Opt. Instrum. 2010, 479631 (2010).
[CrossRef]

Other

A. Authier, Dynamical Theory of X-Ray Diffraction(Oxford University, 2001).

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Figures (4)

Fig. 1.
Fig. 1.

Geometrical layout of curved multilayer mirrors. (a) Confocal ellipses meet the Bragg condition locally, if propagation effects are neglected. (b) The elliptical coordinates (s, t), defined through the distances to the source point S, r0, and to the focal point F, r1; ϑ is the local gracing angle of incidence, 2c is the distance between S and F. Lengths are not to scale.

Fig. 2.
Fig. 2.

Integration scheme used to solve Eqs. (4). Shown are the three-point stencils for the numerical integration of ψ0 and ψ1: Information at the unfilled locations is used to evaluate the field at the filled positions.

Fig. 3.
Fig. 3.

Comparative reflectivity curves using Parratt’s algorithm (red dashes) and a flat version of the presented TT-theory (green line), for parameters shown in Table 1.

Fig. 4.
Fig. 4.

Reflectivity curves using the presented TT-theory for elliptically curved focusing mirrors of indicated lengths. (a) Reflectivity curve as in Fig. 3; (b) reflectivity at the peak. Parameters are as in Table 1.

Tables (1)

Tables Icon

Table 1. Parameters for Simulations of Figs. 3 and 4

Equations (11)

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

t=(r0+r1)/2,s=(r0r1)/2,
2ψ(s,t)+k2[1+χ(s,t)]ψ(s,t)=0
χ(s,t)=χ0+χ1¯exp(2ikt)+χ1exp(2ikt).
ψ(s,t)=ψ0exp[ik(s+t)]+ψ1exp[ik(st)].
χψ(χ0ψ0+χ1ψ1)exp[ik(s+t)]+(χ0ψ1+χ1¯ψ0)exp[ik(st)].
2ψ(s,t)=α2(s2ψ)+β2(t2ψ)+t(tψ)s(sψ)t2s2;α2(s,t)=c2s2t2s2=cos2ϑ,β2(s,t)=sin2ϑ.
(α2s+β2t)ψ0=i(u0ψ0+u1ψ1)ψ0/2(t+s),(α2sβ2t)ψ1=i(u0ψ1+u1¯ψ0)+ψ1/2(ts).
χ=χ0+χ1¯exp[ik(2tφ)]+χ1exp[ik(2tφ)].
(α2sβ2t)ψ1=i(u0ψ1+u1¯ψ0)+ψ1/2(ts)u0u0(α2sφβ2tφ)k.
u0(Δϑ)u02kΔϑsinϑ.
α2sψ=f(ψ)β2tψ,

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