Abstract

We present numerical simulations optimizing the layer shapes of curved focusing x-ray multilayer mirrors deployed at synchrotron radiation facilities using a wave-optical model. The confocal elliptical shapes of the inner layers are corrected for refraction based on the modified Bragg law. Simulated wave amplitudes are further propagated to the focal region, promising nanometer focusing.

© 2013 Optical Society of America

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References

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  1. Ch. Morawe and M. Osterhoff, Nucl. Instrum. Methods A 616, 98 (2010).
    [CrossRef]
  2. H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. C. Kang, Phys. Rev. B 76, 115438 (2007).
    [CrossRef]
  3. S. P. Krüger, K. Giewekemeyer, S. Kalbfleisch, M. Bartels, H. Neubauer, and T. Salditt, Opt. Express 18, 13492 (2010).
    [CrossRef]
  4. A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
    [CrossRef]
  5. H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
    [CrossRef]
  6. F. Döring, A. L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, and H. U. Krebs, Opt. Express 21, 19311 (2013).
    [CrossRef]
  7. M. Osterhoff, Ch. Morawe, C. Ferrero, and J. P. Guigay, Opt. Lett. 37, 3705 (2012).
    [CrossRef]
  8. J. P. Guigay, Ch. Morawe, V. Mocella, and C. Ferrero, Opt. Express 16, 12050 (2008).
    [CrossRef]
  9. Ch. Morawe, J. P. Guigay, V. Mocella, and C. Ferrero, Opt. Express 16, 16138 (2008).
    [CrossRef]
  10. L. G. Parratt, Phys. Rev. 95, 359 (1954).
    [CrossRef]
  11. A. Authier, Dynamical Theory of X-ray Diffraction, IUCr Monographs (Oxford University, 2001).

2013 (1)

2012 (2)

M. Osterhoff, Ch. Morawe, C. Ferrero, and J. P. Guigay, Opt. Lett. 37, 3705 (2012).
[CrossRef]

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

2011 (1)

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

2010 (2)

2008 (2)

2007 (1)

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

1954 (1)

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

Authier, A.

A. Authier, Dynamical Theory of X-ray Diffraction, IUCr Monographs (Oxford University, 2001).

Bartels, M.

Döring, F.

Eberl, C.

Ferrero, C.

Giewekemeyer, K.

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

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

Guigay, J. P.

Hoffmann, S.

Ishikawa, T.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Kalbfleisch, S.

Kang, H. C.

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

Kimura, T.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Koumura, Y.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Krebs, H. U.

F. Döring, A. L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, and H. U. Krebs, Opt. Express 21, 19311 (2013).
[CrossRef]

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

Krüger, S. P.

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

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

Liese, T.

F. Döring, A. L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, and H. U. Krebs, Opt. Express 21, 19311 (2013).
[CrossRef]

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

Macrander, A.

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

Maser, J.

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

Matsuyama, S.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Mimura, H.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Mocella, V.

Morawe, Ch.

Neubauer, H.

Osterhoff, M.

F. Döring, A. L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, and H. U. Krebs, Opt. Express 21, 19311 (2013).
[CrossRef]

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

M. Osterhoff, Ch. Morawe, C. Ferrero, and J. P. Guigay, Opt. Lett. 37, 3705 (2012).
[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]

Radisch, V.

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

Robisch, A. L.

Ruhlandt, A.

F. Döring, A. L. Robisch, C. Eberl, M. Osterhoff, A. Ruhlandt, T. Liese, F. Schlenkrich, S. Hoffmann, M. Bartels, T. Salditt, and H. U. Krebs, Opt. Express 21, 19311 (2013).
[CrossRef]

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

Salditt, T.

Schlenkrich, F.

Shen, Q.

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

Stephenson, G. B.

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

Tamasaku, K.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Vogt, S.

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

Yabashi, M.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Yamauchi, K.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Yan, H.

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

Yokoyama, H.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Yumoto, H.

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

AIP Adv. (1)

A. Ruhlandt, T. Liese, V. Radisch, S. P. Krüger, M. Osterhoff, K. Giewekemeyer, H. U. Krebs, and T. Salditt, AIP Adv. 2, 012175 (2012).
[CrossRef]

AIP Conf. Proc. (1)

H. Mimura, T. Kimura, H. Yokoyama, H. Yumoto, S. Matsuyama, K. Tamasaku, Y. Koumura, M. Yabashi, T. Ishikawa, and K. Yamauchi, AIP Conf. Proc. 1365, 13 (2011).
[CrossRef]

Nucl. Instrum. Methods A (1)

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

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. (1)

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

Phys. Rev. B (1)

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

Other (1)

A. Authier, Dynamical Theory of X-ray Diffraction, IUCr Monographs (Oxford University, 2001).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Geometrical layout of ML mirrors focusing a source S onto a focus F. (b) Refraction of beams due to the averaged index of refraction can be accounted for by optimized layer shapes shown as black lines, while gray dashed curves correspond to confocal ellipses.

Fig. 2.
Fig. 2.

Reflectivity calculations comparing MLs of Bragg type (red dashed curves) and modified Bragg type (green lines). The related numerical parameters are given in Table 1.

Fig. 3.
Fig. 3.

Focus simulations of the ML mirrors for the Δϑ values indicated by vertical lines in Fig. 2. The blue line represents the intensity (multiplied by a factor 5) for an unoptimized mirror (note the prominent sidelobes). The focus for a rotated mirror is shown by the red curves: at the nominal Bragg peak (dashed line) and on the actual peak (solid line). The width (FWHM) of the red curves slightly increases as compared to the blue one, but the intensity gain is substantial. The modified Bragg equation has been used to produce the green curves. The intensity of the green lines is increased further with respect to the red solid curves, while the focus size is decreased down to 1.15 nm (FWHM).

Fig. 4.
Fig. 4.

Reflected intensity inside a curved ML structure for different optimization factors f, see labels in the figure. For no or feeble optimization, standing waves arise in the structure, yielding nodes of zero intensity along the surface and peak intensities far away from the surface. In case of optimal curvature (here for f=0.75, see also Fig. 3), reflectivity becomes homogeneous.

Tables (1)

Tables Icon

Table 1. Parameter Values Used in the Simulations Shown in Figs. 2 and 3a

Equations (9)

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(α2s+β2t)ψ0=i(u0ψ0+u1ψ1)ψ0/2(t+s),(α2sβ2t)ψ1=i(u0ψ1+u1¯ψ0)+ψ1/2(ts),
ΛB=λ2sinϑ(s,t)=λ2β,
ΛmB=λ2n2cos2ϑλ2β22δ.
ΛmB(1+δ/β2)×ΛB,
u0u02(α2sφβ2tφ)k.
u0=u0+2kδ=u02Ru0=u0*,
(α2s+β2t)ψ0=i(u0ψ0+u1ψ1)ψ0/2(t+s),(α2sβ2t)ψ1=i(u0ψ1+u1¯ψ0)+ψ1/2(ts).
u0u0+2δk[12α2Δtβ2(t+s)],
Λ(f)ΛB+f×(ΛmBΛB),fR,

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