Abstract

The effect of layer thickness disorder in periodic multilayers on x-ray reflectivity is investigated numerically and experimentally. We present ensemble calculations, taking into account absorption and interfacial roughness. It is demonstrated that layer thickness disorder yields band broadening and increased integrated reflectivity. For applications we concentrate on extrema of the ensembles, giving the highest integrated reflectivity. We develop global optimization methods that can also be used to generate specified reflection band structures. In a few examples, applications of the optimization methods are discussed. To illustrate the practical applicability of the methods, we compare experimental realizations to the calculation. In one case we achieve a 42% increase in integrated reflectivity in the 130 Å < λ < 190 Å spectral range with respect to a periodic multilayer with its first-order Bragg peak in the center of that range. Accurate control of layer thicknesses is our main experimental obstacle.

© 1996 Optical Society of America

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References

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  1. J. H. Underwood, T. W. Barbee, “Layered synthetic microstructures as Bragg diffractors for x rays and extreme ultraviolet: theory and predicted performance,” Appl. Opt. 20, 3027–3034 (1981).
    [Crossref] [PubMed]
  2. E. Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1994).
    [Crossref]
  3. J. Schelten, K. Mika, “Calculated reflectivities of supermirrors,” Nucl. Instrum. Methods 160, 287–294 (1979).
    [Crossref]
  4. E. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” Opt. Eng. 25, 954–963 (1986).
  5. P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
    [Crossref]
  6. B. A. van Tiggelen, “Multiple scattering and localization of light,” PhD. dissertation, (University of Amsterdam, Amsterdam, The Netherlands, 1992).
  7. K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
    [Crossref]
  8. B.A. van Tiggelen, A. Tip, “Photon localization in disorder-induced periodic multilayers,” J. Phys. France 1, 1145–1154 (1991).
    [Crossref]
  9. S. P. Vernon, D. G. Stearns, R. S. Rosen, “Chirped multilayer coatings for increased x-ray throughput,” Opt. Lett. 18, 672–674 (1993).
    [Crossref] [PubMed]
  10. K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
    [Crossref]
  11. B. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E =50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–3424 (1993).
    [Crossref]
  12. L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95, 359–368 (1954).
    [Crossref]
  13. M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).
  14. R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
    [Crossref]
  15. E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
    [Crossref]
  16. M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

1994 (1)

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

1993 (2)

B. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E =50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–3424 (1993).
[Crossref]

S. P. Vernon, D. G. Stearns, R. S. Rosen, “Chirped multilayer coatings for increased x-ray throughput,” Opt. Lett. 18, 672–674 (1993).
[Crossref] [PubMed]

1992 (1)

E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[Crossref]

1991 (1)

B.A. van Tiggelen, A. Tip, “Photon localization in disorder-induced periodic multilayers,” J. Phys. France 1, 1145–1154 (1991).
[Crossref]

1989 (1)

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[Crossref]

1986 (2)

E. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” Opt. Eng. 25, 954–963 (1986).

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[Crossref]

1981 (1)

1979 (1)

J. Schelten, K. Mika, “Calculated reflectivities of supermirrors,” Nucl. Instrum. Methods 160, 287–294 (1979).
[Crossref]

1954 (1)

L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95, 359–368 (1954).
[Crossref]

Alfano, R. R.

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[Crossref]

Barbee, T. W.

Bartels, W. J.

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

Batson, P. C.

E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[Crossref]

Bruijn, M. P.

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).

Chakraborty, P.

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

Christensen, F. E.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Davis, J. C.

B. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E =50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–3424 (1993).
[Crossref]

Freund, A. K.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Gorenstein, P.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Gullikson, E. M.

B. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E =50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–3424 (1993).
[Crossref]

E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[Crossref]

Henke, B.

B. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E =50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–3424 (1993).
[Crossref]

Høghøj, P.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Joensen, K. D.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Lu, C.

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

Mika, K.

J. Schelten, K. Mika, “Calculated reflectivities of supermirrors,” Nucl. Instrum. Methods 160, 287–294 (1979).
[Crossref]

Nitkin, V.

E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[Crossref]

Papanicolaou, G.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[Crossref]

Parratt, L. G.

L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95, 359–368 (1954).
[Crossref]

Puik, E. J.

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

Rosen, R. S.

Rosenbluth, A. E.

E. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” Opt. Eng. 25, 954–963 (1986).

Schelten, J.

J. Schelten, K. Mika, “Calculated reflectivities of supermirrors,” Nucl. Instrum. Methods 160, 287–294 (1979).
[Crossref]

Schlatmann, R.

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

Sheng, P.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[Crossref]

Spiller, E.

E. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” Opt. Eng. 25, 954–963 (1986).

E. Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1994).
[Crossref]

Stearns, D. G.

Susini, J.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Tip, A.

B.A. van Tiggelen, A. Tip, “Photon localization in disorder-induced periodic multilayers,” J. Phys. France 1, 1145–1154 (1991).
[Crossref]

Underwood, J. H.

E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[Crossref]

J. H. Underwood, T. W. Barbee, “Layered synthetic microstructures as Bragg diffractors for x rays and extreme ultraviolet: theory and predicted performance,” Appl. Opt. 20, 3027–3034 (1981).
[Crossref] [PubMed]

van der Wiel, M. J.

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

van Essen, H. W.

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).

van Tiggelen, B. A.

B. A. van Tiggelen, “Multiple scattering and localization of light,” PhD. dissertation, (University of Amsterdam, Amsterdam, The Netherlands, 1992).

van Tiggelen, B.A.

B.A. van Tiggelen, A. Tip, “Photon localization in disorder-induced periodic multilayers,” J. Phys. France 1, 1145–1154 (1991).
[Crossref]

Verhoeven, J.

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).

Vernon, S. P.

White, B.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[Crossref]

Wood, J. L.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Yoo, K. M.

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[Crossref]

Zhang, Z. Q.

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[Crossref]

Ziegler, E.

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

Appl. Opt. (1)

Appl. Surf. Sci. (1)

R. Schlatmann, C. Lu, J. Verhoeven, E. J. Puik, M. J. van der Wiel, “Modification by Ar and Kr ion bombardment of Mo-Si x-ray multilayers,” Appl. Surf. Sci. 78, 147–157 (1994).
[Crossref]

At. Data Nucl. Data Tables (1)

B. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E =50–30000 eV, Z = 1–92,” At. Data Nucl. Data Tables 54, 181–3424 (1993).
[Crossref]

J. Phys. France (1)

B.A. van Tiggelen, A. Tip, “Photon localization in disorder-induced periodic multilayers,” J. Phys. France 1, 1145–1154 (1991).
[Crossref]

J. X-Ray Sci. Technol. (1)

E. M. Gullikson, J. H. Underwood, P. C. Batson, V. Nitkin, “A soft x-ray/EUV reflectometer based on a laser produced plasma source,” J. X-Ray Sci. Technol. 3, 283–299 (1992).
[Crossref]

Nucl. Instrum. Methods (1)

J. Schelten, K. Mika, “Calculated reflectivities of supermirrors,” Nucl. Instrum. Methods 160, 287–294 (1979).
[Crossref]

Opt. Eng. (1)

E. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” Opt. Eng. 25, 954–963 (1986).

Opt. Lett. (1)

Phys. Rev. (1)

L. G. Parratt, “Surface studies of solids by total reflection of x-rays,” Phys. Rev. 95, 359–368 (1954).
[Crossref]

Phys. Rev. B (2)

K. M. Yoo, R. R. Alfano, “Photon localization in a disordered multilayered system,” Phys. Rev. B 39, 5806–5809 (1989).
[Crossref]

P. Sheng, B. White, Z. Q. Zhang, G. Papanicolaou, “Minimum wave-localization in a one-dimensional random medium,” Phys. Rev. B 34, 4757–4761 (1986).
[Crossref]

Other (5)

B. A. van Tiggelen, “Multiple scattering and localization of light,” PhD. dissertation, (University of Amsterdam, Amsterdam, The Netherlands, 1992).

K. D. Joensen, P. Høghøj, F. E. Christensen, P. Gorenstein, J. Susini, E. Ziegler, A. K. Freund, J. L. Wood, “Multilayered supermirror structures for hard x-ray synchrotron and astrophysics instrumentation,” in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE2011, 360–372 (1993).
[Crossref]

E. Spiller, Soft X-Ray Optics (SPIE Optical Engineering Press, Bellingham, Wash., 1994).
[Crossref]

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, “Automatic deposition of multilayer x-ray coatings with laterally graded d-spacing,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 36–41 (1985).

M. P. Bruijn, P. Chakraborty, H. W. van Essen, J. Verhoeven, M. J. van der Wiel, W. J. Bartels, “E-beam evaporated multilayer soft X-ray coatings, analyzed with Cu-Kα radiation,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 182–194 (1985).

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Figures (8)

Fig. 1
Fig. 1

Averaged spectral reflectivity of an ensemble of 500 randomly generated 40-period Mo/Si multilayers. Corresponding standard deviations in Mo layer thickness are indicated.

Fig. 2
Fig. 2

Integrated reflectivity over the 130–190-Å spectral interval for ensembles of 500 randomly generated Mo/Si multilayers as a function of disorder in Mo layer thicknesses. Plotted are the ensemble average, the average plus and minus of the standard deviation, as well as the highest and lowest individual realizations of each ensemble.

Fig. 3
Fig. 3

Reflectivity as a function of wavelength for the best realizations as found from the ensemble optimization outlined in Subsection 2.A (dashed curve) and by varying individual layer thicknesses as described in Subsection 2.B (solid curve).

Fig. 4
Fig. 4

Period thickness of the best realization in ångstrons as a function of period number counting from the surface toward the substrate as found from the ensemble optimization outlined in Subsection 2.A (open diamonds) and by varying individual layer thicknesses as described in Subsection 2.B (filled circles).

Fig. 5
Fig. 5

Reflectivity as a function of wavelength for a multilayer mirror optimized for constant intensity over the 150–170-Å interval.

Fig. 6
Fig. 6

Reflectivity as a function of wavelength for a multilayer mirror optimized for integrated intensity over the 130–190-Å interval for an optical system consisting of three identical multilayers in series.

Fig. 7
Fig. 7

Reflectivity as a function of grazing angle of incidence at λ = 160 Å. The solid curve shows reflectivity from the ordered stack, and the dashed curve represents a spectrum optimized to uniformity on the angular range from 72 to 90 deg. The gain in integrated intensity equals 18%.

Fig. 8
Fig. 8

Reflectivity as a function of wavelength for the best realization found from ensemble generation (see text, circles and solid curve), and for the ordered multilayer with its peak at the center of the interval (dashed curve and cubes). The solid and dashed curves represent calculations, symbols represent measured values. The disorder yields an increase in integrated reflectivity over the chosen interval of over 40%, in calculation as well as in experiment.

Equations (1)

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d Mo , k ran = d Mo , k ord ( 1 + 3 σ R k ) .

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