Abstract

Extremely broadband grazing-incidence multilayers for hard-x-ray reflection can be obtained by a gradual change of the layer thicknesses down through the structure. Existing approaches for designing similar neutron optics, called supermirrors, are shown to provide respectable performance when applied to x-ray multilayers. However, none of these approaches consider the effects of imperfect layer interfaces and absorption in the overlying layers. Adaptations of neutron designs that take these effects into account are presented, and a thorough analysis of two specific applications (a single hard-x-ray reflector and a hard-x-ray telescope) shows that an improved performance can be obtained. A multilayer whose bilayer thicknesses are given by a power law expression is found to provide the best solution; however, it is only slightly better than some of the adapted neutron designs.

© 1995 Optical Society of America

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  2. O. S. Heavens, H. M. Liddell, “Staggered broad-band reflecting multilayers,” Appl. Opt. 5, 373–376 (1966).
    [CrossRef]
  3. F. Mezei, “Novel polarized neutron devices: supermirror and spin component amplifier,” Commun. Phys. 1, 81–85 (1976).
  4. F. Mezei, P.A. Dagleish, “Corrigendum and first experimental evidence on neutron supermirrors,” Commun. Phys. 2, 41–43 (1977).
  5. J. F. Meekins, R. G. Cruddace, H. Gursky, “Optimization of layered synthetic microstructures for broadband reflectivity at soft x-ray and EUV wavelengths,” Appl. Opt. 26, 990–994 (1987).
    [CrossRef] [PubMed]
  6. A. G. Gukasov, V. A. Ruban, M. N. Bedrisova, “On the feasibility of increasing the region of grazing incidence reflection of neutrons using interference in multilayers,” Sov. Tech. Phys. Lett. 3, 130–135 (1977).
  7. J. Schelten, K. Mika, “Calculated reflectivities of supermirrors,” Nucl. Instrum. Methods 160, 287–294 (1979).
    [CrossRef]
  8. J. B. Hayter, H. A. Mook, “Discrete thin-film multilayer design for x-ray and neutron supermirrors,” J. Appl. Crystallogr. 22, 35–41 (1989).
    [CrossRef]
  9. J. Wood, “Status of supermirror research at OSMC,” in Neutron Optical Devices and Applications, C. F. Majkrzak, J. L. Wood, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1738, 22 (1992).
  10. M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
    [CrossRef]
  11. O. Schaerpf, N. Stuesser, “Recent progress in neutron polarizers,” Nucl. Instrum. Methods A 284, 208–211 (1989).
    [CrossRef]
  12. F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).
  13. K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).
  14. K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).
  15. P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.
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  18. L. Névot, P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
    [CrossRef]
  19. F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
    [CrossRef]
  20. E. Spiller, Soft-X-Ray Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), Chap. 7, pp. 106, 114.
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  22. O. Schaerpf, I. S. Anderson, “The role of surfaces and interfaces in the behavior of nonpolarizing and polarizing supermirrors,” Phys. B 198, 203–212 (1994).
    [CrossRef]
  23. S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).
  24. Several papers in Nucl. Instrum. Methods A 246 (1986) and Nucl. Instrum. Methods, A 266 (1988), and in Rev. Sci. Instrum. 60 (1988) and Rev. Sci. Instrum., 63 (1992).
  25. K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
    [CrossRef]
  26. S. S. Hasnain, J. R. Helliwell, H. Kamitsubo, “Overview on synchrotron radiation and the need for Journal of Synchrotron Radiation,” J. Synchrotron Rad. 1, 1–4 (1994).
    [CrossRef]
  27. E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
    [CrossRef]
  28. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1986), Chap. 10, pp. 274–334.
  29. P. Gorenstein, K. D. Joensen, “Focusing in the hard x-ray band,” in Proceedings of Imaging in High Energy Astronomy (Kluwer, Amsterdam, 1995).
    [CrossRef]
  30. H. Wolter, “Spiegelsystem streifenden einfalls als abbildende optiken fur rontgenstrahlen,” Ann. Physik. 10, 94–114 (1952).
    [CrossRef]
  31. P. Kirkpatrick, A. V. Baez, “Formation of optical images by x-rays,” J. Opt. Soc. Am. 38, 766–774 (1948).
    [CrossRef] [PubMed]

1994 (2)

O. Schaerpf, I. S. Anderson, “The role of surfaces and interfaces in the behavior of nonpolarizing and polarizing supermirrors,” Phys. B 198, 203–212 (1994).
[CrossRef]

S. S. Hasnain, J. R. Helliwell, H. Kamitsubo, “Overview on synchrotron radiation and the need for Journal of Synchrotron Radiation,” J. Synchrotron Rad. 1, 1–4 (1994).
[CrossRef]

1992 (3)

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

1989 (2)

O. Schaerpf, N. Stuesser, “Recent progress in neutron polarizers,” Nucl. Instrum. Methods A 284, 208–211 (1989).
[CrossRef]

J. B. Hayter, H. A. Mook, “Discrete thin-film multilayer design for x-ray and neutron supermirrors,” J. Appl. Crystallogr. 22, 35–41 (1989).
[CrossRef]

1988 (1)

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

1987 (1)

1986 (2)

T. W. Barbee, “Multilayers for x-ray optics,” Opt. Eng. 25, 898–915 (1986).

Several papers in Nucl. Instrum. Methods A 246 (1986) and Nucl. Instrum. Methods, A 266 (1988), and in Rev. Sci. Instrum. 60 (1988) and Rev. Sci. Instrum., 63 (1992).

1980 (1)

L. Névot, P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

1979 (1)

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

1978 (1)

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

1977 (2)

F. Mezei, P.A. Dagleish, “Corrigendum and first experimental evidence on neutron supermirrors,” Commun. Phys. 2, 41–43 (1977).

A. G. Gukasov, V. A. Ruban, M. N. Bedrisova, “On the feasibility of increasing the region of grazing incidence reflection of neutrons using interference in multilayers,” Sov. Tech. Phys. Lett. 3, 130–135 (1977).

1976 (1)

F. Mezei, “Novel polarized neutron devices: supermirror and spin component amplifier,” Commun. Phys. 1, 81–85 (1976).

1966 (1)

1952 (1)

H. Wolter, “Spiegelsystem streifenden einfalls als abbildende optiken fur rontgenstrahlen,” Ann. Physik. 10, 94–114 (1952).
[CrossRef]

1948 (1)

Achiwa, N.

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

Akiyoshi, T.

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

Anderson, I. S.

O. Schaerpf, I. S. Anderson, “The role of surfaces and interfaces in the behavior of nonpolarizing and polarizing supermirrors,” Phys. B 198, 203–212 (1994).
[CrossRef]

Baez, A. V.

Barbee, T. W.

T. W. Barbee, “Multilayers for x-ray optics,” Opt. Eng. 25, 898–915 (1986).

Bedrisova, M. N.

A. G. Gukasov, V. A. Ruban, M. N. Bedrisova, “On the feasibility of increasing the region of grazing incidence reflection of neutrons using interference in multilayers,” Sov. Tech. Phys. Lett. 3, 130–135 (1977).

Berman, L. E.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Chap. 1, p. 40.

Christensen, F. E.

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

Croce, P.

L. Névot, P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

Cruddace, R. G.

Dagleish, P.A.

F. Mezei, P.A. Dagleish, “Corrigendum and first experimental evidence on neutron supermirrors,” Commun. Phys. 2, 41–43 (1977).

Duppich, J.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Ebisawa, T.

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

Faber, W.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1986), Chap. 10, pp. 274–334.

Franco, E. D.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Freund, A.

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

Freund, A. K.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

Göbel, H.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

Gorenstein, P.

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

P. Gorenstein, K. D. Joensen, “Focusing in the hard x-ray band,” in Proceedings of Imaging in High Energy Astronomy (Kluwer, Amsterdam, 1995).
[CrossRef]

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

Graf, W.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Gukasov, A. G.

A. G. Gukasov, V. A. Ruban, M. N. Bedrisova, “On the feasibility of increasing the region of grazing incidence reflection of neutrons using interference in multilayers,” Sov. Tech. Phys. Lett. 3, 130–135 (1977).

Gursky, H.

Hasnain, S. S.

S. S. Hasnain, J. R. Helliwell, H. Kamitsubo, “Overview on synchrotron radiation and the need for Journal of Synchrotron Radiation,” J. Synchrotron Rad. 1, 1–4 (1994).
[CrossRef]

Hayter, J. B.

J. B. Hayter, H. A. Mook, “Discrete thin-film multilayer design for x-ray and neutron supermirrors,” J. Appl. Crystallogr. 22, 35–41 (1989).
[CrossRef]

Heavens, O. S.

Helliwell, J. R.

S. S. Hasnain, J. R. Helliwell, H. Kamitsubo, “Overview on synchrotron radiation and the need for Journal of Synchrotron Radiation,” J. Synchrotron Rad. 1, 1–4 (1994).
[CrossRef]

Henke, B. L.

B. L. Henke, “Scattering factors and mass absorption coefficients,” in X-Ray Data Booklet, D. Vaughan, ed. (Lawrence Berkeley Laboratory, Berkeley, Calif., 1986), pp. 2.28–2.43.

Hill, K. W.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Høghøj, P.

P. Høghøj, “Fabrication and characterization of W/Si multilayers,” Ph.D. dissertation (University of Copenhagen, Denmark, 1995).

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

Hornstrup, A.

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

Hsuan, H.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Hulse, R.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Hustache, R.

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

Iarocci, M.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

Joensen, K. D.

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

P. Gorenstein, K. D. Joensen, “Focusing in the hard x-ray band,” in Proceedings of Imaging in High Energy Astronomy (Kluwer, Amsterdam, 1995).
[CrossRef]

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

Joksch, S.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

Kaiser, W.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Kamitsubo, H.

S. S. Hasnain, J. R. Helliwell, H. Kamitsubo, “Overview on synchrotron radiation and the need for Journal of Synchrotron Radiation,” J. Synchrotron Rad. 1, 1–4 (1994).
[CrossRef]

Kawata, H.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

Kirkpatrick, P.

Krieger, A. S.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Ku, L.-P.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Lengeler, B.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

Liddell, H. M.

Marot, G.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

Meekins, J. F.

Mezei, F.

F. Mezei, P.A. Dagleish, “Corrigendum and first experimental evidence on neutron supermirrors,” Commun. Phys. 2, 41–43 (1977).

F. Mezei, “Novel polarized neutron devices: supermirror and spin component amplifier,” Commun. Phys. 1, 81–85 (1976).

Mika, K.

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

Mook, H. A.

J. B. Hayter, H. A. Mook, “Discrete thin-film multilayer design for x-ray and neutron supermirrors,” J. Appl. Crystallogr. 22, 35–41 (1989).
[CrossRef]

Névot, L.

L. Névot, P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

Okamoto, S.

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

Parker, K.

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

Parsignault, D.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1986), Chap. 10, pp. 274–334.

Rossbach, M.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Ruban, V. A.

A. G. Gukasov, V. A. Ruban, M. N. Bedrisova, “On the feasibility of increasing the region of grazing incidence reflection of neutrons using interference in multilayers,” Sov. Tech. Phys. Lett. 3, 130–135 (1977).

Schaerpf, O.

O. Schaerpf, I. S. Anderson, “The role of surfaces and interfaces in the behavior of nonpolarizing and polarizing supermirrors,” Phys. B 198, 203–212 (1994).
[CrossRef]

O. Schaerpf, N. Stuesser, “Recent progress in neutron polarizers,” Nucl. Instrum. Methods A 284, 208–211 (1989).
[CrossRef]

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Schelten, J.

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

Schirmer, A.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Schnopper, H. W.

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

Schnopper, J.

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

Schuster, M.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

Siddons, D. P.

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

Spiller, E.

E. Spiller, Soft-X-Ray Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), Chap. 7, pp. 106, 114.

Stanglmeier, F.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

Stratton, B. C.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Stuesser, N.

O. Schaerpf, N. Stuesser, “Recent progress in neutron polarizers,” Nucl. Instrum. Methods A 284, 208–211 (1989).
[CrossRef]

Susini, J.

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1986), Chap. 10, pp. 274–334.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1986), Chap. 10, pp. 274–334.

von Goeler, S.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Weber, W.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

Westergaard, N. J.

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Chap. 1, p. 40.

Wolter, H.

H. Wolter, “Spiegelsystem streifenden einfalls als abbildende optiken fur rontgenstrahlen,” Ann. Physik. 10, 94–114 (1952).
[CrossRef]

Wood, J.

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

J. Wood, “Status of supermirror research at OSMC,” in Neutron Optical Devices and Applications, C. F. Majkrzak, J. L. Wood, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1738, 22 (1992).

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

Yamada, S.

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

Young, K. M.

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Zeisler, R.

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Ziegler, E.

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

Acta Crystallogr. Sec. A (1)

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, M. Schuster, “Determination of the dispersive correction f′(E) to the atomic form factor from x-ray reflection,” Acta Crystallogr. Sec. A 48, 626–639 (1992).
[CrossRef]

Ann. Physik. (1)

H. Wolter, “Spiegelsystem streifenden einfalls als abbildende optiken fur rontgenstrahlen,” Ann. Physik. 10, 94–114 (1952).
[CrossRef]

Ann. Rep. Res. Reactor Inst. (1)

S. Yamada, T. Ebisawa, N. Achiwa, T. Akiyoshi, S. Okamoto, “Neutron-optical properties of multilayer systems,” Ann. Rep. Res. Reactor Inst. 11, 8–27 (1978).

Appl. Opt. (2)

Commun. Phys. (1)

F. Mezei, P.A. Dagleish, “Corrigendum and first experimental evidence on neutron supermirrors,” Commun. Phys. 2, 41–43 (1977).

Commun. Phys. (1)

F. Mezei, “Novel polarized neutron devices: supermirror and spin component amplifier,” Commun. Phys. 1, 81–85 (1976).

J. Appl. Crystallogr. (1)

J. B. Hayter, H. A. Mook, “Discrete thin-film multilayer design for x-ray and neutron supermirrors,” J. Appl. Crystallogr. 22, 35–41 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Synchrotron Rad. (1)

S. S. Hasnain, J. R. Helliwell, H. Kamitsubo, “Overview on synchrotron radiation and the need for Journal of Synchrotron Radiation,” J. Synchrotron Rad. 1, 1–4 (1994).
[CrossRef]

Nucl. Instrum. Methods (2)

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

Several papers in Nucl. Instrum. Methods A 246 (1986) and Nucl. Instrum. Methods, A 266 (1988), and in Rev. Sci. Instrum. 60 (1988) and Rev. Sci. Instrum., 63 (1992).

Nucl. Instrum. Methods A (1)

O. Schaerpf, N. Stuesser, “Recent progress in neutron polarizers,” Nucl. Instrum. Methods A 284, 208–211 (1989).
[CrossRef]

Nucl. Instrum. Methods B (1)

M. Rossbach, O. Schaerpf, W. Kaiser, W. Graf, A. Schirmer, W. Faber, J. Duppich, R. Zeisler, “The use of focusing supermirror neutron guides to enhance cold neutron fluence rates,” Nucl. Instrum. Methods B 35, 181–190 (1988).
[CrossRef]

Opt. Eng. (1)

T. W. Barbee, “Multilayers for x-ray optics,” Opt. Eng. 25, 898–915 (1986).

Phys. B (1)

O. Schaerpf, I. S. Anderson, “The role of surfaces and interfaces in the behavior of nonpolarizing and polarizing supermirrors,” Phys. B 198, 203–212 (1994).
[CrossRef]

Rev. Phys. Appl. (1)

L. Névot, P. Croce, “Caractérisation des surfaces par réflexion rasante de rayons X. Application à l’étude du polissage de quelques verres silicates,” Rev. Phys. Appl. 15, 761–779 (1980).
[CrossRef]

Rev. Sci. Instrum. (2)

E. Ziegler, G. Marot, A. K. Freund, S. Joksch, H. Kawata, L. E. Berman, M. Iarocci. Rev. Sci. Instrum. 63, 496–500 (1992).
[CrossRef]

K. W. Hill, K. M. Young, S. von Goeler, H. Hsuan, R. Hulse, L.-P. Ku, B. C. Stratton, A. S. Krieger, D. Parsignault, E. D. Franco, “ITER x-ray diagnostics studies,” Rev. Sci. Instrum. 63, 5032–5034 (1992).
[CrossRef]

Sov. Tech. Phys. Lett. (1)

A. G. Gukasov, V. A. Ruban, M. N. Bedrisova, “On the feasibility of increasing the region of grazing incidence reflection of neutrons using interference in multilayers,” Sov. Tech. Phys. Lett. 3, 130–135 (1977).

Other (11)

J. Wood, “Status of supermirror research at OSMC,” in Neutron Optical Devices and Applications, C. F. Majkrzak, J. L. Wood, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1738, 22 (1992).

F. E. Christensen, A. Hornstrup, N. J. Westergaard, J. Schnopper, J. Wood, K. Parker, “A graded d-spacing multilayer telescope for high-energy x-ray astronomy,” in Multilayer and Grazing Incidence X-Ray/EUV Optics, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1546, 160–167 (1992).

K. D. Joensen, F. E. Christensen, H. W. Schnopper, P. Gorenstein, J. Susini, P. Høghøj, R. Hustache, J. Wood, K. Parker, “Medium-sized grazing incidence high-energy x-ray telescopes employing continuously graded multilayers,” in X-Ray Detector Physics and Applications, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1736, 239–248 (1992).

K. D. Joensen, P. Gorenstein, F. E. Christensen, P. Høghøj, E. Ziegler, J. Susini, A. Freund, D. P. Siddons, J. Wood, “Prospects for supermirrors in hard x-ray spectroscopy,” in X-Ray and Ultraviolet Spectroscopy and Polarimetry, S. Fines-chi, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2283, 130–139 (1994).

P. Høghøj, E. Ziegler, J. Susini, A. K. Freund, K. D. Joensen, P. Gorenstein, “Broad-band focusing of hard x-rays using a supermirror,” in Physics of X-ray Multilayer Structures, Vol. 6 of OSA 1994 Technical Digest Series (Optical Society ofAmerica, Washington, D.C., 1994), pp. 142–145.

B. L. Henke, “Scattering factors and mass absorption coefficients,” in X-Ray Data Booklet, D. Vaughan, ed. (Lawrence Berkeley Laboratory, Berkeley, Calif., 1986), pp. 2.28–2.43.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1989), Chap. 1, p. 40.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, London, 1986), Chap. 10, pp. 274–334.

P. Gorenstein, K. D. Joensen, “Focusing in the hard x-ray band,” in Proceedings of Imaging in High Energy Astronomy (Kluwer, Amsterdam, 1995).
[CrossRef]

E. Spiller, Soft-X-Ray Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994), Chap. 7, pp. 106, 114.

P. Høghøj, “Fabrication and characterization of W/Si multilayers,” Ph.D. dissertation (University of Copenhagen, Denmark, 1995).

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Figures (8)

Fig. 1
Fig. 1

Schematic illustration of the supermirror concept. The multilayer would also allow for the simultaneous reflection of a monochromatic divergent beam over a band of angles of incidence.

Fig. 2
Fig. 2

Schematic illustration of the multilayer structure and the corresponding notation used in the text.

Fig. 3
Fig. 3

Calculated reflectivity at 3 mrad of three coatings: A, a 500-bilayer Ni/C supermirror with perfect interfaces designed with the Mezei algorithm [Eq. (14)]; B, same as A but an interface roughness of σ = 5.0 Å has been included, and the reflectivity has been convoluted with a 1.5-keV (FWHM) Gaussian resolution; C, a 300-Å-thick, bulk density gold coating.

Fig. 4
Fig. 4

Calculated reflectivities for W/Si supermirrors with roughness σ = 5 Å designed according to the models and parameters in Table 1. The curves numbered a1, b1, and d1 are calculated with neutron mirror designs. The other curves are calculated with the adapted or empirical designs and are seen to be superior to the neutron designs. The drop at ≈70 keV is due to the absorption edge of tungsten.

Fig. 5
Fig. 5

Normalized d-spacing change, as defined in Eq. (23), as a function of d spacing for the best designs in each category in Table 1. The most successful designs (g, e, c, and f) are seen to change very slowly at the small d spacings, only reaching a minimum of 23 Å.

Fig. 6
Fig. 6

Focusing scheme of a KB focusing telescope, showing one-dimensional focusing, A, and the orthogonal sets of nested reflectors making up the two-dimensional telescope, B. The dimensions are severely distorted for clarity.

Fig. 7
Fig. 7

Effective area for a number of the designs given in Table 2, normalized with respect to the effective area of the design obtained by the use of the Mezei algorithm. The uppermost curves are the designs obtained when the neutron algorithms are used. The lowermost are for the adapted designs and are seen to improve the performance energy range above 25 keV.

Fig. 8
Fig. 8

Normalized d-spacing change, as defined in Eq. (23), for the best design in each category in Table 2.

Tables (2)

Tables Icon

Table 1 Optimization Results for a 0–70 keV X-Ray Reflector with Coatings Whose d-Spacings Have Been Determined with the Different Designs Given in Sections 4 and 5

Tables Icon

Table 2 Optimization Results for a 0–100 keV Hard X-Ray KB Telescope with Coatings in which the d Spacings Have Been Determined with the Methods Given in Sections 4 and 5

Equations (25)

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n = 1 - δ - i β ,
δ = ρ A K f 1 ,             β = ρ A K f 2 ,             K = r e λ 2 N A 2 π ,
r j s = n j sin θ j - n j + 1 sin θ j + 1 n j sin θ j + n j + 1 sin θ j + 1 ,
r j p = n j sin θ j + 1 - n j sin θ j n j sin θ j + 1 + n j + 1 sin θ j ,
r j = 2 ( δ j - δ j + 1 ) 4 sin 2 θ .
χ j = r j + χ j + 1 exp ( - i 2 ϕ j ) 1 + r j χ j + 1 exp ( - i 2 ϕ j ) , ϕ j = 2 π λ t j n j sin θ j ,
F NC = exp [ - 8 π 2 λ 2 ( n j sin θ j ) ( n j + 1 sin θ j + 1 ) σ 2 ] ,
m λ = 2 [ t j ( n j 2 - cos 2 θ ) 1 / 2 + t j + 1 ( n j + 1 2 - cos 2 θ ) 1 / 2 ] .
m λ = 2 d sin θ [ 1 - 2 ( Γ h δ h + ( 1 - Γ ) δ l ) sin 2 θ ] 1 / 2 .
α d ( k ) 4 = l = - [ N ( k ) / 2 ] N ( k ) / 2 d ( k + l ) - d ( k ) = - d ( k ) k l = - N ( k ) / 2 N ( k ) / 2 l = - d ( k ) k [ N 2 ( k ) 4 + N ( k ) 2 ] ,
[ 1 - exp ( - i 2 π γ k ) ] 2 ( δ k h - δ k l ) 4 sin 2 θ ,
R = N 2 ( k ) ( δ h - δ l sin 2 θ ) 2 = N 2 ( k ) 4 [ d ( k ) d c ] 4 ,
δ d ( k ) δ k = - d ( k ) 5 4 d c 4 ,
d ( k ) = d c k 1 / 4 ,
d ( k ) = R 1 / 4 d c { [ 1 + α ( k + b ) ] 1 / 2 - 1 } 1 / 2 ,
θ k + Δ θ k - = θ k - 1 + Δ θ k - 1 + ,
d k = d k - 1 - a ( b d k - 1 ) - c ,
d k = a ( b + k ) - c ,
Γ = 1 2 π arccos ( L 2 μ h 2 - 4 π 2 L 2 μ h 2 + 4 π 2 ) .
R = N 2 ( k ) 4 [ d ( k ) d c ] 4 exp [ - 4 π 2 σ 2 d 2 ( k ) ] ,
k + b = 2 R d c 4 4 π 2 σ 2 α ( 1 d 2 ( k ) - 1 4 π 2 σ 2 ) × exp [ 4 π 2 σ 2 d 2 ( k ) ] + 4 R d c 2 4 π 2 σ 2 α exp [ 2 π 2 σ 2 d 2 ( k ) ] ,
FOM = d E R ( E , 3.5 mrad ) E d E E ,
Δ d ( k ) Δ d M ( k M ) = d ( k ) - d ( k + 1 ) d M ( k M ) - d M ( k M + 1 ) ,
P 1 = 2 F 1 D 0 D / 2 F 1 d θ R ( E , θ ) B ( θ , L , T ) ,
FOM = d E A ( E ) E d E E ,

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