Abstract

A formula is derived for the total width of the angle spread function in an imaging optical system, the performance of which is degraded by surface roughness. The derivation is done in the framework of the scalar theory of diffraction, but it does not rely on the small roughness approximation. The contribution of scattering from surface roughness to the total width of the angle spread function is found to depend only on the variance of the roughness slope. It is also shown that the common rule of obtaining the total width of the angle spread function as a square sum of the pure scattering component and the pure specular component is actually incorrect and does not fully account for the contribution of the specular component.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (2)

M. J. L. Turner and K. A. Flanagan, eds., Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray, Proc. SPIE 7011 (2008).

F. M. Schellenberg, ed., Emerging Lithographic Technologies XII, Proc. SPIE 6921 (2008).

2007 (1)

D. Spiga, “Analytical evaluation of the X-ray scattering contribution to imaging degradation in grazing-incidence X-ray telescopes,” Astron. Astrophys. 468, 775-784 (2007).
[CrossRef]

2006 (2)

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

F. E. Zocchi, “High efficiency collector design for extreme ultra-violet and X-ray applications,” Appl. Opt. 45 (35), 8882-8888 (2006).
[CrossRef]

2004 (2)

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

E. Spiller, “High performance multilayer coatings for EUV lithography,” Proc. SPIE 5193, 89-97 (2004).
[CrossRef]

2003 (1)

P. Zhao and L. P. Van Spreybroeck, “New method to model X-ray scattering from random rough surfaces,” Proc. SPIE 4851, 124-139 (2003).
[CrossRef]

2000 (1)

1999 (1)

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

1998 (1)

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular X-ray scattering in a multiplayer-coated imaging system,” J. Appl. Phys. 84, 1003-1028 (1998).
[CrossRef]

1996 (1)

J. E. Harvey, K. L. Lewotsky, and A. K. Thompson, “Performance predictions of a Schwarzschild imaging microscope for soft X-ray applications,” Opt. Eng. 35, 2423-2436 (1996).
[CrossRef]

1995 (5)

J. E. Harvey, “Fourier treatment of near field scalar diffraction theory,” Proc. SPIE 2515, 246-272 (1995).
[CrossRef]

J. E. Harvey, “Scattering effects in X-ray imaging systems,” Proc. SPIE 2515, 246-272 (1995).
[CrossRef]

E. L. Church and P. Z. Takacs, “Specification of glancing- and normal-incidence X-ray mirrors,” Opt. Eng. 34, 353-360(1995).
[CrossRef]

J. E. Harvey, “Modeling the image quality of enhanced reflectance X-ray multilayers as a surface power spectral density filter function,” Appl. Opt. 34, 3715-3726 (1995).

K. D. Joensen, P. Voutov, A. Szentgyorgyi, J. Roll, P. Gorenstein, P. Hoghoj, and F. E. Christensen, “Design of grazing-incidence multilayer supermirrors for hard X-ray reflectors,” Appl. Opt. 34, 7935-7944 (1995).

1993 (1)

1991 (1)

R. B. Hoover and B. C. Walker, eds., X-ray/EUV Optics for Astronomy, Microscopy, Polarimetry, and Projection Lithography, Proc. SPIE 1343 (1991).

1988 (3)

1986 (1)

T. W. Barbee, “Multilayer for X-ray optics,” Opt. Eng. 25, 898-915 (1986).

1980 (1)

J. K. Silk, “A grazing incident microscope for X-ray imaging applications,” Ann. N.Y. Acad. Sci. 342, 116-129 (1980).
[CrossRef]

1973 (1)

1969 (1)

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
[CrossRef]

1963 (1)

1952 (1)

H. Wolter, “Mirror systems with glancing incidence as image-producing optics for X-rays,” Ann. Phys. 445, 94-114 (1952).
[CrossRef]

Barbee, T. W.

T. W. Barbee, “Multilayer for X-ray optics,” Opt. Eng. 25, 898-915 (1986).

Bergmann, K.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Braun, S.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Chase, R. C.

Christensen, F. E.

Church, E. L.

Flanagan, K. A.

M. J. L. Turner and K. A. Flanagan, eds., Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray, Proc. SPIE 7011 (2008).

Foltyn, T.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Früke, R.

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

Gaines, D. P.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular X-ray scattering in a multiplayer-coated imaging system,” J. Appl. Phys. 84, 1003-1028 (1998).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Gawlitza, P.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Giacconi, R.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gorenstein, P.

Goto, A.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Gullikson, E. M.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular X-ray scattering in a multiplayer-coated imaging system,” J. Appl. Phys. 84, 1003-1028 (1998).
[CrossRef]

Haga, K.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Harvey, J. E.

J. E. Harvey, K. L. Lewotsky, and A. K. Thompson, “Performance predictions of a Schwarzschild imaging microscope for soft X-ray applications,” Opt. Eng. 35, 2423-2436 (1996).
[CrossRef]

J. E. Harvey, “Scattering effects in X-ray imaging systems,” Proc. SPIE 2515, 246-272 (1995).
[CrossRef]

J. E. Harvey, “Fourier treatment of near field scalar diffraction theory,” Proc. SPIE 2515, 246-272 (1995).
[CrossRef]

J. E. Harvey, “Modeling the image quality of enhanced reflectance X-ray multilayers as a surface power spectral density filter function,” Appl. Opt. 34, 3715-3726 (1995).

J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527-1533 (1988).

Hidaka, Y.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Hoghoj, P.

Hoover, R. B.

R. B. Hoover and B. C. Walker, eds., X-ray/EUV Optics for Astronomy, Microscopy, Polarimetry, and Projection Lithography, Proc. SPIE 1343 (1991).

Hornstrup, A.

Ichimaru, S.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Joensen, K. D.

Karabacak, T.

Kito, H.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Kuneida, H.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Leson, A.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Lewotsky, K. L.

J. E. Harvey, K. L. Lewotsky, and A. K. Thompson, “Performance predictions of a Schwarzschild imaging microscope for soft X-ray applications,” Opt. Eng. 35, 2423-2436 (1996).
[CrossRef]

Lu, T. M.

Moran, E. C.

Neff, W.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Nisius, T.

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

Ogasaka, Y.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Okajima, T.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

Quayle, B.

Rausch, S.

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

Reidy, W. P.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
[CrossRef]

Roll, J.

Schäfer, D.

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

Schellenberg, F. M.

F. M. Schellenberg, ed., Emerging Lithographic Technologies XII, Proc. SPIE 6921 (2008).

Schnopper, H. W.

Silk, J. K.

J. K. Silk, “A grazing incident microscope for X-ray imaging applications,” Ann. N.Y. Acad. Sci. 342, 116-129 (1980).
[CrossRef]

Smith, F. D.

Spiga, D.

D. Spiga, “Analytical evaluation of the X-ray scattering contribution to imaging degradation in grazing-incidence X-ray telescopes,” Astron. Astrophys. 468, 775-784 (2007).
[CrossRef]

Spiller, E.

E. Spiller, “High performance multilayer coatings for EUV lithography,” Proc. SPIE 5193, 89-97 (2004).
[CrossRef]

E. Spiller, Soft X-Ray Optics (SPIE, 1994).

Stearns, D. G.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular X-ray scattering in a multiplayer-coated imaging system,” J. Appl. Phys. 84, 1003-1028 (1998).
[CrossRef]

Stowe, M.

Sweeney, D. W.

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, “Nonspecular X-ray scattering in a multiplayer-coated imaging system,” J. Appl. Phys. 84, 1003-1028 (1998).
[CrossRef]

Szentgyorgyi, A.

Takacs, P. Z.

E. L. Church and P. Z. Takacs, “Specification of glancing- and normal-incidence X-ray mirrors,” Opt. Eng. 34, 353-360(1995).
[CrossRef]

E. L. Church and P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32, 3344-3353 (1993).

TakaHashi, S.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Takeda, S.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Tamura, K.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Tawara, Y.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Thompson, A. K.

J. E. Harvey, K. L. Lewotsky, and A. K. Thompson, “Performance predictions of a Schwarzschild imaging microscope for soft X-ray applications,” Opt. Eng. 35, 2423-2436 (1996).
[CrossRef]

Tsusaka, Y.

K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

Turner, M. J. L.

M. J. L. Turner and K. A. Flanagan, eds., Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray, Proc. SPIE 7011 (2008).

Vaiana, G. S.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
[CrossRef]

Van Speybroeck, L. P.

R. C. Chase and L. P. Van Speybroeck, “Wolter-Schwarzshild telescopes for X-ray astronomy,” Appl. Opt. 12, 1042-1044(1973).
[CrossRef]

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
[CrossRef]

Van Spreybroeck, L. P.

P. Zhao and L. P. Van Spreybroeck, “New method to model X-ray scattering from random rough surfaces,” Proc. SPIE 4851, 124-139 (2003).
[CrossRef]

Vogt, U.

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

Voutov, P.

Walker, B. C.

R. B. Hoover and B. C. Walker, eds., X-ray/EUV Optics for Astronomy, Microscopy, Polarimetry, and Projection Lithography, Proc. SPIE 1343 (1991).

Walter, K.

T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

Wang, G. C.

Wieland, M.

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
[CrossRef]

Wilhein, T.

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[CrossRef]

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K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
[CrossRef]

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R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
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P. Zhao and L. P. Van Spreybroeck, “New method to model X-ray scattering from random rough surfaces,” Proc. SPIE 4851, 124-139 (2003).
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J. K. Silk, “A grazing incident microscope for X-ray imaging applications,” Ann. N.Y. Acad. Sci. 342, 116-129 (1980).
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Ann. Phys. (1)

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[CrossRef]

Appl. Opt. (10)

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J. E. Harvey, E. C. Moran, and W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527-1533 (1988).

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K. D. Joensen, P. Voutov, A. Szentgyorgyi, J. Roll, P. Gorenstein, P. Hoghoj, and F. E. Christensen, “Design of grazing-incidence multilayer supermirrors for hard X-ray reflectors,” Appl. Opt. 34, 7935-7944 (1995).

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K. Yamashita, H. Kuneida, Y. Tawara, K. Tamura, Y. Ogasaka, K. Haga, T. Okajima, Y. Hidaka, S. Ichimaru, S. TakaHashi, A. Goto, H. Kito, Y. Tsusaka, K. Yokoyama, and S. Takeda, “New design concept of multilayer supermirrors for hard X-ray optics,” Proc. SPIE 3766, 327-335 (1999).
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T. Foltyn, K. Bergmann, S. Braun, P. Gawlitza, A. Leson, W. Neff, and K. Walter, “Design and development of an optical system for EUV microscopy,” Proc. SPIE 5533, 37-46 (2004).
[CrossRef]

D. Schäfer, T. Nisius, R. Früke, S. Rausch, M. Wieland, U. Vogt, and T. Wilhein, “Compact X-ray microscopes for EUV- and soft X-radiation with spectral imaging capabilities,” Proc. SPIE 6317, 631704 (2006).
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M. J. L. Turner and K. A. Flanagan, eds., Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray, Proc. SPIE 7011 (2008).

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[CrossRef]

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R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. Van Speybroeck, and T. Zehnpfennig, “Grazing incidence telescopes for X-ray astronomy,” Space Sci. Rev. 9, 3-57 (1969).
[CrossRef]

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E. Spiller, Soft X-Ray Optics (SPIE, 1994).

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Figures (3)

Fig. 1
Fig. 1

Normalized beam width in the normal incidence case as a function of the normalized wavelength for a power-law expression of the roughness power spectral density. m is the exponent of the power-law spectral density.

Fig. 2
Fig. 2

Relative difference ( δ s c δ ) / δ as a function of ρ / δ 0 for some values of the normalized rms value σ / λ of surface roughness.

Fig. 3
Fig. 3

Normalized beam width in the off-normal incidence case as a function of the normalized wavelength for a power-law expression of the roughness power spectral density with m = 2 for different incidence angles θ.

Equations (35)

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H ( f ) = H 0 ( f ) e 16 π 2 λ 2 ( σ 2 C ( λ f ) ) ,
H 0 ( f ) = H ^ 0 ( λ f ) = A ( r ) A * ( r + λ f ) d r | A ( r ) | 2 d r .
P 0 ( α ) d α = H 0 ( 0 ) = 1 .
I ( α ) d Ω = P ( α ) d α = 1 ,
δ = ( α 2 P ( α ) d α ) 1 / 2 .
α 2 P ( α ) d α = 1 4 π 2 Δ H ( f ) | f = 0 ,
δ = 1 2 π Δ H ( f ) | f = 0 .
C ( r ) | r = 0 = 2 π i S ( ν ) ν d ν .
Δ H ( f ) | f = 0 = Δ H 0 ( f ) | f = 0 + 16 π 2 λ 2 Δ C ( λ f ) | f = 0 .
Δ C ( λ f ) | f = 0 = 4 π 2 λ 2 S ( ν ) ν 2 d ν = λ 2 ρ 2 ,
ρ 2 = 4 π 2 S ( ν ) ν 2 d ν
δ = δ 0 2 + 4 ρ 2 .
δ δ 0 = 1 + ( λ λ 0 ) m 6 ,
H ( f ) = H ¯ sp ( f ) + H ¯ sc ( f ) = H 0 ( f ) e 16 π 2 λ 2 σ 2 + H 0 ( f ) e 16 π 2 λ 2 σ 2 [ e 16 π 2 λ 2 C ( λ f ) 1 ] ,
H sc ( f ) = H ¯ sc ( f ) 1 e 16 π 2 λ 2 σ 2 = H 0 ( f ) e 16 π 2 λ 2 σ 2 1 [ e 16 π 2 λ 2 C ( λ f ) 1 ] .
Δ H sc ( f ) | f = 0 = Δ H 0 ( f ) | f = 0 + 16 π 2 λ 2 1 1 e 16 π 2 λ 2 σ 2 Δ C ( λ f ) | f = 0 ,
δ sc = δ 0 2 + 4 ρ 2 1 e 16 π 2 λ 2 σ 2 .
δ 2 = δ sp 2 e 16 π 2 λ 2 σ 2 + δ sc 2 ( 1 e 16 π 2 λ 2 σ 2 ) = δ 0 2 + 4 ρ 2 .
δ sc = δ 0 2 + λ 2 ρ 2 4 π 2 σ 2 .
P ¯ sc ( α ; k λ ) = 1 k 4 P ¯ sc ( α k ; λ ) ,
P sc ( α ; k λ ) = 1 k 2 P sc ( α k ; λ ) .
H ( f ) = H 0 ( f ) e 16 π 2 cos 2 θ λ 2 ( σ 2 C ( Q f ) ) ,
Q = λ [ g 1 cos 1 θ 0 0 g 2 ] .
I [ C ( Q f ) ] = 1 det Q S ( Q 1 ν )
Δ H ( f ) | f = 0 = Δ H 0 ( f ) | f = 0 + 16 π 2 cos 2 θ λ 2 Δ C ( Q f ) | f = 0 ,
Δ C ( Q f ) | f = 0 = 4 π 2 det Q S ( Q 1 ν ) ν 2 d ν .
Δ C ( Q f ) | f = 0 = 4 π 2 S ( ν ) ν T Q T Q ν d ν = 4 π 2 λ 2 S ( ν ) ( ν 1 2 g 1 2 cos 2 θ + g 2 2 ν 2 2 ) d ν ,
ρ i 2 = 4 π 2 S ( ν ) ν i 2 d ν ,
δ = δ 0 2 + 4 ( g 1 2 ρ 1 2 + g 2 2 ρ 2 2 cos 2 θ ) .
ρ 1 2 4 π 2 S 0 ν 1 2 ν 2 m d v 1 d v 2 k 1 cos 3 θ λ 4 m ,
ρ 2 2 4 π 2 S 0 ν 2 2 m d v 1 d v 2 k 2 cos θ λ 4 m ,
δ = δ 0 2 + 4 cos 3 θ λ 4 m ( k 1 g 1 2 + k 2 g 2 2 ) .
δ δ 0 = 1 + cos 3 θ ( λ λ 0 ) m 6 ,
H ( f ) = H 0 ( f ) e 16 π 2 λ 2 k = 1 N cos 2 θ k ( σ k 2 C k ( Q k f ) ) ,
δ = δ 0 2 + 4 k = 1 N ( g k , 1 2 ρ k , 1 2 + g k , 2 2 ρ k , 2 2 cos 2 θ k ) .

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