Abstract

Residual surface roughness over the entire range of relevant spatial frequencies must be specified and controlled in many high-performance optical systems. This is particularly true for enhanced reflectance multilayers if both high reflectance and high spatial resolution are desired. If we assume that the interfaces making up a multilayer coating are uncorrelated at high spatial frequencies (microroughness) and perfectly correlated at low spatial and midspatial frequencies, then the multilayer can be thought of as a surface power spectral density (PSD) filter function. Multilayer coatings thus behave as a low-pass spatial frequency filter acting on the substrate PSD, with the exact location and shape of this cutoff being material and process dependent. This concept allows us to apply conventional linear systems techniques to the evaluation of image quality and to the derivation of optical fabrication tolerances for applications utilizing enhanced reflectance x-ray multilayers.

© 1995 Optical Society of America

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References

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  1. E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
    [CrossRef]
  2. E. Spiller, “Multilayer interference coatings for the vacuum ultraviolet,” in Space Optics, B. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences—National Research Council, Washington, D.C., 1974), pp. 581–597.
  3. E. Spiller, “Reflective multilayer coatings in the far UV region,” Appl. Opt. 15, 2333–2338 (1976).
    [CrossRef] [PubMed]
  4. T. W. Barbee, D. L. Kieth, “Synthetic structure layered on the atomic scale,” in Workshop on X-ray Instrumentation for Synchrotron Radiation Research, H. Winick, G. Brown, eds. (Stanford University, Stanford, Calif., 1978), p. III–26.
  5. T. W. Barbee, “Multilayers for x-ray optics,” Opt. Eng. 25, 898–915 (1986).
  6. N. M. Ceglio, “Revolution in x-ray optics,” J. X-ray Sci. Technol. 1, 7–78 (1989).
    [CrossRef]
  7. Ten papers on x-ray multilayer optics, special issue, Opt. Eng. 25, 897–978 (1986).
  8. More than 40 papers presented at the 1985 SPIE Conference, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563 (1985).
  9. More than 30 papers presented at the 1988 SPIE Conference, in X-Ray Multilayers in Diffractometers, Monochromators, and Spectrometers, F. E. Christensen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.984, 1–271 (1988).
  10. More than 20 papers presented at the 1989 SPIE Conference, in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 187–314 (1989).
  11. More than 30 papers presented at the 1991 SPIE Conference, in Multilayer Optics for Advanced X-Ray Applications, N. M. Ceglio, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1547 (1991).
  12. More than 50 papers presented at the 1992 OSA Topical Meeting, in Physics of X-Ray Multilayer Structures, Vol. 7 of 1992 OSA Technical Digest Series, (Optical Society of America, Washington, D.C., 1992).
  13. More than 40 papers presented at the 1992 OSA Topical Meeting, in Soft X-Ray Projection Lithography, Vol. 8 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1992).
  14. More than 40 papers presented at the 1992 SPIE Conference, in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1741 (1992).
  15. More than 25 papers presented at the 1992 SPIE Conference, in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1742 (1992).
  16. More than 40 papers presented at the 1993 OSA Topical Meeting, in Soft X-Ray Projection Lithography, Vol. 18 of OSA Proceedings (Optical Society of America, Washington, D.C., 1993).
  17. More than 20 papers presented at the 1993 SPIE Conference, in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2011 (1993).
  18. P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5, pp. 70–98.
  19. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  20. J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 87–99 (1989).
  21. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 4, pp. 38–56.
  22. J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990), Chap. 4, pp. 67–88.
  23. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976, available from University Microfilms, Ann Arbor, Mich. 48106).
  24. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  25. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 4, pp. 51–68.
  26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 13–17.
  27. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 3, pp. 40–98.
  28. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
  29. J. E. Harvey, E. C. Moran, W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
    [CrossRef] [PubMed]
  30. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.
  31. J. M. Eastman, “Surface scattering in optical interface coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y, 1974; available from University Microfilms, Ann Arbor, Mich. 48106).
  32. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
  33. J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
    [CrossRef] [PubMed]
  34. 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).
  35. D. G. Stearns, “The scattering of x-rays from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
    [CrossRef]
  36. C. Amra, J. H. Apfel, E. Pelletier, “The role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
    [CrossRef] [PubMed]
  37. J. B. Kortright, “Nonspecular x-ray scattering from multilayer structures,” J. Appl. Phys. 70, 3620–3625 (1991).
    [CrossRef]
  38. E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
    [CrossRef]
  39. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [CrossRef] [PubMed]
  40. E. L. Church, P. Z. Takacs, “Specification of surface figure and finish in terms of system performance,” Appl. Opt. 32, 3344–3353 (1993).
    [CrossRef] [PubMed]
  41. A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
    [CrossRef] [PubMed]
  42. E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
    [CrossRef]
  43. K. L. Lewotsky, “Performance limitations of imaging microscopes for soft x-ray applications,” M.S. thesis (University of Central Florida, Orlando, Fla., 1992).
  44. M. Kado, “Development of Schwarzschild x-ray microscope and its application to ICF experiments,” M.S. Thesis (Osaka University, Osaka, Japan, 1990).
  45. E. Spiller, “Enhancement of the reflectivity of multilayer x-ray mirrors by ion polishing,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 271–279 (1989).

1993 (2)

E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
[CrossRef]

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

1992 (2)

C. Amra, J. H. Apfel, E. Pelletier, “The role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992).
[CrossRef] [PubMed]

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

1991 (1)

J. B. Kortright, “Nonspecular x-ray scattering from multilayer structures,” J. Appl. Phys. 70, 3620–3625 (1991).
[CrossRef]

1989 (2)

D. G. Stearns, “The scattering of x-rays from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
[CrossRef]

N. M. Ceglio, “Revolution in x-ray optics,” J. X-ray Sci. Technol. 1, 7–78 (1989).
[CrossRef]

1988 (3)

A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
[CrossRef] [PubMed]

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
[CrossRef] [PubMed]

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

1986 (3)

Ten papers on x-ray multilayer optics, special issue, Opt. Eng. 25, 897–978 (1986).

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

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).

1980 (1)

1979 (3)

J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

1976 (1)

1972 (1)

E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
[CrossRef]

1961 (1)

Amra, C.

Apfel, J. H.

Barbee, T. W.

A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
[CrossRef] [PubMed]

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

T. W. Barbee, D. L. Kieth, “Synthetic structure layered on the atomic scale,” in Workshop on X-ray Instrumentation for Synchrotron Radiation Research, H. Winick, G. Brown, eds. (Stanford University, Stanford, Calif., 1978), p. III–26.

Beckman, P.

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5, pp. 70–98.

Bennett, H. E.

Bennett, J. M.

J. M. Elson, J. P. Rahn, J. M. Bennett, “Light scattering from multilayer optics: comparison of theory and experiment,” Appl. Opt. 19, 669–679 (1980).
[CrossRef] [PubMed]

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 4, pp. 38–56.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 4, pp. 51–68.

Carniglia, C. K.

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).

Ceglio, N. M.

N. M. Ceglio, “Revolution in x-ray optics,” J. X-ray Sci. Technol. 1, 7–78 (1989).
[CrossRef]

Church, E. L.

Eastman, J. M.

J. M. Eastman, “Surface scattering in optical interface coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y, 1974; available from University Microfilms, Ann Arbor, Mich. 48106).

Elson, J. M.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 3, pp. 40–98.

Golub, L.

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 13–17.

Harvey, J. E.

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

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 87–99 (1989).

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976, available from University Microfilms, Ann Arbor, Mich. 48106).

Hoover, R. B.

A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
[CrossRef] [PubMed]

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Kado, M.

M. Kado, “Development of Schwarzschild x-ray microscope and its application to ICF experiments,” M.S. Thesis (Osaka University, Osaka, Japan, 1990).

Kieth, D. L.

T. W. Barbee, D. L. Kieth, “Synthetic structure layered on the atomic scale,” in Workshop on X-ray Instrumentation for Synchrotron Radiation Research, H. Winick, G. Brown, eds. (Stanford University, Stanford, Calif., 1978), p. III–26.

Kortright, J. B.

J. B. Kortright, “Nonspecular x-ray scattering from multilayer structures,” J. Appl. Phys. 70, 3620–3625 (1991).
[CrossRef]

Krumrey, M.

E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
[CrossRef]

Lewotsky, K. L.

K. L. Lewotsky, “Performance limitations of imaging microscopes for soft x-ray applications,” M.S. thesis (University of Central Florida, Orlando, Fla., 1992).

Lindblom, J. E.

A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
[CrossRef] [PubMed]

Mattsson, L.

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 4, pp. 38–56.

Moran, E. C.

Noll, J.

J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).

Nystrom, G.

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

Pelletier, E.

Porteus, J. O.

Rahn, J. P.

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).

Spiller, E.

E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
[CrossRef]

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

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, “Reflective multilayer coatings in the far UV region,” Appl. Opt. 15, 2333–2338 (1976).
[CrossRef] [PubMed]

E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
[CrossRef]

E. Spiller, “Multilayer interference coatings for the vacuum ultraviolet,” in Space Optics, B. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences—National Research Council, Washington, D.C., 1974), pp. 581–597.

E. Spiller, “Enhancement of the reflectivity of multilayer x-ray mirrors by ion polishing,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 271–279 (1989).

Spizzichino, A.

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5, pp. 70–98.

Stearns, D.

E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
[CrossRef]

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

Stearns, D. G.

D. G. Stearns, “The scattering of x-rays from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
[CrossRef]

Stover, J. C.

J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990), Chap. 4, pp. 67–88.

Takacs, P. Z.

Walker, A. B. C.

A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
[CrossRef] [PubMed]

Wilczynski, J.

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Zmek, W. P.

Appl. Opt. (6)

Appl. Phys. Lett. (2)

E. Spiller, “Low-loss reflection coatings using absorbing materials,” Appl. Phys. Lett. 20, 365–367 (1972).
[CrossRef]

E. Spiller, J. Wilczynski, D. Stearns, L. Golub, G. Nystrom, “Imaging performance of multilayer x-ray mirrors,” Appl. Phys. Lett. 61, 1481–1483 (1992).
[CrossRef]

J. Appl. Phys. (3)

D. G. Stearns, “The scattering of x-rays from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
[CrossRef]

J. B. Kortright, “Nonspecular x-ray scattering from multilayer structures,” J. Appl. Phys. 70, 3620–3625 (1991).
[CrossRef]

E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

N. M. Ceglio, “Revolution in x-ray optics,” J. X-ray Sci. Technol. 1, 7–78 (1989).
[CrossRef]

Opt. Eng. (6)

Ten papers on x-ray multilayer optics, special issue, Opt. Eng. 25, 897–978 (1986).

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).

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).

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

J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).

Science (1)

A. B. C. Walker, T. W. Barbee, R. B. Hoover, J. E. Lindblom, “Soft x-ray images of the solar corona with a normal-incidence Cassegrain multilayer telescope,” Science 241, 1781–1787 (1988).
[CrossRef] [PubMed]

Other (25)

K. L. Lewotsky, “Performance limitations of imaging microscopes for soft x-ray applications,” M.S. thesis (University of Central Florida, Orlando, Fla., 1992).

M. Kado, “Development of Schwarzschild x-ray microscope and its application to ICF experiments,” M.S. Thesis (Osaka University, Osaka, Japan, 1990).

E. Spiller, “Enhancement of the reflectivity of multilayer x-ray mirrors by ion polishing,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 271–279 (1989).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

J. M. Eastman, “Surface scattering in optical interface coatings,” Ph.D. dissertation (University of Rochester, Rochester, N.Y, 1974; available from University Microfilms, Ann Arbor, Mich. 48106).

T. W. Barbee, D. L. Kieth, “Synthetic structure layered on the atomic scale,” in Workshop on X-ray Instrumentation for Synchrotron Radiation Research, H. Winick, G. Brown, eds. (Stanford University, Stanford, Calif., 1978), p. III–26.

E. Spiller, “Multilayer interference coatings for the vacuum ultraviolet,” in Space Optics, B. J. Thompson, R. R. Shannon, eds. (National Academy of Sciences—National Research Council, Washington, D.C., 1974), pp. 581–597.

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1165, 87–99 (1989).

J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 4, pp. 38–56.

J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990), Chap. 4, pp. 67–88.

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976, available from University Microfilms, Ann Arbor, Mich. 48106).

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 4, pp. 51–68.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, pp. 13–17.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 3, pp. 40–98.

More than 40 papers presented at the 1985 SPIE Conference, in Applications of Thin Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. Soc. Photo-Opt. Instrum. Eng.563 (1985).

More than 30 papers presented at the 1988 SPIE Conference, in X-Ray Multilayers in Diffractometers, Monochromators, and Spectrometers, F. E. Christensen, ed., Proc. Soc. Photo-Opt. Instrum. Eng.984, 1–271 (1988).

More than 20 papers presented at the 1989 SPIE Conference, in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 187–314 (1989).

More than 30 papers presented at the 1991 SPIE Conference, in Multilayer Optics for Advanced X-Ray Applications, N. M. Ceglio, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1547 (1991).

More than 50 papers presented at the 1992 OSA Topical Meeting, in Physics of X-Ray Multilayer Structures, Vol. 7 of 1992 OSA Technical Digest Series, (Optical Society of America, Washington, D.C., 1992).

More than 40 papers presented at the 1992 OSA Topical Meeting, in Soft X-Ray Projection Lithography, Vol. 8 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, DC, 1992).

More than 40 papers presented at the 1992 SPIE Conference, in Soft X-Ray Microscopy, C. J. Jacobsen, J. E. Trebes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1741 (1992).

More than 25 papers presented at the 1992 SPIE Conference, in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1742 (1992).

More than 40 papers presented at the 1993 OSA Topical Meeting, in Soft X-Ray Projection Lithography, Vol. 18 of OSA Proceedings (Optical Society of America, Washington, D.C., 1993).

More than 20 papers presented at the 1993 SPIE Conference, in Multilayer and Grazing Incidence X-Ray/EUV Optics II, R. B. Hoover, A. B. Walker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2011 (1993).

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5, pp. 70–98.

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Figures (20)

Fig. 1
Fig. 1

Optical surface irregularities produce a specularly reflected beam with a diffusely reflected component that can degrade optical performance in several different ways.

Fig. 2
Fig. 2

(a) Scattered intensity versus scattering angle; (b) scattered radiance in direction cosine space.

Fig. 3
Fig. 3

Surface profile and the relevant statistical parameters.

Fig. 4
Fig. 4

(a) Surface transfer function and (b) associated angle spread function are related by the Fourier transform operation just as are the OTF and the PSF of modern image-formation theory.

Fig. 5
Fig. 5

Relationship between the optical surface profile, the surface PSD function, and the surface ACV function.

Fig. 6
Fig. 6

In the smooth surface approximation, the PSF is made up of an image core and a scattering function proportional to the filtered PSD. The fractional encircled energy is also indicated.

Fig. 7
Fig. 7

Effect on image quality differs for each spatial frequency regime.

Fig. 8
Fig. 8

PSF, consisting of a narrow image core, a small-angle scatter function, and a wide-angle scattered halo. The shaded area illustrates the sensitivity to midspatial frequency surface errors.

Fig. 9
Fig. 9

PSF of an imaging system is related to the complex pupil function in exactly the same way that the surface PSD is related to the surface profile.

Fig. 10
Fig. 10

Enhanced reflectance multilayer also produces a specularly reflected beam and a diffusely reflected component whose relative strengths depend on the degree of correlation between the various interfaces.

Fig. 11
Fig. 11

Substrate surface PSD is assumed to obey an inverse power law, which spans the entire range of spatial frequencies from low spatial frequency figure errors to high spatial frequency finish.

Fig. 12
Fig. 12

(a) Enhanced reflectance multilayer coatings behave as low-pass spatial frequency filters. (b) The filtered substrate PSD is the effective PSD of the multilayer.

Fig. 13
Fig. 13

Encircled energy predictions for the Stanford/MSFC normal-incidence Cassegrain x-ray telescope. A surface PSD obeying an inverse square law and a band-limited microroughness of 5 Å are assumed.

Fig. 14
Fig. 14

Encircled energy predictions for the Stanford/MSFC normal-incidence Cassegrain x-ray telescope if the proper aspheric figure were achieved to eliminate the residual geometric aberrations.

Fig. 15
Fig. 15

Encircled energy predictions for a 25-cm-diameter Ritchey–Chretien telescope with a PSD slope of −2 and 5 Å of microroughness.

Fig. 16
Fig. 16

Densitometer scan that illustrates the edge response to a step-function object.

Fig. 17
Fig. 17

Edge-response function predicted by actual band-limited metrology data.

Fig. 18
Fig. 18

High spatial frequency surface PSD from metrology data.

Fig. 19
Fig. 19

Assumed inverse power law surface PSD.

Fig. 20
Fig. 20

Edge-response function based on an extrapolation of the measured PSD function through the midspatial frequency regime to the low spatial frequency figure errors.

Equations (32)

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H s ( x ̂ , ŷ ) = exp { ( 4 π σ ̂ s ) 2 [ 1 C ̂ s ( x ̂ l ̂ , ŷ l ̂ ) / σ ̂ s 2 ] } .
H ( x ̂ , ŷ ) = A + B Q ( x ̂ , ŷ ) ,
A = exp [ ( 4 π σ ̂ s ) 2 ] ,
B = 1 A = 1 exp [ ( 4 π σ ̂ s ) 2 ] ,
Q ( x ̂ , ŷ ) = exp [ ( 4 π σ ̂ s ) 2 C ̂ s ( x ̂ l ̂ , ŷ l ̂ ) / σ ̂ s 2 ] 1 exp [ ( 4 π σ ̂ s ) 2 ] 1 .
S ( α , β ) = { H s ( x ̂ , ŷ ) } = A δ ( α , β ) + S ( α , β ) ,
S ( α , β ) = B { Q ( x ̂ , ŷ ) } .
A 1 ( 4 π σ s ) 2 ,
B ( 4 π σ ̂ s ) 2 ,
Q ( x ̂ , ŷ ) C ̂ s ( x ̂ l ̂ , ŷ l ̂ ) / σ ̂ s 2 .
S ( α , β ) = scattering function = B { Q ( x ̂ , ŷ ) } = ( 4 π / λ ) 2 PSD .
I ( θ ) = A I c ( θ ) + ( B / σ s 2 ) PSD ( θ ) ,
H fab = H L H M H H ,
H fab = exp [ ( 4 π σ ̂ L ) 2 ( 1 C ̂ L / σ ̂ L 2 ) ] × exp [ ( 4 π σ ̂ M ) 2 ( 1 C ̂ M / σ ̂ M 2 ) ] × exp [ ( 4 π σ ̂ H ) 2 ( 1 C ̂ H / σ ̂ H 2 ) ] .
C s = C L + C M + C H .
σ s 2 = σ L 2 + σ M 2 + σ H 2 ;
H fab = exp [ ( 4 π σ ̂ s ) 2 ( 1 C ̂ s / σ ̂ s 2 ) ] .
P ( x ̂ , ŷ ) = A ( x ̂ , ŷ ) exp [ 2 π Ŵ ( x ̂ , ŷ ) ] , Ŵ ( x ̂ , ŷ ) = 2 cos θ 0 ĥ ( x ̂ , ŷ ) ,
H ( x ̂ , ŷ ) = H c ( x ̂ , ŷ ) H fab ( x ̂ , ŷ ) .
I ( α , β ) = I c ( α , β ) * S ( α , β ) .
R = ( 1 n ) 2 + κ 2 ( 1 + n ) 2 + κ 2 ,
P s = R A P 0 ,
P d = R B P 0 ,
A = exp [ ( 4 π σ / λ ) 2 ] ,
B = TIS = 1 A ,
σ 2 = σ L 2 + σ M 2 + σ H 2 .
PSD ( f ) = h f 0 α ( f + f 0 ) α .
ACV = σ h 2 exp ( r l h ) ,
σ h 2 = 0 2 π f 2 f 3 b f α f d f d θ = ( 2 . 7 Å ) 2 .
σ T 2 = 0 2 π f 1 f 2 b f α f d f d θ = σ h 2 [ f 3 ( 2 α ) f 1 ( 2 α ) ] [ f 3 ( 2 α ) f 2 ( 2 α ) ] ,
f 1 = 1 D , D = mirror diameter , f 2 = 1 2 L , L = length of profilometer trace , f 3 = 1 2 Δ L , Δ L = data sample width .
σ T 2 = 2.939 σ h 2 .

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