Abstract

The increased availability of advanced synchrotron radiation sources is resulting in a resurgence of activity in the field of x-ray and extreme ultraviolet imaging systems. However, scattering effects caused by residual optical fabrication errors frequently dominate geometrical design errors in the degradation of image quality at these very short wavelengths. Traditional optical design and analysis techniques (geometrical ray tracing) are therefore inadequate for predicting the performance of high-resolution synchrotron beam-line optics. A surface-scattering theory must be implemented to model the image degradation effects of residual surface irregularities over the entire range of relevant spatial frequencies. This includes small-angle scattering effects caused by mid-spatial-frequency surface errors that fall between the traditional figure and finish specifications. Performance predictions are presented parametrically to provide insight into the optical fabrication tolerances necessary to meet the requirements of a specific application.

© 1995 Optical Society of America

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References

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  1. An ALS Handbook, PUB-643 (Lawrence Berkeley Laboratory, Berkeley, Calif., 1988, under U.S. Department of Energy contract DE-AC03-76SF00098).
  2. K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).
  3. A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
    [Crossref]
  4. D. L. Lunt, J. W. Bender, D. G. Ewing, W. R. McKinney, “XUV synchrotron optical components for the Advanced Light Source: fabrication and testing,” in Optics for High-Brightness Synchrotron Radiation Beamlines, J. Arthur, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1740, 161–172 (1992).
  5. P. Kirkpatrick, A. V. Baez, “Formation of optical images by x-rays,” J. Opt. Soc. Am. 38, 766–774 (1948).
    [Crossref] [PubMed]
  6. A. G. Michette, Optical Systems for Soft X-rays (Plenum, New York, 1986), Chap. 2, p. 58.
  7. J. E. Harvey, “Potential pitfalls in the design of x-ray/EUV imaging systems,” in Lens Design (Critical Reviews), W. J. Smith, ed., Vol. 41 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Billingham, Wash., 1992), pp. 192–224.
  8. P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5, p. 70–98.
  9. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  10. 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).
  11. J. M. Bennett, L. Mattsson, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, D.C., 1989), Chap. 4, pp. 38–56.
  12. J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990), Chaps. 1 and 2, pp. 1–44.
  13. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 208.
  14. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D dissertation (University of Arizona, Tucson, Ariz., 1976).
  15. 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]
  16. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), Chap. 4, p. 51.
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2, p. 13.
  18. 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]
  19. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
  20. R. J. Noll, P. E. Glenn, J. Osantowski, “Optical surface analysis code (OSAC),” in Scattering in Optical Materials II, S. Musikant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.362, 78–85 (1982).
  21. R. J. Noll, P. Glenn, “Mirror surface autocovariance functions and their associated visible scattering,” Appl. Opt. 21, 1824–1838 (1982).
    [Crossref] [PubMed]
  22. A. Slomba, R. Babish, P. Glenn, “Mirror surface metrology and polishing for AXAF/TMA,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 40–54 (1985).
  23. E. Spiller, D. Stearns, M. Krumrey, “Multilayer x-ray mirrors: interfacial roughness, scattering, and image quality,” J. Appl. Phys. 74, 107–118 (1993).
    [Crossref]
  24. 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]

1993 (1)

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

1992 (1)

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]

1990 (1)

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

1988 (2)

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]

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

1982 (1)

1979 (2)

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

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

1961 (1)

1948 (1)

Babish, R.

A. Slomba, R. Babish, P. Glenn, “Mirror surface metrology and polishing for AXAF/TMA,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 40–54 (1985).

Baez, A. V.

Beckman, P.

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

Bender, J. W.

D. L. Lunt, J. W. Bender, D. G. Ewing, W. R. McKinney, “XUV synchrotron optical components for the Advanced Light Source: fabrication and testing,” in Optics for High-Brightness Synchrotron Radiation Beamlines, J. Arthur, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1740, 161–172 (1992).

Bennett, H. E.

Bennett, J. M.

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

Berglin, E. J.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

Bracewell, R. N.

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

Church, E. L.

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

de Bergevin, F.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

Ewing, D. G.

D. L. Lunt, J. W. Bender, D. G. Ewing, W. R. McKinney, “XUV synchrotron optical components for the Advanced Light Source: fabrication and testing,” in Optics for High-Brightness Synchrotron Radiation Beamlines, J. Arthur, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1740, 161–172 (1992).

Fantone, S. T.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

Freund, A. K.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

Fuchs, B. A.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 208.

Glenn, P.

R. J. Noll, P. Glenn, “Mirror surface autocovariance functions and their associated visible scattering,” Appl. Opt. 21, 1824–1838 (1982).
[Crossref] [PubMed]

A. Slomba, R. Babish, P. Glenn, “Mirror surface metrology and polishing for AXAF/TMA,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 40–54 (1985).

Glenn, P. E.

R. J. Noll, P. E. Glenn, J. Osantowski, “Optical surface analysis code (OSAC),” in Scattering in Optical Materials II, S. Musikant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.362, 78–85 (1982).

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, p. 13.

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, “Light-scattering characteristics of optical surfaces,” Ph.D dissertation (University of Arizona, Tucson, Ariz., 1976).

J. E. Harvey, “Potential pitfalls in the design of x-ray/EUV imaging systems,” in Lens Design (Critical Reviews), W. J. Smith, ed., Vol. 41 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Billingham, Wash., 1992), pp. 192–224.

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).

Humpal, H. H.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

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).

Karpenko, V. K.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

Kirkpatrick, P.

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]

Kulkarni, S.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

Lunt, D. L.

D. L. Lunt, J. W. Bender, D. G. Ewing, W. R. McKinney, “XUV synchrotron optical components for the Advanced Light Source: fabrication and testing,” in Optics for High-Brightness Synchrotron Radiation Beamlines, J. Arthur, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1740, 161–172 (1992).

Marot, G.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

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.

McKinney, W. R.

D. L. Lunt, J. W. Bender, D. G. Ewing, W. R. McKinney, “XUV synchrotron optical components for the Advanced Light Source: fabrication and testing,” in Optics for High-Brightness Synchrotron Radiation Beamlines, J. Arthur, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1740, 161–172 (1992).

Michette, A. G.

A. G. Michette, Optical Systems for Soft X-rays (Plenum, New York, 1986), Chap. 2, p. 58.

Moran, E. C.

Noll, J.

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

Noll, R. J.

R. J. Noll, P. Glenn, “Mirror surface autocovariance functions and their associated visible scattering,” Appl. Opt. 21, 1824–1838 (1982).
[Crossref] [PubMed]

R. J. Noll, P. E. Glenn, J. Osantowski, “Optical surface analysis code (OSAC),” in Scattering in Optical Materials II, S. Musikant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.362, 78–85 (1982).

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]

Osantowski, J.

R. J. Noll, P. E. Glenn, J. Osantowski, “Optical surface analysis code (OSAC),” in Scattering in Optical Materials II, S. Musikant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.362, 78–85 (1982).

Porteus, J. O.

Riekel, C.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

Slomba, A.

A. Slomba, R. Babish, P. Glenn, “Mirror surface metrology and polishing for AXAF/TMA,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 40–54 (1985).

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]

Spizzichino, A.

P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963), Chap. 5, p. 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]

Stover, J. C.

J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990), Chaps. 1 and 2, pp. 1–44.

Susini, J.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

Tirsell, K. G.

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

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]

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).

Zhang, L.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

Ziegler, E.

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

Zmek, W. P.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

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. (1)

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. (2)

Opt. Eng. (4)

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

K. G. Tirsell, E. J. Berglin, B. A. Fuchs, H. H. Humpal, V. K. Karpenko, S. Kulkarni, S. T. Fantone, “Highly polished, grazing incidence mirrors developed for synchrotron radiation beam lines at Stanford Synchrotron Radiation Laboratory,” Opt. Eng. 27, 985–992 (1988).

A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, E. Ziegler, “X-ray mirrors for the European Sychrotron Radiation Facility,” Opt. Eng. 29, 928–941 (1990).
[Crossref]

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

Other (14)

R. J. Noll, P. E. Glenn, J. Osantowski, “Optical surface analysis code (OSAC),” in Scattering in Optical Materials II, S. Musikant, ed., Proc. Soc. Photo-Opt. Instrum. Eng.362, 78–85 (1982).

A. Slomba, R. Babish, P. Glenn, “Mirror surface metrology and polishing for AXAF/TMA,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 40–54 (1985).

D. L. Lunt, J. W. Bender, D. G. Ewing, W. R. McKinney, “XUV synchrotron optical components for the Advanced Light Source: fabrication and testing,” in Optics for High-Brightness Synchrotron Radiation Beamlines, J. Arthur, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1740, 161–172 (1992).

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

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

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), Chaps. 1 and 2, pp. 1–44.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 7, p. 208.

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D dissertation (University of Arizona, Tucson, Ariz., 1976).

A. G. Michette, Optical Systems for Soft X-rays (Plenum, New York, 1986), Chap. 2, p. 58.

J. E. Harvey, “Potential pitfalls in the design of x-ray/EUV imaging systems,” in Lens Design (Critical Reviews), W. J. Smith, ed., Vol. 41 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Billingham, Wash., 1992), pp. 192–224.

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

An ALS Handbook, PUB-643 (Lawrence Berkeley Laboratory, Berkeley, Calif., 1988, under U.S. Department of Energy contract DE-AC03-76SF00098).

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Figures (17)

Fig. 1
Fig. 1

Lawrence Berkeley Laboratory ALS.

Fig. 2
Fig. 2

Classical Kirkpatrick–Baez grazing-incidence focusing system.

Fig. 3
Fig. 3

Grazing-incidence monochromator configuration: (a) spherical grating, (b) toroidal grating.

Fig. 4
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

Surface profile and the relevant statistical parameters.

Fig. 7
Fig. 7

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

Fig. 8
Fig. 8

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

Fig. 9
Fig. 9

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

Fig. 10
Fig. 10

Grazing incidence reduces the rms wave-front error induced by the surface irregularities and foreshortens the wavefront features in the plane of incidence.

Fig. 11
Fig. 11

Effect on image quality differs for each spatial-frequency regime.

Fig. 12
Fig. 12

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 mid-spatial-frequency surface errors.

Fig. 13
Fig. 13

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. 14
Fig. 14

Rense and Violett spherical grating monochromator: (a) vertical plane, (b) horizontal plane.

Fig. 15
Fig. 15

Image degradation caused by scattering from residual optical fabrication errors for a three-mirror synchrotron beam line operating at a wavelength of 25 Å.

Fig. 16
Fig. 16

Image degradation caused by scattering from residual optical fabrication errors for a five-mirror synchrotron beam line operating at a wavelength of 50 Å.

Fig. 17
Fig. 17

Optical performance predictions for the five-mirror synchrotron beam line for different wavelengths when the optical fabrication tolerances are held constant at a 6-Å rms mid-spatial-frequency roughness and a 1-Å rms microroughness.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

H s ( x ̂ , ŷ ) = exp { ( 4 π σ ̂ s ) 2 [ 1 Ĉ 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 Ĉ 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 ) Ĉ 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 ) ,
σ w = 2 σ s sin ϕ ,
C w ( x ̂ , ŷ ) = 4 sin 2 ϕ C s ( x ̂ l ̂ , ŷ l ̂ sin ϕ ) .
H s ( x ̂ , ŷ ) = exp { ( 4 π sin ϕ σ ̂ s ) 2 [ 1 Ĉ s ( x ̂ l ̂ , ŷ l ̂ sin ϕ ) / σ ̂ s 2 ] } ,
A = exp [ ( 4 π sin ϕ σ ̂ s ) 2 ] ,
B = 1 A = 1 exp [ ( 4 π sin ϕ σ ̂ s ) 2 ] ,
Q ( x ̂ , ŷ ) = exp [ ( 4 π sin ϕ σ ̂ s ) 2 Ĉ s ( x ̂ l ̂ , ŷ l ̂ sin ϕ ) / σ ̂ s 2 ] 1 exp [ ( 4 π sin ϕ σ ̂ s ) 2 ] 1 .
H fab = H L H M H H ,
H fab = exp [ ( 4 π sin ϕ σ ̂ L ) 2 ( 1 Ĉ L / σ ̂ L 2 ) ] × exp [ ( 4 π sin ϕ σ ̂ M ) 2 ( 1 Ĉ M / σ ̂ M 2 ) ] × exp [ ( 4 π sin ϕ σ ̂ H ) 2 ( 1 Ĉ 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 sin ϕ σ ̂ s ) 2 ( 1 Ĉ s / σ ̂ s 2 ) ] .
P ( x ̂ , ŷ ) = A ( x ̂ , ŷ ) exp [ 2 π W ̂ ( x ̂ , ŷ ) ] , W ̂ ( 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 ) .
ACV = σ m 2 exp ( r / l m ) + σ h 2 exp ( r / l h ) ,

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