Abstract

The optical constants of silicon and quartz in the soft x-ray region (400–3750 eV) determined from the incidence-angle dependence of the specular reflectance are reported. The measured reflectance-vs-angle of incidence curves for the lower energies are quantitatively explained using a simple diffractive scattering model including the interference effect between the reflected beams, while those for the higher energies can be interpreted in terms of the microfacet model without the interference effect.

© 1988 Optical Society of America

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References

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  1. E. Spiller, “Soft X-Ray Optics and Microscopy,” in Handbook on Synchrotron Radiation, Vol. 1, E. E. Koch, Ed. (North-Holland, Amsterdam, 1983), p. 1091.
  2. A. G. Michette, Optical Systems for Soft X Rays (Plenum, New York, 1986).
    [CrossRef]
  3. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, Ed. (Academic, New York, 1985), p. 547.
  4. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
    [CrossRef]
  5. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
    [CrossRef]
  6. H. R. Philipp, “Silicon Dioxide (SiO2) (Glass)” in Handbook of Optical Constants of Solids, E. D. Palik, Ed. (Academic, New York, 1985), p. 749.
  7. W. R. Hunter, “Measurement of Optical Properties of Materials in the Vacuum Ultraviolet Spectral Region,” Appl. Opt. 21, 2103 (1982).
    [CrossRef] [PubMed]
  8. M. Yanagihara et al., “Soft X-Ray Reflection from SiC, TiC, and WC Mirrors,” Appl. Opt. 25, 4586 (1986).
    [CrossRef] [PubMed]
  9. H. E. Bennett, J. O. Porteus, “Relation Between Surface Roughness and Specular Reflectance at Normal Incidence,” J. Opt. Soc. Am. 51, 123 (1961).
    [CrossRef]
  10. B. Bhushan, J. C. Wyant, C. L. Koliopoulos, “Measurement of Surface Topography of Magnetic Tapes by Mirau Interferometry,” Appl. Opt. 24, 1489 (1985).
    [CrossRef] [PubMed]
  11. S. Sato, National Laboratory for High Energy Physics, Japan; private communication.
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  13. J. A. Bearden, A. F. Burr, “Reevaluation of X-Ray Atomic Energy Levels,” Rev. Mod. Phys. 39, 125 (1967).
    [CrossRef]
  14. T. Matsushita, T. Ishikawa, K. Kohra, “High-Resolution Measurements of Angle-Resolved X-Ray Scattering from Optically Flat Mirrors,” J. Appl. Crystallogr. 17, 257 (1984).
    [CrossRef]
  15. C. C. Sung, W. D. Eberhardt, “Scattering of an Electromagnetic Wave from a Very Rough Semi-Infinite Dielectric Plane (Exact Treatment of the Boundary Conditions),” J. Appl. Phys. 49, 994 (1978).
    [CrossRef]
  16. J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface. II,” J. Appl. Phys. 49, 1002 (1978).
    [CrossRef]

1986

1985

1984

T. Matsushita, T. Ishikawa, K. Kohra, “High-Resolution Measurements of Angle-Resolved X-Ray Scattering from Optically Flat Mirrors,” J. Appl. Crystallogr. 17, 257 (1984).
[CrossRef]

1982

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

W. R. Hunter, “Measurement of Optical Properties of Materials in the Vacuum Ultraviolet Spectral Region,” Appl. Opt. 21, 2103 (1982).
[CrossRef] [PubMed]

1981

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

1978

C. C. Sung, W. D. Eberhardt, “Scattering of an Electromagnetic Wave from a Very Rough Semi-Infinite Dielectric Plane (Exact Treatment of the Boundary Conditions),” J. Appl. Phys. 49, 994 (1978).
[CrossRef]

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface. II,” J. Appl. Phys. 49, 1002 (1978).
[CrossRef]

1967

J. A. Bearden, A. F. Burr, “Reevaluation of X-Ray Atomic Energy Levels,” Rev. Mod. Phys. 39, 125 (1967).
[CrossRef]

1961

Bearden, J. A.

J. A. Bearden, A. F. Burr, “Reevaluation of X-Ray Atomic Energy Levels,” Rev. Mod. Phys. 39, 125 (1967).
[CrossRef]

Bennett, H. E.

Bhushan, B.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Burr, A. F.

J. A. Bearden, A. F. Burr, “Reevaluation of X-Ray Atomic Energy Levels,” Rev. Mod. Phys. 39, 125 (1967).
[CrossRef]

Eberhardt, W. D.

C. C. Sung, W. D. Eberhardt, “Scattering of an Electromagnetic Wave from a Very Rough Semi-Infinite Dielectric Plane (Exact Treatment of the Boundary Conditions),” J. Appl. Phys. 49, 994 (1978).
[CrossRef]

Edwards, D. F.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, Ed. (Academic, New York, 1985), p. 547.

Fujikawa, B. K.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

Henke, B. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

Holzer, J. A.

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface. II,” J. Appl. Phys. 49, 1002 (1978).
[CrossRef]

Hunter, W. R.

Ishikawa, T.

T. Matsushita, T. Ishikawa, K. Kohra, “High-Resolution Measurements of Angle-Resolved X-Ray Scattering from Optically Flat Mirrors,” J. Appl. Crystallogr. 17, 257 (1984).
[CrossRef]

Kohra, K.

T. Matsushita, T. Ishikawa, K. Kohra, “High-Resolution Measurements of Angle-Resolved X-Ray Scattering from Optically Flat Mirrors,” J. Appl. Crystallogr. 17, 257 (1984).
[CrossRef]

Koliopoulos, C. L.

Lee, P.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

Matsushita, T.

T. Matsushita, T. Ishikawa, K. Kohra, “High-Resolution Measurements of Angle-Resolved X-Ray Scattering from Optically Flat Mirrors,” J. Appl. Crystallogr. 17, 257 (1984).
[CrossRef]

Michette, A. G.

A. G. Michette, Optical Systems for Soft X Rays (Plenum, New York, 1986).
[CrossRef]

Philipp, H. R.

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass)” in Handbook of Optical Constants of Solids, E. D. Palik, Ed. (Academic, New York, 1985), p. 749.

Porteus, J. O.

Sato, S.

S. Sato, National Laboratory for High Energy Physics, Japan; private communication.

Shimabukuro, R. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

Spiller, E.

E. Spiller, “Soft X-Ray Optics and Microscopy,” in Handbook on Synchrotron Radiation, Vol. 1, E. E. Koch, Ed. (North-Holland, Amsterdam, 1983), p. 1091.

Sung, C. C.

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface. II,” J. Appl. Phys. 49, 1002 (1978).
[CrossRef]

C. C. Sung, W. D. Eberhardt, “Scattering of an Electromagnetic Wave from a Very Rough Semi-Infinite Dielectric Plane (Exact Treatment of the Boundary Conditions),” J. Appl. Phys. 49, 994 (1978).
[CrossRef]

Tanaka, T. J.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Wyant, J. C.

Yanagihara, M.

AIP Conf. Proc.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “The Atomic Scattering Factor, f1 + if2, for 94 Elements and for the 100 to 2000 eV Photon Energy Region.” AIP Conf. Proc. 75, 340 (1981).
[CrossRef]

Appl. Opt.

At. Data Nucl. Data Tables

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low Energy X-Ray Interaction Coefficients: Photoabsorption, Scattering and Reflection E = 100–2000 eV Z = 1–94,” At. Data Nucl. Data Tables 27, 1 (1982).
[CrossRef]

J. Appl. Crystallogr.

T. Matsushita, T. Ishikawa, K. Kohra, “High-Resolution Measurements of Angle-Resolved X-Ray Scattering from Optically Flat Mirrors,” J. Appl. Crystallogr. 17, 257 (1984).
[CrossRef]

J. Appl. Phys.

C. C. Sung, W. D. Eberhardt, “Scattering of an Electromagnetic Wave from a Very Rough Semi-Infinite Dielectric Plane (Exact Treatment of the Boundary Conditions),” J. Appl. Phys. 49, 994 (1978).
[CrossRef]

J. A. Holzer, C. C. Sung, “Scattering of Electromagnetic Waves from a Rough Surface. II,” J. Appl. Phys. 49, 1002 (1978).
[CrossRef]

J. Opt. Soc. Am.

Rev. Mod. Phys.

J. A. Bearden, A. F. Burr, “Reevaluation of X-Ray Atomic Energy Levels,” Rev. Mod. Phys. 39, 125 (1967).
[CrossRef]

Other

S. Sato, National Laboratory for High Energy Physics, Japan; private communication.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass)” in Handbook of Optical Constants of Solids, E. D. Palik, Ed. (Academic, New York, 1985), p. 749.

E. Spiller, “Soft X-Ray Optics and Microscopy,” in Handbook on Synchrotron Radiation, Vol. 1, E. E. Koch, Ed. (North-Holland, Amsterdam, 1983), p. 1091.

A. G. Michette, Optical Systems for Soft X Rays (Plenum, New York, 1986).
[CrossRef]

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, Ed. (Academic, New York, 1985), p. 547.

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Figures (2)

Fig. 1
Fig. 1

Optical constants as a function of photon energy for silicon: ●, δ = 1 − n; ○, k; + and Δ show δ and k obtained by Edwards (Ref. 3). The energy position of the onset of the silicon K excitation is indicated.

Fig. 2
Fig. 2

Optical constants as a function of photon energy for quartz: ●, δ = 1 − n; ○, k; + and Δ show δ and k obtained by Philipp (Ref. 6). The energy positions of the onset of the oxygen K and the silicon K excitations are indicated.

Tables (2)

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Table I Optical Constants of Silicon in the Soft X-Ray Region

Tables Icon

Table II Optical Constants of Quartz in the Soft X-Ray Region

Equations (2)

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R 0 = | cos θ n ˜ cos θ t | 2 | cos θ + n ˜ cos θ t | 2 ,
R c = R 0 exp [ ( 4 π σ cos θ ) 2 / λ 2 ] ,

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