Abstract

A procedure is described for simultaneously fitting reflectance data obtained for various photon energies and angles of incidence using a simple physical model for ε(ω), the complex frequency-dependent dielectric constant. The mutual consistency of the model and of the experimental data is tested, within the accuracy of each measurement. As an example of the technique, reflectance data in the vacuum ultraviolet obtained by Toots, Fowler, and Marton for Be, Ge, Sb, and Bi have been satisfactorily fitted. The model parameters have been used to derive the optical constants <i>n</i> and <i>k</i> (in satisfactory agreement with conventional determinations) and can be readily related to other relevant experimental results or theoretical calculations of ε(ω) if these are available.

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  1. See, for example, R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.
  2. H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).
  3. C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).
  4. C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).
  5. W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).
  6. L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).
  7. J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).
  8. J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).
  9. M. H. Cohen, Phil. Mag. 3, 762 (1958).
  10. R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).
  11. D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.
  12. The program used here11 supplies several estimates of the uncertainty of each parameter as briefly described in the Appendix to paper II3.
  13. J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).
  14. K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).
  15. K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).
  16. R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).
  17. See Figs. 5 and 6 of Ref. 8.
  18. The uncertainties of the parameters in the Tables are the one-parameter uncertainties defined in the Appendix to paper II3. These uncertainties indicate the limits on a single parameter (evaluated in this paper at a confidence level of 95%) if all other parameters are assumed to be correctly estimated.
  19. The standard errors of n and k (shown in the captions of Figs. 2, 4, 6, and 8) were computed in each case from the standard errors of the parameters of the fits to the reflectance data, the elements of the parameter correlation matrix, and the partial derivatives of n and k with respect to the parameters evaluated at representative energies3.
  20. C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).
  21. R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).
  22. B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).
  23. J. Toots, private communication.
  24. J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).
  25. W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).
  26. C. J. Powell, Phys. Rev. Letters 15, 852 (1965).
  27. M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).
  28. M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).

Angel, D. W.

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).

Bearden, J. A.

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).

Burr, A. F.

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).

Canfield, L. R.

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).

Cardona, M.

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).

Cohen, M. H.

M. H. Cohen, Phil. Mag. 3, 762 (1958).

Collins, J. G.

J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).

Feuerbacher, B.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).

Fowler, H. A.

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).

Fuchs, R.

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).

Godwin, R. P.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).

Greenaway, D. L.

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).

Hersey, M. D.

M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).

Hunter, W. R.

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).

Kliewer, K. L.

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).

LaVilla, R. E.

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).

Madden, R. P.

See, for example, R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).

Marquardt, D . W.

D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.

Marton, L.

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).

Mendlowitz, H.

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).

Powell, C. J.

C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).

C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).

C. J. Powell, Phys. Rev. Letters 15, 852 (1965).

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).

Rabinovitch, K.

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).

Ritchie, R. H.

R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).

Sasaki, T.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).

Skibowski, M.

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).

Toots, J.

J. Toots, private communication.

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).

Tousey, R .

R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).

Tousey, R.

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).

Verleur, H. W.

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).

Other (28)

See, for example, R. P. Madden, in Physics of Thin Films, Vol. 1, G. Hass, Ed. (Academic Press Inc., New York, 1963), p. 123; O. S. Heavens, in Physics of Thin Films, Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press Inc., New York, 1964), p. 193.

H. W. Verleur, J. Opt. Soc. Am. 58, 1356 (1968).

C. J. Powell, J. Opt. Soc. Am. 60, 78 (1970).

C. J. Powell, J. Opt. Soc. Am. 59, 738 (1969).

W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965); Appl. Opt. 6, 2140 (1967).

L. Marton and J. Toots, Phys. Rev. 160, 602 (1967).

J. Toots, H. A. Fowler, and L. Marton, Phys. Rev. 172, 670 (1968).

J. Toots and L. Marton, J. Opt. Soc. Am. 59, 1305 (1969).

M. H. Cohen, Phil. Mag. 3, 762 (1958).

R . Tousey, J. Opt. Soc. Am. 29, 235 (1939).

D . W. Marquardt, J. Soc. Indust. Appl. Math. 11, 431 (1963); T. Baumeister and D. W. Marquardt, Least-Squares Estimation of Nonlinear Parameters, Share General Purpose Library, Distribution No. 1428, December 1962. A revised version of the program is available with the same title as Share Library Distribution No. 309401, August 1966.

The program used here11 supplies several estimates of the uncertainty of each parameter as briefly described in the Appendix to paper II3.

J. G. Collins, Appl. Sci. Res. Sec. B, 7, 1 (1958).

K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968).

K. Rabinovitch, L. R. Canfield, and R. P. Madden, Appl. Opt. 4, 1005 (1965); D. C. Hinson, J. Opt. Soc. Am. 56, 408 (1966); R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, Phys. Rev. Letters 21, 1530 (1968).

R. H. Ritchie, Surface Sci. 3, 497 (1965); P. Dobberstein, A. Hampe, and G. Sauerbrey, Phys. Letters 27A, 256 (1968); J. L. Stanford, H. E. Bennett, J. M. Bennett, E. J. Ashley, and E. T. Arakawa, Bull. Am. Phys. Soc. 13, 989 (1968); S. E. Schnatterly, 13, 989 (1968).

See Figs. 5 and 6 of Ref. 8.

The uncertainties of the parameters in the Tables are the one-parameter uncertainties defined in the Appendix to paper II3. These uncertainties indicate the limits on a single parameter (evaluated in this paper at a confidence level of 95%) if all other parameters are assumed to be correctly estimated.

The standard errors of n and k (shown in the captions of Figs. 2, 4, 6, and 8) were computed in each case from the standard errors of the parameters of the fits to the reflectance data, the elements of the parameter correlation matrix, and the partial derivatives of n and k with respect to the parameters evaluated at representative energies3.

C. J. Powell, Proc. Phys. Soc. (London) 76, 593 (1960).

R. E. LaVilla and H. Mendlowitz, J. Appl. Phys. 40, 3297 (1969).

B. Feuerbacher, M. Skibowski, R. P. Godwin, and T. Sasaki, J. Opt. Soc. Am. 58, 1434 (1968).

J. Toots, private communication.

J. A. Bearden and A. F. Burr, Rev. Mod. Phys. 39, 125 (1967).

W. R. Hunter, D. W. Angel, and R. Tousey, Appl. Opt. 4, 891 (1965); C. J. Powell, Phys. Rev. 175, 972 (1968).

C. J. Powell, Phys. Rev. Letters 15, 852 (1965).

M. Cardona and D. L. Greenaway, Phys. Rev. 133, A1685 (1964).

M. D. Hersey, J. Res. Natl. Bur. Std. (U. S.) 69B, 139 (1965); J. E. Monahan and A. Langsdorf, Ann. Phys. (N. Y.) 34, 238 (1965).

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