Abstract

The reflectances of inhomogeneous layers are usually calculated by numerical solution of Maxwell's equations. This requires a specific model for the layer structure. We are interested here in the inverse problem: finding the refractive-index profile <i>n</i>(<i>z</i>) from ellipsometric data (ψ and Δ). We have calculated the reflectances explicitly in a first Born approximation [i.e., to first order in <i>n</i>(<i>z</i>) – <i>n</i><sub>0</sub>, where <i>n</i><sub>0</sub> is the index of the pure liquid]. The effect of the reflecting wall at <i>z</i> = 0 is incorporated exactly. Finally, we express ψ and Δ in terms of the Fourier transform of the profile Γ(2<b>q</b>), where q is the normal component of the incident wave vector. The equation Γ(2<b>q</b>) = Γ′+<i>i</i>Γ″ is complex; one can construct Γ′(2<b>q</b>) and Γ″(2<b>q</b>) in terms of the experimental ψ and Δ for all the accessible span of <b>q</b> vectors. For thick diffuse layers of thickness e » λ/4π, this should allow for a complete reconstruction of the profile. For thin layers, e « λ/4π, what are really measured are the moments Γ<sub>0</sub> and Γ<sub>1</sub> (of orders 0 and 1) of the index profile. To illustrate these methods, we discuss two specific examples of a slowly decreasing index profile: (1) wall effects in critical binary mixtures and (2) polymer adsorption from a good solvent.

© 1983 Optical Society of America

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  1. See, for instance, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  2. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 61–66.
  3. M. Fisher and P. G. de Gennes, C. R. Acad. Sci. Ser. B 287, 207 (1978); H. Au-Yang and M. E. Fisher, "Wall effects in critical systems: scaling in Ising model strips," Phys. Rev. B 21, 3956–3970 (1980); H. Au-Yang and M. E. Fisher, "Critical wall perturbations and a local free energy functional," Physica 101A, 255–264 (1980).
  4. C. Frank and S. E. Schnatterly, "New critical anomaly induced in a binary liquid mixture by a selectively absorbing wall," Phys. Rev. Lett. 48, 763–766 (1982).
  5. P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell U. Press, Ithaca, N.Y., 1979).
  6. P. G. de Gennes, "Polymer solutions near an interface. 1, Adsorption and depletion layers," Macromolecules 14, 1637–1644 (1981).
  7. For a lucid description of the method and some applications, see B. Law and D. Beaglehole, "Model calculations of the ellipsometric properties of inhomogeneous dielectric surfaces," J. Phys, D 14, 115–126 (1981).
  8. See, for instance, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1064–1095.
  9. J. Lekner, "Reflection of long waves by interfaces," Physica 112A, 544–556 (1982).
  10. g (r)↔g(q).
  11. J. Hilgevoord, Dispersion Relations and Causal Description (North-Holland, Amsterdam, 1960).
  12. T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).
  13. D. Beaglehole, "Ellipsometry of liquid surfaces," presented at the International Conference on Ellipsometry and Other Optical Methods for Surface and Thin Film Analysis, Paris, June 7–10, 1983.
  14. B. Sheldon, J. S. Haggerty, and A. G. Emslie, "Exact computation of the reflectance of a surface layer of arbitrary refractive-index profile and an approximate solution of the inverse problem," J. Opt. Soc. Am. 72, 1049–1055 (1982).
  15. H. Kaiser and H. C. Kaiser, "Identification of stratified media based on the Bremmer series representation of the reflection coefficient," Appl. Opt. 22, 1337–1345 (1983).
  16. See, for instance, I. S. Gradshteyn and I. M. Rydhik, Table of Integrals, Series and Products (Academic, New York, 1965).

1983 (1)

1982 (3)

B. Sheldon, J. S. Haggerty, and A. G. Emslie, "Exact computation of the reflectance of a surface layer of arbitrary refractive-index profile and an approximate solution of the inverse problem," J. Opt. Soc. Am. 72, 1049–1055 (1982).

C. Frank and S. E. Schnatterly, "New critical anomaly induced in a binary liquid mixture by a selectively absorbing wall," Phys. Rev. Lett. 48, 763–766 (1982).

J. Lekner, "Reflection of long waves by interfaces," Physica 112A, 544–556 (1982).

1981 (3)

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

P. G. de Gennes, "Polymer solutions near an interface. 1, Adsorption and depletion layers," Macromolecules 14, 1637–1644 (1981).

For a lucid description of the method and some applications, see B. Law and D. Beaglehole, "Model calculations of the ellipsometric properties of inhomogeneous dielectric surfaces," J. Phys, D 14, 115–126 (1981).

1978 (1)

M. Fisher and P. G. de Gennes, C. R. Acad. Sci. Ser. B 287, 207 (1978); H. Au-Yang and M. E. Fisher, "Wall effects in critical systems: scaling in Ising model strips," Phys. Rev. B 21, 3956–3970 (1980); H. Au-Yang and M. E. Fisher, "Critical wall perturbations and a local free energy functional," Physica 101A, 255–264 (1980).

Acad, C. R.

M. Fisher and P. G. de Gennes, C. R. Acad. Sci. Ser. B 287, 207 (1978); H. Au-Yang and M. E. Fisher, "Wall effects in critical systems: scaling in Ising model strips," Phys. Rev. B 21, 3956–3970 (1980); H. Au-Yang and M. E. Fisher, "Critical wall perturbations and a local free energy functional," Physica 101A, 255–264 (1980).

Azzam, R. M. A.

See, for instance, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Barnett, K. G.

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

Bashara, N. M.

See, for instance, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Beaglehole, D.

D. Beaglehole, "Ellipsometry of liquid surfaces," presented at the International Conference on Ellipsometry and Other Optical Methods for Surface and Thin Film Analysis, Paris, June 7–10, 1983.

For a lucid description of the method and some applications, see B. Law and D. Beaglehole, "Model calculations of the ellipsometric properties of inhomogeneous dielectric surfaces," J. Phys, D 14, 115–126 (1981).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 61–66.

Cosgrove, T.

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

Crowley, T. L.

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

de Gennes, P. G.

P. G. de Gennes, "Polymer solutions near an interface. 1, Adsorption and depletion layers," Macromolecules 14, 1637–1644 (1981).

M. Fisher and P. G. de Gennes, C. R. Acad. Sci. Ser. B 287, 207 (1978); H. Au-Yang and M. E. Fisher, "Wall effects in critical systems: scaling in Ising model strips," Phys. Rev. B 21, 3956–3970 (1980); H. Au-Yang and M. E. Fisher, "Critical wall perturbations and a local free energy functional," Physica 101A, 255–264 (1980).

P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell U. Press, Ithaca, N.Y., 1979).

Emslie, A. G.

Feshbach, H.

See, for instance, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1064–1095.

Fisher, M.

M. Fisher and P. G. de Gennes, C. R. Acad. Sci. Ser. B 287, 207 (1978); H. Au-Yang and M. E. Fisher, "Wall effects in critical systems: scaling in Ising model strips," Phys. Rev. B 21, 3956–3970 (1980); H. Au-Yang and M. E. Fisher, "Critical wall perturbations and a local free energy functional," Physica 101A, 255–264 (1980).

Frank, C.

C. Frank and S. E. Schnatterly, "New critical anomaly induced in a binary liquid mixture by a selectively absorbing wall," Phys. Rev. Lett. 48, 763–766 (1982).

Gradshteyn, I. S.

See, for instance, I. S. Gradshteyn and I. M. Rydhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Haggerty, J. S.

Hilgevoord, J.

J. Hilgevoord, Dispersion Relations and Causal Description (North-Holland, Amsterdam, 1960).

Kaiser, H.

Kaiser, H. C.

Law, B.

For a lucid description of the method and some applications, see B. Law and D. Beaglehole, "Model calculations of the ellipsometric properties of inhomogeneous dielectric surfaces," J. Phys, D 14, 115–126 (1981).

Lekner, J.

J. Lekner, "Reflection of long waves by interfaces," Physica 112A, 544–556 (1982).

Morse, P. M.

See, for instance, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1064–1095.

Rydhik, I. M.

See, for instance, I. S. Gradshteyn and I. M. Rydhik, Table of Integrals, Series and Products (Academic, New York, 1965).

Schnatterly, S. E.

C. Frank and S. E. Schnatterly, "New critical anomaly induced in a binary liquid mixture by a selectively absorbing wall," Phys. Rev. Lett. 48, 763–766 (1982).

Sheldon, B.

Tadros, T. F.

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

Vincent, B.

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 61–66.

Appl. Opt. (1)

Faraday Symp. Chem. Soc. (1)

T. Cosgrove, T. L. Crowley, B. Vincent, K. G. Barnett, and T. F. Tadros, "The configuration of adsorbed polymers at the solidsolution interface," Faraday Symp. Chem. Soc. 16, 101–108 (1981).

J. Opt. Soc. Am. (1)

Macromolecules (1)

P. G. de Gennes, "Polymer solutions near an interface. 1, Adsorption and depletion layers," Macromolecules 14, 1637–1644 (1981).

Phys. Rev. Lett. (1)

C. Frank and S. E. Schnatterly, "New critical anomaly induced in a binary liquid mixture by a selectively absorbing wall," Phys. Rev. Lett. 48, 763–766 (1982).

Physica (1)

J. Lekner, "Reflection of long waves by interfaces," Physica 112A, 544–556 (1982).

Other (10)

g (r)↔g(q).

J. Hilgevoord, Dispersion Relations and Causal Description (North-Holland, Amsterdam, 1960).

For a lucid description of the method and some applications, see B. Law and D. Beaglehole, "Model calculations of the ellipsometric properties of inhomogeneous dielectric surfaces," J. Phys, D 14, 115–126 (1981).

See, for instance, P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), pp. 1064–1095.

P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell U. Press, Ithaca, N.Y., 1979).

See, for instance, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 61–66.

M. Fisher and P. G. de Gennes, C. R. Acad. Sci. Ser. B 287, 207 (1978); H. Au-Yang and M. E. Fisher, "Wall effects in critical systems: scaling in Ising model strips," Phys. Rev. B 21, 3956–3970 (1980); H. Au-Yang and M. E. Fisher, "Critical wall perturbations and a local free energy functional," Physica 101A, 255–264 (1980).

D. Beaglehole, "Ellipsometry of liquid surfaces," presented at the International Conference on Ellipsometry and Other Optical Methods for Surface and Thin Film Analysis, Paris, June 7–10, 1983.

See, for instance, I. S. Gradshteyn and I. M. Rydhik, Table of Integrals, Series and Products (Academic, New York, 1965).

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