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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 73, Iss. 12 — Dec. 1, 1983
  • pp: 1777–1784

Ellipsometric formulas for an inhomogeneous layer with arbitrary refractive-index profile

J. C. Charmet and P. G. de Gennes  »View Author Affiliations

JOSA, Vol. 73, Issue 12, pp. 1777-1784 (1983)

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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 n(z) from ellipsometric data (ψ and Δ). We have calculated the reflectances explicitly in a first Born approximation [i.e., to first order in n(z) – n0, where n0 is the index of the pure liquid]. The effect of the reflecting wall at z = 0 is incorporated exactly. Finally, we express ψ and Δ in terms of the Fourier transform of the profile Γ(2q), where q is the normal component of the incident wave vector. The equation Γ(2q) = Γ′+iΓ″ is complex; one can construct Γ′(2q) and Γ″(2q) in terms of the experimental ψ and Δ for all the accessible span of q 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 Γ0 and Γ1 (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

J. C. Charmet and P. G. de Gennes, "Ellipsometric formulas for an inhomogeneous layer with arbitrary refractive-index profile," J. Opt. Soc. Am. 73, 1777-1784 (1983)

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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).

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