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Applied Optics

Applied Optics


  • Vol. 22, Iss. 24 — Dec. 15, 1983
  • pp: 4155–4165

Pseudo-Brewster and second-Brewster angles of an absorbing substrate coated by a transparent thin film

R. M. A. Azzam and T. F. Thonn  »View Author Affiliations

Applied Optics, Vol. 22, Issue 24, pp. 4155-4165 (1983)

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The pseudo-Brewster angle of minimum reflectance for the p polarization, the corresponding angle for the s polarization, and the second-Brewster angle of minimum ratio of the p and s reflectances are all determined as functions of the thickness of a transparent film coating an absorbing substrate by numerical solution of the exact equations that govern such angles of the form Re(Z′/Z) = 0, where Z = Rp, Rs, or ρ represent the complex amplitude-reflection coefficients for the p and s polarizations and their ratio (ρ = Rp/Rs), respectively, and Z′ is the angle-of-incidence derivative of Z. Results that show these angles and their associated reflectance and reflectance-ratio minima are presented for the SiO2–Si film–substrate system at wavelength λ = 0.6328 μm and film thickness of up to four periods (≃1.2 μm). Applications of these results are proposed in film-thickness measurement and control.

© 1983 Optical Society of America

Original Manuscript: August 9, 1983
Published: December 15, 1983

R. M. A. Azzam and T. F. Thonn, "Pseudo-Brewster and second-Brewster angles of an absorbing substrate coated by a transparent thin film," Appl. Opt. 22, 4155-4165 (1983)

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  1. R. M. A. Azzam, M. E. R. Khan, Appl. Opt. 22, 253 (1983). [CrossRef] [PubMed]
  2. W. A. Pliskin, in Progress in Analytic Chemistry, Vol. 2, E. M. Murt, W. G. Guldner, Eds. (Plenum, New York, 1969), pp. 1–34.
  3. A direct search method is used. An initial iteration step Δϕin(1°) is selected, and ϕ is scanned in one direction (from 90° to 0). A root of Eq. (2) exists between two successive ϕ values whenever its left-hand side changes sign. The root is refined by successive reduction of Δϕ (0.1,0.01,…) until the absolute value of the left-hand side of Eq. (2) falls below 10−6. The value of the function |Z| is evaluated at the root, and the search is resumed for another root.
  4. R. M. A. Azzam, J. Opt. Soc. Am. 72, 1187 (1982). [CrossRef]
  5. R. M. A. Azzam, Opt. Acta 30, 1113 (1983). [CrossRef]
  6. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 316–317.

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