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Applied Optics

Applied Optics


  • Vol. 22, Iss. 2 — Jan. 15, 1983
  • pp: 253–264

Complex reflection coefficients for the parallel and perpendicular polarizations of a film–substrate system

R. M. A. Azzam and M. Emdadur Rahman Khan  »View Author Affiliations

Applied Optics, Vol. 22, Issue 2, pp. 253-264 (1983)

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The complex reflection coefficients Rν(ϕ,ζ) of a film–substrate system for the parallel (ν = p) and perpendicular (ν = s) polarizations are examined in detail as functions of the angle of incidence ϕ(0 ≤ ϕ ≤ 90°) and the reduced normalized film thickness ζ(0 ≤ ζ < 1). For definiteness, the reflection of light of wavelength λ = 0.6328 μm by the air–SiO2–Si system is assumed. Families of circles that represent the constant-angle-of-incidence contours, their envelopes, and the associated constant-thickness contours of Rp and Rs are all presented in the complex plane. Furthermore, the amplitude-reflectance and phase-shift functions, |Rν|(ϕ,ζ) and argRν(ϕ,ζ), are plotted vs ζ with ϕ constant and vs ϕ with ζ constant. It is shown that Rp or Rs can assume the same complex value at two different angles of incidence (i.e., the film–substrate system can have identical reflection characteristics for a given polarization at two angles) for certain ranges of film thickness. The distinct case of internal reflection is represented by a separate example.

© 1983 Optical Society of America

Original Manuscript: August 27, 1982
Published: January 15, 1983

R. M. A. Azzam and M. Emdadur Rahman Khan, "Complex reflection coefficients for the parallel and perpendicular polarizations of a film–substrate system," Appl. Opt. 22, 253-264 (1983)

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  1. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, J. Opt. Soc. Am. 65, 252 (1975). [CrossRef]
  2. A.-R. M. Zaghloul, R. M. A. Azzam, Appl. Opt. 21, 739 (1982). [CrossRef] [PubMed]
  3. G. Gergely, Ed., Ellipsometric Tables of the Si–SiO2System for Mercury and He–Ne Laser Spectral Lines (Akademiai Kaido, Budapest, 1971).
  4. A. B. Winterbottom, in The Royal Norwegian Scientific Society Report 1 (F. Burns, Trondheim, 1955).
  5. K. D. Naegele, J. Phys. Paris Colloq. C5 38, C5-225 (1977). This paper discusses some of the characteristics of the complex ratio Rν/r02ν of the reflection coefficient of the system Rν to that of the ambient–substrate interface r02ν. [CrossRef]
  6. M. D. Williams, Appl. Opt. 21, 747 (1982). [CrossRef] [PubMed]
  7. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.3.
  8. See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Sec. 8.4.
  9. F. Abelès, J. Phys. Radium 11, 310 (1950). [CrossRef]
  10. The envelope is determined as follows: A given value of Re(Rp) > − 1 is assumed, and ϕ is swept in search of a circle that intersects the line Re(Rp) = constant at a point that is most distant from the real axis [i.e., of maximum possible Im(Rp) for the given value of Re(Rp)]. We have found that this procedure works better for this problem than an established method described by G. Zwikker, The Advanced Geometry of Plane Curves and their Applications (Dover, New York, 1963), Chap. 13.
  11. W. L. Wolfe, G. J. Zissis, Eds., The Infrared Handbook (Office of Naval Research, Department of the Navy, Arlington, Va., 1978), p. 7–18.
  12. M. E. Pedinoff, M. Braunstein, O. M. Stafsudd, in Optical Polarimetry, R. M. A. Azzam, D. L. Coffeen, Eds., Proc. Soc. Photo-Opt. Instrum. Eng.112, 74 (1977). [CrossRef]
  13. R. M. A. Azzam, Surf. Sci. 96, 67 (1980). [CrossRef]

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