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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 67, Iss. 8 — Aug. 1, 1977
  • pp: 1058–1065

Principal angle, principal azimuth, and principal-angle ellipsometry of film-substrate systems

R. M. A. Azzam and A.-R. M. Zaghloul  »View Author Affiliations

JOSA, Vol. 67, Issue 8, pp. 1058-1065 (1977)

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When the film thickness is considered as a parameter, a system composed of a transparent film on an absorbing substrate (in a transparent ambient) is characterized by a range of principal angle ø¯min ≤ ø¯ ≤ ø¯max over which the associated principal azimuth ψ¯ varies between 0° and 90° (i.e., 0° ≤ ψ¯ ≤ 90°) and the reflection phase difference Δ assumes either one of the two values: +π/2 or −π/2. We determine the principal angle ø¯(d) and principal azimuth ψ¯(d) as functions of film thickness d for the vacuum-SiO2-Si system at several wavelengths as a concrete example. When the film thickness exceeds a certain minimum value, more than one principal angle becomes possible, as can be predicted by a simple graphical construction. We apply the results to principal-angle ellipsometry. (PAE) of film-substrate systems; the relationship between ø¯ and ψ¯ during film growth is particularly interesting.

© 1977 Optical Society of America

R. M. A. Azzam and A.-R. M. Zaghloul, "Principal angle, principal azimuth, and principal-angle ellipsometry of film-substrate systems," J. Opt. Soc. Am. 67, 1058-1065 (1977)

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  1. These definitions of the principal angle and the principal azimuth can be extended to include the case of an arbitrary transparent ambient. We should also mention that the value of Δ = + π/2, instead of − π/2, is based on the choice of conventions discussed in the paper by R. H. Muller, "Definitions and Conventions in Ellipsometry," Surface Sci. 16, 14–33 (1969) [also in Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, (1969)].
  2. See, for example, K. Kinosita and M. Yamamoto, "Principal-Angle-of-Incidence Ellipsometry," Surface Sci. 56, 64–75 (1976) [also in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976)].
  3. P is measured from the plane of incidence, positive in a counterclockwise sense when looking into the beam.
  4. C. V. Kent and J. Lawson, "A photoelectric method for the determination of the parameters of elliptically polarized light," J. Opt. Soc. Am. 27, 117–144 (1937).
  5. H. M. O'Bryan, "The Optical Constants of Several Metals in Vacuum," J. Opt. Soc. Am. 26, 122–127 (1936).
  6. At this uv spectral line of mercury, the refractive indices of SiO2 and Si are assumed to be 1.5 and (1.67-j3. 59), respectively. [Ellipsometric Tables of the Si-SiO2 System for Mercury and He-Ne Laser Spectral Lines, edited by G. Gergely (Akademiai Kiado, Budapest, 1971).]
  7. R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, "Ellipsometric function of a film-substrate system: Applications to the design of reflection-type optical devices and to ellipsometry," J. Opt. Soc. Am. 65, 252–260 (1975).
  8. See Eq. (13) and Fig. 3 of Ref. 7 and also Fig. 3 (b) of this paper.
  9. All CAIC's between 66.3° and 84.7° that appear in Fig. 2 intersect the imaginary axis of the complex ρ plane at four points. It is clear, however, that two and three points of intersection (and tangency) will occur, e.g., at angles of incidence between 66.3° and 75°.
  10. A. -R. M. Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Design of film-substrate single-reflection retarders," J. Opt. Soc. Am. 65, 1043–1049 (1975).
  11. R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, "Design of film-substrate single-reflection linear partial polarizers," J. Opt. Soc. Am. 65, 1472–1474 (1975).
  12. R. M. A. Azzam, A. -R. M. Zaghloul, and N. M. Bashara, "Polarizer-surface-analyzer null ellipsometry for film-substrate systems," J. Opt. Soc. Am. 65, 1464–1471 (1975).
  13. At these five wavelengths, the refractive indices of SiO2 are 1.48, 1.475, 1.47, 1.46, and 1.46, and those of Si are (5. 06-j3. 04), (6. 63-j2. 74), (5. 63-j0. 29), (4. 83-j0. 116), and (3. 85-j0. 02), respectively (after Gergely, Ref. 6).
  14. Exact symmetry of ø¯(d) and ψ¯(d) around the line d=ds, occurs in the limit of zero absorption in the substrate.
  15. The reduced-thickness curve (RTC) is obtained by subtracting from the ordinate of each point on the line d = const the proper multiple of the thickness period Dø that is required to bring that point vertically down below the Dø boundary curve of the reduced-thickness zone (RTZ) (see the discussion in Ref. 12).
  16. The equations that give d when m1 = m2 and m1 = m2 + 1 are the same as Eqs. (18a) and (18b), respectively, in Ref. 12.
  17. See, for example, the method discussed in Sec. IV of Ref. 7.
  18. See, for example, M. M. Ibrahim and N. M. Bashara, "Parameter correlation and computational considerations in multiple-angle ellipsometry," J. Opt. Soc. Am. 61, 1622–1629 (1971).
  19. This contour is readily derived from Fig. 4 by plotting ø¯(d) vs ψ¯(d) for different values of d.
  20. It is interesting to observe that when the film-substrate system acts as a p or s reflection polarizer, such a condition is detected experimentally by the extinction of the reflected beam (the sample under measurement and the polarizer of the ellipsometer now operate as a pair of crossed polarizers). The null in both O'Bryan and Kent and Lawson's ellipsometers becomes due to the reflected beam being extinguished, and not because it is circularly polarized.
  21. Furthermore, the coincident branches also become exactly symmetrical around the ψ¯ =45° line.

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