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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 57, Iss. 4 — Apr. 1, 1967
  • pp: 466–471

Formulas for Using Wave Plates in Ellipsometry

D. A. HOLMES and D. L. FEUCHT  »View Author Affiliations

JOSA, Vol. 57, Issue 4, pp. 466-471 (1967)

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Formulas are presented for the use of optical compensators in ellipsometers. Nothing need be known about the compensator characteristics other than the directions of “fast” and “slow” axes. The formulas can be used, without modification, when the compensator is not an exact quarter-wave plate, when multiple internal reflections are present, and when the compensator is absorbing. Also described is a method for calibrating the compensator at the same time it is being used in ellipsometry measurements.

D. A. HOLMES and D. L. FEUCHT, "Formulas for Using Wave Plates in Ellipsometry," J. Opt. Soc. Am. 57, 466-471 (1967)

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  1. R. J. Archer, J. Electrochemical Soc. 104, 619 (1957); Phys. Rev. 110, 354 (1958); J. Opt. Soc. Am. 52, 970 (1962). A. B. Winterbottom, Kgl. Norske Videnskab. Selskab. Skrifter 1, 53 (1955). R. C. Menard, J. Opt. Soc. Am. 52, 427 (1962). 1. L. McCrackin, E. Passaglia, R. R. Stromberg, and H. L. Steinberg, J. Res. Natl. Bur. Stds. 67A, 363 (1963). A. Rothen, Rev. Sci. Instr. 16, 26 (1945); 19, 839 (1948); 28, 283 (1957); Ann. N. Y. Acad. Sci. 53, 1054 (1951). J. B. Bateman and M. W. Harris, Annals of the New York Academy of Sciences 53, 1064 (1951). R. W. Ditchburn and G. A. J. Orchard, Proc. Phys. Soc. (London) 57B, 608 (1954). J. A. IFaucher, G. M. McManus, and H. J. Trurnit, J. Opt. Soc. Am. 48, 51 (1958). F. Partovi, J. Opt. Soc. Am. 52, 918 (1962). 1F. P. Mertens, P. Theroux, and R. C. Plumb, J. Opt. Soc. Am. 53, 788 (1963). K. H. Zaininger and A. G. Revesz, RCA Rev. XXV, 85 (1964). D. K. Burge and H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964). D. W. Peterson and N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965). L. E. Smith and R. R. Stromberg, J. Opt. Soc. Am. 56, 1539 (1966).
  2. Various techniques for using wave-plates in conjunction with rotary analyzers are described by: M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, New York, Paris, Los Angeles, 1959), pp. 688–691. G. N. Ramachandran and S. Ramaseshan, Encyclopedia of Physics, S. Flügge, Ed. (Springer- Verlag, Berlin, Göttingen, Heidelberg, 1961), Vol. XXV/1, pp. 34–53. E. N. Cameron, Economic Geology 52, 252 (1957). M. Richartz, Z. Instrumentenk. 73, 205 (1965). R. C. Plumb, J. Opt. Soc. Am. 50, 892 (1960). D. Bergman, J. Opt. Soc. Am. 52, 1080 (1962). A. C. Hall, J. Opt. Soc. Am. 53, 801 (1963). H. G. Jerrard, J. Opt. Soc. Am. 38, 35 (1948). C. A. Skinner, J. Opt. Soc. Am. and Rev. Sci. Instr. 10, 491 (1925). M. Richartz, J. Opt. Soc. Am. 56, 198 (1966).
  3. H. G. Jerrard, J. Opt. Soc. Am. 42, 159 (1952). This article is extremely well documented.
  4. F. Gabler and P. Sokob, Z. Physik 116, 47 (1940).
  5. H. Weinberger and J. Harris, J. Opt. Soc. Am. 54, 552 (1964).
  6. D. A. Holmes, J. Opt. Soc. Am. 54, 1115 (1964).
  7. R. Bünnagel, Z. Instrumentenk. 69, 79 (1961).
  8. D. A. Holmes, J. Opt. Soc. Am. 54, 1340 (1964); 55, 209 (1965).
  9. A concise description of the various ellipsometer measuring techniques is given by A. Vašíček, in E. Passaglia, R. R. Stromberg, and J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Symposium Proceedings (U. S. Department of Commerce, National Bureau of Standards Miscellaneous Publication 256, Washington, D. C., 15 Sept 1964), pp. 33–36.
  10. H. Jacobs, D. A. Holmes, L. Hatkin, and F. A. Brand, J. Appl. Phys. 34, 2617 (1963).
  11. When converted to a real number, ρoxy can be either positive or negative.
  12. See the papers by Bergman and Hall in Ref. 2.
  13. D. A. Holmes and D. L. Feucht, J. Opt. Soc. Am. 55, 577 (1965).
  14. Recall that 0≤ψi≤π/2, 0≤Δi<2π, 0≤ψr<π/2, 0≤Δr<2π, 0≤c<π, -π/2<a≤+π/2, Tc>0, 0≤Δc<2π, 0≤ψ≤π/2, 0≤Δ<2π.
  15. By "calibrated" we simply mean that Tc exp (jΔc) is determined at the particular wavelength of interest.
  16. For example, a Glan-Thompson prism.
  17. Actually, Eq. (20) gives four solutions for p; however, all physically distinct positions of the PP axis in Fig. 2(a) can be obtained by limiting p to the range -π/2<p≤ +π/2 Equation (20) was derived by setting the imaginary part of ρτ equal to zero and solving for tanp.
  18. An interesting consequence of Eq. (20) is that when light is passed through a tandem arrangement of polarizer, wave-plate, wave-plate, it is always possible to rotate the polarizer to two physically distinct settings for which linearly polarized light will emerge from the last wave-plate. This is true regardless of the respective retardations of the two plates and regardless of the relative orientation (the angle c) of their principal axes.
  19. R. R. Alfano and W. H. Woodruff, App1. Optics 5, 352 (1966), have discussed calibration of a wave plate by ellipsometry; however, they ignored the possibility that. Tc can differ from unity, as established experimentally by H. Weinberger and J. Harris, J. Opt. Soc. Am, 54, 552 (1964). When using monochromatic, well-collimated light, a complete calibration of the optical compensator requires determination of both Δc and Tc.
  20. We are thus able to compute two "polarization-state transfer functions," Tc exp(jΔc) and tanψ exp(jΔ), from the measured angles c, a1, p1, a2, and P2. A fringe benefit from our work is that we could set the ellipsometer arms in the straight-through position, replace the reflecting specimen with a transmitting birefringent plate whose principal axes are the sp axes, perform the measurements c, a, p1, a2, P2 and then we could calibrate both plates.
  21. For a film-covered surface, Δ can lie anywhere in the interval 0≤Δ≤2π. See Ref. 13.
  22. A. B. Winterbottom on p. 102 of the Symposium Proceedings described in Ref. 9.
  23. F. P. Mertens and R. C. Plumb, J. Opt. Soc. Am. 54, 1063 (1964).
  24. Theoretical calculations were presented in Ref. 8 which showed that an isotropic lossless plate could be used as a rotary compensator in the infrared. The present work indicates that an isotropic absorbing plate could also be used as a rotary compensator, provided, of course, that sufficient light is transmitted to permit meaningful extinction settings.
  25. A. L. Bloom, Appl. Opt. 5, 1500 (1966).
  26. T. J. Bridges and J. W. Klüver, Appl. Opt. 4, 1121 (1965).

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