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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 70, Iss. 7 — Jul. 1, 1980
  • pp: 880–883

Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations

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

JOSA, Vol. 70, Issue 7, pp. 880-883 (1980)

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The complex ordinary (No) and extraordinary (Ne) refractive indices of an absorbing uniaxial crystal can be determined using reflection ellipsometry. The measurements are taken with the optic axis parallel and perpendicular to the crystal’s surface. The equations obtained are solved without resort to iterative methods; No and Ne are determined separately. Sixteen solution sets (No, Ne) are obtained and the correct solution can be easily identified. We present an optimum angle of incidence that minimizes the relative errors in No and Ne.

© 1980 Optical Society of America

M. Elshazly-Zaghloul and R. M. A. Azzam, "Inversion of the nonlinear equations of reflection ellipsometry for uniaxial crystals in symmetrical orientations," J. Opt. Soc. Am. 70, 880-883 (1980)

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  1. A computer program based on the Automated Taylor Series method of Dr. Chang of the Computer Department of the University of Nebraska-Lincoln was used. It yields highly accurate results in a very short time (better than 12 significant figures). The program is in FORTRAN IV and can be obtained by writing to Dr. Chang or to the authors. The closeness of No and Ne should cause no problem because they are separated in two polynomials.
  2. The suggested inversion method is superior to the previously used numerical methods in several aspects. First, it does not require previous knowledge of the optical properties to be measured. Of course this is essential whenever we have an unknown crystal to be characterized. Second, its accuracy does not depend on the starting values of No and Ne (it needs no starting values). Third, it separates the determination of No and Ne.
  3. R. A. W. Graves, "Determination of the optical constants of anisotropic crystals," J. Opt. Soc. Am. 59, 1225–1228 (1969).
  4. A. Wünsche, "Neue formeln für die reflexion und brechung des lichtes an anisotropen medien," Ann. Phys. (Leipzig) 25, 201–214 (1970).
  5. D. den Engelsen, "Transmission ellipsometry and polarization spectrometry of thin layers," J. Phys. Chem. 76, 3390–3397 (1972).
  6. M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, "Explicit solution for the optical properties of a uniaxial crystal in generalized ellipsometry," in Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1977), pp. 281–292.
  7. F. Meyer, E. E. de Kluizenaar, and D. den Engelsen, "Ellipsometric determination of the optical anisotropy of gallium selenide," J. Opt. Soc. Am. 63, 529–552 (1973).
  8. J. M. Bennett and H. E. Bennett, "Polarization," in Handbook of Optics, edited by W. G. Driscoll and W. Vaughan (McGraw-Hill, New York, 1978), pp. 10-1 to 10-164.
  9. This range of ø for minimum percent error of No and Ne (caused by 0.1° error of ø) is the same as we have previously found6 in the case of generalized ellipsometry applied to uniaxial crystals with the optic axis in the plane of the surface but is neither parallel nor perpendicular to the plane of incidence.

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