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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 62, Iss. 5 — May. 1, 1972
  • pp: 700–701

Choice of Compensator Azimuth and Position in Ellipsometry

R. M. A. AZZAM and N. M. BASHARA  »View Author Affiliations

JOSA, Vol. 62, Issue 5, pp. 700-701 (1972)

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R. M. A. AZZAM and N. M. BASHARA, "Choice of Compensator Azimuth and Position in Ellipsometry," J. Opt. Soc. Am. 62, 700-701 (1972)

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  1. E. Schmidt, J. Opt. Soc. Am. 60, 490 (1970).
  2. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 600 (1971).
  3. An analytical solution can be developed using the two relations, tanψ sinΔ =tanA tan P* sec2C/(l +tan2C tan2P*) and sinΔ = -tan2P*/sin2C, which are obtained from Eq. (3) (after equating the real and imaginary parts of both sides) and represent another form of the constraint. The above relations together with the explicit expression of the coupling coefficient for a particular imperfection lead to the same conclusions as derived numerically.
  4. In Ref. 2 the coefficient of tan2P* in the numerator of the coupling coefficient for a compensator azimuth-angle error (denoted by γc′) should be ρC2 instead of -1.
  5. The Jones matrix of an imperfect component can be written as Tk = Tk0Tk, where Tk0 describes the ideal behavior of the kth element and δTk is its 2 × 2 imperfection matrix. The effect of δTk is determined by a 2 × 2 array of coupling coefficients of the form [Equations] such that, δρ=Σij γijδTkij gives the error in the specimen reflectance ratio ρ due to δTdk. For the compensator, the effect of δTC can be determined using the imperfection-plate representation of Ref. 7. This yields γ11=-ρcγ22=ρc tanP* sec2C/(1-ρc tanC tanP*)2, γ12=γ11 tanP*, and γ21=-γ11 cotP*/ρc.
  6. D. E. Aspnes, J. Opt. Soc. Am. 61, 1077 (1971).
  7. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 61, 1236 (1971).
  8. R. J. Archer, Manual on Ellipsometry (Gaertner Scientific Corporation, Chicago, III., 1968), p. 4, states that "From the principle of reciprocity, extinction is maintained if the detector and the light source are interchanged." It can be shown that the above statement applies only if the system between the polarizer and the analyzer (in this case, the CWSW′ sequence of elements) is characterized by a unitary Jones matrix with orthogonal eigenpolarizations. This condition is not satisfied simply when ψ ≠45°, i.e., almost for any surface under measurement.
  9. The first partial derivative of this function with respect to any one of its arguments gives the coupling coefficient for the imperfection that corresponds to the deviation of the value of this argument from an ideal value. This includes the coupling coefficients for both azimuth-angle errors and component imperfections. The explicit form of this function when the properties of all of the optical elements are included is very complicated. Special cases can he found in Eqs. (12) and (25) of Ref. 2.
  10. This means that the definition of δρp, as "the deviation of the complex relative transmittance of the polarizer from zero" in Ref. 2 is only a simplification. A more-general analysis using Mueller calculus that accounts for the properties of the source beam-polarizer combination can be found in the authors paper [J. Opt. Soc. Am. 61, 1380 (1971)].

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