Abstract

We present a unified two-step analytical inversion of reflectometric and ellipsometric data of absorbing media. Instead of a direct determination of the optical constants n, κ from reflectometric or ellipsometric measurements, we first calculate the real and the imaginary part η, γ of the normal component of the wave vector in the absorbing medium. New and simple analytical formulas are obtained for η and γ in terms of ρs, ρp or tan ψ, δ, respectively, where ρs, ρp and tan ψ, δ are the measured reflectometric and ellipsometric parameters of the absorbing medium. The optical constants are then easily determined analytically again from η, γ. We use the new formulas to compare the sensitivity with experimental errors due to the inversion of reflectometric and ellipsometric data.

© 2000 Optical Society of America

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References

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  1. L. Ward, The Optical Constants of Bulk Materials and Films (Hilger, Bristol, UK, (1988), Secs. 2.6 and 2.8.
  2. J. Lekner, “Inversion of the s and p reflectances of absorbing media,” J. Opt. Soc. Am. A 14, 1355–1358 (1997).
    [CrossRef]
  3. R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
    [CrossRef]
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  5. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).
  6. S. P. M. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
    [CrossRef]
  7. D. W. Berreman, “Simple relation between reflectances of polarized components of a beam when the angle of incidence is 45 degrees,” J. Opt. Soc. Am. 56, 1784 (1966).
    [CrossRef]
  8. R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta. 26, 113–115 (1979).
    [CrossRef]
  9. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

1997 (1)

1983 (1)

R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
[CrossRef]

1979 (1)

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta. 26, 113–115 (1979).
[CrossRef]

1966 (1)

1961 (1)

S. P. M. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
[CrossRef]

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta. 26, 113–115 (1979).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Berreman, D. W.

Humphreys-Owen, S. P. M.

S. P. M. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Lekner, J.

J. Lekner, “Inversion of the s and p reflectances of absorbing media,” J. Opt. Soc. Am. A 14, 1355–1358 (1997).
[CrossRef]

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).

Ward, L.

L. Ward, The Optical Constants of Bulk Materials and Films (Hilger, Bristol, UK, (1988), Secs. 2.6 and 2.8.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta. (1)

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta. 26, 113–115 (1979).
[CrossRef]

Proc. Phys. Soc. London (1)

S. P. M. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Rev. Sci. Instrum. (1)

R. M. A. Azzam, “Grazing-incidence differential-reflectance method for explicit determination of the complex dielectric function of an isotropic absorbing medium,” Rev. Sci. Instrum. 54, 853–855 (1983).
[CrossRef]

Other (4)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987).

L. Ward, The Optical Constants of Bulk Materials and Films (Hilger, Bristol, UK, (1988), Secs. 2.6 and 2.8.

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985).

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Figures (9)

Fig. 1
Fig. 1

Light incident on an absorbing medium.

Fig. 2
Fig. 2

Dependence of the error of the calculated refractive index on the error of the experimental reflectances ρ s , ρ p for Cu.

Fig. 3
Fig. 3

Dependence of the error of the calculated extinction coefficient on the error of the experimental reflectances ρ s , ρ p for Cu.

Fig. 4
Fig. 4

Dependence of the error of the calculated refractive index on the error of the experimental amplitude ratio tan ψ and phase angle δ for Cu.

Fig. 5
Fig. 5

Dependence of the error of the calculated extinction coefficient on the error of the experimental amplitude ratio tan ψ and phase angle δ for Cu.

Fig. 6
Fig. 6

Dependence of the error of the calculated refractive index on the error of the experimental reflectances ρ s , ρ p for Si.

Fig. 7
Fig. 7

Dependence of the error of the calculated extinction coefficient on the error of the experimental reflectances ρ s , ρ p for Si.

Fig. 8
Fig. 8

Dependence of the error of the calculated refractive index on the error of the experimental amplitude ratio tan ψ and phase angle δ for Si.

Fig. 9
Fig. 9

Dependence of the error of the calculated extinction coefficient on the error of the experimental amplitude ratio tan ψ and phase angle δ for Si.

Tables (2)

Tables Icon

Table 1 Optical Constants9 and Their Ratio, Reflectances ρ p , ρ s , and Ratio and Ellipsometric Parameters tan ψ and δ for Cu and Si at a Wavelength of 633 nm

Tables Icon

Table 2 Maximum Error and Average Error in Percent of the Calculated Optical Constants in the Error Interval ±0.2% of the Measured Reflectometric r and Ellipsometric e Quantities

Equations (24)

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ρs=rsrs*,  ρp=rprp*,
tan ψ=|rp||rs|=ρpρs1/2,  δ=argrp-argrs,
rs=kz0-kzkz0+kz,  rp=N2·kz0-kzN2 · kz0+kz,
kz0=-k02-kx021/2,  kz=-k2-kx021/2
k0=ωc,  k=ωc N,
N=n-i·κ,
kx0=k0 sin α0.
kz0=-k0 cos α0,  kz=-k0N2-sin2 α01/2.
rs=cos α0-wcos α0+w,  rp=N2 cos α0-wN2 cos α0+w,
w=N2-sin2 α01/2.
rprs=sin α0·tan α0-wsin α0·tan α0+w.
w=η-i·γ
ρs=cos α0-η2+γ2cos α0+η2+γ2,
ρpρs=sin α0·tan α0-η2+γ2sin α0·tan α0+η2+γ2.
η=12 cos α0cos2 α0-sin2 α0q·cos2 α0-p·sin2 α0
γ=2qη·cos α0-η2-cos2 α01/2
q=1+ρs1-ρs,  p=1+ρp/ρs1-ρp/ρs.
1-rp/rs1+rp/rs=wsin α0·tan α0.
η=sin2 α0cos α01-tan2 ψ1+tan2 ψ+2 cos δ·tan ψ,
γ=2 sin δ·tan ψ1-tan2 ψ η.
n=b+b2+c21/21/2,
κ=cb+b2+c21/2-1/2
b=12η2-γ2+sin2 α0,  c=ηγ.
n-nPaliknPalik 100,  κ-κPalikκPalik 100

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