Abstract

We have studied the grazing-incidence differential-reflectance method for obtaining the dielectric function of absorbing media in terms of the derivatives Rp and Rs of the polarized light reflectances and found that it does not guarantee adequate accuracy for almost any values of the optical parameters. Therefore we modify that approach and describe what we believe is a novel method for the unambiguous determination of the optical constants n and k of a metal and other absorbing materials in terms of the ratio of the derivatives α=Rp/Rs at the grazing incidence and the normal incidence reflection coefficient R. Moreover, it is possible to express α through the logarithmic derivatives (1/R)R in the vicinity of the grazing angle. The possibility of performing measurements at the unspecified angle without knowledge of the explicit value of this angle is an evident advantage of this technique. For the great majority of metals and semiconductors the relative errors in the optical constants are comparable to or less than the relative errors in the experimentally measured parameters.

© 1999 Optical Society of America

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References

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  1. J. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Secs. 42.1–42.130.
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. M. R. Querry, “Direct solution of the generalized Fresnel reflection equations,” J. Opt. Soc. Am. 59, 876–877 (1969).
    [CrossRef]
  6. R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
    [CrossRef]
  7. J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 10.
  8. R. F. Potter, “Analytical determination of optical constants based on the polarized reflectance at a dielectric–conductor interface,” J. Opt. Soc. Am. 54, 904–908 (1964); “Pseudo-Brewster angle technique for determining optical constants,” in Optical Properties of Solids, S. Nudelman, S. S. Mitra, eds. (Plenum, New York, 1969), Chap. 16, pp. 489–513.
    [CrossRef]
  9. T. E. Darcie, M. S. Whalen, “Determination of optical constants using pseudo-Brewster angle and normal incidence reflectance measurements,” Appl. Opt. 23, 1130–1131 (1984).
    [CrossRef] [PubMed]
  10. P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Ambiguities in determining the optical constants for two reflection methods,” Appl. Opt. 35, 117–119 (1996).
    [CrossRef]
  11. D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
    [CrossRef]
  12. K. Lamprecht, W. Papousek, G. Leising, “Problem of ambiguity in the determination of optical constants of thin absorbing films from spectroscopic reflectance and transmittance measurements,” Appl. Opt. 36, 6364–6371 (1997).
    [CrossRef]
  13. J. Lekner, “Inversion of the s and p reflectances of absorbing media,” J. Opt. Soc. Am. A 14, 1355–1358 (1997).
    [CrossRef]
  14. P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
    [CrossRef]
  15. F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1968), Vol. II, pp. 251–288.
  16. 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]
  17. P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
    [CrossRef]
  18. G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. Durrant, V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65–78 (1998).
    [CrossRef]
  19. O. Hunderi, “New method for accurate determination of optical constants,” Appl. Opt. 11, 1572–1578 (1972).
    [CrossRef] [PubMed]
  20. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

1998

1997

1996

D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Ambiguities in determining the optical constants for two reflection methods,” Appl. Opt. 35, 117–119 (1996).
[CrossRef]

1993

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

1984

1983

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]

1982

1972

1970

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

1969

1964

1961

S. P. F. 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]

Abelès, F.

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1968), Vol. II, pp. 251–288.

Apostol, D.

D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Ambiguities in determining the optical constants for two reflection methods,” Appl. Opt. 35, 117–119 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
[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, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bagnato, V. S.

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

Baranauskas, V.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Basmaji, P.

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

Bennett, H. E.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 10.

Bennett, J. M.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 10.

Damian, V.

D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Ambiguities in determining the optical constants for two reflection methods,” Appl. Opt. 35, 117–119 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
[CrossRef]

Darcie, T. E.

Dobroiu, A.

D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
[CrossRef]

Dobrowolski, J. A.

J. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Secs. 42.1–42.130.

Durrant, S.

Ghiner, A. V.

Griviskas, V.

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

Humphreys-Owen, S. P. F.

S. P. F. 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]

Hunderi, O.

Hunter, W. R.

Julien, L. S.

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Lamprecht, K.

Leising, G.

Lekner, J.

Logofatu, P. C.

D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Ambiguities in determining the optical constants for two reflection methods,” Appl. Opt. 35, 117–119 (1996).
[CrossRef]

Miller, R. F.

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985); Handbook of Optical Constants of Solids, II (Academic, Boston, Mass., 1991).

Papousek, W.

Potter, R. F.

Querry, M. R.

Surdutovich, G. I.

G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. Durrant, V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65–78 (1998).
[CrossRef]

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

Taylor, A. J.

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Tumbar, R.

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Ambiguities in determining the optical constants for two reflection methods,” Appl. Opt. 35, 117–119 (1996).
[CrossRef]

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
[CrossRef]

Vitlina, R. Z.

G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. Durrant, V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65–78 (1998).
[CrossRef]

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

Whalen, M. S.

Appl. Opt.

Infrared Phys. Technol.

P. C. Logofǎtu, D. Apostol, V. Damian, R. Tumbar, “Determination of optical constants of metals by near grazing incidence reflectivity measurements,” Infrared Phys. Technol. 37, 335–341 (1996).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D

R. F. Miller, A. J. Taylor, L. S. Julien, “The optimum angle of incidence for determining optical constants from reflectance measurements,” J. Phys. D 3, 1957–1961 (1970).
[CrossRef]

Opt. Eng. (Bellingham)

D. Apostol, P. C. Logofǎtu, V. Damian, A. Dobroiu, “Sensitivity analysis of parameter determination from measurements of directly measurable quantities using the Jacobian,” Opt. Eng. (Bellingham) 35, 1288–1291 (1996).
[CrossRef]

Proc. Phys. Soc. London

S. P. F. 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.

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]

Thin Solid Films

P. Basmaji, V. S. Bagnato, V. Griviskas, G. I. Surdutovich, R. Z. Vitlina, “Determination of porous silicon film parameters by polarized light reflectance measurement,” Thin Solid Films 233, 131–136 (1993).
[CrossRef]

Other

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1968), Vol. II, pp. 251–288.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), Chap. 10.

J. A. Dobrowolski, “Optical properties of films and coatings,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Secs. 42.1–42.130.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, San Diego, Calif., 1985); Handbook of Optical Constants of Solids, II (Academic, Boston, Mass., 1991).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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Figures (3)

Fig. 1
Fig. 1

(a) Distribution of ε loci in terms of the ratio α=Rp/Rs and the derivative Rs. Curves A (ε=0) and B (|ε|=1) outline the domain of physical significance, assuming that the incidence medium is a vacuum. The horizontally hatched region around the curve C (α-1=4/Rs2) corresponds to the condition that prefactor of dα/α in Eq. (7) is of the order of unity or less. A bunching effect takes place at α1, Rs1. The points represent some metals in the visible. (b) Distribution of ε loci in terms of α and Rs. Dashed curves C and D are given by Eq. (9). In the horizontally hatched regions, prefactors of dα/α and dRs/Rs in Eq. (8) are not greater than unity. A strong bunching occurs near a perfect dielectric boundary.

Fig. 2
Fig. 2

(a) Distribution of n loci in terms of α and R within the physical region delineated by curves A (k=0) and B (|ε|=1). There is no visible bunching in this case. The points represent some metals in the visible. (b) Distribution of k loci within the physical region [the same as in (a)]. The bunching effect is noticeable near the dielectric boundary at k1. The dashed curve α-1=[4R(1+R)]/(1-R)2 corresponds to the condition (k/α)R=0.

Fig. 3
Fig. 3

Distribution of n and k loci in the (α, R) plane. The sizes of the parallelopipeds show the value of the Jacobian [Eq. (13)] at a given point.

Tables (2)

Tables Icon

Table 1 Accuracy of the Determination of α Expressed As the Ratio of the Logarithmic Derivatives from Eq. (18) at θ=89°

Tables Icon

Table 2 Errors in ε and ε in Percent Introduced by a 1% Error in Parameters R, α, Rp(θ), θB

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

Rp=ε cos θ-ε-sin2 θε cos θ+ε-sin2 θ2,
Rs=cos θ-ε-sin2 θcos θ+ε-sin2 θ2,
Rp=2εε-1+ε*ε*-1
Rs=21ε-1+1ε*-1,
α=Rp/Rs=1+|ε-1|.
ε=Rs2 (α-1)28-α+2,
ε=(α-1)2Rs816α-1-Rs21/2.
(A)ε=0,(α-1=16/Rs2)
(B)|ε|=1,α-1=21+Rs2/4.
J(ε, ε)=2ε(ε-1)2+ε2[ε-1+(ε-1)2+4ε2]-1/2>0.
dεε=-α1-(α-1)Rs24(α-1)2Rs28-α+2dαα+(α-1)2Rs24(α-1)2Rs28-α+2dRsRs,
dεε=3α1-(α-1)Rs2122(α-1)1-(α-1)Rs216dαα+1-(α-1)Rs281-(α-1)Rs216dRsRs.
(C)α-1=12Rs2
(D)α-1=8Rs2,
n2=(α-1)2(1-R)216R,
k2=(α-1)R-(α-1)(1-R)4R+12,
(A)k=0,α-1=4R(1-R)2,
(B)|ε|=1,α-1=4R1+R.
J(n, k)=Rknαnk-Rnkαkn=8k(n+1)2+k2(n-1)2+k2(n+1)2+k21/2>0
dnn=αα-1dαα-1+R2(1-R)dRR.
dkk=α(1+R)4k2R1-(α-1)(1-R)24R(1+R) dαα+(α-1)(1-R)8k2R(α-1)(1+R)4R-1 dRR.
Rp,s=Rp,s2
α(θ)=Rp(π/2)-ΔθRp(π/2)Rs(π/2)-ΔθRs(π/2)=Rp(π/2)[1-ΔθRp(π/2)]Rs(π/2)[1-ΔθRs(π/2)]=α 1-[1-Rp(θ)]1-[1-Rs(θ)]=α Rp(θ)Rs(θ).
α=Rs(θ)Rp(θ)α(θ)=Rp/RpRs/Rsθ=(ln Rp)(ln Rs)θ
k2=4n+(n4+14n2+1)1/2
k2=(n2-1)(2-n)n+2,
α=RP(π/2)RS(π/2)
α(exp)=RP/RPRS/RSθ=89°
Δαα=α(exp)-αα

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