Abstract

A method for determining the optical properties of a film on an isotropic substrate is proposed. The method is based on the existence of two specific incidence angles in the angular interference pattern of the p-polarized light where oscillations of the reflection coefficient cease. The first of these angles, θB1, is the well-known Abelès angle, i.e., the ambient–film Brewster angle, and the second angle θB2 is the film–substrate Brewster angle. In the conventional planar geometry and in a vacuum ambient there is a rigorous constraint ε1 + ε > ε1ε on the film and the substrate dielectric permittivities ε1 and ε, respectively, for the existence of the second angle θB2. The limitation may be removed in an experiment by use of a cylindrical lens as an ambient with ε0 > 1, so that both angles become observable. This, contrary to general belief, allows one to adopt the conventional Abelès method not only for films with ε1 close to the substrate’s value ε but also for any value of ε1. The method, when applied to a wedge-shaped film or to any film of unknown variable thickness, permits one to determine (i) the refractive index of a film on an unknown substrate, (ii) the vertical and the horizontal optical anisotropies of a film on an isotropic substrate, (iii) the weak absorption of a moderately thick film on a transparent or an absorbing isotropic substrate.

© 1999 Optical Society of America

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References

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  1. F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1968), Vol. 2, pp. 251–289.
  2. R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
    [CrossRef]
  3. T. Pisarkiewicz, “Reflection spectrum for a thin film with non-uniform thickness,” J. Phys. D 27, 160–164 (1994).
    [CrossRef]
  4. R. M. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, New York, 1977).
  5. M. Hacskaylo, “Determination of the refractive index of thin dielectric films,” J. Opt. Soc. Am. 54, 198–203 (1964).
    [CrossRef]
  6. J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), pp. 10–17.
  7. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).
  8. I. Hodgkinson, Q. H. Wu, C. Rawle, “Common-index thin film polarizers for light at normal incidence,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 173–175.
  9. I. Hodgkinson, J. Hazel, Q. H. Wu, “In situ measurement of principal refractive indices of thin films by two-angle ellipsometry,” Thin Solid Films 313–314, 368–372 (1998).
    [CrossRef]
  10. I. Hodgkinson, Q. H. Wu, J. Hazel, “Empirical equations for the principal refractive indices and column angle of obliquely deposited films of tantalum oxide, titanium oxide, and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998); I. Hodgkinson, Q. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), Chap. 8, pp. 147–149.
    [CrossRef]
  11. G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
    [CrossRef]
  12. G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. F. Durrant, V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65–78 (1998).
    [CrossRef]
  13. O. S. Heavens, H. M. Liddell, “Influence of absorption on measurement of refractive index of films,” Appl. Opt. 4, 629–630 (1965).
    [CrossRef]
  14. H. A. Macleod, “Monitoring of optical coatings,” Appl. Opt. 20, 82–89 (1981).
    [CrossRef] [PubMed]
  15. Rusli, G. A. J. Amaratunga, “Determination of the optical constants and thickness of thin films on slightly absorbing substrates,” Appl. Opt. 34, 7914–7924 (1995).
  16. 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]
  17. C. K. Carniglia, “Ellipsometric calculations for nonabsorbing thin films with linear refractive-index gradients,” J. Opt. Soc. Am. A 7, 848–856 (1990).
    [CrossRef]
  18. G. Parjadis de Lariviere, J. M. Frigerio, J. Rivory, F. Abelès, “Estimate of the degree of inhomogeneity of the refractive index of dielectric films from spectroscopic ellipsometry,” Appl. Opt. 31, 6056–6061 (1992).
    [CrossRef]

1998 (3)

1997 (1)

1996 (1)

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

1995 (1)

1994 (1)

T. Pisarkiewicz, “Reflection spectrum for a thin film with non-uniform thickness,” J. Phys. D 27, 160–164 (1994).
[CrossRef]

1992 (1)

1990 (1)

1984 (1)

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

1981 (1)

1965 (1)

1964 (1)

Abelès, F.

G. Parjadis de Lariviere, J. M. Frigerio, J. Rivory, F. Abelès, “Estimate of the degree of inhomogeneity of the refractive index of dielectric films from spectroscopic ellipsometry,” Appl. Opt. 31, 6056–6061 (1992).
[CrossRef]

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1968), Vol. 2, pp. 251–289.

Amaratunga, G. A. J.

Azzam, R. M.

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

Baranauskas, V.

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

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Bashara, N. M.

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

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), pp. 10–17.

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), pp. 10–17.

Carniglia, C. K.

Durrant, S. F.

Fragalli, J. F.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Frigerio, J. M.

Ghiner, A. V.

Hacskaylo, M.

Hazel, J.

Heavens, O. S.

Hodgkinson, I.

I. Hodgkinson, Q. H. Wu, J. Hazel, “Empirical equations for the principal refractive indices and column angle of obliquely deposited films of tantalum oxide, titanium oxide, and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998); I. Hodgkinson, Q. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), Chap. 8, pp. 147–149.
[CrossRef]

I. Hodgkinson, J. Hazel, Q. H. Wu, “In situ measurement of principal refractive indices of thin films by two-angle ellipsometry,” Thin Solid Films 313–314, 368–372 (1998).
[CrossRef]

I. Hodgkinson, Q. H. Wu, C. Rawle, “Common-index thin film polarizers for light at normal incidence,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 173–175.

Kolenda, J.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Lamprecht, K.

Leising, G.

Liddell, H. M.

Macleod, H. A.

Misoguti, L.

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Papousek, W.

Parjadis de Lariviere, G.

Pisarkiewicz, T.

T. Pisarkiewicz, “Reflection spectrum for a thin film with non-uniform thickness,” J. Phys. D 27, 160–164 (1994).
[CrossRef]

Rawle, C.

I. Hodgkinson, Q. H. Wu, C. Rawle, “Common-index thin film polarizers for light at normal incidence,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 173–175.

Rivory, J.

Rusli,

Surdutovich, G. I.

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

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Swanepoel, R.

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Vitlina, R. Z.

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

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Wu, Q. H.

I. Hodgkinson, Q. H. Wu, J. Hazel, “Empirical equations for the principal refractive indices and column angle of obliquely deposited films of tantalum oxide, titanium oxide, and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998); I. Hodgkinson, Q. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), Chap. 8, pp. 147–149.
[CrossRef]

I. Hodgkinson, J. Hazel, Q. H. Wu, “In situ measurement of principal refractive indices of thin films by two-angle ellipsometry,” Thin Solid Films 313–314, 368–372 (1998).
[CrossRef]

I. Hodgkinson, Q. H. Wu, C. Rawle, “Common-index thin film polarizers for light at normal incidence,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 173–175.

Appl. Opt. (7)

J. Opt. Soc. Am. (1)

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

J. Phys. D (1)

T. Pisarkiewicz, “Reflection spectrum for a thin film with non-uniform thickness,” J. Phys. D 27, 160–164 (1994).
[CrossRef]

J. Phys. E (1)

R. Swanepoel, “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Thin Solid Films (2)

I. Hodgkinson, J. Hazel, Q. H. Wu, “In situ measurement of principal refractive indices of thin films by two-angle ellipsometry,” Thin Solid Films 313–314, 368–372 (1998).
[CrossRef]

G. I. Surdutovich, J. Kolenda, J. F. Fragalli, L. Misoguti, R. Z. Vitlina, V. Baranauskas, “An interference method for the determination of thin film anisotropy,” Thin Solid Films 279, 119–123 (1996).
[CrossRef]

Other (5)

F. Abelès, “Methods for determining optical parameters of thin films,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1968), Vol. 2, pp. 251–289.

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

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), pp. 10–17.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

I. Hodgkinson, Q. H. Wu, C. Rawle, “Common-index thin film polarizers for light at normal incidence,” in Optical Interference Coatings, Vol. 9 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 173–175.

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Figures (7)

Fig. 1
Fig. 1

Diagram showing the SBA regions as a function of ε and ε1 for the different values ε0. Curve E (ε1 = Eε, where E = 1 + 2) corresponds to the maximum possible difference 9.88° between the angles θ B1 and θ B2 B1 = 49.94°, θ B2 = 40.06°) for any values of ε and ε1, with the optimum value ε0 = ε0max = (E/E - 1) = 1.707, thus implying ε0 < ε. For ε0 > ε the maximum divergence of the SBA is reached when ε → ∞ and ε1 → 1, so that θ B1 → (π/4) and θ B2 → (π/2).

Fig. 2
Fig. 2

One-SBA interference patterns of a wedge-shaped film (ε1 = 2.25) on a substrate (ε = 16) for four different values of parameter d/λ: (a) 5, (b) 5.125, (c) 5.25, (d) 5.375. Expression (6a) has the solution θ B1 = 56.31°. Curve s corresponds to the ambient–substrate reflection, and for ε > ε1 it is the bending of all the pattern curves from above for θ < θ B1 and from below for θ > θ B1.

Fig. 3
Fig. 3

(a) Patterns similar to those in Fig. 2 for the film with ε1 = 1.44 and substrate ε = 2.25. Now both expressions (6) have solutions θ B1 = 50.19° and θ B2 = 69.56°. Unfortunately, under large values of angle θ B2 intersection of all the curves takes place at acute angles so that the measurement of this angle becomes difficult. By use of a cylindrical lens [Fig. 4(b)] with ε0 > 1 one can shift angle θ B2 to the left-hand side. Curves ad correspond to the same thicknesses as in Fig. 2. (b) Region of the SBA is shown on a larger scale.

Fig. 4
Fig. 4

Possible geometries of the experiment: (a) convential setup with ambient ε0 = 1, (b) setup with a cylindrical lens ε0 > 1.

Fig. 5
Fig. 5

Domains of existence of the SBA θ B1 and θ B2 in the (ε1, ε0) plane for a given substrate ε = 3.

Fig. 6
Fig. 6

(a) Blurring of the intersection point θ B1 = 56.31° (the same as in Fig. 2) in the case of a weakly absorbing film ñ1 = 1.5 - i0.002 on a transparent substrate ε = 16. Curves ad correspond to the same thicknesses as in Fig. 2. (b) Vicinity of the intercrossing point shown for transparent (ñ1 = 1.5, upper bunch of curves) and absorbing (ñ1 = 1.5 - i0.002, lower group of curves) films on the same transparent substrate ε = 16.

Fig. 7
Fig. 7

Vicinity of the SBA θ B1 for an absorbing film ñ1 = 1.5 - i0.001 deposited on transparent (ñ = 4, lower bunch of curves) and opaque (ñ = 4 - i4, upper bunch of the curves) substrates. In each case the position of the intercrossing point for a transparent film ñ1 = 1.5 is shown.

Equations (30)

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Rθ=r01+r12 exp-iδ1+r01r12 exp-iδ2,
r01θB1=0,
r12θB2=0,
RθB1=|r12θB1|2,
RθB2=|r01θB2|2.
RθB1=|r02θB1|2,
RθB2=|r02θB2|2.
r01θ=ε1 cos θ-ε0ε1-ε0 sin2 θ1/2ε1 cos θ+ε0ε1-ε0 sin2 θ1/2,
r12θ=εε1-ε0 sin2 θ1/2-ε1ε-ε0 sin2 θ1/2εε1-ε0 sin2 θ1/2+ε1ε-ε0 sin2 θ1/2,
sin2 θB1=ε1ε0+ε11,
sin2 θB2=ε1εε0ε1+ε1.
ε0min=εε1ε+ε1,  ε0max=εε1ε1-ε.
ε1=ε0 tan2 θB1,  ε=ε0 sin2 θB21-sin2 θB2 cotan2 θB1.
ε0optε0min+ε0max2εε12ε12-ε2.
δn1n1=2sin 2θB1 δθB1,  δnn=-cotan θB1sin2 θB11-sin2 θB2 cotan2 θB1 δθB1+cotan θB21-sin2 θB2 cotan2 θB1 δθB2.
sin2 θB1=εzzεxx-ε0εxxεzz-ε02,  sin2 θB2=εεzzε0εxx-εεxxεzz-ε2.
βzx=4ε+ε0ε2θB1-π4sin2 θB2-sin2 θB0.
βyx=ε0+εε2-ε02sin2 θB2x-sin2 θB2yε0ε2ε0+εε2 sin2 θB2x-ε0ε,
βxy=εxx-εyyεyy=ε0-εε1-ε02ε1ε0 δRxy,  δRxy=RθB2x-RθB2yRθB2y.
δ=4π dλε1-sin2 θ-iε11/24π dλε1-sin2 θ1/2-2π dλiε1ε1-sin2 θ1/2.
RθB1=|r02θB1|2 exp-4π dλε1ε1-sin2 θB11/2=|r02θB1|2 exp-8π dλn1k1n12-sin2 θB11/2,
ε1d=-λ4πε1ε1+11/2 ln q,  k1d=-λ8πn1n12+11/2 ln q, or
q=RθB1|r02θB1|2.
γ1=dRθ, δ=0dθθB1-dRθ,  δ=πdθθB1,  γ2=dRθ, δ=0dθθB2-dRθ, δ=πdθθB2,
γ1=8εε-ε0ε-ε1ε12-ε02εε1+ε0ε+ε11/2ε13/2ε01/2ε+εε1+ε0ε-ε01/24,
γ2=8ε0ε-ε0ε1-ε0ε2-ε12εε1-ε0ε+ε1ε3/2ε13/2ε0+ε0ε+ε1-εε11/24.
ε1ε,  γ1=ε0ε5/2ε-ε12ε1,  γ2=32 ε-ε1ε1,
ε0ε,  γ1γ2=12ε1ε3/2ε-ε0ε1,
ε0ε1,  γ10  for ε0εmax,  γ2=12ε1ε1/2ε1-ε0ε1.
εxx=ε2ε0sin2 θB2-sin2 θB1ε2 sin2 θB2 cos2 θB1-ε0 sin2 θB1ε-ε0 sin2 θB2,  εzz=ε0ε2 cos2 θB1 sin2 θB2-ε0 sin2 θB1ε-ε0 sin2 θB2ε2 cos2 θB1-εε0+ε02 sin2 θB2.

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