Abstract

For what is the first time, to our knowledge, we report on the extension of spectroscopic rotating-analyzer ellipsometry to generalized ellipsometry to define and to determine three essentially normalized elements of the optical Jones matrix J [ R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1521 ( 1972)]. These elements are measured in reflection over the spectral range of 3.5–4.5 eV on different surface orientations of uniaxial TiO2 cut from the same bulk crystal. With a wavelength-by-wavelength regression and a 4 × 4 generalized matrix algebra, both refractive and absorption indices for the ordinary and the extraordinary waves, no, ko, ne, and ke, are determined. The inclinations and the azimuths of the optic axes with respect to the sample normal and plane of incidence were determined as well. The latter are confirmed by x-ray diffraction and polarization microscopy. Hence the spectrally dependent dielectric function tensor in laboratory coordinates is obtained. Very good agreement between measured and calculated data for the normalized Jones elements for the respective sample orientations and positions are presented. This technique may become an important tool for investigating layered systems with nonscalar dielectric susceptibilities.

© 1996 Optical Society of America

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References

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  1. R. M. A. Azzam, N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).
    [CrossRef]
  2. See, e.g.: R. M. A. Azzam, N. M. Bashara, “Polarization transfer function of a biaxial system as a bilinear transformation,” J. Opt. Soc. Am. 62, 222–229 (1972).
    [CrossRef]
  3. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  4. Depending on what causes the optical anisotropy; see, e.g., H. Wöhler, M. Fritsch, G. Haas, D. A. Mlynski, “Characteristic matrix method for stratified anisotropic media: optical properties of special configurations,” J. Opt. Soc. Am. A 8, 536–540 (1991); K. Eidner, “Light propagation in stratified anisotropic media: orthogonality and symmetry properties of the 4 × 4 matrix formalism,” J. Opt. Soc. Am. A 6, 1657–1660 (1989); Ref. 5.
    [CrossRef]
  5. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  6. M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53(7) (1996).
    [CrossRef]
  7. P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976); see also D. J. De Smet, “Generalized ellipsometry and the 4 × 4 matrix formalism,” Surf. Sci. 56, 293–306 (1976); M. Elshazly-Zaghloul, R. M. A. Azzam, N. M. Bashara, “Explicit solutions for the optical properties of a uniaxial crystal in generalized ellipsometry,” Surf. Sci. 56, 281–292 (1976).
    [CrossRef]
  8. P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
    [CrossRef]
  9. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 798–804.
  10. K. Vos, H. J. Krusemayer, “Reflectance and electroreflectance of TiO2single crystals: I. Optical spectra,” Phys. C 10, 3893–3915 (1977).
    [CrossRef]
  11. M. Cardona, G. Harbeke, “Optical properties and band structure of wurtzite-type crystals and rutile,” Phys. Rev. A 137, 1467–1476 (1965).

1996 (1)

M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53(7) (1996).
[CrossRef]

1991 (1)

1980 (1)

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

1977 (1)

K. Vos, H. J. Krusemayer, “Reflectance and electroreflectance of TiO2single crystals: I. Optical spectra,” Phys. C 10, 3893–3915 (1977).
[CrossRef]

1976 (1)

P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976); see also D. J. De Smet, “Generalized ellipsometry and the 4 × 4 matrix formalism,” Surf. Sci. 56, 293–306 (1976); M. Elshazly-Zaghloul, R. M. A. Azzam, N. M. Bashara, “Explicit solutions for the optical properties of a uniaxial crystal in generalized ellipsometry,” Surf. Sci. 56, 281–292 (1976).
[CrossRef]

1972 (3)

1965 (1)

M. Cardona, G. Harbeke, “Optical properties and band structure of wurtzite-type crystals and rutile,” Phys. Rev. A 137, 1467–1476 (1965).

Azzam, R. M. A.

Bashara, N. M.

Berreman, D. W.

Cardona, M.

M. Cardona, G. Harbeke, “Optical properties and band structure of wurtzite-type crystals and rutile,” Phys. Rev. A 137, 1467–1476 (1965).

Fritsch, M.

Haas, G.

Harbeke, G.

M. Cardona, G. Harbeke, “Optical properties and band structure of wurtzite-type crystals and rutile,” Phys. Rev. A 137, 1467–1476 (1965).

Hauge, P. S.

P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976); see also D. J. De Smet, “Generalized ellipsometry and the 4 × 4 matrix formalism,” Surf. Sci. 56, 293–306 (1976); M. Elshazly-Zaghloul, R. M. A. Azzam, N. M. Bashara, “Explicit solutions for the optical properties of a uniaxial crystal in generalized ellipsometry,” Surf. Sci. 56, 281–292 (1976).
[CrossRef]

Krusemayer, H. J.

K. Vos, H. J. Krusemayer, “Reflectance and electroreflectance of TiO2single crystals: I. Optical spectra,” Phys. C 10, 3893–3915 (1977).
[CrossRef]

Mlynski, D. A.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 798–804.

Schubert, M.

M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53(7) (1996).
[CrossRef]

Vos, K.

K. Vos, H. J. Krusemayer, “Reflectance and electroreflectance of TiO2single crystals: I. Optical spectra,” Phys. C 10, 3893–3915 (1977).
[CrossRef]

Wöhler, H.

Yeh, P.

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Phys. C (1)

K. Vos, H. J. Krusemayer, “Reflectance and electroreflectance of TiO2single crystals: I. Optical spectra,” Phys. C 10, 3893–3915 (1977).
[CrossRef]

Phys. Rev. A (1)

M. Cardona, G. Harbeke, “Optical properties and band structure of wurtzite-type crystals and rutile,” Phys. Rev. A 137, 1467–1476 (1965).

Phys. Rev. B (1)

M. Schubert, “Polarization dependent parameters of arbitrarily anisotropic homogeneous epitaxial systems,” Phys. Rev. B 53(7) (1996).
[CrossRef]

Surf. Sci. (2)

P. S. Hauge, “Generalized rotating-compensator ellipsometry,” Surf. Sci. 56, 148–160 (1976); see also D. J. De Smet, “Generalized ellipsometry and the 4 × 4 matrix formalism,” Surf. Sci. 56, 293–306 (1976); M. Elshazly-Zaghloul, R. M. A. Azzam, N. M. Bashara, “Explicit solutions for the optical properties of a uniaxial crystal in generalized ellipsometry,” Surf. Sci. 56, 281–292 (1976).
[CrossRef]

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Other (2)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985), pp. 798–804.

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

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Figures (9)

Fig. 1
Fig. 1

Incident and emerging electromagnetic plane waves in a nondepolarizing optical system. The Jones matrix connects both plane waves with respect to a chosen Cartesian coordinate system, such as the p and s planes in Fig. 2.

Fig. 2
Fig. 2

Definition of the plane of incidence (p plane) and the incidence angle Φa through the wave vectors of the incident and the emerging (reflected here) plane waves. Ap, As, Bp, and Bs denote the complex amplitudes of the p and the s modes (parallel to the p and the s plane) before and after the sample, respectively. P and A are the azimuth angles of the linear polarizers used in the standard arrangement of RAE. P or A is equal to zero if its selected direction is parallel to the p plane. P or A move counterclockwise with respect to the light propagation; see also comments in Section 3.

Fig. 3
Fig. 3

Differences between simulated Fourier coefficients α(P) and β(P), where −π/2 < P < π/2, with and without Rps and Rsp as examples. (α[Rps, Rsp ≠ 0] − α[Rps, Rsp = 0]: short-dashed curve; β[Rps, Rsp ≠ 0] − β[Rps, Rsp = 0]: long-dashed curve); Rpp{Ψ = 19.14°; |Δ| = 50.13°}, Rps{Ψ = 1.33°; |Δ| = 125.4°}, Rsp{Ψ = 0.82°; |Δ| = 153.4°}.

Fig. 4
Fig. 4

Simulated three-dimensional plot of Rpp versus wavelength and orientation angle ϕ for the extracted ordinary and extraordinary optical constants of TiO2 from sample 380 ([100] surface, Φa 3 60°). The anisotropic reflectivity expressed in terms of normalized Jones matrix elements is equal for 180°-opposite sample orientations ϕ, as is seen from Figs. 46. Note that Rps and Rsp vanish when the optic axis is oriented parallel to the x or the y axis of the laboratory coordinate system.

Fig. 5
Fig. 5

Same as Fig. 4 for Rps.

Fig. 6
Fig. 6

Same as Fig. 4 for Rsp.

Fig. 7
Fig. 7

Experimental and generated data obtained from best fit for No, Ne, optic axis inclination θ, and azimuth ϕ by use of experimental data from many sample azimuth angles and one incidence angle (see Table 1). The values represented by the symbols indicate the azimuths ϕ; [100] TiO2 sample, Φa = 70°. Here the optic axes are parallel to the sample surface and hence the inclinations θ were fixed at 0° during the regression.

Fig. 8
Fig. 8

Same as Fig. 7 for [110] TiO2 sample, Φa = 70°. The differences Δϕ between adjacent azimuths ϕi for sample 375 ([110] TiO2) were intentionally 90°. Hence the results in each Rpp are twofold degenerate. The Rsp values demonstrate the experimental limits for the determination of vanishing off-diagonal elements as discussed in Section 4.

Fig. 9
Fig. 9

Ordinary and extraordinary optical constants from TiO2 extracted for our three samples with different surface orientations. Note that no overlayer correction was performed during the model fit. The relative error for each value is less than 1% (380, 588) or 3% (375).

Tables (2)

Tables Icon

Table 1 Measurement Setup and Results for the Euler Angles

Tables Icon

Table 2 Ordinary and Extraordinary Optical Constants at Selected Spectral Positions in Comparison with Results Reviewed in Ref. 9

Equations (17)

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[ B p B s ] = J [ A p A s ] = [ j p p j s p j p s j s s ] [ A p A s ] ,
ρ tan Ψ exp ( i Δ ) ζ / χ ,             χ = A p / A s ,             ζ = B p / B s ,
ρ = ζ χ = r p p + r s p χ - 1 r s s + r p s χ ,
ρ = ( r p p / r s s ) + ( r s p / r s s ) χ - 1 1 + ( r p p / r s s ) ( r p s / r p p ) χ .
r p p r s s R p p = tan Ψ p p exp ( i Δ p p ) , r p s r p p R p s = tan Ψ p s exp ( i Δ p s ) , r s p r s s R s p = tan Ψ s p exp ( i Δ s p ) .
ρ = R p p + R s p χ - 1 1 + R p p R p s χ .
E det = R ( A ) · Π · R ( - A ) · r · R ( P ) · Π · R ( - P ) · E i .
Π = [ 1 0 0 0 ] ,             R ( α ) [ cos α - sin α sin α cos α ] .
r = [ r p p r s p r p s r s s ] .
( E ˜ p E ˜ s ) det = E ˜ p i ( R 1 cos Ω t + R 2 sin Ω t 0 ) ,
E ˜ det R ( - A ) · E det ,             E ˜ i R ( - P ) · E i , R 1 = r p p cos P + r s p sin P , R 2 = r p s cos P + r s s sin P .
I ( Ω t ) = I 0 { 1 + α cos 2 Ω t + β sin 2 Ω t } ,
α R 1 2 - R 2 2 R 1 2 + R 2 2 ,             β ( R 1 R ¯ 2 + R 2 R ¯ 1 ) R 1 2 + R 2 2 ,
tan Ψ = ( 1 + α 1 - α ) 1 / 2 tan P ,             cos Δ = β ( 1 - α 2 ) 1 / 2 .
α = R p p + R s p tan P 2 - R p p R p s + tan P 2 R p p + R s p tan P 2 + R p p R p s + tan P 2 , β = 2 Re [ R p p + R s p tan P ) * ( R p p R s p + tan P ) ] R p p + R s p tan P 2 + R p p R p s + tan P 2 ,
MSE = 1 N i = 1 N { [ α i m - α i calc ( R p p , R p s , R s p ; P ) δ α i m ] 2 + [ β i m - β i calc ( R p p , R p s , R s p ; P ) δ β i m ] 2 } .
MSE = 1 N i = 1 N [ ( R p p - R p p calc δ R p p ) i 2 + ( R p s - R p s calc δ R p s ) i 2 + ( R s p - R s p calc δ R s p ) i 2 ] .

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