Abstract

The reflection of s- and p-polarized electromagnetic plane waves from an anisotropic ultrathin dielectric film on transparent isotropic substrate is investigated in the long-wavelength limit. The analytical approximate formulas are obtained for the reflection coefficients and ellipsometric angles that agree with the exact computer solution of the reflection problem for anisotropic systems. The possibilities of using the obtained expressions for resolving the inverse problem for ultrathin anisotropic dielectric films upon isotropic dielectric substrates are discussed. It is shown that a promising technique for determining the optical constants of anisotropic dielectric films on transparent substrates is the integration of ellipsometry and differential reflectivity.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Robertson, “High dielectric constant gate oxides for metal oxide Si transistors,” Rep. Prog. Phys. 69, 327-396 (2006).
    [CrossRef]
  2. G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
    [CrossRef]
  3. M. Yamamoto and T. Namioka, “In situ ellipsometric study of optical properties of ultrathin films,” Appl. Opt. 31, 1612-1621 (1992).
    [CrossRef] [PubMed]
  4. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, 1987).
  5. I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
    [CrossRef]
  6. F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
    [CrossRef] [PubMed]
  7. H. Wang, “Propagation and reflection of plane waves in a medium with the 3-dimensional columnar structure induced anisotropy,” Optik (Jena) 106, 140-146 (1997).
  8. I. Hodgkinson, Q. H. Wu, and 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).
    [CrossRef]
  9. G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. F. Durrant, and V. Baranauskas, “Three polarization reflectometry methods for determination of optical anisotropy,” Appl. Opt. 37, 65-78(1998).
    [CrossRef]
  10. M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
    [CrossRef]
  11. Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. 244, 269-277 (2005).
    [CrossRef]
  12. Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constant determination of an anisotropic thin film via polarization conversion,” Opt. Express 15, 4445-4451 (2007).
    [CrossRef] [PubMed]
  13. A. N. Saxena, “Changes in the phase and amplitude of polarized light reflected from a film-covered surface and their relations with the film thickness,” J. Opt. Soc. Am. 55, 1061-1067(1965).
    [CrossRef]
  14. J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434(1971).
    [CrossRef]
  15. P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. I. Absorbing substrate,” J. Opt. Soc. Am. B 20, 752-759 (2003).
    [CrossRef]
  16. P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. II. Transparent substrate,” J. Opt. Soc. Am. B 21, 645-654 (2004).
    [CrossRef]
  17. M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the optical response of surfaces measured by spectroscopic ellipsometry: application to Si and Ge,” Surf. Sci. 285, 282-294(1993).
    [CrossRef]
  18. K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of reflectance difference spectroscopy and surface photoabsorption used for the investigation of anisotropic surfaces,” Surf. Sci. 287/288, 686-692 (1993).
    [CrossRef]
  19. Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
    [CrossRef]
  20. H. Goldstein, Classical Mechanics (Addison-Wesley, 1965).
  21. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
    [CrossRef]
  22. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  23. R. M. A. Azzam and N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521-1523 (1972).
    [CrossRef]
  24. M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
    [CrossRef]
  25. W. Xu, L. T. Wood, and T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740-1743 (2000).
    [CrossRef]

2007 (1)

Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constant determination of an anisotropic thin film via polarization conversion,” Opt. Express 15, 4445-4451 (2007).
[CrossRef] [PubMed]

2006 (1)

J. Robertson, “High dielectric constant gate oxides for metal oxide Si transistors,” Rep. Prog. Phys. 69, 327-396 (2006).
[CrossRef]

2005 (1)

Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. 244, 269-277 (2005).
[CrossRef]

2004 (2)

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. II. Transparent substrate,” J. Opt. Soc. Am. B 21, 645-654 (2004).
[CrossRef]

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

2003 (1)

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. I. Absorbing substrate,” J. Opt. Soc. Am. B 20, 752-759 (2003).
[CrossRef]

2000 (1)

W. Xu, L. T. Wood, and T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740-1743 (2000).
[CrossRef]

1999 (1)

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

1998 (2)

I. Hodgkinson, Q. H. Wu, and 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).
[CrossRef]

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

1997 (1)

H. Wang, “Propagation and reflection of plane waves in a medium with the 3-dimensional columnar structure induced anisotropy,” Optik (Jena) 106, 140-146 (1997).

1996 (1)

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

1995 (1)

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

1993 (3)

F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
[CrossRef] [PubMed]

M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the optical response of surfaces measured by spectroscopic ellipsometry: application to Si and Ge,” Surf. Sci. 285, 282-294(1993).
[CrossRef]

K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of reflectance difference spectroscopy and surface photoabsorption used for the investigation of anisotropic surfaces,” Surf. Sci. 287/288, 686-692 (1993).
[CrossRef]

1992 (1)

M. Yamamoto and T. Namioka, “In situ ellipsometric study of optical properties of ultrathin films,” Appl. Opt. 31, 1612-1621 (1992).
[CrossRef] [PubMed]

1987 (1)

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, 1987).

1985 (1)

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

1977 (1)

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

1972 (2)

R. M. A. Azzam and N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521-1523 (1972).
[CrossRef]

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
[CrossRef]

1971 (1)

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434(1971).
[CrossRef]

1965 (2)

A. N. Saxena, “Changes in the phase and amplitude of polarized light reflected from a film-covered surface and their relations with the film thickness,” J. Opt. Soc. Am. 55, 1061-1067(1965).
[CrossRef]

H. Goldstein, Classical Mechanics (Addison-Wesley, 1965).

Adamson, P.

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. II. Transparent substrate,” J. Opt. Soc. Am. B 21, 645-654 (2004).
[CrossRef]

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. I. Absorbing substrate,” J. Opt. Soc. Am. B 20, 752-759 (2003).
[CrossRef]

Aspnes, D. E.

K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of reflectance difference spectroscopy and surface photoabsorption used for the investigation of anisotropic surfaces,” Surf. Sci. 287/288, 686-692 (1993).
[CrossRef]

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434(1971).
[CrossRef]

Azzam, R. M. A.

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

R. M. A. Azzam and N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521-1523 (1972).
[CrossRef]

Baranauskas, V.

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

Bashara, N. M.

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

R. M. A. Azzam and N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521-1523 (1972).
[CrossRef]

Berreman, D. W.

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
[CrossRef]

Cardona, M.

M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the optical response of surfaces measured by spectroscopic ellipsometry: application to Si and Ge,” Surf. Sci. 285, 282-294(1993).
[CrossRef]

Chang, H. H.

Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constant determination of an anisotropic thin film via polarization conversion,” Opt. Express 15, 4445-4451 (2007).
[CrossRef] [PubMed]

Cho, K.

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

Durrant, S. F.

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

Endelema, D.

F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
[CrossRef] [PubMed]

Ferré, J.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Flory, F.

F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
[CrossRef] [PubMed]

Ghiner, A. V.

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

Golding, T. D.

W. Xu, L. T. Wood, and T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740-1743 (2000).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1965).

Gonze, X.

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

Gottschalch, V.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

Hazel, J.

I. Hodgkinson, Q. H. Wu, and 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).
[CrossRef]

Herzinger, C. M.

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

Hingerl, K.

K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of reflectance difference spectroscopy and surface photoabsorption used for the investigation of anisotropic surfaces,” Surf. Sci. 287/288, 686-692 (1993).
[CrossRef]

Hodgkinson, I.

I. Hodgkinson, Q. H. Wu, and 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).
[CrossRef]

F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
[CrossRef] [PubMed]

Hodgkinson, I. J.

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

Hofmann, T.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

Horowitz, F.

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

Hsieh, C. H.

Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. 244, 269-277 (2005).
[CrossRef]

Jen, Y. J.

Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constant determination of an anisotropic thin film via polarization conversion,” Opt. Express 15, 4445-4451 (2007).
[CrossRef] [PubMed]

Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. 244, 269-277 (2005).
[CrossRef]

Johs, B.

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

Jun, G.

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

Kamiya, I.

K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of reflectance difference spectroscopy and surface photoabsorption used for the investigation of anisotropic surfaces,” Surf. Sci. 287/288, 686-692 (1993).
[CrossRef]

Kelly, M. K.

M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the optical response of surfaces measured by spectroscopic ellipsometry: application to Si and Ge,” Surf. Sci. 285, 282-294(1993).
[CrossRef]

Krishnan, R.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Lekner, J.

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, 1987).

Lo, T. S.

Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. 244, 269-277 (2005).
[CrossRef]

Lopušník, R.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Macleod, H. A.

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

McIntyre, J. P. E.

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434(1971).
[CrossRef]

Namioka, T.

M. Yamamoto and T. Namioka, “In situ ellipsometric study of optical properties of ultrathin films,” Appl. Opt. 31, 1612-1621 (1992).
[CrossRef] [PubMed]

Nývlt, M.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Pasquarello, A.

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

Pelletier, E.

F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
[CrossRef] [PubMed]

Peng, C. Y.

Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constant determination of an anisotropic thin film via polarization conversion,” Opt. Express 15, 4445-4451 (2007).
[CrossRef] [PubMed]

Pénissard, G.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Pietzonka, I.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

Prosser, V.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Renard, D.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Rheinländer, B.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

Rignanese, G.-M.

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

Robertson, J.

J. Robertson, “High dielectric constant gate oxides for metal oxide Si transistors,” Rep. Prog. Phys. 69, 327-396 (2006).
[CrossRef]

Sass, T.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

Saxena, A. N.

A. N. Saxena, “Changes in the phase and amplitude of polarized light reflected from a film-covered surface and their relations with the film thickness,” J. Opt. Soc. Am. 55, 1061-1067(1965).
[CrossRef]

Schubert, M.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

Sikkens, M.

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

Surdutovich, G. I.

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

Urban, R.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Višnovský, Š.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Vitlina, R. Z.

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

Wang, H.

H. Wang, “Propagation and reflection of plane waves in a medium with the 3-dimensional columnar structure induced anisotropy,” Optik (Jena) 106, 140-146 (1997).

Wharton, J. J.

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

Wood, L. T.

W. Xu, L. T. Wood, and T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740-1743 (2000).
[CrossRef]

Woollam, J. A.

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

Wu, Q. H.

I. Hodgkinson, Q. H. Wu, and 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).
[CrossRef]

Xu, W.

W. Xu, L. T. Wood, and T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740-1743 (2000).
[CrossRef]

Yamamoto, M.

M. Yamamoto and T. Namioka, “In situ ellipsometric study of optical properties of ultrathin films,” Appl. Opt. 31, 1612-1621 (1992).
[CrossRef] [PubMed]

Zollner, S.

M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the optical response of surfaces measured by spectroscopic ellipsometry: application to Si and Ge,” Surf. Sci. 285, 282-294(1993).
[CrossRef]

Appl. Opt. (4)

M. Yamamoto and T. Namioka, “In situ ellipsometric study of optical properties of ultrathin films,” Appl. Opt. 31, 1612-1621 (1992).
[CrossRef] [PubMed]

F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2 films,” Appl. Opt. 32, 5649-5659 (1993).
[CrossRef] [PubMed]

I. Hodgkinson, Q. H. Wu, and 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).
[CrossRef]

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

J. Opt. Soc. Am. (3)

A. N. Saxena, “Changes in the phase and amplitude of polarized light reflected from a film-covered surface and their relations with the film thickness,” J. Opt. Soc. Am. 55, 1061-1067(1965).
[CrossRef]

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, “Generalized ellipsometry for surfaces with directional preference: application to diffraction gratings,” J. Opt. Soc. Am. 62, 1521-1523 (1972).
[CrossRef]

J. Opt. Soc. Am. A (2)

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, and C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875-883 (1996).
[CrossRef]

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, “Measurement of the principal refractive indices of thin films deposited at oblique incidence,” J. Opt. Soc. Am. A 2, 1693-1697 (1985).
[CrossRef]

J. Opt. Soc. Am. B (2)

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. I. Absorbing substrate,” J. Opt. Soc. Am. B 20, 752-759 (2003).
[CrossRef]

P. Adamson, “Reflection of light in a long-wavelength approximation from an N-layer system of inhomogeneous dielectric films and optical diagnostics of ultrathin layers. II. Transparent substrate,” J. Opt. Soc. Am. B 21, 645-654 (2004).
[CrossRef]

Opt. Commun. (1)

Y. J. Jen, C. H. Hsieh, and T. S. Lo, “Optical constant determination of an anisotropic thin film via surface plasmon resonance: analyzed by sensitivity calculation,” Opt. Commun. 244, 269-277 (2005).
[CrossRef]

Opt. Express (1)

Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constant determination of an anisotropic thin film via polarization conversion,” Opt. Express 15, 4445-4451 (2007).
[CrossRef] [PubMed]

Optik (Jena) (1)

H. Wang, “Propagation and reflection of plane waves in a medium with the 3-dimensional columnar structure induced anisotropy,” Optik (Jena) 106, 140-146 (1997).

Phys. Rev. B (4)

M. Schubert, T. Hofmann, B. Rheinländer, I. Pietzonka, T. Sass, V. Gottschalch, and J. A. Woollam, “Near-band-gap CuPt-order-induced birefringence in Al0.48Ga0.52InP2,” Phys. Rev. B 60, 16618-16634 (1999).
[CrossRef]

G.-M. Rignanese, X. Gonze, G. Jun, K. Cho, and A. Pasquarello, “First-principles investigation of high-k dielectrics: comparison between the silicates and oxides of hafnium and zirconium,” Phys. Rev. B 69, 184301 (2004).
[CrossRef]

W. Xu, L. T. Wood, and T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740-1743 (2000).
[CrossRef]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090-1106 (1995).
[CrossRef]

Rep. Prog. Phys. (1)

J. Robertson, “High dielectric constant gate oxides for metal oxide Si transistors,” Rep. Prog. Phys. 69, 327-396 (2006).
[CrossRef]

Surf. Sci. (3)

J. P. E. McIntyre and D. E. Aspnes, “Differential reflection spectroscopy of very thin surface films,” Surf. Sci. 24, 417-434(1971).
[CrossRef]

M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the optical response of surfaces measured by spectroscopic ellipsometry: application to Si and Ge,” Surf. Sci. 285, 282-294(1993).
[CrossRef]

K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of reflectance difference spectroscopy and surface photoabsorption used for the investigation of anisotropic surfaces,” Surf. Sci. 287/288, 686-692 (1993).
[CrossRef]

Other (3)

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, 1987).

H. Goldstein, Classical Mechanics (Addison-Wesley, 1965).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Relative errors of approximate formulas for (a)  Δ R p p / R p ( 0 ) (solid curves), Δ R s s / R s ( 0 ) (dashed curves), and for (b)  R p s (solid curves), R s p (dashed curves) as functions of λ for anisotropic ultrathin films with d = 3 nm , n x x = 1.6 (1), 3.5 (2), n y y = 1.8 (1), 3.2 (2), n z z = 1.5 (1), 3.8 (2), θ = 60 ° , φ = 50 ° , and ψ = 20 ° at n s = 2.5 and ϕ a = 45 ° . Numbers in parentheses are curve labels.

Fig. 2
Fig. 2

Relative errors of approximate formulas for (a)  δ Δ p p ( r ) (solid curve), δ Ψ p p ( r ) (dashed curve), and for (b)  tan Ψ p s ( r ) (solid curve), tan Ψ s p ( r ) (dashed curve) as functions of incident angle ϕ a for an anisotropic film with d = 5 nm , n x x = 2.5 , n y y = 2.4 , n z z = 2.3 , θ = 80 ° , φ = 30 ° , and ψ = 60 ° at n s = 1.5 and λ = 630 nm .

Fig. 3
Fig. 3

Comparison of long-wavelength limit (dashed curves) and exact computer calculation (solid curves) for the reflectances (a)  R p p , R s s and (b)  R p s , R s p as functions of n s for an anisotropic ultrathin film with d = 6 nm , n x x = 3.1 , n y y = 3.15 , n z z = 3.1 , θ = 10 ° , and φ = ψ = 45 ° at ϕ a = 60 ° and λ = 630 nm .

Fig. 4
Fig. 4

Comparison of long-wavelength limit (dashed curves) and exact computer calculation (solid curves) for the reflectances (a)  R p s and (b)  R s p as a function of n x x when n y y = 2.6 , n z z = 2.55 and of n z z when n x x = 2.55 , n y y = 2.6 for an anisotropic film with d = 6 nm , θ = 70 ° , φ = 60 ° , and ψ = 40 ° at n s = 1.5 , ϕ a = 5 ° , and λ = 630 nm .

Fig. 5
Fig. 5

Relative error of Eq. (42) as a function of λ when d = 3 nm , n x x = 2.2 , n y y = 2.6 , n z z = 2.0 , θ = 40 ° , φ = 10 ° , ψ = 70 ° , n s = 3.5 , ϕ a ( 1 ) = 40 ° , ϕ a ( 2 ) = 70 ° , v 1 = v 2 = 0 (dashed curve), v 1 = 0.5 % (1), 0.5% (2), 1% (3), 2% (4), and v 2 = 1 % (1), 4% (2), 2 % (3), 0.5% (4). Numbers in parentheses are curve labels.

Fig. 6
Fig. 6

Relative errors of Eqs. (49)+(52) (solid curves) and Eqs. (50)+(52) (dashed curves) as functions of λ when d = 3 nm , n x x = 1.7 , n y y = 1.8 , n z z = 2.0 , θ = 10 ° , φ = 80 ° , ψ = 50 ° n s = 1.5 , ϕ a = 50 ° , and μ = 0.5 % (1), 1% (2), 3% (3), 3 % (4). Numbers in parentheses are curve labels.

Fig. 7
Fig. 7

Relative error of Eq. (55) as a function of d / λ when n x x = n y y = 3.5 , n z z = 3.8 , θ = 10 ° , φ = 30 ° , n s = 2.5 , v 1 = v 2 = τ = 0 (dashed curves), v 1 = 2 % , v 2 = 4 % , τ = 1 % (solid curves), and ρ = 0 (1), 5% (2), 5 % (3). Numbers in parentheses are curve labels.

Fig. 8
Fig. 8

Relative error of Eq. (58) as a function of d / λ when v 1 = v 2 = τ = 0 (dashed curves), v 1 = 0.2 % , v 2 = 0.4 % , τ = 0.1 % (solid curves), and ρ = 0 (1), 0.3% (2). The values of the remaining parameters are the same as in Fig. 7. Numbers in parentheses are curve labels.

Equations (64)

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

[ ε 11 ε 12 ε 13 ε 21 ε 22 ε 23 ε 31 ε 32 ε 33 ] = A [ ε x x 0 0 0 ε y y 0 0 0 ε z z ] A 1 ,
ε 11 = Γ 1 sin 2 φ sin 2 ψ cos θ + Γ 2 cos 2 φ + Γ 3 sin 2 φ cos 2 θ + ε z z sin 2 φ sin 2 θ ,
ε 22 = Γ 1 sin 2 φ sin 2 ψ cos θ + Γ 2 sin 2 φ + cos 2 φ ( Γ 3 cos 2 θ + ε z z sin 2 θ ) ,
ε 33 = Γ 3 sin 2 θ + ε z z cos 2 θ ,
ε 12 = ε 21 = Γ 1 cos 2 φ sin 2 ψ cos θ + sin 2 φ ( Γ 2 Γ 3 cos 2 θ ε z z sin 2 θ ) / 2 ,
ε 13 = ε 31 = sin θ [ Γ 1 cos φ sin 2 ψ sin φ cos θ ( Γ 3 ε z z ) ] ,
ε 23 = ε 32 = sin θ [ Γ 1 sin φ sin 2 ψ + cos φ cos θ ( Γ 3 ε z z ) ] ,
Γ 1 ( ε x x ε y y ) / 2 , Γ 2 ε x x cos 2 ψ + ε y y sin 2 ψ , Γ 3 ε x x sin 2 ψ + ε y y cos 2 ψ .
r p p r p ( 0 ) { 1 + i 4 π n a cos ϕ a [ ( ϵ 11 ϵ 13 2 ϵ 33 1 ) cos 2 ϕ s ( 1 ϵ a ϵ 33 1 sin 2 ϕ a ) ϵ s ] ( ϵ a cos 2 ϕ s ϵ s cos 2 ϕ a ) ( d λ ) + [ ( 1 ε a ε 33 1 sin 2 ϕ a ) ε s ( ε 11 ε 13 2 ε 33 1 ) cos 2 ϕ s ] [ ( 1 ε a ε 33 1 sin 2 ϕ a ) n a n s + ( ε 11 ε 13 2 ε 33 1 ) cos ϕ a cos ϕ s ] ( n a cos ϕ s + n s cos ϕ a ) ( d λ ) 2 + [ ε a ε s ε 23 2 ε 33 2 sin 2 ϕ a ( ε 12 ε 13 ε 23 ε 33 1 ) 2 cos 2 ϕ s ] ( n a cos ϕ a + n s cos ϕ s ) ( d λ ) 2 } ,
r s s = r s ( 0 ) { 1 + i 4 π n a cos ϕ a ( ϵ 22 ϵ 23 2 ϵ 33 1 ϵ a sin 2 ϕ a ϵ s cos 2 ϕ s ) ( ϵ a ϵ s ) ( d λ ) + [ ε s cos 2 ϕ s + ε a sin 2 ϕ a ( ε 22 ε 23 2 ε 33 1 ) ] [ ε 22 ε 23 2 ε 33 1 ε a sin 2 ϕ a + n a n s cos ϕ a cos ϕ s ] ( n a cos ϕ a + n s cos ϕ s ) ( d λ ) 2 [ n a 3 n s ε 23 2 ε 33 2 sin 2 ϕ a + ( ε 12 ε 13 ε 23 ε 33 1 ) 2 cos ϕ a cos ϕ s ] ( n a cos ϕ s + n s cos ϕ a ) ( d λ ) 2 } ,
r σ 4 n a cos ϕ a ( n a cos ϕ a + n s cos ϕ s ) ( n a cos ϕ s + n s cos ϕ a ) { i π [ ( ε 12 ε 13 ε 23 ε 33 1 ) cos ϕ s P σ n a n s ε 23 ε 33 1 sin ϕ a ] ( d λ ) + π 2 [ n s ( 1 ε a ε 33 1 sin 2 ϕ a ) ( ε 12 ε 13 ε 23 ε 33 1 ) P σ n a ε 23 ε 33 1 ( ε 11 ε 13 2 ε 33 1 ) sin ϕ a cos ϕ s + ( ( ε 12 ε 13 ε 23 ε 33 1 ) cos ϕ s P σ n a n s ε 23 ε 33 1 sin ϕ a ) ( n s cos ϕ s + P σ n a ε 13 ε 33 1 sin ϕ a ) 2 ( ( ε 12 ε 13 ε 23 ε 33 1 ) cos ϕ s P σ n a n s ε 23 ε 33 1 sin ϕ a ) × [ ( ε 22 ε 23 2 ε 33 1 ε a sin 2 ϕ a + n a n s cos ϕ a cos ϕ s ) ( n a cos ϕ a + n s cos ϕ s ) + ( n a n s ( 1 ε a ε 33 1 sin 2 ϕ a ) + ( ε 11 ε 13 2 ε 33 1 ) cos ϕ a cos ϕ s ) ( n a cos ϕ s + n s cos ϕ a ) ] ] ( d λ ) 2 } ,
r p ( 0 ) = ( n a cos ϕ s n s cos ϕ a ) / ( n a cos ϕ s + n s cos ϕ a ) ,
r s ( 0 ) = ( n a cos ϕ a n s cos ϕ s ) / ( n a cos ϕ a + n s cos ϕ s ) ,
R p p R p ( 0 ) { 1 + 16 π 2 n a n s cos ϕ a cos ϕ s ( ε a cos 2 ϕ s ε s cos 2 ϕ a ) × [ ( ( ε 11 ε 13 2 ε 33 1 ) cos 2 ϕ s ( 1 ε a ε 33 1 sin 2 ϕ a ) ε s ) ( ( ε 11 ε 13 2 ε 33 1 ) cos 2 ϕ a ( 1 ε a ε 33 1 sin 2 ϕ a ) ε a ) ( ε a cos 2 ϕ s ε s cos 2 ϕ a ) + ( ε a ε s ε 23 2 ε 33 2 sin 2 ϕ a + ( ε 12 ε 13 ε 23 ε 33 1 ) 2 cos 2 ϕ s ) ( n a n s cos ϕ a cos ϕ s + ε s cos 2 ϕ s ) ] ( d λ ) 2 } ,
R s s R s ( 0 ) { 1 + 16 π 2 n a n s cos ϕ a cos ϕ s ( ε a ε s ) × [ ( ε 22 ε 23 2 ε 33 1 ε a sin 2 ϕ a ε s cos 2 ϕ s ) ( ε 22 ε 23 2 ε 33 1 ε a sin 2 ϕ a ε a cos 2 ϕ a ) ( ε a ε s ) ( n a 3 n s ε 23 2 ε 33 2 sin 2 ϕ a + ( ε 12 ε 13 ε 23 ε 33 1 ) 2 cos ϕ a cos ϕ s ) ( n a n s cos 2 ϕ s + ε s cos ϕ a cos ϕ s ) ] ( d λ ) 2 } ,
R σ = 16 π 2 ε a cos 2 ϕ a [ ( ε 12 ε 13 ε 23 ε 33 1 ) cos ϕ s P σ n a n s ε 23 ε 33 1 sin ϕ a ) ] 2 ( n a cos ϕ a + n s cos ϕ s ) 2 ( n a cos ϕ s + n s cos ϕ a ) 2 ( d λ ) 2 ,
r p p / r s s tan Ψ p p ( r ) exp ( i Δ p p ( r ) ) ,
r p s / r s s tan Ψ p s ( r ) exp ( i Δ p s ( r ) ) ,
r s p / r s s tan Ψ s p ( r ) exp ( i Δ s p ( r ) ) .
δ Δ p p ( r ) = 4 π n a cos ϕ a ( ε a ε s ) [ ( ( ε 11 ε 13 2 ε 33 1 ) cos 2 ϕ s ( 1 ε a ε 33 1 sin 2 ϕ a ) ε s ) ( cos 2 ϕ a ε a ε s 1 sin 2 ϕ a ) ( ε 22 ε 23 2 ε 33 1 ) + ε a sin 2 ϕ a + ε s cos 2 ϕ s ] ( d λ ) ,
δ Ψ p p ( r ) = π ( ε a + ε s ) 1 / 2 | [ ε 11 ε 13 2 ε 33 1 ( 1 ε a ε 33 1 sin 2 ϕ a ) ( ε a + ε s ) ] ( ε a ε s ) | ( d λ ) ,
tan Ψ σ ( r ) = 4 π n a cos ϕ a ( n a cos ϕ a + n s cos ϕ s ) ( n a cos ϕ s + n s cos ϕ a ) | ( ε 12 ε 13 ε 23 ε 33 1 ) cos ϕ s P σ n a n s ε 23 ε 33 1 sin ϕ a ) ( ε a cos 2 ϕ a ε s cos 2 ϕ s ) | ( d λ ) .
t p p / t s s tan Ψ p p ( t ) exp ( i Δ p p ( t ) ) ,
t p s / t s s tan Ψ p s ( t ) exp ( i Δ p s ( t ) ) ,
t s p / t s s tan Ψ s p ( t ) exp ( i Δ s p ( t ) ) ,
δ Δ p p ( t ) = 2 π [ ( ε 11 ε 13 2 ε 33 1 ) cos ϕ a cos ϕ s + n a n s ( 1 ε a ε 33 1 sin 2 ϕ a ) ( n a cos ϕ s + n s cos ϕ a ) ( n a n s cos ϕ a cos ϕ s + ε 22 ε 23 2 ε 33 1 ε a sin 2 ϕ a ) ( n a cos ϕ a + n s cos ϕ s ) n a ε 13 ε 33 1 sin ϕ a ] ( d λ ) ,
tan Ψ p s ( t ) = 2 π cos ϕ s ( n a cos ϕ s + n s cos ϕ a ) | ε 12 ε 13 ε 23 ε 33 1 n a n s ε 23 ε 33 1 sin ϕ a cos 1 ϕ s | ( d λ ) ,
tan Ψ s p ( t ) = 2 π cos ϕ a ( n a cos ϕ s + n s cos ϕ a ) | ε 12 ε 13 ε 23 ε 33 1 ε a ε 23 ε 33 1 sin ϕ a cos 1 ϕ a | ( d λ ) ,
Δ I p p , s s / I p , s = ( I p p , s s I p , s ) / I p , s = ( R p p , s s I p , s ( in ) R p , s ( 0 ) I p , s ( in ) ) / R p , s ( 0 ) I p , s ( in ) Δ R p p , s s / R p , s ( 0 ) ,
x = ( ε a + ε s ) / 2 ± ( ( ε a + ε s ) 2 / 4 + P 0 ε a ε s ) 1 / 2 ,
P 0 = [ Δ R p p ( 0 ° ) R p ( 0 ) ( 0 ° ) ( ε a ε s ) 2 16 π 2 n a n s + S 0 ( n a n s ) n s ] ( λ d ) 2 ,
S 0 = ± ( n a + n s ) 4 16 π 2 ε a [ R p s ( 0 ° ) R s p ( 0 ° ) ] 1 / 2 ,
y = [ ( a i 3 x + a i 5 ) ± ( ( a i 3 x + a i 5 ) 2 4 a i 2 ( a i 1 x 2 + a i 4 x + a i 6 ) ) 1 / 2 ] / ( 2 a i 2 ) ,
a i 1 = cos 2 ϕ s ( i ) cos 2 ϕ a ( i ) ,
a i 2 = ε a 3 ε s sin 4 ϕ a ( i ) ,
a i 3 = ε a sin 2 ϕ a ( i ) ( ε a cos 2 ϕ s ( i ) + ε s cos 2 ϕ a ( i ) ) ,
a i 4 = ( ε a cos 2 ϕ s ( i ) + ε s cos 2 ϕ a ( i ) ) ,
a i 5 = 2 ε a 2 ε s sin 2 ϕ a ( i ) ,
a i 6 = ε a ε s P i ,
P i = [ ( ε a cos 2 ϕ s ( i ) ε s cos 2 ϕ a ( i ) ) 2 16 π 2 n a n s cos ϕ a ( i ) cos ϕ s ( i ) Δ R p p ( ϕ a ( i ) ) R p ( 0 ) ( ϕ a ( i ) ) + S i ( ε a cos 2 ϕ s ( i ) ε s cos 2 ϕ a ( i ) ) ( n a n s cos ϕ a ( i ) cos ϕ s ( i ) + ε s cos 2 ϕ s ( i ) ) ] ( λ d ) 2 ,
S i = ± ( n a cos ϕ a ( i ) + n s cos ϕ s ( i ) ) 2 ( n a cos ϕ s ( i ) + n s cos ϕ a ( i ) ) 2 16 π 2 ε a cos 2 ϕ a ( i ) [ R p s ( ϕ a ( i ) ) R s p ( ϕ a ( i ) ) ] 1 / 2 .
a i 1 x 2 + a i 2 y 2 + a i 3 x y + a i 4 x + a i 5 y + a i 6 = 0 ,
A y 4 + B y 3 + C y 2 + D y + F = 0 ,
A = a 11 f 1 2 + a 12 f 4 2 a 13 f 1 f 4 ,
B = 2 ( a 11 f 1 f 2 + a 12 f 4 f 5 ) a 13 ( f 2 f 4 + f 1 f 5 ) a 14 f 1 f 4 + a 15 f 4 2 ,
C = a 11 ( f 2 2 + 2 f 1 f 3 ) + a 12 f 5 2 a 13 ( f 3 f 4 + f 2 f 5 ) a 14 ( f 2 f 4 + f 1 f 5 ) + 2 a 15 f 4 f 5 + a 16 f 4 2 ,
D = 2 a 11 f 2 f 3 a 13 f 3 f 5 a 14 ( f 3 f 4 + f 2 f 5 ) + a 15 f 5 2 + 2 a 16 f 4 f 5 ,
F = a 11 f 3 2 a 14 f 3 f 5 + a 16 f 5 2 ,
ε 22 ε 23 2 ε 33 = ε a sin 2 ϕ a + ε s cos 2 ϕ s + ( ε 11 ε 13 2 ε 33 1 ) cos 2 ϕ s ( 1 ε a sin 2 ϕ a ε 33 1 ) ε s cos 2 ϕ a ε a ε s 1 sin 2 ϕ a δ Δ p p ( r ) ( ε a ε s ) 4 π n a cos ϕ a λ d ,
ε 12 ε 13 ε 23 ε 33 = K p s + K s p 2 cos ϕ s ,
ε 23 ε 33 = K s p K p s 2 n a n s sin ϕ a ,
K σ = ± R σ 1 / 2 ( n a cos ϕ a + n s cos ϕ s ) ( n a cos ϕ s + n s cos ϕ a ) 4 π n a cos ϕ a λ d ,
K σ = ± tan ψ σ ( r ) ( n a cos ϕ a n s cos ϕ s ) ( n a cos ϕ s + n s cos ϕ a ) 4 π n a cos ϕ a λ d .
ε 13 ε 33 = 1 n a sin ϕ a [ c 21 cos ϕ a cos ϕ s + c 12 n a n s n a cos ϕ s + n s cos ϕ a n a n s cos ϕ a cos ϕ s + c 43 n a cos ϕ a + n s cos ϕ s δ Δ p p ( t ) 2 π λ d ] .
ε o sin 2 θ + ε e cos 2 θ = ε 33 , ε o sin 2 φ + ( ε o + ε e ε 33 ) cos 2 φ = ε 22 , ( ε o ε e ) cos θ sin θ cos φ = ε 23 , ε o + ( ε e ε 33 ) sin 2 φ ε 23 3 ε 33 1 tan 2 φ = ε p .
ε o = { ε 22 + ε 33 ± [ ( ε 22 ε 33 ) 2 + 4 ε 23 2 ] 1 / 2 } / 2 ,
ε e = ε 33 ( ε 22 ε o ) ( ε p + ε 22 + ε 33 2 ε o ) + ε 23 2 ( ε 0 ε 22 ε 33 ) ε 33 ( ε 22 ε o ) ε 23 2 ,
φ = arccos [ ( ε 22 ε o ) / ( ε e ε 33 ) ] 1 / 2 ,
θ = arccos [ ( ε 33 ε o ) / ( ε e ε o ) ] 1 / 2 .
ε o + ( ε e ε o ) sin 2 φ sin 2 θ = ε 11 , ε e ( ε e ε o ) sin 2 θ = ε 33 , ( ε e ε o ) sin φ sin θ cos θ = ε 13 , ε o + ( ε e ε o ) cos 2 φ sin 2 θ + ε 13 2 ε 33 1 ( ε e ε o ) 2 ε 33 1 sin 2 θ cos 2 θ = ε r .
ε o = { ε 11 + ε 33 ± [ ( ε 11 ε 33 ) 2 + 4 ε 13 2 ] 1 / 2 } / 2 ,
ε e = ε 33 [ ε o 1 ( ε r ε 13 2 ε 33 1 + ε 11 ) 1 ] ,
θ = arcsin [ ( ε e ε 33 ) / ( ε e ε o ) ] 1 / 2 ,
φ = arcsin [ ( ε 11 ε o ) / ( ε e ε 33 ) ] 1 / 2 .

Metrics