Abstract

The possibilities of determining the parameters of uniaxially anisotropic ultrathin nonabsorbing dielectric films on absorbing or transparent isotropic substrates by surface differential reflectance measurements are analyzed. The analysis is based on analytical reflection formulas obtained in the framework of a long-wavelength approximation. It is shown that, in the case of transparent substrates, it is always possible to determine the thickness of a uniaxially ultrathin film and its four parameters of anisotropy (optical constants no and ne and angles θ and φ) simultaneously. However, for such films on absorbing substrates, it is possible to decouple the thickness and optical constants by differential reflectance measurements only if θ0. The accuracy of the obtained analytic formulas for determining the parameters of ultrathin films is estimated by computer simulations where the reflection problem was solved numerically on the basis of the rigorous electromagnetic theory for anisotropic layered systems.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
    [CrossRef]
  2. J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
    [CrossRef]
  3. R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of a thin metal film by optical means,” Appl. Surf. Sci. 142, 451–454 (1999).
    [CrossRef]
  4. H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
    [CrossRef]
  5. S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
    [CrossRef]
  6. L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
    [CrossRef]
  7. H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, 1993).
  8. H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, 1999).
  9. I. K. Kim and D. E. Aspnes, “Analytic determination of n, κ, and d of an absorbing film from polarimetric data in the thin-film limit,” J. Appl. Phys. 101, 033109 (2007).
    [CrossRef]
  10. J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Martinus Nijhoff, 1987).
  11. D. Bedeaux and J. Vlieger, Optical Properties of Surfaces(Imperial College Press, 2004).
  12. 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]
  13. 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]
  14. H. Wang, “Propagation and reflection of plane waves in a medium with the 3-dimensional columnar structure induced anisotropy,” Optik 106, 140–146 (1997).
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. B. Lecourt, D. Blaudez, and J. M. Turlet, “Specific approach of generalized ellipsometry for the determination of weak in-plane anisotropy: application to Langmuir–Blodgett ultrathin films,” J. Opt. Soc. Am. A 15, 2769–2782 (1998).
    [CrossRef]
  21. S. Visnovsky, Optics in Magnetic Multilayers and Nanostructures (Taylor & Francis, 2006).
  22. M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
    [CrossRef]
  23. P. Adamson, “Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates,” Surf. Sci. 603, 3227–3233 (2009).
    [CrossRef]
  24. P. Adamson, “Optical diagnostics of anisotropic nanoscale films on transparent isotropic materials by integrating reflectivity and ellipsometry,” Appl. Opt. 48, 5906–5916 (2009).
    [CrossRef] [PubMed]
  25. H. Goldstein, Classical Mechanics (Addison-Wesley, 1965).
  26. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  27. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  28. P. J. Lin-Chung and S. Teitler, “4×4-matrix formalisms for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
    [CrossRef]
  29. P. Yeh, Optical Waves in Layered Media (Wiley, 2005).
  30. P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on absorbing medium,” Waves Random Complex Media 18, 651–668 (2008).
    [CrossRef]
  31. P. Adamson, “Differential reflection photometry of ultrathin dielectric layers on strongly absorbing materials,” Opt. Spectrosc. 86, 408–414 (1999).
  32. L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
    [CrossRef]
  33. R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer, “Adsorption of poly(propylene imine) dendrimers on glass. An interplay between surface and particle properties,” Langmuir 16, 7713–7719 (2000).
    [CrossRef]
  34. P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on non-absorbing isotropic medium,” Waves Random Complex Media 20, 443–471(2010).
    [CrossRef]
  35. P. Landry, J. Gray, M. K. O’Toole, and X. D. Zhu, “Incidence-angle dependence of optical reflectivity difference from an ultrathin film on solid surface,” Opt. Lett. 31531–533(2006).
    [CrossRef] [PubMed]

2010

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on non-absorbing isotropic medium,” Waves Random Complex Media 20, 443–471(2010).
[CrossRef]

2009

P. Adamson, “Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates,” Surf. Sci. 603, 3227–3233 (2009).
[CrossRef]

P. Adamson, “Optical diagnostics of anisotropic nanoscale films on transparent isotropic materials by integrating reflectivity and ellipsometry,” Appl. Opt. 48, 5906–5916 (2009).
[CrossRef] [PubMed]

2008

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on absorbing medium,” Waves Random Complex Media 18, 651–668 (2008).
[CrossRef]

2007

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

I. K. Kim and D. E. Aspnes, “Analytic determination of n, κ, and d of an absorbing film from polarimetric data in the thin-film limit,” J. Appl. Phys. 101, 033109 (2007).
[CrossRef]

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

2005

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

2000

R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer, “Adsorption of poly(propylene imine) dendrimers on glass. An interplay between surface and particle properties,” Langmuir 16, 7713–7719 (2000).
[CrossRef]

1999

P. Adamson, “Differential reflection photometry of ultrathin dielectric layers on strongly absorbing materials,” Opt. Spectrosc. 86, 408–414 (1999).

R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of a thin metal film by optical means,” Appl. Surf. Sci. 142, 451–454 (1999).
[CrossRef]

1998

1997

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

1996

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
[CrossRef]

1995

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

1993

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]

1985

1984

P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
[CrossRef]

P. J. Lin-Chung and S. Teitler, “4×4-matrix formalisms for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
[CrossRef]

1972

Adamson, P.

P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on non-absorbing isotropic medium,” Waves Random Complex Media 20, 443–471(2010).
[CrossRef]

P. Adamson, “Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates,” Surf. Sci. 603, 3227–3233 (2009).
[CrossRef]

P. Adamson, “Optical diagnostics of anisotropic nanoscale films on transparent isotropic materials by integrating reflectivity and ellipsometry,” Appl. Opt. 48, 5906–5916 (2009).
[CrossRef] [PubMed]

P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on absorbing medium,” Waves Random Complex Media 18, 651–668 (2008).
[CrossRef]

P. Adamson, “Differential reflection photometry of ultrathin dielectric layers on strongly absorbing materials,” Opt. Spectrosc. 86, 408–414 (1999).

Antad, V.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Aspnes, D. E.

I. K. Kim and D. E. Aspnes, “Analytic determination of n, κ, and d of an absorbing film from polarimetric data in the thin-film limit,” J. Appl. Phys. 101, 033109 (2007).
[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]

Azzam, R. M. A.

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

Babonneau, D.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Baranauskas, V.

Bashara, N. M.

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

Bedeaux, D.

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces(Imperial College Press, 2004).

Berreman, D. W.

Blaudez, D.

Böhmer, M. R.

R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer, “Adsorption of poly(propylene imine) dendrimers on glass. An interplay between surface and particle properties,” Langmuir 16, 7713–7719 (2000).
[CrossRef]

Borensztein, Y.

R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of a thin metal film by optical means,” Appl. Surf. Sci. 142, 451–454 (1999).
[CrossRef]

Camelio, S.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[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.

Chiaradia, P.

P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
[CrossRef]

Chiarotti, G.

P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
[CrossRef]

Cricenti, A.

P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
[CrossRef]

Dienel, T.

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

Durrant, S. F.

En Naciri, A.

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Endelema, D.

Flory, F.

Fritz, T.

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

Ghiner, A. V.

Gilliot, M.

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Goldstein, H.

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

Gray, J.

Grob, J. J.

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Guérin, P.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Hazel, J.

Heinrich, L.

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (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.

Hodgkinson, I. J.

Horowitz, F.

Jen, Y. J.

Johann, L.

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Jupille, J.

R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of a thin metal film by optical means,” Appl. Surf. Sci. 142, 451–454 (1999).
[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]

Kim, I. K.

I. K. Kim and D. E. Aspnes, “Analytic determination of n, κ, and d of an absorbing film from polarimetric data in the thin-film limit,” J. Appl. Phys. 101, 033109 (2007).
[CrossRef]

Kobayashi, H.

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

Koper, G. J. M.

R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer, “Adsorption of poly(propylene imine) dendrimers on glass. An interplay between surface and particle properties,” Langmuir 16, 7713–7719 (2000).
[CrossRef]

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
[CrossRef]

Lamongie, B.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Landry, P.

Lantiat, D.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Lazzari, R.

R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of a thin metal film by optical means,” Appl. Surf. Sci. 142, 451–454 (1999).
[CrossRef]

Lecourt, B.

Lekner, J.

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

Leo, K.

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

Lin-Chung, P. J.

Macleod, H. A.

Mann, E. K.

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
[CrossRef]

McGahan, W. A.

H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, 1999).

McGilp, J. F.

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

Mitobe, F.

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

Muller, D.

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Nitsche, R.

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

O’Toole, M. K.

Ohno, S.

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

Pelletier, E.

Peng, C. Y.

Proehl, H.

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

Schaaf, P.

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
[CrossRef]

Selci, S.

P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
[CrossRef]

Shudo, K.

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

Sikkens, M.

Simonot, L.

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Stoquert, J. P.

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Surdutovich, G. I.

Suzuki, T.

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

Tanaka, M.

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

Teitler, S.

Tompkins, H. G.

H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, 1993).

H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, 1999).

Turlet, J. M.

van Duijvenbode, R. C.

R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer, “Adsorption of poly(propylene imine) dendrimers on glass. An interplay between surface and particle properties,” Langmuir 16, 7713–7719 (2000).
[CrossRef]

Visnovsky, S.

S. Visnovsky, Optics in Magnetic Multilayers and Nanostructures (Taylor & Francis, 2006).

Vitlina, R. Z.

Vlieger, J.

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces(Imperial College Press, 2004).

Voegel, J. C.

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
[CrossRef]

Wang, H.

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

Wharton, J. J.

Wu, Q. H.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

Zhu, X. D.

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.

Appl. Surf. Sci.

R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of a thin metal film by optical means,” Appl. Surf. Sci. 142, 451–454 (1999).
[CrossRef]

J. Appl. Phys.

I. K. Kim and D. E. Aspnes, “Analytic determination of n, κ, and d of an absorbing film from polarimetric data in the thin-film limit,” J. Appl. Phys. 101, 033109 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Langmuir

L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P. Schaaf, “Scanning angle reflectometry study of the structure of antigen-antibody layers adsorbed on silica surfaces,” Langmuir 12, 4857–4865 (1996).
[CrossRef]

R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer, “Adsorption of poly(propylene imine) dendrimers on glass. An interplay between surface and particle properties,” Langmuir 16, 7713–7719 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

P. Adamson, “Differential reflection photometry of ultrathin dielectric layers on strongly absorbing materials,” Opt. Spectrosc. 86, 408–414 (1999).

Optik

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

Phys. Rev. B

H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situ differential reflectance spectroscopy of thin crystalline films of PTCDA on different substrates,” Phys. Rev. B 71, 165207 (2005).
[CrossRef]

S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, and M. Tanaka, “Monolayer oxidation on Si(001)-(2×1) studied by means of reflectance difference spectroscopy,” Phys. Rev. B 77, 085319 (2008).
[CrossRef]

M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob, and D. Muller, “Optical anisotropy of shaped oriented cobalt nanoparticles by generalized spectroscopic ellipsometry,” Phys. Rev. B 76, 045424 (2007).
[CrossRef]

Phys. Rev. Lett.

P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differential reflectivity of Si(111)2×1 surface with polarized light: a test for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).
[CrossRef]

Prog. Surf. Sci.

J. F. McGilp, “Optical characterization of semiconductor surfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).
[CrossRef]

Surf. Sci.

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]

P. Adamson, “Reflection characterization of anisotropic ultrathin dielectric films on absorbing isotropic substrates,” Surf. Sci. 603, 3227–3233 (2009).
[CrossRef]

Thin Solid Films

L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin, B. Lamongie, and V. Antad, “In situ optical spectroscopy during deposition of Ag:Si3N4 nanocomposite films by magnetron sputtering,” Thin Solid Films 518, 2637–2643 (2010).
[CrossRef]

Waves Random Complex Media

P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on non-absorbing isotropic medium,” Waves Random Complex Media 20, 443–471(2010).
[CrossRef]

P. Adamson, “Reflection of electromagnetic plane waves in a long-wavelength approximation from a multilayer system of anisotropic transparent films on absorbing medium,” Waves Random Complex Media 18, 651–668 (2008).
[CrossRef]

Other

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

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

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

H. G. Tompkins, A User’s Guide to Ellipsometry (Academic, 1993).

H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsometry and Reflectometry: A User’s Guide (Wiley, 1999).

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

D. Bedeaux and J. Vlieger, Optical Properties of Surfaces(Imperial College Press, 2004).

S. Visnovsky, Optics in Magnetic Multilayers and Nanostructures (Taylor & Francis, 2006).

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 (6)

Fig. 1
Fig. 1

Relative errors of (a)  ε e and (b) φ (solid curves) and d (dashed curves) determined by Eqs. (31, 29, 32), respectively, as functions of λ for an anisotropic ultrathin film with d = 1 nm , n o = 3.4 , n e = 3.0 , θ = φ = 40 ° at n ^ s = 2.5 + i 2 , v p p = v s s = 0 (curves 1, 3) and 2% (curves 2), and v s p = v p s = 0 (curves 1, 2) and 5 % (curves 3). Incident angles ϕ a p ( 1 ) = 20 ° , ϕ a p ( 2 ) = 70 ° , ϕ a s = 20 ° , and ϕ a p s = ϕ a s p = 70 ° .

Fig. 2
Fig. 2

Relative error (a) of the quantity ( ε o ε a ) d determined by Eq. (49) (solid curves) or (47) (dashed curves) and (b) of the quantity ( ε e 1 ε a 1 ) d determined by Eq. (50) (solid curves) or Eq. (48) (dashed curves) as functions of d / λ for an anisotropic ultrathin film with n o = 2.4 , n e = 2.2 , and θ = φ = 0 ° at n ^ s = 3.5 + i 1.5 , v p p = v s s = 0 (curves 1), and v p p = v s s = 1 % (curves 2). Incident angles ϕ a p ( 1 ) = 45 ° , ϕ a p ( 2 ) = 70 ° , and ϕ a s = 45 ° .

Fig. 3
Fig. 3

(a) Differential reflectances Δ R p p / R p (solid curves) and Δ R s s / R s (dashed curves) and (b) reflectances R p s (solid curves) and R s p (dashed curves) as functions of λ for an absorbing substrate at two different incident angles ϕ a = 20 ° (curves 1) and 70 ° (curves 2). The other parameters are the same as in Fig. 1.

Fig. 4
Fig. 4

Relative errors of (a)  ε o and (b)  ε e determined by Eqs. (77, 78), respectively, as functions of λ for an anisotropic ultrathin film with d = 2 nm , n o = 2.49 , n e = 2.86 , θ = 40 ° , φ = 30 ° at n s = 1.46 , v p p = v p s = v s p = 0 (solid curves), v p p = 1 % , v p s = v s p = 0 (dotted curves), and v p p = 0 , v p s = v s p = 4 % (dashed curves). Incident angles ϕ a ( 1 ) = 0 ° , ϕ a ( 2 ) = 30 ° , and ϕ a ( 3 ) = 60 ° .

Fig. 5
Fig. 5

Relative errors of (a) θ and (b) φ determined by Eqs. (79, 80), respectively, as functions of λ for an anisotropic ultrathin film with the same parameters as in Fig. 4.

Fig. 6
Fig. 6

(a) Differential reflectances Δ R p p / R p (solid and dashed–dotted curves) and Δ R s s / R s (dashed and dotted curves) and (b) reflectances R p s (solid and dashed–dotted curves) and R s p (dashed and dotted curves) as functions of λ for a transparent substrate at two different incident angles ϕ a = 30 ° (curves 1) and 60 ° (curves 2). The other parameters are the same as in Fig. 4 (solid and dashed curves) and Fig. 1 [dashed–dotted and dotted curves ( n s I = 0 )].

Equations (80)

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 o 0 0 0 ε o 0 0 0 ε e ] A 1 ,
ε 11 = ε o cos 2 φ + ( ε o cos 2 θ + ε e sin 2 θ ) sin 2 φ ,
ε 12 = ε 21 = ( ε o ε e ) sin 2 θ sin φ cos φ ,
ε 22 = ε o sin 2 φ + ( ε o cos 2 θ + ε e sin 2 θ ) cos 2 φ ,
ε 13 = ε 31 = ( ε e ε o ) sin θ cos θ sin φ ,
ε 23 = ε 32 = ( ε o ε e ) sin θ cos θ cos φ ,
ε 33 = ε o sin 2 θ + ε e cos 2 θ ,
Δ R s s / R s 8 π K 0 ε s I n a cos ϕ a ( ε a + ε 23 2 ε 33 1 ε 22 ) ( d / λ ) ,
Δ R p p / R p 8 π K ε s I n a cos ϕ a [ ε a ( 1 ε a ε 33 1 sin 2 ϕ a ) ( ε 11 ε 13 2 ε 33 1 ) cos 2 ϕ a ] ( d / λ ) ,
R s = | ( n a cos ϕ a n ^ s cos ϕ s ) / ( n a cos ϕ a + n ^ s cos ϕ s ) | 2 ,
R p = | ( n a cos ϕ s n ^ s cos ϕ a ) / ( n a cos ϕ s + n ^ s cos ϕ a ) | 2 ,
K = [ 1 2 ε a α sin 2 ϕ a ] / [ { ε a ( 1 ε a α sin 2 ϕ a ) ε s R cos 2 ϕ a } 2 + ε s I 2 ( cos 2 ϕ a ε a 2 | ε ^ s | 2 sin 2 ϕ a ) 2 ] , cos ϕ s = ( 1 ε a ε ^ s 1 sin 2 ϕ a ) 1 / 2 , α = ε s R / | ε ^ s | 2 , | ε ^ s | 2 = ε s R 2 + ε s I 2 , K 0 = K ( ϕ a = 0 ) .
R σ 16 π 2 ε a cos 2 ϕ a | ( n a cos ϕ s + n ^ s cos ϕ a ) | 2 | ( n a cos ϕ a + n ^ s cos ϕ s ) | 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 | 2 ( d / λ ) 2 ,
( ε 11 ε 13 2 ε 33 1 ε a ) d = p 1 ,
( ε 33 1 ε a 1 ) d = p 2 ,
( ε 22 ε 23 2 ε 33 1 ε a ) d = p 3 ,
( ε 23 ε 33 1 ) d = p 4 ,
( ε 12 ε 13 ε 23 ε 33 1 ) d = p 5 .
p 1 = P 2 sin 2 ϕ a p ( 1 ) P 1 sin 2 ϕ a p ( 2 ) sin 2 ϕ a p ( 2 ) cos 2 ϕ a p ( 1 ) sin 2 ϕ a p ( 1 ) cos 2 ϕ a p ( 2 ) ,
p 2 = [ P 1 cos 2 ϕ a p ( 2 ) P 2 cos 2 ϕ a p ( 1 ) sin 2 ϕ a p ( 2 ) cos 2 ϕ a p ( 1 ) sin 2 ϕ a p ( 1 ) cos 2 ϕ a p ( 2 ) ] 1 ε a 2 ,
P i = λ 8 π n a K cos ϕ a p ( i ) ε s I Δ R p p ( ϕ a p ( i ) ) R p ( ϕ a p ( i ) ) ,
p 3 = λ 8 π n a K 0 cos ϕ a s ε s I Δ R s s ( ϕ a s ) R s ( ϕ a s ) ,
p 4 = ± ( ( η p s + η s p 4 β ) ( 1 ± ( 1 4 α β γ 2 ( η p s η s p η p s + η s p ) 2 ) 1 / 2 ) ) 1 / 2 ,
p 5 = η s p η p s 2 γ p 4 ,
η σ = R σ | n ^ s cos ϕ a σ + n a cos ϕ s σ | 2 | n a cos ϕ a σ + n ^ s cos ϕ s σ | 2 16 π 2 ε a cos 2 ϕ a σ λ 2 ,
( ε o sin 2 θ + ε e cos 2 θ ) [ 2 ( ε o ε a ) + ( ε e ε o ) sin 2 θ ] sin 2 θ cos 2 θ ( ε e ε o ) 2 = ( p 1 + p 3 ) p 4 1 cos φ sin θ cos θ ( ε o ε e ) ,
1 ε a 1 ( ε o sin 2 θ + ε e cos 2 θ ) = p 2 p 4 1 cos φ sin θ cos θ ( ε o ε e ) ,
ε o = ( p 1 p 3 ) p 4 1 cos φ ( cos 2 φ tan θ ) 1 = p 5 p 4 1 ( sin φ tan θ ) 1 .
tan 2 φ = 2 p 5 ( p 1 p 3 ) 1 ,
tan θ = p 5 ( p 4 ε o sin φ ) 1 ,
ε e = p 5 2 + p 4 2 ε o 2 sin 2 φ p 5 2 ε a 1 ε o p 2 p 5 ε o 2 sin φ cos φ p 4 2 ε a 1 ε o 2 sin 2 φ p 2 p 5 ε o sin φ cos φ .
d = p 2 / ( ε 33 1 ε a 1 ) .
ε 11 = ε o ,
ε 22 = ε o + ( ε e ε o ) sin 2 θ ,
ε 33 = ε e ( ε e ε o ) sin 2 θ ,
ε 23 = ( ε o ε e ) sin θ cos θ ,
ε 12 = ε 13 = 0 ,
( ε o ε a ) d = p 1 ,
( 1 ε e ( ε e ε o ) sin 2 θ 1 ε a ) d = p 2 ,
( ε o + ( ε e ε o ) sin 2 θ sin 2 θ cos 2 θ ( ε e ε o ) 2 ε e ( ε e ε o ) sin 2 θ ε a ) d = p 3 ,
( sin θ cos θ ( ε o ε e ) ε e ( ε e ε o ) sin 2 θ ) d = p 4 ,
p 4 = λ | n a cos ϕ s σ + n ^ s cos ϕ a σ | | n a cos ϕ a σ + n ^ s cos ϕ s σ | 2 π ε a | n ^ s | sin 2 ϕ a σ R σ ( ϕ a σ ) ,
ε o = 1 ε a [ p 1 ( p 1 p 3 ) p 4 2 + p 2 ( p 3 p 1 ) ] ,
ε e = 1 ε o [ p 3 ( ε o ε a ) + p 1 ε a p 2 ( ε o ε a ) + p 1 ε a 1 ] ,
sin 2 θ = ε e ( ε e ε o ) [ 1 ε o ε a + ( ε o ε a ) p 3 p 1 1 ] .
d = p 1 / ( ε o ε a ) .
( ε o ε a ) d = p 1 ,
( ε e 1 ε a 1 ) d = p 2 .
( ε o ε a ) d = λ 8 π n a K 0 cos ϕ a s ε s I Δ R s s ( ϕ a s ) R s ( ϕ a s ) ,
( ε e 1 ε a 1 ) d = λ 8 π n a ε a 2 sin 2 ϕ a p ε s I [ cos 2 ϕ a p K 0 cos ϕ a s Δ R s s ( ϕ a s ) R s ( ϕ a s ) 1 K cos ϕ a p Δ R p p ( ϕ a p ) R p ( ϕ a p ) ] .
Δ R s s / R s 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 s ) ( ε 22 ε 23 2 ε 33 1 ε 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 ,
Δ R p p / R p 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 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 ,
a 11 x 2 + a 12 y 2 + a 13 x y + a 14 x + a 15 y + a 16 = 0 , a 21 x 2 + a 22 y 2 + a 23 x y + a 24 x + a 25 y + a 26 = 0 ,
a i 1 = cos 2 ϕ s ( i ) cos 2 ϕ a ( i ) P 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 + ε s ) P i 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 ( 1 P i ) ,
P i = ( ε a cos 2 ϕ s ( i ) ε s cos 2 ϕ a ( i ) ) 2 ( ε a ε s ) 2 cos ϕ a ( i ) cos ϕ s ( i ) [ Δ R p p ( ϕ a ( i ) ) R p ( ϕ a ( i ) ) + S i ] [ Δ R p p ( ϕ a = 0 ) R p ( ϕ a = 0 ) + S 0 ] 1 ,
S i = ± ( n a cos ϕ a ( i ) + n s cos ϕ s ( i ) ) ( n a cos ϕ s ( i ) + n s cos ϕ a ( i ) ) n a ( n a cos ϕ s ( i ) n s cos ϕ a ( i ) ) cos ϕ a ( i ) [ R p s ( ϕ a ( i ) ) R s p ( ϕ a ( i ) ) ] 1 / 2 ,
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 ,
( d / λ ) 2 = ( ε a ε s ) 2 [ 16 π 2 n a n s ( x ε s ) ( x ε a ) ] 1 [ Δ R p p ( ϕ a = 0 ) R p ( ϕ a = 0 ) + S 0 ] .
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 ,
ε o sin 2 θ + ε e cos 2 θ = ε 33 ,
( ε o ε e ) sin θ cos θ cos φ = ε 23 ,
ε o cos 2 φ + ( ε o + ε e ε 33 ) sin 2 φ ε 23 2 ε 33 1 tan 2 φ = x ,
( ε o ε e ) sin 2 θ sin φ cos φ + ε 23 2 ε 33 1 tan φ = t .
ε o = 1 2 ( x + t 2 ε 33 ε 23 2 ± ( x + t 2 ε 33 ε 23 2 ) 2 4 ( t ε 33 ε 23 ) 2 ) ,
ε e = x ε 33 ε o ε 23 2 ε o ε 33 ,
sin 2 θ = ε 33 ε e ε o ε e ,
cos 2 φ = ε 23 2 ( ε 33 ε e ) ( ε o ε 33 ) .

Metrics