Abstract

Until today dispersion analysis could not be successfully applied to evaluate oscillator parameters of modes that have their transition moments perpendicular to the surface of an anisotropic crystal or a layered medium. The main reason for this failure is that while such modes generate maxima in the external reflection spectra, which are obtained with polarized light parallel to the plane of incidence under nonzero angles of incidence, the positions of these maxima do not allow us to unambiguously trace back the oscillator positions. In contrast, total internal reflection of parallel polarized light generates minima at spectral positions close to the oscillator frequency. Starting from this observation, we found that a combined evaluation of external and total internal reflection spectra by dispersion analysis allows us to gain the oscillator parameters of perpendicular modes unambiguously.

© 2011 Optical Society of America

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  1. W. G. Spitzer and D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
    [CrossRef]
  2. E. E. Koch, A. Otto, and K. L. Kliewer, “Reflection spectroscopy on monoclinic crystals,” Chem. Phys. 3, 362–369 (1974).
    [CrossRef]
  3. V. F. Pavinich and M. V. Belousov, “Dispersion analysis of reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 881–883 (1978).
  4. A. G. Emslie and J. R. Aronson, “Determination of the complex dielectric tensor of triclinic crystals: theory,” J. Opt. Soc. Am. 73, 916–919 (1983).
    [CrossRef]
  5. J. R. Aronson, A. G. Emslie, and P. F. Strong, “Optical constants of triclinic anisotropic crystals: blue vitriol,” Appl. Opt. 24, 1200–1203 (1985).
    [CrossRef] [PubMed]
  6. B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
    [CrossRef]
  7. Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
    [CrossRef]
  8. G. Leveque, Y. Bertrand, and S. Robin-Kandare, “Determination of optical anisotropy of layered crystals in VUV range,” Solid State Commun. 38, 759–764 (1981).
    [CrossRef]
  9. P. Drude, “Über die Gesetze der Reflexion und Brechung des Lichtes an der Grenze absorbirender Krystalle,” Ann. Phys. Chem. 268, 584–625 (1887).
    [CrossRef]
  10. We would find it more consistent to name the main tensor component along the normal to the surface ε⊥ but decided to employ ε∥ instead. The latter term is generally used in the literature, which usually deals only with optical uniaxial materials, where ε∥ stands for the main or principal tensor component along the optical axis.
  11. L. P. Mosteller, Jr., and F. Wooten, “Optical properties and reflectance of uniaxial absorbing crystals,” J. Opt. Soc. Am. 58, 511–518 (1968).
    [CrossRef]
  12. F. Abelès, H. A. Washburn, and H. H. Soonpaa, “Calculating optical constants of anisotropic materials from reflectivity data,” J. Opt. Soc. Am. 63, 104–105 (1973).
    [CrossRef]
  13. C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
    [CrossRef]
  14. D. W. Berreman, “Infrared absorption at longitudinal optic frequency in cubic crystal films,” Phys. Rev. 130, 2193–2198 (1963).
    [CrossRef]
  15. J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
    [CrossRef]
  16. For such measurements incidence media are used that have a higher refractive index than the sample; as a consequence, light is totally reflected for incidence angles higher than the so-called critical angle except at frequencies at which the sample material is absorbing.
  17. K. R. Kirov and H. E. Assender, “Quantitative ATR-IR analysis of anisotropic polymer films: extraction of optical constants,” Macromolecules 37, 894–904 (2004).
    [CrossRef]
  18. K. Yamamoto and H. Ishida, “Complex refractive index determination for uniaxial anisotropy with the use of Kramers-Kronig analysis,” Appl. Spectrosc. 51, 1287–1293 (1997).
    [CrossRef]
  19. To be more precise, one of the main axes of the tensor ellipsoid is, independent of frequency, always oriented perpendicular to the surface of the material of interest.
  20. In monoclinic materials only one of the principal axes has an orientation that is, independent of frequency, always perpendicular to the other two axes. Many monoclinic materials, e.g., phlogopite, can easily be cleaved perpendicular to this axis, called the monoclinic b-axis.
  21. T. G. Mayerhöfer and J. Popp, “Modeling IR-spectra of polycrystalline materials in the large crystallites limit—quantitative determination of orientation,” J. Opt. A: Pure Appl. Opt. 8, 657–671 (2006).
    [CrossRef]
  22. D. W. Berreman, “Optics in stratified and anisotropic media—4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510(1972).
    [CrossRef]
  23. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  24. T. G. Mayerhöfer, S. Weber, and J. Popp, “Simplified formulas for non-normal reflection from monoclinic crystals,” Opt. Commun. 284, 719–723 (2011).
    [CrossRef]
  25. The formulas presented in this paper allow for anisotropic substrate materials down to monoclinic symmetry, where the monoclinic b-axis is oriented perpendicular to the substrate surface, which covers to our best knowledge all common substrate materials. Generally valid formulas can be found in .
  26. The X- and Y-components of the wave vector k, kX and kY, are the same as for the incidence medium and are given by kX=0 and kY=nisin⁡α, wherein α is the angle of incidence and n the index of refraction of the isotropic incidence medium provided that the plane of incidence is the Y–Z plane (cf. Figure ). All components are dimensionless quantities.
  27. For absorbing media, a positive imaginary part of γi determines a forward travelling wave, whereas the same wave in a nonabsorbing medium is characterized by a positive real part of γi.
  28. For a semi-infinte medium the Ψ2,l and the Ψ4,l are zero .
  29. For the wavenumber, traditionally the symbol ν˜ is also in use, especially in the context of spectroscopy [see, e.g., Eq. ].
  30. If for both the sample medium and the substrate the dynamical matrix can be calculated according to Eq. , then the waves are uncoupled and rij=tij=0 holds.
  31. Equation  does not directly transform under the condition of Eq. .
  32. Note that to compute the reflectance at TIR conditions [e.g., Rp,calc(n=4,α=50°)], the index of refraction of the incidence medium (i.e., the ATR crystal) must be inserted into the wave vector and Eq. .
  33. V. Ivanovski, T. G. Mayerhöfer, and J. Popp, “Employing polyethylene as contacting agent between ATR-crystals and solid samples with hard surfaces,” J. Mol. Struct. 924–926, 571–576 (2009).
    [CrossRef]

2011 (1)

T. G. Mayerhöfer, S. Weber, and J. Popp, “Simplified formulas for non-normal reflection from monoclinic crystals,” Opt. Commun. 284, 719–723 (2011).
[CrossRef]

2009 (1)

V. Ivanovski, T. G. Mayerhöfer, and J. Popp, “Employing polyethylene as contacting agent between ATR-crystals and solid samples with hard surfaces,” J. Mol. Struct. 924–926, 571–576 (2009).
[CrossRef]

2007 (1)

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

2006 (2)

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

T. G. Mayerhöfer and J. Popp, “Modeling IR-spectra of polycrystalline materials in the large crystallites limit—quantitative determination of orientation,” J. Opt. A: Pure Appl. Opt. 8, 657–671 (2006).
[CrossRef]

2005 (1)

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

2004 (1)

K. R. Kirov and H. E. Assender, “Quantitative ATR-IR analysis of anisotropic polymer films: extraction of optical constants,” Macromolecules 37, 894–904 (2004).
[CrossRef]

1997 (1)

1994 (1)

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

1985 (1)

1983 (1)

1981 (1)

G. Leveque, Y. Bertrand, and S. Robin-Kandare, “Determination of optical anisotropy of layered crystals in VUV range,” Solid State Commun. 38, 759–764 (1981).
[CrossRef]

1979 (1)

1978 (1)

V. F. Pavinich and M. V. Belousov, “Dispersion analysis of reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 881–883 (1978).

1974 (1)

E. E. Koch, A. Otto, and K. L. Kliewer, “Reflection spectroscopy on monoclinic crystals,” Chem. Phys. 3, 362–369 (1974).
[CrossRef]

1973 (1)

1972 (1)

1968 (1)

1963 (1)

D. W. Berreman, “Infrared absorption at longitudinal optic frequency in cubic crystal films,” Phys. Rev. 130, 2193–2198 (1963).
[CrossRef]

1961 (1)

W. G. Spitzer and D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

1887 (1)

P. Drude, “Über die Gesetze der Reflexion und Brechung des Lichtes an der Grenze absorbirender Krystalle,” Ann. Phys. Chem. 268, 584–625 (1887).
[CrossRef]

Abelès, F.

Aronson, J. R.

Asghar, A.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Assender, H. E.

K. R. Kirov and H. E. Assender, “Quantitative ATR-IR analysis of anisotropic polymer films: extraction of optical constants,” Macromolecules 37, 894–904 (2004).
[CrossRef]

Belousov, M. V.

V. F. Pavinich and M. V. Belousov, “Dispersion analysis of reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 881–883 (1978).

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]

D. W. Berreman, “Infrared absorption at longitudinal optic frequency in cubic crystal films,” Phys. Rev. 130, 2193–2198 (1963).
[CrossRef]

Bertrand, Y.

G. Leveque, Y. Bertrand, and S. Robin-Kandare, “Determination of optical anisotropy of layered crystals in VUV range,” Solid State Commun. 38, 759–764 (1981).
[CrossRef]

Bundesmann, C.

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

Dietz, N.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Drude, P.

P. Drude, “Über die Gesetze der Reflexion und Brechung des Lichtes an der Grenze absorbirender Krystalle,” Ann. Phys. Chem. 268, 584–625 (1887).
[CrossRef]

Emslie, A. G.

Feenstra, B. J.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Ferguson, I. T.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Gerrits, A. M.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Gorshunov, B. P.

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

Grundmann, M.

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

Hu, Z. G.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Ishida, H.

Ivanovski, V.

V. Ivanovski, T. G. Mayerhöfer, and J. Popp, “Employing polyethylene as contacting agent between ATR-crystals and solid samples with hard surfaces,” J. Mol. Struct. 924–926, 571–576 (2009).
[CrossRef]

Kane, M. H.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Kim, J.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Kirov, K. R.

K. R. Kirov and H. E. Assender, “Quantitative ATR-IR analysis of anisotropic polymer films: extraction of optical constants,” Macromolecules 37, 894–904 (2004).
[CrossRef]

Kleinmann, D. A.

W. G. Spitzer and D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

Kliewer, K. L.

E. E. Koch, A. Otto, and K. L. Kliewer, “Reflection spectroscopy on monoclinic crystals,” Chem. Phys. 3, 362–369 (1974).
[CrossRef]

Koch, E. E.

E. E. Koch, A. Otto, and K. L. Kliewer, “Reflection spectroscopy on monoclinic crystals,” Chem. Phys. 3, 362–369 (1974).
[CrossRef]

Kutskov, I.

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

Lee, W. Y.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Lemanov, V. V.

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

Leveque, G.

G. Leveque, Y. Bertrand, and S. Robin-Kandare, “Determination of optical anisotropy of layered crystals in VUV range,” Solid State Commun. 38, 759–764 (1981).
[CrossRef]

Lorenz, M.

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

Mayerhöfer, T. G.

T. G. Mayerhöfer, S. Weber, and J. Popp, “Simplified formulas for non-normal reflection from monoclinic crystals,” Opt. Commun. 284, 719–723 (2011).
[CrossRef]

V. Ivanovski, T. G. Mayerhöfer, and J. Popp, “Employing polyethylene as contacting agent between ATR-crystals and solid samples with hard surfaces,” J. Mol. Struct. 924–926, 571–576 (2009).
[CrossRef]

T. G. Mayerhöfer and J. Popp, “Modeling IR-spectra of polycrystalline materials in the large crystallites limit—quantitative determination of orientation,” J. Opt. A: Pure Appl. Opt. 8, 657–671 (2006).
[CrossRef]

Mosteller, L. P.

Otto, A.

E. E. Koch, A. Otto, and K. L. Kliewer, “Reflection spectroscopy on monoclinic crystals,” Chem. Phys. 3, 362–369 (1974).
[CrossRef]

Pavinich, V. F.

V. F. Pavinich and M. V. Belousov, “Dispersion analysis of reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 881–883 (1978).

Perera, A. G. U.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Popp, J.

T. G. Mayerhöfer, S. Weber, and J. Popp, “Simplified formulas for non-normal reflection from monoclinic crystals,” Opt. Commun. 284, 719–723 (2011).
[CrossRef]

V. Ivanovski, T. G. Mayerhöfer, and J. Popp, “Employing polyethylene as contacting agent between ATR-crystals and solid samples with hard surfaces,” J. Mol. Struct. 924–926, 571–576 (2009).
[CrossRef]

T. G. Mayerhöfer and J. Popp, “Modeling IR-spectra of polycrystalline materials in the large crystallites limit—quantitative determination of orientation,” J. Opt. A: Pure Appl. Opt. 8, 657–671 (2006).
[CrossRef]

Pronin, A. V.

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

Rahm, A.

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

Robin-Kandare, S.

G. Leveque, Y. Bertrand, and S. Robin-Kandare, “Determination of optical anisotropy of layered crystals in VUV range,” Solid State Commun. 38, 759–764 (1981).
[CrossRef]

Schubert, M.

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

Somal, H. S.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Soonpaa, H. H.

Spitzer, W. G.

W. G. Spitzer and D. A. Kleinmann, “Infrared lattice bands of quartz,” Phys. Rev. 121, 1324–1335 (1961).
[CrossRef]

Strassburg, M.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Strong, P. F.

Torgashev, V. I.

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

van der Marel, D.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Volkov, A. A.

B. P. Gorshunov, A. V. Pronin, I. Kutskov, A. A. Volkov, V. V. Lemanov, and V. I. Torgashev, “Polar phonons and specific features of the ferroelectric states in cadmium titanate,” Phys. Solid State 47, 547–555 (2005).
[CrossRef]

Washburn, H. A.

Weber, S.

T. G. Mayerhöfer, S. Weber, and J. Popp, “Simplified formulas for non-normal reflection from monoclinic crystals,” Opt. Commun. 284, 719–723 (2011).
[CrossRef]

Weerasekara, A. B.

Z. G. Hu, A. B. Weerasekara, N. Dietz, A. G. U. Perera, M. Strassburg, M. H. Kane, A. Asghar, and I. T. Ferguson, “Infrared optical anisotropy of diluted magnetic Ga1−xMnxN/c-sapphire epilayers grown with a GaN buffer layer by metalorganic chemical vapor deposition,” Phys. Rev. B 75, 205320 (2007).
[CrossRef]

Wittlin, A.

J. Kim, B. J. Feenstra, H. S. Somal, W. Y. Lee, A. M. Gerrits, A. Wittlin, and D. van der Marel, “c-axis infrared response of Tl2Ba2Ca2Cu3O10 studied by oblique-incidence polarized-reflectivity measurements,” Phys. Rev. B 49, 13065–13069(1994).
[CrossRef]

Wooten, F.

Yamamoto, K.

Yeh, P.

Ann. Phys. Chem. (1)

P. Drude, “Über die Gesetze der Reflexion und Brechung des Lichtes an der Grenze absorbirender Krystalle,” Ann. Phys. Chem. 268, 584–625 (1887).
[CrossRef]

Appl. Opt. (1)

Appl. Spectrosc. (1)

Chem. Phys. (1)

E. E. Koch, A. Otto, and K. L. Kliewer, “Reflection spectroscopy on monoclinic crystals,” Chem. Phys. 3, 362–369 (1974).
[CrossRef]

J. Appl. Phys. (1)

C. Bundesmann, A. Rahm, M. Lorenz, M. Grundmann, and M. Schubert, “Infrared optical properties of MgxZn1−xO thin films (0≤x≤1): long-wavelength optical phonons and dielectric constants,” J. Appl. Phys. 99, 113504 (2006).
[CrossRef]

J. Mol. Struct. (1)

V. Ivanovski, T. G. Mayerhöfer, and J. Popp, “Employing polyethylene as contacting agent between ATR-crystals and solid samples with hard surfaces,” J. Mol. Struct. 924–926, 571–576 (2009).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

T. G. Mayerhöfer and J. Popp, “Modeling IR-spectra of polycrystalline materials in the large crystallites limit—quantitative determination of orientation,” J. Opt. A: Pure Appl. Opt. 8, 657–671 (2006).
[CrossRef]

J. Opt. Soc. Am. (5)

Macromolecules (1)

K. R. Kirov and H. E. Assender, “Quantitative ATR-IR analysis of anisotropic polymer films: extraction of optical constants,” Macromolecules 37, 894–904 (2004).
[CrossRef]

Opt. Commun. (1)

T. G. Mayerhöfer, S. Weber, and J. Popp, “Simplified formulas for non-normal reflection from monoclinic crystals,” Opt. Commun. 284, 719–723 (2011).
[CrossRef]

Opt. Spectrosc. (1)

V. F. Pavinich and M. V. Belousov, “Dispersion analysis of reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 881–883 (1978).

Phys. Rev. (2)

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Other (12)

For such measurements incidence media are used that have a higher refractive index than the sample; as a consequence, light is totally reflected for incidence angles higher than the so-called critical angle except at frequencies at which the sample material is absorbing.

We would find it more consistent to name the main tensor component along the normal to the surface ε⊥ but decided to employ ε∥ instead. The latter term is generally used in the literature, which usually deals only with optical uniaxial materials, where ε∥ stands for the main or principal tensor component along the optical axis.

The formulas presented in this paper allow for anisotropic substrate materials down to monoclinic symmetry, where the monoclinic b-axis is oriented perpendicular to the substrate surface, which covers to our best knowledge all common substrate materials. Generally valid formulas can be found in .

The X- and Y-components of the wave vector k, kX and kY, are the same as for the incidence medium and are given by kX=0 and kY=nisin⁡α, wherein α is the angle of incidence and n the index of refraction of the isotropic incidence medium provided that the plane of incidence is the Y–Z plane (cf. Figure ). All components are dimensionless quantities.

For absorbing media, a positive imaginary part of γi determines a forward travelling wave, whereas the same wave in a nonabsorbing medium is characterized by a positive real part of γi.

For a semi-infinte medium the Ψ2,l and the Ψ4,l are zero .

For the wavenumber, traditionally the symbol ν˜ is also in use, especially in the context of spectroscopy [see, e.g., Eq. ].

If for both the sample medium and the substrate the dynamical matrix can be calculated according to Eq. , then the waves are uncoupled and rij=tij=0 holds.

Equation  does not directly transform under the condition of Eq. .

Note that to compute the reflectance at TIR conditions [e.g., Rp,calc(n=4,α=50°)], the index of refraction of the incidence medium (i.e., the ATR crystal) must be inserted into the wave vector and Eq. .

To be more precise, one of the main axes of the tensor ellipsoid is, independent of frequency, always oriented perpendicular to the surface of the material of interest.

In monoclinic materials only one of the principal axes has an orientation that is, independent of frequency, always perpendicular to the other two axes. Many monoclinic materials, e.g., phlogopite, can easily be cleaved perpendicular to this axis, called the monoclinic b-axis.

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Figures (8)

Fig. 1
Fig. 1

Optical models and coordinate systems used throughout this work. Medium 0: semi-infinite, nonabsorbing incidence medium; medium 1: either semi-infinite exit medium (left side) or layer (right side); medium 2: semi-infinite exit medium (right side). For simplicity, all media are considered isotropic in this figure.

Fig. 2
Fig. 2

Real [ Re ( ε i ) ] and imaginary [ Im ( ε i ) ] parts as well as dielectric loss [ Im ( 1 / ε i ) ] of the principal dielectric functions generated by Eq. (14) and the values provided by Table 1.

Fig. 3
Fig. 3

Upper part: parallel polarized reflectance spectra of a model material based on the principal dielectric functions generated by Eq. (14) and the values provided by Table 1. The model material is assumed to be semi-infinite like the incidence medium, which is characterized by an index of refraction n = 1 . Lower part: real [ Re ( ε i ) ] and imaginary [ Im ( ε i ) ] parts as well as dielectric loss [ Im ( 1 / ε i ) ] of the principal dielectric functions generated by Eq. (14).

Fig. 4
Fig. 4

Plot of the error function of the external reflection ( n = 1 ) at an angle of incidence α = 60 ° for different pairs of oscillator strength S and position ν ˜ , keeping the damping fixed at its true value of 10 cm 1 . The plot represents 10,000 data points calculated according to Eq. (15). (a) Error function employing the true value of the dielectric background ε , = 2.0 , (b) error function resulting from the assumption of an erroneous dielectric background ε , = 1.9 .

Fig. 5
Fig. 5

Plot of the error function of the total internal reflection ( n = 4 ) at an angle of incidence α = 50 ° for different pairs of oscillator strength S and position ν ˜ , keeping the damping fixed at the true value of 10 cm 1 . The plot represents 10,000 data points calculated according to Eq. (16). (a) Error function employing the true value of the dielectric background ε , = 2.0 , (b) error function resulting from the assumption of an erroneous dielectric background ε , = 1.9 .

Fig. 6
Fig. 6

Upper part: parallel polarized reflectance spectra of a model material based on the principal dielectric functions generated by Eq. (14) and the values provided by Table 1. The model material is assumed to be semi-infinite like the incidence medium, which is characterized by an index of refraction n = 4 . Lower part: real [ Re ( ε i ) ] and imaginary [ Im ( ε i ) ] parts as well as dielectric loss [ Im ( 1 / ε i ) ] of the principal dielectric functions generated by Eq. (14).

Fig. 7
Fig. 7

Dispersion analysis of the spectra of the model semi-infinite medium employing the error function according to Eq. (16). Upper part: comparison between R p ( n = 4 , α = 50 ° ) and R p , calc ( n = 4 , α = 50 ° ) . Lower part: comparison between R p ( n = 1 , α = 60 ° ) and R p , calc ( n = 1 , α = 60 ° ) .

Fig. 8
Fig. 8

Dispersion analysis of the spectra of a layered medium characterized by the same oscillator parameters as the semi-infinite medium in Fig. 7 (medium 2 is vacuum, layer thickness is 1 μm ), employing the error function according to Eq. (16). Upper part: comparison between R p ( n = 4 , α = 50 ° ) and R p , calc ( n = 4 , α = 50 ° ) . Lower part: comparison between R p ( n = 1 , α = 60 ° ) and R p , calc ( n = 1 , α = 60 ° ) .

Tables (1)

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Table 1 Oscillator Parameters Employed to Generate ε = ε X = ε Y and ε = ε Z

Equations (16)

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ε = ( ε X X ε X Y 0 ε X Y ε Y Y 0 0 0 ε Z ) ,
γ 1 = 1 2 ε Z ( K 1 + K 1 2 + K 2 ) , γ 2 = γ 1 , γ 3 = 1 2 ε Z ( K 1 + K 1 2 + K 2 ) , γ 4 = γ 3 , K 1 = ε Z ( ε X X + ε Y Y ) + k Y 2 ( ε Y Y + ε Z ) , K 2 = 4 ( ε X Y 2 + ε Y Y ( k Y 2 ε X X ) ) ( k Y 2 ε Z ) ε Z .
D Ψ ( 1 ) = ( Ψ 1 , 1 Ψ 2 , 1 Ψ 3 , 1 Ψ 4 , 1 Ψ 1 , 2 Ψ 2 , 2 Ψ 3 , 2 Ψ 4 , 2 Ψ 1 , 3 Ψ 2 , 3 Ψ 3 , 3 Ψ 4 , 3 Ψ 1 , 4 Ψ 2 , 4 Ψ 3 , 4 Ψ 4 , 4 ) , Ψ i = ( ε X Y ( 1 k Y 2 ε Z ) ε X Y ( 1 k Y 2 ε Z ) γ i ( 1 k Y 2 ε Z ) ( γ i 2 ( ε X X k Y 2 ) ) γ i 3 γ i ( ε X X k Y 2 ) ) .
D Ψ ( i ) = ( 1 1 0 0 n i cos α n i cos α 0 0 0 0 cos α cos α 0 0 n i n i ) .
P ( 1 ) = ( exp ( i k 0 d γ 1 ) 0 0 0 0 exp ( i k 0 d γ 2 ) 0 0 0 0 exp ( i k 0 d γ 3 ) 0 0 0 0 exp ( i k 0 d γ 4 ) ) ,
M ˜ = D Ψ 1 ( 0 ) D Ψ ( 1 ) P ( 1 ) D Ψ 1 ( 1 ) D Ψ ( 2 ) ,
M ˜ = D Ψ 1 ( 0 ) D Ψ ( 1 ) ,
( A s B s A p B p ) = ( M ˜ 11 M ˜ 12 M ˜ 13 M ˜ 14 M ˜ 21 M ˜ 22 M ˜ 23 M ˜ 24 M ˜ 31 M ˜ 32 M ˜ 33 M ˜ 34 M ˜ 41 M ˜ 42 M ˜ 43 M ˜ 44 ) ( C s 0 C p 0 ) .
r s s = ( B s A s ) A p = 0 = M ˜ 21 M ˜ 33 M ˜ 23 M ˜ 31 d , t s s = ( C s A s ) A p = 0 = M ˜ 33 d , r s p = ( B p A s ) A p = 0 = M ˜ 41 M ˜ 33 M ˜ 43 M ˜ 31 d , t s p = ( C p A s ) A p = 0 = M ˜ 31 d , r p s = ( B s A p ) A s = 0 = M ˜ 11 M ˜ 23 M ˜ 21 M ˜ 13 d , t p s = ( C s A p ) A s = 0 = M ˜ 13 d , r p p = ( B p A p ) A s = 0 = M ˜ 11 M ˜ 43 M ˜ 41 M ˜ 13 d , t p p = ( C p A p ) A s = 0 = M ˜ 11 d .
R i = R i i + R i j = | r i i | 2 + | r i j | 2 , T i = T i i + T i j = | t i i | 2 + | t i j | 2 .
ε = ( ε X 0 0 0 ε Y 0 0 0 ε Z ) ,
D Ψ ( 1 ) = ( 1 1 0 0 k y 2 + ε X k y 2 + ε X 0 0 0 0 k y 2 + ε Z k y 2 + ε Z 0 0 ε Y ε Z ε Y ε Z ) .
γ 1 = k y 2 + ε X , γ 2 = γ 1 , γ 3 = ε Y ( k y 2 ε Z ) ε Z , γ 4 = γ 3 .
ε j ( ν ˜ ) = ε j , + S j 2 ( ν ˜ j 2 ν ˜ 2 ) i ν ˜ γ j ; j = , .
d 2 ¯ = 1 2 ( R p ( n = 1 , α = 50 ° ) R p , calc ( n = 1 , α = 50 ° ) ) 2 + 1 2 ( R p ( n = 1 , α = 60 ° ) R p , calc ( n = 1 , α = 60 ° ) ) 2 ,
d 2 ¯ = 1 2 ( R p ( n = 4 , α = 50 ° ) R p , calc ( n = 4 , α = 50 ° ) ) 2 + 1 2 ( R p ( n = 1 , α = 60 ° ) R p , calc ( n = 1 , α = 60 ° ) ) 2 .

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