Abstract

The angular dependence of the reflectance from an absorbing randomly oriented polydomain medium consisting of domains either small or large compared with the wavelength is investigated. Besides the two conventional cases, where the refractive index of the incidence medium is either smaller or larger than the averaged index of refraction (small-domain limit) or every principal index of refraction of the domains (large-domain limit), we also discuss a third principal case, which exists only in the large-domain limit. In this third case, only one of the principal indices of refraction is larger than that of the incidence medium, while the averaged index of refraction is smaller. Thus, in contrast to the small-domain limit, total reflection is completely suppressed even in the absence of absorption. A characteristic property of such a polydomain medium is its ability to considerably depolarize linear polarized light in spite of being optically isotropic. Additionally, the parallel polarized reflectance Rp can exceed the perpendicular polarized reflectance Rs over certain angle of incidence ranges. Absorption decreases these domain-size-dependent properties, even under the assumption of constant anisotropy. Nevertheless, for materials with low absorption indices, these effects can affect the optical properties significantly.

© 2005 Optical Society of America

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References

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  1. G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
    [CrossRef]
  2. P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
    [CrossRef]
  3. Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
    [CrossRef]
  4. References [2, 3] do not explicitly state the existence of an optical crystallite size effect. Its existence can be deduced, however, from the need to apply different methods of averaging (effective-medium approximation in Ref. [2] and averaging over the reflectances in Ref. [3]). Doll’s approach has been used earlier [R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976)] but without a statement about the range of applicability with regard to the domain size.
    [CrossRef]
  5. T. G. Mayerhöfer, “New method of modeling infrared spectra of non-cubic single-phase polycrystalline materials with random orientation,” Appl. Spectrosc. 56, 1194–1205 (2002).
    [CrossRef]
  6. The critical size of the crystallites may be one tenth of a wavelength. Below this size, it is assumed that light is unable to detect the anisotropic nature of the individual crystallite so that an averaged index of refraction results.[5] In the case of the effective-medium approach, the crystallites must be small compared with wavelength to justify this quasi-static approach. Accordingly, the critical size is usually presumed to be λ∕10.[1] Up to now the exact critical size has not been experimentally determined. On the basis of the experiments carried out so far,[5] the actual critical size lies between about λ∕5 and λ∕20.
  7. As an alternative, average refractive index theory[5] can be used.
  8. P. Yeh, Optical Waves in Layered Media (Wiley, 1988)..
  9. I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
    [CrossRef]
  10. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  11. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  12. T. G. Mayerhöfer, “Modelling IR spectra of single-phase polycrystalline materials with random orientation in the large crystallites limit—extension to arbitrary crystal symmetry,” J. Opt. A, Pure Appl. Opt. 4, 540–548 (2002).
    [CrossRef]
  13. T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
    [CrossRef]
  14. T. G. Mayerhöfer, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—a unified approach,” Vib. Spectrosc. 35, 67–76 (2004).
    [CrossRef]
  15. T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
    [CrossRef]
  16. T. G. Mayerhöfer, J. Musfeldt, “Angular dependence of the reflectance from an isotropic medium: surprising results regarding Brewster’s angle,” J. Opt. Soc. Am. A 22, 185–189 (2005).
    [CrossRef]
  17. T. G. Mayerhöfer, J. Popp, “Angular dependence of the reflectance from an isotropic polydomain medium: effect of large domain size on total reflection,” J. Opt. Soc. Am. A 22, 569–573 (2005).
    [CrossRef]
  18. A comparison of different orientation representations and their usefulness for orientational averaging can be found in T. G. Mayerhöfer, “Symmetric Euler orientation representations for Orientational averaging,” Spectrochim. Acta, Part A 61, 2611–2621 (2005).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, 1999).
    [CrossRef]
  20. The average has to be carried out over the reflection coefficients r(Ω) instead of the reflectances R(Ω) if a coherent light source is used.
  21. M. V. Belousov, V. F. Pavinich, “Infrared reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 771–774 (1978).
  22. Francis M. Mirabella, ed., Internal Reflection Spectroscopy (Marcel Dekker, 1993).
  23. P. Parayanthal, F. H. Pollak, “Raman scattering in alloy semiconductors: ‘Spatial correlation’ model,” Phys. Rev. Lett. 52, 1822–1825 (1984).
    [CrossRef]

2005

T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
[CrossRef]

A comparison of different orientation representations and their usefulness for orientational averaging can be found in T. G. Mayerhöfer, “Symmetric Euler orientation representations for Orientational averaging,” Spectrochim. Acta, Part A 61, 2611–2621 (2005).
[CrossRef]

T. G. Mayerhöfer, J. Musfeldt, “Angular dependence of the reflectance from an isotropic medium: surprising results regarding Brewster’s angle,” J. Opt. Soc. Am. A 22, 185–189 (2005).
[CrossRef]

T. G. Mayerhöfer, J. Popp, “Angular dependence of the reflectance from an isotropic polydomain medium: effect of large domain size on total reflection,” J. Opt. Soc. Am. A 22, 569–573 (2005).
[CrossRef]

2004

T. G. Mayerhöfer, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—a unified approach,” Vib. Spectrosc. 35, 67–76 (2004).
[CrossRef]

2003

T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
[CrossRef]

2002

T. G. Mayerhöfer, “Modelling IR spectra of single-phase polycrystalline materials with random orientation in the large crystallites limit—extension to arbitrary crystal symmetry,” J. Opt. A, Pure Appl. Opt. 4, 540–548 (2002).
[CrossRef]

T. G. Mayerhöfer, “New method of modeling infrared spectra of non-cubic single-phase polycrystalline materials with random orientation,” Appl. Spectrosc. 56, 1194–1205 (2002).
[CrossRef]

1999

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
[CrossRef]

1987

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
[CrossRef]

Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
[CrossRef]

1984

P. Parayanthal, F. H. Pollak, “Raman scattering in alloy semiconductors: ‘Spatial correlation’ model,” Phys. Rev. Lett. 52, 1822–1825 (1984).
[CrossRef]

1978

M. V. Belousov, V. F. Pavinich, “Infrared reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 771–774 (1978).

1976

References [2, 3] do not explicitly state the existence of an optical crystallite size effect. Its existence can be deduced, however, from the need to apply different methods of averaging (effective-medium approximation in Ref. [2] and averaging over the reflectances in Ref. [3]). Doll’s approach has been used earlier [R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976)] but without a statement about the range of applicability with regard to the domain size.
[CrossRef]

1972

Abdulhalim, I.

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Belousov, M. V.

M. V. Belousov, V. F. Pavinich, “Infrared reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 771–774 (1978).

Berreman, W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1999).
[CrossRef]

Collins, R. T.

Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
[CrossRef]

Doll, G. L.

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

Dresselhaus, G.

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

Dresselhaus, M. S.

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

Engler, E. M.

Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
[CrossRef]

Frech, R.

References [2, 3] do not explicitly state the existence of an optical crystallite size effect. Its existence can be deduced, however, from the need to apply different methods of averaging (effective-medium approximation in Ref. [2] and averaging over the reflectances in Ref. [3]). Doll’s approach has been used earlier [R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976)] but without a statement about the range of applicability with regard to the domain size.
[CrossRef]

Höche, T.

T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
[CrossRef]

Keding, R.

T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
[CrossRef]

T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
[CrossRef]

Mayerhöfer, T. G.

T. G. Mayerhöfer, J. Musfeldt, “Angular dependence of the reflectance from an isotropic medium: surprising results regarding Brewster’s angle,” J. Opt. Soc. Am. A 22, 185–189 (2005).
[CrossRef]

T. G. Mayerhöfer, J. Popp, “Angular dependence of the reflectance from an isotropic polydomain medium: effect of large domain size on total reflection,” J. Opt. Soc. Am. A 22, 569–573 (2005).
[CrossRef]

T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
[CrossRef]

A comparison of different orientation representations and their usefulness for orientational averaging can be found in T. G. Mayerhöfer, “Symmetric Euler orientation representations for Orientational averaging,” Spectrochim. Acta, Part A 61, 2611–2621 (2005).
[CrossRef]

T. G. Mayerhöfer, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—a unified approach,” Vib. Spectrosc. 35, 67–76 (2004).
[CrossRef]

T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
[CrossRef]

T. G. Mayerhöfer, “New method of modeling infrared spectra of non-cubic single-phase polycrystalline materials with random orientation,” Appl. Spectrosc. 56, 1194–1205 (2002).
[CrossRef]

T. G. Mayerhöfer, “Modelling IR spectra of single-phase polycrystalline materials with random orientation in the large crystallites limit—extension to arbitrary crystal symmetry,” J. Opt. A, Pure Appl. Opt. 4, 540–548 (2002).
[CrossRef]

McWhirter, J. T.

P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
[CrossRef]

Musfeldt, J.

T. G. Mayerhöfer, J. Musfeldt, “Angular dependence of the reflectance from an isotropic medium: surprising results regarding Brewster’s angle,” J. Opt. Soc. Am. A 22, 185–189 (2005).
[CrossRef]

T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
[CrossRef]

Noh, T. W.

P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
[CrossRef]

Parayanthal, P.

P. Parayanthal, F. H. Pollak, “Raman scattering in alloy semiconductors: ‘Spatial correlation’ model,” Phys. Rev. Lett. 52, 1822–1825 (1984).
[CrossRef]

Pavinich, V. F.

M. V. Belousov, V. F. Pavinich, “Infrared reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 771–774 (1978).

Pollak, F. H.

P. Parayanthal, F. H. Pollak, “Raman scattering in alloy semiconductors: ‘Spatial correlation’ model,” Phys. Rev. Lett. 52, 1822–1825 (1984).
[CrossRef]

Popp, J.

Schlesinger, Z.

Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
[CrossRef]

Shaver, M. W.

Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
[CrossRef]

Shen, Z.

T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
[CrossRef]

T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
[CrossRef]

Sievers, A. J.

P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
[CrossRef]

Steinbeck, J.

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

Strauss, A. J.

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

Sulewski, P. E.

P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, 1999).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988)..

Zeiger, H. J.

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

Appl. Spectrosc.

J. Opt. A, Pure Appl. Opt.

I. Abdulhalim, “Analytic propagation matrix method for linear optics of arbitrary biaxial layered media,” J. Opt. A, Pure Appl. Opt. 1, 646–653 (1999).
[CrossRef]

T. G. Mayerhöfer, “Modelling IR spectra of single-phase polycrystalline materials with random orientation in the large crystallites limit—extension to arbitrary crystal symmetry,” J. Opt. A, Pure Appl. Opt. 4, 540–548 (2002).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Spectrosc.

M. V. Belousov, V. F. Pavinich, “Infrared reflection spectra of monoclinic crystals,” Opt. Spectrosc. 45, 771–774 (1978).

Optik (Stuttgart)

T. G. Mayerhöfer, Z. Shen, R. Keding, T. Höche, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—supplementations and refinements for optically uniaxial crystallites,” Optik (Stuttgart) 114, 351–359 (2003).
[CrossRef]

Phys. Rev. B

G. L. Doll, J. Steinbeck, G. Dresselhaus, M. S. Dresselhaus, A. J. Strauss, H. J. Zeiger, “Infrared anisotropy of La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 8884–8887 (1987).
[CrossRef]

P. E. Sulewski, T. W. Noh, J. T. McWhirter, A. J. Sievers, “Far-infrared composite-medium study of sintered La2NiO4 and La1.85Sr0.15CuO4−y,” Phys. Rev. B 36, 5735–5738 (1987).
[CrossRef]

Z. Schlesinger, R. T. Collins, M. W. Shaver, E. M. Engler, “Normal-state reflectivity and superconducting energy-gap measurement of La2−xSrxCuO4,” Phys. Rev. B 36, 5275–5278 (1987).
[CrossRef]

References [2, 3] do not explicitly state the existence of an optical crystallite size effect. Its existence can be deduced, however, from the need to apply different methods of averaging (effective-medium approximation in Ref. [2] and averaging over the reflectances in Ref. [3]). Doll’s approach has been used earlier [R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976)] but without a statement about the range of applicability with regard to the domain size.
[CrossRef]

T. G. Mayerhöfer, Z. Shen, R. Keding, J. Musfeldt, “Optical isotropy in polycrystalline Ba2TiSi2O8: testing the limits of a well-established concept,” Phys. Rev. B 71, 184116 (2005).
[CrossRef]

Phys. Rev. Lett.

P. Parayanthal, F. H. Pollak, “Raman scattering in alloy semiconductors: ‘Spatial correlation’ model,” Phys. Rev. Lett. 52, 1822–1825 (1984).
[CrossRef]

Spectrochim. Acta, Part A

A comparison of different orientation representations and their usefulness for orientational averaging can be found in T. G. Mayerhöfer, “Symmetric Euler orientation representations for Orientational averaging,” Spectrochim. Acta, Part A 61, 2611–2621 (2005).
[CrossRef]

Vib. Spectrosc.

T. G. Mayerhöfer, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—a unified approach,” Vib. Spectrosc. 35, 67–76 (2004).
[CrossRef]

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

The critical size of the crystallites may be one tenth of a wavelength. Below this size, it is assumed that light is unable to detect the anisotropic nature of the individual crystallite so that an averaged index of refraction results.[5] In the case of the effective-medium approach, the crystallites must be small compared with wavelength to justify this quasi-static approach. Accordingly, the critical size is usually presumed to be λ∕10.[1] Up to now the exact critical size has not been experimentally determined. On the basis of the experiments carried out so far,[5] the actual critical size lies between about λ∕5 and λ∕20.

As an alternative, average refractive index theory[5] can be used.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988)..

M. Born, E. Wolf, Principles of Optics (Pergamon, 1999).
[CrossRef]

The average has to be carried out over the reflection coefficients r(Ω) instead of the reflectances R(Ω) if a coherent light source is used.

Francis M. Mirabella, ed., Internal Reflection Spectroscopy (Marcel Dekker, 1993).

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Figures (5)

Fig. 1
Fig. 1

Angular dependence of the reflectances R s , R p of an isotropic semi-infinite medium consisting of either optically small ( d λ 10 ) or large ( d λ 10 ) uniaxial domains ( n ord = 2.2 and n extraord = 1.6 ) for several principal absorption indices ( k i = 0 , 1 2 n i , n i , 2 n i , i = ordinary , extraordinary). The incidence medium is characterized by an index of refraction n inc = 1 .

Fig. 2
Fig. 2

Upper panel: angular dependence of the cross-polarization terms R s p and R p s . Middle panel: difference between R p (large domains) and R p (small domains). Lower panel: difference between R s (large domains) and R s (small domains). ( n inc = 1 , n ord = 2.2 , and n extraord = 1.6 ; k i = 0 , 1 2 n i , n i , 2 n i , i = ordinary , extraordinary).

Fig. 3
Fig. 3

Angular dependence of the reflectances R s , R p and the cross-polarized reflectance terms R s p and R p s ( R s p = R p s ) of an isotropic semi-infinite medium consisting of optically large uniaxial domains ( n ord = 2.2 and n extraord = 1.6 ) for several principal absorption indices ( k i = 0 , 1 2 n i , n i , 2 n i , i = ordinary , extraordinary) and comparison with R s and R p of the corresponding small-domain medium. The incidence medium is characterized by an index of refraction higher than any of the principal indices n ord and n extraord of the polydomain medium ( n inc = 3 ) .

Fig. 4
Fig. 4

Angular dependence of the reflectances R s , R p and the cross-polarized reflectance terms R s p and R p s ( R s p = R p s ) of an isotropic semi-infinite medium consisting of optically large uniaxial domains ( n ord = 2.2 and n extraord = 1.6 ) for several principal absorption indices ( k i = 0 , 1 2 n i , n i , 2 n i , i = ordinary , extraordinary) and comparison with R s and R p of the corresponding small-domain medium. The incidence medium is characterized by an index of refraction that lies between the principal indices n ord and n extraord of the polydomain medium ( n inc = 2 ) .

Fig. 5
Fig. 5

Comparison of the infrared reflectance of polydomain media consisting of optically uniaxial domains (either large or small). The principal dielectric functions are generated by the classical damped harmonic oscillator model with the following oscillator parameters: oscillator strengths S ord = S extraord = 1000 cm 1 , damping constants γ ord = γ extraord = 10 cm 1 , oscillator positions ν ̃ ord = 900 cm 1 and ν ̃ extraord = 1000 cm 1 , and dielectric background constants ε , ord = 3.1 and ε , extraord = 2.9 . Column 1: wavenumber dependence of the principal indices of refraction n i and indices of absorption k i of the domains, wavenumber dependence of the averaged index of refraction n and averaged index of absorption k (small domains); cross-polarization terms of the reflectance R s p and R p s ( R s p = R p s ) for two angles of incidence α = 20 ° and α = 70 ° (large domains). Column 2: Comparison of the reflectance for s- and p-polarized incident radiation and two angles of incidence α = 20 ° and 70°.

Tables (2)

Tables Icon

Table 1 Parameters Used to Compute the Principal Dielectric Functions ε ord and ε extraord

Tables Icon

Table 2 Comparison of n ord , n extraord , and n inc in the Spectral Range from 800 1400 cm 1

Equations (6)

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

n = N ( 3 ) Ω ( 3 ) ( n 1 ( Ω ) 2 + n 2 ( Ω ) 2 ) d Ω ,
R = 1 2 R s + 1 2 R p = N ( 3 ) Ω ( 3 ) ( R s ( Ω ) 2 + R p ( Ω ) 2 ) d Ω .
R s ( Ω ) = R s s ( Ω ) + R s p ( Ω ) ,
R p ( Ω ) = R p p ( Ω ) + R p s ( Ω ) .
ε i ( ν ̃ ) = ε , i + S i 2 ( ν ̃ i 2 ν ̃ 2 ) i ν ̃ γ i ,
i = ordinary , extraordinary .

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