Abstract

The angular dependence of the reflectance from an isotropic medium with large, randomly oriented, ordered domains is investigated and compared with that of an isotropic medium in which the optical properties can be characterized by a scalar dielectric function. Employing average reflectance and transmittance theory, we demonstrate that an isotropic medium consisting of optically large ordered domains exhibits only a parallel-polarized reflectance minimum, not the formal zero expected at Brewster’s angle. We attribute this result to the existence of nonzero cross-polarization terms despite the random orientation and the application of an incoherent averaging scheme, which is used to mimic traditional light sources.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
  2. 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]
  3. 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 (Munich) 114, 351–359 (2003).
  4. T. G. Mayerhöfer, “Modelling IR-spectra of single-phase polycrystalline materials with random orientation—a unified approach,” Vib. Spectrosc. 35, 67–76 (2004).
    [CrossRef]
  5. S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
    [CrossRef] [PubMed]
  6. Note that this size effect is purely optical in nature, independent of material properties with the exception that the optical constants are anisotropic in the ordered or single-crystalline form and that the size effect clearly depends on the wavelength. (In the visible spectral range, e.g., ordered regions must be considered large if their diameter exceeds 30–70 nm.)
  7. 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]
  8. D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
    [CrossRef]
  9. 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]
  10. C. Pecharromán, J. E. Iglesias, “Effect of particle shape on the IR reflectance spectra of pressed powders of anisotropic materials,” Appl. Spectrosc. 54, 634–638 (2000).
    [CrossRef]
  11. T. G. Mayerhöfer, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (2004).
    [CrossRef]
  12. R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976).
    [CrossRef]
  13. On the basis of the models presented in Refs. 2-4 and 7, it is also possible to calculate the properties of randomly oriented materials with lower symmetry of the ordered regions.
  14. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  15. 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]
  16. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  17. Because of the excellent resemblance between the experimental and the theoretical curves, we have not considered scattering or coherence-related effects in our model for the large-domain case. The assumption of coherence-related effects, in particular, would lower the agreement between modeled and measured spectra.4
  18. E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
    [CrossRef]
  19. Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
    [CrossRef]
  20. J. A. Zvirgzd, “Refractive-index of polycrystalline ferroelectrics determined by Brewster angle method,” Phys. Status Solidi A 32, K181–K185 (1975).
    [CrossRef]
  21. H. J. Lewerenz, N. Dietz, “Brewster-angle spectroscopy: a new method for characterization of defect levels in semiconductors,” Appl. Phys. Lett. 59, 1470–1472 (1991).
    [CrossRef]
  22. As a consequence and on the basis of Eqs. (4), however, Rppmust be zero at αMin.
  23. Note that such contributions do not occur if a coherent averaging scheme is applied. An example of a coherent light source is a laser.

2004 (2)

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, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (2004).
[CrossRef]

2003 (2)

E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
[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 (Munich) 114, 351–359 (2003).

2002 (2)

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]

2000 (1)

1999 (1)

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]

1998 (1)

S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
[CrossRef] [PubMed]

1991 (2)

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[CrossRef]

H. J. Lewerenz, N. Dietz, “Brewster-angle spectroscopy: a new method for characterization of defect levels in semiconductors,” Appl. Phys. Lett. 59, 1470–1472 (1991).
[CrossRef]

1987 (1)

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]

1976 (1)

R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976).
[CrossRef]

1975 (2)

J. A. Zvirgzd, “Refractive-index of polycrystalline ferroelectrics determined by Brewster angle method,” Phys. Status Solidi A 32, K181–K185 (1975).
[CrossRef]

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[CrossRef]

1972 (1)

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]

Baba, A.

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[CrossRef]

Berreman, W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Clarke, S. M.

S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
[CrossRef] [PubMed]

Diaz-Marti, E.

E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
[CrossRef]

Dietz, N.

H. J. Lewerenz, N. Dietz, “Brewster-angle spectroscopy: a new method for characterization of defect levels in semiconductors,” Appl. Phys. Lett. 59, 1470–1472 (1991).
[CrossRef]

Dunken, H. H.

T. G. Mayerhöfer, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (2004).
[CrossRef]

Finkelmann, H.

S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
[CrossRef] [PubMed]

Frech, R.

R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976).
[CrossRef]

Goto, T.

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[CrossRef]

Hirai, T.

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[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 (Munich) 114, 351–359 (2003).

Iglesias, J. E.

Keding, R.

T. G. Mayerhöfer, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (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 (Munich) 114, 351–359 (2003).

Kundler, I.

S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
[CrossRef] [PubMed]

Lewerenz, H. J.

H. J. Lewerenz, N. Dietz, “Brewster-angle spectroscopy: a new method for characterization of defect levels in semiconductors,” Appl. Phys. Lett. 59, 1470–1472 (1991).
[CrossRef]

Loste, E.

E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
[CrossRef]

Masuda, Y.

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[CrossRef]

Masumoto, H.

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[CrossRef]

Mayerhöfer, T. G.

T. G. Mayerhöfer, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (2004).
[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 (Munich) 114, 351–359 (2003).

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]

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]

Meldrum, F. C.

E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
[CrossRef]

Minakata, M.

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[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]

Pecharromán, C.

Rüssel, C.

T. G. Mayerhöfer, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (2004).
[CrossRef]

Shen, Z.

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 (Munich) 114, 351–359 (2003).

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]

Stroud, D.

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[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]

Terentjev, E. M.

S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Yeh, P.

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

Zarbakhsh, A.

E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
[CrossRef]

Zvirgzd, J. A.

J. A. Zvirgzd, “Refractive-index of polycrystalline ferroelectrics determined by Brewster angle method,” Phys. Status Solidi A 32, K181–K185 (1975).
[CrossRef]

Appl. Phys. Lett. (1)

H. J. Lewerenz, N. Dietz, “Brewster-angle spectroscopy: a new method for characterization of defect levels in semiconductors,” Appl. Phys. Lett. 59, 1470–1472 (1991).
[CrossRef]

Appl. Spectrosc. (2)

J. Non-Cryst. Solids (1)

T. G. Mayerhöfer, H. H. Dunken, R. Keding, C. Rüssel, “Interpretation and modelling of IR-reflectance spectra of glasses considering medium range order,” J. Non-Cryst. Solids 333, 172–181 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

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]

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]

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

Y. Masuda, A. Baba, H. Masumoto, T. Goto, M. Minakata, T. Hirai, “Preparation and dielectric and electrooptic properties of Bi4Ti3O12 films by electron cyclotron resonance plasma sputtering deposition,” Jpn. J. Appl. Phys., Part 1 30, 2212–2215 (1991).
[CrossRef]

Langmuir (1)

E. Loste, E. Diaz-Marti, A. Zarbakhsh, F. C. Meldrum, “Study of calcium carbonate precipitation under a series of fatty acid Langmuir monolayers using Brewster angle microscopy,” Langmuir 19, 2830–2837 (2003).
[CrossRef]

Macromolecules (1)

S. M. Clarke, E. M. Terentjev, I. Kundler, H. Finkelmann, “Texture evolution during the polydomain–monodomain transition in nematic elastomers,” Macromolecules 31, 4862–4872 (1998).
[CrossRef] [PubMed]

Optik (Munich) (1)

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 (Munich) 114, 351–359 (2003).

Phys. Rev. B (3)

R. Frech, “Infrared reflectivity of uniaxial microcrystalline powders,” Phys. Rev. B 13, 2342–2348 (1976).
[CrossRef]

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[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]

Phys. Status Solidi A (1)

J. A. Zvirgzd, “Refractive-index of polycrystalline ferroelectrics determined by Brewster angle method,” Phys. Status Solidi A 32, K181–K185 (1975).
[CrossRef]

Vib. Spectrosc. (1)

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 (7)

Note that this size effect is purely optical in nature, independent of material properties with the exception that the optical constants are anisotropic in the ordered or single-crystalline form and that the size effect clearly depends on the wavelength. (In the visible spectral range, e.g., ordered regions must be considered large if their diameter exceeds 30–70 nm.)

On the basis of the models presented in Refs. 2-4 and 7, it is also possible to calculate the properties of randomly oriented materials with lower symmetry of the ordered regions.

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Because of the excellent resemblance between the experimental and the theoretical curves, we have not considered scattering or coherence-related effects in our model for the large-domain case. The assumption of coherence-related effects, in particular, would lower the agreement between modeled and measured spectra.4

As a consequence and on the basis of Eqs. (4), however, Rppmust be zero at αMin.

Note that such contributions do not occur if a coherent averaging scheme is applied. An example of a coherent light source is a laser.

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Figures (5)

Fig. 1
Fig. 1

Experimental and modeled reflectances of polycrystalline fresnoite (Ba2TiSi2O8) consisting of exclusively large (left panel) or small domains (right panel), compared with wavelength.4-7 The simulations were carried out according to Eqs. (1) (dλ/10) and (3) (dλ/10) based on the single-crystal dielectric function tensor.4

Fig. 2
Fig. 2

Optical model and coordinate system used in this paper. Medium A: semi-infinite, nonabsorbing incidence medium (n0=1); medium B: semi-infinite exit medium with optically uniaxial ordered domains (nord=2.2 and nextraord=1.6).

Fig. 3
Fig. 3

(a) Calculated angular dependence of the perpendicular and parallel-polarized reflectances, Rs and Rp, of an isotropic medium consisting of optically small (dλ/10) and large (dλ/10) ordered domains. (b) Close-up view of the calculated angular dependence of the parallel-polarized reflectance, Rp, for an isotropic medium consisting of optically small (dλ/10) and large (dλ/10) ordered regions near αBrew and αMin, respectively. For comparison, we also show the cross-polarization reflectance (Rsp=Rps).

Fig. 4
Fig. 4

Angular dependence of Rsp, Rps (thin curves), and Rp (thick curves) calculated for an anisotropic medium for different orientations Ω=Ω(φ, θ). (a) φ=15°, 45°, and 75° and θ=45° (for these orientations, RspRps). (b) φ=15°, 45°, and 75° and θ=90° (in case of these orientations, Rsp=Rps).

Fig. 5
Fig. 5

Angular dependence of Rp calculated for an anisotropic medium for both φ=0°, θ=0° (optical axis perpendicular to the interface) and φ=90°, θ=90° (optical axis in the interface plane) oriented parallel to the X axis. The (incoherent) average of these quantities, [Rp(0°, 0°)+Rp(90°, 90°)]/2, is also shown.

Equations (4)

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

n=N(2)Ω(2)n1(Ω)2+n2(Ω)2dΩ.
n=23nord+13nextraord=2
R=12Rs+12Rp=N(2)Ω(2)Rs(Ω)2+Rp(Ω)2dΩ.
Rs(Ω)=Rss(Ω)+Rsp(Ω),Rp(Ω)=Rpp(Ω)+Rps(Ω),Rsp=Rps.

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