Abstract

The transmittance T and reflectance R of s- and p-polarized light, incident at various angles onto thin films which contain metal and voids in an oblique columnar structure, are analyzed using suitably modified thin film equations. Columnar aluminum is treated for various column angles in uniaxial and biaxial models to demonstrate that the model can predict the novel features found in recent experiments. Dielectric constants from quasistatic effective medium theory are used. The p-wave transmittance can be very asymmetric as incident angle θ varies about the normal, but Rp, Rs, and Ts are symmetric. It is differences in the forward and reverse imaginary part of the complex p-wave phase shift for each θ that causes Tp to be asymmetric. This difference leads to a modification to the standard thin film equations, or transfer matrix elements, which do not vanish when intensity amplitudes are calculated. Angular selective transmittance of luminous and solar radiation then becomes possible, which is important for several energy related applications.

© 1990 Optical Society of America

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References

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  1. R. T. Kivaisi, “Optical Properties of Obliquely Evaporated Aluminum Films,” Thin Solid Films 97, 153–163 (1982).
    [CrossRef]
  2. R. T. Kivaisi, “Spectral and Angular Selectivity of Obliquely Deposited Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 428, 32–38 (1983).
  3. G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
    [CrossRef]
  4. G. W. Mbise, G. B. Smith, C. G. Granqvist, “High Resolution Studies of Columnar Growth in Obliquely Deposited Metal Films on Glass,” Thin Solid Films 174, L123–L127 (1989).
    [CrossRef]
  5. O. S. Heavens, Optical Properties of Thin Films (Butterworth, London, 1955).
  6. I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, 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]
  7. G. B. Smith, “Effective Medium Theory and Angular Dispersion of Optical Constants in Films with Oblique Columnar Structure,” Opt. Commun. 71, 279–284 (1989).
    [CrossRef]
  8. N. G. Nakhodkin, A. I. Shaldervan, “Effect of Vapor Incidence Angle on Profile and Properties of Condensed Films,” Thin Solid Films 10, 109–124 (1972).
    [CrossRef]
  9. K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
    [CrossRef]
  10. G. B. Smith, “The Scope of Effective Medium Theory for Fine Metal Particle Solar Absorbers,” Appl. Phys. Lett. 35, 668–670 (1979).
    [CrossRef]
  11. D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249–262 (1982).
    [CrossRef]
  12. S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York, 1965).
  13. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1983).
  15. D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4—Matrix Formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  16. G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
    [CrossRef]
  17. C. G. Granqvist, “Energy Efficient Windows: Options with Present and Forthcoming Technology” in Electricity: Efficient End Use and New Generation Technologies, and Their Planning Implications, T. B. Johansson, B. Bodlund, R. H. Williams, Eds. (Lund U.P., Lund, Sweden, 1989), pp. 89–123.
  18. G. W. Mbise, Physics Department Chalmes University of Technology Gothenburg, Sweden unpublished.
  19. A. G. Dirks, H. J. Leany, “Columnar Microstructure in Vapour Deposited Thin Films,” Thin Solid Films 47, 219–230 (1977).
    [CrossRef]
  20. J. C. Maxwell-Garnett, “Colours in Metallic Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).
  21. R. Landauer, “Electrical Conduction in Inhomogeneous Media,” AIP Conf. Proc. 40, 2–43 (1978).
    [CrossRef]
  22. G. A. Niklasson, C. G. Granqvist, “Optical Properties and Solar Selectivity of Coevaporated Co-Al2O3 Composite Films,” J. Appl. Phys. 55, 3382–3410 (1984).
    [CrossRef]
  23. D. E. Aspnes, “Analysis of Cermet Films with Large Metal Packing Fractions,” Phys. Rev. B 33, 677–682 (1986).
    [CrossRef]

1989 (3)

G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, “High Resolution Studies of Columnar Growth in Obliquely Deposited Metal Films on Glass,” Thin Solid Films 174, L123–L127 (1989).
[CrossRef]

G. B. Smith, “Effective Medium Theory and Angular Dispersion of Optical Constants in Films with Oblique Columnar Structure,” Opt. Commun. 71, 279–284 (1989).
[CrossRef]

1986 (2)

G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
[CrossRef]

D. E. Aspnes, “Analysis of Cermet Films with Large Metal Packing Fractions,” Phys. Rev. B 33, 677–682 (1986).
[CrossRef]

1985 (2)

I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, 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]

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

1984 (1)

G. A. Niklasson, C. G. Granqvist, “Optical Properties and Solar Selectivity of Coevaporated Co-Al2O3 Composite Films,” J. Appl. Phys. 55, 3382–3410 (1984).
[CrossRef]

1983 (1)

R. T. Kivaisi, “Spectral and Angular Selectivity of Obliquely Deposited Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 428, 32–38 (1983).

1982 (2)

R. T. Kivaisi, “Optical Properties of Obliquely Evaporated Aluminum Films,” Thin Solid Films 97, 153–163 (1982).
[CrossRef]

D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249–262 (1982).
[CrossRef]

1979 (1)

G. B. Smith, “The Scope of Effective Medium Theory for Fine Metal Particle Solar Absorbers,” Appl. Phys. Lett. 35, 668–670 (1979).
[CrossRef]

1978 (1)

R. Landauer, “Electrical Conduction in Inhomogeneous Media,” AIP Conf. Proc. 40, 2–43 (1978).
[CrossRef]

1977 (1)

A. G. Dirks, H. J. Leany, “Columnar Microstructure in Vapour Deposited Thin Films,” Thin Solid Films 47, 219–230 (1977).
[CrossRef]

1976 (1)

K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
[CrossRef]

1972 (2)

N. G. Nakhodkin, A. I. Shaldervan, “Effect of Vapor Incidence Angle on Profile and Properties of Condensed Films,” Thin Solid Films 10, 109–124 (1972).
[CrossRef]

D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4—Matrix Formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, “Colours in Metallic Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).

Aspnes, D. E.

D. E. Aspnes, “Analysis of Cermet Films with Large Metal Packing Fractions,” Phys. Rev. B 33, 677–682 (1986).
[CrossRef]

D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249–262 (1982).
[CrossRef]

Berreman, D. W.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1983).

Dirks, A. G.

A. G. Dirks, H. J. Leany, “Columnar Microstructure in Vapour Deposited Thin Films,” Thin Solid Films 47, 219–230 (1977).
[CrossRef]

Fujiwara, H.

K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
[CrossRef]

Gaylord, T. K.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Granqvist, C. G.

G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, “High Resolution Studies of Columnar Growth in Obliquely Deposited Metal Films on Glass,” Thin Solid Films 174, L123–L127 (1989).
[CrossRef]

G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, “Optical Properties and Solar Selectivity of Coevaporated Co-Al2O3 Composite Films,” J. Appl. Phys. 55, 3382–3410 (1984).
[CrossRef]

C. G. Granqvist, “Energy Efficient Windows: Options with Present and Forthcoming Technology” in Electricity: Efficient End Use and New Generation Technologies, and Their Planning Implications, T. B. Johansson, B. Bodlund, R. H. Williams, Eds. (Lund U.P., Lund, Sweden, 1989), pp. 89–123.

Hara, K.

K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
[CrossRef]

Hashimoto, T.

K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Films (Butterworth, London, 1955).

Hodgkinson, I. J.

Horowitz, F.

Kivaisi, R. T.

R. T. Kivaisi, “Spectral and Angular Selectivity of Obliquely Deposited Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 428, 32–38 (1983).

R. T. Kivaisi, “Optical Properties of Obliquely Evaporated Aluminum Films,” Thin Solid Films 97, 153–163 (1982).
[CrossRef]

Knoesen, A.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Landauer, R.

R. Landauer, “Electrical Conduction in Inhomogeneous Media,” AIP Conf. Proc. 40, 2–43 (1978).
[CrossRef]

Leany, H. J.

A. G. Dirks, H. J. Leany, “Columnar Microstructure in Vapour Deposited Thin Films,” Thin Solid Films 47, 219–230 (1977).
[CrossRef]

Macleod, H. A.

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in Metallic Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).

Mbise, G. W.

G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, “High Resolution Studies of Columnar Growth in Obliquely Deposited Metal Films on Glass,” Thin Solid Films 174, L123–L127 (1989).
[CrossRef]

G. W. Mbise, Physics Department Chalmes University of Technology Gothenburg, Sweden unpublished.

Moharam, M. G.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Nakhodkin, N. G.

N. G. Nakhodkin, A. I. Shaldervan, “Effect of Vapor Incidence Angle on Profile and Properties of Condensed Films,” Thin Solid Films 10, 109–124 (1972).
[CrossRef]

Niklasson, G. A.

G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, “Optical Properties and Solar Selectivity of Coevaporated Co-Al2O3 Composite Films,” J. Appl. Phys. 55, 3382–3410 (1984).
[CrossRef]

Okamoto, K.

K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
[CrossRef]

Ramo, S.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York, 1965).

Shaldervan, A. I.

N. G. Nakhodkin, A. I. Shaldervan, “Effect of Vapor Incidence Angle on Profile and Properties of Condensed Films,” Thin Solid Films 10, 109–124 (1972).
[CrossRef]

Sikkens, M.

Smith, G. B.

G. B. Smith, “Effective Medium Theory and Angular Dispersion of Optical Constants in Films with Oblique Columnar Structure,” Opt. Commun. 71, 279–284 (1989).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, “High Resolution Studies of Columnar Growth in Obliquely Deposited Metal Films on Glass,” Thin Solid Films 174, L123–L127 (1989).
[CrossRef]

G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
[CrossRef]

G. B. Smith, “The Scope of Effective Medium Theory for Fine Metal Particle Solar Absorbers,” Appl. Phys. Lett. 35, 668–670 (1979).
[CrossRef]

Svensson, J. S. E. M.

G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
[CrossRef]

van Duzer, T.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York, 1965).

Wharton, J. J.

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York, 1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1983).

AIP Conf. Proc. (1)

R. Landauer, “Electrical Conduction in Inhomogeneous Media,” AIP Conf. Proc. 40, 2–43 (1978).
[CrossRef]

Appl. Phys. B (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic Propagation at Interfaces and in Waveguides in Uniaxial Crystals: Surface Impedance/Admittance Approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Appl. Phys. Lett. (2)

G. B. Smith, “The Scope of Effective Medium Theory for Fine Metal Particle Solar Absorbers,” Appl. Phys. Lett. 35, 668–670 (1979).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, G. A. Niklasson, “Angular Selective Optical Properties of Cr Films Made by Oblique Angle Evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

J. Appl. Phys. (2)

G. B. Smith, G. A. Niklasson, J. S. E. M. Svensson, C. G. Granqvist, “Noble Metal Based Transparent Infra-red Reflectors. Preparation and Analysis of Thin Gold Films,” J. Appl. Phys. 59, 571–581 (1986).
[CrossRef]

G. A. Niklasson, C. G. Granqvist, “Optical Properties and Solar Selectivity of Coevaporated Co-Al2O3 Composite Films,” J. Appl. Phys. 55, 3382–3410 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys. Soc. Jpn. (1)

K. Okamoto, K. Hara, H. Fujiwara, T. Hashimoto, “Dependence of Columnar Structure on Film Thickness in Iron Film Evaporation at Oblique Incidence,” J. Phys. Soc. Jpn. 40, 293–294 (1976).
[CrossRef]

Opt. Commun. (1)

G. B. Smith, “Effective Medium Theory and Angular Dispersion of Optical Constants in Films with Oblique Columnar Structure,” Opt. Commun. 71, 279–284 (1989).
[CrossRef]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell-Garnett, “Colours in Metallic Glasses and in Metallic Films,” Philos. Trans. R. Soc. London 203, 385–420 (1904).

Phys. Rev. B (1)

D. E. Aspnes, “Analysis of Cermet Films with Large Metal Packing Fractions,” Phys. Rev. B 33, 677–682 (1986).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. T. Kivaisi, “Spectral and Angular Selectivity of Obliquely Deposited Films,” Proc. Soc. Photo-Opt. Instrum. Eng. 428, 32–38 (1983).

Thin Solid Films (5)

A. G. Dirks, H. J. Leany, “Columnar Microstructure in Vapour Deposited Thin Films,” Thin Solid Films 47, 219–230 (1977).
[CrossRef]

R. T. Kivaisi, “Optical Properties of Obliquely Evaporated Aluminum Films,” Thin Solid Films 97, 153–163 (1982).
[CrossRef]

N. G. Nakhodkin, A. I. Shaldervan, “Effect of Vapor Incidence Angle on Profile and Properties of Condensed Films,” Thin Solid Films 10, 109–124 (1972).
[CrossRef]

D. E. Aspnes, “Optical Properties of Thin Films,” Thin Solid Films 89, 249–262 (1982).
[CrossRef]

G. W. Mbise, G. B. Smith, C. G. Granqvist, “High Resolution Studies of Columnar Growth in Obliquely Deposited Metal Films on Glass,” Thin Solid Films 174, L123–L127 (1989).
[CrossRef]

Other (5)

O. S. Heavens, Optical Properties of Thin Films (Butterworth, London, 1955).

S. Ramo, J. R. Whinnery, T. van Duzer, Fields and Waves in Communication Electronics, (Wiley, New York, 1965).

C. G. Granqvist, “Energy Efficient Windows: Options with Present and Forthcoming Technology” in Electricity: Efficient End Use and New Generation Technologies, and Their Planning Implications, T. B. Johansson, B. Bodlund, R. H. Williams, Eds. (Lund U.P., Lund, Sweden, 1989), pp. 89–123.

G. W. Mbise, Physics Department Chalmes University of Technology Gothenburg, Sweden unpublished.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1983).

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Figures (7)

Fig. 1
Fig. 1

Angular selective transmittance through coated glass; a schematic illustrating the principle and example aims for T vs angle of incidence.

Fig. 2
Fig. 2

Idealized columnar system used to model thin film optical properties. The film parallels the x-y plane. The x′,y′,z′ system has each axis parallel to one of the optical axes defined by the columns.

Fig. 3
Fig. 3

Reflectance and transmittance amplitude coefficients and associated (phase amplitude) rays. Ray paths inside the film are not plane polarized and are oversimplified for illustration purposes.

Fig. 4
Fig. 4

Tp vs angle of incidence in columnar aluminum at λ = 5220 Å in a 100-Å thick film. Br models are used with uniaxial symmetry (Lz = 0.05): (a) for 100-Å film thickness and various column angles β. Fill factor f is correlated to β from experimental data.1 (b) For various thicknesses at β = 60°, f = 0.717. One curve is for f = 0.47 near the percolation limits of f = 0.475 in Br theory for the Lz used.

Fig. 5
Fig. 5

Transmittance (Tp and Ts) vs angle of incidence at λ = 5220 Å for columnar aluminum. Column angle, void fill factor, and film thickness are as shown. Uniaxial U and biaxial B results are given for the stated combinations of Lx,Ly,Lz: (a) Bruggeman theory; (b) Maxwell Garnett theory.

Fig. 6
Fig. 6

Tp vs angle of incidence at λ = 5220 Å in columnar aluminum. The MG model is used with a constant void fill factor of 0.7. Curves refer to the indicated column angles.

Fig. 7
Fig. 7

Tp vs wavelength at angles of incidence of ±70, ±35, and 0°. In both models β = 60°, f = 0.7, Lz = 0.05, and thickness = 200 Å. Symmetry—uniaxial. (a) Aluminum—Bruggeman; (b) aluminum—Maxwell Garnett. Included are data at +70° (⋯) and −70° (- - - -) angle of incidence18 for a thicker film, ~400 Å thick deposited at an 80° angle of incidence.

Equations (35)

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T u ( θ , ϕ ) = ½ [ T s ( θ , ϕ ) + T p ( θ , ϕ ) ] .
T p ( θ ) T p ( - θ ) ,
T s ( θ ) = T s ( - θ ) .
z z = x sin 2 β + z cos 2 β ,
x z = ( z - x ) cos β sin β ,
t 12 s = 2 Y 1 s Y 1 s + Y 2 s ,
r 12 s = Y 1 s - Y 2 s Y 1 s + Y 2 s ,
Y 1 s = - 1 cos θ ,
Y 2 s = y - 1 sin 2 θ .
t 12 p = 2 Z 1 p Z 1 p + Z 2 p ,
r 12 p = Z 1 p - Z 2 p Z 1 p + Z 2 p ,
Z 1 p = cos θ 1 .
Z 2 p = 1 2 2 - 1 sin 2 θ .
Z 2 p = z z - 1 sin 2 θ x z ,
k 2 = k x 2 + k z 2 = k 0 2
k z s = k 0 ( n z s + i K z s ) = k 0 y - 1 sin 2 θ ,
k z p = k 0 ( n z p + i K z p ) = k 0 ( x z sin θ ± x z Z 2 p z z ) .
t 13 s = t 12 s t 23 s exp ( i k z s t ) 1 - r 23 s r 21 s exp ( 2 i k z s t ) ,
r 13 s = r 12 s + t 12 s r 23 s t 21 s exp ( 2 i k z s t ) 1 - r 23 s r 21 s exp ( 2 i k z s t ) .
t 13 p = t 12 p t 23 p exp ( i k z + p t ) 1 - r 23 p r 21 p exp [ i ( k z + p - k z - p ) t ] ,
r 13 p = r 12 p + t 12 p r 23 p t 21 p exp [ i ( k z + p - k z - p ) t ] 1 - r 23 p r 21 p exp [ i ( k z + p - k z - p ) t ] .
k z + p ( - θ ) = - k z - p ( + θ ) ,
k z + p ( + θ ) k z + p ( - θ ) ,
t 13 p ( + θ ) t 13 p ( - θ ) .
K z + p ( + θ ) K z + p ( - θ )
T 13 p ( - θ ) T 13 p ( + θ ) = exp [ - 2 k 0 γ ( θ ) t ] ,
γ ( θ ) = - [ K z + p ( θ ) + K z - p ( θ ) ] .
F p ( z + t ) = M p ( t ) F p ( z )
F p ( z ) = [ H y ( z ) E x ( z ) ] ,
M p ( t ) = exp ( i Φ p + / 2 ) [ cos ( Φ p - / 2 ) - i Z p sin ( Φ p - / 2 ) - i Z p sin ( Φ p - / 2 ) cos ( Φ p - / 2 ) ] .
Φ p + ( θ ) = ( k z + p + k z - p ) t ,
Φ p - ( θ ) = ( k z + p - k z - p ) t .
tan β = ½ tan δ .
f a [ a - Br Br + L a ( a - Br ) ] + f b [ b - β Br Br + L b ( b - Br ) ] = 0.
MG - b b + L b ( MG - b ) = f a [ a - b b + L a ( a - b ) ] .

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