Abstract

Two different types of inhomogeneity of optical thin films resulting from the deposition conditions are considered to be the origin of volume scattering. With the use of a theory based on a first-order Born approximation, both structure types can be treated easily. In the case of an oblique columnar structure, the light-scattering patterns show a pronounced anisotropy and a strong dependence on the side at which the light enters the sample. As explanation of these facts, the role of the four partial scattering processes inside the film is revealed, leading to a simple diffraction model that reproduces the main scatter features and makes the individual physical processes involved more obvious. Inclusions without any preferred orientation only contribute to other isotropic background scatter sources such as roughness, allowing the use of anisotropic scatter for investigations of structure.

© 1995 Optical Society of America

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    [CrossRef]
  4. A. Duparré, S. Kassam, “Relations between light scattering and microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1993).
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  5. P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
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  8. R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 490–495 (1986).
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    [CrossRef]
  12. M. Sikkens, I. J. Hodgkinson, F. Horowitz, H. A. Macleod, J. J. Wharton, “Computer simulation of thin film growth: applying the results to optical coatings,” Opt. Eng. 25, 142–147 (1986).
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    [CrossRef]
  15. S. Müller-Pfeiffer, H. Van Kranenburg, J. C. Lodder, “A two-dimensional Monte Carlo model for thin film growth by oblique evaporation: simulation of two-component systems for the example of Co–Cr,” Thin Solid Films 213, 143–153 (1992).
    [CrossRef]
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  21. T. Motohiro, Y. Taga, “Thin film retardation plate by oblique deposition,” Appl. Opt. 28, 2466–2482 (1989).
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  23. I. J. Hodgkinson, “Optical anisotropy in thin films deposited obliquely: in situobservations and computer modeling,” Appl. Opt. 30, 1303–1312 (1991).
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  24. F. Flory, D. Endelema, E. Pelletier, I. J. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2films,” Appl. Opt. 32, 5649–5659 (1993).
    [CrossRef] [PubMed]
  25. I. J. Hodgkinson, Q. H. Wu, “Optical properties of single layer and multilayer anisotropic coatings,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 882–892 (1994).
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  26. I. J. Hodgkinson, P. I. Bowmar, Q. H. Wu, “Scatter from tilted-columnar birefringent thin films: observation and measurement of anisotropic scatter distributions,” Appl. Opt. 33, 163–168 (1994).
  27. S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
    [CrossRef] [PubMed]
  28. P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Media 1, 207–221 (1991).
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    [CrossRef]
  31. A. Duparré, S. Gliech, K. Hehl, S. Pichlmaier, U. Schuhmann, “Interface and volume inhomogeneities in optical thin films investigated by light scattering methods,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 1060–1067 (1994).
    [CrossRef]
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  35. G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  36. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
    [CrossRef]
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  38. C. Amra, “Light scattering from multilayer optics. II. Application to experiment,” J. Opt. Soc. Am. A 11, 211–226 (1994).
    [CrossRef]
  39. F. Horowitz, S. B. Mendes, “Envelope and waveguide methods: a comparative study of PbF2and CeO2birefringent films,” Appl. Opt. 33, 2659–2663 (1994).
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    [CrossRef]

1994 (4)

1993 (5)

1992 (2)

S. Müller-Pfeiffer, H. Van Kranenburg, J. C. Lodder, “A two-dimensional Monte Carlo model for thin film growth by oblique evaporation: simulation of two-component systems for the example of Co–Cr,” Thin Solid Films 213, 143–153 (1992).
[CrossRef]

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

1991 (2)

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Media 1, 207–221 (1991).
[CrossRef]

I. J. Hodgkinson, “Optical anisotropy in thin films deposited obliquely: in situobservations and computer modeling,” Appl. Opt. 30, 1303–1312 (1991).
[CrossRef] [PubMed]

1989 (3)

G. Mbise, G. B. Smith, G. A. Niklasson, C. G. Granquist, “Angular-selective optical properties of Cr films made by oblique-angle evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

T. Motohiro, Y. Taga, “Thin film retardation plate by oblique deposition,” Appl. Opt. 28, 2466–2482 (1989).
[CrossRef] [PubMed]

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]

1988 (2)

K. Hara, M. Kamyia, T. Hashimoto, K. Okamoto, H. Fujiwara, “Columnar structure of obliquely deposited iron films prepared at low substrate temperatures,” Thin Solid Films 158, 239–244 (1988).
[CrossRef]

I. J. Hodgkinson, P. W. Wilson, “Microstructural-induced anisotropy in thin films for optical applications,” CRC Crit. Rev. Solid State Mater. Sci. 15, 27–61 (1988).
[CrossRef]

1986 (4)

P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
[CrossRef]

R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 490–495 (1986).
[CrossRef]

M. Sikkens, I. J. Hodgkinson, F. Horowitz, H. A. Macleod, J. J. Wharton, “Computer simulation of thin film growth: applying the results to optical coatings,” Opt. Eng. 25, 142–147 (1986).
[CrossRef]

S. Lichter, J. Chen, “Model for columnar microstructure of thin solid films,” Phys. Rev. Lett. 56, 1396–1399 (1986).
[CrossRef] [PubMed]

1984 (1)

J. M. Elson, “Theory of light scattering from a rough surface in an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

1977 (1)

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapour-deposited thin films,” Thin Solid Films 47, 219–233 (1977).
[CrossRef]

1976 (1)

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[CrossRef]

1975 (1)

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

1972 (1)

N. G. Nakhodkin, A. I. Shaldervan, “Effect of vapour incidence angles on profile and properties of condensed films,” Thin Solid Films 10, 109–122 (1972).
[CrossRef]

1969 (2)

B. A. Movchan, A. V. Demchishin, “Study of the structure and properties of thin vacuum condensates of nickel, titanium, tungsten, aluminium oxide and zirconium dioxide,” Fiz. Met. Metalloved. 28, 653–660 (1969).

E. Neugebauer, C. von Fragstein, “Doppelbrechung schräg aufgedampfter Schichten,” Optik 29, 150–161 (1969).

1966 (1)

J. M. Nieuwenhuizen, H. B. Haanstra, “Mikrofraktographie dünner Schichten,” Philips Tech. Rundsch. 27, 177–181 (1966).

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

1886 (1)

A. Kundt, “Über Doppelbrechung des Lichtes in Metallschichten, welche durch Zerstäuben einer Kathode hergestellt sind,” Wied. Ann. 27, 59–71 (1886).
[CrossRef]

Amra, C.

Black, J. P.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Bowmar, P. I.

I. J. Hodgkinson, P. I. Bowmar, Q. H. Wu, “Scatter from tilted-columnar birefringent thin films: observation and measurement of anisotropic scatter distributions,” Appl. Opt. 33, 163–168 (1994).

Brett, M. J.

R. N. Tait, T. Smy, M. J. Brett, “Modelling and characterization of columnar growth in evaporated films,” Thin Solid Films 226, 196–201 (1993).
[CrossRef]

Bussemer, P.

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Media 1, 207–221 (1991).
[CrossRef]

Carniglia, C. K.

Chen, J.

S. Lichter, J. Chen, “Model for columnar microstructure of thin solid films,” Phys. Rev. Lett. 56, 1396–1399 (1986).
[CrossRef] [PubMed]

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

Demchishin, A. V.

B. A. Movchan, A. V. Demchishin, “Study of the structure and properties of thin vacuum condensates of nickel, titanium, tungsten, aluminium oxide and zirconium dioxide,” Fiz. Met. Metalloved. 28, 653–660 (1969).

Dirks, A. G.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapour-deposited thin films,” Thin Solid Films 47, 219–233 (1977).
[CrossRef]

Duparré, A.

A. Duparré, S. Kassam, “Relations between light scattering and microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1993).
[CrossRef]

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

A. Duparré, S. Gliech, K. Hehl, S. Pichlmaier, U. Schuhmann, “Interface and volume inhomogeneities in optical thin films investigated by light scattering methods,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 1060–1067 (1994).
[CrossRef]

A. Duparré, S. Kassam, “Determination of thin film roughness and volume structure parameters from light scattering investigations,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 283–286 (1991).
[CrossRef]

Elson, J. M.

J. M. Elson, “Theory of light scattering from a rough surface in an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
[CrossRef]

Endelema, D.

Flory, F.

Fujiwara, H.

K. Hara, M. Kamyia, T. Hashimoto, K. Okamoto, H. Fujiwara, “Columnar structure of obliquely deposited iron films prepared at low substrate temperatures,” Thin Solid Films 158, 239–244 (1988).
[CrossRef]

Gliech, S.

A. Duparré, S. Gliech, K. Hehl, S. Pichlmaier, U. Schuhmann, “Interface and volume inhomogeneities in optical thin films investigated by light scattering methods,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 1060–1067 (1994).
[CrossRef]

Granquist, C. G.

G. Mbise, G. B. Smith, G. A. Niklasson, C. G. Granquist, “Angular-selective optical properties of Cr films made by oblique-angle evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

Gruber, H. L.

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[CrossRef]

Guenther, K. H.

K. H. Guenther, H. L. Gruber, H. K. Pulker, “Morphology and light scattering of dielectric multilayer systems,” Thin Solid Films 34, 363–367 (1976).
[CrossRef]

Haanstra, H. B.

J. M. Nieuwenhuizen, H. B. Haanstra, “Mikrofraktographie dünner Schichten,” Philips Tech. Rundsch. 27, 177–181 (1966).

Hara, K.

K. Hara, M. Kamyia, T. Hashimoto, K. Okamoto, H. Fujiwara, “Columnar structure of obliquely deposited iron films prepared at low substrate temperatures,” Thin Solid Films 158, 239–244 (1988).
[CrossRef]

Hashimoto, T.

K. Hara, M. Kamyia, T. Hashimoto, K. Okamoto, H. Fujiwara, “Columnar structure of obliquely deposited iron films prepared at low substrate temperatures,” Thin Solid Films 158, 239–244 (1988).
[CrossRef]

Hehl, K.

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Media 1, 207–221 (1991).
[CrossRef]

A. Duparré, S. Gliech, K. Hehl, S. Pichlmaier, U. Schuhmann, “Interface and volume inhomogeneities in optical thin films investigated by light scattering methods,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 1060–1067 (1994).
[CrossRef]

Hodgkinson, I. J.

I. J. Hodgkinson, P. I. Bowmar, Q. H. Wu, “Scatter from tilted-columnar birefringent thin films: observation and measurement of anisotropic scatter distributions,” Appl. Opt. 33, 163–168 (1994).

F. Flory, D. Endelema, E. Pelletier, I. J. Hodgkinson, “Anisotropy in thin films: modeling and measurement of guided and nonguided optical properties: application to TiO2films,” Appl. Opt. 32, 5649–5659 (1993).
[CrossRef] [PubMed]

I. J. Hodgkinson, “Optical anisotropy in thin films deposited obliquely: in situobservations and computer modeling,” Appl. Opt. 30, 1303–1312 (1991).
[CrossRef] [PubMed]

I. J. Hodgkinson, P. W. Wilson, “Microstructural-induced anisotropy in thin films for optical applications,” CRC Crit. Rev. Solid State Mater. Sci. 15, 27–61 (1988).
[CrossRef]

M. Sikkens, I. J. Hodgkinson, F. Horowitz, H. A. Macleod, J. J. Wharton, “Computer simulation of thin film growth: applying the results to optical coatings,” Opt. Eng. 25, 142–147 (1986).
[CrossRef]

I. J. Hodgkinson, Q. H. Wu, “Optical properties of single layer and multilayer anisotropic coatings,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 882–892 (1994).
[CrossRef]

Horowitz, F.

F. Horowitz, S. B. Mendes, “Envelope and waveguide methods: a comparative study of PbF2and CeO2birefringent films,” Appl. Opt. 33, 2659–2663 (1994).
[CrossRef] [PubMed]

M. Sikkens, I. J. Hodgkinson, F. Horowitz, H. A. Macleod, J. J. Wharton, “Computer simulation of thin film growth: applying the results to optical coatings,” Opt. Eng. 25, 142–147 (1986).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kamyia, M.

K. Hara, M. Kamyia, T. Hashimoto, K. Okamoto, H. Fujiwara, “Columnar structure of obliquely deposited iron films prepared at low substrate temperatures,” Thin Solid Films 158, 239–244 (1988).
[CrossRef]

Kassam, S.

A. Duparré, S. Kassam, “Relations between light scattering and microstructure of optical thin films,” Appl. Opt. 32, 5475–5480 (1993).
[CrossRef]

S. Kassam, A. Duparré, K. Hehl, P. Bussemer, J. Neubert, “Light scattering from the volume of optical thin films: theory and experiment,” Appl. Opt. 31, 1304–1313 (1992).
[CrossRef] [PubMed]

P. Bussemer, K. Hehl, S. Kassam, “Theory of light scattering from rough surfaces and interfaces and from volume inhomogeneities in an optical layer stack,” Waves Random Media 1, 207–221 (1991).
[CrossRef]

A. Duparré, S. Kassam, “Determination of thin film roughness and volume structure parameters from light scattering investigations,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 283–286 (1991).
[CrossRef]

Kundt, A.

A. Kundt, “Über Doppelbrechung des Lichtes in Metallschichten, welche durch Zerstäuben einer Kathode hergestellt sind,” Wied. Ann. 27, 59–71 (1886).
[CrossRef]

Leamy, H. J.

A. G. Dirks, H. J. Leamy, “Columnar microstructure in vapour-deposited thin films,” Thin Solid Films 47, 219–233 (1977).
[CrossRef]

Lichter, S.

S. Lichter, J. Chen, “Model for columnar microstructure of thin solid films,” Phys. Rev. Lett. 56, 1396–1399 (1986).
[CrossRef] [PubMed]

Lodder, J. C.

S. Müller-Pfeiffer, H. Van Kranenburg, J. C. Lodder, “A two-dimensional Monte Carlo model for thin film growth by oblique evaporation: simulation of two-component systems for the example of Co–Cr,” Thin Solid Films 213, 143–153 (1992).
[CrossRef]

Macleod, H. A.

M. Sikkens, I. J. Hodgkinson, F. Horowitz, H. A. Macleod, J. J. Wharton, “Computer simulation of thin film growth: applying the results to optical coatings,” Opt. Eng. 25, 142–147 (1986).
[CrossRef]

H. A. Macleod, “The microstructure of optical thin films,” in Optical Thin Films, R. I. Seddon, ed., Proc. Soc. Photo-Opt. Instrum. Eng.325, 21–28 (1982).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Martin, P. J.

P. J. Martin, “Ion-based methods for optical thin film deposition,” J. Mater. Sci. 21, 1–25 (1986).
[CrossRef]

Mattsson, L.

L. Mattsson, “Light scattering and characterization of thin films,” in Thin Film Technologies II, J. R. Jacobsson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.652, 215–222 (1986).
[CrossRef]

Mbise, G.

G. Mbise, G. B. Smith, G. A. Niklasson, C. G. Granquist, “Angular-selective optical properties of Cr films made by oblique-angle evaporation,” Appl. Phys. Lett. 54, 987–989 (1989).
[CrossRef]

Mendes, S. B.

Messier, R.

R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 490–495 (1986).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Mills, D. L.

A. A. Maradudin, D. L. Mills, “Scattering and absorption of electromagnetic radiation by a semi-infinite medium in the presence of surface roughness,” Phys. Rev. B 11, 1392–1415 (1975).
[CrossRef]

Motohiro, T.

Movchan, B. A.

B. A. Movchan, A. V. Demchishin, “Study of the structure and properties of thin vacuum condensates of nickel, titanium, tungsten, aluminium oxide and zirconium dioxide,” Fiz. Met. Metalloved. 28, 653–660 (1969).

Müller-Pfeiffer, S.

S. Müller-Pfeiffer, H. Van Kranenburg, J. C. Lodder, “A two-dimensional Monte Carlo model for thin film growth by oblique evaporation: simulation of two-component systems for the example of Co–Cr,” Thin Solid Films 213, 143–153 (1992).
[CrossRef]

Nakhodkin, N. G.

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A. Duparré, S. Gliech, K. Hehl, S. Pichlmaier, U. Schuhmann, “Interface and volume inhomogeneities in optical thin films investigated by light scattering methods,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 1060–1067 (1994).
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[CrossRef]

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[CrossRef]

A. Duparré, S. Gliech, K. Hehl, S. Pichlmaier, U. Schuhmann, “Interface and volume inhomogeneities in optical thin films investigated by light scattering methods,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 1060–1067 (1994).
[CrossRef]

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M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

I. J. Hodgkinson, Q. H. Wu, “Optical properties of single layer and multilayer anisotropic coatings,” in Optical Interference Coatings, F. Abeles, A. Duparré, G. Emiliani, J.-P. Gailliard, K. H. Guenther, R. P. Netterfield, E. P. Pelletier, H. Rudigier, A. MacLeod, C. Boccara, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2253, 882–892 (1994).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Configuration used in this paper.

Fig. 2
Fig. 2

Incidence and observation direction parameters.

Fig. 3
Fig. 3

Scattering by a PbF2 layer backed by an Al substrate (parameter set 1). (a) Scattering in the column plane for different column angles. (b) Azimuth dependence for different column angles (TE, θ = 30°): ⋯, α = 0°;– – –, α = 10°;– · –, α = 20°; ●●●, α = 30°; ——, α = 40°. (c) Azimuth dependence for different scattering angles θ and α = 40° (numbers indicate θ in degrees). Here and in subsequent figures, norm. diff. means normalized differential.

Fig. 4
Fig. 4

Four parts of Eq. (29) and their corresponding processes. Scattering: b, back; f, forward. Reflection: r, incident wave; R, scattered wave.

Fig. 5
Fig. 5

Composition of the total scattering for bottom incidence (parameter set 2): (a) angular scattering in the column plane, (b) polar plot for θ = 80° (——, total scattering; ⋯, mp part).

Fig. 6
Fig. 6

Same as Fig. 5 but for top incidence: (a) angular scattering in the column plane, (b) polar plot for θ = 80°. ——, Total scattering; ⋯, mp part; +++, pm part. 8e − 05 stands for 8 × 10−5, and so on.

Fig. 7
Fig. 7

Scattering in the column plane for different film thicknesses (parameter set 3).– – –, d = 0.01 μm; ——, d = 2.4 μm; ⋯, d = 10 μm.

Fig. 8
Fig. 8

Scattering distribution projected into a hemisphere for parameter set 3. Central region: θ = 0°; margin, θ = 90°;– –, column plane;– > –, column side.

Fig. 9
Fig. 9

(a) Scattering distribution projected into a hemisphere (for parameter set 2, except that nb = 1.52) for bottom incidence [calculated with Eqs. (27) and (28) compared with the cone model described by Eq. (31) for θ = 0°(10)90°]. – –,– > –as in Fig. 8. + stands for Smp. (b) Same as Fig. 9(a) but for top incidence. + stands for Smp, ° stands for Spm.

Fig. 10
Fig. 10

Phase relations of the scattered light that depend on the depth z from which the light is emitted.

Fig. 11
Fig. 11

Angular scattering of a granular structured film for parameter set 4. 1 stands for τg = 0.08 μm; 2 stands for τg = 0.04 μm; b, bottom incidence; t, top incidence.

Fig. 12
Fig. 12

Relative polar scatter distributions (actual values divided by τg3) for different grain sizes and parameter set 5. (Curve 1, τg = 0.01 μm; curve 2, τg = 0.02 μm; curve 3, τg = 0.04 μm; curve 4, τg = 0.08 μm; curve 5, τg = 0.16 μm; curve 6, τg = 0.32 μm; curve 7, τg = 0.64 μm).

Equations (59)

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× × E ( r ) - ( ω / c ) 2 ɛ E ( r ) = ( ω / c ) 2 Δ ɛ ( r ) E ( r ) ,
× × E ( r ) - ( ω / c ) 2 ɛ E ( r ) = ( ω / c ) 2 Δ ɛ ( r ) E 0 ( r ) .
E ( r ) = ( ω / c ) 2 d 3 r Δ ɛ ( r ) G ^ ( r , r ) E 0 ( r ) .
E 0 ( r ) = e ( z ) exp ( i k 0 ρ ) ,
E ( r ) = 1 ( 2 π ) 2 ( ω c ) 2 d 3 r d 2 k Δ ɛ ( r ) g ^ ( z , z , k ) × exp [ i k ( ρ - ρ ) ] e ( z ) exp ( i k 0 ρ ) .
P = 1 2 ω μ 0 Re ( d 2 ρ K z E 2 ) ,
P = 1 2 ω μ 0 1 2 ( π ) 2 ( ω c ) 4 Re { K z d 3 r d 2 k d 3 r × Δ ɛ ( r ) Δ ɛ * ( r ) g ^ ( z , z , k ) e ( z ) g ^ * ( z , z , k ) e ( z ) × exp [ - i k ( ρ - ρ ) ] exp [ i k 0 ( ρ - ρ ) ] } .
P 0 = 1 2 ω μ 0 L 2 ( ω c ) n in cos θ 0 E 0 2 ,
n in = { n t for top incidence n b for bottom incidence .
1 P 0 d P d Ω = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 cos 2 θ n in cos θ 0 { d 3 r d 3 r × f c ( r , r ) L 2 g ^ ( z , z , k ) e ( z ) g ^ * ( z , z , k ) e ( z ) × exp [ i ( k 0 - k ) ( ρ - ρ ) ] } ,
f c ( r , r ) Δ ɛ ( r ) Δ ɛ * ( r )
tan α = ½ tan δ ,
f c ( r , r ) = Δ ɛ ( r ) Δ ɛ * ( r ) = Δ ɛ 2 f ( ρ - ρ ) ,
f c ( r , r ) = Δ ɛ 2 exp [ - ( Δ x 2 + Δ y 2 ) τ 2 ] ,
Δ ɛ 2 = f fill ( 1 - f fill ) ɛ void - ɛ h 2 .
F c ( k - k 0 ) = Δ ɛ 2 F ( Δ k x , Δ k y ) F * ( Δ k x , Δ k y ) ,
F ( Δ k x , Δ k y ) = π τ exp [ - ( Δ k ) 2 τ 2 8 ] .
f c ( r - r ) = Δ ɛ 2 × exp [ - ( Δ x cos α cos ϕ + Δ y cos α sin ϕ - Δ z sin α ) 2 τ 2 ] × exp [ - ( Δ y cos ϕ - Δ x sin ϕ ) 2 τ 2 ] .
F c [ ( k - k 0 ) , Δ z , α , ϕ ] = Δ ɛ 2 π τ 2 cos α exp [ - i tan α × ( Δ k x cos ϕ + Δ k y sin ϕ ) Δ z ] × exp [ - ( Δ k y cos ϕ - Δ k x sin ϕ ) 2 τ 2 4 ] × exp [ - ( Δ k x cos ϕ + Δ k y sin ϕ ) 2 τ 2 4 cos 2 α ] .
F c ( Δ k , Δ z , α , ϕ ) = Δ ɛ 2 F ( Δ k x , Δ k y , z , α , ϕ ) × F * ( Δ k x , Δ k y , z , α , ϕ ) ,
F ( Δ k x , Δ k y , z , α , ϕ ) = ( π τ 2 cos α ) 1 / 2 exp [ - i tan α ( Δ k x cos ϕ + Δ k y sin ϕ ) z ] × exp [ - ( Δ k y cos ϕ - Δ k x sin ϕ ) 2 τ 2 8 ] × exp [ - ( Δ k x cos ϕ + Δ k y sin ϕ ) 2 τ 2 8 cos 2 α ]
F ( Δ k x , Δ k y , z , α , ϕ ) = π τ x τ y cos α exp [ - i tan α ( Δ k x cos ϕ + Δ k y sin ϕ ) z ] exp [ - ( Δ k y cos ϕ - Δ k x sin ϕ ) 2 τ y 2 8 ] × exp [ - ( Δ k x cos ϕ + Δ k y sin ϕ ) 2 τ x 2 8 cos 2 α ] ,
1 P 0 d P d Ω = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 cos 2 θ n in cos θ 0 Δ ɛ 2 × [ | d z F [ ( k - k 0 ) , z , α , ϕ ] g ^ ( z , z , k ) e ( z ) | 2 ] .
1 P 0 d P d Ω TE = 1 ( 2 π ) 2 ( ω c ) 6 cos 2 θ n t 3 n in Δ ɛ 2 × [ | d z F ( k , z , α , ϕ ) g y y ( z , z , k ) e y ( z ) | 2 ] , 1 P 0 d P d Ω TM = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 n in Δ ɛ 2 × [ | d z F ( k , z , α , ϕ ) g x x ( z , z , k ) e x ( z ) | 2 ] .
F ( k , z , α , ϕ ) f [ k ( θ ) , τ , α , ϕ ] exp ( - i C f z )
f [ k ( θ ) , τ , α , ϕ ] = π τ cos α exp [ - ( k τ sin ϕ ) 2 8 ] × exp [ - ( k τ cos ϕ ) 2 8 cos 2 α ] , C f k tan α cos ϕ = ω c n t tan α sin θ cos ϕ ,
1 P 0 d P d Ω TE = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 n in cos 2 θ Δ ɛ 2 × f [ k ( θ ) , τ , α , ϕ ] 2 C y 2 D y 2 × { | d z exp ( - i C f z ) [ exp ( - i Q 1 z ) + C y + exp ( i Q 1 z ) ] × [ exp ( - i q 1 z ) + D + exp ( i q 1 z ) ] | 2 } = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 n in cos 2 θ Δ ɛ 2 f [ k ( θ ) , τ , α , ϕ ] 2 × C y 2 D y 2 { | [ 1 - exp { - i d [ ( - Q 1 - q 1 ) - C f ] } i [ ( - Q 1 - q 1 ) - C f ] + C y + ( 1 - exp { - i d [ ( Q 1 - q 1 ) - C f ] } ) i [ ( Q 1 - q 1 ) - C f ] + D + ( 1 - exp { - i d [ ( q 1 - Q 1 ) - C f ] } ) i [ ( q 1 - Q 1 ) - C f ] + C y + D + ( 1 - exp { - i d [ ( q 1 + Q 1 ) - C f ] } ) i [ ( q 1 + Q 1 ) - C f ] ] | 2 } ,
1 P 0 d P d Ω TM = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 n in Δ ɛ 2 f [ k ( θ ) , τ , α , ϕ ] 2 × C x 2 D x 2 { | [ 1 - exp { - i d [ ( - Q 1 - q 1 ) - C f ] } i [ ( - Q 1 - q 1 ) - C f ] + C x + ( 1 - exp { - i d [ ( Q 1 - q 1 ) - C f ] } ) i [ ( Q 1 - q 1 ) - C f ] + D + ( 1 - exp { - i d [ ( q 1 - Q 1 ) - C f ] } ) i [ ( q 1 - Q 1 ) - C f ] + C x + D + ( 1 - exp { - i d [ ( q 1 + Q 1 ) - C f ] } ) i [ ( q 1 + Q 1 ) - C f ] ] | 2 } ,
1 P 0 d P d Ω S m m + S p m + S m p + S p p 2 ,
r · c = const . = cos α = sin θ 1 cos ϕ sin α + cos θ 1 cos α ,
ϕ = arccos { 1 - [ 1 - ( n t / n 1 sin θ ) 2 ] 1 / 2 n t / n 1 sin θ } cot α .
I ( θ ) 1 d 2 | - d 0 d z exp [ i Φ ( z ) ] | 2 ,
Φ ( z ) = s 1 + s 2 + s 3 z ω c n 1 [ 1 - cos ( θ 1 - α ) cos α ] = z ( q 1 - Q 1 - C f ) ,
I ( θ ) f ϕ = 1 - exp [ - i d ( q 1 - Q 1 - C f ) ] 2 d 2 q 1 - Q 1 - C f 2 = 2 { 1 - cos [ Φ ( d ) ] } [ Φ ( d ) ] 2 ,
I ( θ ) f dip = { 1 for TE cos 2 θ 1 for TM .
I ( θ ) f T out = { T 1 t s = 4 Q 1 Q t C y 2 for TE T 1 t p = 4 ( ω c ) 4 ɛ 1 ɛ t Q 1 Q t C x 2 for TM ,
I ( θ ) f Ω = d Ω 1 d Ω t = ɛ t cos θ ɛ 1 cos θ 1 .
I ( θ ) f cor = exp [ - ( k τ sin ϕ ) 2 4 ] exp [ - ( k τ cos ϕ ) 2 4 cos 2 α ] ,
I ( θ ) = c f ϕ f dip f T out f Ω f cor .
c = f T in ( ω c ) 4 Δ ɛ 2 V col d 1 ( 4 π ) 2 f T in c ˜ ,
f T in = n 1 n b D 0 2 D + 2 .
Q sca ( set ) = C sca ( set ) A film = N C sca ( 1 ) A film = f fill V film V col A film C sca ( 1 ) = f fill d V col C sca ( 1 ) ,
C sca ( 1 ) = 8 π 3 ( ω c ) 4 ɛ h 2 ( V col 4 π ) 2 | ɛ void - ɛ h ɛ h + L ( ɛ void - ɛ h ) | 2 8 π 3 ( ω c ) 4 ( V col 4 π ) 2 ɛ void - h 2 ,
1 P 0 d P d Ω m p = f T in f fill 1 P 0 d P d Ω 1 dipole f Ω f T out F cor ,
f c ( r - r ) = Δ ɛ 2 exp [ - ( Δ x 2 + Δ y 2 + Δ z 2 ) τ g 2 ] ,
F c [ ( k - k 0 ) , Δ z ] = Δ ɛ 2 π τ g 2 exp [ - ( k - k 0 ) 2 τ g 2 4 ] × exp ( - Δ z 2 τ g 2 ) .
1 P 0 d P d Ω = 1 ( 2 π ) 2 ( ω c ) 6 n t 3 cos 2 θ n in cos θ 0 d z d z × F c [ ( k - k 0 ) , z , z ] g ^ ( z , z , k ) × e ( z ) g ^ * ( z , z , k ) e * ( z ) .
1 P 0 d P d Ω 1 ( 2 π ) 2 ( ω c ) 6 n t 3 cos 2 θ n in cos θ 0 Δ ɛ 2 d z × F ˜ c [ ( k - k 0 ) , z ] g ^ ( z , z , k ) e ( z ) 2 ,
Δ ɛ 2 F ˜ c [ ( k - k 0 ) , z ] = - d 0 d z F c [ ( k - k 0 ) , z , z ] .
F ˜ c [ ( k - k 0 ) , z ] π 3 τ g 3 exp [ - ( k - k 0 ) 2 τ g 2 4 ] .
E 0 ( r ) = e ( z ) = ( e x , e y , 0 ) ,
e x D x [ exp ( - i q 1 z ) + D + exp ( i q 1 z ) ] = D 0 cos σ 0 [ exp ( - i q 1 z ) + D + exp ( i q 1 z ) ] , e y D y [ exp ( - i q 1 z ) + D + exp ( i q 1 z ) ] = D 0 sin σ 0 [ exp ( - i q 1 z ) + D + exp ( i q 1 z ) ] ,
top incidence : D 0 = t t 1 1 - r 1 t r 1 b exp ( i 2 q 1 d ) , D + = r 1 b exp ( i 2 q 1 d ) , bottom incidence : D 0 = t b 1 r 1 t exp [ i ( q 1 - q b ) d ] 1 - r 1 t r 1 b exp ( i 2 q 1 d ) , D + = 1 r 1 t ,
r 1 t = q 1 - q t q 1 + q t , t t 1 = 1 - r 1 t , r 1 b = q 1 - q b q 1 + q b , t b 1 = 1 - r 1 b , q i = ω c n i ( i = 1 , b , t ) .
D x 2 = D y 2 = ½ D 0 2 .
g ^ ( z , z , k ) = [ g x x 0 g x z 0 g y y 0 g z x 0 g z z ] .
g x x exp ( i Q t z ) C x [ exp ( - i Q 1 z ) + C x + exp ( i Q 1 z ) ] , g y y exp ( i Q t z ) C y [ exp ( - i Q 1 z ) + C y + exp ( i Q 1 z ) ] ,
C x = i Q 1 Q t ( ω / c ) 2 ( ɛ t Q 1 + ɛ 1 Q t ) 1 1 - R 1 t x R 1 b x exp ( i 2 Q 1 d ) , C x + = - R 1 b x exp ( i 2 Q 1 d ) , C y = i ( Q 1 + Q t ) 1 1 - R 1 t y R 1 b y exp ( i 2 Q 1 d ) , C y + = R 1 b y exp ( i 2 Q 1 d ) ,
R 1 t x = ( ɛ t Q 1 - ɛ 1 Q t ) ( ɛ t Q 1 + ɛ 1 Q t ) ,             R 1 t y = ( Q 1 - Q t ) ( Q 1 + Q t ) , R 1 b x = ( ɛ b Q 1 - ɛ 1 Q b ) ( ɛ b Q 1 + ɛ 1 Q b ) ,             R 1 b y = ( Q 1 - Q b ) ( Q 1 + Q b ) , Q i = ω c ( ɛ i - ɛ t sin 2 θ ) 1 / 2             ( i = 1 , b , t ) .

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