Abstract

A theoretical model is presented that describes the volume scattering in thin optical films, particularly in typical columnar structures. It is based on a first-order perturbation theory that concerns the fluctuation of the dielectric permittivity in the film. For evaporated PbF2 films that show a pronounced columnar morphology, angular as well as total integrated scattering measurements at λ = 633 nm have been performed on a special layer design to suppress roughness-induced scattering. A comparison of the predicted theoretical and the measured experimental values leads to such structural parameters as packing density and the evolutionary exponent of the columns.

© 1992 Optical Society of America

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References

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  1. J. M. Bennett, “Optical scattering and absorption losses at interfaces and in thin films,” Thin Solid Films 123, 27–44 (1985).
    [CrossRef]
  2. J. Ebert, H. Pannhorst, H. Küster, H. Welling, “‘Scatter losses of broadband interference coatings,” ’ Appl. Opt. 18, 818–822 (1979).
    [CrossRef] [PubMed]
  3. 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).
  4. G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
    [CrossRef]
  5. C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
    [CrossRef]
  6. A. Duparré, R. Dohle, H. Müller, “Relation between light scattering and morphology of columnar structured optical thin films,” J. Mod. Opt. 37, 1383–1390 (1990).
    [CrossRef]
  7. E. Pelletier, P. Roche, C. Grèzes-Besset, “Measurement of scattering curves of coated or uncoated surfaces: experimental techniques for determining surface roughness,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 98–110 (1988).
  8. A. Duparré, H.-G. Walter, “Surface smoothing and roughening by dielectric thin film deposition,” Appl. Opt. 27, 1393–1395 (1988).
    [CrossRef] [PubMed]
  9. A. Roos, M. Bergkvist, C.-R. Ribbing, “Optical scattering from oxidized metals. 1: Model formulation and properties,” Appl. Opt. 28, 1360–1364 (1989).
    [CrossRef] [PubMed]
  10. J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977).
    [CrossRef] [PubMed]
  11. P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  12. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–114 (1979).
    [CrossRef]
  13. J. M. Elson, “Theory of light scattering from a rough surface with an inhomogeneous dielectric permittivity,” Phys. Rev. B 30, 5460–5480 (1984).
    [CrossRef]
  14. K. H. Guenther, “Microstructure of vapor-deposited optical coatings,” Appl. Opt. 23, 3806–3816 (1984).
    [CrossRef] [PubMed]
  15. 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).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 1.6
  17. 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]
  18. A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
    [CrossRef]
  19. A. Duparré, S. Kassam, Verfahren zur Bestimmung der Volumenstreuung von transparenten, insbesondere optischen Schichten, German PatentG01N/3407553 (17May1990).
  20. J. Neubert, “Beitrag zur Untersuchung der Grenzflächen-Korrelationseigen-schaften dünner optischer Schichten nach der Streulichtmethode,” Ph.D. dissertation (University of Jena, Jena, Germany, 1991).
  21. A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).
  22. R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 496–499 (1986).
    [CrossRef]
  23. 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 (to be published).

1990 (2)

A. Duparré, R. Dohle, H. Müller, “Relation between light scattering and morphology of columnar structured optical thin films,” J. Mod. Opt. 37, 1383–1390 (1990).
[CrossRef]

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

1989 (1)

1988 (1)

1987 (1)

1986 (2)

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 496–499 (1986).
[CrossRef]

1985 (2)

G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
[CrossRef]

J. M. Bennett, “Optical scattering and absorption losses at interfaces and in thin films,” Thin Solid Films 123, 27–44 (1985).
[CrossRef]

1984 (2)

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

K. H. Guenther, “Microstructure of vapor-deposited optical coatings,” Appl. Opt. 23, 3806–3816 (1984).
[CrossRef] [PubMed]

1981 (1)

1979 (2)

1977 (1)

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]

1969 (1)

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).

Al-Jumaily, G. A.

G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
[CrossRef]

Amra, C.

Bennett, J. M.

J. M. Bennett, “Optical scattering and absorption losses at interfaces and in thin films,” Thin Solid Films 123, 27–44 (1985).
[CrossRef]

Bergkvist, M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 1.6

Bousquet, P.

Bussemer, P.

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 (to be published).

Carniglia, C. K.

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–114 (1979).
[CrossRef]

Darr, C.-P.

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

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).

Dohle, R.

A. Duparré, R. Dohle, H. Müller, “Relation between light scattering and morphology of columnar structured optical thin films,” J. Mod. Opt. 37, 1383–1390 (1990).
[CrossRef]

Duparré, A.

A. Duparré, R. Dohle, H. Müller, “Relation between light scattering and morphology of columnar structured optical thin films,” J. Mod. Opt. 37, 1383–1390 (1990).
[CrossRef]

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

A. Duparré, H.-G. Walter, “Surface smoothing and roughening by dielectric thin film deposition,” Appl. Opt. 27, 1393–1395 (1988).
[CrossRef] [PubMed]

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

A. Duparré, S. Kassam, Verfahren zur Bestimmung der Volumenstreuung von transparenten, insbesondere optischen Schichten, German PatentG01N/3407553 (17May1990).

Ebert, J.

Elson, J. M.

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

J. M. Elson, “Infrared light scattering from surfaces covered with multiple dielectric overlayers,” Appl. Opt. 16, 2872–2881 (1977).
[CrossRef] [PubMed]

Flory, F.

Grèzes-Besset, C.

E. Pelletier, P. Roche, C. Grèzes-Besset, “Measurement of scattering curves of coated or uncoated surfaces: experimental techniques for determining surface roughness,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 98–110 (1988).

Guenther, K. H.

Hacker, E.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

Hehl, K.

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 (to be published).

Herrmann, W. C.

G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
[CrossRef]

Kaiser, N.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

Kassam, S.

A. Duparré, S. Kassam, Verfahren zur Bestimmung der Volumenstreuung von transparenten, insbesondere optischen Schichten, German PatentG01N/3407553 (17May1990).

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 (to be published).

Küster, H.

Lauth, H.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[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]

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).

McNally, J. J.

G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
[CrossRef]

McNeil, J. R.

G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
[CrossRef]

Messier, R.

R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 496–499 (1986).
[CrossRef]

Meyer, J.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[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]

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, H.

A. Duparré, R. Dohle, H. Müller, “Relation between light scattering and morphology of columnar structured optical thin films,” J. Mod. Opt. 37, 1383–1390 (1990).
[CrossRef]

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

Neubert, J.

J. Neubert, “Beitrag zur Untersuchung der Grenzflächen-Korrelationseigen-schaften dünner optischer Schichten nach der Streulichtmethode,” Ph.D. dissertation (University of Jena, Jena, Germany, 1991).

Pannhorst, H.

Pelletier, E.

C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
[CrossRef]

E. Pelletier, P. Roche, C. Grèzes-Besset, “Measurement of scattering curves of coated or uncoated surfaces: experimental techniques for determining surface roughness,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 98–110 (1988).

Ribbing, C.-R.

Roche, P.

C. Amra, P. Roche, E. Pelletier, “Interface roughness cross-correlation laws deduced from scattering diagram measurements on optical multilayers: effect of material grain size,” J. Opt. Soc. Am. B 4, 1087–1093 (1987).
[CrossRef]

P. Bousquet, F. Flory, P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
[CrossRef]

E. Pelletier, P. Roche, C. Grèzes-Besset, “Measurement of scattering curves of coated or uncoated surfaces: experimental techniques for determining surface roughness,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 98–110 (1988).

Roos, A.

Schirmer, G.

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

Truckenbrodt, H.

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

Walter, H.-G.

Walther, H.-G.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

Weiss, M.

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

Weissbrodt, P.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

Welling, H.

Welsch, E.

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 1.6

Appl. Opt. (5)

Fiz. Met. Metalloved. (1)

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).

J. Mod. Opt. (1)

A. Duparré, R. Dohle, H. Müller, “Relation between light scattering and morphology of columnar structured optical thin films,” J. Mod. Opt. 37, 1383–1390 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Vac. Sci. Technol. A (2)

G. A. Al-Jumaily, J. J. McNally, J. R. McNeil, W. C. Herrmann, “Effect of ion assisted deposition on optical scatter and surface microstructure of thin films,” J. Vac. Sci. Technol. A 3, 651–655 (1985).
[CrossRef]

R. Messier, “Toward quantification of thin film morphology,” J. Vac. Sci. Technol. A 4, 496–499 (1986).
[CrossRef]

Opt. Eng. (1)

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–114 (1979).
[CrossRef]

Optik (1)

A. Duparré, C.-P. Darr, G. Schirmer, H. Truckenbrodt, M. Weiss, “The influence of angle limitation in light scattering measurements on the determination of the reflectance of multilayer dielectric mirrors,” Optik 72, 153–156 (1986).

Phys. Rev. B (2)

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

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]

Thin Solid Films (2)

A. Duparré, E. Welsch, H.-G. Walther, N. Kaiser, H. Müller, E. Hacker, H. Lauth, J. Meyer, P. Weissbrodt, “Structure-related bulk losses in ZrO2 optical thin films,” Thin Solid Films 187, 275–288 (1990).
[CrossRef]

J. M. Bennett, “Optical scattering and absorption losses at interfaces and in thin films,” Thin Solid Films 123, 27–44 (1985).
[CrossRef]

Other (6)

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).

E. Pelletier, P. Roche, C. Grèzes-Besset, “Measurement of scattering curves of coated or uncoated surfaces: experimental techniques for determining surface roughness,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 98–110 (1988).

A. Duparré, S. Kassam, Verfahren zur Bestimmung der Volumenstreuung von transparenten, insbesondere optischen Schichten, German PatentG01N/3407553 (17May1990).

J. Neubert, “Beitrag zur Untersuchung der Grenzflächen-Korrelationseigen-schaften dünner optischer Schichten nach der Streulichtmethode,” Ph.D. dissertation (University of Jena, Jena, Germany, 1991).

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 (to be published).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964), Chap. 1.6

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Figures (9)

Fig. 1
Fig. 1

(a) Configuration, (b) scattering geometry.

Fig. 2
Fig. 2

Angular dependence of volume scattering at normal incidence and λ = 633 nm, which is calculated for a PbF2 layer (n1 = 1.78) on a high-reflecting substrate (n3 = 100 is chosen), assuming a columnar structure that is represented by ξ = 0.05; τc = 30 and 60 nm, respectively. Note that a variation of τc does not affect the shape of the scattering distribution. It only shifts the curve that is parallel to the vertical axis.

Fig. 3
Fig. 3

(a) Same as Fig. 2 except that the dependence on the layer thickness parameter m for the TE case is shown. (b) Same as (a) except that the TM case is shown.

Fig. 4
Fig. 4

Same as Fig. 2 except that the dependence on the columnar structure parameter A for an (m = 10) layer in the TE case is demonstrated.

Fig. 5
Fig. 5

Variation of TIS with the layer thickness parameter m (m = 1, 2,…). The angle of incidence is 0°, and the scattering medium is PbF2. The solid curves illustrate the dependence on parameter A in the case of a high-reflecting substrate (n3 = 100). The dashed curve shows that the qualitative behavior of TIS does not change in the case of an Al substrate (n3 = 1.39 + i7.67/interpolated).

Fig. 6
Fig. 6

C–Pt transmission electron micrograph of a PbF2-film cross section (film thickness m = 4).

Fig. 7
Fig. 7

Angle-resolved scattering measured on PbF2 layers that are deposited onto aluminized BK7 substrates (normally incident light, λ = 633 nm). Reduced film thickness (a) m = 4, (b) m = 10.

Fig. 8
Fig. 8

Theoretical volume scattering versus scattering angle for a PbF2 film (n1 = 1.78) on an aluminum substrate. The structural parameters that we chose are ξ = 0.05, τc = 60 nm. Reduced film thickness (a) m = 4, (b) m = 10.

Fig. 9
Fig. 9

Calculated angular scattering of a PbF2 film (m = 10) assuming an ensemble of five different parameters A (−1, −0.5, 0, 0.5, 1) demonstrating the smearing effect.

Equations (36)

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

( ρ , z ) = { 2 0 < z , 1 + Δ ( ρ ) exp ( α z ) d z 0 , 3 z < d .
× H = i ( ω c ) E , × E = i ( ω c ) H ,
× × E ( ρ , z ) ( w c ) 2 E ( ρ , z ) = ( w c ) 2 Δ ( ρ ) exp ( α z ) E ( ρ , z ) ,
E ( ρ , z ) = E 0 ( ρ , z ) + ( w c ) 2 d 2 ρ d z Δ ( ρ , z ) G ( r , r ) E ( ρ , z )
E 1 ( ρ , z ) = ( w 2 π c ) 2 d 2 k d 2 ρ d z g ( 2 , 1 ) ( z , z ) × E 0 ( ρ , z ) Δ ( ρ ) exp ( α z ) exp ( i k [ ρ ρ ] ) ,
G ( 2 , 1 ) ( r , r ) = ( 1 2 π ) 2 d 2 k g ( 2 , 1 ) ( k , z , z ) exp ( i k [ ρ ρ ] ) .
E 0 ( ρ , z ) = e 0 ( k 0 , z ) exp ( i k 0 ρ ) ,
Δ ˜ ( k ) = d 2 ρ Δ ( ρ ) exp ( i k ρ ) ,
E 1 ( ρ , z ) = ( ω 2 π c ) 2 d 2 k d z g ( 2 , 1 ) ( z , z ) e 0 ( k 0 , z ) × Δ ˜ ( k 0 k ) exp ( α z ) exp ( i k ρ ) ,
E 1 ( ρ , z ) = 1 ( 2 π ) 2 d 2 k Δ ˜ ( k 0 k ) exp ( i [ k ρ + Q 2 z ] ) × [ ( x ˆ Q 2 z ˆ k ) γ p ( x ˆ × z ˆ ) ( w c ) γ s ] ,
γ s = ω c Y ( E 1 y exp ( i Q 1 d ) ( q 1 + Q 1 + i α ) { 1 exp [ ( α + i [ Q 1 + q 1 ] ) d ] } + E 2 y exp ( i Q 1 d ) ( q 1 Q 1 i α ) { 1 exp [ ( α i [ q 1 Q 1 ] ) d ] } R 13 y E 1 y exp ( i Q 1 d ) ( q 1 Q 1 + i α ) { 1 exp [ ( α + i [ q 1 Q 1 ] ) d ] } + R 13 y E 2 y exp ( i Q 1 d ) ( q 1 + Q 1 i α ) { 1 exp [ ( α i [ q 1 + Q 1 ] ) d ] } ) ,
γ p = X ( ( E 1 x + k Q 1 E 1 z ) exp ( i Q 1 d ) ( q 1 + Q 1 + i α ) × { 1 exp ( α + i [ q 1 + Q 1 ] ) d } + ( E 2 x k Q 1 E 2 z ) exp ( i Q 1 d ) ( q 1 Q 1 i α ) × { 1 exp ( α i [ q 1 Q 1 ] ) d } + ( E 1 x + k Q 1 E 1 z ) R 13 x exp ( i Q 1 d ) ( q 1 Q 1 + i α ) × { 1 exp ( α + i [ q 1 Q 1 ] ) d } + ( E 2 x k Q 1 E 2 z ) R 13 x exp ( i Q 1 d ) ( q 1 + Q 1 i α ) × { 1 exp ( α i ( q 1 + Q 1 ) ) d } ) .
H 1 ( ρ , z ) = 1 ( 2 π ) 2 d 2 k Δ ˜ ( k 0 k ) exp ( i [ k ρ + Q 2 z ] ) × [ ( x ˆ Q 2 z ˆ k ) γ s + ( x ˆ × z ˆ ) ( ω c ) γ p ] .
S 1 = c 8 π Re ( E 1 × H 1 * ) ,
P = d 2 ρ ( S 1 z ˆ ) ,
P = w 4 ( 2 π ) 3 d 2 k Q 2 | Δ ˜ ( k 0 k ) | 2 ( | γ p | 2 + | γ s | 2 ) exp { 2 z [ Im ( Q 2 ) ] } .
1 P 0 d P d Ω = ( ω c ) 4 cos 2 ϑ ( 2 π ) 2 cos ϑ 0 [ | γ p | 2 + | γ s | 2 ] | Δ ˜ ( k 0 k ) | 2 L 2 .
| Δ ˜ ( k ) | 2 L 2 = 1 L 2 d 2 ρ d 2 τ Δ ( ρ + τ ) Δ ( ρ ) exp ( i k τ ) = d 2 τ G ( τ ) exp ( i k τ ) = g ( k ) ,
1 P 0 d P d Ω = ( ω c ) 4 cos 2 ϑ ( 2 π ) 2 cos ϑ 0 [ | γ p | 2 + | γ s | 2 ] g ( k 0 k ) ,
TIS = 0 π/ 2 d ϑ sin ϑ 0 π/ 2 d φ 1 P 0 d P d Ω ,
| γ p | 2 = cos 2 σ ( 2 w c sin 2 ϑ ) 2 q 1 2 ( 1 Q 2 cot Q 1 d ) 2 + Q 1 2 cos 2 σ ( 2 w c sin 2 ϑ ) 2 | γ ˜ p | 2 ,
| γ s | 2 = sin 2 σ ( 2 sin 2 ϑ ) 2 1 ( Q 1 cot Q 1 d ) 2 + Q 2 2 sin 2 σ ( 2 w c sin 2 ϑ ) 2 | γ ˜ s | 2 ,
G ( τ ) = ξ 2 exp ( [ τ / τ c ] 2 ) ,
1 P 0 d P d Ω = x 2 exp ( [ x 2 sin ϑ ] 2 ) × ξ 2 1 π cos 2 ϑ sin 4 ϑ ( | γ ˜ p | 2 cos 2 σ + | γ ˜ | 2 sin 2 σ ) ,
sin ϑ min = n 1 [ 1 ( N m ) 2 ] 1 / 2 .
1 P 0 d P d Ω ϑ 0 π 4 1 3 x 2 ξ 2 m 2 { [ 1 exp ( A ) ] 4 A ( 4 + A 2 m 2 π 2 ) } 2 .
1 P 0 d P d Ω ϑ 0 = | A | 1 π 4 1 3 x 2 ξ 2 m 2 ( 1 A 2 ) 2 .
TIS = ( τ c ξ λ ) 2 F ( 1 , 3 , a ) m ν ,
ξ 2 = ( void 1 ) 2 p ( 1 p ) ,
β = ln ( κ · m m ) 2 1 n ( m m ) , where κ = TIS ( m ) TIS ( m ) .
E 0 y = [ B 1 y exp ( i q 1 z ) + B 2 y exp ( i q 1 z ) ] exp ( i k 0 ρ ) ( TE ) , E 0 x = [ B 1 x exp ( i q 1 z ) + B 2 x exp ( i q 1 z ) ] exp ( i k 0 ρ ) E 0 z = [ B 1 z exp ( i q 1 z ) + B 2 z exp ( i q 1 z ) ] exp ( i k 0 ρ ) , } ( TM ) ,
B 1 y = t 21 1 + r 21 r 13 exp ( i 2 q 1 d ) sin σ 0 , B 2 y = r 13 exp ( i 2 q 1 d ) B 1 y , B 1 x = q 1 2 q 2 1 t 21 1 + r 21 r 13 exp ( i 2 q 1 d ) cos ϑ 0 cos σ 0 , B 2 x = r 13 exp ( i 2 q 1 d ) B 1 x , B 1 z = 2 1 t 21 1 + r 21 r 13 exp ( i 2 q 1 d ) sin ϑ 0 cos σ 0 B 2 z = r 13 exp ( i 2 q 1 d ) B 1 z , r 21 = q 2 q 1 q 2 + q 1 , t 21 = 1 + r 21 , r 13 = q 1 q 3 q 1 + q 3 , r 21 = 1 q 2 2 q 1 1 q 2 + 2 q 1 , t 21 = 1 + r 21 r 13 = 3 q 1 1 q 3 3 q 1 + 1 q 3 , q i = w c ( i sin 2 ϑ 0 ) 1 / 2 ( i = 1 , 2 , 3 ) .
E 1 x = cos φ B 1 x sin φ B 1 y , E 2 x = cos φ B 2 x sin φ B 2 y , E 1 y = sin φ B 1 x + cos φ B 1 y , E 2 y = sin φ B 2 x + cos φ B 2 y , E 1 z = B 1 z , E 2 z = B 2 z .
[ ( ω c ) 2 2 z 2 0 i k z 0 k 2 ( ω c ) 2 2 z 2 0 i k z 0 k 2 ( ω c ) 2 ] × ( g x x g x y g x z g y x g y y g y z g z x g z y g z z ) = δ ( z z ) ( 100 010 001 ) ,
g x x ( 2 , 1 ) ( z , z ) = i Q 2 ( ω c ) 2 exp ( i Q 2 z ) X × { exp [ i Q 1 ( z + d ) ] R 13 x exp [ i Q 1 ( z + d ) ] } , g y y ( 2 , 1 ) ( z , z ) = i exp ( i Q 2 z ) Y × { exp [ i Q 1 ( z + d ) ] + R 13 y exp [ i Q 1 ( z + d ) ] } , g x z ( 2 , 1 ) ( z , z ) = i Q 2 k ( ω c ) 2 Q 1 exp ( i Q 2 z ) X × { exp [ i Q 1 ( z + d ) ] + R 13 x exp [ i Q 1 ( z + d ) ] } , g z x ( 2 , 1 ) ( z , z ) = k Q 2 g x x ( 2 , 1 ) ( z , z ) , g z z ( 2 , 1 ) ( z , z ) = k Q 2 g x z ( 2 , 1 ) ( z , z ) ,
X = Q 1 1 Q 2 + 2 Q 1 exp ( i Q 1 d ) 1 1 + R 21 x R 13 x exp ( i 2 Q 1 d ) , Y = 1 Q 1 + Q 2 exp ( i Q 1 d ) 1 1 + R 21 y R 13 y exp ( i 2 Q 1 d ) , R 21 x = 1 Q 2 2 Q 1 1 Q 2 + 2 Q 1 , R 13 x = 3 Q 1 1 Q 3 3 Q 1 + 1 Q 3 , R 21 y = Q 2 Q 1 Q 2 + Q 1 R 13 y = Q 1 Q 3 Q 1 + Q 3 , Q i = w c ( i sin 2 ϑ ) 1 / 2 ( i = 1 , 2 , 3 ) .

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