Abstract

Fresnel reflectance and transmittance coefficients of a thin film system consisting of an arbitrary number of layers are expressed explicitly in the form of a power series of Fresnel coefficients for individual boundaries and phase terms for the individual films. The series is based on the evaluation of all possible paths light can pass through the system. However, the series is written as consolidated, i. e. all paths corresponding to the same powers are represented using a single term in the series, with multiplicity which is a simple product of binomial coefficients. This result is used to express the normal reflectance of a thin film system with arbitrarily correlated randomly rough boundaries and it is shown that such approach can be computationally efficient in practice.

© 2014 Optical Society of America

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2013 (1)

2011 (2)

S. Schröder, T. Herffurth, H. Blaschke, A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. 50, C164–C171 (2011).
[CrossRef] [PubMed]

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

2007 (1)

J. Meunier, “Optical reflectivity of thin rough films: Application to ellipsometric measurements of liquid films,” Phys. Rev. E 75, 061601 (2007).
[CrossRef]

2005 (1)

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

2002 (2)

J. E. Harvey, S. Schröder, N. Choi, A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2002).
[CrossRef]

A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171 (2002).
[CrossRef] [PubMed]

1999 (1)

I. Ohlídal, F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. 46, 2043–2062 (1999).
[CrossRef]

1998 (2)

D. Franta, I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45, 903–934 (1998).
[CrossRef]

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

1997 (1)

1994 (2)

1993 (2)

1992 (1)

1991 (1)

1990 (1)

J. T. Butler, “On the number of propagation paths in multilayer media,” Fibonacci Q. 28, 334–339 (1990).

1989 (1)

I. Ohlídal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
[CrossRef]

1988 (1)

L. Névot, B. Pardo, J. Corno, “Characterization of X-UV multilayers by grazing incidence X-ray reflectometry,” Rev. Phys. Appl. 23, 1675–1686 (1988).
[CrossRef]

1987 (1)

1985 (1)

J. Szczyrbowski, K. Schmalzbauer, H. Hoffmann, “Optical properties of rough thin films,” Thin Solid Films 130, 57–73 (1985).
[CrossRef]

1979 (2)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
[CrossRef]

1977 (1)

J. Bauer, L. Biste, D. Bolze, “Optical-properties of aluminum nitride prepared by chemical and plasmachemical vapor-deposition,” Phys. Status Solidi A 39, 173–181 (1977).
[CrossRef]

1974 (2)

J. M. Eastman, P. W. Baumeist, “Measurement of microtopography of optical surfaces using a scanning Fizeau interferometer,” J. Opt. Soc. Am. 64, 1369 (1974).

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

1972 (1)

I. Ohlídal, F. Lukeš, “Ellipsometric parameters of rough surfaces and of a system substrate-thin film with rough boundaries,” Opt. Acta 19, 817–843 (1972).
[CrossRef]

1971 (2)

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light on a system of non-absorbing isotropic film–non-absorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[CrossRef]

1967 (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. 15, 552–557 (1967).
[CrossRef]

1963 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

1948 (1)

1947 (1)

1937 (1)

P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291–384 (1937).

Amra, C.

Apfel, J. H.

Aspnes, D. E.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Bauer, J.

J. Bauer, L. Biste, D. Bolze, “Optical-properties of aluminum nitride prepared by chemical and plasmachemical vapor-deposition,” Phys. Status Solidi A 39, 173–181 (1977).
[CrossRef]

Baumeist, P. W.

J. M. Eastman, P. W. Baumeist, “Measurement of microtopography of optical surfaces using a scanning Fizeau interferometer,” J. Opt. Soc. Am. 64, 1369 (1974).

Bennett, H. E.

Bennett, J. M.

Biste, L.

J. Bauer, L. Biste, D. Bolze, “Optical-properties of aluminum nitride prepared by chemical and plasmachemical vapor-deposition,” Phys. Status Solidi A 39, 173–181 (1977).
[CrossRef]

Blaschke, H.

Bolze, D.

J. Bauer, L. Biste, D. Bolze, “Optical-properties of aluminum nitride prepared by chemical and plasmachemical vapor-deposition,” Phys. Status Solidi A 39, 173–181 (1977).
[CrossRef]

Bonanni, A.

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

Bottomley, H.

H. Bottomley, “The on-line encyclopedia of integer sequences,” http://oeis.org/A080937 . Sequence A080937.

Bruel, L.

Butler, J. T.

J. T. Butler, “On the number of propagation paths in multilayer media,” Fibonacci Q. 28, 334–339 (1990).

Carniglia, C. K.

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
[CrossRef]

Choi, N.

J. E. Harvey, S. Schröder, N. Choi, A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2002).
[CrossRef]

Coriand, L.

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

Corno, J.

L. Névot, B. Pardo, J. Corno, “Characterization of X-UV multilayers by grazing incidence X-ray reflectometry,” Rev. Phys. Appl. 23, 1675–1686 (1988).
[CrossRef]

Crook, A. W.

Drévillon, B.

Duparré, A.

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

S. Schröder, T. Herffurth, H. Blaschke, A. Duparré, “Angle-resolved scattering: an effective method for characterizing thin-film coatings,” Appl. Opt. 50, C164–C171 (2011).
[CrossRef] [PubMed]

A. Duparré, J. Ferre-Borrull, S. Gliech, G. Notni, J. Steinert, J. M. Bennett, “Surface characterization techniques for determining the root-mean-square roughness and power spectral densities of optical components,” Appl. Opt. 41, 154–171 (2002).
[CrossRef] [PubMed]

J. E. Harvey, S. Schröder, N. Choi, A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2002).
[CrossRef]

Eastman, J. M.

J. M. Eastman, P. W. Baumeist, “Measurement of microtopography of optical surfaces using a scanning Fizeau interferometer,” J. Opt. Soc. Am. 64, 1369 (1974).

J. M. Eastman, “Scattering in all-dielectric multilayer bandpass filters and mirrors for lasers,” in Physics of Thin Films, G. Hass, H. M. Francombe, eds. (Academic, 1978), Vol. 10, p. 167.

Ferre-Borrull, J.

Franta, D.

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

D. Franta, I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45, 903–934 (1998).
[CrossRef]

I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2000), vol. 41, pp. 181–282.
[CrossRef]

Freilikher, V.

Gliech, S.

Grèzes-Besset, C.

Harvey, J. E.

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

J. E. Harvey, S. Schröder, N. Choi, A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2002).
[CrossRef]

Herffurth, T.

Herzinger, C. M.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Hoffmann, H.

J. Szczyrbowski, K. Schmalzbauer, H. Hoffmann, “Optical properties of rough thin films,” Thin Solid Films 130, 57–73 (1985).
[CrossRef]

Hottier, F.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Hunderi, O.

Jacobson, R.

R. Jacobson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1966), Vol. 5, pp. 249–286.

Johs, B.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Kildemo, M.

Killough, L. D.

L. D. Killough, “The on-line encyclopedia of integer sequences,” http://oeis.org/A011782 . Sequence A011782.

Knittl, Z.

Z. Knittl, Optics of Thin Films (John Wiley, 1976).

Krywonos, A.

A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. thesis, University of Central Florida, Orlando (2006).

Lukeš, F.

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

I. Ohlídal, F. Lukeš, “Ellipsometric parameters of rough surfaces and of a system substrate-thin film with rough boundaries,” Opt. Acta 19, 817–843 (1972).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light on a system of non-absorbing isotropic film–non-absorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[CrossRef]

Mallows, C.

C. Mallows, N. J. A. Sloane, S. Plouffe, R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051 . Sequence A007051.

McGahan, W. A.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Meunier, J.

J. Meunier, “Optical reflectivity of thin rough films: Application to ellipsometric measurements of liquid films,” Phys. Rev. E 75, 061601 (2007).
[CrossRef]

Montaigne-Ramil, A.

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

Navrátil, K.

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light on a system of non-absorbing isotropic film–non-absorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[CrossRef]

I. Ohlídal, K. Navrátil, M. Ohlídal, “Scattering of light from multilayer systems with rough boundaries,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1995), Vol. 34, pp. 249–331.
[CrossRef]

Névot, L.

L. Névot, B. Pardo, J. Corno, “Characterization of X-UV multilayers by grazing incidence X-ray reflectometry,” Rev. Phys. Appl. 23, 1675–1686 (1988).
[CrossRef]

Notni, G.

Ohlídal, I.

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

I. Ohlídal, F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. 46, 2043–2062 (1999).
[CrossRef]

D. Franta, I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45, 903–934 (1998).
[CrossRef]

I. Ohlídal, “Approximate formulas for the reflectances, transmittances, and scattering losses of nonabsorbing multilayers systems with randomly rough boundaries,” J. Opt. Soc. Am. A 10, 158–170 (1993).
[CrossRef]

I. Ohlídal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
[CrossRef]

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

I. Ohlídal, F. Lukeš, “Ellipsometric parameters of rough surfaces and of a system substrate-thin film with rough boundaries,” Opt. Acta 19, 817–843 (1972).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light on a system of non-absorbing isotropic film–non-absorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light by a system of nonabsorbing isotropic film–nonabsorbing isotropic substrate with randomly rough boundaries,” J. Opt. Soc. Am. 61, 1630–1639 (1971).
[CrossRef]

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[CrossRef]

Ohlídal, M.

I. Ohlídal, K. Navrátil, M. Ohlídal, “Scattering of light from multilayer systems with rough boundaries,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1995), Vol. 34, pp. 249–331.
[CrossRef]

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L. Névot, B. Pardo, J. Corno, “Characterization of X-UV multilayers by grazing incidence X-ray reflectometry,” Rev. Phys. Appl. 23, 1675–1686 (1988).
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C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

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[CrossRef]

H. R. Philipp, “Silicon dioxide (SiO2) (glass),” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), Vol. I, pp. 749–763.
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C. Mallows, N. J. A. Sloane, S. Plouffe, R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051 . Sequence A007051.

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J. Szczyrbowski, K. Schmalzbauer, H. Hoffmann, “Optical properties of rough thin films,” Thin Solid Films 130, 57–73 (1985).
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Schröder, S.

T. Herffurth, S. Schröder, M. Trost, A. D. A. Tünnermann, “Comprehensive nanostructure and defect analysis using a simple 3D light-scatter sensor,” Appl. Opt. 52, 3279–3287 (2013).
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S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

J. E. Harvey, S. Schröder, N. Choi, A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2002).
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M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

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M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

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N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A024175 . Sequence A024175.

C. Mallows, N. J. A. Sloane, S. Plouffe, R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051 . Sequence A007051.

N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A000108 . Sequence A000108.

N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A001519 . Sequence A001519.

Steinert, J.

Stifter, D.

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

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A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, 1966).

Szczyrbowski, J.

J. Szczyrbowski, K. Schmalzbauer, H. Hoffmann, “Optical properties of rough thin films,” Thin Solid Films 130, 57–73 (1985).
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D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

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Tünnermann, A.

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

Tünnermann, A. D. A.

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G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. 15, 552–557 (1967).
[CrossRef]

Vašícek, A.

Vižd’a, F.

I. Ohlídal, F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. 46, 2043–2062 (1999).
[CrossRef]

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C. Mallows, N. J. A. Sloane, S. Plouffe, R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051 . Sequence A007051.

Woollam, J. A.

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

Yurkevich, I.

Zavislan, J. M.

Ann. Phys. (1)

P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291–384 (1937).

Appl. Opt. (7)

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Fibonacci Q. (1)

J. T. Butler, “On the number of propagation paths in multilayer media,” Fibonacci Q. 28, 334–339 (1990).

IEEE Trans. Antennas Propag. (1)

G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. 15, 552–557 (1967).
[CrossRef]

J. Appl. Phys. (1)

C. M. Herzinger, B. Johs, W. A. McGahan, J. A. Woollam, W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83, 3323–3336 (1998).
[CrossRef]

J. Mod. Opt. (3)

I. Ohlídal, F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. 46, 2043–2062 (1999).
[CrossRef]

M. Šiler, I. Ohlídal, D. Franta, A. Montaigne-Ramil, A. Bonanni, D. Stifter, H. Sitter, “Optical characterization of double layers containing epitaxial ZnSe and ZnTe films,” J. Mod. Opt. 52, 583–602 (2005).
[CrossRef]

D. Franta, I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45, 903–934 (1998).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

I. Ohlídal, F. Lukeš, “Ellipsometric parameters of rough surfaces and of a system substrate-thin film with rough boundaries,” Opt. Acta 19, 817–843 (1972).
[CrossRef]

Opt. Commun. (2)

I. Ohlídal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989).
[CrossRef]

I. Ohlídal, K. Navrátil, F. Lukeš, “Reflection of light on a system of non-absorbing isotropic film–non-absorbing isotropic substrate with rough boundaries,” Opt. Commun. 3, 40–44 (1971).
[CrossRef]

Opt. Eng. (3)

J. E. Harvey, S. Schröder, N. Choi, A. Duparré, “Total integrated scatter from surfaces with arbitrary roughness, correlation widths, and incident angles,” Opt. Eng. 51, 013402 (2002).
[CrossRef]

C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979).
[CrossRef]

S. Schröder, A. Duparré, L. Coriand, A. Tünnermann, D. H. Penalver, J. E. Harvey, “Modeling of light scattering in different regimes of surface roughness,” Opt. Eng. 19, 9820–9835 (2011).

Opt. Lett. (1)

Phys. Rev. B (1)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Phys. Rev. E (1)

J. Meunier, “Optical reflectivity of thin rough films: Application to ellipsometric measurements of liquid films,” Phys. Rev. E 75, 061601 (2007).
[CrossRef]

Phys. Status Solidi A (1)

J. Bauer, L. Biste, D. Bolze, “Optical-properties of aluminum nitride prepared by chemical and plasmachemical vapor-deposition,” Phys. Status Solidi A 39, 173–181 (1977).
[CrossRef]

Rev. Phys. Appl. (1)

L. Névot, B. Pardo, J. Corno, “Characterization of X-UV multilayers by grazing incidence X-ray reflectometry,” Rev. Phys. Appl. 23, 1675–1686 (1988).
[CrossRef]

Surf. Sci. (1)

I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974).
[CrossRef]

Thin Solid Films (1)

J. Szczyrbowski, K. Schmalzbauer, H. Hoffmann, “Optical properties of rough thin films,” Thin Solid Films 130, 57–73 (1985).
[CrossRef]

Other (17)

I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2000), vol. 41, pp. 181–282.
[CrossRef]

R. Jacobson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1966), Vol. 5, pp. 249–286.

Z. Knittl, Optics of Thin Films (John Wiley, 1976).

J. M. Eastman, “Scattering in all-dielectric multilayer bandpass filters and mirrors for lasers,” in Physics of Thin Films, G. Hass, H. M. Francombe, eds. (Academic, 1978), Vol. 10, p. 167.

A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2010).

J. W. S. Rayleigh, Theory of Sound (Macmillan, 1877), Vol. 2.

A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. thesis, University of Central Florida, Orlando (2006).

L. D. Killough, “The on-line encyclopedia of integer sequences,” http://oeis.org/A011782 . Sequence A011782.

N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A001519 . Sequence A001519.

C. Mallows, N. J. A. Sloane, S. Plouffe, R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051 . Sequence A007051.

H. Bottomley, “The on-line encyclopedia of integer sequences,” http://oeis.org/A080937 . Sequence A080937.

N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A024175 . Sequence A024175.

N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A000108 . Sequence A000108.

I. Ohlídal, K. Navrátil, M. Ohlídal, “Scattering of light from multilayer systems with rough boundaries,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1995), Vol. 34, pp. 249–331.
[CrossRef]

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, 1966).

H. R. Philipp, “Silicon dioxide (SiO2) (glass),” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), Vol. I, pp. 749–763.
[CrossRef]

H. R. Philipp, “Silicon nitride (Si3N4) (noncrystalline),” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, 1985), Vol. I, pp. 771–774.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Illustration of equivalent light paths through a two-layer system. Paths (a) and (b) lead to identical powers of both the Fresnel coefficients and phase terms, whereas (c) leads only to the same phase terms powers as (a) and (b); the Fresnel coefficients differ.

Fig. 2
Fig. 2

Reflection and transmission coefficient and phase term notation.

Fig. 3
Fig. 3

Relations between medium and visit counts connected with medium j and the numbers of reflections within this medium and refractions to or from this medium.

Fig. 4
Fig. 4

Schema of the system of three thin films studied in the numerical example.

Fig. 5
Fig. 5

Calculated spectral dependence of normal reflectance for the three-layer system. The curve labelled ‘series, integration or ray tracing’ represents the result of all three methods when calculated to a sufficient precision.

Fig. 6
Fig. 6

Trade-off between computation speed and precision for individual calculation procedures. The approximate formulae are represented with a single point since their precision is not tunable.

Fig. 7
Fig. 7

Dependence of computation efficiencies of the series and direct integration on the magnitude of roughness. The mean times correspond to computations achieving the precision at least 10−6.

Tables (1)

Tables Icon

Table 1 Number of terms in the series for individual path lengths p and two to five-layer systems. Columns labelled ‘all’ represent all possible paths and correspond to Table 2 in [39]. Columns labelled ‘cons.’ represent the consolidated number of terms corresponding to the series presented in this work.

Equations (38)

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

X ^ j = 2 π λ N ^ j h j cos ϑ ^ j .
m = 2 p + 1 .
p = j = 1 d m j
v j > 0 , m j > 0 ,
m j max ( v j , v j + 1 ) or v j min ( m j , m j 1 ) .
p = 1 P d = 1 min ( L , p ) m v
m = m 1 = 1 p ( d 1 ) M 0 m 2 = 1 p ( d 2 ) M 1 m d 1 = 1 p 1 M d 2
v = v 2 = 1 min ( m 1 , m 2 ) v 3 = 1 min ( m 2 , m 3 ) v d = 1 min ( m d 1 , m d ) .
M j = l = 1 j m l , M 0 0
( t ^ j t ^ j ) v j ( r ^ j ) m j v j r ^ j + 1 m j v j + 1 exp ( 2 i m j X ^ j )
P 1 ( m 1 ) = ( t ^ 1 t ^ 1 ) v 1 ( r ^ 1 ) m 1 v 1 r ^ 2 m 1 exp ( 2 i m 1 X ^ 1 ) = t ^ 1 t ^ 1 ( r ^ 1 ) m 1 1 r ^ 2 m 1 exp ( 2 i m 1 X ^ 1 ) ,
P 2 ( m 1 , m 2 , v 2 ) = t ^ 1 t ^ 1 ( r ^ 1 ) m 1 1 r ^ 2 m 1 v 2 ( t ^ 2 t ^ 2 ) v 2 ( r ^ 2 ) m 2 v 2 r ^ 3 m 2 exp ( 2 i m 2 X ^ 2 ) exp ( 2 i m 1 X ^ 1 ) .
P 3 ( m 1 , m 2 , m 3 , v 2 , v 3 ) = t ^ 1 t ^ 1 ( r ^ 1 ) m 1 1 r ^ 2 m 1 v 2 ( t ^ 2 t ^ 2 ) v 2 ( r ^ 2 ) m 2 v 2 r ^ 3 m 2 v 3 ( t ^ 3 t ^ 3 ) v 3 ( r ^ 3 ) m 3 v 3 r ^ 4 m 3 × exp ( 2 i m 3 X ^ 3 ) exp ( 2 i m 2 X ^ 2 ) exp ( 2 i m 1 X ^ 1 ) .
( m j 1 v j )
( m j 1 v j 1 )
F ( v j , m j , m j 1 ) = ( m j 1 v j ) ( m j 1 v j 1 )
C d ( m 1 , m 2 , , m d 1 ; v 2 , v 3 , , v d ) = C d ( m , v ) = j = 1 d F ( v j , m j , m j 1 ) .
C 0 1 , C 1 ( m 1 ) = C 0 F ( v 1 , m 1 , m 0 ) = ( 1 1 ) ( m 1 1 0 ) = 1 , C 2 ( m 1 , m 2 ; v 2 ) = C 1 ( m 1 ) F ( v 2 , m 2 , m 1 ) = ( m 1 v 2 ) ( m 2 1 v 2 1 ) , C 3 ( m 1 , m 2 , m 3 ; v 2 , v 3 ) = C 2 ( m 1 , m 2 ; v 2 ) F ( v 3 , m 3 , m 2 ) = ( m 1 v 2 ) ( m 2 1 v 2 1 ) ( m 2 v 3 ) ( m 3 1 v 3 1 ) .
r ^ = r ^ 1 + p = 1 P d = 1 min ( L , p ) m exp ( 2 i j = 1 d m j X ^ j ) Q ^ ( m ) ,
Q ^ ( m ) = t ^ 1 t ^ 1 ( r ^ 1 ) m 1 1 r ^ d + 1 m d j = 2 d v j = 1 min ( m j , m j 1 ) F ( v j , m j , m j 1 ) r ^ j m j 1 v j ( t ^ j t ^ j ) v j ( r ^ j ) m j v j .
p 2 ( p p 2 ) p 2 4 .
t ^ j v j ( t ^ j ) v j 1 ( r ^ j ) m j v j r ^ j + 1 m j v j + 1 exp ( i ( 2 m j 1 ) X ^ j )
F ( v j , m j , m j 1 ) = ( m j 1 1 v j 1 ) ( m j 1 v j 1 ) .
R = r ^ ( u ) exp ( i v t 1 ) r ^ * ( u ) exp ( i v t 1 ) = | r ^ ( u ) exp ( i v t 1 ) | 2 ,
v = 4 π λ N 0 ,
r ^ ( u ) exp ( i v t 1 ) = r ^ ( u ) exp ( i v t 1 ) ρ ( u ) d u .
ρ ( u ) = 1 det S 1 ( 2 π ) ( L + 1 ) / 2 exp ( 1 2 u T S 1 u ) ,
S i , j = u i u j = σ i σ j C i , j ,
σ j 2 = u j 2
C i , j = u i u j σ i σ j .
1 det S 1 ( 2 π ) ( L + 1 ) / 2 exp ( i f T u ) exp ( 1 2 u T S 1 u ) d u = exp ( 1 2 f T S f ) .
r ^ ( u ) exp ( i v t 1 ) = r ^ 1 exp ( v 2 σ 1 2 2 ) + p = 1 P d = 1 min ( L , p ) m exp ( 2 i j = 1 d m j X ^ j ) Q ^ ( m ) H ^ ( m ) .
X ^ j = 2 π λ N ^ j h j .
H ^ ( m ) = exp ( 1 2 q = 1 d + 1 s = 1 d + 1 D ^ q D ^ s S q , s ) ,
D ^ j = 4 π λ ( m j N ^ j m j 1 N ^ j 1 ) .
u k ( x , y ) = u k + 1 ( x , y ) + ε k ( x , y ) for 1 k L ,
u L + 1 ( x , y ) u L + 1 ( x , y ) = σ L + 1 2 , ε k ( x , y ) ε j ( x , y ) = δ j k σ 2 , ε k ( x , y ) u L + 1 ( x , y ) = 0 .
S j k = σ L + 1 2 + ( L + 1 max ( j , k ) ) σ 2 .

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