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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4499–4515

Consolidated series for efficient calculation of the reflection and transmission in rough multilayers

David Nečas and Ivan Ohlídal  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 4499-4515 (2014)
http://dx.doi.org/10.1364/OE.22.004499


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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

OCIS Codes
(000.3860) General : Mathematical methods in physics
(240.0310) Optics at surfaces : Thin films
(240.5770) Optics at surfaces : Roughness
(310.6805) Thin films : Theory and design

ToC Category:
Thin Films

History
Original Manuscript: September 24, 2013
Revised Manuscript: January 10, 2014
Manuscript Accepted: January 13, 2014
Published: February 20, 2014

Citation
David Nečas and Ivan Ohlídal, "Consolidated series for efficient calculation of the reflection and transmission in rough multilayers," Opt. Express 22, 4499-4515 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4499


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References

  1. I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2000), vol. 41, pp. 181–282. [CrossRef]
  2. 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.
  3. Z. Knittl, Optics of Thin Films (John Wiley, 1976).
  4. P. Rouard, “Etudes des propriétés optiques des lames métalliques très minces,” Ann. Phys. 7, 291–384 (1937).
  5. A. Vašíček, “The reflection of light from glass with double and multiple films,” J. Opt. Soc. Am. 37, 623–634 (1947). [CrossRef]
  6. A. W. Crook, “The reflection and transmission of light by any system of parallel isotropic films,” J. Opt. Soc. Am. 38, 954–963 (1948). [CrossRef] [PubMed]
  7. 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]
  8. J. Szczyrbowski, K. Schmalzbauer, H. Hoffmann, “Optical properties of rough thin films,” Thin Solid Films 130, 57–73 (1985). [CrossRef]
  9. 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]
  10. H. E. Bennett, “Specular reflectance of aluminized ground glass and the height distribution of surface irregularities,” J. Opt. Soc. Am. 53, 1389–1394 (1963). [CrossRef]
  11. 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]
  12. 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]
  13. J. M. Eastman, P. W. Baumeist, “Measurement of microtopography of optical surfaces using a scanning Fizeau interferometer,” J. Opt. Soc. Am. 64, 1369 (1974).
  14. 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.
  15. C. K. Carniglia, “Scalar scattering theory for multilayer optical coatings,” Opt. Eng. 18, 104–115 (1979). [CrossRef]
  16. 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]
  17. J. M. Zavislan, “Angular scattering from optical interference coatings: scalar scattering predictions and measurements,” Appl. Opt. 30, 2224–2244 (1991). [CrossRef] [PubMed]
  18. C. Amra, J. H. Apfel, E. Pelltier, “Role of interface correlation in light scattering by a multilayer,” Appl. Opt. 31, 3134–3151 (1992). [CrossRef] [PubMed]
  19. C. Amra, C. Grèzes-Besset, L. Bruel, “Comparison of surface and bulk scattering in optical multilayers,” Appl. Opt. 32, 5492–5503 (1993). [CrossRef] [PubMed]
  20. C. Amra, “Light scattering from multilayer optics I. tools of investigation,” J. Opt. Soc. Am. A 11, 197–210 (1994). [CrossRef]
  21. 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]
  22. J. Meunier, “Optical reflectivity of thin rough films: Application to ellipsometric measurements of liquid films,” Phys. Rev. E 75, 061601 (2007). [CrossRef]
  23. A. A. Maradudin, ed., Light Scattering and Nanoscale Surface Roughness (Springer, 2010).
  24. 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]
  25. 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]
  26. 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). [CrossRef] [PubMed]
  27. J. W. S. Rayleigh, Theory of Sound (Macmillan, 1877), Vol. 2.
  28. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951). [CrossRef]
  29. G. R. Valenzuela, “Depolarization of EM waves by slightly rough surfaces,” IEEE Trans. Antennas Propag. 15, 552–557 (1967). [CrossRef]
  30. R. Schiffer, “Reflectivity of a slightly rough surface,” Appl. Opt. 26, 704–712 (1987). [CrossRef] [PubMed]
  31. V. Freilikher, M. Pustilnik, I. Yurkevich, V. I. Tatarskii, “Polarization of light scattered from slightly rough dielectric film,” Opt. Lett. 19, 1382–1384 (1994). [CrossRef] [PubMed]
  32. D. Franta, I. Ohlídal, “Ellipsometric parameters and reflectances of thin films with slightly rough boundaries,” J. Mod. Opt. 45, 903–934 (1998). [CrossRef]
  33. A. Krywonos, “Predicting surface scatter using a linear systems formulation of non-paraxial scalar diffraction,” Ph.D. thesis, University of Central Florida, Orlando (2006).
  34. 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).
  35. I. Ohlídal, F. Vižd’a, “Optical quantities of multilayer systems with correlated randomly rough boundaries,” J. Mod. Opt. 46, 2043–2062 (1999). [CrossRef]
  36. 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]
  37. 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]
  38. I. Ohlídal, F. Lukeš, K. Navrátil, “Rough silicon surfaces studied by optical methods,” Surf. Sci. 45, 91–116 (1974). [CrossRef]
  39. J. T. Butler, “On the number of propagation paths in multilayer media,” Fibonacci Q. 28, 334–339 (1990).
  40. L. D. Killough, “The on-line encyclopedia of integer sequences,” http://oeis.org/A011782 . Sequence A011782.
  41. N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A001519 . Sequence A001519.
  42. C. Mallows, N. J. A. Sloane, S. Plouffe, R. G. Wilson, “The on-line encyclopedia of integer sequences,” http://oeis.org/A007051 . Sequence A007051.
  43. H. Bottomley, “The on-line encyclopedia of integer sequences,” http://oeis.org/A080937 . Sequence A080937.
  44. N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A024175 . Sequence A024175.
  45. N. J. A. Sloane, “The on-line encyclopedia of integer sequences,” http://oeis.org/A000108 . Sequence A000108.
  46. M. Kildemo, O. Hunderi, B. Drévillon, “Approximation of reflection coefficients for rapid real-time calculation of inhomogeneous films,” J. Opt. Soc. Am. A 14, 931–939 (1997). [CrossRef]
  47. I. Ohlídal, “Reflectance of multilayer systems with randomly rough boundaries,” Opt. Commun. 71, 323–326 (1989). [CrossRef]
  48. 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]
  49. 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]
  50. A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, 1966).
  51. 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]
  52. 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]
  53. 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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