OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 41, Iss. 13 — May. 1, 2002
  • pp: 2521–2531

Matrix description of coherent and incoherent light reflection and transmission by anisotropic multilayer structures

Kamil Postava, Tomuo Yamaguchi, and Roman Kantor  »View Author Affiliations

Applied Optics, Vol. 41, Issue 13, pp. 2521-2531 (2002)

View Full Text Article

Enhanced HTML    Acrobat PDF (211 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We propose a matrix method for the description of light reflection and transmission by an anisotropic multilayer system consisting of thin and thick layers. A method based on partial-wave matrix summations is applicable in the field of reflection and transmission photometry and ellipsometry. In the case of a thin anisotropic film, the interference effects were described by use of a coherent summation of Jones matrices. Incoherent intensity summations for a thick weakly anisotropic layer were characterized by use of the coherency vector formalism. Observable quantities or Mueller matrix components were obtained from the matrix describing transformation of the coherence vectors.

© 2002 Optical Society of America

OCIS Codes
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(260.1180) Physical optics : Crystal optics
(260.1440) Physical optics : Birefringence
(260.2130) Physical optics : Ellipsometry and polarimetry

Original Manuscript: January 11, 2001
Revised Manuscript: October 11, 2001
Published: May 1, 2002

Kamil Postava, Tomuo Yamaguchi, and Roman Kantor, "Matrix description of coherent and incoherent light reflection and transmission by anisotropic multilayer structures," Appl. Opt. 41, 2521-2531 (2002)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).
  2. I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41. [CrossRef]
  3. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972). [CrossRef]
  4. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–755 (1979). [CrossRef]
  5. P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980). [CrossRef]
  6. S̆. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986). [CrossRef]
  7. Š. Višňovský, “Optics of magnetic multilayers,” Czech. J. Phys. B 41, 663–694 (1991). [CrossRef]
  8. M. Mansuripur, “Analysis of multilayer thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2 × 2 matrices,” J. Appl. Phys. 67, 6466–6475 (1990). [CrossRef]
  9. E. Cojocaru, “Simple recurrence matrix relations for multilayer anisotropic thin films,” Appl. Opt. 39, 141–148 (2000). [CrossRef]
  10. I. Abdulhalim, “2 × 2 matrix summation method for multiple reflections and transmissions in a biaxial slab between two anisotropic media,” Opt. Commun. 163, 9–14 (1999). [CrossRef]
  11. S. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering, and polarization from a very rough back surface,” in Optical Scattering: Applications, measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993). [CrossRef]
  12. S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996). [CrossRef]
  13. A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. A 9, 1124–1131 (1992). [CrossRef]
  14. M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986). [CrossRef] [PubMed]
  15. G. Bader, P. V. Ashrit, F. E. Girouard, V.-V. Truong, “Reflection-transmission photoellipsometry: theory and experiments,” Appl. Opt. 34, 1684–1691 (1995). [CrossRef] [PubMed]
  16. G. Bader, P. V. Ashrit, V.-V. Truong, “Transmission and reflection ellipsometry of thin films and multilayer systems,” Appl. Opt. 37, 1146–1151 (1998). [CrossRef]
  17. Y. H. Yang, J. R. Abelson, “Spectroscopic ellipsometry of thin films on transparent substrates: a formalism for data interpretation,” J. Vac. Sci. Technol. A 13, 1145–1149 (1995). [CrossRef]
  18. K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997). [CrossRef]
  19. R. Joerger, K. Forcht, A. Gombert, M. Köhl, W. Graf, “Influence of incoherent superposition of light on ellipsometric coefficients,” Appl. Opt. 36, 319–327 (1997). [CrossRef] [PubMed]
  20. M. Kildemo, P. Buklin, B. Drévillon, O. Hunderi, “Real-time control by multiwavelength ellipsometry of plasma-deposited multilayers on glass by use of an incoherent-reflection model,” Appl. Opt. 36, 6352–6359 (1997). [CrossRef]
  21. M. Kildemo, R. Ossikovski, M. Stchakovsky, “Measurement of the absorption edge of thick transparent substrates using the incoherent reflection model and spectroscopic UV-visible-near IR ellipsometry,” Thin Solid Films 313–314, 108–113 (1998). [CrossRef]
  22. R. Ossikovski, M. Kildemo, M. Stchakovsky, M. Mooney, “Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate,” Appl. Opt. 39, 2071–2077 (2000). [CrossRef]
  23. M. Kildemo, M. Mooney, C. Sudre, P. V. Kelly, “Investigation of a half-wave method for birefringence or thickness measurements of a thick, semitransparent, uniaxial, anisotropic substrate by use of spectroscopic ellipsometry,” Appl. Opt. 39, 4649–4657 (2000). [CrossRef]
  24. K. Postava, J. Pištora, Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49, 1185–1204 (1999). [CrossRef]
  25. M. Shubert, T. E. Tiwald, J. A. Woollam, “Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry,” Appl. Opt. 38, 177–187 (1999). [CrossRef]
  26. K. Postava, J. F. Bobo, M. D. Ortega, B. Raquet, H. Jaffres, E. Snoeck, M. Goiran, A. R. Fert, J. P. Redoules, J. Pištora, J. C. Ouset, “Magneto-optical measurements of magnetization reversal in nanometer scale sputtered Fe thin films,” J. Magn. Magn. Mater. 163, 8–20 (1996). [CrossRef]
  27. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982). [CrossRef]
  28. C. Gu, P. Yeh, “Extended Jones matrix method II,” J. Opt. Soc. Am. A 10, 966–973 (1993). [CrossRef]
  29. A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990). [CrossRef]
  30. M. Kildemo, M. B. Mooney, P. V. Kelly, C. Sudre, G. M. Crean, “Anisotropic dielectric function properties of semi-insulating 4H-SiC determined from spectroscopic ellipsometry,” Mater. Sci. Forum 338–342, 571–574 (2000). [CrossRef]
  31. M. Kildemo, O. Hunderi, “Spectroscopic Fourier methods for thickness measurements of thick uniaxial wafers, with dispersive birefrigence, using polarimetric techniques,” J. Opt. A. Pure Appl. Opt. 2, L33–L37 (2000). [CrossRef]
  32. S.-M. F. Nee, “Depolarization and retardation of a birefringent slab,” J. Opt. Soc. Am. A 17, 2067–2073 (2000). [CrossRef]
  33. K. Postava, T. Yamaguchi, T. Nakano, “Characterization of organic low-dielectric-constant materials using optical spectroscopy,” Opt. Express 9, 141–151 (2001), http://www.opticsexpress.org/oearchive/source/32940.htm . [CrossRef] [PubMed]
  34. S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999). [CrossRef]
  35. S.-M. F. Nee, T. Cole, “Effects of depolarization of polarimetric components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998). [CrossRef]
  36. I. Ohlídal, D. Franta, “Matrix formalism for imperfect thin films,” Acta Phys. Slova. 50, 489–500 (2000).
  37. K. Postava, J. Pištora, P. Hlubina, “Effect of permeability on guided modes in planar structures,” in OPTIKA “98: 5th congress on modern optics, G. Akos, G. Lupkovics, A. Podmaniczky, Proc. SPIE3573, 524–527 (1998). [CrossRef]
  38. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, London, 1984), Chap. 4.
  39. J. Pištora, K. Postava, R. Šebesta, “Optical guided modes in sandwiches with ultrathin metallic films,” J. Magn. Magn. Mater. 168–9, 683–685 (1999). [CrossRef]
  40. Ch. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996). [CrossRef]
  41. Ch. Brosseau, Fundamentals of Polarized Light—A Statistical Optics Approach (Wiley, New York, 1998).
  42. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975).
  43. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995). [CrossRef]
  44. A. Gerrard, J. M. Burch, Introduction to Matrix Method in Optics (Wiley, London, 1975).
  45. Š. Višňovský, V. Prosser, R. Krishnan, “Effect of multiple internal reflections on Faraday rotation in multilayer structures,” J. Appl. Phys. 49, 403–408 (1978). [CrossRef]
  46. Š. Višňovský, R. Krishnan, “Complex Faraday effect in multilayer structures,” J. Opt. Soc. Am. 71, 315–319 (1981). [CrossRef]
  47. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981). [CrossRef]
  48. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982). [CrossRef]
  49. D. G. M. Anderson, R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305–2319 (1994). [CrossRef]
  50. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987). [CrossRef]
  51. R. Kantor, K. Postava, D. Ciprian, J. Pištora, “Optics of anisotropic media: Jones 2 × 2 matrix method for description of multilayerd structures,” in Proceedings of International Conference VŠB-TU Ostrava, Czech Republic (University Press Centrum, Ostrava, Czech Republic, 1995), pp. 49–54.
  52. J. Pištora, A. Domaňski, O. Bárta, F. Staněk, K. Postava, D. Ciprian, I. Kopřiva, “Waveguiding in thin films with quadratic magneto-optical medium,” Acta Phys. Pol. A 99, 17–23 (2001).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


Fig. 1 Fig. 2 Fig. 3
Fig. 4 Fig. 5

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited