Abstract

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

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    [CrossRef]
  9. E. Cojocaru, “Simple recurrence matrix relations for multilayer anisotropic thin films,” Appl. Opt. 39, 141–148 (2000).
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  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).
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  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).
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    [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).
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    [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).

2001 (2)

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]

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).

2000 (7)

I. Ohlídal, D. Franta, “Matrix formalism for imperfect thin films,” Acta Phys. Slova. 50, 489–500 (2000).

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]

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]

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]

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]

S.-M. F. Nee, “Depolarization and retardation of a birefringent slab,” J. Opt. Soc. Am. A 17, 2067–2073 (2000).
[CrossRef]

E. Cojocaru, “Simple recurrence matrix relations for multilayer anisotropic thin films,” Appl. Opt. 39, 141–148 (2000).
[CrossRef]

1999 (5)

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]

K. Postava, J. Pištora, Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49, 1185–1204 (1999).
[CrossRef]

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]

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]

S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
[CrossRef]

1998 (3)

S.-M. F. Nee, T. Cole, “Effects of depolarization of polarimetric components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

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]

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]

1997 (3)

1996 (3)

S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
[CrossRef]

Ch. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

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]

1995 (2)

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]

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]

1994 (1)

1993 (1)

1992 (1)

1991 (1)

Š. Višňovský, “Optics of magnetic multilayers,” Czech. J. Phys. B 41, 663–694 (1991).
[CrossRef]

1990 (2)

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]

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

1987 (1)

1986 (2)

1982 (2)

P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

1981 (2)

Š. Višňovský, R. Krishnan, “Complex Faraday effect in multilayer structures,” J. Opt. Soc. Am. 71, 315–319 (1981).
[CrossRef]

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

1980 (1)

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

1979 (1)

1978 (1)

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]

1972 (1)

Abdulhalim, I.

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]

Abelson, J. R.

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]

Anderson, D. G. M.

Archibald, P. C.

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]

Ashrit, P. V.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

Bader, G.

Barakat, R.

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]

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

Bárta, O.

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).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

Bennett, J. M.

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]

Berreman, D. W.

Bobo, J. F.

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]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975).

Brosseau, Ch.

Ch. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

Ch. Brosseau, Fundamentals of Polarized Light—A Statistical Optics Approach (Wiley, New York, 1998).

Buklin, P.

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Method in Optics (Wiley, London, 1975).

Ciprian, D.

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).

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.

Cojocaru, E.

Cole, T.

S.-M. F. Nee, T. Cole, “Effects of depolarization of polarimetric components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Crean, G. M.

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]

Domanski, A.

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).

Drévillon, B.

Fert, A. R.

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]

Forcht, K.

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

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]

Franta, D.

I. Ohlídal, D. Franta, “Matrix formalism for imperfect thin films,” Acta Phys. Slova. 50, 489–500 (2000).

I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41.
[CrossRef]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Method in Optics (Wiley, London, 1975).

Girouard, F. E.

Goiran, M.

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]

Gombert, A.

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

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]

Graf, W.

Gu, C.

Hlubina, P.

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]

Hunderi, O.

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]

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]

Jaffres, H.

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]

Joerger, R.

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]

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

Kantor, R.

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.

Kelly, P. V.

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]

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]

Kildemo, M.

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]

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]

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]

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]

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]

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]

Kim, K.

Köhl, M.

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]

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

Kopriva, I.

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).

Krishnan, R.

Š. Višňovský, R. Krishnan, “Complex Faraday effect in multilayer structures,” J. Opt. Soc. Am. 71, 315–319 (1981).
[CrossRef]

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]

Lien, A.

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

Mandel, L.

Mansuripur, M.

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]

Mooney, M.

Mooney, M. B.

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]

Nakano, T.

Nee, S. F.

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]

Nee, S.-M. F.

Ohlídal, I.

I. Ohlídal, D. Franta, “Matrix formalism for imperfect thin films,” Acta Phys. Slova. 50, 489–500 (2000).

I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41.
[CrossRef]

Ortega, M. D.

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]

Ossikovski, R.

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]

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]

Ouset, J. C.

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]

Pištora, J.

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).

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]

K. Postava, J. Pištora, Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49, 1185–1204 (1999).
[CrossRef]

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]

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]

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.

Postava, K.

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).

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]

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]

K. Postava, J. Pištora, Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49, 1185–1204 (1999).
[CrossRef]

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]

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]

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.

Prosser, V.

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]

Raquet, B.

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]

Redoules, J. P.

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]

Röseler, A.

Šebesta, R.

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]

Shubert, M.

Simon, R.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Snoeck, E.

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]

Stanek, F.

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).

Stchakovsky, M.

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]

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]

Sudre, C.

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]

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]

Tiwald, T. E.

Truong, V.-V.

Višnovský, Š.

K. Postava, J. Pištora, Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49, 1185–1204 (1999).
[CrossRef]

Š. Višňovský, “Optics of magnetic multilayers,” Czech. J. Phys. B 41, 663–694 (1991).
[CrossRef]

Š. Višňovský, R. Krishnan, “Complex Faraday effect in multilayer structures,” J. Opt. Soc. Am. 71, 315–319 (1981).
[CrossRef]

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]

Višnovský, S?.

S̆. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
[CrossRef]

Williams, M. W.

Wolf, E.

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]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975).

Woollam, J. A.

Yamaguchi, T.

Yang, Y. H.

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]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, London, 1984), Chap. 4.

Yeh, P.

Acta Phys. Pol. A (1)

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).

Acta Phys. Slova. (1)

I. Ohlídal, D. Franta, “Matrix formalism for imperfect thin films,” Acta Phys. Slova. 50, 489–500 (2000).

Appl. Opt. (11)

E. Cojocaru, “Simple recurrence matrix relations for multilayer anisotropic thin films,” Appl. Opt. 39, 141–148 (2000).
[CrossRef]

M. W. Williams, “Depolarization and cross polarization in ellipsometry of rough surfaces,” Appl. Opt. 25, 3616–3622 (1986).
[CrossRef] [PubMed]

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]

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]

S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
[CrossRef]

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]

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]

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]

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]

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]

S.-M. F. Nee, “Error analysis of null ellipsometry with depolarization,” Appl. Opt. 38, 5388–5398 (1999).
[CrossRef]

Appl. Phys. Lett. (1)

A. Lien, “Extended Jones matrix representation for the twisted nematic liquid-crystal display at oblique incidence,” Appl. Phys. Lett. 57, 2767–2769 (1990).
[CrossRef]

Czech. J. Phys. B (3)

K. Postava, J. Pištora, Š. Višňovský, “Magneto-optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. B 49, 1185–1204 (1999).
[CrossRef]

S̆. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
[CrossRef]

Š. Višňovský, “Optics of magnetic multilayers,” Czech. J. Phys. B 41, 663–694 (1991).
[CrossRef]

J. Appl. Phys. (2)

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]

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]

J. Magn. Magn. Mater. (2)

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]

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]

J. Opt. A. Pure Appl. Opt. (1)

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]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (5)

J. Vac. Sci. Technol. A (1)

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]

Mater. Sci. Forum (1)

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]

Opt. Commun. (4)

Ch. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

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]

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1982).
[CrossRef]

Opt. Express (1)

Surf. Sci. (1)

P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
[CrossRef]

Thin Solid Films (3)

K. Forcht, A. Gombert, R. Joerger, M. Köhl, “Incoherent superposition in ellipsometric measurements,” Thin Solid Films 302, 43–50 (1997).
[CrossRef]

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]

S.-M. F. Nee, T. Cole, “Effects of depolarization of polarimetric components on null ellipsometry,” Thin Solid Films 313–314, 90–96 (1998).
[CrossRef]

Other (10)

Ch. Brosseau, Fundamentals of Polarized Light—A Statistical Optics Approach (Wiley, New York, 1998).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).
[CrossRef]

A. Gerrard, J. M. Burch, Introduction to Matrix Method in Optics (Wiley, London, 1975).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed. (North-Holland, Amsterdam, 1987).

I. Ohlídal, D. Franta, “Ellipsometry of thin film systems,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 2000), Vol. 41.
[CrossRef]

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]

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.

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]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, London, 1984), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1

Schematic definition of a thin film (l c ≫ 2N z d) and a thick weakly anisotropic layer (l c ≪ 2N z d, l c ≫ 2ΔN z d). d and l c denote the layer thickness and the coherence length of the light source, respectively. The z component of the effective refractive index N z = k z /k 0 is a function of the layer optical constants and the incidence angle.

Fig. 2
Fig. 2

Four eigenmodes in an anisotropic layer.

Fig. 3
Fig. 3

Reflected and transmitted waves by an interface or by a thin-film system.

Fig. 4
Fig. 4

Partial waves reflected and transmitted by a layer.

Fig. 5
Fig. 5

Recurrence for multilayer system consisting of thin and thick layers.

Equations (65)

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

P1,3=exp-ikz 1,3d, P2,4=expikz 2,4d,
E2rE4r=r12r32r14r34E1iE3i=RE1iE3i,
EprEsr=rpprsprpsrssEpiEsi=REpiEsi.
E1tE3t=t11t31t13t33E1iE3i=TE1iE3i,
EptEst=tpptsptpstssEpiEsi=TEpiEsi.
R˜=r˜21r˜41r˜23r˜43, T˜=t˜22t˜42t˜24t˜44,
P13=P100P3, P˜24=P200P4,
T=T12P13I+R˜10P˜24R12P13+R˜10P˜24R12P13R˜10P˜24R12P13+T01 =T12P13I+Q+Q2+Q3+T01,
Q=R˜10P˜24R12P13.
I+Q+Q2+Q3+=I-Q-1
limnQn=0.
T=T12P13I-Q-1T01=T12P13I-R˜10P˜24R12P13-1T01.
R=R01+T˜10P˜24R12P13I-Q-1T01=R01+T˜10P˜24R12P13×I-R˜10P˜24R12P13-1T01.
T˜=T˜10P24I-R12P13R˜10P˜24-1T˜21,
R˜=R˜21+T12P13R˜10P˜24×(I-R12P13R˜10P˜24]-1T˜(21).
detI - R˜10P˜24R12P13=0
rj=rj01+tj01t˜j10rj12 exp-i2kzjd1-r˜j10rj12 exp-i2kzjd=rj01+rj12 exp-i2kzjd1+rj01rj12 exp-i2kzjd,
tj=tj01tj12 exp-ikzjd1-r˜j10rj12 exp-i2kzjd,
tan ψ expiΔ=rprs,
Rs=|rs|2, Rp=|rp|2.
2kzjd  k0lc,
2Δkzd  k0lc,
T=T12P13I-Q-1T01=T12P13I-R˜10P˜24R12P13-1T01,
Q=R˜10P˜24R12P13
R=R01+T˜10P˜24R12P13I-Q-1T01=R01+T˜10P˜24R12P13×I-R˜10P˜24R12P13-1T01.
P13=P˜24=exp2 Imkzpd0000exp-ikzp-kzs*d0000exp-ikzs-kzp*d0000exp2 Imkzsd,
rjrk*=rj01rk01*+tj01tk01*tj10tk10*rj12rk12*exp-i2kzj-kzk*d1-rj10rk10*rj12rk12*exp-i2kzj-kzk*d,
|rj|2=rj012+tj012tj102rj122 exp4Imkzjd1-rj102rj122 exp4Imkzjd,j, k=s, p.
tjtk*=tj01tk01*tj12tk12*exp-ikzj-kzk*d1-rj10rk10*rj12rk12*exp-i2kzj-kzk*d,
|tj|2=tj012tj122 exp2Imkzjd1-rj102rj122 exp4Imkzjd,j, k=s, p,
R11=R1-P cos 2ψ,R22=RP sin 2ψ exp+iΔ,R33=RP sin 2ψ exp-iΔ,R44=R1+P cos 2ψ,
Rs=R44, Rp=R11, R=Rs+Rp/2,
P=R11-R442+4R22R331/2R11+R44,
cos 2ψ=R44-R11PR11+R44,
tan Δ=-iR22-R33R22+R33.
T1=T12P13T01.
R2=T˜10P˜24R12P13T01.
tan ψ expiΔ=rprs=tp01ts01tp10ts10rp12rs12× exp-i2kzp-kzsd.
Nˆ H=-0μ01/2ˆE, H=0μ01/2μˆ-1Nˆ E,
k0Nˆ E=k00-NzNyNz00-Ny00E=k × E,
Nˆμˆ-1Nˆ + ˆE=0.
E=ExEy=T11T12T21T22ExEy=T E,
T=TNTN-1  T2T1.
Jc=E  E+=ExEx*ExEy*EyEx*EyEy*=JxxJxyJyxJyy,
I=tr Jc=Jxx+Jyy,
P=1-4 det Jctr Jc21/2.
J=JxxJxyJyxJyy=TJxxJxyJyxJyy=TJ.
S=AJ,
A=1001100-101100-ii0.
S=MS,
M=ATA-1.
T=T  T*=T11T11*T11T12*T12T11*T12T12*T11T21*T11T22*T12T21*T12T22*T21T11*T21T12*T22T11*T22T12*T21T21*T21T22*T22T21*T22T22*.
AB  CD=A  CB  D,
A  B-1=A-1  B-1.
T=Tnondep + Tdep.
r12=r1201+t1101t˜2210r1212P1P2+t1101t˜4210r1412P1P4+t1301t˜2210r3212P2P3+t1301t˜4210r3412P3P4+t1101t˜2210r˜4310-t1101t˜4210r˜2310-t1301t˜2210r˜4110+t1301t˜4210r˜2110r3212r1412-r1212r3412P1P2P3P4/w,
r14=r1401+t1101t˜2410r1212P1P2+t1101t˜4410t1412P1P4+t1301t˜2410r3212P2P3+t1301t˜4410r3412P3P4+t1101t˜2410r˜4310-t1101t˜4410r˜2310-t1301t˜2410r˜4110+t1301t˜4410r˜2110×r3212r1412-r1212r3412P1P2P3P4/w,
r32=r3201+t3101t˜2210r1212P1P2+t3101t˜4210r1412P1P4+t3301t˜2210r3212P2P3+t3301t˜4210r3412P3P4+t3101t˜2210r˜4310-t3101t˜4210r˜2310-t3301t˜2210r˜4110+t3301t˜4210r˜2110r3212r1412-r1212r3412P1P2P3P4/w,
r34=r3401+t3101t˜2410r1212P1P2+t3101t˜4410r1412P1P4+t3301t˜2410r3212P2P3+t3301t˜4410r3412P3P4+t3101t˜2410r˜4310-t3101t˜4401r˜2310-t3301t˜2410r˜4110+t3301t˜4410r˜2110r3212r1412-r1212r3412P1P2P3P4/w,
t11=t1101t1112P1+t1301t3112P3+t1301r˜2110-t1101r˜2310t1112r3212-t3112r1212P1P2P3+t1301r˜4110-t1101r˜4310t1112r3412-t3112r1412P1P3P4/w,
t13=t1101t1312P1+t1301t3312P3+t1301r˜2110-t1101r˜2310t1312r3212-t3312r1212P1P2P3+t1301r˜4110-t1101r˜4310t1312r3412-t3312r1412P1P3P4/w,
t31=t3101t1112P1+t3301t3112P3+t3301r˜2110-t3101r˜2310t1112r3212-t3112r1212P1P2P3+t3301r˜4110-t3101r˜4310t1112r3412-t3112r1412P1P3P4/w,
t33=t3101t1312P1+t3301t3312P3+t3301r˜2110-t3101r˜2310t1312r3212-t3312r1212P1P2P3+t3301r˜4110-t3101r˜4310t1312r3412-t3312r1412P1P3P4/w,
w=1-r˜2110r1212P1P2-r˜4310r3412P3P4-r˜2310r3212P2P3-r˜4110r1412P1P4+r˜2310r˜4110-r˜4310r˜2110r3212r1412-r1212r3412P1P2P3P4.
θKs+iKs=rsprss, θKp+iKp=- rpsrpp.

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