Abstract

Magnetooptic (MO) effects in magnetic multilayers with periodically stratified regions are analyzed for the case of normal light wave incidence and polar magnetization (Faraday and polar Kerr effects). From the universal 4×4-matrix formalism simplified analytical representations restricted to terms linear in the off-diagonal permittivity tensor elements are obtained with no loss in accuracy. The MO effects are expressed as weighted sums of contributions from individual layers. Approximate expressions useful for the evaluation of trends in MO effects are given for periodic multilayers consisting of blocks with ultrathin magnetic films. The procedure is illustrated on periodic systems built of symmetric units. Limits on the ultrathin approximation are discussed.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. Heinrich and J. A. C. Bland, eds., Ultrathin Magnetic Structures, (Springer Verlag, Berlin Heidelberg, 1994).
  2. M. Schubert, T. E. Tiwald, and J. A. Woollam, “Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry,” Appl. Opt. 38, 177–187 (1999).
    [Crossref]
  3. P. Yeh, “Optics of anisotropic layered media: a new 4×4 matrix algebra,” Surf. Sci.,  96, 41–53 (1980).
    [Crossref]
  4. Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
    [Crossref]
  5. K. Balasubramian, A. Marathay, and H. A. Macleod, “Modelling magneto-optical thin-film media for optical data storage,” Thin Solid Films 164, 122–128 (1988).
  6. F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusöidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 596–640 (1950).
  7. M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).
  8. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri, and K. Driss-Khodja, “Effective medium equivalent to a symmetric multilayer at oblique incidence,” Appl. Opt. 29, 2460–2465 (1990).
    [Crossref] [PubMed]
  9. Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
    [Crossref]
  10. M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
    [Crossref]
  11. Z. Q. Qiu and S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mater. 200, 664–78 (1999).
    [Crossref]
  12. J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
    [Crossref]
  13. R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolar izedL ight. North Holland, Elsevier, Amsterdam, 1987.
  14. J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
    [Crossref]
  15. G. E. Jellison and F. A. Modine, “Two channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
    [Crossref] [PubMed]
  16. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
    [Crossref]
  17. Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
    [Crossref]
  18. J. H. Weaver, “Optical properties of Rh, Pd, Ir and Pt,” Phys. Rev. B 11, 1416–1425 (1975).
    [Crossref]
  19. D. W. Lynch and W. R. Hunter, “Comments on the optical constants of metals and an introduction to the several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Inc., Orlando, 1985) pp. 275–367.
  20. J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater. 89, 107–123 (1990).
    [Crossref]

1999 (2)

1995 (3)

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

1993 (1)

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

1990 (3)

1988 (1)

K. Balasubramian, A. Marathay, and H. A. Macleod, “Modelling magneto-optical thin-film media for optical data storage,” Thin Solid Films 164, 122–128 (1988).

1986 (1)

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

1980 (1)

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

1977 (1)

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
[Crossref]

1975 (1)

J. H. Weaver, “Optical properties of Rh, Pd, Ir and Pt,” Phys. Rev. B 11, 1416–1425 (1975).
[Crossref]

1974 (1)

P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[Crossref]

1950 (1)

F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusöidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 596–640 (1950).

Abelés, F.

F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusöidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 596–640 (1950).

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolar izedL ight. North Holland, Elsevier, Amsterdam, 1987.

Bader, S. D.

Z. Q. Qiu and S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mater. 200, 664–78 (1999).
[Crossref]

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater. 89, 107–123 (1990).
[Crossref]

Badoz, J.

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
[Crossref]

Balasubramian, K.

K. Balasubramian, A. Marathay, and H. A. Macleod, “Modelling magneto-optical thin-film media for optical data storage,” Thin Solid Films 164, 122–128 (1988).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolar izedL ight. North Holland, Elsevier, Amsterdam, 1987.

Bichri, A.

Billardon, M.

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
[Crossref]

Boher, P.

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).

Canit, J. C.

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
[Crossref]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[Crossref]

Driss-Khodja, K.

Ferré, J.

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Frigerio, J. M.

Houdy, P.

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Hunter, W. R.

D. W. Lynch and W. R. Hunter, “Comments on the optical constants of metals and an introduction to the several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Inc., Orlando, 1985) pp. 275–367.

Jamet, J.-P.

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Jellison, G. E.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[Crossref]

Kielar, P.

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

Krishnan, R.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

Lafait, J.

Liu, C.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater. 89, 107–123 (1990).
[Crossref]

Lopušník, R.

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

Lynch, D. W.

D. W. Lynch and W. R. Hunter, “Comments on the optical constants of metals and an introduction to the several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Inc., Orlando, 1985) pp. 275–367.

Macleod, H. A.

K. Balasubramian, A. Marathay, and H. A. Macleod, “Modelling magneto-optical thin-film media for optical data storage,” Thin Solid Films 164, 122–128 (1988).

Marathay, A.

K. Balasubramian, A. Marathay, and H. A. Macleod, “Modelling magneto-optical thin-film media for optical data storage,” Thin Solid Films 164, 122–128 (1988).

Modine, F. A.

Moog, E. R.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater. 89, 107–123 (1990).
[Crossref]

Nývlt, M.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

Parízek, V.

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

Pénissard, G.

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

Prosser, V.

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

Qiu, Z. Q.

Z. Q. Qiu and S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mater. 200, 664–78 (1999).
[Crossref]

Renard, D.

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

Russel, M. F.

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
[Crossref]

Schubert, M.

Tiwald, T. E.

Urban, R.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Višnovský, Š.

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

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

Weaver, J. H.

J. H. Weaver, “Optical properties of Rh, Pd, Ir and Pt,” Phys. Rev. B 11, 1416–1425 (1975).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).

Woollam, J. A.

Yamaguchi, T.

Yeh, P.

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

Zak, J.

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater. 89, 107–123 (1990).
[Crossref]

Ann. Phys. Paris (1)

F. Abelés, “Recherches sur la propagation des ondes électromagnétiques sinusöidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 596–640 (1950).

Appl. Opt. (3)

Czech. J. Phys. B (1)

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

IEEE Trans. Magn. (1)

Š. Višňovský, M. Nývlt, V. Pařízek, P. Kielar, V. Prosser, and R. Krishnan, “Magneto-Optical Studies of Pt/Co Multilayers and Pt-Co Alloy Thin Films,” IEEE Trans. Magn. 29, 3390–3392 (1993).
[Crossref]

J. Magn. Magn. Mater. (4)

J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Universal approach to magneto-optics,” J. Magn. Magn. Mater. 89, 107–123 (1990).
[Crossref]

M. Nývlt, J. Ferré, J.-P. Jamet, P. Houdy, P. Boher, Š. Višňovský, R. Urban, and R. Lopušník “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches,” J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

Z. Q. Qiu and S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mater. 200, 664–78 (1999).
[Crossref]

J. Ferré, M. Nývlt, G. Pénissard, V. Prosser, D. Renard, and Š. Višňovský, “MO Kerr and Faraday studies of Au/Co ultrathin film sandwiches, J. Magn. Magn. Mater. 148, 281–282 (1995).
[Crossref]

J. Optics (1)

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of a light using a birefringent modulator,” J. Optics,  8, 373–384 (1977).
[Crossref]

Phys. Rev. B (3)

P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9, 5056–5070 (1974).
[Crossref]

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, and R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
[Crossref]

J. H. Weaver, “Optical properties of Rh, Pd, Ir and Pt,” Phys. Rev. B 11, 1416–1425 (1975).
[Crossref]

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 (1)

K. Balasubramian, A. Marathay, and H. A. Macleod, “Modelling magneto-optical thin-film media for optical data storage,” Thin Solid Films 164, 122–128 (1988).

Other (4)

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1959).

R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolar izedL ight. North Holland, Elsevier, Amsterdam, 1987.

D. W. Lynch and W. R. Hunter, “Comments on the optical constants of metals and an introduction to the several metals,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic Press, Inc., Orlando, 1985) pp. 275–367.

B. Heinrich and J. A. C. Bland, eds., Ultrathin Magnetic Structures, (Springer Verlag, Berlin Heidelberg, 1994).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Symmetric A/B/A unit.

Fig. 2.
Fig. 2.

Magnetooptic azimuth rotation (full lines), θK , and ellipticity (dashed), K , in a periodic multilayer consistingo f symmetric A/B/A blocks, Pt(0.4 nm)/Co(0.4 nm)/Pt(0.4 nm), Pt(1.2 nm)/Co(0.4 nm)/Pt(1.2 nm), and Cu(1.2 nm)/Co(0.4 nm)/Cu(1.2 nm), as a function of number of the blocks. The curves 1 correspond to the ultrathin film approximation applied to all layers (Eq. (46)), the curves 2 were obtained with the ultrathin film approximation applied selectively to the magnetic layers using Eq. (45). The curves 3 were computed with Eq. (31).

Fig. 3.
Fig. 3.

The effect of the Co film thickness, d (Co), on the reflection MO azimuth rotation, θK (full lines), and ellipticity, K , (dashed) at the wavelength λ=632.8 nm. The Co film is deposited on Pt or Cu substrate. The curves 1, 2 and, 3 correspond to the approximations limited to first, second (Eq.(48)), and third orders in β (Co)=(2π/λ)N (Co) d (Co). The curves 4 were obtained without restriction on the magnetic layer thickness (Eq.(47)).

Equations (71)

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

ε ( n ) = ( ε xx ( n ) ε xy ( n ) 0 ε xy ( n ) ε xx ( n ) 0 0 0 ε zz ( n ) ) .
[ ( c ω γ z ( n ) ) 2 ε xx ( n ) ε xy ( n ) 0 ε xy ( n ) ( c ω γ z ( n ) ) 2 ε xx ( n ) 0 0 0 ε zz ( n ) ] [ E 0 x ( n ) E 0 y ( n ) E 0 z ( n ) ] = 0 ,
E 0 ( 0 ) = [ E 01 ( 0 ) E 02 ( 0 ) E 03 ( 0 ) E 04 ( 0 ) ] = [ M 11 M 12 0 0 M 21 M 22 0 0 0 0 M 33 M 34 0 0 M 43 M 44 ] [ E 01 ( 𝓝 + 1 ) E 02 ( 𝓝 + 1 ) E 03 ( 𝓝 + 1 ) E 04 ( 𝓝 + 1 ) ] = M E 0 ( 𝓝 + 1 ) .
M = [ D ( 0 ) ] i n = 1 𝓝 S ( n ) D ( 𝓝 + 1 ) ,
S ( n ) = D ( n ) P ( n ) [ D ( n ) ] 1 =
= [ cos β + ( n ) i N + ( n ) 1 sin β + ( n ) 0 0 i N + ( n ) sin β + ( n ) cos β + ( n ) 0 0 0 0 cos β ( n ) i N ( n ) 1 sin β ( n ) 0 0 i N ( n ) sin β ( n ) cos β ( n ) ]
β ± ( n ) = ω c N ± ( n ) d n ,
D ( n ) = [ 1 1 0 0 N + ( n ) N + ( n ) 0 0 0 0 1 1 0 0 N ( n ) N ( n ) ]
P ( n ) = [ exp ( i β + ( n ) ) 0 0 0 0 exp ( i β + ( n ) ) 0 0 0 0 exp ( i β ( n ) ) 0 0 0 0 exp ( i β ( n ) ) ] ,
[ E + ( t ) E ( t ) ] = [ t + 0 0 t ] [ E + ( i ) E ( i ) ] ,
[ E + ( r ) E ( r ) ] = [ r + 0 0 r ] [ E + ( i ) E ( i ) ] ,
[ t ] = [ t xx t xy t xy t xx ] = 1 2 [ ( t + t + ) i ( t t + ) i ( t t + ) ( t + t + ) ] ,
[ r ] = [ r xx r xy r xy r xx ] = 1 2 [ ( r + r + ) i ( r r + ) i ( r r + ) ( r + r + ) ] .
[ E x ( t ) E y ( t ) ] = [ t ] [ E x ( i ) E y ( i ) ] ,
[ E x ( r ) E y ( r ) ] = [ r ] [ E x ( i ) E y ( i ) ]
χ t = t xy t xx = i t + t t + + t = i M 11 M 33 M 11 + M 33 ,
χ r = r xy r xx = i r + r r + + r = i M 43 M 11 M 21 M 33 M 43 M 11 + M 21 M 33 .
t xy ( 0 , 𝓝 + 1 ) = i 2 ( t + ( 0 , 𝓝 + 1 ) t ( 0 , 𝓝 + 1 ) ) ,
r xy ( 0 , 𝓝 + 1 ) = i 2 ( r + ( 0 , 𝓝 + 1 ) r ( 0 , 𝓝 + 1 ) ) .
Δ N ( n ) = 1 2 ( N + ( n ) N ( n ) ) = i ε xy ( n ) 2 N ( n ) ,
t xx ( 0 , 𝓝 + 1 ) = 1 2 ( t + ( 0 , 𝓝 + 1 ) + t ( 0 , 𝓝 + 1 ) ) ,
r xx ( 0 , 𝓝 + 1 ) = 1 2 ( r + ( 0 , 𝓝 + 1 ) + r ( 0 , 𝓝 + 1 ) ) ,
t xy ( 0 , 𝓝 + 1 ) = i Δ t xx ( 0 , 𝓝 + 1 ) = i n = 1 𝓝 t xx ( 0 , 𝓝 + 1 ) N n Δ N n ,
r xy ( 0 , 𝓝 + 1 ) = i Δ r xx ( 0 , 𝓝 + 1 ) = i n = 1 𝓝 r xx ( 0 , 𝓝 + 1 ) N n Δ N n ,
t xx ( 0 , 𝓝 + 1 ) = 1 M 11 , t xy ( 0 , 𝓝 + 1 ) = i Δ ( M 11 ) M 11 2 = i M 11 2 n = 1 𝓝 M 11 N n Δ N n ,
r xx ( 0 , 𝓝 + 1 ) = M 21 M 11 ,
r xy ( 0 , 𝓝 + 1 ) = i Δ ( M 21 M 11 ) = i M 11 2 ( M 21 n = 1 𝓝 + 1 M 11 N n M 11 n = 1 𝓝 + 1 M 21 N n ) Δ N n
χ t ( 0 , 𝓝 + 1 ) = t xy t xx = i Δ ( M 11 ) M 11 = i n = 1 𝓝 + 1 ln ( M 11 ) N n Δ N n
χ r ( 0 , 𝓝 + 1 ) = r xy r xx = i M 11 Δ ( M 21 ) M 21 Δ ( M 11 ) M 11 M 21 = i n = 1 𝓝 + 1 N n [ ln ( M 21 M 11 ) ] Δ N n .
M = C L q W ,
L = [ m 11 + m 12 + 0 0 m 21 + m 22 + 0 0 0 0 m 11 m 12 0 0 m 21 m 22 ] .
L q = [ m 11 + p q + p q 1 + m 12 + p q + 0 0 m 21 + p q + m 11 + p q + p q 1 + 0 0 0 0 m 11 p q p q 1 m 12 p q 0 0 m 21 p q m 11 p q p q 1 ] .
L = S ( A ) S ( B ) S ( A ) ,
m 11 ± = cos β ± ( A ) cos β ± ( B ) 1 2 ( N ± ( A ) N ± ( B ) + N ± ( B ) N ± ( A ) ) sin β ± ( A ) sin β ± ( B ) ,
m 12 ( ± ) = i N ± ( A ) [ sin β ± ( A ) cos β ± ( B ) + 1 2 ( N ± ( A ) N ± ( B ) + N ± ( B ) N ± ( A ) ) cos β ± ( A ) sin β ± ( B )
+ 1 2 ( N ± ( A ) N ± ( B ) N ± ( B ) N ± ( A ) = sin β ± ( B ) ] ,
m 21 ± = i N ± ( A ) [ sin β ± ( A ) cos β ± ( B ) + 1 2 ( N ± ( A ) N ± ( B ) + N ± ( B ) N ± ( A ) ) cos β ± ( A ) sin β ± ( B )
1 2 ( N ± ( A ) N ± ( B ) N ± ( B ) N ± ( A ) ) sin β ± ( B ) ] .
M = C L q W = 1 2 N 0 ( N 0 1 N 0 1 ) ( L 11 ( q ) L 12 ( q ) L 21 ( q ) L 11 ( q ) ) ( 1 1 N 2 N 2 ) .
χ t ( 0 , q L , 2 ) = i { p q [ N 0 ( m 11 + N 2 m 12 ) + ( N 2 m 11 + m 21 ) ] p q 1 ( N 0 + N 2 ) } 1
× { [ ( p q p q 1 ) ( N 0 + N 2 ) + p q N 0 ( m 11 + N 2 m 12 )
+ p q ( N 2 m 11 + m 21 ) ] Δ m 11 + p q ( N 0 N 2 Δ m 12 + Δ m 21 ) } ,
χ r ( 0 , q L , 2 ) = 2 i N 0 { p q [ N 0 ( m 11 + N 2 m 12 ) + ( N 2 m 11 + m 21 ) ] p q 1 ( N 0 + N 2 ) } 1
× { p q [ N 0 ( m 11 + N 2 m 12 ) ( N 2 m 11 + m 21 ) ] p q 1 ( N 0 N 2 ) } 1
× { Δ m 11 ( N 2 2 m 12 m 21 ) ( p q 2 + p q p q 1 p q p q 1 ) Δ m 12 N 2 p q
× [ ( N 2 m 11 + m 21 ) p q N 2 p q 1 ] + Δ m 21 p q [ p q ( m 11 + N 2 m 12 ) p q 1 ] } .
m 11 = U AB [ 1 r AB 2 e 2 i β A + e 2 i β B ( e 2 i β A r AB 2 ) ] ,
m 12 = U AB N A [ ( 1 + r AB e i β A ) 2 e 2 i β B ( r AB + e i β A ) 2 ] ,
m 21 = U AB N A [ ( 1 r AB e i β A ) 2 e 2 i β B ( r AB e i β A ) 2 ] .
r n 1 , n = N n 1 N n N n 1 + N n , and t n 1 , n = 2 N n 1 N n 1 + N n .
Δ m 11 = i Δ N B N B U AB { β B [ 1 r AB 2 e 2 i β A e 2 i β B ( e 2 i β A r AB 2 ) ]
+ i r AB ( 1 e 2 i β A ) ( 1 e 2 i β B ) } ,
Δ m 12 = i Δ N B N B U AB 1 N A { β B [ ( 1 + r AB e i β A ) 2 + e 2 i β B ( r AB + e i β A ) 2 ]
+ i ( 1 + r AB e i β A ) ( r AB + e i β A ) ( 1 e 2 i β B ) } ,
Δ m 21 = i Δ N B N B U AB N A { β B [ ( 1 r AB e i β A ) 2 + e 2 i β B ( r AB e i β A ) 2 ]
+ i ( 1 r AB e i β A ) ( r AB e i β A ) ( 1 e 2 i β B ) } .
m 11 cos β A , m 12 i N A sin β A , m 21 i N A sin β A .
Δ m 11 β B sin β A Δ N B N A ,
Δ m 12 i β B ( 1 cos β A ) Δ N B N A 2 ,
Δ m 21 i β B ( 1 + cos β A ) Δ N B .
χ r ( 0 , q L , A ) 4 N 0 β B Δ N B p q ( N 0 2 N A 2 ) ( p q e i β A p q 1 ) ,
e i q β A = p q e i β A p q 1
p q = e i ( q 1 ) β A ( 1 + e 2 i β A + e 4 i β A + + e 2 i ( q 1 ) β A ) ,
χ r ( 0 , qL , A ) 4 N 0 β B Δ N B e i β A ( 1 e 2 i q β A ) ( N 0 2 N A 2 ) ( 1 e 2 i β A ) .
χ r ( 0 , qL , 2 ) 4 q β B N 0 Δ N B N 0 2 N 2 2 .
χ r ( 0 , 1 , 2 ) = Δ N 1 2 N 1 ( 1 + e 2 i β 1 r 01 r 12 ) 1 ( r 01 + e 2 i β 1 r 12 ) 1
× ( 1 r 01 2 ) [ 4 β 1 r 12 e 2 i β 1 i ( 1 e 2 i β 1 ) ( 1 + r 12 2 e 2 i β 1 ) ] .
χ r ( 0 , 1 , 2 ) 4 ( Δ N 1 ) N 0 N 0 2 N 2 2 [ β 1 + N 2 N 1 ( 1 2 N 0 2 N 1 2 N 0 2 N 2 2 ) β 1 2 ] .
p 0 = 0 , p 6 = 32 m 11 5 32 m 11 3 + 6 m 11 , p 1 = 1 , p 7 = 64 m 11 6 80 m 11 4 + 24 m 11 2 1 , p 2 = 2 m 11 , p 8 = 128 m 11 7 192 m 11 5 + 80 m 11 3 8 m 11 , p 3 = 4 m 11 2 1 , p 9 = 256 m 11 8 448 m 11 6 + 240 m 11 4 40 m 11 2 + 1 , p 4 = 8 m 11 3 4 m 11 , p 10 = 512 m 11 9 1024 m 11 7 + 672 m 11 5 p 5 = 16 m 11 4 12 m 11 2 + 1 , 160 m 11 3 + 10 m 11 ,
[ p 0 ] = 0 , [ p 4 ] = 6 p 3 + 2 p 1 , [ p 8 ] = 14 p 7 + 10 p 5 + 6 p 3 + 2 p 1 , [ p 1 ] = 0 , [ p 5 ] = 8 p 4 + 4 p 2 , [ p 9 ] = 16 p 8 + 12 p 6 + 8 p 4 + 4 p 2 , [ p 2 ] = 2 p 1 = 2 , [ p 6 ] = 10 p 5 + 6 p 3 + 2 p 1 , [ p 10 ] = 18 p 9 + 14 p 7 + 10 p 5 + [ p 3 ] = 4 p 2 , [ p 7 ] = 12 p 6 + 8 p 4 + 4 p 2 , + 6 p 3 + 2 p 1 .
p q + 1 = 2 m 11 p q p q 1 , p q + 1 2 2 m 11 p q p q + 1 + p q 2 = 1 , p q + 2 = 2 ( q + 1 ) p q + 1 + p q , p q 2 p q + 1 p q 1 1 = 0 , p q + 1 = p q [ m 11 p q + 1 p q ] + p q p q + 1 p q + 1 m 11 p q .

Metrics