Abstract

Analytical representations of Jones matrices for the magneto-optic (MO) reflection in (a) a magnetic film on a magnetic substrate separated by a nonmagnetic spacer and (b) two ultrathin magnetic films separated by a nonmagnetic spacer, sandwiched between a nonmagnetic cover and a nonmagnetic substrate with arbitrary and independent orientations of magnetization, are provided. Originally isotropic media subjected to a uniform magnetization are considered. The discussion of MO response is restricted to the terms linear in the off-diagonal permittivity tensor elements, and typical situations are illustrated numerically. The results are useful for the analysis of the experiments in MO ellipsometry and MO magnetometry of exchange-coupled magnetic ultrathin-film structures.

© 2002 Optical Society of America

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  1. Z. Q. Qiu, S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mat. 200, 664–678 (1999).
    [CrossRef]
  2. M. R. Pufall, C. Platt, A. Berger, “Layer resolved magnetometry of magnetic bilayer using the magneto-optical Kerr effect with varying angle of incidence,” J. Appl. Phys. 85, 4818–4820 (1999).
    [CrossRef]
  3. M. Schubert, 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]
  4. W. A. McGahan, “Variable angle of incidence analysis of magneto-optic multilayers,” J. Appl. Phys. 67, 4801–4802 (1990).
    [CrossRef]
  5. Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, R. Krishnan, “Polar magneto-optics in simple ultrathin-magnetic-film structures,” Phys. Rev. B 52, 1090–1106 (1995).
    [CrossRef]
  6. Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
    [CrossRef]
  7. P. Yeh, “Optics of anisotropic layered media: a new 4 × 4 matrix algebra,” Surf. Sci. 96, 41–53 (1980).
    [CrossRef]
  8. Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
    [CrossRef]
  9. J. Ferré, P. Meyer, M. Nývlt, “Magnetooptic depth sensitivity in a simple ultrathin film structure,” J. Magn. Magn. Mater. 165, 92–95 (1997).
    [CrossRef]
  10. G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
    [CrossRef]
  11. K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
    [CrossRef]
  12. K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
    [CrossRef]
  13. R. M. Osgood, S. D. Bader, B. M. Clemens, R. L. White, H. Matsuyama, “Second order magneto-optic effects in anisotropic thin films,” J. Magn. Magn. Mater. 182, 297–323 (1998).
    [CrossRef]
  14. S.-S. Yan, R. Schreiber, P. Grünberg, R. Schäfer, “Magnetization reversal in (001)Fe thin film studied by combining domain images and MOKE hysteresis loops,” J. Magn. Magn. Mater. 210, 309–315 (2000).
    [CrossRef]
  15. B. Heinrich, “Magnetic nanostructures. From physical principles to spintronics,” Can. J. Phys. 78, 161–199 (2000).
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  16. B. Heinrich, J. A. C. Bland, eds., Ultrathin Magnetic Structures (Springer-Verlag, Berlin, 1994).
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    [CrossRef]
  18. D. T. Pierce, J. Unguris, R. J. Celotta, M. D. Stiles, “Effect of roughness, frustration, and antiferromagnetic order on magnetic coupling of Fe/Cr multilayers,” J. Magn. Magn. Mat. 200, 290–321 (1999).
    [CrossRef]
  19. J. Zak, E. R. Moog, C. Liu, S. D. Bader, “Magneto-optics of multilayers with arbitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
    [CrossRef]
  20. W. Wettling, “Magneto-optics of ferrites,” J. Magn. Magn. Mat. 3, 147–160 (1976).
    [CrossRef]
  21. Š. Višňovský, “Magneto-optical permittivity tensor in crystals,” Czech. J. Phys. B 36, 1424–1433 (1986).
    [CrossRef]
  22. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North Holland, Amsterdam, 1987).
  23. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 596–640 (1950).
  24. M. Born, E. Wolf, Principles of Optics (Cambridge University, Cambridge, UK, 1997) pp. 51–70.
  25. J. Lafait, T. Yamaguchi, J. M. Frigerio, A. Bichri, K. Driss-Khodja, “Effective medium equivalent to a symmetric multilayer at oblique incidence,” Appl. Opt. 29, 2460–2465 (1990).
    [CrossRef] [PubMed]
  26. P. B. Johnson, 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]
  27. Š. Višňovský, R. Krishnan, M. Nývlt, V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20, Suppl. S1 41–46 (1996).
  28. D. W. Lynch, W. R. Hunter, Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
    [CrossRef]

2000 (2)

S.-S. Yan, R. Schreiber, P. Grünberg, R. Schäfer, “Magnetization reversal in (001)Fe thin film studied by combining domain images and MOKE hysteresis loops,” J. Magn. Magn. Mater. 210, 309–315 (2000).
[CrossRef]

B. Heinrich, “Magnetic nanostructures. From physical principles to spintronics,” Can. J. Phys. 78, 161–199 (2000).
[CrossRef]

1999 (5)

B. Heinrich, J. F. Cochran, T. Monchesky, R. Urban, “Exchange coupling through spin-density waves in Cr(001) structures: Fe-whisker/Cr/Fe(001) studies,” Phys. Rev. B 59, 14520–14532 (1999).
[CrossRef]

D. T. Pierce, J. Unguris, R. J. Celotta, M. D. Stiles, “Effect of roughness, frustration, and antiferromagnetic order on magnetic coupling of Fe/Cr multilayers,” J. Magn. Magn. Mat. 200, 290–321 (1999).
[CrossRef]

Z. Q. Qiu, S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mat. 200, 664–678 (1999).
[CrossRef]

M. R. Pufall, C. Platt, A. Berger, “Layer resolved magnetometry of magnetic bilayer using the magneto-optical Kerr effect with varying angle of incidence,” J. Appl. Phys. 85, 4818–4820 (1999).
[CrossRef]

M. Schubert, 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]

1998 (2)

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

R. M. Osgood, S. D. Bader, B. M. Clemens, R. L. White, H. Matsuyama, “Second order magneto-optic effects in anisotropic thin films,” J. Magn. Magn. Mater. 182, 297–323 (1998).
[CrossRef]

1997 (2)

J. Ferré, P. Meyer, M. Nývlt, “Magnetooptic depth sensitivity in a simple ultrathin film structure,” J. Magn. Magn. Mater. 165, 92–95 (1997).
[CrossRef]

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

1996 (1)

Š. Višňovský, R. Krishnan, M. Nývlt, V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20, Suppl. S1 41–46 (1996).

1995 (1)

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

1993 (1)

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

1991 (1)

J. Zak, E. R. Moog, C. Liu, S. D. Bader, “Magneto-optics of multilayers with arbitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[CrossRef]

1990 (2)

1986 (2)

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

Š. Višňovský, “Magneto-optical permittivity tensor in crystals,” Czech. J. Phys. B 36, 1424–1433 (1986).
[CrossRef]

1980 (1)

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

1976 (1)

W. Wettling, “Magneto-optics of ferrites,” J. Magn. Magn. Mat. 3, 147–160 (1976).
[CrossRef]

1974 (1)

P. B. Johnson, 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. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. Paris 5, 596–640 (1950).

Abelès, F.

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

Azzam, R. M. A.

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

Bader, S. D.

Z. Q. Qiu, S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mat. 200, 664–678 (1999).
[CrossRef]

R. M. Osgood, S. D. Bader, B. M. Clemens, R. L. White, H. Matsuyama, “Second order magneto-optic effects in anisotropic thin films,” J. Magn. Magn. Mater. 182, 297–323 (1998).
[CrossRef]

J. Zak, E. R. Moog, C. Liu, S. D. Bader, “Magneto-optics of multilayers with arbitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[CrossRef]

Bashara, N. M.

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

Berger, A.

M. R. Pufall, C. Platt, A. Berger, “Layer resolved magnetometry of magnetic bilayer using the magneto-optical Kerr effect with varying angle of incidence,” J. Appl. Phys. 85, 4818–4820 (1999).
[CrossRef]

Bichri, A.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge University, Cambridge, UK, 1997) pp. 51–70.

Celotta, R. J.

D. T. Pierce, J. Unguris, R. J. Celotta, M. D. Stiles, “Effect of roughness, frustration, and antiferromagnetic order on magnetic coupling of Fe/Cr multilayers,” J. Magn. Magn. Mat. 200, 290–321 (1999).
[CrossRef]

Christy, R. W.

P. B. Johnson, 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]

Ciprian, D.

K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
[CrossRef]

Clemens, B. M.

R. M. Osgood, S. D. Bader, B. M. Clemens, R. L. White, H. Matsuyama, “Second order magneto-optic effects in anisotropic thin films,” J. Magn. Magn. Mater. 182, 297–323 (1998).
[CrossRef]

Cochran, J. F.

B. Heinrich, J. F. Cochran, T. Monchesky, R. Urban, “Exchange coupling through spin-density waves in Cr(001) structures: Fe-whisker/Cr/Fe(001) studies,” Phys. Rev. B 59, 14520–14532 (1999).
[CrossRef]

Driss-Khodja, K.

Ferré, J.

J. Ferré, P. Meyer, M. Nývlt, “Magnetooptic depth sensitivity in a simple ultrathin film structure,” J. Magn. Magn. Mater. 165, 92–95 (1997).
[CrossRef]

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

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

Fert, A. R.

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
[CrossRef]

Frigerio, J. M.

Goiran, M.

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

Grünberg, P.

S.-S. Yan, R. Schreiber, P. Grünberg, R. Schäfer, “Magnetization reversal in (001)Fe thin film studied by combining domain images and MOKE hysteresis loops,” J. Magn. Magn. Mater. 210, 309–315 (2000).
[CrossRef]

Heinrich, B.

B. Heinrich, “Magnetic nanostructures. From physical principles to spintronics,” Can. J. Phys. 78, 161–199 (2000).
[CrossRef]

B. Heinrich, J. F. Cochran, T. Monchesky, R. Urban, “Exchange coupling through spin-density waves in Cr(001) structures: Fe-whisker/Cr/Fe(001) studies,” Phys. Rev. B 59, 14520–14532 (1999).
[CrossRef]

Hrabovský, D.

K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
[CrossRef]

Hunter, W. R.

D. W. Lynch, W. R. Hunter, Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
[CrossRef]

Jaffres, H.

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

Jellison, G. E.

G. E. Jellison, “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313–314, 33–39 (1998).
[CrossRef]

Johnson, P. B.

P. B. Johnson, 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]

Krishnan, R.

Š. Višňovský, R. Krishnan, M. Nývlt, V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20, Suppl. S1 41–46 (1996).

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

Lafait, J.

Lesnák, M.

K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
[CrossRef]

Liu, C.

J. Zak, E. R. Moog, C. Liu, S. D. Bader, “Magneto-optics of multilayers with arbitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[CrossRef]

Lopušník, R.

Š. Višňovský, M. Nývlt, V. Prosser, R. Lopušník, R. Urban, J. Ferré, G. Pénissard, D. Renard, 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, W. R. Hunter, Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 275–367.
[CrossRef]

Matsuyama, H.

R. M. Osgood, S. D. Bader, B. M. Clemens, R. L. White, H. Matsuyama, “Second order magneto-optic effects in anisotropic thin films,” J. Magn. Magn. Mater. 182, 297–323 (1998).
[CrossRef]

McGahan, W. A.

W. A. McGahan, “Variable angle of incidence analysis of magneto-optic multilayers,” J. Appl. Phys. 67, 4801–4802 (1990).
[CrossRef]

Meyer, P.

J. Ferré, P. Meyer, M. Nývlt, “Magnetooptic depth sensitivity in a simple ultrathin film structure,” J. Magn. Magn. Mater. 165, 92–95 (1997).
[CrossRef]

Monchesky, T.

B. Heinrich, J. F. Cochran, T. Monchesky, R. Urban, “Exchange coupling through spin-density waves in Cr(001) structures: Fe-whisker/Cr/Fe(001) studies,” Phys. Rev. B 59, 14520–14532 (1999).
[CrossRef]

Moog, E. R.

J. Zak, E. R. Moog, C. Liu, S. D. Bader, “Magneto-optics of multilayers with arbitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[CrossRef]

Nguyen Van Dau, F.

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

Nývlt, M.

J. Ferré, P. Meyer, M. Nývlt, “Magnetooptic depth sensitivity in a simple ultrathin film structure,” J. Magn. Magn. Mater. 165, 92–95 (1997).
[CrossRef]

Š. Višňovský, R. Krishnan, M. Nývlt, V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20, Suppl. S1 41–46 (1996).

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

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

Osgood, R. M.

R. M. Osgood, S. D. Bader, B. M. Clemens, R. L. White, H. Matsuyama, “Second order magneto-optic effects in anisotropic thin films,” J. Magn. Magn. Mater. 182, 297–323 (1998).
[CrossRef]

Pénissard, G.

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

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

Pierce, D. T.

D. T. Pierce, J. Unguris, R. J. Celotta, M. D. Stiles, “Effect of roughness, frustration, and antiferromagnetic order on magnetic coupling of Fe/Cr multilayers,” J. Magn. Magn. Mat. 200, 290–321 (1999).
[CrossRef]

Pištora, J.

K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
[CrossRef]

Platt, C.

M. R. Pufall, C. Platt, A. Berger, “Layer resolved magnetometry of magnetic bilayer using the magneto-optical Kerr effect with varying angle of incidence,” J. Appl. Phys. 85, 4818–4820 (1999).
[CrossRef]

Postava, K.

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

K. Postava, J. Pištora, D. Ciprian, D. Hrabovský, M. Lesňák, A. R. Fert, “Linear and quadratic magneto-optical effects in reflection from a medium with an arbitrary direction of magnetization,” in 11th Slovak–Czech–Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba, W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
[CrossRef]

Prosser, V.

Š. Višňovský, R. Krishnan, M. Nývlt, V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20, Suppl. S1 41–46 (1996).

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

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

Pufall, M. R.

M. R. Pufall, C. Platt, A. Berger, “Layer resolved magnetometry of magnetic bilayer using the magneto-optical Kerr effect with varying angle of incidence,” J. Appl. Phys. 85, 4818–4820 (1999).
[CrossRef]

Qiu, Z. Q.

Z. Q. Qiu, S. D. Bader, “Surface magneto-optic Kerr effect (SMOKE),” J. Magn. Magn. Mat. 200, 664–678 (1999).
[CrossRef]

Renard, D.

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

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

Schäfer, R.

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[CrossRef]

Schreiber, R.

S.-S. Yan, R. Schreiber, P. Grünberg, R. Schäfer, “Magnetization reversal in (001)Fe thin film studied by combining domain images and MOKE hysteresis loops,” J. Magn. Magn. Mater. 210, 309–315 (2000).
[CrossRef]

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Schuhl, A.

K. Postava, H. Jaffres, A. Schuhl, F. Nguyen Van Dau, M. Goiran, A. R. Fert, “Linear and quadratic magneto-optical measurements of the spin reorientation in epitaxial Fe films on MgO,” J. Magn. Magn. Mater. 172, 199–208 (1997).
[CrossRef]

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Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Š. Višňovský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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S.-S. Yan, R. Schreiber, P. Grünberg, R. Schäfer, “Magnetization reversal in (001)Fe thin film studied by combining domain images and MOKE hysteresis loops,” J. Magn. Magn. Mater. 210, 309–315 (2000).
[CrossRef]

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[CrossRef]

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J. Zak, E. R. Moog, C. Liu, S. D. Bader, “Magneto-optics of multilayers with arbitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

B. Heinrich, J. F. Cochran, T. Monchesky, R. Urban, “Exchange coupling through spin-density waves in Cr(001) structures: Fe-whisker/Cr/Fe(001) studies,” Phys. Rev. B 59, 14520–14532 (1999).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Diagram of a magnetic film on a magnetic substrate separated by a nonmagnetic spacer with arbitrary magnetization in magnetic media.

Fig. 2
Fig. 2

Diagram of two magnetic films with arbitrary and independent orientations of magnetization separated by a nonmagnetic spacer, sandwiched between a nonmagnetic cover and a nonmagnetic substrate.

Fig. 3
Fig. 3

Amplitude and phase of the perturbation Δrpp1¯,3¯ (solid curves) to the reflection coefficient r pp originating from the transverse magnetization in the Fe(2 nm)–Cu(2 nm)–Fe system plotted as a function of the angle of incidence θ(0). The contributions originating from the magnetic film (dashed), Δrpp1¯, and the magnetic substrate (dashed-dotted), Δrpp3¯ are also shown.

Fig. 4
Fig. 4

Amplitude and phase of the polar component of the off-diagonal Jones matrix element rps1¯,3¯ (solid curves) in the Fe(2 nm)–Cu(2 nm)–Fe system as a function of the angle of incidence, θ(0). The contributions originating from the magnetic film rps1¯ (pol) (dashed) and magnetic substrate rps3¯ (pol) (dashed-dotted) are also shown.

Fig. 5
Fig. 5

Same as in Fig. 4 for longitudinal component.

Fig. 6
Fig. 6

Amplitude and phase of r ss (solid curves) and r pp (dashed) at zero magnetization in the Fe(2 nm)–Cu(2 nm)–Fe system as a function of the angle of incidence, θ(0).

Fig. 7
Fig. 7

Geometry used for modeling of general magnetization directions. The magnetization vector rotates on the helix from the origin in the longitudinal (y) direction to the final polar (z) direction perpendicular to the interface.

Fig. 8
Fig. 8

Amplitude and phase of the perturbation Δrpp1¯,3¯ to the reflection coefficient r pp originating from the spiral motion of the magnetizations (Fig. 7) in the Fe(2 nm)–Cu(2 nm)–Fe system as a function of the relative polar magnetization component expressed as cos θ M . The magnetizations in the film and in the substrate rotate simultaneously in unison. The incident angle of 60° was chosen. The MO response Δr pp is sensitive to the transverse magnetization component. The 180° jump in the phase corresponds to the Δr pp sign change when the magnetization vector traverses the plane of incidence.

Fig. 9
Fig. 9

Amplitude and phase of the off-diagonal Jones matrix elements rps1¯,3¯ (solid curves) and rsp1¯,3¯ (dashed) originating from the spiral motion of the magnetizations (Fig. 7) in the Fe(2 nm)–Cu(2 nm)–Fe system are shown. The magnetizations in the film and in the substrate rotate simultaneously. The incident angle of 60° was chosen. The MO response originate from the sum of the complex polar and longitudinal contributions.

Fig. 10
Fig. 10

Amplitude and phase of the off-diagonal element of the Jones reflection matrix rps1¯,3¯ (solid curves) in the Fe(2 nm)–Cu[d(Cu)]–Fe system, at the angle of incidence of 65°, as a function of the spacer thickness, d (Cu). The magnetization vectors M(3) and M(1) are fixed at the longitudinal direction [θM(1) = θM(3) = π/2 and ϕM(1) = ϕM(3) = π/2]. The effect of the spacer thickness on the contributions originating from the magnetic film (dashed), rps1¯, and magnetic substrate (dashed-dotted), rps3¯, are also shown.

Fig. 11
Fig. 11

Same as in Fig. 10 for Cr spacer.

Equations (75)

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n=0ni1n cos θMn-i1n sin θMn sin ϕMn-i1n cos θMn0ni1n sin θMn cos ϕMni1n sin θMn sin ϕMn-i1n sin θMn cos ϕMn0n.
γn2E0n-γnγn · E0n=ω2c2 nE0n.
Nz1,3n=Nz0n1-1n240nNz0n2±1n20n1/2Nz0n×Nz0n cos θMn+Ny sin θMn sin ϕMn+1n280nNz0n3Nz0n2cos2 θMn-Ny2 sin θMn2 sin ϕMn2
Nz2,4n=-Nz0n1-1n240nNz0n21n20n1/2Nz0n×Nz0n cos θMn-Ny sin θMn sin ϕMn-1n280nNz0n3Nz0n2cos2 θMn-Ny2 sin θMn2 sin ϕMn2
Nzn2±=Nzn2-1220±120-1Ny2sin2 ϕM+122021/2,
Nzn2±=Nzn2-12sin2 θM20±120-1Nz02cos2 θM+12sin2 θM2021/2.
E00=ME0N+1,
M=D0-1D1P1D1-1  DNPN×DN-1DN+1.
E0srE0pr=rssrsprpsrppE0siE0pi.
rss=E0srE0siE0pi=0=M21M33-M23M31M11M33-M13M31,
rps=E0prE0siE0pi=0=M41M33-M43M31M11M33-M13M31,
rsp=E0srE0piE0si=0=M11M23-M13M21M11M33-M13M31,
rpp=E0prE0piE0si=0=M11M43-M13M41M11M33-M13M31.
E00=E010 E020 E030 E040T=E0si E0sr E0pi E0prT,
E0N+1=E01N+1 E02N+1 E03N+1 E04N+1T=E0st 0 E0pt 0T.
D1jn=-i1nNz0n2 cos θMn-i1nNyNzjn sin θMn sin ϕMn-1n2 sin2 θMn cos ϕMn sin ϕMn,
D2jn=NzjnD1jn,
D3jn=Nz0n2Nz0n2-Nzjn2-1n2 sin2 θMn sin2 ϕMn,
D4jn=-0nNzjn-i1nNy sin θMn cos ϕMn×Nz0n2-Nzjn2,+1n2 sin θMn sin ϕMnNzjn sin θMn sin ϕMn-Ny cos θMn.
Pjjn=expiω/cNzjndn, j=1,  , 4,
Dn=1100Nz0n-Nz0n0000Nz0n0n-1/2Nz0n0n-1/200-0n-1/20n-1/2.
Sn=S11nS12nS13nS14nS21nS11nS23nS24nS24nS14nS33nS34nS23nS13nS43nS44n,
S11n=dn,
S12n=Nz0n-1cn,
S21n=Nz0ncn,
S34n=-Nz0n0n-1cn,
S43n=-Nz0n-10ncn,
S33n=dn+iqncn,
S44n=dn-iqncn,
S13n=iNz0n-10n1/2an-lncn,
S14n=-i0n-1/2bn-pncn,
S23n=i0n1/2bn+pncn,
S24n=-iNz0n0n-1/2an+lncn.
an=¼expiβnexpiΔn+-exp-iΔn+-exp-iβnexpiΔn--exp-iΔn-,
bn=¼expiβnexpiΔn+-exp-iΔn++exp-iβnexpiΔn--exp-iΔn-,
cn=¼expiβnexpiΔn++exp-iΔn+-exp-iβnexpiΔn-+exp-iΔn-,
dn=¼expiβnexpiΔn++exp-iΔn++exp-iβnexpiΔn-+exp-iΔn-,
βn=ω/cdnNz0n,
Δn±=ω/2cdn1n0n-1/2Nz0n-1×Nz0n cos θMn±Ny sin θMn sin ϕMn.
pn=1nNz0n cos θMn20n1/2Nz0n2,
ln=1nNy sin θMn sin ϕMn20n1/2Nz0n2,
qn=1nNy sin θMn cos ϕMn0nNz0n.
Δn±=ωc dnNz0npn±ln=βnpn±ln.
M=D0-1S1S2D3
rpp11=13=0=rpp01+rpp123 exp-2iβ11+rpp01rpp123 exp-2iβ1,
rpp123=rpp12+rpp23 exp-2iβ21+rpp12rpp23 exp-2iβ2
rssij=Nz0i-Nz0jNz0i+Nz0j,
rppij=0iNz0j-0jNz0i0iNz0j+0jNz0i,
rpp=rpp11=13=0+i/2tpp01tpp10q1×1-exp-2iβ11-rpp1232 exp-2iβ1+tpp12tpp21 exp-2iβ1+β2q3tpp23tpp32×1+rpp12rpp23 exp-2iβ2-2×1+rpp01rpp123 exp-2iβ1-2.
rps,sp=tss01tpp10[β1 exp-2iβ1p1rss123+rpp123+l1rss123-rpp123+i21-exp-2iβ1×±p11+rss123rpp123 exp-2iβ1-l1×1-rss123rpp123 exp-2iβ1+i2 tss12tpp21× exp-2iβ1+β2±p3-l3tss23tpp32×1+rss12rss23 exp-2iβ2-11+rpp12rpp23×exp-2iβ2-1]1+rss01rss123 exp-2iβ1×1+rpp01rpp123 exp-2iβ1-1,
tssij=1+rssij,
tppij=0i/0j1/21-rppij.
rps,sp01¯=i2±p1-l1tss01tpp10,
Δrpp01¯=i2 q1tpp01tpp10=i2 q11-rpp012.
rps,sp=4N0 cos θ0β1N0 cos θ0+Nz02N0Nz0202-1/2+021/2 cos θ01+rss23 exp-2iβ2p1011/2Nz0202-1/2×1+rpp23 exp-2iβ2+l1Nz0101-1/2021/21-rpp23 exp-2iβ2+i2±p3-l3tss23tpp32tss02tpp20×exp-2iβ21+rss02rss23 exp-2iβ21+rpp02rpp23 exp-2iβ2-1,
rpp=rpp02+rpp23 exp-2iβ21+rpp02rpp23 exp-2iβ2+tpp02tpp201+rpp02rpp23 exp-2iβ22-β1q11-rpp232 exp-4iβ2+12 iq3tpp23tpp32 exp-2iβ2,
rss=rss02+rss23 exp-2iβ21+rss02rss23 exp-2iβ2
M=D0-1S1S2S3S4D5.
rps,sp=tss01tpp01tss13tpp132N0 cos θ0-1 exp-2iβ11+rss35 exp-2iβ3ωc d21202-1N3Ny sin θM2 sin ϕM2×1-rpp35 exp-2iβ312 cos θM2 cos θ31+rpp35 exp-2iβ3+ωc d41404-1N5Ny sin θM4 sin ϕM414 cos θM4 cos θ5tss35tpp35 exp-2iβ31+rss13rss35 exp-2iβ3+rss01 exp-2iβ1rss13+rss35 exp-2iβ3-11+rpp13rpp35 exp-2iβ3+rpp01 exp-2iβ1×rpp13+rpp35 exp-2iβ3-1,
rpp=rpp011+rpp13rpp35 exp-2iβ3+exp-2iβ1rpp13+rpp35 exp-2iβ31+rpp13rpp35 exp-2iβ3+rpp01 exp-2iβ1rpp13+rpp35 exp-2iβ3-tpp01tpp10tpp13tpp31×exp-2iβ1ωc Nyd21202-1 sin θM2 cos ϕM21-rpp352 exp-4iβ3+d41404-1 sin θM4 cos ϕM4×exp-2iβ3tpp35tpp531+rpp13rpp35 exp-2iβ3+rpp01 exp-4iβ1rpp13+rpp35 exp-2iβ3-2,
rss=rss011+rss13rss35 exp-2iβ3+exp-2iβ1rss13+rss35 exp-2iβ31+rss13rss35 exp-4iβ3+rss01 exp-4iβ1rss13+rss35 exp-4iβ3.
Δrpp1¯,3¯=rpp110, 130-rpp11=13=0,
Δrpp1¯,3¯=Δrpp1¯ sin θM1 cos ϕM1+Δrpp3¯ sin θM3 cos ϕM3.
rps,sp1¯,3¯=rps,sp1¯+rps,sp3¯,
rps1¯=rps1¯polcos θM1+rps1¯lonsin θM1 sin ϕM1,
rps3¯=rps3¯polcos θM3+rps3¯lonsin θM3 sin ϕM3.
E0stE0pt=tsstsptpstppE0siE0pi.
tss=E0stE0siE0pi=0=M33M11M33-M13M31,
tps=E0ptE0siE0pi=0=-M31M11M33-M13M31,
tsp=E0stE0piE0si=0=-M13M11M33-M13M31,
tpp=E0ptE0piE0si=0=M11M11M33-M13M31.
tss=tss01tss12 exp-iβ11+rss01rss12 exp-2iβ1,
tps=-exp-iβ1tss01tpp12β1p11-rpp01rss12×exp-2iβ1+l11+rpp01rss12 exp-2iβ1+i21-exp-2iβ1p1rpp01-rss12-l1×rpp01+rss121+rss01rss12 exp-2iβ1×1+rpp01rpp12 exp-2iβ1-1,
tsp=exp-iβ1tpp01tss12[β1p11-rss01rpp12×exp-2iβ1+l11+rss01rpp12 exp-2iβ1+i21-exp-2iβ1p1rss01-rpp12+l1×rss01+rpp12)](1+rss01rss12 exp-2iβ1×1+rpp01rpp12 exp[-2iβ1)-1,
tpp=tpp01tpp12 exp-iβ11+rpp01rpp12 exp-2iβ1-iq1Nz0101N0Nz02-02 cos θ0N0Nz01+01 cos θ001Nz02+02Nz01×tpp01tpp12 exp-iβ11-exp-2iβ11+rpp01rpp12 exp-2iβ12,

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