Abstract

The Yeh’s 4×4 matrix formalism is applied to determine the electromagnetic wave response in multilayers with arbitrary magnetization. With restriction to magneto–optic (MO) effects linear in the off–diagonal permittivity tensor elements, a simplified characteristic matrix for a magnetic layer is obtained. For a magnetic film–substrate system analytical representations of the MO response expressed in terms of the Jones reflection matrix are provided. These are numerically evaluated for cases when the magnetization develops in three mutually perpendicular planes.

© 2001 Optical Society of America

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References

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  1. Z. Q. Qiu and S. D. Bader, “Surface magneto–optic Kerr effect (SMOKE),” J. Magn. Magn. Mat. 200, 664–78 (1999).
    [Crossref]
  2. M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
    [Crossref]
  3. A. Berger and M. R. Pufall, “Generalized magneto–optical ellipsometry,” Appl. Phys. Lett. 71, 965–967 (1997).
    [Crossref]
  4. A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto–optical ellipsometry,” J. Appl. Phys. 85, 4583–4585 (1999).
    [Crossref]
  5. 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]
  6. P. Yeh, “Optics of anisotropic layered media: a new 4×4 matrix algebra,” Surface Sci. 96, 41–53 (1980).
    [Crossref]
  7. Š. Višňovský, “Magneto–optical ellipsometry,” Czech. J. Phys. B 36, 625–650 (1986).
    [Crossref]
  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. K. Rokushima and J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73901–908 (1983).
    [Crossref]
  10. K. Postava, J. Píštora, D. Ciprian, D. Hrabovský, M. Lesňák, and A. R. Fert“Linear and quadratic magneto–optical effects in reflection from a medium withan arbitrary direction of magnetization,” in 11th Slovak-Czech-Polish Optical Conference on Wave and Quantum Aspects of Contemporary Optics, M. Hrabovsky, A. Strba and W. Urbanczyk, eds., Proc. SPIE3820, 412–422 (1999).
  11. I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).
  12. J. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Magneto–optics of multilayers withar bitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
    [Crossref]
  13. H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
    [Crossref]
  14. W. A. McGahan, Liang-Yao Chen, and J. A. Woollam, “Variable angle of incidence analysis of magneto–optic multilayers,” J. Appl. Phys. 67, 4801–4802 (1990).
    [Crossref]
  15. Š. Viš?novský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, and G. Sczigel, “Magnetooptical effects in Au/Co/Au ultrathin film sandwiches,” J. Magn. Magn. Mater. 128, 179–189 (1993).
    [Crossref]
  16. Š. Viš?novský, 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]
  17. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthH olland, Elsevier, Amsterdam-Lausanne-New York-Oxford-Shannon–Tokyo, 1987).
  18. G. E. Jellison, , “Spectroscopic ellipsometry data analysis: measured versus calculated quantities,” Thin Solid Films 313– 314, 33–39 (1998).
    [Crossref]
  19. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusodales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys., Paris 5, 596–640 (1950). M. Born and E.Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), pp. 51–70.
  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).
  22. Š. Višňovský, “Magneto–optical longitudinal and transverse Kerr and birefringence effects in orthorhombic crystals,” Czech. J. Phys. B 34, 969–980 (1984).
    [Crossref]
  23. Š. Višňovský, “Magneto–optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
    [Crossref]
  24. K. Postava, J. Pištora, and Š. Višňovský, “Magneto–optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. 491185–1204 (1999).
    [Crossref]
  25. V. Doormann, J.-P. Krumme, and H. Lenz, “Optical and magneto–optical tensor spectra of bismuth-substituted yttrium-iron-garnet films,” J. Appl. Phys. 68, 3544–3553 (1990).
    [Crossref]
  26. W. Gunsser, U. Wolfmeier, and J. Fleischhauer, “Non–iron garnets,” in Landolt-Börnstein Numerical Data and Functional Relationship in Science and Technology, vol 12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege and A. M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p. 307.

2000 (2)

M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
[Crossref]

H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
[Crossref]

1999 (4)

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto–optical ellipsometry,” J. Appl. Phys. 85, 4583–4585 (1999).
[Crossref]

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]

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

K. Postava, J. Pištora, and Š. Višňovský, “Magneto–optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. 491185–1204 (1999).
[Crossref]

1998 (2)

Š. Višňovský, “Magneto–optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
[Crossref]

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

1997 (1)

A. Berger and M. R. Pufall, “Generalized magneto–optical ellipsometry,” Appl. Phys. Lett. 71, 965–967 (1997).
[Crossref]

1995 (1)

Š. Viš?novský, 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]

1993 (1)

Š. Viš?novský, M. Nývlt, V. Prosser, J. Ferré, G. Pénissard, D. Renard, and 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, and S. D. Bader, “Magneto–optics of multilayers withar bitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[Crossref]

1990 (3)

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

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]

V. Doormann, J.-P. Krumme, and H. Lenz, “Optical and magneto–optical tensor spectra of bismuth-substituted yttrium-iron-garnet films,” J. Appl. Phys. 68, 3544–3553 (1990).
[Crossref]

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).

1984 (1)

Š. Višňovský, “Magneto–optical longitudinal and transverse Kerr and birefringence effects in orthorhombic crystals,” Czech. J. Phys. B 34, 969–980 (1984).
[Crossref]

1983 (1)

1980 (1)

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

1976 (1)

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

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusodales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys., Paris 5, 596–640 (1950). M. Born and E.Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), pp. 51–70.

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthH olland, Elsevier, Amsterdam-Lausanne-New York-Oxford-Shannon–Tokyo, 1987).

Bader, S. D.

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

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

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (NorthH olland, Elsevier, Amsterdam-Lausanne-New York-Oxford-Shannon–Tokyo, 1987).

Bauer, M.

M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
[Crossref]

Berger, A.

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto–optical ellipsometry,” J. Appl. Phys. 85, 4583–4585 (1999).
[Crossref]

A. Berger and M. R. Pufall, “Generalized magneto–optical ellipsometry,” Appl. Phys. Lett. 71, 965–967 (1997).
[Crossref]

Bichri, A.

Chen, Liang-Yao

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

Ciprian, D.

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

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

Ding, H. F.

H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
[Crossref]

Doormann, V.

V. Doormann, J.-P. Krumme, and H. Lenz, “Optical and magneto–optical tensor spectra of bismuth-substituted yttrium-iron-garnet films,” J. Appl. Phys. 68, 3544–3553 (1990).
[Crossref]

Driss-Khodja, K.

Fassbender, J.

M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
[Crossref]

Ferré, J.

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

Fert, A. R.

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

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

Fleischhauer, J.

W. Gunsser, U. Wolfmeier, and J. Fleischhauer, “Non–iron garnets,” in Landolt-Börnstein Numerical Data and Functional Relationship in Science and Technology, vol 12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege and A. M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p. 307.

Frigerio, J. M.

Gunsser, W.

W. Gunsser, U. Wolfmeier, and J. Fleischhauer, “Non–iron garnets,” in Landolt-Börnstein Numerical Data and Functional Relationship in Science and Technology, vol 12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege and A. M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p. 307.

Hillebrands, B.

M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
[Crossref]

Hrabovsky, M.

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

Hrabovský, D.

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

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

Jellison, G. E.

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

Kirschner, J.

H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
[Crossref]

Kopriva, I.

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

Krishnan, R.

Š. Viš?novský, 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]

Krumme, J.-P.

V. Doormann, J.-P. Krumme, and H. Lenz, “Optical and magneto–optical tensor spectra of bismuth-substituted yttrium-iron-garnet films,” J. Appl. Phys. 68, 3544–3553 (1990).
[Crossref]

Lafait, J.

Lenz, H.

V. Doormann, J.-P. Krumme, and H. Lenz, “Optical and magneto–optical tensor spectra of bismuth-substituted yttrium-iron-garnet films,” J. Appl. Phys. 68, 3544–3553 (1990).
[Crossref]

Lesnák, M.

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

Liu, C.

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

Lopusnik, R.

M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
[Crossref]

Lopušník, R.

Š. Viš?novský, 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]

McGahan, W. A.

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

Moog, E. R.

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

Nývlt, M.

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

Oepen, H. P.

H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
[Crossref]

Pénissard, G.

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

Pištora, J.

K. Postava, J. Pištora, and Š. Višňovský, “Magneto–optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. 491185–1204 (1999).
[Crossref]

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

Píštora, J.

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

Postava, K.

K. Postava, J. Pištora, and Š. Višňovský, “Magneto–optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. 491185–1204 (1999).
[Crossref]

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

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

Prosser, V.

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

Pufall, M. R.

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto–optical ellipsometry,” J. Appl. Phys. 85, 4583–4585 (1999).
[Crossref]

A. Berger and M. R. Pufall, “Generalized magneto–optical ellipsometry,” Appl. Phys. Lett. 71, 965–967 (1997).
[Crossref]

Pütter, S.

H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
[Crossref]

Qiu, Z. Q.

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

Renard, D.

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

Rokushima, K.

Schubert, M.

Sczigel, G.

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

Strba, A.

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

Tiwald, T. E.

Urban, R.

Š. Viš?novský, 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]

Urbanczyk, W.

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

Viš?novský, Š.

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

Višnovský, Š.

K. Postava, J. Pištora, and Š. Višňovský, “Magneto–optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. 491185–1204 (1999).
[Crossref]

Š. Višňovský, “Magneto–optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
[Crossref]

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

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

Š. Višňovský, “Magneto–optical longitudinal and transverse Kerr and birefringence effects in orthorhombic crystals,” Czech. J. Phys. B 34, 969–980 (1984).
[Crossref]

Wettling, W.

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

Wolfmeier, U.

W. Gunsser, U. Wolfmeier, and J. Fleischhauer, “Non–iron garnets,” in Landolt-Börnstein Numerical Data and Functional Relationship in Science and Technology, vol 12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege and A. M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p. 307.

Woollam, J. A.

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]

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

Yamaguchi, T.

Yamakita, J.

Yeh, P.

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

Zak, J.

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

Appl. Opt. (2)

Appl. Phys. Lett. (2)

M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, “Suppression of magnetic field pulse induced magnetization precession by pulse tailoring,” Appl. Phys. Lett. 76, 2758–60 (2000).
[Crossref]

A. Berger and M. R. Pufall, “Generalized magneto–optical ellipsometry,” Appl. Phys. Lett. 71, 965–967 (1997).
[Crossref]

Czech. J. Phys. (2)

Š. Višňovský, “Magneto–optic effects in ultrathin structures at longitudinal and polar magnetizations,” Czech. J. Phys. 48, 1083–1104 (1998).
[Crossref]

K. Postava, J. Pištora, and Š. Višňovský, “Magneto–optical effects in ultrathin structures at transversal magnetization,” Czech. J. Phys. 491185–1204 (1999).
[Crossref]

Czech. J. Phys. B (3)

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

Š. Višňovský, “Magneto–optical longitudinal and transverse Kerr and birefringence effects in orthorhombic crystals,” Czech. J. Phys. B 34, 969–980 (1984).
[Crossref]

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

J. Appl. Phys. (3)

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

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto–optical ellipsometry,” J. Appl. Phys. 85, 4583–4585 (1999).
[Crossref]

V. Doormann, J.-P. Krumme, and H. Lenz, “Optical and magneto–optical tensor spectra of bismuth-substituted yttrium-iron-garnet films,” J. Appl. Phys. 68, 3544–3553 (1990).
[Crossref]

J. Magn. Magn. Mat. (3)

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

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

H. F. Ding, S. Pütter, H. P. Oepen, and J. Kirschner, “Experimental method for separating longitudinal and polar Kerr signals,” J. Magn. Magn. Mat. 212, L5–L11 (2000).
[Crossref]

J. Magn. Magn. Mater. (1)

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

J. Opt. Soc. Am. (1)

Phys. Rev. B (2)

Š. Viš?novský, 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. Zak, E. R. Moog, C. Liu, and S. D. Bader, “Magneto–optics of multilayers withar bitrary magnetization directions,” Phys. Rev. B 43, 6423–6429 (1991).
[Crossref]

Surface Sci. (1)

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

Thin Solid Films (1)

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

Other (5)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusodales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys., Paris 5, 596–640 (1950). M. Born and E.Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), pp. 51–70.

W. Gunsser, U. Wolfmeier, and J. Fleischhauer, “Non–iron garnets,” in Landolt-Börnstein Numerical Data and Functional Relationship in Science and Technology, vol 12a (Magnetic and Other Properties of Oxides and Related Compounds) K.-H. Hellwege and A. M. Hellwege, eds. (Springer Verlag, Berlin, Heidelberg, New York, 1978), p. 307.

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

I. Kopřiva, D. Hrabovský, K. Postava, D. Ciprian, J. Pištora, and A. R. Fert, “Anisotropy of the quadratic magneto–optical effects in a cubic crystal,” in Photonics, Devices, and Systems, M. Hrabovsky, P. Tomanek, and M. Miler, eds., Proc. SPIE4016, 54–59 (2000).

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Figures (5)

Fig. 1.
Fig. 1.

The magnetization M displayed as a cartesian vector sum of polar, M P , longitudinal, M L , and transverse M T . In the spherical coordinates M is specified by its magnitude | M | and the angles θM and ϕM .

Fig. 2.
Fig. 2.

The geometry used in the modelling. The magnetization vector evolves on cone shaped surfaces about polar (a), longitudinal (b), and transverse (c) axes.

Fig. 3.
Fig. 3.

Magneto–optical Kerr effect for an interface between vacuum and Bi0.96Lu2.04Fe5O12 magnetic garnet expressed in terms of the real part of the ratio r ps/r ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕM (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕy (b), and normal to the plane of incidence specified by an angle ϕx (c). The initial position of M is given by an angle θM between M and the interface normal at a fixed azimuth ϕM =0 deg (a), ϕM =90 deg (b), and ϕM =0 deg (c). The incident radiation is s-polarized.

Fig. 4.
Fig. 4.

Magneto–optical reflection characteristics at a film/substrate system consisting of a Bi0.96Lu2.04Fe5O12 magnetic garnet film 1.5 µm thick deposited on a Gd3Ga5O12 substrate expressed in terms of the real part of the ratio r ps/r ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕM (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕy (b), and normal to the plane of incidence specified by an angle ϕx (c). The initial position of M is given by an angle θM between M and the interface normal at a fixed azimuth ϕM =0 deg (a), ϕM =90 deg (b), and ϕM =0 deg (c). The incident radiation is s-polarized. Note that the MO effect values are of two orders in magnitude higher than in the case of a single vacuum/BiLuIG interface.

Fig. 5.
Fig. 5.

The effect on ℜ(r ps/r ss) of the angle of incidence, θ 0, ranging from -90 deg to +90 deg at an interface between vacuum and Bi0.96Lu2.04Fe5O12 magnetic garnet (a) and in a film/substrate system consisting of a Bi0.96Lu2.04Fe5O12 magnetic garnet film 1.5 µm thick deposited on a Gd3Ga5O12 substrate (b). The magnetization vector M is restricted to the plane of incidence. Its orientation is specified by an angle θM between M and interface normal. The incident radiation is s-polarized.

Tables (1)

Tables Icon

Table 1. The permittivity tensor elements of the materials used in modelling.

Equations (62)

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ε ( n ) = ( ε 0 ( n ) i ε 1 ( n ) cos θ M ( n ) i ε 1 ( n ) sin θ M ( n ) sin ϕ M ( n ) i ε 1 ( n ) cos θ M ( n ) ε 0 ( n ) i ε 1 ( n ) sin θ M ( n ) cos ϕ M ( n ) i ε 1 ( n ) sin θ M ( n ) sin ϕ M ( n ) i ε 1 ( n ) sin θ M ( n ) cos ϕ M ( n ) ε 0 ( n ) )
γ ( n ) 2 E 0 ( n ) γ ( n ) ( γ ( n ) · E 0 ( n ) ) = ω 2 c 2 ε ( n ) E 0 ( n )
ε 0 ( n ) N z ( n ) 4 ( 2 ε 0 ( n ) N z 0 ( n ) 2 ε 1 ( n ) 2 sin 2 θ M ( n ) ) N z ( n ) 2 2 ε 1 ( n ) 2 sin θ M ( n ) cos θ M ( n ) sin ϕ M ( n ) N y N z ( n )
+ ε 0 ( n ) ( N z 0 ( n ) 4 ε 1 ( n ) 2 ) + N y 2 ε 1 ( n ) 2 ( 1 sin 2 θ M ( n ) sin 2 ϕ M ( n ) ) = 0
N z 1,3 ( n ) = N z 0 ( n ) ( 1 ε 1 ( n ) 2 4 ε 0 ( n ) N z 0 ( n ) 2 ) ± ε 1 ( n ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) ( N z 0 ( n ) cos θ M ( n ) + N y sin θ M ( n ) sin ϕ M ( n ) )
+ ε 1 ( n ) 2 8 ε 0 ( n ) N z 0 ( n ) 3 ( N z 0 ( n ) 2 cos 2 θ M ( n ) N y 2 sin θ M ( n ) 2 sin ϕ M ( n ) 2 )
N z 2,4 ( n ) = N z 0 ( n ) ( 1 ε 1 ( n ) 2 4 ε 0 ( n ) N z 0 ( n ) 2 ) ε 1 ( n ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) ( N z 0 ( n ) cos θ M ( n ) N y sin θ M ( n ) sin ϕ M ( n ) )
ε 1 ( n ) 2 8 ε 0 ( n ) N z 0 ( n ) 3 ( N z 0 ( n ) 2 cos 2 θ M ( n ) N y 2 sin θ M ( n ) 2 sin ϕ M ( n ) 2 )
E 0 ( 0 ) = M E 0 ( 𝓝 + 1 )
M = [ D ( 0 ) ] 1 D ( 1 ) P ( 1 ) [ ( D ) 1 ] 1 D ( 𝓝 ) P ( 𝓝 ) [ D ( 𝓝 ) ] 1 [ D ( 𝓝 + 1 ) ]
D 1 j ( n ) = i ε 1 ( n ) N z 0 ( n ) 2 cos θ M ( n ) i ε 1 ( n ) N y N zj ( n ) sin θ M ( n ) sin ϕ M ( n )
ε 1 ( n ) 2 sin 2 θ M ( n ) cos ϕ M ( n ) sin ϕ M ( n )
D 2 j ( n ) = N zj ( n ) D 1 j ( n )
D 3 j ( n ) = N z 0 ( n ) 2 ( N z 0 ( n ) 2 N zj ( n ) 2 ) ε 1 ( n ) 2 sin 2 θ M ( n ) sin 2 ϕ M ( n )
D 4 j ( n ) = ( ε 0 ( n ) N zj ( n ) i ε 1 ( n ) N y sin θ M ( n ) cos ϕ M ( n ) ) ( N z 0 ( n ) 2 N zj ( n ) 2 )
+ ε 1 ( n ) 2 sin θ M ( n ) sin ϕ M ( n ) ( N zj ( n ) sin θ M ( n ) sin ϕ M ( n ) N y cos θ M ( n ) )
P ( n ) = [ exp ( i ω c N z 1 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 2 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 3 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 4 ( n ) d ( n ) ) ]
D ( n ) = [ 1 1 0 0 N z 0 ( n ) N z 0 ( n ) 0 0 0 0 N z 0 ( n ) ( ε 0 ( n ) ) 1 / 2 N z 0 ( n ) ( ε 0 ( n ) ) 1 / 2 0 0 ( ε 0 ( n ) ) 1 / 2 ( ε 0 ( n ) ) 1 / 2 ]
[ E 0 s ( r ) E 0 p ( r ) ] = [ r ss r sp r ps r pp ] [ E 0 s ( i ) E 0 p ( i ) ]
r ss = [ E 0 s ( r ) E 0 s ( i ) ] E 0 p ( i ) = 0 = M 21 M 33 M 23 M 31 M 11 M 33 M 13 M 31
r ps = [ E 0 p ( r ) E 0 s ( i ) ] E 0 p ( i ) = 0 = M 41 M 33 M 43 M 31 M 11 M 33 M 13 M 31
r sp = [ E 0 s ( r ) E 0 p ( i ) ] E 0 s ( i ) = 0 = M 11 M 23 M 13 M 21 M 11 M 33 M 13 M 31
r pp = [ E 0 p ( r ) E 0 p ( i ) ] E 0 s ( i ) = 0 = M 11 M 43 M 13 M 41 M 11 M 33 M 13 M 31
E 0 ( 0 ) = [ E 01 ( 0 ) E 02 ( 0 ) E 03 ( 0 ) E 04 ( 0 ) ] T = [ E 0 s ( i ) E 0 s ( r ) E 0 p ( i ) E 0 p ( r ) ] T
E 0 ( 𝓝 + 1 ) = [ E 01 ( 𝓝 + 1 ) E 02 ( 𝓝 + 1 ) E 03 ( 𝓝 + 1 ) E 04 ( 𝓝 + 1 ) ] T = [ E 0 s ( t ) 0 E 0 p ( t ) 0 ] T
S ( n ) = D ( n ) P ( n ) ( D ( n ) ) 1
S ( n ) = [ S 11 ( n ) S 12 ( n ) S 13 ( n ) S 14 ( n ) S 21 ( n ) S 11 ( n ) S 23 ( n ) S 24 ( n ) S 24 ( n ) S 14 ( n ) S 33 ( n ) S 34 ( n ) S 23 ( n ) S 13 ( n ) S 43 ( n ) S 44 ( n ) ]
S 11 ( n ) = cos β ( n )
S 12 ( n ) = i N z 0 ( n ) 1 sin β ( n )
S 21 ( n ) = i N z 0 ( n ) sin β ( n )
S 34 ( n ) = i N z 0 ( n ) ε 0 ( n ) 1 sin β ( n )
S 43 ( n ) = i N z 0 ( n ) 1 ε 0 ( n ) sin β ( n )
S 33 ( n ) = cos β ( n ) q ( n ) sin β ( n )
S 44 ( n ) = cos β ( n ) + q ( n ) sin β ( n )
S 13 ( n ) = N z 0 ( n ) 1 ε 0 ( n ) 1 / 2 ( l ( n ) sin β ( n ) + i a n )
S 14 ( n ) = ε 0 ( n ) 1 / 2 ( p ( n ) sin β ( n ) + i b n )
S 23 ( n ) = ε 0 ( n ) 1 / 2 ( p ( n ) sin β ( n ) i b n )
S 24 ( n ) = N z 0 ( n ) ε 0 ( n ) 1 / 2 ( l ( n ) sin β ( n ) i a n )
a n = i 2 ( e i β ( n ) Δ ( n ) + e i β ( n ) Δ ( n ) )
b n = i 2 ( e i β ( n ) Δ ( n ) + + e i β ( n ) Δ ( n ) )
Δ ( n ) ± = ω 2 c d ( n ) ε 1 ( n ) ε 0 ( n ) 1 / 2 N z 0 ( n ) 1 ( N z 0 ( n ) cos θ M ( n ) ± N y sin θ M ( n ) sin ϕ M ( n ) )
β ( n ) = ω c d ( n ) N z 0 ( n )
p ( n ) = ε 1 ( n ) ( N z 0 ( n ) cos θ M ( n ) ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) 2
l ( n ) = ε 1 ( n ) ( N y sin θ M ( n ) sin ϕ M ( n ) ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) 2
q ( n ) = ε 1 ( n ) ( N y sin θ M ( n ) cos ϕ M ( n ) ) ε 0 ( n ) N z 0 ( n )
ω c d ( n ) N z 0 ( n ) 1
M = ( D ( 0 ) ) 1 S ( 1 ) D ( 2 )
r ss = r ss ( 01 ) + r ss ( 12 ) e 2 i β ( 1 ) 1 + r ss ( 01 ) r ss ( 12 ) e 2 i β ( 1 )
r ps , sp = t ss ( 01 ) t pp ( 10 ) { β 1 e 2 i β ( 1 ) [ p ( 1 ) ( r ss ( 12 ) + r pp ( 12 ) ) + l ( 1 ) ( r ss ( 12 ) r pp ( 12 ) ) ]
+ i 2 ( 1 e 2 i β ( 1 ) ) [ ± p ( 1 ) ( 1 + r ss ( 12 ) r pp ( 12 ) e 2 i β ( 1 ) ) l ( 1 ) ( 1 r ss ( 12 ) r pp ( 12 ) e 2 i β ( 1 ) ) ] }
× [ ( 1 + r ss ( 01 ) r ss ( 12 ) e 2 i β ( 1 ) ) ( 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( i ) ) ] 1
r pp = r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) + i 2 q ( 1 ) ( 1 e 2 i β ( 1 ) ) t pp ( 01 ) t pp ( 10 ) 1 r pp ( 12 ) 2 e 2 i β ( 1 ) ( 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) ) 2
r ss ( ij ) = N z 0 ( i ) N z 0 ( j ) N z 0 ( i ) + N z 0 ( j )
r pp ( ij ) = ε 0 ( i ) N z 0 ( j ) ε 0 ( j ) N z 0 ( i ) ε 0 ( i ) N z 0 ( j ) + ε 0 ( j ) N z 0 ( i )
t ss ( ij ) = 1 + r ss ( ij )
t pp ( ij ) = ( ε 0 ( i ) / ε 0 ( j ) ) 1 / 2 ( 1 r pp ( ij ) )
r ps ( 01 , pol ) cos θ M ( 1 ) = i ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) cos θ M ( 1 ) N z 0 ( 1 ) N z 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 1 ) ) ( N ( 0 ) N z 0 ( 1 ) + ε 0 ( 1 ) cos θ ( 0 ) )
= i 2 p ( 1 ) t ss ( 01 ) t pp ( 10 )
r ps ( 01 , lon ) sin θ M ( 1 ) sin ϕ M ( 1 ) = i ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) sin θ M ( 1 ) sin ϕ M ( 1 ) N y N z 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 1 ) ) ( N ( 0 ) N z 0 ( 1 ) + ε 0 ( 1 ) cos θ ( 0 ) )
= i 2 l ( 1 ) t ss ( 01 ) t pp ( 10 )
r ps , sp = 2 ω c d ( 1 ) ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) ( ε 0 ( 1 ) N z 0 ( 2 ) cos θ M ( 1 ) + ε 0 ( 2 ) N y sin θ M ( 1 ) sin ϕ M ( 1 ) ) ε 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 2 ) ) ( N ( 0 ) N z 0 ( 2 ) + ε 0 ( 2 ) cos θ ( 0 ) )
r pp = r pp ( ε 1 ( 1 ) = 0 ) t pp ( 02 ) t pp ( 20 ) ω c d ( 1 ) ε 1 ( 1 ) ε 0 ( 1 ) 1 N y sin θ M ( 1 ) cos ϕ M ( 1 )

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