Abstract

The rigorous coupled-wave analysis with Airy-like internal-reflection series and Fourier-factorization for the calculation of the diffracted magneto-optical (MO) effects from polar and longitudinally magnetized gyrotropic gratings are fully described. For both gratings the numerical and experimental results are in good agreement, and the enhancement of Kerr rotation in higher orders compared to that of the 0th order diffraction is calculated as a function of grating depth. At last, this numerical method can be applied to many other applications such as extraordinary optical transmission from metallic gratings either through surface plasmon or cavity mode, and MO hysteresis loops.

© 2008 Optical Society of America

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  1. A. Zvezdin and V. Kotov, Modern magneto-optics and magneto-optical materials (IOP, Bristol, 1997).
    [Crossref]
  2. T. McDaniel, “Magneto-optical data storage,” Comm. of the ACM 43, 57–63 (2000).
    [Crossref]
  3. M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
    [Crossref]
  4. J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
    [Crossref]
  5. J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
    [Crossref]
  6. Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, “Magneto-optical enhancement through gyrotropic gratings,” Opt. Express 16, 5378–5384 (2008).
    [Crossref] [PubMed]
  7. M. G. Moharam and T. G. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  8. K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
    [Crossref]
  9. R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
    [Crossref]
  10. R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
    [Crossref]
  11. R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
    [Crossref]
  12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
    [Crossref]
  13. A. B. Khanikaev, A. V. Baryshev, A. A. Fedyanin, A. B. Granovsky, and M. Inoue, “Anomalous Faraday effect of a system with extraordinary optical transmittance,” Opt. Express 15, 6612–6622 (2007).
    [Crossref] [PubMed]
  14. V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).
    [Crossref] [PubMed]
  15. M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, (NIST, Gaithersburg, MD, 1999)
  16. A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
    [Crossref]
  17. H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
    [Crossref]
  18. D. F. Edwards, “Silicon (Si),” in Handbook of optical constants of solids, E. D. Palik, eds. (Academic, New York, 1998); H. R. Philipp, “Silicon dioxide (SiO2) (glass),” ibid.
  19. P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999).
    [Crossref]
  20. G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
    [Crossref]
  21. T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
    [Crossref]

2008 (1)

2007 (4)

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

A. B. Khanikaev, A. V. Baryshev, A. A. Fedyanin, A. B. Granovsky, and M. Inoue, “Anomalous Faraday effect of a system with extraordinary optical transmittance,” Opt. Express 15, 6612–6622 (2007).
[Crossref] [PubMed]

V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).
[Crossref] [PubMed]

A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
[Crossref]

2006 (4)

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

2005 (1)

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

2003 (2)

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

2000 (2)

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

T. McDaniel, “Magneto-optical data storage,” Comm. of the ACM 43, 57–63 (2000).
[Crossref]

1999 (2)

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
[Crossref]

P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999).
[Crossref]

1996 (1)

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
[Crossref]

1981 (1)

Abe, M.

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
[Crossref]

Antos, R.

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

Arai, K.

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
[Crossref]

Backstrom, J.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Baryshev, A. V.

Belotelov, V. I.

V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).
[Crossref] [PubMed]

Cho, M. H.

Demokrritov, S. O.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

Diserens, M.

P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999).
[Crossref]

Donahue, M. J.

M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, (NIST, Gaithersburg, MD, 1999)

Doskolovich, L. L.

V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).
[Crossref] [PubMed]

Edwards, D. F.

D. F. Edwards, “Silicon (Si),” in Handbook of optical constants of solids, E. D. Palik, eds. (Academic, New York, 1998); H. R. Philipp, “Silicon dioxide (SiO2) (glass),” ibid.

Egawa, M.

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

Fedyanin, A. A.

Fujii, T.

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
[Crossref]

Gaylord, T. G.

Granovsky, A. B.

Hillerbrands, B.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

Hones, P.

P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999).
[Crossref]

Horie, M.

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

Inoue, M.

A. B. Khanikaev, A. V. Baryshev, A. A. Fedyanin, A. B. Granovsky, and M. Inoue, “Anomalous Faraday effect of a system with extraordinary optical transmittance,” Opt. Express 15, 6612–6622 (2007).
[Crossref] [PubMed]

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
[Crossref]

Kato, H.

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

Khanikaev, A. B.

Kim, J. B.

Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, “Magneto-optical enhancement through gyrotropic gratings,” Opt. Express 16, 5378–5384 (2008).
[Crossref] [PubMed]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

Kim, K. W.

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

Kimura, T.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

Korn, T.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Kotov, V.

A. Zvezdin and V. Kotov, Modern magneto-optics and magneto-optical materials (IOP, Bristol, 1997).
[Crossref]

Kunze, J.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Kunze, U.

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

Lee, G. J.

Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, “Magneto-optical enhancement through gyrotropic gratings,” Opt. Express 16, 5378–5384 (2008).
[Crossref] [PubMed]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

Lee, M. S.

A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
[Crossref]

Lee, Y. P.

Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, “Magneto-optical enhancement through gyrotropic gratings,” Opt. Express 16, 5378–5384 (2008).
[Crossref] [PubMed]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

Levy, F.

P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999).
[Crossref]

Li, L.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
[Crossref]

Liskova, E.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

Lu, Y. H.

Matsushita, T.

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

McDaniel, T.

T. McDaniel, “Magneto-optical data storage,” Comm. of the ACM 43, 57–63 (2000).
[Crossref]

Meier, G.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Merkt, U.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Mistrik, J.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

Moharam, M. G.

Neuber, G.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Nishimura, K.

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

Otani, Y.

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

Pels, C.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Pistora, J.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

Porter, D. G.

M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, (NIST, Gaithersburg, MD, 1999)

Postora, J.

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

Rauer, P.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Remhof, A.

A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
[Crossref]

Rhee, J. Y.

Y. H. Lu, M. H. Cho, J. B. Kim, G. J. Lee, Y. P. Lee, and J. Y. Rhee, “Magneto-optical enhancement through gyrotropic gratings,” Opt. Express 16, 5378–5384 (2008).
[Crossref] [PubMed]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

Rokushima, K.

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

Rubhausen, M.

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Schädler, K.

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

Schemberg, O.

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

Shimitte, T.

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

Takayama, A.

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

Veis, M.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

Visnovsky, S.

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

Westerholt, K.

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

Westphalen, A.

A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
[Crossref]

Yamaguchi, S.

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

Yamaguchi, T.

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

Yoon, C. S.

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

Zabel, H.

A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
[Crossref]

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

Zvezdin, A.

A. Zvezdin and V. Kotov, Modern magneto-optics and magneto-optical materials (IOP, Bristol, 1997).
[Crossref]

Zvezdin, A. K.

V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).
[Crossref] [PubMed]

Appl. Phys. Lett. (3)

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, K. W. Kim, and C. S. Yoon, “One-dimensional magnetic grating structure made easy,” Appl. Phys. Lett. 89, 151111 (2006).
[Crossref]

R. Antos, J. Mistrik, T. Yamaguchi, S. Visnovsky, S. O. Demokrritov, and B. Hillerbrands, “Evidence of native oxides on the capping and substrate of Permalloy gratings by magneto-optical spectroscopy in the zeroth- and first diffraction orders,” Appl. Phys. Lett. 86, 231101 (2005).
[Crossref]

G. Neuber, P. Rauer, J. Kunze, T. Korn, C. Pels, G. Meier, U. Merkt, J. Backstrom, and M. Rubhausen, “Temperature-dependent spectral generalized magneto-optical ellipsometry,” Appl. Phys. Lett. 83, 4509–4511 (2003).
[Crossref]

Comm. of the ACM (1)

T. McDaniel, “Magneto-optical data storage,” Comm. of the ACM 43, 57–63 (2000).
[Crossref]

Czech. J. Phys. (1)

K. Rokushima, R. Antos, J. Mistrik, S. Visnovsky, and T. Yamaguchi, “Optics of anisotropic nanostructures,” Czech. J. Phys. 56, 665–764 (2006).
[Crossref]

J. Appl. Phys. (5)

R. Antos, J. Postora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006).
[Crossref]

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Schädler, and U. Kunze “Magneto-optical Kerr effects of ferromagnetic Ni-graings,” J. Appl. Phys. 87, 5630–5632 (2000).
[Crossref]

M. Inoue, K. Arai, T. Fujii, and M. Abe, “One-dimensional magnetophotonic crystals,” J. Appl. Phys. 85, 5768–5770 (1999).
[Crossref]

J. B. Kim, G. J. Lee, Y. P. Lee, J. Y. Rhee, and C. S. Yoon, “Enhancement of magneto-optical properties of magnetic grating,” J. Appl. Phys. 101, 09C518 (2007).
[Crossref]

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue“Theoretical analysis of optical and magneto-optical properties of one-dimensional magnetophotonic crystals,” J. Appl. Phys. 93, 3906–3911 (2003).
[Crossref]

J. Magn. Soc. Jpn. (1)

R. Antos, J. Mistrik, T. Yamaguchi, M. Veis, E. Liskova, S. Visnovsky, J. Pistora, B. Hillerbrands, S. O. Demokrritov, T. Kimura, and Y. Otani “Magneto-optical spectroscopic scatterometry for analyzing patterned magnetic nanostructures,” J. Magn. Soc. Jpn. 30, 630–636 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 13, 1870–1876 (1996).
[Crossref]

M. G. Moharam and T. G. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]

Opt. Express (2)

Phys. Rev. Lett. (1)

V. I. Belotelov, L. L. Doskolovich, and A. K. Zvezdin, “Extraordinary magneto-optical effects and transmission through metal-dielectric plasmonic systems,” Phys. Rev. Lett. 98, 077401 (2007).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

A. Westphalen, M. S. Lee, A. Remhof, and H. Zabel, “Vector and Bragg magneto-optical Kerr effect for the analysis if nanostructured magnetic arrays,” Rev. Sci. Instrum. 78, 121301 (2007).
[Crossref]

Surf. Coat. Technol. (1)

P. Hones, M. Diserens, and F. Levy, “Characterization of sputter-deposited chromium oxide thin films,” Surf. Coat. Technol. 120, 277–283 (1999).
[Crossref]

Other (3)

D. F. Edwards, “Silicon (Si),” in Handbook of optical constants of solids, E. D. Palik, eds. (Academic, New York, 1998); H. R. Philipp, “Silicon dioxide (SiO2) (glass),” ibid.

A. Zvezdin and V. Kotov, Modern magneto-optics and magneto-optical materials (IOP, Bristol, 1997).
[Crossref]

M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376, (NIST, Gaithersburg, MD, 1999)

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Figures (8)

Fig. 1.
Fig. 1.

Gyrotropic grating with period d and grating depth h, and schematics for Airy like internal reflection series expansions in the grating.

Fig. 2.
Fig. 2.

Theoretical and experimental Kerr rotation spectra in (a) the 0th and (b) the -1st order diffractions for the s-polarized incidence. Incidence angle was 7° for the 0th order. For the -1st order diffraction, the angle between incident and reflected beams was fixed to be 20°. Nmax =10.

Fig. 3.
Fig. 3.

Relative errors as a function of grating depth.

Fig. 4.
Fig. 4.

Simulated Kerr rotation as a function of grating depth. Incident angle was set to be 7° for (a) the -2nd, (b) the -1st, (c) the 0th, (d) the 1st, (e) the 2nd. and (f) the 3rd orders with a truncation order Nmax =50.

Fig. 5.
Fig. 5.

Logarithmic magnification of (a) the -2nd order A (-2), (b) the -1st order A (-1), (c) the 1st order A (1), and (d) the 2nd order A (2) as a function of grating depth.

Fig. 6.
Fig. 6.

Absolute errors as a function of grating depth.

Fig. 7.
Fig. 7.

Kerr rotations of the -3rd, the 0th, the 1st, and the 9th orders from the numerical calculation (black triangular dots) and the experiments (red square dots).

Fig. 8.
Fig. 8.

(a) The 0th, (b) the ± 1st, (c) the ± 2nd, and (d) the ± 3rd order diffracted Kerr rotations and (e) MO enhancement A (n) as a function of grating depth h for Ni grating.

Equations (109)

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× E = i H ˜ ,
× H ˜ = i ε ( y ̅ ) E ,
ε ( y ̅ ) = ( ε xx ε xy 0 ε yx ε yy 0 0 0 ε zz ) for 0 y ̅ < w ,
ε ( y ̅ ) = I 3 × 3 for w < y ̅ d
ε ( y ̅ ) = ( ε xx 0 ε xz 0 ε yy 0 ε zx 0 ε zz ) for 0 y ̅ < w ,
ε ( y ̅ ) = I 3 × 3 for w < y ̅ d
n k = λ d y ̂ = q y ̂ ,
E k ( y ̅ , z ̅ ) = n = + e k , n ( z ̅ ) exp ( i q n y ̅ ) , k = x , y , z ,
H ˜ k ( y ̅ , z ̅ ) = n = + h k , n ( z ̅ ) exp ( i q n y ̅ ) , k = x , y , z
q n = n 0 sin θ i + n q = q 0 + n q .
ε α β ( y ̅ ) = m = + ε α β , m exp ( i m q y ̅ ) ,
y ̅ E z ( y ̅ , z ̅ ) z ̅ E y ( y ̅ , z ̅ ) = i H ˜ x ( y ̅ , z ̅ ) ,
z ̅ E x ( y ̅ , z ̅ ) x ̅ E z ( y ̅ , z ̅ ) = i H ˜ y ( y ̅ , z ̅ ) ,
x ̅ E y ( y ̅ , z ̅ ) y ̅ E x ( y ̅ , z ̅ ) = i H ˜ z ( y ̅ , z ̅ ) ,
y ̅ H ˜ z ( y ̅ , z ̅ ) z ̅ H ˜ y ( y ̅ , z ̅ ) = i ε xx E x ( y ̅ , z ̅ ) + i ε xy E y ( y ̅ , z ̅ ) ,
z ̅ H ˜ x ( y ̅ , z ̅ ) x ̅ H ˜ z ( y ̅ , z ̅ ) = i ε yx E x ( y ̅ , z ̅ ) + i ε yy E y ( y ̅ , z ̅ ) ,
x ̅ H ˜ y ( y ̅ , z ̅ ) y ̅ H ˜ x ( y ̅ , z ̅ ) = i ε zz E z ( y ̅ , z ̅ ) .
y ̅ n = + e z , n exp ( i q n y ̅ ) z ̅ n = + e y , n exp ( i q n y ̅ ) = i n = + h x , n exp ( i q n y ̅ ) ,
i n = + q n e z , n exp ( i q n y ̅ ) n = + d d z ̅ e y , n exp ( i q n y ̅ ) = i n = + h x , n exp ( i q n y ̅ ) .
d d z ̅ e y , n = i ( q n e z , n + h x , n ) .
d d z ̅ e x , n = i h y , n ,
q n e x , n = h z , n .
y ̅ n = + h z , n exp ( i q n y ̅ ) z ̅ n = + h y , n exp ( i q n y ̅ )
= i m = + ε xx , m exp ( i m q y ̅ ) n = + e x , n exp ( i q n y ̅ )
+ i m = + ε xy , m exp ( i m q y ̅ ) n = + e y , n exp ( i q n y ̅ ) ,
n = + i q n h z , n exp ( i q n y ̅ ) n = + d d z ̅ h y , n exp ( i q n y ̅ )
= i m = + n = + ε xx , m e x , n exp ( i ( m n ) q y ̅ ) exp ( i q 0 y ̅ )
+ i m = + n = + ε xy , m e y , n exp ( i ( m n ) q y ̅ ) exp ( i q 0 y ̅ )
( i q n ' ) h z , n ' d d z ̅ h y , n ' = i n = + ε xx , n n ' e x , n + i n = + ε xy , n n ' e y , n .
d d z ̅ h y , n = i q n h z , n i l = + ε xx , l n e x , l i l = + ε xy , l n e y , l .
d d z ̅ h x , n = i l = + ε yx , l n e x , l + i l = + ε yy , l n e y , l ,
q n h x , n = l = + ε zz , l n e z , l .
d d z ̅ ( e y ) = i [ q ] ( e z ) + i ( h x ) ,
d d z ̅ ( e x ) = i ( h y ) ,
[ q ] ( e x ) = ( h z ) ,
d d z ̅ ( h y ) = i [ q ] ( h z ) i [ ε xx ] ( e x ) i [ ε xy ] ( e y ) ,
d d z ̅ ( h x ) = i [ ε yx ] ( e x ) + i [ ε yy ] ( e y ) ,
[ q ] ( h x ) = [ ε zz ] ( e z ) ,
( e j ) = ( , e j , 1 , e j , 0 , e j , 1 , , ) T , j = x , y , z ,
( h j ) = ( , h j , 1 , h j , 0 , h j , 1 , , ) T , j = x , y , z .
[ q ] = ( 0 0 0 0 0 q 1 0 0 0 0 0 q 0 0 0 0 0 0 q 1 0 0 0 0 0 )
[ ε α β ] = ( ε α β , 0 ε α β , 1 ε α β , 2 ε α β , 1 ε α β , 0 ε α β , 1 ε α β , 2 ε α β , 1 ε α β , 0 ) .
( h z ) = [ q ] ( e x ) ,
( e z ) = [ ε zz ] 1 [ q ] ( h x ) .
d d z ̅ ( e x e y h x h y ) = ( 0 0 0 i 0 0 i ( 1 [ q ] [ ε zz ] 1 [ q ] ) 0 i [ ε yx ] i [ ε yy ] 0 0 i ( [ q ] 2 [ ε xx ] ) i [ ε xy ] 0 0 ) ( e x e y h x h y )
( e z h z ) = ( 0 0 [ ε zz ] 1 [ q ] 0 [ q ] 0 0 0 ) ( e x e y h x h y ) .
d d z ̅ ( e x e y ) = i ( 0 1 ( 1 [ q ] [ ε zz ] 1 [ q ] ) 0 ) ( h x h y ) ,
d d z ̅ ( h x h y ) = i ( [ ε yx ] [ ε yy ] ( [ q ] 2 [ ε xx ] ) [ ε xy ] ) ( e x e y ) .
d 2 d z ̅ 2 f = A f ,
d d z ̅ g = i B f ,
A = ( [ ε xx ] [ q ] 2 [ ε xy ] ( 1 [ q ] [ ε zz ] 1 [ q ] ) [ ε yx ] ( 1 [ q ] [ ε zz ] 1 [ q ] ) [ ε yy ] ) ,
B = ( [ ε yx ] [ ε yy ] ( [ q ] 2 [ ε xx ] ) [ ε xy ] ) .
A = T Λ T 1 ,
d 2 d z ̅ 2 f ˜ = Λ f ˜ ,
f ˜ ( z ̅ ) = exp ( i Λ ( z ̅ z ̅ 0 ) ) f ˜ ( z ̅ 0 ) .
f ( z ̅ ) = T exp ( i Λ ( z ̅ z ̅ 0 ) ) T 1 f ( z ̅ 0 ) = P f ( z ̅ 0 ) ,
P = T exp ( i Λ ( z ̅ z ̅ 0 ) ) T 1 .
g = D f = B T S 1 T 1 f ,
f i + f r = f t ,
D 0 ( f i f r ) = D 1 f t ,
f r = R 01 f i ,
f t = T 01 f i ,
R 01 = [ 1 + ( D 1 ) 1 D 0 ] 1 [ 1 ( D 1 ) 1 D 0 ] ,
T 01 = 1 + R 01 .
f r = f 0 r + f 1 r + f 2 r + ,
f t = f 1 t + f 2 t + f 3 t + ,
f 0 r = R 01 f i ,
f 1 r = T 10 P 1 R 12 P 1 T 01 f i ,
f 2 r = T 10 P 1 R 12 ( P 1 R 10 P 1 R 12 ) P 1 T 01 f i ,
f n r = T 10 P 1 R 12 ( P 1 R 10 P 1 R 12 ) ( n 1 ) P 1 T 01 f i ,
f 1 t = T 12 P 1 T 01 f i ,
f 2 t = T 12 ( P 1 R 10 P 1 R 12 ) P 1 T 01 f i ,
f n t = T 12 ( P 1 R 10 P 1 R 12 ) ( n 1 ) P 1 T 01 f i .
f r = R p f i = ( R 01 + T 10 P 1 R 12 [ k = 0 + Q k ] P 1 T 01 ) f i
= ( R 01 + T 10 P 1 R 12 ( 1 Q ) 1 P 1 T 01 ) f i ,
f t = T p f i = ( T 12 [ k = 0 + q k ] P 1 T 01 ) f i
= ( T 12 ( 1 Q ) 1 P 1 T 01 ) f i ,
k = 0 + Q k = ( 1 Q ) 1 .
f r = R p f i ,
f t = T p f i ,
R p = R 01 + T 10 P 1 R 12 ( 1 Q ) 1 P 1 T 01 ,
T p = T 12 ( 1 Q ) 1 P 1 T 01 .
y ̅ E z ( y ̅ , z ̅ ) z ̅ E y ( y ̅ , z ̅ ) = i H ˜ x ( y ̅ , z ̅ ) ,
z ̅ E x ( y ̅ , z ̅ ) x ̅ E z ( y ̅ , z ̅ ) = i H ˜ y ( y ̅ , z ̅ ) ,
x ̅ E y ( y ̅ , z ̅ ) y ̅ E x ( y ̅ , x ̅ ) = i H ˜ z ( y ̅ , z ̅ ) ,
y ̅ H ˜ z ( y ̅ , z ̅ ) z ̅ H ̅ y ( y ̅ , z ̅ ) = i ε xx E x ( y ̅ , z ̅ ) + i ε xz E z ( y ̅ , z ̅ ) ,
z ̅ H ˜ x ( y ̅ , z ̅ ) x ̅ H ˜ z ( y ̅ , z ̅ ) = i ε yy E y ( y ̅ , z ̅ ) ,
x ̅ H ˜ y ( y ̅ , z ̅ ) y ̅ H ˜ x ( y ̅ , z ̅ ) = i ε zx E x ( y ̅ , z ̅ ) + i ε zz E z ( y ̅ , z ̅ ) .
d d z ̅ ( e y ) = i [ q ] ( e z ) + i ( h x ) ,
d d z ̅ ( e x ) = i ( h y ) ,
[ q ] ( e x ) = ( h z ) ,
d d z ̅ ( h y ) = i [ q ] ( h z ) i [ ε xx ] ( e x ) i [ ε xz ] ( e z ) ,
d d z ̅ ( h x ) = i [ ε yy ] ( e y ) ,
[ q ] ( h x ) = [ ε zx ] ( e x ) + [ ε zz ] ( e zz ) .
( h z ) = [ q ] ( e x ) ,
( e z ) = [ ε zz ] 1 [ q ] ( h x ) [ ε zz ] 1 [ ε zx ] ( e x ) .
d d z ̅ ( e x e y h x h y ) = ( 0 0 0 i i [ q ] [ ε zz ] 1 [ ε zx ] 0 i i [ q ] [ ε zz ] 1 [ q ] 0 0 i [ ε yy ] 0 0 i [ q ] 2 i [ ε xx ] + i [ ε xz ] [ ε zz ] 1 [ ε zx ] 0 i [ ε xz ] [ ε zz ] 1 [ q ] 0 ) ( e x e y h x h y )
( e z h z ) = ( [ ε zz ] 1 [ ε zx ] 0 [ ε zz ] 1 [ q ] 0 [ q ] 0 0 0 ) ( e x e y h x h y ) .
d d z ̅ ( e x h x ) = ( 0 i i [ 1 ε yy ] 1 0 ) ( e y h y ) ,
d d z ̅ ( e y h y ) = i ( [ q ] [ ε zz ] 1 [ ε zx ] 1 [ q ] [ ε zz ] 1 [ q ] [ q ] 2 [ ε xx ] + [ ε xz ] [ ε zz ] 1 [ ε zx ] [ ε xz ] [ ε zz ] 1 [ q ] ) ( e x h x )
d 2 d z ̅ 2 f = Af ,
d d z ̅ g = iBf ,
A = ( [ ε xx ] [ q ] 2 [ ε xz ] [ ε zz ] 1 [ ε zx ] [ ε xz ] [ ε zz ] 1 [ q ] [ 1 ε yy ] 1 [ q ] [ ε zz ] 1 [ ε zx ] [ 1 ε yy ] 1 [ 1 ε yy ] 1 [ q ] [ ε zz ] 1 [ q ] ) ,
B = ( [ q ] [ ε zz ] 1 [ ε zx ] 1 [ q ] [ ε zz ] 1 [ q ] [ q ] 2 [ ε zz ] + [ ε xz ] [ ε zz ] 1 [ ε zx ] [ ε xz ] [ ε zz ] 1 [ q ] ) .
f r = R l f i = ( R 01 + T 10 P 1 R 12 ( 1 Q ) 1 P 1 P 01 ) f i ,
f t = T l f i = ( T 12 ( 1 Q ) 1 P 1 T 01 ) f i .
θ K ( n ) = 1 2 tan 1 ( 2 Re ( χ ( n ) ) 1 χ ( n ) 2 ) ,
A ( n ) = log ( θ K ( n ) θ K ( 0 ) ) ,
Relative error : = θ K ( 0 ) , N max i + 1 θ K ( 0 ) , N max i θ K ( 0 ) , N max i

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