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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    [CrossRef]

2008

2007

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

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]

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]

2005

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

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]

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]

2000

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Sch¨adler, and U. Kunze "Magneto-optical Kerr effects of ferromagnetic Ni-graings," J. Appl. Phys. 87, 5630-5632 (2000).
[CrossRef]

1999

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

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. 13, 1870-1876 (1996).
[CrossRef]

1981

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, 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]

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]

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, 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, 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]

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]

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. Sch¨adler, 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, 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, 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]

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]

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.

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]

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]

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]

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, 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]

Sch¨adler, K.

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Sch¨adler, 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. Sch¨adler, 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. Sch¨adler, 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. 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. Sch¨adler, 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.

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]

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]

Yoon, C. S.

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. Sch¨adler, and U. Kunze "Magneto-optical Kerr effects of ferromagnetic Ni-graings," J. Appl. Phys. 87, 5630-5632 (2000).
[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.

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]

Czech. J. Phys.

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.

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]

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

T. Shimitte, O. Schemberg, K. Westerholt, H. Zabel, K. Sch¨adler, and U. Kunze "Magneto-optical Kerr effects of ferromagnetic Ni-graings," J. Appl. Phys. 87, 5630-5632 (2000).
[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.

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).
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L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. 13, 1870-1876 (1996).
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Phys. Rev. Lett.

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.

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]

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

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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.

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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).
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[CrossRef]

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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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