Abstract

An analytical formulation for the band structure and Bloch modes in elliptically birefringent magnetophotonic crystals is presented. The model incorporates both the effects of gyrotropy and linear birefringence generally present in magneto-optic thin-film devices. Full analytical expressions are obtained for the dispersion relation and Bloch modes in a layered-stack photonic crystal, and their properties are analyzed. It is shown that other models recently discussed in the literature are contained as special limiting cases of the formulation presented herein.

© 2007 Optical Society of America

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  1. A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609 (2001).
    [CrossRef]
  2. A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
    [CrossRef]
  3. A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
    [CrossRef]
  4. A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
    [CrossRef]
  5. M. Inoue, K. Arai, T. Fuji, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (1999).
    [CrossRef]
  6. M. Levy, H. Yang, M. Steel, and J. Fujita, "Flat-top response in one-dimensional magnetic photonic bandgap structures with Faraday rotation enhancement," J. Lightwave Technol. 19, 1964-1969 (2001).
    [CrossRef]
  7. M. Steel, M. Levy, and R. M. Osgood, "Photonic bandgaps with defects and the enhancement of Faraday rotation," J. Lightwave Technol. 18, 1297-1308 (2000).
    [CrossRef]
  8. M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
    [CrossRef]
  9. S. Kahl and A. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
    [CrossRef]
  10. A. Figotin, Y. A. Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Phys. Rev. B 57, 2841-2848 (1998).
    [CrossRef]
  11. A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
    [CrossRef]
  12. M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
    [CrossRef]
  13. H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
    [CrossRef]
  14. X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
    [CrossRef]
  15. The use of TE/TM terminology in this context is meant to illustrate the fact that different waveguide modes with different polarization states possess different modal refractive indices. The authors are aware that the polarization state of light in gyrotropic films, especially under material stresses and the geometrical confinement generated by asymmetric optical waveguide channels should be described in terms of waveguide modes that account for elliptical birefringence. A full treatment of the polarization state of Bloch modes is presented in this work.
  16. M. Levy, "Normal modes and birefringent magnetophotonic crystals," J. Appl. Phys. 99, 073104 (2006).
    [CrossRef]
  17. R. Li and M. Levy, "Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
    [CrossRef]
  18. L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).
  19. P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  20. S. Visnovsky, K. Postava, and T. Yamaguchi, "Magneto-optic polar Kerr and Faraday effects in periodic multilayers," Opt. Express 9, 158-171 (2001).
    [CrossRef] [PubMed]
  21. We use the greek letter λ to represent the eigenvalue of the Bloch equation. It is not used as a symbol for wavelength.
  22. T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
    [CrossRef]
  23. Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
    [CrossRef]
  24. S. I. Khartsev and A. M. Grishin, "Heteroepitaxial Bi3Fe5O12/La3Ga5O12 films for magneto-optical photonic crystals," Appl. Phys. Lett. 86, 141108 (2005).
    [CrossRef]

2007 (1)

T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
[CrossRef]

2006 (4)

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
[CrossRef]

X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
[CrossRef]

M. Levy, "Normal modes and birefringent magnetophotonic crystals," J. Appl. Phys. 99, 073104 (2006).
[CrossRef]

2005 (3)

R. Li and M. Levy, "Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
[CrossRef]

S. I. Khartsev and A. M. Grishin, "Heteroepitaxial Bi3Fe5O12/La3Ga5O12 films for magneto-optical photonic crystals," Appl. Phys. Lett. 86, 141108 (2005).
[CrossRef]

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

2004 (4)

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

S. Kahl and A. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
[CrossRef]

2003 (1)

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

2001 (3)

2000 (1)

1999 (1)

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

1998 (2)

M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
[CrossRef]

A. Figotin, Y. A. Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Phys. Rev. B 57, 2841-2848 (1998).
[CrossRef]

1979 (1)

Abe, M.

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

M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
[CrossRef]

Aktsipetrov, O.

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

Arai, K.

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

M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
[CrossRef]

Bandyopadhyay, A.

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

Baryshev, A.

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

Cho, S.

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

Fedyanin, A.

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609 (2001).
[CrossRef]

A. Figotin, Y. A. Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Phys. Rev. B 57, 2841-2848 (1998).
[CrossRef]

Fuji, T.

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

Fujii, T.

M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
[CrossRef]

Fujita, J.

Godin, Y. A.

A. Figotin, Y. A. Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Phys. Rev. B 57, 2841-2848 (1998).
[CrossRef]

Granovsky, A.

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

Grishin, A.

S. Kahl and A. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
[CrossRef]

Grishin, A. M.

S. I. Khartsev and A. M. Grishin, "Heteroepitaxial Bi3Fe5O12/La3Ga5O12 films for magneto-optical photonic crystals," Appl. Phys. Lett. 86, 141108 (2005).
[CrossRef]

Guo, X.

T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
[CrossRef]

Gutierrez, C.

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

Huang, X.

X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
[CrossRef]

Hwang, Y.

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

Inoue, M.

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

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

M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
[CrossRef]

Jeen, G.

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

Kahl, S.

S. Kahl and A. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
[CrossRef]

Khanikaev, A.

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

Khartsev, S. I.

S. I. Khartsev and A. M. Grishin, "Heteroepitaxial Bi3Fe5O12/La3Ga5O12 films for magneto-optical photonic crystals," Appl. Phys. Lett. 86, 141108 (2005).
[CrossRef]

Kim, H.

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

Kobayashi, D.

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

Kodama, T.

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

Landau, L.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Levy, M.

M. Levy, "Normal modes and birefringent magnetophotonic crystals," J. Appl. Phys. 99, 073104 (2006).
[CrossRef]

M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
[CrossRef]

X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
[CrossRef]

R. Li and M. Levy, "Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
[CrossRef]

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

M. Levy, H. Yang, M. Steel, and J. Fujita, "Flat-top response in one-dimensional magnetic photonic bandgap structures with Faraday rotation enhancement," J. Lightwave Technol. 19, 1964-1969 (2001).
[CrossRef]

M. Steel, M. Levy, and R. M. Osgood, "Photonic bandgaps with defects and the enhancement of Faraday rotation," J. Lightwave Technol. 18, 1297-1308 (2000).
[CrossRef]

Li, R.

X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
[CrossRef]

M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
[CrossRef]

R. Li and M. Levy, "Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
[CrossRef]

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

Lifshitz, E.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Moran, P.

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

Nishimura, K.

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

Osgood, R. M.

Park, H.

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

Pitaevskii, L.

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Postava, K.

Ram, R. J.

T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
[CrossRef]

Steel, M.

Uchida, H.

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

Um, Y.

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

Vinogradov, A.

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

Visnovsky, S.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Vitebsky, I.

A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609 (2001).
[CrossRef]

A. Figotin, Y. A. Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Phys. Rev. B 57, 2841-2848 (1998).
[CrossRef]

Yamaguchi, T.

Yang, H.

X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
[CrossRef]

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

M. Levy, H. Yang, M. Steel, and J. Fujita, "Flat-top response in one-dimensional magnetic photonic bandgap structures with Faraday rotation enhancement," J. Lightwave Technol. 19, 1964-1969 (2001).
[CrossRef]

Yeh, P.

Zaman, T. R.

T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
[CrossRef]

Appl. Phys. Lett. (5)

M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
[CrossRef]

S. Kahl and A. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
[CrossRef]

R. Li and M. Levy, "Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
[CrossRef]

T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
[CrossRef]

S. I. Khartsev and A. M. Grishin, "Heteroepitaxial Bi3Fe5O12/La3Ga5O12 films for magneto-optical photonic crystals," Appl. Phys. Lett. 86, 141108 (2005).
[CrossRef]

IEEE Trans. Magn. (1)

H. Yang, M. Levy, R. Li, P. Moran, C. Gutierrez, and A. Bandyopadhyay, "Linear birefringence control and magnetization in sputter-deposited magnetic garnet films," IEEE Trans. Magn. 40, 3533-3537 (2004).
[CrossRef]

J. Appl. Phys. (5)

M. Inoue, K. Arai, T. Fujii, and M. Abe, "Magneto-optical properties of one-dimensional photonic crystals composed of magnetic and dielectric layers," J. Appl. Phys. 83, 6768-6770 (1998).
[CrossRef]

M. Levy, "Normal modes and birefringent magnetophotonic crystals," J. Appl. Phys. 99, 073104 (2006).
[CrossRef]

Y. Hwang, H. Kim, S. Cho, Y. Um, H. Park, and G. Jeen, "Magneto-optical properties for bulk Cd0.63−yMn0.37HgyTe single crystals," J. Appl. Phys. 100, 063509 (2006).
[CrossRef]

A. Baryshev, T. Kodama, K. Nishimura, H. Uchida, and M. Inoue, "Three-dimensional magnetophotonic crystals based on artificial opals," J. Appl. Phys. 95, 7336-7338 (2004).
[CrossRef]

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

J. Lightwave Technol. (2)

J. Magn. Magn. Mater. (2)

A. Fedyanin, O. Aktsipetrov, D. Kobayashi, K. Nishimura, H. Uchida, and M. Inoue, "Enhanced Faraday and nonlinear magneto-optical Kerr effects in magnetophotonic crystals," J. Magn. Magn. Mater. 282, 256-259 (2004).
[CrossRef]

X. Huang, R. Li, H. Yang, and M. Levy, "Multimodal and birefringence effects in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 112-116 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Phys. Rev. B (3)

A. Figotin, Y. A. Godin, and I. Vitebsky, "Two-dimensional tunable photonic crystals," Phys. Rev. B 57, 2841-2848 (1998).
[CrossRef]

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

A. Khanikaev, A. Baryshev, M. Inoue, A. Granovsky, and A. Vinogradov, "Two-dimensional magnetophotonic crystal: exactly solvable model," Phys. Rev. B 72, 035123 (2005).
[CrossRef]

Phys. Rev. E (1)

A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609 (2001).
[CrossRef]

Other (3)

L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

We use the greek letter λ to represent the eigenvalue of the Bloch equation. It is not used as a symbol for wavelength.

The use of TE/TM terminology in this context is meant to illustrate the fact that different waveguide modes with different polarization states possess different modal refractive indices. The authors are aware that the polarization state of light in gyrotropic films, especially under material stresses and the geometrical confinement generated by asymmetric optical waveguide channels should be described in terms of waveguide modes that account for elliptical birefringence. A full treatment of the polarization state of Bloch modes is presented in this work.

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Figures (4)

Fig. 1
Fig. 1

Schematic of a one-dimensional birefringent magnetophotonic crystal with period of Λ. The magnetophotonic crystal extends indefinitely in the x and y directions. A plane wave is incident from the left onto the structure.

Fig. 2
Fig. 2

Variation of elements of the B matrix versus layer thickness d ( n ) in the unit cell for a typical [ Bi , Lu ] 3 Fe 5 O 12 layer. Solid curves with open and solid circles are curves for B 11 and B 21 from block-diagonal elements, respectively. Dashed and dotted curves are curves for B 23 and B 13 from off-block diagonal elements, respectively.

Fig. 3
Fig. 3

Band structure for a periodic stack of Bi : LuIG and LaGG with d ( n ) = 0.3 Λ and d ( n + 1 ) = 0.7 Λ , respectively. The dashed curve shows the K + wave branch and the solid curve shows the K wave branch.

Fig. 4
Fig. 4

Band structure for a periodic stack of Bi : LuIG and LaGG with d ( n ) = d ( n + 1 ) = 0.5 Λ . The dashed curve shows the K + wave branch and the solid curve shows the K wave branch.

Equations (39)

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( k 0 2 ϵ ̃ k 2 I + kk ) E 0 = 0 ,
ϵ ̃ = ( ϵ x x i ϵ x y 0 i ϵ x y ϵ y y 0 0 0 ϵ z z ) .
e ̂ ± = 1 2 ( cos α ± sin α ± i cos α i sin α 0 ) ,
sin γ = Δ Δ 2 + ϵ x y 2 ,
cos γ = ϵ x y Δ 2 + ϵ x y 2 .
( e ̂ + e ̂ ) = ( cos γ 2 sin γ 2 sin γ 2 cos γ 2 ) ( c ̂ + c ̂ ) ,
c ̂ ± = 1 2 ( 1 ± i ) .
E ( z ) = [ E 01 exp ( i ω c n + ( z z n ) ) + E 02 exp ( i ω c n + ( z z n ) ) ] e ̂ + + [ E 03 exp ( i ω c n ( z z n ) ) + E 04 exp ( i ω c n ( z z n ) ) ] e ̂ .
T ( n 1 , n ) = ( D ( n 1 ) ) 1 D ( n ) P ( n ) ,
D ( n ) = ( cos α ( n ) cos α ( n ) sin α ( n ) sin α ( n ) n + ( n ) cos α ( n ) n + ( n ) cos α ( n ) n ( n ) sin α ( n ) n ( n ) sin α ( n ) sin α ( n ) sin α ( n ) cos α ( n ) cos α ( n ) n + ( n ) sin α ( n ) n + ( n ) sin α ( n ) n ( n ) cos α ( n ) n ( n ) cos α ( n ) ) ,
P ( n ) = ( e i β + ( n ) 0 0 0 0 e i β + ( n ) 0 0 0 0 e i β ( n ) 0 0 0 0 e i β ( n ) ) ,
E ( z ) = E K ( z ) exp ( i K z ) ,
T ( n 1 , n + 1 ) E = λ E .
S ( n ) = ( U ( n ) ) 1 S c ( n ) U ( n ) ,
U ( n ) = ( cos α ( n ) 0 sin α ( n ) 0 0 cos α ( n ) 0 sin α ( n ) sin α ( n ) 0 cos α ( n ) 0 0 sin α ( n ) 0 cos α ( n ) ) ,
S c ( n ) = ( cos β + ( n ) i n + ( n ) sin β + ( n ) 0 0 i n + ( n ) sin β + ( n ) cos β + ( n ) 0 0 0 0 cos β ( n ) i n ( n ) sin β ( n ) 0 0 i n ( n ) sin β ( n ) cos β ( n ) ) ,
T ( n 1 , n + 1 ) = ( Φ ( n , n + 1 ) ) 1 S c ( n ) Φ ( n , n + 1 ) P ( n + 1 ) ,
Φ ( n , n + 1 ) = U ( n ) D ( n + 1 ) .
( cos χ ( n , n + 1 ) cos χ ( n , n + 1 ) sin χ ( n , n + 1 ) sin χ ( n , n + 1 ) n + ( n + 1 ) cos χ ( n , n + 1 ) n + ( n + 1 ) cos χ ( n , n + 1 ) n ( n + 1 ) sin χ ( n , n + 1 ) n ( n + 1 ) sin χ ( n , n + 1 ) sin χ ( n , n + 1 ) sin χ ( n , n + 1 ) cos χ ( n , n + 1 ) cos χ ( n , n + 1 ) n + ( n + 1 ) sin χ ( n , n + 1 ) n + ( n + 1 ) sin χ ( n , n + 1 ) n ( n + 1 ) cos χ ( n , n + 1 ) n ( n + 1 ) cos χ ( n , n + 1 ) ) ,
S c ( n ) E = λ B E ,
S c ( n ) λ B = 0 .
B 1 , 1 = B 2 , 2 = cos 2 χ ( n , n + 1 ) cos β + ( n + 1 ) + sin 2 χ ( n , n + 1 ) cos β ( n + 1 ) ,
B 1 , 2 = i sin 2 χ ( n , n + 1 ) sin β ( n + 1 ) n ( n + 1 ) i cos 2 χ ( n , n + 1 ) sin β + ( n + 1 ) n + ( n + 1 ) ,
B 1 , 3 = B 3 , 1 = B 2 , 4 = B 4 , 2 = 1 2 sin 2 χ ( n , n + 1 ) ( cos β ( n + 1 ) cos β + ( n + 1 ) ) 0 ,
B 1 , 4 = B 3 , 2 = i sin 2 χ ( n , n + 1 ) 2 n ( n + 1 ) n + ( n + 1 ) ( n ( n + 1 ) sin β + ( n + 1 ) n + ( n + 1 ) sin β ( n + 1 ) ) 0 ,
B 2 , 1 = i n ( n + 1 ) sin 2 χ ( n , n + 1 ) sin β ( n + 1 ) i n + ( n + 1 ) cos 2 χ ( n , n + 1 ) sin β + ( n + 1 ) ,
B 2 , 3 = B 4 , 1 = i 1 2 sin 2 χ ( n , n + 1 ) ( n + ( n + 1 ) sin β + ( n + 1 ) n ( n + 1 ) sin β ( n + 1 ) ) 0 ,
B 3 , 3 = B 4 , 4 = cos 2 χ ( n , n + 1 ) cos β ( n + 1 ) + sin 2 χ ( n , n + 1 ) cos β + ( n + 1 ) ,
B 3 , 4 = i cos 2 χ ( n , n + 1 ) sin β ( n + 1 ) n ( n + 1 ) i sin 2 χ ( n , n + 1 ) sin β + ( n + 1 ) n + ( n + 1 ) ,
B 4 , 3 = i n ( n + 1 ) cos 2 χ ( n , n + 1 ) sin β ( n + 1 ) i n + ( n + 1 ) sin 2 χ ( n , n + 1 ) sin β + ( n + 1 ) .
( 5.369 i 0.00274 0 i 0.00274 5.373 0 0 0 ϵ z z ) ,
( 0.000681 i 0.431 0.000196 i 0.0000539 i i 2.32 0.000681 i 0.00029 0.000196 0.000196 i 0.0000539 0.000681 i 0.432 i 0.00029 0.000196 i 2.32 0.000681 ) ,
( 0.501 i 0.373 0.000227 i 0.000103 i 2.01 0.501 i 0.0000525 0.000227 0.000227 i 0.000103 0.499 i 0.374 i 0.0000525 0.000227 i 2.01 0.499 ) ,
cos K Λ = B ± cos β ± ( n ) 1 2 C ± sin β ± ( n ) ,
C ± = i n ± ( n ) B i , j + i n ± ( n ) B j , i ,
cos K Λ cos k 0 Λ 1 2 ( m ± 1 2 Δ ) k 0 sin k 0 Λ ,
cos K Λ = cos ( β ± ( n ) + β ± ( n + 1 ) ) ,
K = ω c ( n ± ( n ) d ( n ) + n ± ( n + 1 ) d ( n + 1 ) d ( n ) + d ( n + 1 ) ) ω c n ¯ ± ,
E = ( Φ P ( n + 1 ) ) 1 ( i n + ( n ) sin β + ( n ) + B 1 , 2 e ± i K ± Λ cos β + ( n ) B + e ± i K ± Λ 1 i n ( n ) sin β ( n ) + B 3 , 4 e ± i K ± Λ cos β ( n ) B e ± i K ± Λ 1 ) .

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