Abstract

An analysis is presented of wave-vector dispersion in elliptically birefringent stratified magneto-optic media having one-dimensional periodicity. It is found that local normal-mode polarization-state differences between adjacent layers lead to mode coupling and affect the wave-vector dispersion and the character of the Bloch states of the system. This coupling produces extra terms in the dispersion relation not present in uniform circularly birefringent magneto-optic stratified media. Normal-mode coupling lifts the degeneracy at frequency band crossover points under certain conditions and induces a magnetization-dependent optical bandgap. This study examines the conditions for bandgap formation in the system. It shows that such a frequency split can be characterized by a simple coupling parameter that depends on the relation between polarization states of local normal modes in adjacent layers. The character of the Bloch states and conditions for maximizing the strength of the band splitting in these systems are analyzed.

© 2008 Optical Society of America

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References

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  1. P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979).
    [CrossRef]
  2. A. Mandatori, C. Sibilia, M. Centini, G. D'Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. Bowden, "Birefringence in one-dimensional finite photonic bandgap structure," J. Opt. Soc. Am. B 20, 504-513 (2003).
    [CrossRef]
  3. M. Levy and A. A. Jalali, "Band structure and Bloch states in birefringent one-dimensional magnetophotonic crystals: an analytical approach," J. Opt. Soc. Am. B 24, 1603-1609 (2007).
    [CrossRef]
  4. L. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).
  5. M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
    [CrossRef]
  6. R. Li and M. Levy, "Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
    [CrossRef]
  7. R. Li and M. Levy, "Erratum: `Bragg grating magnetic photonic crystal waveguides' [Appl. Phys. Lett. 86, 251102 (2005)]," Appl. Phys. Lett. 87, 269901 (2005).
    [CrossRef]
  8. M. Inoue, K. Arai, T. Fuji, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (1999).
    [CrossRef]
  9. A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609 (2001).
    [CrossRef]
  10. 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]
  11. S. Kahl and A. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
    [CrossRef]
  12. S. Khartsev and A. Grishin, "High performance [Bi3Fe5O12/Sm3Ga5O12]m magneto-optical photonic crystals," J. Appl. Phys. 101, 053906 (2007).
    [CrossRef]
  13. A. Merzlikin, A. Vinogradov, A. Dorofeenko, M. Inoue, M. Levy, and A. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B 394, 277-280 (2007).
    [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]

2007 (3)

M. Levy and A. A. Jalali, "Band structure and Bloch states in birefringent one-dimensional magnetophotonic crystals: an analytical approach," J. Opt. Soc. Am. B 24, 1603-1609 (2007).
[CrossRef]

S. Khartsev and A. Grishin, "High performance [Bi3Fe5O12/Sm3Ga5O12]m magneto-optical photonic crystals," J. Appl. Phys. 101, 053906 (2007).
[CrossRef]

A. Merzlikin, A. Vinogradov, A. Dorofeenko, M. Inoue, M. Levy, and A. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B 394, 277-280 (2007).
[CrossRef]

2006 (2)

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]

2005 (3)

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

R. Li and M. Levy, "Erratum: `Bragg grating magnetic photonic crystal waveguides' [Appl. Phys. Lett. 86, 251102 (2005)]," Appl. Phys. Lett. 87, 269901 (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 (1)

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)

2001 (1)

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

1999 (1)

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

1979 (1)

Appl. Phys. Lett. (4)

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]

R. Li and M. Levy, "Erratum: `Bragg grating magnetic photonic crystal waveguides' [Appl. Phys. Lett. 86, 251102 (2005)]," Appl. Phys. Lett. 87, 269901 (2005).
[CrossRef]

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

J. Appl. Phys. (2)

S. Khartsev and A. Grishin, "High performance [Bi3Fe5O12/Sm3Ga5O12]m magneto-optical photonic crystals," J. Appl. Phys. 101, 053906 (2007).
[CrossRef]

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

J. Magn. Magn. Mater. (1)

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)

J. Opt. Soc. Am. B (2)

Phys. Rev. B (1)

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]

Physica B (1)

A. Merzlikin, A. Vinogradov, A. Dorofeenko, M. Inoue, M. Levy, and A. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B 394, 277-280 (2007).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram 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 normally to the layered structure. A unit cell spans the region between z n 1 and z n + 1 .

Fig. 2
Fig. 2

Energy band diagram for BiIG periodic structure (fifth branch) with d ( n ) = 0.4 and d ( n + 1 ) = 0.6 . The dashed lines correspond to the case where χ ( n , n + 1 ) = 0 in the transfer matrix. The solid curves correspond to a realistic case with χ = 0.14 .

Fig. 3
Fig. 3

Width of the gyrotropic degenerate bandgap versus χ ( n , n + 1 ) . The bandgap was calculated for the fifth branch of the band structure of the periodic structure of BiIG with d ( n ) = 0.4 and d ( n + 1 ) = 0.6 . In one case the average in the refractive indices is kept constant while χ ( n , n + 1 ) is allowed to change (solid curve). In the other, the diagonal elements of the dielectric tensors of adjacent layers are kept constant while the off-diagonal elements are allowed to change simultaneously (dashed curve).

Fig. 4
Fig. 4

Polarization of Bloch wave traveling through a MO birefringent periodic structure. The Bloch wave polarization is depicted on the boundary of each layer in a unit cell (solid ellipses) just before the gyrotropic degenerate bandgap in the band structure of the medium. Dashed ellipses show the local eigenmodes polarizations e ̂ + for each layer.

Equations (39)

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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 ) ,
n ± 2 = ϵ ¯ ± Δ 2 + ϵ x y 2 .
β ± = ( ω c ) n ± ,
E ( n ) ( z ) = ( E 01 ( n ) e i β + ( n ) ( z z n ) + E 02 ( n ) e i β + ( n ) ( z z n ) ) e ̂ + ( n ) + ( E 03 ( n ) e i β ( n ) ( z z n ) + E 04 ( n ) e i β ( n ) ( z z n ) ) e ̂ ( n ) .
T ( n 1 , n + 1 ) E = exp ( i K Λ ) E ,
( f 1 , 1 + g 1 , 1 sin 2 χ ( n , n + 1 ) f 1 , 2 + g 1 , 2 sin 2 χ ( n , n + 1 ) g 1 , 3 sin 2 χ ( n , n + 1 ) g 1 , 4 sin 2 χ ( n , n + 1 ) f 2 , 1 + g 2 , 1 sin 2 χ ( n , n + 1 ) f 2 , 2 + g 2 , 2 sin 2 χ ( n , n + 1 ) g 2 , 3 sin 2 χ ( n , n + 1 ) g 2 , 4 sin 2 χ ( n , n + 1 ) g 3 , 1 sin 2 χ ( n , n + 1 ) g 3 , 2 sin 2 χ ( n , n + 1 ) f 3 , 3 + g 3 , 3 sin 2 χ ( n , n + 1 ) f 3 , 4 + g 3 , 4 sin 2 χ ( n , n + 1 ) g 4 , 1 sin 2 χ ( n , n + 1 ) g 4 , 2 sin 2 χ ( n , n + 1 ) f 4 , 3 + g 4 , 3 sin 2 χ ( n , n + 1 ) f 4 , 4 + g 4 , 4 sin 2 χ ( n , n + 1 ) ) ,
f 1 , 1 = exp ( i β + ( n + 1 ) d ( n + 1 ) ) [ cos ( β + ( n ) d ( n ) ) i 2 ( n + ( n ) n + ( n + 1 ) + n + ( n + 1 ) n + ( n ) ) sin ( β + ( n ) d ( n ) ) ] ,
g 1 , 1 = exp ( i β + ( n + 1 ) d ( n + 1 ) ) [ cos ( β ( n ) d ( n ) ) cos ( β + ( n ) d ( n ) ) i 2 ( n ( n ) n + ( n + 1 ) + n + ( n + 1 ) n ( n ) ) sin ( β ( n ) d ( n ) ) + i 2 ( n + ( n ) n + ( n + 1 ) + n + ( n + 1 ) n + ( n ) ) sin ( β + ( n ) d ( n ) ) ] ,
f 1 , 2 = i 2 exp ( i β + ( n + 1 ) d ( n + 1 ) ) [ n + ( n + 1 ) n + ( n ) n + ( n ) n + ( n + 1 ) ] sin ( β + ( n ) d ( n ) ) ,
g 1 , 2 = i 2 exp ( i β + ( n + 1 ) d ( n + 1 ) ) × [ ( n + ( n ) n + ( n + 1 ) n + ( n + 1 ) n + ( n ) ) sin ( β + ( n ) d ( n ) ) + ( n + ( n + 1 ) n ( n ) n ( n ) n + ( n + 1 ) ) sin ( β ( n ) d ( n ) ) ] ,
g 1 , 3 = exp ( i β ( n + 1 ) d ( n + 1 ) ) [ ( n ( n + 1 ) + n + ( n + 1 ) 4 n + ( n + 1 ) ) ( cos ( β + ( n ) d ( n ) ) cos ( β ( n ) d ( n ) ) ) + i 4 ( n ( n + 1 ) n ( n ) + n ( n ) n + ( n + 1 ) ) sin ( β ( n ) d ( n ) ) i 4 ( n ( n + 1 ) n + ( n ) + n + ( n ) n + ( n + 1 ) ) sin ( β + ( n ) d ( n ) ) ] ,
g 1 , 4 = exp ( i β ( n + 1 ) d ( n + 1 ) ) [ ( n + ( n + 1 ) n ( n + 1 ) 4 n + ( n + 1 ) ) ( cos ( β + ( n ) d ( n ) ) cos ( β ( n ) d ( n ) ) ) i 4 ( n ( n + 1 ) n ( n ) n ( n ) n + ( n + 1 ) ) sin ( β ( n ) d ( n ) ) + i 4 ( n ( n + 1 ) n + ( n ) n + ( n ) n + ( n + 1 ) ) sin ( β + ( n ) d ( n ) ) ] .
f 2 , 1 = f 1 , 2 * , g 2 , 1 = g 1 , 2 * ;
f 2 , 2 = f 1 , 1 * , g 2 , 2 = g 1 , 1 * ;
g 2 , 3 = g 1 , 4 * ;
g 2 , 4 = g 1 , 3 * ;
g 3 , 1 = g 1 , 3 ( n + ( n + 1 ) n ( n + 1 ) ) ;
g 3 , 2 = g 1 , 4 ( n + ( n + 1 ) n ( n + 1 ) ) ;
f 3 , 3 = f 1 , 1 , g 3 , 3 = g 1 , 1 ( n + ( n + 1 ) n ( n + 1 ) and n + ( n ) n ( n ) ) ;
f 3 , 4 = f 1 , 2 , g 3 , 4 = g 1 , 2 ( n + ( n + 1 ) n ( n + 1 ) and n + ( n ) n ( n ) ) ;
g 4 , 1 = g 3 , 2 * ;
g 4 , 2 = g 3 , 1 * ;
f 4 , 3 = f 3 , 4 * , g 4 , 3 = g 3 , 4 * ;
f 4 , 4 = f 3 , 3 * , g 4 , 4 = g 3 , 3 * .
g ( λ ) = f ( λ ) + h ( λ ) sin 2 χ ( n , n + 1 ) + j ( λ ) sin 2 2 χ ( n , n + 1 ) + k ( λ ) sin 2 3 χ ( n , n + 1 ) + l ( λ ) sin 2 4 χ ( n , n + 1 ) = 0 ,
g ( λ ) = g ( λ 0 ) + g ( λ 0 ) ( λ λ 0 ) h ( λ 0 ) sin 2 χ ( n , n + 1 ) + [ f ( λ 0 ) + h ( λ 0 ) sin 2 χ ( n , n + 1 ) ] ( λ λ 0 ) .
λ 0 λ 0 = h ( λ 0 ) sin 2 χ ( n , n + 1 ) f ( λ 0 ) + h ( λ 0 ) sin 2 χ ( n , n + 1 ) ,
λ 0 λ 0 h ( λ 0 ) f ( λ 0 ) sin 2 χ ( n , n + 1 ) ,
cos K ± Λ = cos ( β ± ( n + 1 ) d ( n + 1 ) ) cos ( β ± ( n ) d ( n ) ) 1 2 N ± sin ( β ± ( n + 1 ) d ( n + 1 ) ) sin ( β ± ( n ) d ( n ) ) .
cos K ± Λ = [ cos ( β ± ( n + 1 ) d ( n + 1 ) ) cos ( β ± ( n ) d ( n ) ) 1 2 N ± sin ( β ± ( n + 1 ) d ( n + 1 ) ) sin ( β ± ( n ) d ( n ) ) ] R ( h ( λ 0 ± ) f ( λ 0 ± ) ) sin 2 χ ( n , n + 1 ) .
cos 2 β ¯ Λ < 4 + 4 u ( λ 0 + ) sin 2 χ ( n , n + 1 ) 2 + N + ,
cos 2 β ¯ Λ > 4 + 4 u ( λ 0 + ) sin 2 χ ( n , n + 1 ) 2 + N + ,
ϵ ̃ ( n ) = ( 6.5411 i 0.018 0 i 0.018 6.611 0 0 0 ϵ z z ) ,
ϵ ̃ ( n + 1 ) = ( 5.9859 i 0.018 0 i 0.018 6.1699 0 0 0 ϵ z z ) .
E K = c K ( A 0 B 0 C 0 D 0 ) K ,
E K ( n + 1 ) ( z ) = ( A 0 e i β + ( n + 1 ) ( z n Λ ) + B 0 e i β + ( n + 1 ) ( z n Λ ) ) e i K n Λ e ̂ + ( n + 1 ) + ( C 0 e i β ( n + 1 ) ( z n Λ ) + D 0 e i β ( n + 1 ) ( z n Λ ) ) e i K n Λ e ̂ ( n + 1 ) .
ϵ ̃ ( n ) = ( 4 i 0.1 0 i 0.1 2.5 0 0 0 ϵ z z ) ,
ϵ ̃ ( n + 1 ) = ( 6 i 2 0 i 2 5 0 0 0 ϵ z z ) .

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