Abstract

Birefringent magnetophotonic crystals are found to exhibit degeneracy breaking for asymmetric contradirectional coupling in planar waveguides. Fundamental to high-order local normal mode coupling leads to partially overlapping gyrotropic bandgaps inside the Brillouin zone and partial suppression of Bloch mode propagation. A large magneto-optically active reorientation in polarization state is found for allowed Bloch modes at bandgap edges.

© 2008 Optical Society of America

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References

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  1. M. Levy and R. Li, "Polarization rotation enhancement and scattering mechanisms in waveguide magnetophotonic crystals," Appl. Phys. Lett. 89, 121113 (2006).
    [CrossRef]
  2. R. Li and M. Levy,"Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 86, 251102 (2005).
    [CrossRef]
  3. R. Li and M. Levy, "Erratum: Bragg grating magnetic photonic crystal waveguides," Appl. Phys. Lett. 87, 269901 (2005).
    [CrossRef]
  4. S. Kahl and A. M. Grishin, "Enhanced Faraday rotation in all-garnet magneto-optical photonic crystal," Appl. Phys. Lett. 84, 1438-1440 (2004).
    [CrossRef]
  5. M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
    [CrossRef]
  6. S. Visnovsky, K. Postava, and T. Yamaguchi, "Magneto-optic polar Kerr and Faraday effects in periodic multilayers," Opt. Express 9, 158-171 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-3-158.
    [CrossRef] [PubMed]
  7. A. A. Jalali and M. Levy, "Local normal mode coupling and energy band splitting in elliptically birefringent 1D magnetophotonic crystals," J. Opt. Soc. Am. B 25, 119-125 (2008).
    [CrossRef]
  8. A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
    [CrossRef]
  9. M. Levy and A. A. Jalali, "Band structure and Bloch states in birefringent 1D magnetophotonic crystals: An analytical approach," J. Opt. Soc. Am. B 24, 1603-1609 (2007).
    [CrossRef]
  10. A. Erdmann and P. Hertel, "Beam-propagation in magnetooptic waveguides," IEEE J. Quantum Electron. 31, 1510-1516 (1995).
    [CrossRef]
  11. R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, "Elimination of birefringence in garnet films for magnetooptic waveguide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
    [CrossRef]
  12. T. R. Zaman, X. Guo, and R. J. Ram, "Faraday rotation in an InP waveguide," Appl. Phys. Lett. 90, 023514 (2007).
    [CrossRef]
  13. D. Marcuse, "Coupled Mode Theory," in Theory of Dielectric Optical Waveguides, (Academic Press, 1991), pp. 95-131.
  14. A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 117-121 (2006).
    [CrossRef]
  15. M. J. Steel, M. Levy, and R. M. Osgood, Jr., "Photonic bandgaps with defects and the enhancement of Faraday rotation," J. Lightwave Technol. 18, 1297 (2000).
    [CrossRef]

2008

2007

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

M. Levy and A. A. Jalali, "Band structure and Bloch states in birefringent 1D magnetophotonic crystals: An analytical approach," J. Opt. Soc. Am. B 24, 1603-1609 (2007).
[CrossRef]

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

2006

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 117-121 (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

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. 87, 269901 (2005).
[CrossRef]

2004

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

2001

2000

1999

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
[CrossRef]

1995

A. Erdmann and P. Hertel, "Beam-propagation in magnetooptic waveguides," IEEE J. Quantum Electron. 31, 1510-1516 (1995).
[CrossRef]

1987

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, "Elimination of birefringence in garnet films for magnetooptic waveguide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
[CrossRef]

Abe, M.

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
[CrossRef]

Arai, K.

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
[CrossRef]

Dorofeenko, A.V.

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

Erdmann, A.

A. Erdmann and P. Hertel, "Beam-propagation in magnetooptic waveguides," IEEE J. Quantum Electron. 31, 1510-1516 (1995).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 117-121 (2006).
[CrossRef]

Fratello, V. J.

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, "Elimination of birefringence in garnet films for magnetooptic waveguide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
[CrossRef]

Fujii, T.

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
[CrossRef]

Granovsky, A. B.

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

Grishin, A. M.

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

Hertel, P.

A. Erdmann and P. Hertel, "Beam-propagation in magnetooptic waveguides," IEEE J. Quantum Electron. 31, 1510-1516 (1995).
[CrossRef]

Inoue, M.

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
[CrossRef]

Jalali, A. A.

Kahl, S.

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

Levy, M.

A. A. Jalali and M. Levy, "Local normal mode coupling and energy band splitting in elliptically birefringent 1D magnetophotonic crystals," J. Opt. Soc. Am. B 25, 119-125 (2008).
[CrossRef]

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

M. Levy and A. A. Jalali, "Band structure and Bloch states in birefringent 1D magnetophotonic crystals: An analytical approach," J. Opt. Soc. Am. B 24, 1603-1609 (2007).
[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]

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

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

Li, R.

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. 87, 269901 (2005).
[CrossRef]

McGlashan-Powell, M.

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, "Elimination of birefringence in garnet films for magnetooptic waveguide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
[CrossRef]

Merzlikin, A.M.

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

Osgood, R. M.

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. J.

Vinogradov, A.P.

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

Visnovsky, S.

Vitebskiy, I.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 117-121 (2006).
[CrossRef]

Wolfe, R.

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, "Elimination of birefringence in garnet films for magnetooptic waveguide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
[CrossRef]

Yamaguchi, T.

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.

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. 87, 269901 (2005).
[CrossRef]

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

R. Wolfe, V. J. Fratello, and M. McGlashan-Powell, "Elimination of birefringence in garnet films for magnetooptic waveguide devices," Appl. Phys. Lett. 51, 1221-1223 (1987).
[CrossRef]

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

IEEE J. Quantum Electron.

A. Erdmann and P. Hertel, "Beam-propagation in magnetooptic waveguides," IEEE J. Quantum Electron. 31, 1510-1516 (1995).
[CrossRef]

J. Appl. Phys.

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-70 (1999).
[CrossRef]

J. Lightwave Technol.

J. Magn. Magn. Mater.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magn. Magn. Mater. 300, 117-121 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Physica B: Condensed Matter

A. M. Merzlikin, A. P. Vinogradov, A.V. Dorofeenko, M. Inoue, M. Levy and A. B. Granovsky, "Controllable Tamm states in magnetophotonic crystal," Physica B: Condensed Matter 94, 277-280 (2007).
[CrossRef]

Other

D. Marcuse, "Coupled Mode Theory," in Theory of Dielectric Optical Waveguides, (Academic Press, 1991), pp. 95-131.

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Figures (6)

Fig. 1.
Fig. 1.

(a) SEM micrograph of one-dimensional waveguide Bragg-filter with phase shift step. (b) SEM micrograph of one-dimensional waveguide Bragg-filter without phase shift step

Fig. 2.
Fig. 2.

The figure plots the measured transmittance and polarization response [right panel], and calculated band structure and polarization rotation (left panel) of a one-dimensional photonic crystal with phase shift step patterned on a 2.86 µm-thick (Bi,Lu)2.8Fe4.7O12.1 film (sample set A). Separate curves for the calculated semi-major axis orientation correspond to different Bloch states. The red and blue data points on the right panel describe the orientation of the semi-major axis of the polarization ellipse for magnetization in opposite directions collinear with the ridge waveguide axis.

Fig. 3.
Fig. 3.

Measured transmittance and polarization response (right panel), and calculated band structure and polarization rotation [left panel] of a one-dimensional Bragg filter without phase shift step patterned on a 2.7 µm-thick Bi0.8Gd0.2Lu2.0Fe5O12 film (sample set B). Separate curves for the calculated semi-major axis orientation correspond to different Bloch states. The red and blue data points on the right describe the orientation of the semi-major axis of the output polarization ellipse for magnetization in opposite directions collinear with the ridge waveguide axis.

Fig. 4.
Fig. 4.

Measured transmittance and polarization response for a one-dimensional photonic crystal with a single phase shift step patterned on a (Bi,Lu,Nd)3(Fe,Ga,Al)5O12 film of thickness 1.8µm. Black triangles and solid circles plot the orientation of the semi-major axis of the output polarization ellipse relative to the linear input polarization for opposite magnetization directions colinear with the ridge waveguide axis.

Fig. 5.
Fig. 5.

Measured ellipticity in the output polarization for Bragg filters patterned on (Bi,Lu)2.8Fe4.7O12.1 film (sample set A), Bi0.8Gd0.2Lu2.0Fe5O12 film (sample set B) and (Bi,Lu,Nd)3(Fe,Ga,Al)5O12 film (samples set C). Ellipticity is defined as the ratio of the semi-minor to semi-major axes of the polarization ellipse in the optical electric field amplitude. The horizontal double-tipped arrows indicate the locations of the stopbands in each case.

Fig. 6.
Fig. 6.

Model structure consisting of a bilayer unit cell of length Λ. The transfer matrix operator is denoted by . The forward and backward normal mode refractive indices nf,b ± are allowed to differ from each other. Similarly, the normal mode indices in adjacent layers (n) and (n+1) are different from each other.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

ε ˜ = ( ε x x i ε x y 0 i ε x y ε y y 0 0 0 ε z z ) ,
E ( z , t ) = [ ( E 01 exp ( i ω c n + f ( z z n ) ) ) e ̂ + f + ( E 02 exp ( i ω c n + b ( z z n ) ) ) e ̂ + b ] exp ( i ω t )
+ [ ( E 03 exp ( i ω c n f ( z z n ) ) ) e ̂ f + ( E 04 exp ( i ω c n b ( z z n ) ) ) e ̂ b ] exp ( i ω t )
e ̂ ± f , b = 1 2 ( cos α f , b ± sin α f , b ± i cos α f , b i sin α f , b 0 ) ,

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