Abstract

We develop an exact theory for light propagation in transversely-magnetized low-symmetry magnetic photonic crystals. We investigate the nature of nonreciprocal dispersion and unidirectionality in these systems and show that it is associated with boundary effects rather than propagation. We calculate the nonreciprocal response of finite structures and propose an asymmetric magneto-optical cavity as a practical building block for one-way optical components.

© 2009 Optical Society of America

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  1. MInoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina and A. Granovsky, "Magnetophotonic crystals," J. Phys. D: Appl. Phys. 39, R151-R161 (2006).
    [CrossRef]
  2. Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
    [CrossRef]
  3. Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
    [CrossRef]
  4. Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007).
    [CrossRef]
  5. V. Dmitriev, "Symmetry properties of 2D magnetic photonic crystals with square lattice," Eur. Phys. J. Appl. Phys. 32, 159-165 (2005).
    [CrossRef]
  6. A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001).
    [CrossRef]
  7. A. K. Zvezdin, V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997).
    [CrossRef]
  8. P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).
  9. A. A. Jalali and M. Levy, "Local normal-mode coupling and energy band splitting in elliptically birefringent one-dimensional magnetophotonic crystals," J. Opt. Soc. Am. B 25, 119-125, (2008).
    [CrossRef]
  10. A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
    [CrossRef]
  11. 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]
  12. M. Inoue, K. Arai, T. Fujii and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (1999).
    [CrossRef]
  13. Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005).
    [CrossRef] [PubMed]
  14. Z. Wang and S. Fan, "Magneto-optical defects in two-dimensional photonic crystals," Appl. Phys. B: Lasers Opt. 81, 369-375 (2005).
    [CrossRef]

2008

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
[CrossRef]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
[CrossRef]

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

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

2007

Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007).
[CrossRef]

2006

MInoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina and A. Granovsky, "Magnetophotonic crystals," J. Phys. D: Appl. Phys. 39, R151-R161 (2006).
[CrossRef]

2005

V. Dmitriev, "Symmetry properties of 2D magnetic photonic crystals with square lattice," Eur. Phys. J. Appl. Phys. 32, 159-165 (2005).
[CrossRef]

Z. Wang and S. Fan, "Magneto-optical defects in two-dimensional photonic crystals," Appl. Phys. B: Lasers Opt. 81, 369-375 (2005).
[CrossRef]

Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005).
[CrossRef] [PubMed]

2001

A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001).
[CrossRef]

1999

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

1998

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]

Abe, M.

M. Inoue, K. Arai, T. Fujii 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.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Arai, K.

M. Inoue, K. Arai, T. Fujii 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]

Baryshev, A. B.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Chong, Y. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
[CrossRef]

Dmitriev, V.

V. Dmitriev, "Symmetry properties of 2D magnetic photonic crystals with square lattice," Eur. Phys. J. Appl. Phys. 32, 159-165 (2005).
[CrossRef]

Fan, S.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
[CrossRef]

Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007).
[CrossRef]

Z. Wang and S. Fan, "Magneto-optical defects in two-dimensional photonic crystals," Appl. Phys. B: Lasers Opt. 81, 369-375 (2005).
[CrossRef]

Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005).
[CrossRef] [PubMed]

Fedyanin, A. A.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001).
[CrossRef]

Fujii, T.

M. Inoue, K. Arai, T. Fujii 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]

Inoue, M.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

M. Inoue, K. Arai, T. Fujii 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]

Jalali, A. A.

Joannopoulos, J. D.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
[CrossRef]

Khanikaev, A. B.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Levy, M.

Lim, P. B.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Maydykovskiy, A. I.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Soljacic, M.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
[CrossRef]

Uchida, H.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
[CrossRef]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001).
[CrossRef]

Wang, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
[CrossRef]

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
[CrossRef]

Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007).
[CrossRef]

Z. Wang and S. Fan, "Magneto-optical defects in two-dimensional photonic crystals," Appl. Phys. B: Lasers Opt. 81, 369-375 (2005).
[CrossRef]

Z. Wang and S. Fan, "Optical circulators in two-dimensional magneto-optical photonic crystals," Opt. Lett. 30, 1989-1991 (2005).
[CrossRef] [PubMed]

Yu, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
[CrossRef]

Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007).
[CrossRef]

Zhdanov, A. G.

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Appl. Phys. B: Lasers Opt.

Z. Wang and S. Fan, "Magneto-optical defects in two-dimensional photonic crystals," Appl. Phys. B: Lasers Opt. 81, 369-375 (2005).
[CrossRef]

Appl. Phys. Lett.

Z. Yu, and Z. Wang, and S. Fan, "One-way total reflection with one-dimensional magneto-optical photonic crystals," Appl. Phys. Lett. 90, 121133:1-3 (2007).
[CrossRef]

Eur. Phys. J. Appl. Phys.

V. Dmitriev, "Symmetry properties of 2D magnetic photonic crystals with square lattice," Eur. Phys. J. Appl. Phys. 32, 159-165 (2005).
[CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am. B

J. Phys. D: Appl. Phys.

MInoue, R. Fujikawa, A. Baryshev, A. Khanikaev, P. B. Lim, H. Uchida, O. Aktsipetrov, A. Fedyanin, T. Murzina and A. Granovsky, "Magnetophotonic crystals," J. Phys. D: Appl. Phys. 39, R151-R161 (2006).
[CrossRef]

Opt. Lett.

Phys. Rev. B

A. B. Khanikaev, A. B. Baryshev, P. B. Lim, H. Uchida, M. Inoue, A. G. Zhdanov, A. A. Fedyanin, A. I. Maydykovskiy, and O. A. Aktsipetrov, "Nonlinear Verdet law in magnetophotonic crystals: Interrelation between Faraday and Borrmann effects," Phys. Rev. B 78, 193102:1-4 (2008).
[CrossRef]

Phys. Rev. E

A. Figotin and I. Vitebskiy, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609:1-17 (2001).
[CrossRef]

Phys. Rev. Lett.

Z. Wang, Y. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal," Phys. Rev. Lett. 100, 01390501-01390504 (2008).
[CrossRef]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, "One-Way Electromagnetic Waveguide Formed at the Interface between a Plasmonic Metal under a Static Magnetic Field and a Photonic Crystal," Phys. Rev. Lett. 100, 02390201-02390204 (2008).
[CrossRef]

Other

A. K. Zvezdin, V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials, (Institute of Physics Pub., Bristol, 1997).
[CrossRef]

P. Yeh, Optical Waves in Layered Media, (Wiley Interscience, New York, 1988).

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Figures (5)

Fig. 1.
Fig. 1.

Geometry of the problem: (a) ternary MPC and (b) microcavity-type nonreciprocal Fabry-Perot resonator. Magnetization M is along . Magnetic layers are in green.

Fig. 2.
Fig. 2.

Photonic band structure (a) and its nonreciprocity (b) in 1D MPC with three-layer elementary cell formed from layers of SiO2 (a 1=220 nm, ε 1=2.1), Ce:BIG (a 2=130 nm, ε 2=6.25, Δ2=0.06) and Ta2O5 (a 3=110 nm, ε 3=4.7), respectively.

Fig. 3.
Fig. 3.

Photonic band structure of nonmagnetic (a) and nonreciprocity of magnetic (b) ternary 1D PCs, respectively. The structure parameters are the same as in Fig. 2(a) and the angle of incidence is 30°. Thin blue lines correspond to the exact result and the thick red line is obtained from the approximate expression Eq. (5). Insets in (b) show the nonreciprocity in the proximity of the band edges.

Fig. 4.
Fig. 4.

Transmittance of nonmagnetic (a) and nonreciprocal transmittance of magnetic (b) ternary 1D PCs, respectively. Figure (a) shows the whole bandgap region, while (b) shows the transmittance in the proximity of the short-wavelength band edge. The structure parameters are the same as in Fig. 2(a) and the angle of incidence is 30°.

Fig. 5.
Fig. 5.

Transmittance of nonmagnetic (a) and nonreciprocal transmittance of magnetic (b) microcavities, respectively. Figure (a) shows the whole bandgap region, while (b) shows the proximity of the resonance only. The structure consists of a single Ce:BIG layer (aM =155 nm, εM =6.25, Δ M =0.06) sandwiched between two Bragg mirrors formed from a sequence often SiO2 (a 1=267 nm, ε 1=2.1) and Ta2O5 (a 2=179 nm, ε 2=4.7) layers. The asymmetry is achieved through bounding the Ce:BIG layer with different layers of the Bragg mirrors, i.e. SiO2 and Ta2O5 for left and right boundaries, respectively. The angle of incidence is 30°.

Equations (8)

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

ρ a = ρ 0 a + i k x ρ 1 a ,
ρ 0 r = ( d 1 ε 1 k 2 z ) 2 ( d 2 ε 1 k 1 z ) 2 + k x 2 ( d 1 Δ 2 d 2 Δ 1 ) 2 ( d 1 ε 2 k 2 z + d 2 ε 1 k 1 z ) 2 + k x 2 ( d 1 Δ 2 d 2 Δ 1 ) 2 ,
ρ 0 t = 2 d 2 ε 1 k 1 z ( d 1 ε 2 k 2 z + d 2 ε 1 k 1 z ) ( d 1 ε 2 k 2 z + d 2 ε 1 k 1 z ) 2 + k x 2 ( d 1 Δ 2 d 2 Δ 1 ) 2 ,
ρ 1 t = ρ 1 r = ( d 1 Δ 1 d 2 Δ 2 ) 2 d 2 ε 1 k 1 z ( d 1 ε 1 k 2 z + d 2 ε 1 k 1 z ) 2 + k x 2 ( d 1 Δ 2 d 2 Δ 1 ) 2 ,
[ H y + H y ] j = M ̂ ij [ H y + H y ] i = d 2 ε j k zj [ F j * + F i F j * F i * F j F i F j + F i * ] [ H y + H y + ] i ,
cos ( K M a ) cos ( K 0 a ) + ε 3 k 3 z ε 1 k 1 z k x k 2 x Δ 2 ε 2 ( k 1 z 2 ε 1 2 k 3 z 2 ε 3 2 ) sin ( k 1 z a 1 ) sin ( k 2 z a 2 ) sin ( k 3 z a 3 ) ,
δ ε 3 k 3 z ε 1 k 1 z k x k 2 z Δ 2 ε 2 ( k 1 z 2 ε 1 2 k 3 z 2 ε 3 2 ) sin ( k 1 z a 1 ) sin ( k 2 z a 2 ) sin ( k 3 z a 3 ) a sin ( K 0 a ) .
ϕ M = ϕ 0 + δ ϕ 1 ( Δ 1 k x ) + δ ϕ 2 ( Δ 2 , k x ) = 2 πn ,

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