Abstract

This report presents the first three-dimensional characterization of nonreciprocal phase shifts in magneto-photonic crystal (MPC) slab waveguides. We model MPC waveguides using a three-dimensional finite element method with curvilinear tetrahedral edge elements. This study investigates the dependence of nonreciprocal phase shifts on the width and the thickness of the waveguides, and we investigate the dependence of losses on the air hole depth, leading to a guideline for the design of optical isolators. Simulations show that waveguides with reduced width and deep air holes exhibit high nonreciprocal phase shifts and low losses. The study also shows that, compared with two-dimensional calculations, nonreciprocal phase shifts express key similarities, although the frequencies of the guided modes shift.

© 2005 Optical Society of America

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Appl. Opt. (1)

Appl. Phys. B (1)

S. G. Johnson, M. L. Povinelli, M. Solja�?i�?, A. Karalis, S. Jacobs, and J. D. Joannopoulos, �??Roughness losses and volume-current methods in photonic-crystal waveguides,�?? Appl. Phys. B 81, 283-293 (2005).
[CrossRef]

Appl. Phys. Lett. (4)

J. P. Krumme and P. Hansen, �??A new type of magnetic domain wall in nearly compensated Ga-substituted YIG,�?? Appl. Phys. Lett. 22, 312-314 (1973).
[CrossRef]

L. Wilkens. D. Trager, H. Dötsch, A. F. Popkov, and A. M. Alekseev, �??Nonreciprocal phase shift of TE modes induced by compensation wall in a magneto-optic rib waveguide,�?? Appl. Phys. Lett. 79, 4292-4294 (2001).
[CrossRef]

L. C. Andreani and M. Agio, �??Intrinsic diffraction losses in photonic crystal waveguides with line defects,�?? Appl. Phys. Lett. 82, 2011-2013 (2003).
[CrossRef]

J. Fujita, M. Levy, R. M. Osgood Jr., L. Wilkens, and H. Dötsch, �??Waveguide optical isolator based on Mach-Zehnder interferometer,�?? Appl. Phys. Lett. 76, 2158-2160 (2000).
[CrossRef]

Eur. Phys. J. (1)

A. K. Zvezdin and V. I. Belotelov, �??Magnetooptical properties of two dimensional photonic crystals,�?? Eur. Phys. J. B 37, 479-487 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Notomi, A. Shinya, K. Yamada, J. Tahashi, C. Takahashi, and I. Yokohama, �??Structural tuning of guideing modes of line-defect waveguides of silicon-on insulator photonic crystal slabs,�?? IEEE J. Quantum Electron. 38, 736-742 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Levy, �??The on-chip integration of magnetooptic waveguide isolators,�?? IEEE J. Sel. Top. Quantum Electron. 8, 1300-1306 (2001).

IEEE Microwave Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, �??General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media,�?? IEEE Microwave Guided Wave Lett. 8, 223-225 (1998).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

W. Zaets and K. Ando, �??Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,�?? IEEE Photonics Technol. Lett. 11, 1012-1014 (1999).
[CrossRef]

S. K. Mondal and B. J. H. Stadler, �??Novel designs for integrating YIG/air photonic crystal slab polarizers with waveguide Faraday rotators,�?? IEEE Photonics Technol. Lett. 17, 127-129 (2005).
[CrossRef]

N. Kono and M. Koshiba, �??General finite-element modeling of 2-D magnetophotonic crystal waveguides,�?? IEEE Photonics Technol. Lett. 17, 1432-1434 (2005).
[CrossRef]

IEEE Trans. Magn. (2)

J. S. Wang and N. Ida, �??Curvilinear and higher order 'edge' finite elements in electromagnetic field computation,�?? IEEE Trans. Magn. 29, 1491-1494 (1993).
[CrossRef]

J. F. Lee, D. K. Sun, and Z. J. Cendes, �??Tangential vector finite elements for electromagnetic field,�?? IEEE Trans. Magn. 27, 4032-4035 (1991).
[CrossRef]

Indium Phosphide and Related Mater. 1999 (1)

M. Takenaka and Y. Nakano, �??Proposal of a Novel Semiconductor Optical Waveguide Isolator,�?? in Proceedings of IEEE Eleventh International Conference on Indium Phosphide and Related Materials (Institute of Electrical and Electronics Engineers, Davos, Switzerland, 1999), pp. 289-292.

J. Appl. Phys. (1)

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

J. Fut. Gen. Comput. Syst. (1)

O. Schenk, and K. Gärtner, �??Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO,�?? J. Fut. Gen. Comput. Syst. 20, 475-487 (2004).
[CrossRef]

J. Lightwave Technol. (6)

J. Magn. Magn. Mater. (2)

Y. Ikezawa, K. Nishimura, H. Uchida, and M. Inoue, �??Preparation of two-dimensional magneto-photonic crystals of bismuth substitute yttrium iron garnet materials,�?? J. Magn. Magn. Mater. 272-276, 1690-1691, (2004).
[CrossRef]

A. A. Fedyanin, O. A. 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, 256-269 (2004).
[CrossRef]

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

J. Quantum Electron. (1)

N. Bahlmann, M. Lohmeyer, H. Dötch, and P. Hertel, �??Finite-element analysis of nonreciprocal phase shift for TE modes in magnetooptic rib waveguides with a compensation wall,�?? J. Quantum Electron. 35, 250- 253 (1999).
[CrossRef]

Opt. Commun. (3)

H. Kato, T. Matsushita, A. Takayama, M. Egawa, K. Nishimura, and M. Inoue, �??Effect of optical losses on optical and magneto-optical properties of one-dimensional magnetophotonic crystals for use in optical isolator devices,�?? Opt. Commun. 219, 271-276 (2003).
[CrossRef]

A. A. Jalali and A. T. Friberg, �??Faraday rotation in two-dimensional magneto-optic photonic crystal,�?? Opt. Commun. 253, 145-150 (2005).
[CrossRef]

N. Bahlmann, M. Lohmeyer, O. Zhuromskyy, H. Dötsch, and P. Hertel, �??Nonreciprocal coupled waveguides for integrated optical isolators and circulators for TM-modes,�?? Opt. Commun. 161, 330-337 (1999).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. B (3)

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, �??Nature of lossy Bloch states in polaritonic photonic crystals,�?? Phys. Rev. B 69, 195111 (2004).
[CrossRef]

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

T. Ochiai and K. Sakoda, �??Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,�?? Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, �??Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,�?? Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

SIAM J. Scient. Comput. (1)

G. Karypis and V. Kumar, �??A fast and high quality multilevel scheme for partitioning irregular graphs,�?? SIAM J. Scient. Comput. 1, 359-392 (1998).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

(a) Schematic representation, (b) top cross sectional view, and (c) side cross sectional view of a 2-D MPC-slab waveguide with a compensation wall. The parameters used in sec. 2 are the hole radius r = 0.25a and the Faraday rotation ΘF ≈ 3000°/cm; the refractive indices of the core layer and the cladding layer are n = 2.3 and 1.45, respectively.

Fig. 2.
Fig. 2.

(a) Nonreciprocal phase shifts and (b) dispersion curves for an MPC waveguide with air holes of infinite depth, a constant slab thickness, h 2 = 0.8a, and different waveguide widths, W. In the calculation of (a), the lattice constant a is adjusted to set the wavelength as λ = 1.3 μm.

Fig. 3.
Fig. 3.

(a), (b), (c), and (d) Cross-sectional distributions of the x-component of the magnetic fields on the y = 0 plane (shown in the upper row) and the x = 0 plane (shown in the lower row) for each parameter set identified in Fig. 2. (b). (e) and (f) Normalized magnetic field amplitude |Hx | at x = 0 and z = 0, respectively, on the y = 0 plane.

Fig. 4.
Fig. 4.

(a) Nonreciprocal phase shifts and (b) dispersion curves for an MPC waveguide with air holes of infinite depth, a constant waveguide width, W = 0.56W 0, and different slab thicknesses, h 2. h 2 → ∞ denotes the results of 2-D calculations.

Fig. 5.
Fig. 5.

(a) Losses and (b) dispersion curves for an MPC waveguide with air holes of infinite depth, the slab thickness h 2 = 0.8a, the waveguide width W = 0.56W 0, and upper cladding layers of different thicknesses, h 1.

Fig. 6.
Fig. 6.

(a) Losses and (b) dispersion curves for an MPC waveguide with the upper cladding thickness h 1 = 0.3a, the slab thickness h 2 = 0.8a, the waveguide width W = 0.56W 0, and air holes of different depths.

Fig. 7.
Fig. 7.

(a) Nonreciprocal phase shifts, figures of merit, and (b) dispersion curves for an MPC waveguide with the waveguide width W = 0.56W 0 and the slab thickness h 2 = 0.8a. The inset in (b) is a conceptual diagram representing the mechanism of the increments in nonreciprocal phase shifts near the light line.

Equations (14)

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

[ ε ] = [ n 2 0 0 0 n 2 0 n 2 ] ,
ξ 2 n Θ F / k 0 ,
× ( [ p ] × Φ ) k 0 2 [ q ] Φ = 0
[ p ] = [ μ ] PML 1 , [ q ] = [ ε ] PML for Φ = E
[ p ] = [ ε ] PML 1 , [ q ] = [ μ ] PML for Φ = H
Φ = ϕ ( x , y , z ) exp ( jβz ) ,
ϕ = { N } T { ϕ } e ,
[ K ] { ϕ } k 0 2 [ M ] { ϕ } = i = 1 2 Σ e i { N } · n × ( [ p ] × { ϕ } i ) d Γ i
[ K ] = Σ e ( × { N } ) · ( [ p ] × { N } ) T d Ω
[ M ] = Σ e { N } · [ p ] · { N } T d Ω
[ K ˜ ] { ϕ ˜ } k 0 2 [ M ˜ ] { ϕ ˜ } = { 0 }
{ ϕ ˜ } = [ { ϕ } 0 { ϕ } 1 ]
[ A ˜ ] = [ [ A ] 00 [ A ] 01 + [ A ] 02 [ A ] 10 + [ A ] 20 [ A ] 11 + [ A ] 22 + [ A ] 12 + [ A ] 21 ]
Im ( β ) = Im ( ω ) / v g ,

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