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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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]

2005 (10)

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]

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]

A. A. Jalali and A. T. Friberg , “ Faraday rotation in two-dimensional magneto-optic photonic crystal ,” Opt. Commun.   253 , 145 – 150 ( 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]

Z. Wang and S. Fan , “ Optical circulators in two-dimensional magneto-optical photonic crystals ,” Opt. Lett. 1989 – 1991 ( 2005 ).
[Crossref] [PubMed]

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]

D. Gerace and L. C. Andreani , “ Low-loss guided modes in photonic crystal waveguides ,” Opt. Express.   13 , 4939 – 4951 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4939.
[Crossref] [PubMed]

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

M. Skorobogatiy , G. Bégin , and A. Talneau , “ Statistical analysis of geometrical imperfections from the images of 2D photonic crystals ,” Opt. Express   13 , 2487 – 2502 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2487.
[Crossref] [PubMed]

A. A. Jalali and A. T. Friberg , “ Faraday rotation in a two-dimensional photonic crystal with a magneto-optic defect ,” Opt. Lett.   30 , 1213 – 1215 ( 2005 ).
[Crossref] [PubMed]

2004 (9)

W. J. Kim and J. D. O’Brien , “ Optimization of a two-dimensional photonic-crystal waveguide branch by simulated annealing and the finite-element method ,” J. Opt. Soc. Am. B   21 , 289 – 295 ( 2004 ).
[Crossref]

D. Gerace and L. C. Andreani , “ Disorder-induced losses in photonic crystal waveguides with line defects ,” Opt. Lett.   29 , 1897 – 1899 ( 2004 ).
[Crossref] [PubMed]

N. Kono and Y. Tsuji , “ A novel finite-element method for nonreciprocal magneto-photonic crystal waveguides ,” J. Lightwave Technol.   22 , 1741 – 1747 ( 2004 ).
[Crossref]

Y. Tanaka , T Asano , R. Hatsuta , and S. Noda , “ Analysis of a line-defect waveguide on a silicon-on-insulator two-dimensional photonic-crystal slab ,” J. Lightwave Technol.   22 , 2787 – 2792 ( 2004 ).
[Crossref]

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. 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]

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. K. Zvezdin and V. I. Belotelov , “ Magnetooptical properties of two dimensional photonic crystals ,” Eur. Phys. J. B   37 , 479 – 487 ( 2004 ).
[Crossref]

O Schenk and K. Gärtner , “ Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO ,” J. Fut. Gen. Comput. Syst.   20 , 475 – 487 ( 2004 ).
[Crossref]

2003 (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]

L. C. Andreani and M. Agio , “ Intrinsic diffraction losses in photonic crystal waveguides with line defects ,” Appl. Phys. Lett.   82 , 2011 – 2013 ( 2003 ).
[Crossref]

W. Bogaerts , P. Bienstman , and R. Baets , “ Scattering at sidewall roughness in photonic crystal slabs ,” Opt. Lett.   28 , 689 – 691 ( 2003 ).
[Crossref] [PubMed]

2002 (3)

2001 (4)

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

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

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

L. Wilkens , D. Trager , H. DÖtsch , A. F. opkov , 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]

2000 (3)

1999 (5)

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]

A. Sakaguchi and N. Sugimoto , “ Transmisson properties of multilayer films composed of magneto-optical and dielectric materials ,” J. Lightwave Technol.   17 , 1087 – 1092 ( 1999 ).
[Crossref]

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]

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]

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

1998 (2)

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]

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]

1993 (1)

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]

1991 (1)

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]

1984 (1)

1973 (1)

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]

Abe, M.

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

Agio, M.

L. C. Andreani and M. Agio , “ Intrinsic diffraction losses in photonic crystal waveguides with line defects ,” Appl. Phys. Lett.   82 , 2011 – 2013 ( 2003 ).
[Crossref]

Aktsipetrov, O. A.

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]

Alekseev, A. M.

L. Wilkens , D. Trager , H. DÖtsch , A. F. opkov , 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]

Ando, K.

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]

Andreani, L. C.

D. Gerace and L. C. Andreani , “ Low-loss guided modes in photonic crystal waveguides ,” Opt. Express.   13 , 4939 – 4951 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4939.
[Crossref] [PubMed]

D. Gerace and L. C. Andreani , “ Disorder-induced losses in photonic crystal waveguides with line defects ,” Opt. Lett.   29 , 1897 – 1899 ( 2004 ).
[Crossref] [PubMed]

L. C. Andreani and M. Agio , “ Intrinsic diffraction losses in photonic crystal waveguides with line defects ,” Appl. Phys. Lett.   82 , 2011 – 2013 ( 2003 ).
[Crossref]

Arai, K.

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

Arentoft, J.

Asano, T

Baets, R.

Bahlmann, N.

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]

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]

Baryshev, A. V.

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]

Bégin, G.

Belotelov, V. I.

A. K. Zvezdin and V. I. Belotelov , “ Magnetooptical properties of two dimensional photonic crystals ,” Eur. Phys. J. B   37 , 479 – 487 ( 2004 ).
[Crossref]

Bienstman, P.

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]

W. Bogaerts , P. Bienstman , and R. Baets , “ Scattering at sidewall roughness in photonic crystal slabs ,” Opt. Lett.   28 , 689 – 691 ( 2003 ).
[Crossref] [PubMed]

Bogaerts, W.

Cao, J. R.

Cendes, Z. J.

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]

Chew, W. C.

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]

Dapkus, P. D.

Doll, T.

DÖtch, H.

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]

DÖtsch, H.

L. Wilkens , D. Trager , H. DÖtsch , A. F. opkov , 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]

J. Fujita , M. Levy , R. M. Osgood , L. Wilkens , and H. DÖtsch , “ Waveguide optical isolator based on Mach-Zehnder interferometer ,” Appl. Phys. Lett.   76 , 2158 – 2160 ( 2000 ).
[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]

Egawa, M.

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]

Fan, S.

Z. Wang and S. Fan , “ Optical circulators in two-dimensional magneto-optical photonic crystals ,” Opt. Lett. 1989 – 1991 ( 2005 ).
[Crossref] [PubMed]

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]

Fedyanin, A. A.

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]

Figotin, A.

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

Friberg, A. T.

A. A. Jalali and A. T. Friberg , “ Faraday rotation in two-dimensional magneto-optic photonic crystal ,” Opt. Commun.   253 , 145 – 150 ( 2005 ).
[Crossref]

A. A. Jalali and A. T. Friberg , “ Faraday rotation in a two-dimensional photonic crystal with a magneto-optic defect ,” Opt. Lett.   30 , 1213 – 1215 ( 2005 ).
[Crossref] [PubMed]

Fujii, T.

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

Fujita, J.

J. Fujita , M. Levy , R. M. Osgood , L. Wilkens , and H. DÖtsch , “ Waveguide optical isolator based on Mach-Zehnder interferometer ,” Appl. Phys. Lett.   76 , 2158 – 2160 ( 2000 ).
[Crossref]

Gärtner, K.

O Schenk and K. Gärtner , “ Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO ,” J. Fut. Gen. Comput. Syst.   20 , 475 – 487 ( 2004 ).
[Crossref]

Gerace, D.

D. Gerace and L. C. Andreani , “ Low-loss guided modes in photonic crystal waveguides ,” Opt. Express.   13 , 4939 – 4951 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-13-4939.
[Crossref] [PubMed]

D. Gerace and L. C. Andreani , “ Disorder-induced losses in photonic crystal waveguides with line defects ,” Opt. Lett.   29 , 1897 – 1899 ( 2004 ).
[Crossref] [PubMed]

Granovsky, A. B.

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]

Hansen, P.

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]

Hatsuta, R.

Hertel, P.

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]

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]

Huang, K. C.

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]

Hughes, S.

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]

Ida, N.

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]

Ikezawa, Y.

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]

Inoue, M.

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]

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]

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]

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

Jacobs, S.

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

Jalali, A. A.

A. A. Jalali and A. T. Friberg , “ Faraday rotation in a two-dimensional photonic crystal with a magneto-optic defect ,” Opt. Lett.   30 , 1213 – 1215 ( 2005 ).
[Crossref] [PubMed]

A. A. Jalali and A. T. Friberg , “ Faraday rotation in two-dimensional magneto-optic photonic crystal ,” Opt. Commun.   253 , 145 – 150 ( 2005 ).
[Crossref]

Jiang, X.

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]

Joannopoulos, J. D.

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

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]

Johnson, S. G.

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

Karalis, A.

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

Karypis, G.

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]

Kato, H.

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]

Khanikaev, A. B.

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]

Kim, C.

Kim, W. J.

Kobayashi, D.

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]

Kono, N.

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

N. Kono and Y. Tsuji , “ A novel finite-element method for nonreciprocal magneto-photonic crystal waveguides ,” J. Lightwave Technol.   22 , 1741 – 1747 ( 2004 ).
[Crossref]

Koshiba, M.

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

Kristensen, M.

Krumme, J. P.

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]

Kumar, V.

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]

Lee, J. F.

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]

Levy, M.

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

M. J. 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]

J. Fujita , M. Levy , R. M. Osgood , L. Wilkens , and H. DÖtsch , “ Waveguide optical isolator based on Mach-Zehnder interferometer ,” Appl. Phys. Lett.   76 , 2158 – 2160 ( 2000 ).
[Crossref]

Lidorikis, E.

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]

Lohmeyer, M.

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]

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]

Loncar, M.

Matsushita, T.

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]

Mondal, S. K.

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]

Nakano, Y.

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 .

Negami, T.

Nelson, K. A.

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]

Nishimura, K.

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]

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]

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]

Noda, S.

Notomi, M.

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]

O’Brien, J. D.

Ochiai, T.

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

Okamura, Y.

opkov, A. F.

L. Wilkens , D. Trager , H. DÖtsch , A. F. opkov , 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]

Osgood, R. M.

J. Fujita , M. Levy , R. M. Osgood , L. Wilkens , and H. DÖtsch , “ Waveguide optical isolator based on Mach-Zehnder interferometer ,” Appl. Phys. Lett.   76 , 2158 – 2160 ( 2000 ).
[Crossref]

M. J. 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]

Povinelli, M. L.

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

Ramunno, L.

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]

Sakaguchi, A.

Sakoda, K.

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

Schenk, O

O Schenk and K. Gärtner , “ Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO ,” J. Fut. Gen. Comput. Syst.   20 , 475 – 487 ( 2004 ).
[Crossref]

Scherer, A.

Shinya, A.

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]

Sipe, J. E.

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]

Skorobogatiy, M.

Soljacic, M.

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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]

SØndergaard, T.

Stadler, B. J. H.

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]

Stapleton, A.

Steel, M. J.

Sugimoto, N.

Sun, D. K.

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]

Tahashi, J.

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]

Takahashi, C.

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]

Takayama, A.

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]

Takenaka, M.

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 .

Talneau, A.

Tanaka, Y.

Teixeira, F. L.

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]

Trager, D.

L. Wilkens , D. Trager , H. DÖtsch , A. F. opkov , 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]

Tsuji, Y.

Uchida, H.

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]

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]

Vinogradov, A. P.

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]

Vitebsky, I.

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

Vuckovic, J.

Wang, J. S.

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]

Wang, Z.

Z. Wang and S. Fan , “ Optical circulators in two-dimensional magneto-optical photonic crystals ,” Opt. Lett. 1989 – 1991 ( 2005 ).
[Crossref] [PubMed]

Wilkens, L.

L. Wilkens , D. Trager , H. DÖtsch , A. F. opkov , 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]

J. Fujita , M. Levy , R. M. Osgood , L. Wilkens , and H. DÖtsch , “ Waveguide optical isolator based on Mach-Zehnder interferometer ,” Appl. Phys. Lett.   76 , 2158 – 2160 ( 2000 ).
[Crossref]

Yamada, K.

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]

Yamamoto, S.

Yokohama, I.

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]

Young, J. F.

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]

Zaets, W.

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]

Zhuromskyy, O.

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]

Zvezdin, A. K.

A. K. Zvezdin and V. I. Belotelov , “ Magnetooptical properties of two dimensional photonic crystals ,” Eur. Phys. J. B   37 , 479 – 487 ( 2004 ).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

S. G. Johnson , M. L. Povinelli , M. Soljacic , 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. opkov , 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 , L. Wilkens , and H. DÖtsch , “ Waveguide optical isolator based on Mach-Zehnder interferometer ,” Appl. Phys. Lett.   76 , 2158 – 2160 ( 2000 ).
[Crossref]

Eur. Phys. J. B (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]

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 ).
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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)

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[ 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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