Abstract

By introducing nonreciprocal phase shifts into microresonators, we propose new designs for the miniaturization of optical waveguide isolators and circulators. We present detailed design procedures, and numerically demonstrate the operation of these magneto-optical devices. The device sizes can be reduced down to several tens of micrometers. The nonreciprocal function of these devices is due to nonreciprocal resonance shifts. Next, the operation bandwidth can be expanded by increasing the number of resonators (the filter order). This is demonstrated by comparing the characteristics of a single-resonator structure with those of a three-resonator structure. This paper furthermore presents the nonreciprocal characteristics of three-dimensional resonators with finite heights, leading to a guideline for the design of nonreciprocal optical circuits. This involves a demonstration of how the resonators with selected parameters are practical for miniaturized nonreciprocal circuits.

© 2007 Optical Society of America

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References

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  1. Y. Okamura, T. Negami, and S. Yamamoto, "Integrated optical isolator and circulator using nonreciprocal phase shifters: a proposal," Appl. Opt. 23, 1886-1889 (1984).
    [CrossRef] [PubMed]
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    [CrossRef]
  3. 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]
  4. M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
    [CrossRef]
  5. 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," IEEE J. Quantum Electron. 35, 250-253 (1999).
    [CrossRef]
  6. H. Yokoi, T. Mizumoto, and Y. Shoji, "Optical nonreciprocal devices with a silicon guiding layer fabricated by wafer bonding," Appl. Opt. 42, 6605-6612 (2003).
    [CrossRef] [PubMed]
  7. N. Kono and Y. Tsuji, "A novel finite-element method for nonreciprocal magneto-photonic crystal waveguides," J. Lightwave Technol. 22, 1741-1747 (2004).
    [CrossRef]
  8. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Express 14, 11128-11141 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. N. N. Dadoenkova, I. L. Lyubchanskii, M. I. Lyubchanskii, E. A. Shapovalov, A. E. Zabolotin, and T. Rasing, "Influence of magnetic field on nonlinear magneto-optical diffraction on two-dimensional hexagonal magnetic bubble lattice," J. Opt. Soc. Am. B 22, 215-219 (2005).
    [CrossRef]
  11. F. L. Teixeira and W. C. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).
    [CrossRef]
  12. M. Shamonin and P. Hertel, "Analysis of nonreciprocal mode propagation in magneto-optic rib-waveguide structures with the spectral-index method," Appl. Opt. 33, 6415-6421 (1994).
    [CrossRef] [PubMed]
  13. N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F.-J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magnetooptic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
    [CrossRef]
  14. C.-S. Ma, Y.-Z, Xu, X. Yan, Z.-K. Qin, and X.-Y. Wang, "Effect of ring spacing on spectral response of parallel-cascaded mircroring resonator arrays," Opt. Quantum Electron. 37, 561-574 (2005).
    [CrossRef]
  15. R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
    [CrossRef]
  16. A. Melloni, "Synthesis of a parallel-coupled ring-resonator filter," Opt. Lett. 26, 917-919 (2001).
    [CrossRef]
  17. O. Schwelb and I. Frigyes, "A design for a high finesse parallel-coupled microring resonator filter," Microw. Opt. Technol. Lett. 38, 125-129 (2003).
    [CrossRef]
  18. M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
    [CrossRef]
  19. H. Yokoi and T. Mizumoto, "Proposed configuration of integrated optical isolator employing wafer-direct bounding technique," Electron. Lett. 33, 1787-1788 (1997).
    [CrossRef]
  20. T. Uno and S. Noge, "Growth of magneto-optic Ce:YIG thin films on amorphous silica substrates," J. Eur. Ceramic Soc. 21, 1957-1960 (2001).
    [CrossRef]
  21. M. Koshiba and Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightwave Technol. 18, 737-743 (2000).
    [CrossRef]

2006 (1)

2005 (2)

2004 (1)

2003 (2)

O. Schwelb and I. Frigyes, "A design for a high finesse parallel-coupled microring resonator filter," Microw. Opt. Technol. Lett. 38, 125-129 (2003).
[CrossRef]

H. Yokoi, T. Mizumoto, and Y. Shoji, "Optical nonreciprocal devices with a silicon guiding layer fabricated by wafer bonding," Appl. Opt. 42, 6605-6612 (2003).
[CrossRef] [PubMed]

2002 (1)

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

2001 (2)

A. Melloni, "Synthesis of a parallel-coupled ring-resonator filter," Opt. Lett. 26, 917-919 (2001).
[CrossRef]

T. Uno and S. Noge, "Growth of magneto-optic Ce:YIG thin films on amorphous silica substrates," J. Eur. Ceramic Soc. 21, 1957-1960 (2001).
[CrossRef]

2000 (2)

M. Koshiba and Y. Tsuji, "Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems," J. Lightwave Technol. 18, 737-743 (2000).
[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]

1999 (2)

M. Inoue, K. Arai, T. Fujii, and M. Abe, "One-dimensional magnetophotonic crystals," J. Appl. Phys. 85, 5768-5770 (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," IEEE J. Quantum Electron. 35, 250-253 (1999).
[CrossRef]

1998 (1)

1997 (3)

F. L. Teixeira and W. C. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).
[CrossRef]

M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
[CrossRef]

H. Yokoi and T. Mizumoto, "Proposed configuration of integrated optical isolator employing wafer-direct bounding technique," Electron. Lett. 33, 1787-1788 (1997).
[CrossRef]

1995 (1)

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

1994 (1)

1984 (1)

1979 (1)

J. W. Nielsen, "Magnetic bubble materials," Ann. Rev. Mater. Sci. 9, 87-121 (1979).
[CrossRef]

Abe, M.

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

Absil, P. P.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Arai, K.

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

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," IEEE J. Quantum Electron. 35, 250-253 (1999).
[CrossRef]

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F.-J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magnetooptic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[CrossRef]

Bakhru, H.

M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
[CrossRef]

Calhoun, L. C.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Chandrasekhara, V.

Chew, W. C.

F. L. Teixeira and W. C. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).
[CrossRef]

Dadoenkova, N. N.

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," IEEE J. Quantum Electron. 35, 250-253 (1999).
[CrossRef]

Dötsch, H.

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]

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F.-J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magnetooptic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[CrossRef]

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

Erdmann, A.

Frigyes, I.

O. Schwelb and I. Frigyes, "A design for a high finesse parallel-coupled microring resonator filter," Microw. Opt. Technol. Lett. 38, 125-129 (2003).
[CrossRef]

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, Jr., L. Wilkens, and H. Dötsch, "Waveguide optical isolator based on Mach-Zehnder interferometer," Appl. Phys. Lett. 76, 2158-2160 (2000).
[CrossRef]

Gather, B.

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

Gerhardt, R.

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F.-J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magnetooptic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[CrossRef]

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

Grover, R.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Hertel, P.

Hertel, R.

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

Ho, P.-T.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Hryniewicz, J. V.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Ibrahim, T. A.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Inoue, M.

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

Johnson, F. G.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Kakihara, K.

Kono, N.

Koshiba, M.

Kumar, A.

M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
[CrossRef]

Lehmann, R.

Levy, M.

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]

M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
[CrossRef]

Lohmeyer, M.

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," IEEE J. Quantum Electron. 35, 250-253 (1999).
[CrossRef]

Lyubchanskii, I. L.

Lyubchanskii, M. I.

Ma, C.-S.

C.-S. Ma, Y.-Z, Xu, X. Yan, Z.-K. Qin, and X.-Y. Wang, "Effect of ring spacing on spectral response of parallel-cascaded mircroring resonator arrays," Opt. Quantum Electron. 37, 561-574 (2005).
[CrossRef]

Melloni, A.

Mizumoto, T.

H. Yokoi, T. Mizumoto, and Y. Shoji, "Optical nonreciprocal devices with a silicon guiding layer fabricated by wafer bonding," Appl. Opt. 42, 6605-6612 (2003).
[CrossRef] [PubMed]

H. Yokoi and T. Mizumoto, "Proposed configuration of integrated optical isolator employing wafer-direct bounding technique," Electron. Lett. 33, 1787-1788 (1997).
[CrossRef]

Negami, T.

Nielsen, J. W.

J. W. Nielsen, "Magnetic bubble materials," Ann. Rev. Mater. Sci. 9, 87-121 (1979).
[CrossRef]

Niemöller, M.

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

Noge, S.

T. Uno and S. Noge, "Growth of magneto-optic Ce:YIG thin films on amorphous silica substrates," J. Eur. Ceramic Soc. 21, 1957-1960 (2001).
[CrossRef]

Okamura, Y.

Osgood, R. M.

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]

M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
[CrossRef]

Rasing, T.

Saitoh, K.

Salz, D.

Schröteler, F.-J.

Schwelb, O.

O. Schwelb and I. Frigyes, "A design for a high finesse parallel-coupled microring resonator filter," Microw. Opt. Technol. Lett. 38, 125-129 (2003).
[CrossRef]

Shamonin, M.

Shapovalov, E. A.

Shoji, Y.

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).
[CrossRef]

Tsuji, Y.

Uno, T.

T. Uno and S. Noge, "Growth of magneto-optic Ce:YIG thin films on amorphous silica substrates," J. Eur. Ceramic Soc. 21, 1957-1960 (2001).
[CrossRef]

Van, V.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightwave Technol. 20, 900-905 (2002).
[CrossRef]

Wallenhorst, M.

N. Bahlmann, V. Chandrasekhara, A. Erdmann, R. Gerhardt, P. Hertel, R. Lehmann, D. Salz, F.-J. Schröteler, M. Wallenhorst, and H. Dötsch, "Improved design of magnetooptic rib waveguides for optical isolators," J. Lightwave Technol. 16, 818-823 (1998).
[CrossRef]

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

Wilkens, L.

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]

Yamamoto, S.

Yokoi, H.

H. Yokoi, T. Mizumoto, and Y. Shoji, "Optical nonreciprocal devices with a silicon guiding layer fabricated by wafer bonding," Appl. Opt. 42, 6605-6612 (2003).
[CrossRef] [PubMed]

H. Yokoi and T. Mizumoto, "Proposed configuration of integrated optical isolator employing wafer-direct bounding technique," Electron. Lett. 33, 1787-1788 (1997).
[CrossRef]

Zabolotin, A. E.

Ann. Rev. Mater. Sci. (1)

J. W. Nielsen, "Magnetic bubble materials," Ann. Rev. Mater. Sci. 9, 87-121 (1979).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

M. Levy, R. M. Osgood, Jr., A. Kumar, and H. Bakhru, "Epitaxial liftoff of thin oxide layers: Yttrium iron garnets onto GaAs," Appl. Phys. Lett. 71, 2617-2619 (1997).
[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]

Electron. Lett. (1)

H. Yokoi and T. Mizumoto, "Proposed configuration of integrated optical isolator employing wafer-direct bounding technique," Electron. Lett. 33, 1787-1788 (1997).
[CrossRef]

IEEE 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," IEEE J. Quantum Electron. 35, 250-253 (1999).
[CrossRef]

IEEE Microw. Guided Wave Lett. (1)

F. L. Teixeira and W. C. Chew, "PML-FDTD in cylindrical and spherical grids," IEEE Microw. Guided Wave Lett. 7, 285-287 (1997).
[CrossRef]

J. Appl. Phys. (2)

M. Wallenhorst, M. Niemöller, H. Dötsch, R. Hertel, R. Gerhardt, and B. Gather, "Enhancement of the nonreciprocal magneto-optic effect of TM modes using iron garnet double layers with opposite Faraday rotation," J. Appl. Phys. 77, 2902-2905 (1995).
[CrossRef]

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

J. Eur. Ceramic Soc. (1)

T. Uno and S. Noge, "Growth of magneto-optic Ce:YIG thin films on amorphous silica substrates," J. Eur. Ceramic Soc. 21, 1957-1960 (2001).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. Soc. Am. B (1)

Microw. Opt. Technol. Lett. (1)

O. Schwelb and I. Frigyes, "A design for a high finesse parallel-coupled microring resonator filter," Microw. Opt. Technol. Lett. 38, 125-129 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

C.-S. Ma, Y.-Z, Xu, X. Yan, Z.-K. Qin, and X.-Y. Wang, "Effect of ring spacing on spectral response of parallel-cascaded mircroring resonator arrays," Opt. Quantum Electron. 37, 561-574 (2005).
[CrossRef]

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Figures (20)

Fig. 1.
Fig. 1.

Schematic representation of a nonreciprocal microresoantor coupled with two straight waveguides.

Fig. 2.
Fig. 2.

Dependence of the radiation losses per circulation, γ, on disk radius R D, for the TM mode at the wavelength of λ = 1300 nm.

Fig. 3.
Fig. 3.

Mode profiles for different radii, R D = 1, 2, 3, 4, and 5 μm.

Fig. 4.
Fig. 4.

Dependence of Δφ, the nonreciprocal phase shifts per circulation, on R D - R B, the difference of the disk radius and the magnetic bubble radius.

Fig. 5.
Fig. 5.

Wavelength dispersion of the nonreciprocal phase shifts per circulation, Δφ.

Fig. 6.
Fig. 6.

Wavelength dispersion of the phase shifts per circulation, φ.

Fig. 7.
Fig. 7.

Wavelength dispersion of the radiation losses per circulation, γ.

Fig. 8.
Fig. 8.

Dependence of the isolation on λ and L, for R D = 1 μm.

Fig. 9.
Fig. 9.

Dependence of (a) the isolation and (b) the insertion loss on λ and L, for R D = 2 μm.

Fig. 10.
Fig. 10.

Dependence of (a) the isolation and (b) the insertion loss on λ and L, for R D = 3 μm.

Fig. 11.
Fig. 11.

Dependency of κ, the coupling coefficient, on D, the gap width between the waveguide and disk resonator.

Fig. 12.
Fig. 12.

Transmission spectra evaluated with the finite element method (a) for D = 310 μm and (b) for D = 240 μm.

Fig. 13.
Fig. 13.

Schematic representation of parallel-coupled nonreciprocal microdisk resonators.

Fig. 14.
Fig. 14.

Dependence of (a) the isolation and (b) insertion loss on λ and L, for B = 100 GHz and N = 3.

Fig. 15.
Fig. 15.

Dependence of (a) the isolation and (b) insertion loss on λ and B, for L = 5.82 μm and N= 3.

Fig. 16.
Fig. 16.

Transmission spectra of the parallel-coupled nonreciprocal microresonators, identified with the finite element method.

Fig. 17.
Fig. 17.

(a) Schematic representation of the 3-D magneto-optical resonator. (b) The computational window for finite-element analysis in the cylindrical coordinate system.

Fig. 18.
Fig. 18.

Dependence of the radiation losses per circulation, γ, on disk radius R D, for the TM mode at the wavelength of λ = 1300 nm.

Fig. 19.
Fig. 19.

Dependence of Δφ, the nonreciprocal phase shifts per circulation, on R D - R B, the difference of the disk radius and the magnetic bubble radius, for (a) H = 300 nm, (b) H = 400 nm, and (c) H = 500 nm.

Fig. 20.
Fig. 20.

Transmission spectra of the 3-D parallel-coupled nonreciprocal microresonators.

Tables (1)

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Table 1. Parameters at the resonances.

Equations (7)

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Δ φ = 2 π R eff Δ β eff = 2 π R eff ( H x x ) H x ( ξ n 4 ) d ρ H x 2 n 2 d ρ ,
λ m = φ k 0 m ,
Δ λ m = φ a k 0 m φ c k 0 m = Δ φ k 0 m .
κ i = π 2 Q [ ( 1 + 4 Q i 2 π 2 ) 1 2 1 ] 1 2
Q i = FSR g i B
g i = 2 sin ( 2 i 1 2 N π )
FSR = 2 πc ( m + 1 λ m + 1 φ m + 1 m λ m φ m ) .

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