Abstract

In this work, a dielectric waveguide mode solver is presented considering a general nonreciprocal permittivity tensor. The proposed method allows us to investigate important cases of practical interest in the field of integrated optics, such as magneto-optical isolators and anisotropic waveguides. Unlike the earlier developed mode solver, our approach allows for the precise computation of both forward and backward propagating modes in the nonreciprocal case, ensuring high accuracy and computational efficiency. As a result, the nonreciprocal loss/phase shift can be directly computed, avoiding the use of the perturbation method. To compute the electromagnetic modes, the Rayleigh-Ritz functional is derived for the non-self adjoint case, it is discretized using the node-based finite element method and the penalty function is added to remove the spurious solutions. The resulting quadratic eigenvalue problem is linearized and solved in terms of the propagation constant for a given frequency (i.e., γ–formulation). The main benefits of this formulation are that it avoids the time-consuming iterations and preserves the matrix sparsity. Finally, the method is used to study two examples of integrated optical isolators based on nonreciprocal phase shift and nonreciprocal loss effect, respectively. The developed method is then compared with the perturbation approach and its simplified formulation based on semivectorial approximation.

© 2014 Optical Society of America

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  70. V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
    [CrossRef]

2013 (2)

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
[CrossRef]

P. Pintus, F. Di Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express21, 5041–5052 (2013).
[CrossRef] [PubMed]

2012 (2)

M. C. Sekhar, M. R. Singh, S. Basu, and S. Pinnepalli, “Giant faraday rotation in BixCe3-xFe5O12 epitaxial garnet films,” Opt. Express20, 9624–9639 (2012).
[CrossRef]

T. Mizumoto, R. Takei, and Y. Shoji, “Waveguide optical isolators for integrated optics,” IEEE J. Quantum Electron.48, 252–260 (2012).
[CrossRef]

2011 (5)

M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express19, 11740–11745 (2011).
[CrossRef] [PubMed]

P. Pintus, M.-C. Tien, and J. E. Bowers, “Design of magneto-optical ring isolator on SOI based on the finite element method,” IEEE Photonic. Tech. L.23, 1670–1672 (2011).
[CrossRef]

P. Pintus, S. Faralli, and F. Di Pasquale, “Integrated 2.8μm laser source in Al2O3:Er3+ slot waveguide on SOI,” J. Lightwave Technol.29, 1206–1212 (2011).
[CrossRef]

P. Pintus, F. Di Pasquale, and J. E. Bowers, “Design of TE ring isolators for ultra low loss Si3N4 waveguides based on the finite element method,” Opt. Lett.36, 4599–4601 (2011).
[CrossRef] [PubMed]

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

2010 (3)

D. Jalas, A. Petrov, M. Krause, J. Hampe, and M. Eich, “Resonance splitting in gyrotropic ring resonators,” Opt. Lett.35, 3438–3440 (2010).
[CrossRef] [PubMed]

P. Pintus, S. Faralli, and F. Di Pasquale, “Low threshold pump power and high integration in Al2O3:Er3+ slot waveguide laser on SOI,” IEEE Photonic. Tech. L.22, 1428–1430 (2010).
[CrossRef]

R. Takei and T. Mizumoto, “Design and simulation of silicon waveguide optical circulator employing nonreciprocal phase shift,” Jpn. J. Appl. Phys.49, 052203 (2010).
[CrossRef]

2009 (1)

H. Shimizu and G. Syunsuke, “InGaAsP/InP evanescent mode waveguide optical isolators and their application to InGaAsP/InP/Si hybrid evanescent optical isolators,” Opt.Quant. Electron.41, 653–660 (2009).

2008 (4)

H. Yokoi, “Calculation of nonreciprocal phase shift in magneto-optic waveguides with Ce:YIG layer,” Opt. Mater.31, 189–192 (2008).
[CrossRef]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol.26, 1423–1431 (2008).
[CrossRef]

Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
[CrossRef]

N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey, “Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems,” Int. J. Numer. Meth. Eng.73, 344–360 (2008).
[CrossRef]

2006 (2)

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol.24, 38–43 (2006).
[CrossRef]

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
[CrossRef]

2005 (3)

2004 (1)

H. Shimizu and Y. Nakano, “First demonstration of TE mode nonreciprocal propagation in an InGaAsP/InP active waveguide for an integratable optical isolator,” Jpn. J. Appl. Phys.43, 1561–1563 (2004).
[CrossRef]

2001 (3)

O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, “Magnetooptical waveguides with polarization-independent nonreciprocal phase-shift,” J. Lightwave Technol.19, 214–221 (2001).
[CrossRef]

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt.Quant. Electron.33, 359–371 (2001).

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev.43, 235–286 (2001).
[CrossRef]

2000 (2)

P. Quéffélec, M. Le Floc’h, and P. Gelin, “New method for determining the permeability tensor of magnetized ferrites in a wide frequency range,” IEEE Trans. Microwave Theory Tech.48, 1344–1351 (2000).
[CrossRef]

H. Yokoi, T. Mizumoto, Y. Shoji, N. Futakuchi, and Y. Nakano, “Demonstration of an optical isolator with a semiconductor guiding layer that was obtained by use of a nonreciprocal phase shift,” Appl. Opt.39, 6158–6164 (2000).
[CrossRef]

1999 (3)

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photonic. Tech. L.11, 1012–1014 (1999).
[CrossRef]

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
[CrossRef]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, Dötsch, P. Hertel, and A. F. Popkov, “Analysis of polarization independent Mach-Zehnder-type integrated optical isolator,” J. Lightwave Technol.17, 1200–1205 (1999).
[CrossRef]

1997 (3)

V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
[CrossRef]

G. Pan and J. Tan, “General edge element approach to lossy and dispersive structures in anisotropic media,” IEE Proc.-Microw. Antennas Propag.144, 81–90 (1997).
[CrossRef]

S. Selleri and M. Zoboli, “Performance comparison of finite-element approaches for electromagnetic waveguides,” J. Opt. Soc. Am. A14, 1460–1466 (1997).
[CrossRef]

1996 (3)

P. Berini and K. Wu, “Modeling lossy anisotropic dielectric waveguides with the method of lines,” IEEE Trans. Microwave Theory Tech.44, 749–759 (1996).
[CrossRef]

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. A.17, 789–821 (1996).
[CrossRef]

L. Valor and J. Zapata, “An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials,” IEEE Trans. Microwave Theory Tech.44, 291–296 (1996).
[CrossRef]

1995 (4)

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag.43, 1460–1463 (1995).
[CrossRef]

J. Tan and G. Pan, “A new edge element analysis of dispersive waveguiding structures,” IEEE Trans. Microwave Theory Tech.43, 2600–2607 (1995).
[CrossRef]

S. Selleri and M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech.43, 887–892 (1995).
[CrossRef]

M. Wallenhorst, M. Niemöller, H. Dötsch, P. 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 (2)

J.-F. Lee, “Finite element analysis of lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech.42, 1025–1031 (1994).
[CrossRef]

G. Mur, “Edge elements, their advantages and their disadvantages,” IEEE T. Magn.30, 3552–3557 (1994).
[CrossRef]

1993 (3)

A. S. Sudbø, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt.2, 211–233 (1993).
[CrossRef]

Y. Lu and F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech.41, 1215–1223 (1993).
[CrossRef]

T. Shintaku, T. Uno, and M. Kobayashi, “Magneto-optic channel waveguides in Ce-substituted yttrium iron garnet,” J. Appl. Phys.74, 4877–4881 (1993).
[CrossRef]

1992 (1)

D. C. Sorensen, “Implicit application of polynomial filters in a k-step Arnoldi method,” SIAM J. Matrix Anal. A.13, 357–385 (1992).
[CrossRef]

1991 (1)

J.-F. Lee, D.-K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech.39, 1262–1271 (1991).
[CrossRef]

1989 (2)

K. Hayata, K. Miura, and M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech.37, 875–883 (1989).
[CrossRef]

W. C. Chew and M. Nasir, “A variational analysis of anisotropic, inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech.37, 661–668 (1989).
[CrossRef]

1988 (1)

K. Hayata, K. Miura, and M. Koshiba, “Finite-element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech.36, 268–276 (1988).
[CrossRef]

1986 (4)

K. Hayata, M. Koshiba, and M. Suzuki, “Vectorial wave analysis of stress-applied polarization-mantaining optical fiber by finite element method,” J. Lightwave Technol.4, 133–139 (1986).
[CrossRef]

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech.34, 1120–1124 (1986).
[CrossRef]

S. R. Cvetkovic and J. B. Davis, “Self-adjoint variational formulation for lossy anisotropic dielectric waveguide,” IEEE Trans. Microwave Theory Tech.34, 129–134 (1986).
[CrossRef]

R. Hoffman, “Comments on “self-adjoint variational formulation for lossy anisotropic dielectric waveguide”,” IEEE Trans. Microwave Theory Tech.34, 1227–1228 (1986).
[CrossRef]

1984 (1)

B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech.32, 922–928 (1984).
[CrossRef]

1980 (1)

C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microwave Theory Tech.28, 878–886 (1980).
[CrossRef]

1977 (1)

A. Konrad, “High-order triangular finite elements for electromagnetic waves in anisotropic media,” IEEE Trans. Microwave Theory Tech.25, 353–360 (1977).
[CrossRef]

1974 (1)

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
[CrossRef]

1968 (1)

G. Krinchik and V. Artemjev, “Magneto-optic properties of nickel, iron, and cobalt,” J. Appl. Phys.39, 1276–1278 (1968).
[CrossRef]

1967 (1)

P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys.38, 1482–1490 (1967).
[CrossRef]

1965 (1)

G. J. Gabriel and M. E. Brodwin, “The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods,” IEEE Trans. Microwave Theory Tech.13, 364–370 (1965).
[CrossRef]

1946 (1)

C. Kittel, “Theory of the dispersion of magnetic permeability in ferromagnetic materials at microwave frequencies,” Phys. Rev.70, 281–290 (1946).
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Amemiya, T.

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
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Ando, K.

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photonic. Tech. L.11, 1012–1014 (1999).
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G. Krinchik and V. Artemjev, “Magneto-optic properties of nickel, iron, and cobalt,” J. Appl. Phys.39, 1276–1278 (1968).
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Basu, S.

Berini, P.

P. Berini and K. Wu, “Modeling lossy anisotropic dielectric waveguides with the method of lines,” IEEE Trans. Microwave Theory Tech.44, 749–759 (1996).
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Bianconi, M.

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

Bowers, J.

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
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Bowers, J. E.

Boyd, J. T.

V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
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Brodwin, M. E.

G. J. Gabriel and M. E. Brodwin, “The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods,” IEEE Trans. Microwave Theory Tech.13, 364–370 (1965).
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Cendes, Z. J.

J.-F. Lee, D.-K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech.39, 1262–1271 (1991).
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Chen, C. H.

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W. Chew, Waves and Fields in Inhomogenous Media (Wiley-IEEE Press, 1999).
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W. C. Chew and M. Nasir, “A variational analysis of anisotropic, inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech.37, 661–668 (1989).
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S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt.Quant. Electron.33, 359–371 (2001).

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
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S. R. Cvetkovic and J. B. Davis, “Self-adjoint variational formulation for lossy anisotropic dielectric waveguide,” IEEE Trans. Microwave Theory Tech.34, 129–134 (1986).
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B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech.32, 922–928 (1984).
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Davis, J. B.

S. R. Cvetkovic and J. B. Davis, “Self-adjoint variational formulation for lossy anisotropic dielectric waveguide,” IEEE Trans. Microwave Theory Tech.34, 129–134 (1986).
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V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
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De Nicola, P.

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

Di Pasquale, F.

P. Pintus, F. Di Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express21, 5041–5052 (2013).
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P. Pintus, F. Di Pasquale, and J. E. Bowers, “Design of TE ring isolators for ultra low loss Si3N4 waveguides based on the finite element method,” Opt. Lett.36, 4599–4601 (2011).
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P. Pintus, S. Faralli, and F. Di Pasquale, “Integrated 2.8μm laser source in Al2O3:Er3+ slot waveguide on SOI,” J. Lightwave Technol.29, 1206–1212 (2011).
[CrossRef]

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

P. Pintus, S. Faralli, and F. Di Pasquale, “Low threshold pump power and high integration in Al2O3:Er3+ slot waveguide laser on SOI,” IEEE Photonic. Tech. L.22, 1428–1430 (2010).
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H.-Y. Fan, W.-W. Lin, and P. V. Dooren, “Normwise scaling of second order polynomial matrices,” SIAM J. Matrix Anal. A.26, 252–256 (2005).
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Dötsch, H.

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K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech.34, 1120–1124 (1986).
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Fallahkhair, A. B.

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H.-Y. Fan, W.-W. Lin, and P. V. Dooren, “Normwise scaling of second order polynomial matrices,” SIAM J. Matrix Anal. A.26, 252–256 (2005).
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Faralli, S.

P. Pintus, S. Faralli, and F. Di Pasquale, “Integrated 2.8μm laser source in Al2O3:Er3+ slot waveguide on SOI,” J. Lightwave Technol.29, 1206–1212 (2011).
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P. Pintus, S. Faralli, and F. Di Pasquale, “Low threshold pump power and high integration in Al2O3:Er3+ slot waveguide laser on SOI,” IEEE Photonic. Tech. L.22, 1428–1430 (2010).
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Y. Lu and F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech.41, 1215–1223 (1993).
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Gabriel, G. J.

G. J. Gabriel and M. E. Brodwin, “The solution of guided waves in inhomogeneous anisotropic media by perturbation and variational methods,” IEEE Trans. Microwave Theory Tech.13, 364–370 (1965).
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N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey, “Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems,” Int. J. Numer. Meth. Eng.73, 344–360 (2008).
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M. Wallenhorst, M. Niemöller, H. Dötsch, P. 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).
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P. Quéffélec, M. Le Floc’h, and P. Gelin, “New method for determining the permeability tensor of magnetized ferrites in a wide frequency range,” IEEE Trans. Microwave Theory Tech.48, 1344–1351 (2000).
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H. Dötsch, N. Bahlmann, O. Zhuromskyy, H. Hammer, L. Wilkens, R. Gerhardt, P. Hertel, and A. F. Popkov, “Applications of magneto-optical waveguides in integrated optics: review,” J. Opt. Soc. Am. B22, 240–253 (2005).
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M. Wallenhorst, M. Niemöller, H. Dötsch, P. 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).
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A. Nicolet and C. Geuzaine, “Waveguide propagation modes and quadratic eigenvalue problems,” in “6th International Conference on Computational Electromagnetics (CEM), 2006,” (Aachen, Germany, 2006), 1–8 (2006).

Hai, P.

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
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Hammer, H.

Hampe, J.

Hayata, K.

K. Hayata, K. Miura, and M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech.37, 875–883 (1989).
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K. Hayata, K. Miura, and M. Koshiba, “Finite-element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech.36, 268–276 (1988).
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K. Hayata, M. Koshiba, and M. Suzuki, “Vectorial wave analysis of stress-applied polarization-mantaining optical fiber by finite element method,” J. Lightwave Technol.4, 133–139 (1986).
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K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech.34, 1120–1124 (1986).
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Heck, M.

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
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Hertel, P.

Higham, N. J.

N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey, “Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems,” Int. J. Numer. Meth. Eng.73, 344–360 (2008).
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R. Hoffman, “Comments on “self-adjoint variational formulation for lossy anisotropic dielectric waveguide”,” IEEE Trans. Microwave Theory Tech.34, 1227–1228 (1986).
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Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
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Janz, S.

Jin, J.

J. Jin, The Finite Element Method in Electro-magnetics (Wiley, 2002), 2nd ed.

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Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag.43, 1460–1463 (1995).
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Kittel, C.

C. Kittel, “Theory of the dispersion of magnetic permeability in ferromagnetic materials at microwave frequencies,” Phys. Rev.70, 281–290 (1946).
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Kobayashi, M.

T. Shintaku, T. Uno, and M. Kobayashi, “Magneto-optic channel waveguides in Ce-substituted yttrium iron garnet,” J. Appl. Phys.74, 4877–4881 (1993).
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Konrad, A.

A. Konrad, “High-order triangular finite elements for electromagnetic waves in anisotropic media,” IEEE Trans. Microwave Theory Tech.25, 353–360 (1977).
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Koshiba, M.

K. Hayata, K. Miura, and M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech.37, 875–883 (1989).
[CrossRef]

K. Hayata, K. Miura, and M. Koshiba, “Finite-element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech.36, 268–276 (1988).
[CrossRef]

K. Hayata, M. Koshiba, and M. Suzuki, “Vectorial wave analysis of stress-applied polarization-mantaining optical fiber by finite element method,” J. Lightwave Technol.4, 133–139 (1986).
[CrossRef]

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech.34, 1120–1124 (1986).
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Krause, M.

Krinchik, G.

G. Krinchik and V. Artemjev, “Magneto-optic properties of nickel, iron, and cobalt,” J. Appl. Phys.39, 1276–1278 (1968).
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Kromer, H.

Kurczveil, G.

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
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Lamontagne, B.

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P. Quéffélec, M. Le Floc’h, and P. Gelin, “New method for determining the permeability tensor of magnetized ferrites in a wide frequency range,” IEEE Trans. Microwave Theory Tech.48, 1344–1351 (2000).
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Lee, J.-F.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag.43, 1460–1463 (1995).
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J.-F. Lee, “Finite element analysis of lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech.42, 1025–1031 (1994).
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J.-F. Lee, D.-K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech.39, 1262–1271 (1991).
[CrossRef]

Lee, R.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag.43, 1460–1463 (1995).
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C. H. Chen and C.-D. Lien, “The variational principle for non-self-adjoint electromagnetic problems,” IEEE Trans. Microwave Theory Tech.28, 878–886 (1980).
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L. D. Landau and E. M. Lifshits, “Electrodynamics of continuous media,” in “A Course of Theoretical Physics,” vol. 8 (Pergamon, 1960).

Lin, W.-W.

H.-Y. Fan, W.-W. Lin, and P. V. Dooren, “Normwise scaling of second order polynomial matrices,” SIAM J. Matrix Anal. A.26, 252–256 (2005).
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Lu, Y.

Y. Lu and F. A. Fernandez, “An efficient finite element solution of inhomogeneous anisotropic and lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech.41, 1215–1223 (1993).
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Mackey, D. S.

N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey, “Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems,” Int. J. Numer. Meth. Eng.73, 344–360 (2008).
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Makimoto, T.

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
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F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev.43, 235–286 (2001).
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Miura, K.

K. Hayata, K. Miura, and M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech.37, 875–883 (1989).
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K. Hayata, K. Miura, and M. Koshiba, “Finite-element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech.36, 268–276 (1988).
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Mizumoto, T.

T. Mizumoto, R. Takei, and Y. Shoji, “Waveguide optical isolators for integrated optics,” IEEE J. Quantum Electron.48, 252–260 (2012).
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M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express19, 11740–11745 (2011).
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R. Takei and T. Mizumoto, “Design and simulation of silicon waveguide optical circulator employing nonreciprocal phase shift,” Jpn. J. Appl. Phys.49, 052203 (2010).
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Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
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H. Yokoi, T. Mizumoto, Y. Shoji, N. Futakuchi, and Y. Nakano, “Demonstration of an optical isolator with a semiconductor guiding layer that was obtained by use of a nonreciprocal phase shift,” Appl. Opt.39, 6158–6164 (2000).
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Montanari, G. B.

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
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V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
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Nakano, Y.

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol.24, 38–43 (2006).
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T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
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H. Shimizu and Y. Nakano, “First demonstration of TE mode nonreciprocal propagation in an InGaAsP/InP active waveguide for an integratable optical isolator,” Jpn. J. Appl. Phys.43, 1561–1563 (2004).
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H. Yokoi, T. Mizumoto, Y. Shoji, N. Futakuchi, and Y. Nakano, “Demonstration of an optical isolator with a semiconductor guiding layer that was obtained by use of a nonreciprocal phase shift,” Appl. Opt.39, 6158–6164 (2000).
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W. C. Chew and M. Nasir, “A variational analysis of anisotropic, inhomogeneous dielectric waveguides,” IEEE Trans. Microwave Theory Tech.37, 661–668 (1989).
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A. Nicolet and C. Geuzaine, “Waveguide propagation modes and quadratic eigenvalue problems,” in “6th International Conference on Computational Electromagnetics (CEM), 2006,” (Aachen, Germany, 2006), 1–8 (2006).

Niemöller, M.

M. Wallenhorst, M. Niemöller, H. Dötsch, P. 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]

Osgood, R. M.

Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
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A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
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P. S. Pershan, “Magneto-optical effects,” J. Appl. Phys.38, 1482–1490 (1967).
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Peters, J.

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
[CrossRef]

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Pinnepalli, S.

Pintus, P.

P. Pintus, F. Di Pasquale, and J. E. Bowers, “Integrated TE and TM optical circulators on ultra-low-loss silicon nitride platform,” Opt. Express21, 5041–5052 (2013).
[CrossRef] [PubMed]

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
[CrossRef]

P. Pintus, F. Di Pasquale, and J. E. Bowers, “Design of TE ring isolators for ultra low loss Si3N4 waveguides based on the finite element method,” Opt. Lett.36, 4599–4601 (2011).
[CrossRef] [PubMed]

P. Pintus, S. Faralli, and F. Di Pasquale, “Integrated 2.8μm laser source in Al2O3:Er3+ slot waveguide on SOI,” J. Lightwave Technol.29, 1206–1212 (2011).
[CrossRef]

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express19, 11740–11745 (2011).
[CrossRef] [PubMed]

P. Pintus, M.-C. Tien, and J. E. Bowers, “Design of magneto-optical ring isolator on SOI based on the finite element method,” IEEE Photonic. Tech. L.23, 1670–1672 (2011).
[CrossRef]

P. Pintus, S. Faralli, and F. Di Pasquale, “Low threshold pump power and high integration in Al2O3:Er3+ slot waveguide laser on SOI,” IEEE Photonic. Tech. L.22, 1428–1430 (2010).
[CrossRef]

Popkov, A. F.

Prati, G.

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

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P. Quéffélec, M. Le Floc’h, and P. Gelin, “New method for determining the permeability tensor of magnetized ferrites in a wide frequency range,” IEEE Trans. Microwave Theory Tech.48, 1344–1351 (2000).
[CrossRef]

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B. M. A. Rahman and J. B. Davies, “Penalty function improvement of waveguides solution by finite elements,” IEEE Trans. Microwave Theory Tech.32, 922–928 (1984).
[CrossRef]

Sacks, Z. S.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag.43, 1460–1463 (1995).
[CrossRef]

Sekhar, M. C.

Selleri, S.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt.Quant. Electron.33, 359–371 (2001).

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
[CrossRef]

S. Selleri and M. Zoboli, “Performance comparison of finite-element approaches for electromagnetic waveguides,” J. Opt. Soc. Am. A14, 1460–1466 (1997).
[CrossRef]

S. Selleri and M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech.43, 887–892 (1995).
[CrossRef]

Sher, S. M.

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

Shimizu, H.

H. Shimizu and G. Syunsuke, “InGaAsP/InP evanescent mode waveguide optical isolators and their application to InGaAsP/InP/Si hybrid evanescent optical isolators,” Opt.Quant. Electron.41, 653–660 (2009).

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
[CrossRef]

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol.24, 38–43 (2006).
[CrossRef]

H. Shimizu and Y. Nakano, “First demonstration of TE mode nonreciprocal propagation in an InGaAsP/InP active waveguide for an integratable optical isolator,” Jpn. J. Appl. Phys.43, 1561–1563 (2004).
[CrossRef]

Shintaku, T.

T. Shintaku, T. Uno, and M. Kobayashi, “Magneto-optic channel waveguides in Ce-substituted yttrium iron garnet,” J. Appl. Phys.74, 4877–4881 (1993).
[CrossRef]

Shoji, Y.

T. Mizumoto, R. Takei, and Y. Shoji, “Waveguide optical isolators for integrated optics,” IEEE J. Quantum Electron.48, 252–260 (2012).
[CrossRef]

Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
[CrossRef]

H. Yokoi, T. Mizumoto, Y. Shoji, N. Futakuchi, and Y. Nakano, “Demonstration of an optical isolator with a semiconductor guiding layer that was obtained by use of a nonreciprocal phase shift,” Appl. Opt.39, 6158–6164 (2000).
[CrossRef]

Singh, M. R.

Sorensen, D. C.

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. A.17, 789–821 (1996).
[CrossRef]

D. C. Sorensen, “Implicit application of polynomial filters in a k-step Arnoldi method,” SIAM J. Matrix Anal. A.13, 357–385 (1992).
[CrossRef]

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM Publications, Philadelphia (1998).
[CrossRef]

Subramaniam, V.

V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
[CrossRef]

Sudbø, A. S.

A. S. Sudbø, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt.2, 211–233 (1993).
[CrossRef]

Sugliani, S.

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

Sun, D.-K.

J.-F. Lee, D.-K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech.39, 1262–1271 (1991).
[CrossRef]

Suzuki, M.

K. Hayata, M. Koshiba, and M. Suzuki, “Vectorial wave analysis of stress-applied polarization-mantaining optical fiber by finite element method,” J. Lightwave Technol.4, 133–139 (1986).
[CrossRef]

K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech.34, 1120–1124 (1986).
[CrossRef]

Syunsuke, G.

H. Shimizu and G. Syunsuke, “InGaAsP/InP evanescent mode waveguide optical isolators and their application to InGaAsP/InP/Si hybrid evanescent optical isolators,” Opt.Quant. Electron.41, 653–660 (2009).

Takei, R.

T. Mizumoto, R. Takei, and Y. Shoji, “Waveguide optical isolators for integrated optics,” IEEE J. Quantum Electron.48, 252–260 (2012).
[CrossRef]

R. Takei and T. Mizumoto, “Design and simulation of silicon waveguide optical circulator employing nonreciprocal phase shift,” Jpn. J. Appl. Phys.49, 052203 (2010).
[CrossRef]

Tan, J.

G. Pan and J. Tan, “General edge element approach to lossy and dispersive structures in anisotropic media,” IEE Proc.-Microw. Antennas Propag.144, 81–90 (1997).
[CrossRef]

J. Tan and G. Pan, “A new edge element analysis of dispersive waveguiding structures,” IEEE Trans. Microwave Theory Tech.43, 2600–2607 (1995).
[CrossRef]

Tanaka, M.

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
[CrossRef]

Tarr, N. G.

Tien, M.-C.

P. Pintus, M.-C. Tien, and J. E. Bowers, “Design of magneto-optical ring isolator on SOI based on the finite element method,” IEEE Photonic. Tech. L.23, 1670–1672 (2011).
[CrossRef]

M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express19, 11740–11745 (2011).
[CrossRef] [PubMed]

Tisseur, F.

N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey, “Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems,” Int. J. Numer. Meth. Eng.73, 344–360 (2008).
[CrossRef]

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev.43, 235–286 (2001).
[CrossRef]

Uno, T.

T. Shintaku, T. Uno, and M. Kobayashi, “Magneto-optic channel waveguides in Ce-substituted yttrium iron garnet,” J. Appl. Phys.74, 4877–4881 (1993).
[CrossRef]

Valor, L.

L. Valor and J. Zapata, “An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials,” IEEE Trans. Microwave Theory Tech.44, 291–296 (1996).
[CrossRef]

Vincetti, L.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt.Quant. Electron.33, 359–371 (2001).

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
[CrossRef]

Wallenhorst, M.

M. Wallenhorst, M. Niemöller, H. Dötsch, P. 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.

Wu, K.

P. Berini and K. Wu, “Modeling lossy anisotropic dielectric waveguides with the method of lines,” IEEE Trans. Microwave Theory Tech.44, 749–759 (1996).
[CrossRef]

Xu, D.-X.

Yamamoto, S.

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
[CrossRef]

Yang, C.

R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM Publications, Philadelphia (1998).
[CrossRef]

Ye, W. N.

Yokoi, H.

H. Yokoi, “Calculation of nonreciprocal phase shift in magneto-optic waveguides with Ce:YIG layer,” Opt. Mater.31, 189–192 (2008).
[CrossRef]

Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
[CrossRef]

H. Yokoi, T. Mizumoto, Y. Shoji, N. Futakuchi, and Y. Nakano, “Demonstration of an optical isolator with a semiconductor guiding layer that was obtained by use of a nonreciprocal phase shift,” Appl. Opt.39, 6158–6164 (2000).
[CrossRef]

Yokoyama, M.

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
[CrossRef]

Zaets, W.

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photonic. Tech. L.11, 1012–1014 (1999).
[CrossRef]

Zapata, J.

L. Valor and J. Zapata, “An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials,” IEEE Trans. Microwave Theory Tech.44, 291–296 (1996).
[CrossRef]

Zhuromskyy, O.

Zoboli, M.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt.Quant. Electron.33, 359–371 (2001).

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
[CrossRef]

S. Selleri and M. Zoboli, “Performance comparison of finite-element approaches for electromagnetic waveguides,” J. Opt. Soc. Am. A14, 1460–1466 (1997).
[CrossRef]

S. Selleri and M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech.43, 887–892 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

Y. Shoji, T. Mizumoto, H. Yokoi, I. W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett.92, 071117 (2008).
[CrossRef]

T. Amemiya, H. Shimizu, Y. Nakano, P. Hai, M. Yokoyama, and M. Tanaka, “Semiconductor waveguide optical isolator based on nonreciprocal loss induced by ferromagnetic MnAs,” Appl. Phys. Lett.89, 021104 (2006).
[CrossRef]

IEE Proc.-Microw. Antennas Propag. (1)

G. Pan and J. Tan, “General edge element approach to lossy and dispersive structures in anisotropic media,” IEE Proc.-Microw. Antennas Propag.144, 81–90 (1997).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. M. Sher, P. Pintus, F. Di Pasquale, M. Bianconi, G. B. Montanari, P. De Nicola, S. Sugliani, and G. Prati, “Design of 980nm-pumped waveguide laser for continuous wave operation in ion implanted Er:LiNbO3,” IEEE J. Quantum Electron.47, 526–533 (2011).
[CrossRef]

T. Mizumoto, R. Takei, and Y. Shoji, “Waveguide optical isolators for integrated optics,” IEEE J. Quantum Electron.48, 252–260 (2012).
[CrossRef]

IEEE Photonic. Tech. L. (3)

P. Pintus, M.-C. Tien, and J. E. Bowers, “Design of magneto-optical ring isolator on SOI based on the finite element method,” IEEE Photonic. Tech. L.23, 1670–1672 (2011).
[CrossRef]

P. Pintus, S. Faralli, and F. Di Pasquale, “Low threshold pump power and high integration in Al2O3:Er3+ slot waveguide laser on SOI,” IEEE Photonic. Tech. L.22, 1428–1430 (2010).
[CrossRef]

W. Zaets and K. Ando, “Optical waveguide isolator based on nonreciprocal loss/gain of amplifier covered by ferromagnetic layer,” IEEE Photonic. Tech. L.11, 1012–1014 (1999).
[CrossRef]

IEEE Photonics Journal (1)

G. Kurczveil, P. Pintus, M. Heck, J. Peters, and J. Bowers, “Characterization of insertion loss and back reflection in passive hybrid silicon tapers,” IEEE Photonics Journal5, 6600410 (2013).
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IEEE T. Magn. (1)

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K. Hayata, M. Koshiba, M. Eguchi, and M. Suzuki, “Vectorial finite-element method without any spurious solutions for dielectric waveguiding problems using transverse magnetic-field component,” IEEE Trans. Microwave Theory Tech.34, 1120–1124 (1986).
[CrossRef]

P. Berini and K. Wu, “Modeling lossy anisotropic dielectric waveguides with the method of lines,” IEEE Trans. Microwave Theory Tech.44, 749–759 (1996).
[CrossRef]

K. Hayata, K. Miura, and M. Koshiba, “Finite-element formulation for lossy waveguides,” IEEE Trans. Microwave Theory Tech.36, 268–276 (1988).
[CrossRef]

K. Hayata, K. Miura, and M. Koshiba, “Full vectorial finite element formalism for lossy anisotropic waveguides,” IEEE Trans. Microwave Theory Tech.37, 875–883 (1989).
[CrossRef]

S. Selleri and M. Zoboli, “An improved finite element method formulation for the analysis of nonlinear anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech.43, 887–892 (1995).
[CrossRef]

J.-F. Lee, D.-K. Sun, and Z. J. Cendes, “Full-wave analysis of dielectric waveguides using tangential vector finite elements,” IEEE Trans. Microwave Theory Tech.39, 1262–1271 (1991).
[CrossRef]

J.-F. Lee, “Finite element analysis of lossy dielectric waveguides,” IEEE Trans. Microwave Theory Tech.42, 1025–1031 (1994).
[CrossRef]

J. Tan and G. Pan, “A new edge element analysis of dispersive waveguiding structures,” IEEE Trans. Microwave Theory Tech.43, 2600–2607 (1995).
[CrossRef]

L. Valor and J. Zapata, “An efficient finite element formulation to analyze waveguides with lossy inhomogeneous bi-anisotropic materials,” IEEE Trans. Microwave Theory Tech.44, 291–296 (1996).
[CrossRef]

Int. J. Numer. Meth. Eng. (1)

N. J. Higham, D. S. Mackey, F. Tisseur, and S. D. Garvey, “Scaling, sensitivity and stability in the numerical solution of quadratic eigenvalue problems,” Int. J. Numer. Meth. Eng.73, 344–360 (2008).
[CrossRef]

J. Appl. Phys. (5)

T. Shintaku, T. Uno, and M. Kobayashi, “Magneto-optic channel waveguides in Ce-substituted yttrium iron garnet,” J. Appl. Phys.74, 4877–4881 (1993).
[CrossRef]

M. Wallenhorst, M. Niemöller, H. Dötsch, P. 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]

S. Yamamoto and T. Makimoto, “Circuit theory for a class of anisotropic and gyrotropic thin-film optical waveguides and design of nonreciprocal devices for integrated optics,” J. Appl. Phys.45, 882–888 (1974).
[CrossRef]

G. Krinchik and V. Artemjev, “Magneto-optic properties of nickel, iron, and cobalt,” J. Appl. Phys.39, 1276–1278 (1968).
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[CrossRef]

J. Lightwave Technol. (8)

P. Pintus, S. Faralli, and F. Di Pasquale, “Integrated 2.8μm laser source in Al2O3:Er3+ slot waveguide on SOI,” J. Lightwave Technol.29, 1206–1212 (2011).
[CrossRef]

H. Shimizu and Y. Nakano, “Fabrication and characterization of an InGaAsP/InP active waveguide optical isolator with 14.7dB/mm TE mode nonreciprocal attenuation,” J. Lightwave Technol.24, 38–43 (2006).
[CrossRef]

V. Subramaniam, G. N. De Brabander, D. H. Naghski, and J. T. Boyd, “Measurement of mode field profiles and bending and transition losses in curved optical channel waveguides,” J. Lightwave Technol.15, 990–997 (1997).
[CrossRef]

O. Zhuromskyy, M. Lohmeyer, N. Bahlmann, Dötsch, P. Hertel, and A. F. Popkov, “Analysis of polarization independent Mach-Zehnder-type integrated optical isolator,” J. Lightwave Technol.17, 1200–1205 (1999).
[CrossRef]

O. Zhuromskyy, H. Dötsch, M. Lohmeyer, L. Wilkens, and P. Hertel, “Magnetooptical waveguides with polarization-independent nonreciprocal phase-shift,” J. Lightwave Technol.19, 214–221 (2001).
[CrossRef]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol.26, 1423–1431 (2008).
[CrossRef]

W. N. Ye, D.-X. Xu, S. Janz, P. Cheben, M.-J. Picard, B. Lamontagne, and N. G. Tarr, “Birefringence control using stress engineering in silicon-on-insulator (SOI) waveguides,” J. Lightwave Technol.23, 1308–1318 (2005).
[CrossRef]

K. Hayata, M. Koshiba, and M. Suzuki, “Vectorial wave analysis of stress-applied polarization-mantaining optical fiber by finite element method,” J. Lightwave Technol.4, 133–139 (1986).
[CrossRef]

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

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

Jpn. J. Appl. Phys. (2)

H. Shimizu and Y. Nakano, “First demonstration of TE mode nonreciprocal propagation in an InGaAsP/InP active waveguide for an integratable optical isolator,” Jpn. J. Appl. Phys.43, 1561–1563 (2004).
[CrossRef]

R. Takei and T. Mizumoto, “Design and simulation of silicon waveguide optical circulator employing nonreciprocal phase shift,” Jpn. J. Appl. Phys.49, 052203 (2010).
[CrossRef]

Microw. Opt. Techn. Let. (1)

A. Cucinotta, G. Pelosi, S. Selleri, L. Vincetti, and M. Zoboli, “Perfectly matched anisotropic layers for optical waveguide analysis through the finite-element beam-propagation method,” Microw. Opt. Techn. Let.23, 67–69 (1999).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Opt. Mater. (1)

H. Yokoi, “Calculation of nonreciprocal phase shift in magneto-optic waveguides with Ce:YIG layer,” Opt. Mater.31, 189–192 (2008).
[CrossRef]

Opt.Quant. Electron. (2)

H. Shimizu and G. Syunsuke, “InGaAsP/InP evanescent mode waveguide optical isolators and their application to InGaAsP/InP/Si hybrid evanescent optical isolators,” Opt.Quant. Electron.41, 653–660 (2009).

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt.Quant. Electron.33, 359–371 (2001).

Phys. Rev. (1)

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[CrossRef]

Pure Appl. Opt. (1)

A. S. Sudbø, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt.2, 211–233 (1993).
[CrossRef]

SIAM J. Matrix Anal. A. (3)

H.-Y. Fan, W.-W. Lin, and P. V. Dooren, “Normwise scaling of second order polynomial matrices,” SIAM J. Matrix Anal. A.26, 252–256 (2005).
[CrossRef]

R. B. Lehoucq and D. C. Sorensen, “Deflation techniques for an implicitly re-started Arnoldi iteration,” SIAM J. Matrix Anal. A.17, 789–821 (1996).
[CrossRef]

D. C. Sorensen, “Implicit application of polynomial filters in a k-step Arnoldi method,” SIAM J. Matrix Anal. A.13, 357–385 (1992).
[CrossRef]

SIAM Rev. (1)

F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev.43, 235–286 (2001).
[CrossRef]

Other (7)

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[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Cross section of a uniform waveguide loaded with inhomogeneity.

Fig. 2
Fig. 2

Magneto-optic waveguide cross section.

Fig. 3
Fig. 3

Magnetic field H for the TE-mode (first row) and TM-mode (second row) at 1550nm. The fields have been normalized in term of power and the profiles are shown in H/μm.

Fig. 4
Fig. 4

NRPS for the TM mode at 1550nm.

Fig. 5
Fig. 5

Dispersion curves (eigenvalue spectrum) as a function of the wavelength (λ = 2πc/ω) for the waveguide under examination.

Fig. 6
Fig. 6

Relative change of Δβpm 6(a) and Δβsv 6(b) with respect to the NRPS computed directly.

Fig. 7
Fig. 7

NRPS computation assuming hMO = 500nm and varying only the silicon thickness. In the plot we compared the three methods.

Fig. 8
Fig. 8

NRL isolator waveguide cross section.

Fig. 9
Fig. 9

Amplitude of the magnetic field H for the TE-mode at 1550nm. The amplitude of the field components have been normalized in term of power and the profiles are shown in H/μm.

Fig. 10
Fig. 10

Nonreciprocal effects on the TE-mode in the NRL-waveguide.

Tables (1)

Tables Icon

Table 1 Comparison between the two methods.

Equations (56)

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

ε r = ( ε x x ε x y ε x z ε x y ε y y ε y z ε x z ε y z ε z z ) ,
ε r = ( ε x x 0 0 0 ε y y 0 0 0 ε z z ) + K ( 0 M z M y M z 0 M x M y M x 0 ) ,
H ( r , t ) = H ( r ) e j ω t , E ( r , t ) = E ( r ) e j ω t ,
μ 0 μ r ( r ) H ( r ) = 0 ,
ε 0 ε r ( r ) E ( r ) = ρ ( r ) ,
× E ( r ) = j ω μ 0 μ r ( r ) H ( r ) ,
× H ( r ) = j ω ε 0 ε r ( r ) E ( r ) + J ( r ) ,
{ × [ ε r 1 × H ] ( ω c ) 2 μ r H = × ε r 1 J , in V , ( H ) = 0 , on S ,
{ × [ ( ε r a ) 1 × H a ] ( ω c ) 2 μ r a H = × ( ε r a ) 1 J a , in V , a ( H a ) = 0 , on S ,
( × H ) × i n = 0 , PEC ,
H × i n = 0 , PMC ,
F ( H , H a ) = 1 ε 0 V [ × H a ε r 1 × H ] d V ω 2 μ 0 V μ r H H a d V + 1 ε 0 V × ε r 1 J H a d V 1 ε 0 V H × ( ε r T ) 1 J a d V ,
E = E ( x , y ) e j ω t γ z , H = H ( x , y ) e j ω t γ z ,
E a = E a ( x , y ) e j ω t γ a z , H a = H a ( x , y ) e j ω t γ a z ,
( 0 γ y γ 0 x y x 0 ) ( E x E y E z ) = j ω μ 0 ( H x H y H z ) ,
( 0 γ y γ 0 x y x 0 ) ( H x H y H z ) = j ω ε 0 ( ε x x ε x y ε x z ε x y ε y y ε y z ε x z ε y z ε z z ) ( E x E y E z ) ,
( 0 γ y γ 0 x y x 0 ) ( E x a E y a E z a ) = j ω μ 0 ( H x a H y a H z a ) ,
( 0 γ y γ 0 x y x 0 ) ( H x a H y a H z a ) = j ω ε 0 ( ε x x ε x y ε x z ε x y ε y y ε y z ε x z ε y z ε z z ) ( E x a E y a E z a ) .
( E x a E y a E z a ) = ( E x E y E z ) , ( H x a H y a H z a ) = ( H x H y H z ) ,
H n e ( x , y ) = i = 1 m [ h x i e N i e ( x , y ) i x + h y i e N i e ( x , y ) i y + h z i e N i e ( x , y ) i z ] e j ω t γ z ,
H ( x , y ) H n ( x , y ) = e = 1 N H n e ( x , y ) ,
μ 0 H ( r ) = 0 .
F ˜ ( H , H a ) = F ( H , H a ) + α p ε 0 V H a H d V .
h x e = ( h x 1 e h x 2 e h x m e ) , h y e = ( h y 1 e h y 2 e h y m e ) , h z e = ( h z 1 e h z 2 e h z m e ) ,
r i j e = S e N i e N j e d x d y , d i j e = S e N i e y N j e y d x d y , j i j e = S e N i e N j e y d x d y , e i j e = S e N i e x N j e x d x d y , n i j e = S e N i e N j e x d x d y , z i j e = S e N i e y N j e x d x d y ,
S e [ × H a p × H ] d x d y = ( h x e h y e h z e ) T [ γ 2 ( p y y R e p y x R e 0 p x y R e p x x R e 0 0 0 0 ) + + γ ( p z y J e T p y z J e p y z N e p z x J e T p y x J e p y y N e p x z J e p z y N e T p z x N e T p x z N e p x y N e p x x J e p x y J e T p y y N e T p y x N e T p x x J e T 0 ) + ( p z z D e p z z Z e p z y Z e p z x D e p z z Z e T p z z E e p x z Z e T p z y E e p x z D e p y z Z e T p y z E e p x z Z e L z z e ) ] ( h x e h y e h z e ) ,
S e H a H d x d y = ( h x e h y e h z e ) T ( R e 0 0 0 R e 0 0 0 R e ) ( h x e h y e h z e ) ,
S e H a H d x d y = ( h x e h y e h z e ) T [ γ 2 ( 0 0 0 0 0 0 0 0 R e ) γ ( 0 0 N e T 0 0 J e T N e J e 0 ) + ( E e Z e T 0 Z e D e 0 0 0 0 ) ] ( h x e h y e h z e ) .
F ˜ ( H , H a ) | e ( h x e h y e h z e ) T [ γ 2 M e + γ C e + K e ω 2 μ 0 T e ] ( h x e h y e h z e ) .
F ˜ ( H , H a ) = e = 1 N F ˜ ( H , H a ) | e .
h x = ( h x 1 h x 2 h x N ^ ) , h y = ( h y 1 h y 2 h y N ^ ) , h z = ( h z 1 h z 2 h z N ^ ) ,
F ˜ ( H , H a ) ( h x h y h z ) T [ γ 2 M + γ C + K ω 2 μ 0 T ] ( h x h y h z ) ,
[ γ 2 M + γ C + K ω 2 μ 0 T ] h = 0 ,
[ γ 2 M + γ C + K ] h = ω 2 μ 0 T h ,
u = γ h ,
( 0 I K + ω 2 μ 0 T C ) ( h u ) = γ ( I 0 0 M ) ( h u ) .
A v = σ B v .
A v σ 0 B v = σ B v σ 0 B v .
( A σ 0 B ) v = ( σ σ 0 ) B v .
( v , σ ) and ( v , σ σ 0 ) .
n min k 0 β n max k 0 ,
m 2 = M 2 , c 2 = C 2 , s 2 = K ω 2 μ 0 T 2 ,
ξ = s 2 m 2 , δ = 2 s 2 + c 2 ξ .
γ ˜ = γ ξ , M ˜ = ξ 2 δ M , C ˜ = ξ δ C , K ˜ = δ K , T ˜ = δ T .
ε r = ( ε x x ε x y j ε x z ε x y ε y y j ε y z j ε x z j ε y z ε z z ) ,
E = [ E x ( x , y ) i x + E y ( x , y ) i y + j E z ( x , y ) i z ] e j ω t j β z ,
H = [ H x ( x , y ) i x + H y ( x , y ) i y + j H z ( x , y ) i z ] e j ω t j β z ,
ε r = ( ε x x 0 0 0 ε y y j ε y z 0 j ε y z ε z z ) ,
ε y z = 2 n Ce : YIG θ F k 0 .
Δ β = | β + | | β | ,
Δ β p m = 2 ω ε 0 S E * Δ ε E d x d y S [ E * × H + E × H * ] i z d x d y ,
Δ β s v = 2 β ω ε 0 S ε y z y | H x | 2 n 4 d x d y S [ E × H * + E * × H ] i z d x d y ,
ε r = ( ε x x 0 ε x z 0 ε y y 0 ε x z 0 ε z z ) ,
Δ α = | α + | | α | ,
Δ β = | β + | | β | ,
Δ γ p m = Δ α p m + j Δ β p m = 2 ω ε 0 S E a Δ ε E d x d y S [ E a × H E × H a ] i z d x d y ,

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