Abstract

This paper presents a new full-vectorial finite-element method in a local cylindrical coordinate system, to effectively analyze bending losses in photonic wires. The discretization is performed in the cross section of a three-dimensional curved waveguide, using hybrid edge/nodal elements. The solution region is truncated by anisotropic, perfectly matched layers in the cylindrical coordinate system, to deal properly with leaky modes of the waveguide. This approach is used to evaluate bending losses in silicon wire waveguides. The numerical results of the present approach are compared with results calculated with an equivalent straight waveguide approach and with reported experimental data. These comparisons together demonstrate the validity of the present approach based on the cylindrical coordinate system and also clarifies the limited validity of the equivalent straight waveguide approximation.

© 2006 Optical Society of America

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  1. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  6. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
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    [CrossRef]
  23. W. W. Lui, C. L. Xu, T. Hirono, K. Yokoyama, and W. P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightwave Technol. 16, 910-914 (1998).
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  26. F. L. Teixeira and W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
    [CrossRef]
  27. F. L. Teixeira and W. C. Chew, "General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media," IEEE Microw. Guid. Wave Lett. 8, 223-225 (1998).
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  28. 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]
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    [CrossRef]
  30. W. C. Chew and W. Weedon, "A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
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    [CrossRef]
  33. Y. Tsuchida, K. Saitoh, and M. Koshiba, "Design and characterization of single-mode holey fibers with low bending losses," Opt. Express 13, 4770-4779 (2005).
    [CrossRef] [PubMed]
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2006 (1)

2005 (5)

2004 (9)

C. Li, N. Ma, and A. W. Poon, "Waveguide-coupled octagonal microdisk channel add-drop filters," Opt. Lett. 29, 471-473 (2004).
[CrossRef] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

A. Sakai, T. Fukazawa, and T. Baba, "Estimation of polarization crosstalk at a micro-bend in Si-photonic wire waveguide," J. Lightwave Technol. 22, 520-525 (2004).
[CrossRef]

A. Jiang, S. Shi, G. Jin, and D. W. Prather, "Performance analysis of three dimensional high index contrast dielectric waveguides," Opt. Express 12, 633-643 (2004).
[CrossRef] [PubMed]

E. Cassan, L. Vivien, and S. Laval, "Polarization-independent 90°-turns in single-mode micro-waveguides on silicon-on-insulator wafers for telecommunication wavelengths," Opt. Commun. 235, 83-88 (2004).
[CrossRef]

Y. A. Vlasov and S. J. McNab, "Losses in single-mode silicon-on-insulator strip waveguides and bends," Opt. Express 12, 1622-1631 (2004).
[CrossRef] [PubMed]

Y. Tsuji and M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorical finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).
[CrossRef]

S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, "Full vectorial finite element modal solution of curved optical waveguides," Laser Phys. Lett. 2, 131-136 (2004).
[CrossRef]

D. Dai and S. He, "Analysis of characteristics of bent rib waveguides," J. Opt. Soc. Am. A 21, 113-121 (2004).
[CrossRef]

2002 (2)

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

N. N. Feng, G. R. Zhou, C. Xu, and W. P. Huang, "Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers," J. Lightwave Technol. 20, 1976-1980 (2002).
[CrossRef]

2001 (3)

2000 (1)

1998 (3)

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

W. W. Lui, C. L. Xu, T. Hirono, K. Yokoyama, and W. P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightwave Technol. 16, 910-914 (1998).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

1997 (3)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

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

F. L. Teixeira and W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
[CrossRef]

1996 (1)

S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

1994 (2)

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994).
[CrossRef]

W. C. Chew and W. Weedon, "A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

1993 (1)

T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by the finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993).
[CrossRef]

1990 (1)

Y. Cheng, W. Lin, and Y. Fujii, "Local field analysis of bent graded-index planar waveguides," J. Lightwave Technol. 8, 1461-1469 (1990).
[CrossRef]

1983 (1)

I. S. Duff and J. K. Reid, "The multifrontal solution of indefinite sparse symmetric linear equations," ACM Trans. Math. Softw. 9, 302-325 (1983).
[CrossRef]

Ahmad, R. U.

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Baba, T.

K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70×60 μm2 size based on Si photonic wire waveguides," Electron. Lett. 41, 801-802 (2005).
[CrossRef]

A. Sakai, T. Fukazawa, and T. Baba, "Estimation of polarization crosstalk at a micro-bend in Si-photonic wire waveguide," J. Lightwave Technol. 22, 520-525 (2004).
[CrossRef]

A. Sakai, G. Hara, and T. Baba, "Propagation characteristics of ultrahigh-Δ optical waveguide on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 40, L383-L385 (2001).
[CrossRef]

Baets, R.

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Berenger, J.

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994).
[CrossRef]

Bogaerts, W.

Brouckaert, J.

Cassan, E.

E. Cassan, L. Vivien, and S. Laval, "Polarization-independent 90°-turns in single-mode micro-waveguides on silicon-on-insulator wafers for telecommunication wavelengths," Opt. Commun. 235, 83-88 (2004).
[CrossRef]

Cerrina, F.

Cheng, Y.

Y. Cheng, W. Lin, and Y. Fujii, "Local field analysis of bent graded-index planar waveguides," J. Lightwave Technol. 8, 1461-1469 (1990).
[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 Microw. Guid. Wave Lett. 8, 223-225 (1998).
[CrossRef]

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

F. L. Teixeira and W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
[CrossRef]

W. C. Chew and W. Weedon, "A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Christodoulides, D. N.

Chu, S. T.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Dadap, J. I.

Dai, D.

Davies, J. B.

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

Duff, I. S.

I. S. Duff and J. K. Reid, "The multifrontal solution of indefinite sparse symmetric linear equations," ACM Trans. Math. Softw. 9, 302-325 (1983).
[CrossRef]

Dumon, P.

El-Ganainy, R.

Espinola, R. L.

Fan, S.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Feng, N. N.

Fernandez, F. A.

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

Ferrera, J.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Foresi, J. S.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Fujii, Y.

Y. Cheng, W. Lin, and Y. Fujii, "Local field analysis of bent graded-index planar waveguides," J. Lightwave Technol. 8, 1461-1469 (1990).
[CrossRef]

Fukazawa, T.

Fukuda, H.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Gopinath, A.

S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

Grattan, K. T. V.

S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, "Full vectorial finite element modal solution of curved optical waveguides," Laser Phys. Lett. 2, 131-136 (2004).
[CrossRef]

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

Greene, W.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Hara, G.

A. Sakai, G. Hara, and T. Baba, "Propagation characteristics of ultrahigh-Δ optical waveguide on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 40, L383-L385 (2001).
[CrossRef]

Haus, H. A.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

He, S.

Heaton, J. M.

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

Hirono, T.

Huang, W. P.

Ippen, E. P.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Itabashi, S.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Jiang, A.

Jin, G.

Joannopoulos, J. D.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Kim, S.

S. Kim and A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightwave Technol. 14, 2085-2092 (1996).
[CrossRef]

Kimerling, L. C.

K. K. Lee, D. R. Lim, L. C. Kimerling, J. Shin, and F. Cerrina, "Fabrication of ultralow-loss Si/SiO2 waveguides by roughness reduction," Opt. Lett. 26, 1888-1890 (2001).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Koshiba, M.

Y. Tsuchida, K. Saitoh, and M. Koshiba, "Design and characterization of single-mode holey fibers with low bending losses," Opt. Express 13, 4770-4779 (2005).
[CrossRef] [PubMed]

Y. Tsuji and M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorical finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).
[CrossRef]

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]

T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by the finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993).
[CrossRef]

Laval, S.

E. Cassan, L. Vivien, and S. Laval, "Polarization-independent 90°-turns in single-mode micro-waveguides on silicon-on-insulator wafers for telecommunication wavelengths," Opt. Commun. 235, 83-88 (2004).
[CrossRef]

Lee, K. K.

Li, C.

Lim, D. R.

Lin, W.

Y. Cheng, W. Lin, and Y. Fujii, "Local field analysis of bent graded-index planar waveguides," J. Lightwave Technol. 8, 1461-1469 (1990).
[CrossRef]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Little, B. E.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

Lui, W. W.

Ma, N.

Malek, F. A.

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

McNab, S. J.

Mokhov, S.

Morita, H.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Motegi, A.

K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70×60 μm2 size based on Si photonic wire waveguides," Electron. Lett. 41, 801-802 (2005).
[CrossRef]

Nötzel, R.

Obayya, S. S. A.

S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, "Full vectorial finite element modal solution of curved optical waveguides," Laser Phys. Lett. 2, 131-136 (2004).
[CrossRef]

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

Ohno, F.

K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70×60 μm2 size based on Si photonic wire waveguides," Electron. Lett. 41, 801-802 (2005).
[CrossRef]

Osgood, R. M.

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Pizzuto, F.

Poon, A. W.

Prather, D. W.

Rahman, B. M. A.

S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, "Full vectorial finite element modal solution of curved optical waveguides," Laser Phys. Lett. 2, 131-136 (2004).
[CrossRef]

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

Reid, J. K.

I. S. Duff and J. K. Reid, "The multifrontal solution of indefinite sparse symmetric linear equations," ACM Trans. Math. Softw. 9, 302-325 (1983).
[CrossRef]

Roelkens, G.

Saitoh, K.

Sakai, A.

A. Sakai, T. Fukazawa, and T. Baba, "Estimation of polarization crosstalk at a micro-bend in Si-photonic wire waveguide," J. Lightwave Technol. 22, 520-525 (2004).
[CrossRef]

A. Sakai, G. Hara, and T. Baba, "Propagation characteristics of ultrahigh-Δ optical waveguide on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 40, L383-L385 (2001).
[CrossRef]

Sasaki, K.

K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70×60 μm2 size based on Si photonic wire waveguides," Electron. Lett. 41, 801-802 (2005).
[CrossRef]

Shi, S.

Shin, J.

Shoji, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Smit, M.

Smith, H. I.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Steel, M. J.

Steinmeyer, G.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Taillaert, D.

Takahashi, J.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Takahashi, M.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Tamechika, E.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

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 Microw. Guid. Wave Lett. 8, 223-225 (1998).
[CrossRef]

F. L. Teixeira and W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
[CrossRef]

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

Thoen, E. R.

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Tsuchida, Y.

Tsuchizawa, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Tsuji, Y.

Y. Tsuji and M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorical finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).
[CrossRef]

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]

Van Thourhout, D.

Villeneuve, P. R.

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

Vivien, L.

E. Cassan, L. Vivien, and S. Laval, "Polarization-independent 90°-turns in single-mode micro-waveguides on silicon-on-insulator wafers for telecommunication wavelengths," Opt. Commun. 235, 83-88 (2004).
[CrossRef]

Vlasov, Y. A.

Watanabe, T.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Weedon, W.

W. C. Chew and W. Weedon, "A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Xu, C.

Xu, C. L.

Yamada, K.

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Yamamoto, T.

T. Yamamoto and M. Koshiba, "Numerical analysis of curvature loss in optical waveguides by the finite-element method," J. Lightwave Technol. 11, 1579-1583 (1993).
[CrossRef]

Yokoyama, K.

Zhou, G. R.

ACM Trans. Math. Softw. (1)

I. S. Duff and J. K. Reid, "The multifrontal solution of indefinite sparse symmetric linear equations," ACM Trans. Math. Softw. 9, 302-325 (1983).
[CrossRef]

Electromagnetics (1)

Y. Tsuji and M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorical finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).
[CrossRef]

Electron. Lett. (1)

K. Sasaki, F. Ohno, A. Motegi, and T. Baba, "Arrayed waveguide grating of 70×60 μm2 size based on Si photonic wire waveguides," Electron. Lett. 41, 801-802 (2005).
[CrossRef]

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

T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Top. Quantum Electron. 11, 232-240 (2005).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (3)

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

F. L. Teixeira and W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guid. Wave Lett. 7, 371-373 (1997).
[CrossRef]

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

IEEE Photon. Technol. Lett. (2)

F. A. Malek, S. S. A. Obayya, B. M. A. Rahman, J. B. Davies, F. A. Fernandez, J. M. Heaton, and K. T. V. Grattan, "Full vectorial finite-element analysis of sharp optical waveguide corners," IEEE Photon. Technol. Lett. 14, 1527-1529 (2002).
[CrossRef]

B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimerling, and W. Greene, "Ultra-compact Si-SiO2 microring resonator optical channel dropping filters," IEEE Photon. Technol. Lett. 10, 549-551 (1998).
[CrossRef]

J. Comp. Phys. (1)

J. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comp. Phys. 114, 185-200 (1994).
[CrossRef]

J. Lightwave Technol. (7)

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

Jpn. J. Appl. Phys. (1)

A. Sakai, G. Hara, and T. Baba, "Propagation characteristics of ultrahigh-Δ optical waveguide on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 40, L383-L385 (2001).
[CrossRef]

Laser Phys. Lett. (1)

S. S. A. Obayya, B. M. A. Rahman, and K. T. V. Grattan, "Full vectorial finite element modal solution of curved optical waveguides," Laser Phys. Lett. 2, 131-136 (2004).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

W. C. Chew and W. Weedon, "A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
[CrossRef]

Nature (2)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
[CrossRef]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

E. Cassan, L. Vivien, and S. Laval, "Polarization-independent 90°-turns in single-mode micro-waveguides on silicon-on-insulator wafers for telecommunication wavelengths," Opt. Commun. 235, 83-88 (2004).
[CrossRef]

Opt. Express (7)

Opt. Lett. (2)

Other (1)

J. Jin, The Finite Element Method in Electromagnetics (John Wiley & Sons, Inc., 1993), Chap. 4.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic representation of the photonic wire bend and the local cylindrical coordinates: (a) bird’s-eye view and (b) solution region.

Fig. 2.
Fig. 2.

Wavelength dependence of bending losses of the fundamental quasi-TE modes for different radii: (a) R=1 µm, (b) R=2 µm, and (c) R=5 µm. The red lines represent the results of the modal analysis in the CCS and the green lines represent those for the ESW approximation. The experimental data [11] are also represented with the blue lines.

Fig. 3.
Fig. 3.

Wavelength dependence of bending losses of the fundamental quasi-TM modes for different radii: (a) R=1 µm, (b) R=2 µm, and (c) R=5 µm. The red lines represent the results of the modal analysis in the CCS and the green lines represent those for the ESW approximation. The experimental data [11] are also represented with the blue lines.

Fig. 4.
Fig. 4.

Bend radius dependence of bending losses, (a) for the quasi-TE mode and (b) for the quasi-TM mode.

Fig. 5.
Fig. 5.

Poynting vector S distributions at R=2 µm, (a) for the quasi-TE mode and (b) for the quasi-TM mode, where the operating wavelength is λ=1.55 µm.

Fig. 6.
Fig. 6.

Hybridness as a function of bend radius.

Tables (1)

Tables Icon

Table 1. PML parameters.

Equations (52)

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

r ~ = [ x ~ , y ~ , z ~ ] T
= [ 0 x s x ( x ) d x , 0 y s y ( y ) d y , 0 z s z ( z ) d z ] T ,
s i = 1 j ( ρ d PML ) m tan δ
r ~ = [ Γ ] · r
r = [ x , y , z ] T
[ Γ ] = [ x ~ x 0 0 0 y ~ y 0 0 0 z ~ z ] .
x ~ x = s x ( x ) , y ~ y = s y ( y ) , z ~ z = s z ( z ) .
× ( [ S ] 1 A ) = [ 0 R R + x z y R R + x z 0 1 R + x + x y x 0 ] [ s x A x s y A y R + x ~ R + x s z A z ]
= [ R + x ~ R + x s y s z ( R R + x ~ A y z ~ A z y ~ ) R + x ~ R + x s z s x ( R R + x ~ A x z ~ + 1 R + x ~ A z + A z x ~ ) s x s y ( A x y ~ A y x ~ ) ]
= ( det [ S ] ) 1 [ S ] ~ × A
[ S ] = [ 1 s x 0 0 0 1 s y 0 0 0 R + x R + x ~ 1 s z ]
~ × = [ 0 R R + x ~ z ~ y ~ R R + x ~ z ~ 0 1 R + x ~ + x ~ y ~ x ~ 0 ] .
× E ( r ) = j ω μ 0 [ μ ( r ) ] H ( r )
× H ( r ) = j ω ε 0 [ ε ( r ) ] E ( r ) .
~ × E c ( r ~ ) = j ω μ 0 [ μ ( r ~ ) ] H c ( r ~ )
~ × H c ( r ~ ) = j ω ε 0 [ ε ( r ~ ) ] E c ( r ~ ) ,
× ( [ S ] 1 E c ( [ Γ ] · r ) ) = j ω μ 0 ( det [ S ] ) 1 [ S ] [ μ ( [ Γ ] · r ) ] H c ( [ Γ ] · r )
× ( [ S ] 1 H c ( [ Γ ] · r ) ) = j ω ε 0 ( det [ S ] ) 1 [ S ] [ ε ( [ Γ ] · r ) ] E c ( [ Γ ] · r ) .
E a ( r ) = [ S ] 1 E c ( [ Γ ] · r )
H a ( r ) = [ S ] 1 H c ( [ Γ ] · r ) ,
× E a ( r ) = j ω μ 0 [ μ ] PML H a ( r )
× H a ( r ) = j ω ε 0 [ ε ] PML E a ( r )
[ μ ] PML = ( det [ S ] ) 1 [ S ] [ μ ( [ Γ ] · r ) ] [ S ]
[ ε ] PML = ( det [ S ] ) 1 [ S ] [ ε ( [ Γ ] · r ) ] [ S ] .
× ( [ p ] × Φ ) k 0 2 [ q ] Φ = 0
[ p ] = [ p x 0 0 0 p y 0 0 0 p z ] = [ μ ] PML 1 , [ q ] = [ q x 0 0 0 q y 0 0 0 q z ] = [ ε ] PML for Φ = E
[ p ] = [ p x 0 0 0 p y 0 0 0 p z ] = [ ε ] PML 1 , [ q ] = [ q x 0 0 0 q y 0 0 0 q z ] = [ μ ] PML for Φ = H ,
Φ = [ ϕ x ϕ y ϕ z ] exp ( j β z ) = [ N ] T [ { ϕ t } e { ϕ z } e ] exp ( j β z ) ,
[ N ] = [ { U } { V } { 0 } { 0 } { 0 } j β { N } ]
[ [ [ N ] * exp ( j β z ) { × [ N ] T exp ( j β z ) } k 0 2 [ N ] * [ q ] [ N ] T ] d x d y ] [ { ϕ t } e { ϕ z } e ] = [ 0 ]
[ [ { × [ N ] * exp ( j β z ) } { [ p ] × [ N ] T exp ( j β z ) } k 0 2 [ N ] * [ q ] [ N ] T ] d x d y ] [ { ϕ t } e { ϕ z } e ]
[ Γ v [ N ] * exp ( j β z ) [ { [ p ] × [ N ] T exp ( j β z ) } × n ] d Γ
+ Γ u { n × [ N ] * exp ( j β z ) } { [ p ] × [ N ] T exp ( j β z ) } d Γ ] [ { ϕ t } e { ϕ z } e ]
= [ 0 ]
n × [ N ] T exp ( j β z ) = 0 on Γ u
n × { [ p ] × [ N ] T exp ( j β z ) } = 0 on Γ v .
[ [ { × [ N ] * exp ( j β z ) } { [ p ] × [ N ] T exp ( j β z ) } k 0 2 [ N ] * [ q ] [ N ] T ] d x d y ] [ { ϕ t } e { ϕ z } e ] = [ 0 ]
{ × [ N ] * exp ( j β z ) } { [ p ] × [ N ] T exp ( j β z ) }
= [ j β R R + x { V } j β R R + x { U } { U } y { V } x j β { N } y j β ( 1 R + x { N } + { N } x ) { 0 } ] [ p x 0 0 0 p y 0 0 0 p z ] [ j β R R + x { V } T j β { N } T y j β R R + x { U } T j β ( 1 R + x { N } T + { N } T x ) [ U ] T y [ V ] T x { 0 } T ]
[ β 2 ( R R + x ) 2 p x { V } { V } T + β 2 ( R R + x ) 2 p y { U } { U } T + p z ( { U } y { V } x ) ( { U } T y [ V ] T x ) β 2 R R + x p x { V } { N } T y + β 2 R R + x p y { U } ( 1 R + x { N } T + { N } T x ) β 2 R R + x p x { N } y { V } T + β 2 R R + x p x ( 1 R + x { N } + { N } x ) { U } T β 2 p x { N } y { N } T y + β 2 p y ( 1 R + x { N } + { N } x ) ( 1 R + x { N } T + { N } T x ) ] .
[ N ] * [ q ] [ N ] T = [ { U } { V } { 0 } { 0 } { 0 } j β { N } ] [ q x 0 0 0 q y 0 0 0 q z ] [ { U } T { 0 } T { V } T { 0 } T { 0 } T j β { N } T ]
= [ q x { U } { U } T + q y { V } { V } T { 0 } { 0 } β 2 q z { N } { N } T ] .
[ K ] { ϕ } = β 2 [ M ] { ϕ }
[ K ] = [ [ K tt ] [ 0 ] [ 0 ] [ 0 ] ]
[ M ] = [ [ M tt ] [ M tz ] [ M zt ] [ M zz ] ]
[ K tt ] = e e [ k 0 2 ( q x { U } { U } T + q y { V } { V } T )
p z ( { U } y { V } x ) ( { U } T y { V } T x ) ] d x d y
[ M tt ] = e e [ ( R R + x ) 2 p x { V } { V } T + ( R R + x ) 2 p y { U } { U } T ] d x d y
[ M tz ] = [ M zt ] T
= e e [ R R + x p x { V } { N } T y + R R + x p y { U } ( 1 R + x { N } T + { N } T x ) ] d x d y
[ M zz ] = e e [ k 0 2 q z { N } { N } T + p x { N } y { N } T y
+ p y ( 1 R + x { N } + { N } x ) ( 1 R + x { N } T + { N } T x ) ] d x d y ,

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