Abstract

We present rigorous solutions of electromagnetics problems involving 3-D dielectric photonic crystals (PhCs). Problems are formulated with recently developed surface integral equations and solved iteratively using the multilevel fast multipole algorithm (MLFMA). For efficient solutions, iterations are accelerated via robust Schur-complement preconditioners. We show that complicated PhC structures can be analyzed with unprecedented efficiency and accuracy by an effective solver based on the combined tangential formulation, MLFMA, and Schur-complement preconditioners.

© 2011 IEEE

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  3. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
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  7. D. Pissoort, E. Michielssen, D. V. Ginste, F. Olyslager, "Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs," J. Lightw. Technol. 25, 2847-2863 (2007).
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  9. A. Mekis, J. D. Joannopoulos, "Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides," J. Lightw. Technol. 19, 861-865 (2001).
  10. M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightw. Technol. 19, 1970-1975 (2001).
  11. W. Kuang, W. J. Kim, J. D. O'Brien, "Finite-difference time domain method for nonorthogonal unit-cell two-dimensional photonic crystals," J. Lightw. Technol. 25, 2612-2617 (2007).
  12. S. Venakides, M. A. Haider, V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures," J. Appl. Math. 60, 1686-1706 (2000).
  13. T.-L. Wu, J.-S. Chiang, C.-H. Chao, "A novel approach for calculating the dispersions of photonic crystal fibers," IEEE Photon. Technol. Lett. 16, 1492-1494 (2004).
  14. J. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys. 227, 4617-4629 (2008).
  15. A. J. Poggio, E. K. Miller, Computer Techniques for Electromagnetics (Pergamon, 1973).
  16. T. K. Wu, L. L. Tsai, "Scattering from arbitrarily-shaped lossy dielectric bodies of revolution," Radio Sci. 12, 709-718 (1977).
  17. Y. Chang, R. F. Harrington, "A surface formulation for characteristic modes of material bodies," IEEE Trans. Antennas Propagat. AP-25, 789-795 (1977).
  18. C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).
  19. J. Song, C.-C. Lu, W. C. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propagat. 45, 1488-1493 (1997).
  20. P. Ylä-Oijala, M. Taskinen, S. Järvenpää, "Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods," Radio Sci. 40, (2005) doi:10.1029/2004RS003169.
  21. P. Ylä-Oijala, M. Taskinen, "Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects," IEEE Trans. Antennas Propag. 53, 3316-3323 (2005).
  22. P. Ylä-Oijala, M. Taskinen, "Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects," IEEE Trans. Antennas Propag. 53, 1168-1173 (2005).
  23. S. M. Rao, D. R. Wilton, A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag. AP-30, 409-418 (1982).
  24. Ö. Ergül, L. Gürel, "Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm," IEEE Trans. Antennas Prop. 57, 176-187 (2009).
  25. Ö. Ergül, L. Gürel, "Discretization error due to the identity operator in surface integral equations," Comput. Phys. Comm. 180, 1746-1752 (2009).
  26. M. Benzi, G. H. Golub, J. Liesen, "Numerical solution of saddle point problems," Acta Numer. 14, 1-137 (2005).
  27. Ö. Ergül, L. Gürel, "Efficient solution of the electric and magnetic current combined-field integral equation with the multilevel fast multipole algorithm and block-diagonal preconditioning," Radio Sci. 44, RS6001-1-RS6001-15 (2009) RS6001, doi:10.1029/2009RS004143.
  28. E. Chow, Y. Saad, "Approximate inverse techniques for block-partitioned matrices," J. Sci. Comput. 18, 1657-1675 (1997).
  29. P. Ylä-Oijala, M. Taskinen, S. Järvenpää, "Analysis of surface integral equations in electromagnetic scattering and radiation problems," Eng. Anal. Boundary Elem. 32, 196-209 (2008).
  30. C. Siefert, E. de Sturler, "Preconditioners for generalized saddle-point problems," J. Numer. Anal. 44, 1275-1296 (2006).
  31. Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).
  32. T. Malas, L. Gürel, "Incomplete LU preconditioning with multilevel fast multipole algorithm for electromagnetic scattering," J. Sci. Comput. 29, 1476-1494 (2007).
  33. T. Malas, L. Gürel, "Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm," SIAM J. Sci. Comput. (2011).

2011 (1)

T. Malas, L. Gürel, "Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm," SIAM J. Sci. Comput. (2011).

2009 (3)

Ö. Ergül, L. Gürel, "Efficient solution of the electric and magnetic current combined-field integral equation with the multilevel fast multipole algorithm and block-diagonal preconditioning," Radio Sci. 44, RS6001-1-RS6001-15 (2009) RS6001, doi:10.1029/2009RS004143.

Ö. Ergül, L. Gürel, "Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm," IEEE Trans. Antennas Prop. 57, 176-187 (2009).

Ö. Ergül, L. Gürel, "Discretization error due to the identity operator in surface integral equations," Comput. Phys. Comm. 180, 1746-1752 (2009).

2008 (2)

J. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys. 227, 4617-4629 (2008).

P. Ylä-Oijala, M. Taskinen, S. Järvenpää, "Analysis of surface integral equations in electromagnetic scattering and radiation problems," Eng. Anal. Boundary Elem. 32, 196-209 (2008).

2007 (3)

T. Malas, L. Gürel, "Incomplete LU preconditioning with multilevel fast multipole algorithm for electromagnetic scattering," J. Sci. Comput. 29, 1476-1494 (2007).

D. Pissoort, E. Michielssen, D. V. Ginste, F. Olyslager, "Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs," J. Lightw. Technol. 25, 2847-2863 (2007).

W. Kuang, W. J. Kim, J. D. O'Brien, "Finite-difference time domain method for nonorthogonal unit-cell two-dimensional photonic crystals," J. Lightw. Technol. 25, 2612-2617 (2007).

2006 (1)

C. Siefert, E. de Sturler, "Preconditioners for generalized saddle-point problems," J. Numer. Anal. 44, 1275-1296 (2006).

2005 (4)

M. Benzi, G. H. Golub, J. Liesen, "Numerical solution of saddle point problems," Acta Numer. 14, 1-137 (2005).

P. Ylä-Oijala, M. Taskinen, S. Järvenpää, "Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods," Radio Sci. 40, (2005) doi:10.1029/2004RS003169.

P. Ylä-Oijala, M. Taskinen, "Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects," IEEE Trans. Antennas Propag. 53, 3316-3323 (2005).

P. Ylä-Oijala, M. Taskinen, "Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects," IEEE Trans. Antennas Propag. 53, 1168-1173 (2005).

2004 (2)

S. Boscolo, M. Midrio, "Three-dimensional multiple-scattering technique for the analysis of photonic-crystal slabs," J. Lightw. Technol. 22, 2778-2786 (2004).

T.-L. Wu, J.-S. Chiang, C.-H. Chao, "A novel approach for calculating the dispersions of photonic crystal fibers," IEEE Photon. Technol. Lett. 16, 1492-1494 (2004).

2002 (1)

2001 (2)

A. Mekis, J. D. Joannopoulos, "Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides," J. Lightw. Technol. 19, 861-865 (2001).

M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightw. Technol. 19, 1970-1975 (2001).

2000 (2)

R. Stoffer, H. J. W. M. Hoekstra, R. M. De Ridder, E. Van Groesen, F. P. H. Van Beck Um, "Numerical studies of 2-D photonic crystals: Waveguides, coupling between waveguides and filters," Opt. Quantum Electron. 32, 947-961 (2000).

S. Venakides, M. A. Haider, V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures," J. Appl. Math. 60, 1686-1706 (2000).

1999 (2)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).

P. Loschialpo, D. W. Forester, J. Schelleng, "Anomalous transmission through near unit index contrast dielectric photonic crystals," J. Appl. Phys. 86, 5342-5347 (1999).

1997 (2)

J. Song, C.-C. Lu, W. C. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propagat. 45, 1488-1493 (1997).

E. Chow, Y. Saad, "Approximate inverse techniques for block-partitioned matrices," J. Sci. Comput. 18, 1657-1675 (1997).

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).

1982 (1)

S. M. Rao, D. R. Wilton, A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag. AP-30, 409-418 (1982).

1977 (2)

T. K. Wu, L. L. Tsai, "Scattering from arbitrarily-shaped lossy dielectric bodies of revolution," Radio Sci. 12, 709-718 (1977).

Y. Chang, R. F. Harrington, "A surface formulation for characteristic modes of material bodies," IEEE Trans. Antennas Propagat. AP-25, 789-795 (1977).

Acta Numer. (1)

M. Benzi, G. H. Golub, J. Liesen, "Numerical solution of saddle point problems," Acta Numer. 14, 1-137 (2005).

Comput. Phys. Comm. (1)

Ö. Ergül, L. Gürel, "Discretization error due to the identity operator in surface integral equations," Comput. Phys. Comm. 180, 1746-1752 (2009).

Eng. Anal. Boundary Elem. (1)

P. Ylä-Oijala, M. Taskinen, S. Järvenpää, "Analysis of surface integral equations in electromagnetic scattering and radiation problems," Eng. Anal. Boundary Elem. 32, 196-209 (2008).

IEEE Photon. Technol. Lett. (1)

T.-L. Wu, J.-S. Chiang, C.-H. Chao, "A novel approach for calculating the dispersions of photonic crystal fibers," IEEE Photon. Technol. Lett. 16, 1492-1494 (2004).

IEEE Trans. Antennas Propag. (1)

S. M. Rao, D. R. Wilton, A. W. Glisson, "Electromagnetic scattering by surfaces of arbitrary shape," IEEE Trans. Antennas Propag. AP-30, 409-418 (1982).

IEEE Trans. Antennas Propagat. (1)

Y. Chang, R. F. Harrington, "A surface formulation for characteristic modes of material bodies," IEEE Trans. Antennas Propagat. AP-25, 789-795 (1977).

IEEE Trans. Antennas Prop. (1)

Ö. Ergül, L. Gürel, "Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm," IEEE Trans. Antennas Prop. 57, 176-187 (2009).

IEEE Trans. Antennas Propag. (2)

P. Ylä-Oijala, M. Taskinen, "Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects," IEEE Trans. Antennas Propag. 53, 3316-3323 (2005).

P. Ylä-Oijala, M. Taskinen, "Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects," IEEE Trans. Antennas Propag. 53, 1168-1173 (2005).

IEEE Trans. Antennas Propagat. (1)

J. Song, C.-C. Lu, W. C. Chew, "Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects," IEEE Trans. Antennas Propagat. 45, 1488-1493 (1997).

J. Appl. Phys. (1)

P. Loschialpo, D. W. Forester, J. Schelleng, "Anomalous transmission through near unit index contrast dielectric photonic crystals," J. Appl. Phys. 86, 5342-5347 (1999).

J. Comput. Phys. (1)

J. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys. 227, 4617-4629 (2008).

J. Lightw. Technol. (2)

S. Boscolo, M. Midrio, "Three-dimensional multiple-scattering technique for the analysis of photonic-crystal slabs," J. Lightw. Technol. 22, 2778-2786 (2004).

D. Pissoort, E. Michielssen, D. V. Ginste, F. Olyslager, "Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs," J. Lightw. Technol. 25, 2847-2863 (2007).

J. Appl. Math. (1)

S. Venakides, M. A. Haider, V. Papanicolaou, "Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures," J. Appl. Math. 60, 1686-1706 (2000).

J. Lightw. Technol. (3)

A. Mekis, J. D. Joannopoulos, "Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides," J. Lightw. Technol. 19, 861-865 (2001).

M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightw. Technol. 19, 1970-1975 (2001).

W. Kuang, W. J. Kim, J. D. O'Brien, "Finite-difference time domain method for nonorthogonal unit-cell two-dimensional photonic crystals," J. Lightw. Technol. 25, 2612-2617 (2007).

J. Numer. Anal. (1)

C. Siefert, E. de Sturler, "Preconditioners for generalized saddle-point problems," J. Numer. Anal. 44, 1275-1296 (2006).

J. Sci. Comput. (2)

T. Malas, L. Gürel, "Incomplete LU preconditioning with multilevel fast multipole algorithm for electromagnetic scattering," J. Sci. Comput. 29, 1476-1494 (2007).

E. Chow, Y. Saad, "Approximate inverse techniques for block-partitioned matrices," J. Sci. Comput. 18, 1657-1675 (1997).

Opt. Lett. (1)

Opt. Quantum Electron. (1)

R. Stoffer, H. J. W. M. Hoekstra, R. M. De Ridder, E. Van Groesen, F. P. H. Van Beck Um, "Numerical studies of 2-D photonic crystals: Waveguides, coupling between waveguides and filters," Opt. Quantum Electron. 32, 947-961 (2000).

Phys. Rev. B (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).

Phys. Rev. Lett. (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).

Radio Sci. (1)

T. K. Wu, L. L. Tsai, "Scattering from arbitrarily-shaped lossy dielectric bodies of revolution," Radio Sci. 12, 709-718 (1977).

Radio Sci. (2)

P. Ylä-Oijala, M. Taskinen, S. Järvenpää, "Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods," Radio Sci. 40, (2005) doi:10.1029/2004RS003169.

Ö. Ergül, L. Gürel, "Efficient solution of the electric and magnetic current combined-field integral equation with the multilevel fast multipole algorithm and block-diagonal preconditioning," Radio Sci. 44, RS6001-1-RS6001-15 (2009) RS6001, doi:10.1029/2009RS004143.

SIAM J. Sci. Comput. (1)

T. Malas, L. Gürel, "Schur complement preconditioners for surface integral-equation formulations of dielectric problems solved with the multilevel fast multipole algorithm," SIAM J. Sci. Comput. (2011).

Other (4)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals, Molding the Flow of Light (Princeton Univ. Press, 2008).

Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves (Springer-Verlag, 1969).

A. J. Poggio, E. K. Miller, Computer Techniques for Electromagnetics (Pergamon, 1973).

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