Abstract

Optical waveguides with structural variations along the waveguide axes are useful in many applications. One way to realize such a variation is to insert a sequence of holes in the waveguide core. In this paper, an efficient numerical method is developed for 2-D slab waveguides with hole arrays in the core. The method divides the waveguide into a number of segments, then jumps over the holes using the Dirichlet-to-Neumann maps of the segments.

© 2009 IEEE

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  1. T. F. Krauss, B. Vogele, C. R. Stanley, R. M. De la Rue, "Waveguide microcavity based on photonic microstructures," IEEE Photon. Technol. Lett. 9, 176-178 (1997).
  2. J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, R. M. De La Rue, "Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task," Opt. Quantum Electron. 34, 455-470 (2002).
  3. M. Gnan, G. Bellanca, H. M. H. Chong, P. Bassi, R. M. De la Rue, "Modelling of photonic wire Bragg gratings," Opt. Quantum Electron. 38, 133-148 (2006).
  4. P. R. Villeneuve, S. H. Fan, J. D. Joannopoulos, K. Y. Lim, G. S. Petrich, L. A. Kolodziejski, R. Reif, "Air-bridge microcavities," Appl. Phys. Lett. 67, 167-169 (1995).
  5. J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, "Nanofabrication of 1-D photonic bandgap structures along a photonic wire," IEEE Photon. Technol. Lett. 8, 491-493 (1996).
  6. J. C. Chen, H. A. Haus, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightw. Technol. 14, 2575-2580 (1996).
  7. J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).
  8. A. S. Jugessur, P. Pottier, R. M. De La Rue, "Engineering the filter response of photonic crystal microcavity filters," Opt. Expr. 12, 1304-1312 (2004).
  9. C. H. Chen, Y. Fainman, "Photonic bandgap microcavities with flat-top response," IEEE J. Sel. Topics Quantum Electron. 13, 262-269 (2007).
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  11. G. Sztefka, H. P. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
  12. J. Willems, J. Haes, R. Baets, "The bidirectional mode expansion method for 2-dimensional wave-guides—The TM case," Opt. Quantum Electron. 27, 995-1007 (1995).
  13. S. F. Helfert, R. Pregla, "Efficient analysis of periodic structures," J. Lightw. Technol. 16, 1694-1702 (1998).
  14. P. Bienstman, R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).
  15. E. Silberstein, P. Lalanne, J. P. Hugonin, Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Amer. A 18, 2865-2875 (2001).
  16. J. Čtyroký, "Improved bidirectional mode expansion propagation algorithm based on Fourier series," J. Lightw. Technol. 25, 2321-2330 (2007).
  17. H. Rao, R. Scarmozzino, R. M. Osgood, "A bidirectional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
  18. H. El-Refaei, D. Yevick, I. Betty, "Stable and noniterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 12, 389-391 (2000).
  19. P. L. Ho, Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators," IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
  20. P. L. Ho, Y. Y. Lu, "A bidirectional beam propagation method for periodic waveguides," IEEE Photon. Technol. Lett. 14, 325-327 (2002).
  21. Y. Y. Lu, S. H. Wei, "A new iterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
  22. L. Yuan, Y. Y. Lu, "An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps," IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).
  23. L. Yuan, Y. Y. Lu, "A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides," J. Lightw. Technol. 25, 3649-3656 (2007).
  24. E. Popov, M. Nevière, B. Gralak, G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Amer. A 19, 33-42 (2002).
  25. Y. Huang, Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightw. Technol. 24, 3448-3453 (2006).
  26. J. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys. 227, 4617-3629 (2008).
  27. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).
  28. W. C. Chew, W. H. Weedon, "A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).
  29. Y. Huang, Y. Y. Lu, S. Li, "Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps," J. Opt. Soc. America B 24, 2860-2867 (2007).
  30. Y. P. Chiou, Y. C. Chiang, H. C. Chang, "Improved three-points formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).
  31. Y. Y. Lu, "Minimizing the discrete reflectivity of perfectly matched layers," IEEE Photon. Technol. Lett. 18, 487-489 (2006).

2008 (1)

J. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys. 227, 4617-3629 (2008).

2007 (4)

Y. Huang, Y. Y. Lu, S. Li, "Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps," J. Opt. Soc. America B 24, 2860-2867 (2007).

L. Yuan, Y. Y. Lu, "A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides," J. Lightw. Technol. 25, 3649-3656 (2007).

C. H. Chen, Y. Fainman, "Photonic bandgap microcavities with flat-top response," IEEE J. Sel. Topics Quantum Electron. 13, 262-269 (2007).

J. Čtyroký, "Improved bidirectional mode expansion propagation algorithm based on Fourier series," J. Lightw. Technol. 25, 2321-2330 (2007).

2006 (4)

M. Gnan, G. Bellanca, H. M. H. Chong, P. Bassi, R. M. De la Rue, "Modelling of photonic wire Bragg gratings," Opt. Quantum Electron. 38, 133-148 (2006).

L. Yuan, Y. Y. Lu, "An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps," IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).

Y. Huang, Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightw. Technol. 24, 3448-3453 (2006).

Y. Y. Lu, "Minimizing the discrete reflectivity of perfectly matched layers," IEEE Photon. Technol. Lett. 18, 487-489 (2006).

2004 (1)

A. S. Jugessur, P. Pottier, R. M. De La Rue, "Engineering the filter response of photonic crystal microcavity filters," Opt. Expr. 12, 1304-1312 (2004).

2002 (4)

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, R. M. De La Rue, "Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task," Opt. Quantum Electron. 34, 455-470 (2002).

E. Popov, M. Nevière, B. Gralak, G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Amer. A 19, 33-42 (2002).

P. L. Ho, Y. Y. Lu, "A bidirectional beam propagation method for periodic waveguides," IEEE Photon. Technol. Lett. 14, 325-327 (2002).

Y. Y. Lu, S. H. Wei, "A new iterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).

2001 (3)

P. L. Ho, Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators," IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).

P. Bienstman, R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).

E. Silberstein, P. Lalanne, J. P. Hugonin, Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Amer. A 18, 2865-2875 (2001).

2000 (2)

H. El-Refaei, D. Yevick, I. Betty, "Stable and noniterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 12, 389-391 (2000).

Y. P. Chiou, Y. C. Chiang, H. C. Chang, "Improved three-points formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).

1999 (1)

H. Rao, R. Scarmozzino, R. M. Osgood, "A bidirectional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).

1998 (1)

S. F. Helfert, R. Pregla, "Efficient analysis of periodic structures," J. Lightw. Technol. 16, 1694-1702 (1998).

1997 (2)

T. F. Krauss, B. Vogele, C. R. Stanley, R. M. De la Rue, "Waveguide microcavity based on photonic microstructures," IEEE Photon. Technol. Lett. 9, 176-178 (1997).

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

1996 (2)

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, "Nanofabrication of 1-D photonic bandgap structures along a photonic wire," IEEE Photon. Technol. Lett. 8, 491-493 (1996).

J. C. Chen, H. A. Haus, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightw. Technol. 14, 2575-2580 (1996).

1995 (2)

P. R. Villeneuve, S. H. Fan, J. D. Joannopoulos, K. Y. Lim, G. S. Petrich, L. A. Kolodziejski, R. Reif, "Air-bridge microcavities," Appl. Phys. Lett. 67, 167-169 (1995).

J. Willems, J. Haes, R. Baets, "The bidirectional mode expansion method for 2-dimensional wave-guides—The TM case," Opt. Quantum Electron. 27, 995-1007 (1995).

1994 (2)

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).

W. C. Chew, W. H. Weedon, "A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).

1993 (1)

G. Sztefka, H. P. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).

1991 (1)

Q. H. Liu, W. C. Chew, "Analysis of discontinuities in planar dielectric waveguides: An eigenmode propagation method," IEEE Trans. Microw. Theory Tech. 39, 422-430 (1991).

Appl. Phys. Lett. (1)

P. R. Villeneuve, S. H. Fan, J. D. Joannopoulos, K. Y. Lim, G. S. Petrich, L. A. Kolodziejski, R. Reif, "Air-bridge microcavities," Appl. Phys. Lett. 67, 167-169 (1995).

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

C. H. Chen, Y. Fainman, "Photonic bandgap microcavities with flat-top response," IEEE J. Sel. Topics Quantum Electron. 13, 262-269 (2007).

IEEE Photon. Technol. Lett. (10)

T. F. Krauss, B. Vogele, C. R. Stanley, R. M. De la Rue, "Waveguide microcavity based on photonic microstructures," IEEE Photon. Technol. Lett. 9, 176-178 (1997).

J. P. Zhang, D. Y. Chu, S. L. Wu, W. G. Bi, R. C. Tiberio, R. M. Joseph, A. Taflove, C. W. Tu, S. T. Ho, "Nanofabrication of 1-D photonic bandgap structures along a photonic wire," IEEE Photon. Technol. Lett. 8, 491-493 (1996).

H. Rao, R. Scarmozzino, R. M. Osgood, "A bidirectional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).

H. El-Refaei, D. Yevick, I. Betty, "Stable and noniterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 12, 389-391 (2000).

P. L. Ho, Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators," IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).

P. L. Ho, Y. Y. Lu, "A bidirectional beam propagation method for periodic waveguides," IEEE Photon. Technol. Lett. 14, 325-327 (2002).

Y. Y. Lu, S. H. Wei, "A new iterative bidirectional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).

L. Yuan, Y. Y. Lu, "An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps," IEEE Photon. Technol. Lett. 18, 1967-1969 (2006).

G. Sztefka, H. P. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).

Y. Y. Lu, "Minimizing the discrete reflectivity of perfectly matched layers," IEEE Photon. Technol. Lett. 18, 487-489 (2006).

IEEE Trans. Microw. Theory Tech. (1)

Q. H. Liu, W. C. Chew, "Analysis of discontinuities in planar dielectric waveguides: An eigenmode propagation method," IEEE Trans. Microw. Theory Tech. 39, 422-430 (1991).

J. Comput. Phys. (2)

J. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys. 227, 4617-3629 (2008).

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1994).

J. Lightw. Technol. (6)

J. Čtyroký, "Improved bidirectional mode expansion propagation algorithm based on Fourier series," J. Lightw. Technol. 25, 2321-2330 (2007).

S. F. Helfert, R. Pregla, "Efficient analysis of periodic structures," J. Lightw. Technol. 16, 1694-1702 (1998).

Y. P. Chiou, Y. C. Chiang, H. C. Chang, "Improved three-points formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).

Y. Huang, Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightw. Technol. 24, 3448-3453 (2006).

J. C. Chen, H. A. Haus, S. H. Fan, P. R. Villeneuve, J. D. Joannopoulos, "Optical filters from photonic band gap air bridges," J. Lightw. Technol. 14, 2575-2580 (1996).

L. Yuan, Y. Y. Lu, "A recursive doubling Dirichlet-to-Neumann map method for periodic waveguides," J. Lightw. Technol. 25, 3649-3656 (2007).

J. Opt. Soc. Amer. A (2)

E. Popov, M. Nevière, B. Gralak, G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Amer. A 19, 33-42 (2002).

E. Silberstein, P. Lalanne, J. P. Hugonin, Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Amer. A 18, 2865-2875 (2001).

J. Opt. Soc. America B (1)

Y. Huang, Y. Y. Lu, S. Li, "Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps," J. Opt. Soc. America B 24, 2860-2867 (2007).

Microw. Opt. Technol. Lett. (1)

W. C. Chew, W. H. Weedon, "A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates," Microw. Opt. Technol. Lett. 7, 599-604 (1994).

Nature (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, E. P. Ippen, "Photonic-bandgap microcavities in optical waveguides," Nature 390, 143-145 (1997).

Opt. Expr. (1)

A. S. Jugessur, P. Pottier, R. M. De La Rue, "Engineering the filter response of photonic crystal microcavity filters," Opt. Expr. 12, 1304-1312 (2004).

Opt. Quantum Electron. (4)

J. Ctyroky, S. Helfert, R. Pregla, P. Bienstman, R. Baets, R. De Ridder, R. Stoffer, G. Klaasse, J. Petracek, P. Lalanne, J. P. Hugonin, R. M. De La Rue, "Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task," Opt. Quantum Electron. 34, 455-470 (2002).

M. Gnan, G. Bellanca, H. M. H. Chong, P. Bassi, R. M. De la Rue, "Modelling of photonic wire Bragg gratings," Opt. Quantum Electron. 38, 133-148 (2006).

P. Bienstman, R. Baets, "Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers," Opt. Quantum Electron. 33, 327-341 (2001).

J. Willems, J. Haes, R. Baets, "The bidirectional mode expansion method for 2-dimensional wave-guides—The TM case," Opt. Quantum Electron. 27, 995-1007 (1995).

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