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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 22 — Oct. 29, 2007
  • pp: 14454–14466

Computing Photonic Crystal Defect Modes by Dirichlet-to-Neumann Maps

Shaojie Li and Ya Yan Lu  »View Author Affiliations

Optics Express, Vol. 15, Issue 22, pp. 14454-14466 (2007)

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We develop an efficient numerical method for computing defect modes in two dimensional photonic crystals based on the Dirichlet-to-Neumann (DtN) maps of the defect and normal unit cells. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative. The frequencies of the defect modes are solved from a condition that a small matrix is singular. The size of the matrix is related to the number of points used to discretize the boundary of the defect cell. The matrix is obtained by solving a linear system involving only discrete points on the edges of the unit cells in a truncated domain.

© 2007 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

Original Manuscript: September 4, 2007
Revised Manuscript: October 10, 2007
Manuscript Accepted: October 17, 2007
Published: October 18, 2007

Shaojie Li and Ya Yan Lu, "Computing Photonic Crystal Defect Modes by Dirichlet-to-Neumann Maps," Opt. Express 15, 14454-14466 (2007)

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  1. J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, Princeton, NJ. 1995).
  2. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith and S. Schultz, "Microwave Propagation in two dimensional dielectric lattices," Phys. Rev. Lett. 67, 2017-2020 (1991). [CrossRef] [PubMed]
  3. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer and J. D. Joannopoulos, "Donor and acceptor modes in photonic band-structure," Phys. Rev. Lett. 67, 3380-3383 (1991). [CrossRef] [PubMed]
  4. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall and P. M. Platzman, "Photonic band structure and defects in one and two dimensions," J. Opt. Soc. Am. B 10, 314-321 (1993). [CrossRef]
  5. S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E 71, 056606 (2005). [CrossRef]
  6. R. R. Villeneuve, S. H. Fan and J. D. Joannopoulos, "Microcavities in photonic crystals: Mode symmetry, tunability, and coupling efficiency," Phys. Rev. B 54, 7837-7842 (1996). [CrossRef]
  7. X. P. Feng and Y. Arakawa, "Defect modes in two-dimensional triangular photonic crystals," Jpn. J. Appl. Phys. 36, L120-L123, (1997). [CrossRef]
  8. K. M. Ho, C. T. Chan and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Phys. Rev. Lett. 65, 3152-3155 (1990). [CrossRef] [PubMed]
  9. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
  10. D. C. Dobson, "An efficient method for band structure calculations in 2D photonic crystals," J. Comput. Phys. 149, 363-376 (1999). [CrossRef]
  11. W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case," J. Comput. Phys. 150, 468-481 (1999). [CrossRef]
  12. H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996). [CrossRef]
  13. C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two dimensional photonic crystals," Opt. Express 12, 1397-1408 (2004). [CrossRef] [PubMed]
  14. P. J. Chiang and C. P. Yu and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007). [CrossRef]
  15. V. F. Rodr´ıguez-Esquerre, M. Koshiba and H. E. Hern´andez-Figueroa, "Finite-element analysis of photonic crystal cavities: Time and frequency domains," J. Lightwave Technol. 23, 1514-1521 (2005). [CrossRef]
  16. K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997). [CrossRef]
  17. K. Sakoda, "Numerical study on localized defect modes in two-dimensional triangular photonic crystals," J. Appl. Phys. 84, 1210-1214 (1998). [CrossRef]
  18. V. Kuzmiak and A. A. Maradudin, "Localized defect modes in a two-dimensional triangular photonic crystal," Phys. Rev. B 57, 15242-15250 (1998). [CrossRef]
  19. N. Stojíc, J. Glimm, Y. Deng, and J. W. Haus, "Transverse magnetic defect modes in two-dimensional triangular-lattice photonic crystals," Phys. Rev. E 64, 056614 (2001). [CrossRef]
  20. S. P. Guo and S. Albin, "Numerical techniques for excitation and analysis of defect modes in photonic crystals," Opt. Express 11, 1080-1089 (2003). [CrossRef] [PubMed]
  21. V. F. Rodríguez-Esquerre, M. Koshiba and H. E. Hernández-Figueroa, "Finite-element time-domain analysis of 2-D photonic crystal resonant cavities," IEEE Photon. Technol. Lett. 16, 816-818 (2004). [CrossRef]
  22. R. Moussa, L. Salomon, F. de Fornel and H. Aourag, "Numerical study on localized defect modes in twodimensional lattices: a high Q-resonant cavity," Physica B 338, 97-102 (2003). [CrossRef]
  23. J. H. Yuan and Y. Y. Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 23, 3217-3222 (2006). [CrossRef]
  24. J. H. Yuan and Y. Y. Lu, "Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice," Opt. Commun. 273, 114-120 (2007). [CrossRef]
  25. Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006). [CrossRef]
  26. Y. H. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 337-349 (2007).
  27. S. J. Li and Y. Y. Lu, "Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells," J. Opt. Soc. Am. A 24, 2438-2442 (2007). [CrossRef]
  28. J. H. Yuan, Y. Y. Lu, X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," submitted for publication.
  29. Y. Y. Lu and S.-T. Yau, "Eigenvalues of the Laplacian through boundary integral equations," SIAM J. Matrix Anal. Appl. 12, 597-609 (1991). [CrossRef]
  30. T. Lu and D. Yevick, "A vectorial boundary element method analysis of integrated optical waveguides," J. Lightwave Technol. 21, 1793-1807 (2003). [CrossRef]
  31. L. Prkna, M. Hubalek and J. Ctyroky, "Vectorial eigenmode solver for bent waveguides based on mode matching," IEEE Photon. Technol. Lett. 16, 2057-2059 (2004). [CrossRef]
  32. D. Felbacq, G. Tayeb and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994). [CrossRef]
  33. K. B. Dossou, R. C. McPhedran, L. C. Botten, A. A. Asatryan and C. M. de Sterke, "Gap-edge asymptotics of defect modes in two-dimensional photonic crystals," Opt. Express 15, 4753-4762 (2007). [CrossRef] [PubMed]

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