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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Editor: Henry van Driel
  • Vol. 29, Iss. 6 — Jun. 1, 2012
  • pp: 1356–1360

Simple boundary condition for terminating photonic crystal waveguides

Zhen Hu and Ya Yan Lu  »View Author Affiliations

JOSA B, Vol. 29, Issue 6, pp. 1356-1360 (2012)

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Many photonic crystal (PhC) devices are nonperiodic structures due to the introduced defects in an otherwise perfectly periodic PhC, and they are often connected by PhC waveguides that serve as input and output ports. Numerical simulation of a PhC device requires boundary conditions to terminate PhC waveguides that extend to infinity. The rigorous boundary condition for terminating a PhC waveguide is a nonlocal condition that connects the wave field on the entire surface (or line in two-dimensional problems) transverse to the waveguide axis, and it is relatively difficult to use, especially for realistic devices, such as those in PhC slabs. In this paper, a simple approximate boundary condition involving a few points in the waveguide axis direction is introduced. The new boundary condition is used with the Dirichlet-to-Neumann map method to take advantage of the lattice structures and identical unit cells in PhC devices. Comparisons with the rigorous nonlocal boundary condition indicate that the simple boundary condition gives accurate solutions if the computational domain is enlarged by a few lattice constants in each direction.

© 2012 Optical Society of America

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(130.5296) Integrated optics : Photonic crystal waveguides
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Diffraction and Gratings

Original Manuscript: January 23, 2012
Revised Manuscript: March 28, 2012
Manuscript Accepted: March 28, 2012
Published: May 22, 2012

Zhen Hu and Ya Yan Lu, "Simple boundary condition for terminating photonic crystal waveguides," J. Opt. Soc. Am. B 29, 1356-1360 (2012)

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed.(Princeton University, 2008).
  2. D. W. Prather, S. Shi, A. Sharkawy, and G. J. Schneider, Photonic Crystals: Theory, Applications, and Fabrication (Wiley, 2009).
  3. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).
  4. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  5. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]
  6. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretching coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994). [CrossRef]
  7. G. O. Olaofe, “Scattering by an arbitrary configuration of parallel circular cylinders,” J. Opt. Soc. Am. 60, 1233–1236 (1970). [CrossRef]
  8. 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]
  9. G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A 14, 3323–3332 (1997). [CrossRef]
  10. J. Yonekura, M. Ikeda, and T. Baba, “Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method,” J. Lightwave Technol. 17, 1500–1508 (1999). [CrossRef]
  11. P. A. Martin, Multiple Scattering (Cambridge University, 2006).
  12. Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
  13. Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008). [CrossRef]
  14. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006). [CrossRef]
  15. J. Yuan, Y. Y. Lu, and X. Antoine, “Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps,” J. Comput. Phys. 227, 4617–4629 (2008). [CrossRef]
  16. P. Joly, J.-R. Li, and S. Fliss, “Exact boundary conditions for periodic waveguides containing a local perturbation,” Commun. Comput. Phys. 1, 945–973 (2006).
  17. M. Ehrhardt, J. G. Sun, and C. Zheng, “Evaluation of scattering operators for semi-infinite periodic arrays,” Commun. Math. Sci. 7, 347–364 (2009).
  18. Y. Huang, Y. Y. Lu, and S. Li, “Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. B 24, 2860–2867 (2007). [CrossRef]
  19. S. Li and Y. Y. Lu, “Efficient method for computing leaky modes in two-dimensional photonic crystal waveguides,” J. Lightwave Technol. 28, 978–983 (2010). [CrossRef]
  20. L. Yuan and Y. Y. Lu, “An efficient numerical method for analyzing photonic crystal slab waveguides,” J. Opt. Soc. Am. B 28, 2265–2270 (2011). [CrossRef]
  21. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996). [CrossRef]
  22. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. 18, 102–110 (2000). [CrossRef]
  23. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003). [CrossRef]
  24. C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. 36, 145–163 (2004). [CrossRef]
  25. S. H. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162–165 (2001). [CrossRef]
  26. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).
  27. L. Yuan and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing hole arrays in a slab,” J. Opt. Soc. Am. B 27, 2568–2579 (2010). [CrossRef]
  28. M. Ehrhardt, ed., Wave Propagation in Periodic Media—Analysis, Numerical Techniques and Practical Applications, Progress in Computational Physics, Vol. 1 (Bentham Science, 2010).

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