Optical and Quantum Electronics

, Volume 40, Issue 11–12, pp 921–932

Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices



A typical photonic crystal (PhC) device has only a small number of distinct unit cells. The Dirichlet-to-Neumann (DtN) map of a unit cell is an operator that maps the wave field to its normal derivative on the boundary of the cell. Based on the DtN maps of the unit cells, a PhC device can be efficiently analyzed by solving the wave field only on edges of the unit cells. In this paper, the DtN map method is further improved by an operator marching method assuming that a main propagation direction can be identified in at least part of the device. A Bloch mode expansion method is also developed for structures exhibiting partial periodicity. Both methods are formulated on a set of curves for maximum flexibility. Numerical examples are used to illustrate the efficiency of the improved DtN map method.


Photonic crystal Numerical method Dirichlet-to-Neumann map Operator marching Bloch mode expansion 


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  1. Felbacq J.D., Tayeb C., Maystre D.: Scattering by a random set of parallel cylinders. J. Optical Soc. Am. A 11, 2526–2538 (1994)CrossRefADSMathSciNetGoogle Scholar
  2. Fujisawa T., Koshiba T.: Finite-element modeling of nonlinear Mach-Zehnder interferometers based on photonic-crystal waveguides for all-optical signal processing. J. Lightwave Technol. 24, 617–623 (2006)CrossRefADSGoogle Scholar
  3. Hu Z., Lu Y.Y.: Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps. Opt. Express 16, 17282–17399 (2008)CrossRefADSGoogle Scholar
  4. Huang Y., Lu Y.Y.: Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps. J. Lightwave Technol. 24, 3448–3453 (2006)CrossRefADSMathSciNetGoogle Scholar
  5. Huang Y., Lu Y.Y.: Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps. J. Comput. Math. 25, 337–349 (2007)MATHMathSciNetGoogle Scholar
  6. Huang Y., Lu Y.Y., Li S.: Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps. J. Optical Soc. Am. B 24, 2860–2867 (2007)CrossRefADSGoogle Scholar
  7. Joannopoulos, J.D., Meade, R.D., Winn, J.N.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton, NJGoogle Scholar
  8. Li S., Lu Y.Y.: Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells. J. Optical Soc. Am. A 24, 2438–2442 (2007)CrossRefADSMathSciNetGoogle Scholar
  9. Martin P.A.: Multiple Scattering. Cambridge University Press, Cambridge, UK (2006)MATHGoogle Scholar
  10. McPhedran R.C., Botten L.C., Asatryan A.A. et al.: Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders. Phys. Rev. E 60, 7614–7617 (1999)CrossRefADSGoogle Scholar
  11. White T.P., Botten L.C., de Sterke C.M., McPhedran R.C., Asatryan A.A., Langtry T.N.: Block mode scattering matrix methods for modeling exteded photonic crystal structures Applications. Phys. Rev. E 70, 056607 (2004)CrossRefADSGoogle Scholar
  12. Wu Y., Lu Y.Y.: Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice. J. Optical Soc. Am. B 25, 1466–1473 (2008)CrossRefADSGoogle Scholar
  13. Yasumoto K., Toyama H., Kushta T.: Accurate analysis of two-dimensional electromagnetic scattering from multilaye red periodic arrays of circular cylinders using lattice sums technique. IEEE Trans. Antennas Propag. 52, 2603–2611 (2004)CrossRefADSMathSciNetGoogle Scholar
  14. Yonekura J., Ikeda M., Baba T.: Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method. J. Lightwave Technol. 17, 1500–1508 (1999)CrossRefADSGoogle Scholar
  15. Yuan J., Lu Y.Y.: Photonic bandgap calculations using Dirichlet-to-Neumann maps. J Optical Soc Am A 23, 3217–3222 (2006)CrossRefADSMathSciNetGoogle Scholar
  16. Yuan J., Lu Y.Y.: Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice. Optics. Commun 273, 114–120 (2007)CrossRefADSGoogle Scholar
  17. Yuan J., Lu Y.Y., Antoine X.: Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps. J. Comput. Phys 227, 4617–3629 (2008)MATHCrossRefADSMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Joint Advanced Research Center of University of Science and Technology of China and City University of Hong KongSuzhouChina
  2. 2.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  3. 3.Department of MathematicsCity University of Hong KongKowloonHong Kong

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