Abstract
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.
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References
Felbacq J.D., Tayeb C., Maystre D.: Scattering by a random set of parallel cylinders. J. Optical Soc. Am. A 11, 2526–2538 (1994)
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)
Hu Z., Lu Y.Y.: Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps. Opt. Express 16, 17282–17399 (2008)
Huang Y., Lu Y.Y.: Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps. J. Lightwave Technol. 24, 3448–3453 (2006)
Huang Y., Lu Y.Y.: Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps. J. Comput. Math. 25, 337–349 (2007)
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)
Joannopoulos, J.D., Meade, R.D., Winn, J.N.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton, NJ
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)
Martin P.A.: Multiple Scattering. Cambridge University Press, Cambridge, UK (2006)
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)
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)
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)
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)
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)
Yuan J., Lu Y.Y.: Photonic bandgap calculations using Dirichlet-to-Neumann maps. J Optical Soc Am A 23, 3217–3222 (2006)
Yuan J., Lu Y.Y.: Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice. Optics. Commun 273, 114–120 (2007)
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)
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Hu, Z., Lu, Y.Y. Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices. Opt Quant Electron 40, 921–932 (2008). https://doi.org/10.1007/s11082-009-9288-z
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DOI: https://doi.org/10.1007/s11082-009-9288-z