Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T17:25:18.076Z Has data issue: false hasContentIssue false
  • English
  • Français

Dirichlet-to-Neumann Map Method with Boundary Cells for Photonic Crystals Devices

Published online by Cambridge University Press:  20 August 2015

Jianhua Yuan  and
Ya Yan Lu
Jianhua Yuan*
Affiliation:
Department of Mathematics, Beijing University of Posts and Telecommunications, Beijing 100876, China
Ya Yan Lu*
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

Abstract

In a two-dimensional (2D) photonic crystal (PhC) composed of circular cylinders (dielectric rods or air holes) on a square or triangular lattice, various PhC devices can be created by removing or modifying some cylinders. Most existing numerical methods for PhC devices give rise to large sparse or smaller but dense linear systems, all of which are expensive to solve if the device is large. In a previous work [Z. Hu et al., Optics Express, 16 (2008), 17383-17399], an efficient Dirichlet-to-Neumann (DtN) map method was developed for general 2D PhC devices with an infinite background PhC to take full advantage of the underlying lattice structure. The DtN map of a unit cell is an operator that maps the wave field to its normal derivative on the cell boundary and it allows one to avoid computing the wave field in the interior of the unit cell. In this paper, we extend the DtN map method to PhC devices with a finite background PhC. Since there is no bandgap effect to confine the light in a finite PhC, a different technique for truncating the domain is needed. We enclose the finite structure with a layer of empty boundary and corner unit cells, and approximate the DtN maps of these cells based on expanding the scattered wave in outgoing plane waves. Our method gives rise to a relatively small and sparse linear systems that are particularly easy to solve.

Keywords

78M2578M1678A45Photonic crystalDirichlet-to-Neumann mapnumerical simulation
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 1 , January 2011 , pp. 113 - 128
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Joannopoulos, J. D., Meade, R. D. and Winn, J. N., Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NJ. 1995.Google Scholar
[2]Mekis, A., Chen, J. C., Kurland, I., Fan, S., Villeneuve, P. R. and Joannopoulos, J. D., High transmission through sharp bends in photonic crystal waveguides, Phys. Rev. Lett., 77 (1996), 3787–3790.Google Scholar
[3]Koshiba, M., Tsuji, Y. and Hikari, M., Time-domain beam propagation method and its application to photonic crystal circuits, J. Lightw. Technol., 18 (2000), 102–110.Google Scholar
[4]Fan, S., Villeneuve, P. R., Joannopoulos, J. D. and Haus, H. A., Channel drop tunneling through localized states, Phys. Rev. Lett., 80 (1998), 960–963.Google Scholar
[5]Fan, S., Johnson, S. G., Joannopoulos, J. D., Manolatou, C. and Haus, H. A., Waveguide branches in photonic crystals, J. Opt. Soc. Am. B., 18 (2001), 162–165.Google Scholar
[6]Fujisawa, T. and Koshiba, M., Finite-element modeling of nonlinear interferometers based on photonic-crystal waveguides for all-optical signal processing, J. Lightw. Technol., 24 (2006), 617–623.CrossRefGoogle Scholar
[7]Taflove, A. and Hagness, S. C., Computational Electrodynamics: the Finite-Difference TimeDomain Method, 2nd ed., Artech House, 2000.Google Scholar
[8]Jin, J. M., The Finite Element Method in Electromagnetics, 2nd ed., John Wiley & Sons, New York, 2002.Google Scholar
[9]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Springer-Verlag, Berlin, 1998.Google Scholar
[10]Felbacq, D., Tayeb, G. and Maystre, D., Scattering by a random set of parallel cylinders, J. Opt. Soc. Am. A., 11 (1994), 2526–2538.Google Scholar
[11]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 (1999), 7614–7617.CrossRefGoogle ScholarPubMed
[12]Botten, L. C., White, T. P., Asatryan, A. A.et al., Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I: theory, Phys. Rev. E., 70 (2004), 056606.Google Scholar
[13]Yasumoto, K., Toyama, H. and Kushta, T., Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique, IEEE T. Antenn. Propag., 52 (2004), 2603–2611.Google Scholar
[14]Yuan, J. and Lu, Y. Y., Photonic bandgap calculations using Dirichlet-to-Neumann maps, J. Opt. Soc. Am. A., 23 (2006), 3217–3222.Google Scholar
[15]Huang, Y. and Lu, Y. Y., Scattering from periodic arraysof cylinders by Dirichlet-to-Neumann maps, J. Lightw. Technol., 24 (2006), 3448–3453.Google Scholar
[16]Yuan, J. and Lu, Y. Y., Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice, Opt. Commun., 273 (2007), 114–120.CrossRefGoogle Scholar
[17]Huang, Y., Lu, Y. Y. and Li, S., Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps, J. Opt. Soc. Am. B., 24 (2007), 2860–2867.Google Scholar
[18]Li, S. and Lu, Y. Y., Computing photonic crystal defect modes by Dirichlet-to-Neumann maps, Opt. Express., 15 (2007), 14454–14466.Google Scholar
[19]Huang, Y. and Lu, Y. Y., Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps, J. Comput. Math., 25 (2007), 337–349.Google Scholar
[20]Wu, Y. and Lu, Y. Y., Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice, J. Opt. Soc. Am. B., 25 (2008), 1466–1473.Google Scholar
[21]Hu, Z. and Lu, Y. Y., Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps, Opt. Express., 16 (2008), 17383–17399.Google Scholar
[22]Hu, Z. and Lu, Y. Y., Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices, Opt. Quant. Electron., 40 (2008), 921–932.Google Scholar
[23]Li, S. and Lu, Y. Y., Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells, J. Opt. Soc. Am. A., 24 (2007), 2438–2442.Google Scholar
[24]Yuan, J., Lu, Y. Y. and Antoine, X., Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps, J. Comput. Phys., 227 (2008), 4617–3629.Google Scholar