OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 4072–4092

The finite element method applied to the study of two-dimensional photonic crystals and resonant cavities

Imanol Andonegui and Angel J. Garcia-Adeva  »View Author Affiliations

Optics Express, Vol. 21, Issue 4, pp. 4072-4092 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (3555 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A critical assessment of the finite element (FE) method for studying two-dimensional dielectric photonic crystals is made. Photonic band structures, transmission coefficients, and quality factors of various two-dimensional, periodic and aperiodic, dielectric photonic crystals are calculated by using the FE (real-space) method and the plane wave expansion or the finite difference time domain (FDTD) methods and a comparison is established between those results. It is found that, contrarily to popular belief, the FE method (FEM) not only reproduces extremely well the results obtained with the standard plane wave method with regards to the eigenvalue analysis (photonic band structure and density of states calculations) but it also allows to study very easily the time-harmonic propagation of electromagnetic fields in finite clusters of arbitrary complexity and, thus, to calculate their transmission coefficients in a simple way. Moreover, the advantages of using this real space method in the context of point defect cluster quality factor calculations are also stressed by comparing the results obtained with this method with those obtained with the FDTD one. As a result of this study, FEM comes out as an stable, robust, rigorous, and reliable tool to study light propagation and confinement in both periodic and aperiodic dielectric photonic crystals and clusters.

© 2013 OSA

OCIS Codes
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(140.3945) Lasers and laser optics : Microcavities
(160.5293) Materials : Photonic bandgap materials
(160.5298) Materials : Photonic crystals

ToC Category:
Photonic Crystals

Original Manuscript: November 28, 2012
Revised Manuscript: January 4, 2013
Manuscript Accepted: January 15, 2013
Published: February 11, 2013

Imanol Andonegui and Angel J. Garcia-Adeva, "The finite element method applied to the study of two-dimensional photonic crystals and resonant cavities," Opt. Express 21, 4072-4092 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. J. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, New Jersey, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
  3. C. Lopez, “Material aspects of photonic crystals,” Adv. Mater.15, 1679–1704, (2003). [CrossRef]
  4. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures,” Rev. Mod. Phys.78, 455–481, (2006). [CrossRef]
  5. A. J. Garcia-Adeva, “Band gap atlas for photonic crystals having the symmetry of the kagome and pyrochlore lattices,” New J. Phys.8, 86/1–14 (2006). [CrossRef]
  6. A. J. Garcia-Adeva, “Band structure of photonic crystals with the symmetry of a pyrochlore lattice,” Phys. Rev. B.73, 0731071 (2006). [CrossRef]
  7. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2299, (1991). [CrossRef] [PubMed]
  8. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001). [CrossRef] [PubMed]
  9. MPB on-line manual, http://ab-initio.mit.edu/wiki/index.php/MPB\_manual .
  10. J. Jin, The Finite Element Method in Electromagnetism (Wiley–IEEE press, New York, 2002).
  11. A. Sopaheluwakan, Defect States and Defect Modes in 1D Photonic Crystals (MSc Thesis, University of Twente, 2003).
  12. M. -C. Lin and R. -F. Jao, “Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation,” Opt. Express15, 207–218 (2007). [CrossRef] [PubMed]
  13. W. R. Frei and H. T. Johnson, “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B70, 1651161 (2004). [CrossRef]
  14. J. L. Garcia-Pomar and M. Nieto-Vesperinas, “Transmission study of prisms and slabs of lossy negative index media,” Opt. Express12, 2081–2095, (2004). [CrossRef] [PubMed]
  15. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature462, 78–82, (2009). [CrossRef] [PubMed]
  16. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature459, 550–555, (2009). [CrossRef] [PubMed]
  17. A. H. Safavi-Naeini, T. P. M. Alegre, M. Winger, and O. Painter, “Optomechanics in an ultrahigh-Q two-dimensional photonic crystal cavity,” Appl. Phys. Lett.97, 1811061–3, (2010). [CrossRef]
  18. E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106, 2039021 (2011). [CrossRef]
  19. Huang-Ming Lee and Jong-Ching Wua, “Transmittance spectra in one-dimensional superconductor-dielectric photonic crystal,” J. Appl. Phys.107, 09E1491–3, (2010).
  20. V. F. Rodriguez-Esquerre, M. Koshiba, and H. E. Hernandez-Figueroa, “Finite-Element Analysis of Photonic Crystal Cavities: Time and Frequency Domains,” J. Lightwave Technol.23, 1514–1521, (2005). [CrossRef]
  21. J. K. Hwang, S. B. Hyun, H. Y. Ryu, and Y. H. Lee, “Resonant modes of two-dimensional photonic bandgap cavities determined by the finite-element method and by use of the anisotropic perfectly matched layer boundary condition,” J. Opt. Soc. Am. B15, 2316–2324, (1998). [CrossRef]
  22. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B547837–7842, (1996). [CrossRef]
  23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm.181, 687–702, (2010). [CrossRef]
  24. R. P. Brent, Algorithms for Minimization without Derivatives (Courier Dover Publications, 1973).
  25. Elmer – Finite Element Software for Multiphysical Problems, http://www.csc.fi/elmer/index.phtml .
  26. The unofficial numerical electromagnetic code archives, http://www.si-list.org/swindex2.html .
  27. The EMAP Finite Element Modeling Codes, http://www.emclab.umr.edu/emap.html .
  28. Comsol multiphysics and Electromagnetics module, http://www.comsol.com .
  29. T. A. Davis, UMFPACK 4.6: Unsymmetric MultiFrontal sparse LU factorization package, http://www.cise.ufl.edu/research/sparse/umfpack/ .
  30. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  31. 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]
  32. E. Waks and J. Vuckovic, “Coupled mode theory for photonic crystal cavity-waveguide interaction,” Opt. Express13, 5064–5073 (2005). [CrossRef] [PubMed]
  33. C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, Photonic Crystals: Physics and Technology (Springer, Milano, 2008). [CrossRef]
  34. B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys.87, 603–605, (2000). [CrossRef]
  35. Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics1, 49–52, (2007). [CrossRef]
  36. T. Baba, T. Kawasaki, H. Sasaki, J. Adachi, and D. Mori, “Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide,” Opt. Express16, 9245–9253, (2008). [CrossRef] [PubMed]
  37. D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett.80, 3901–3903, (2002). [CrossRef]
  38. R. V. Nair and R. Vijaya, “Photonic crystal sensors: an overview,” Prog. Quant. Electron.34, 89–134, (2010). [CrossRef]
  39. 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. B10, 314–321, (1993). [CrossRef]
  40. S. -H. Kim and Y. -H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron.39, 1081–1085, (2003). [CrossRef]
  41. S. J. Cox and D. C. Dobson, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math.59, 2108–2120, (1999). [CrossRef]
  42. L. F. Shen, Z. Ye, and S. He, “Design of two-dimensional photonic crystals with large absolute band gaps using a genetic algorithm,” Phys. Rev. B68, 035109 (2003). [CrossRef]
  43. J. M. Geremia, J. Williams, and H. Mabuchi, “Inverse-problem approach to designing photonic crystals for cavity QED experiments,” Phys. Rev. E66, 066606 (2002). [CrossRef]
  44. I. Andonegui and A. J. Garcia-Adeva, (unpublished).
  45. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys.114, 185–200, (1994). [CrossRef]
  46. A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, “The failure of perfectly matched layers and towards their redemption by adiabatic absorbers,” Opt. Express16, 113761 (2008). [CrossRef]
  47. Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop.43, 1460–1463, (1995). [CrossRef]
  48. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769, (1997). Erratum, ibid. 109, 4128 (1998). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited