, Volume 13, Issue 2, pp 393–402 | Cite as

Curie’s Symmetry Principle for Selection Rule of Photonic Crystal Defect Modes

  • Juliana Park
  • Wonyl Choi
  • TaeSun Song
  • Wonho JheEmail author


Symmetry, which defines invariant properties under a group of transformations, provides a frame of generalization uncovering regularities from given quantitative descriptions. Based on the Curie’s symmetry principle, connecting between causality and symmetry, we formulate the intuitive but formal selection rules and apply to determine the excitable resonant modes of a photonic crystal defect cavity, which is an important element for plasmonic applications. Quantitative agreement with the numerical simulations demonstrates the effectiveness of the fundamental principle in finding the critical symmetry conditions for the available localized defect states within photonic crystals. Moreover, the principle facilitates analysis of the higher-order or even forbidden modes in the asymmetric excitation configurations regarding the polarizations or positions of the light source, which typically require heavy computations. Our results may be extended similarly to develop the qualitative selection rules in other physical systems with a geometric symmetry.


Photonic crystal Selection rule Localized modes Curie’s Symmetry Principle 


  1. 1.
    Curie P (1894) Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ magnétique. J Phys Theor Appl 3(1):393–415. doi: 10.1051/jphystap:018940030039300 CrossRefGoogle Scholar
  2. 2.
    Brading K, Castellani E (2003) Symmetries in physics. Symmetries in Physics, Edited by Katherine Brading and Elena Castellani, pp 458 ISBN 0521821371 Cambridge. Cambridge University Press, UK, p 1Google Scholar
  3. 3.
    Jaeger FM (1920) Lectures on the Principles of Symmetry and Its Application in All Natural Sciences, 2nd edn. Elsevier, Amsterdam. doi: 10.1002/bbpc.192400056 Google Scholar
  4. 4.
    Ismael J (1997) Curie’s principle. Synthese 110(2):167–190. doi: 10.1023/A:1004929109216 CrossRefGoogle Scholar
  5. 5.
    Earman J (2004) Curie’s Principle and spontaneous symmetry breaking. Int Stud Philos Sci 18(2-3):173–198. doi: 10.1080/0269859042000311299 CrossRefGoogle Scholar
  6. 6.
    Fan S, Joannopoulos J, Winn JN, Devenyi A, Chen J, Meade RD (1995) Guided and defect modes in periodic dielectric waveguides. JOSA B 12(7):1267–1272. doi: 10.1364/JOSAB.12.001267 CrossRefGoogle Scholar
  7. 7.
    Akahane Y, Asano T, Song B-S, Noda S (2003) High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature 425(6961):944–947. doi: 10.1038/nature02063 CrossRefGoogle Scholar
  8. 8.
    Zhang Z, Qiu M (2004) Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs. Opt Express 12(17):3988–3995. doi: 10.1364/OPEX.12.003988 CrossRefGoogle Scholar
  9. 9.
    Yanik MF, Fan S, Soljačić M (2003) High-contrast all-optical bistable switching in photonic crystal microcavities. Appl Phys Lett 83(14):2739–2741. doi: 10.1063/1.1615835 CrossRefGoogle Scholar
  10. 10.
    Soljačić M, Luo C, Joannopoulos JD, Fan S (2003) Nonlinear photonic crystal microdevices for optical integration. Opt Lett 28(8):637–639. doi: 10.1364/OL.28.000637 CrossRefGoogle Scholar
  11. 11.
    Asakawa K, Sugimoto Y, Watanabe Y, Ozaki N, Mizutani A, Takata Y, Kitagawa Y, Ishikawa H, Ikeda N, Awazu K (2006) Photonic crystal and quantum dot technologies for all-optical switch and logic device. J Phys 8(9):208. doi: 10.1088/1367-2630/8/9/208 Google Scholar
  12. 12.
    Xu Q, Lipson M (2007) All-optical logic based on silicon micro-ring resonators. Opt Express 15(3):924–929. doi: 10.1364/OE.15.000924 CrossRefGoogle Scholar
  13. 13.
    Andalib P, Granpayeh N (2009) All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators. JOSA B 26(1):10–16. doi: 10.1364/JOSAB.26.000010 CrossRefGoogle Scholar
  14. 14.
    Liu Y, Qin F, Meng Z-M, Zhou F, Mao Q-H, Li Z-Y (2011) All-optical logic gates based on two-dimensional low-refractive-index nonlinear photonic crystal slabs. Opt Express 19(3):1945–1953. doi: 10.1364/OE.19.001945 CrossRefGoogle Scholar
  15. 15.
    Yanik MF, Fan S, Soljačić M (2003) High-contrast all-optical bistable switching in photonic crystal microcavities. Appl Phys Lett 83(14):2739–2741. doi: 10.1063/1.1615835 CrossRefGoogle Scholar
  16. 16.
    Tanabe T, Notomi M, Mitsugi S, Shinya A, Kuramochi E (2005) All-optical switches on a silicon chip realized using photonic crystal nanocavities. Appl Phys Lett 87(15):151112. doi: 10.1063/1.2089185 CrossRefGoogle Scholar
  17. 17.
    Ren H, Jiang C, Hu W, Gao M, Wang J (2006) Photonic crystal channel drop filter with a wavelength-selective reflection micro-cavity. Opt Express 14(6):2446–2458. doi: 10.1364/OE.14.002446 CrossRefGoogle Scholar
  18. 18.
    Akahane Y, Asano T, Takano H, Song B-S, Takana Y, Noda S (2005) Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response. Opt Express 13(7):2512–2530. doi: 10.1364/OPEX.13.002512 CrossRefGoogle Scholar
  19. 19.
    Chhipa MK, Rewar E (2014) Effect of variable dielectric constant of Si material rods on 2-D photonic crystal ring resonator based channel drop filter for ITU. T. 694.2 CWDM system 2014 International Conference on Computer and Communication Technology (ICCCT). IEEE, , pp 257–262. doi: 10.1109/ICCCT.2014.7001501
  20. 20.
    Bahabady AM, Olyaee S (2015) Two-curve-shaped biosensor for detecting glucose concentration and salinity of seawater based on photonic crystal nano-ring resonator. Sens Lett 13(9):774–777. doi: 10.1166/sl.2015.3517 CrossRefGoogle Scholar
  21. 21.
    Lee MR, Fauchet PM (2007) Two-dimensional silicon photonic crystal based biosensing platform for protein detection. Optics express 15(8):4530–4535. doi: 10.1364/OE.15.004530 CrossRefGoogle Scholar
  22. 22.
    Fan X, White IM, Shopova SI, Zhu H, Suter JD, Sun Y (2008) Sensitive optical biosensors for unlabeled targets: A review. Analytica Chimica Acta 620(1):8–26. doi: 10.1016/j.aca.2008.05.022 CrossRefGoogle Scholar
  23. 23.
    Chutinan A, Noda S (2000) Waveguides and waveguide bends in two-dimensional photonic crystal slabs. Phys Rev B 62(7):4488. doi: 10.1103/PhysRevB.62.4488 CrossRefGoogle Scholar
  24. 24.
    Qiu M (2001) Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method. Microw Opt Technol Lett 30(5):327–330. doi: 10.1002/mop.1304 CrossRefGoogle Scholar
  25. 25.
    Lončar M, Doll T, Vučković J, Scherer A (2000) Design and fabrication of silicon photonic crystal optical waveguides. J Light Technol 18(10):1402. doi: 10.1109/50.887192 CrossRefGoogle Scholar
  26. 26.
    Kuzmiak V, Maradudin AA (2000) Symmetry analysis of the localized modes associated with substitutional and interstitial defects in a two-dimensional triangular photonic crystal. Phys Rev B 61(16):10750. doi: 10.1103/PhysRevB.61.10750 CrossRefGoogle Scholar
  27. 27.
    Lin L-L, Li Z-Y, Ho K-M (2003) Lattice symmetry applied in transfer-matrix methods for photonic crystals. J Appl Phys 94(2):811–821. doi: 10.1063/1.1587011 CrossRefGoogle Scholar
  28. 28.
    Painter O, Srinivasan K (2003) Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis. Phys Rev B 68(3):035110. doi: 10.1103/PhysRevB.68.035110 CrossRefGoogle Scholar
  29. 29.
    Robertson W, Arjavalingam G, Meade R, Brommer K, Rappe A, Joannopoulos J (1993) Measurement of the photon dispersion relation in two-dimensional ordered dielectric arrays. JOSA B 10(2):322–327. doi: 10.1364/JOSAB.10.000322 CrossRefGoogle Scholar
  30. 30.
    Yeung KY, Chee J, Yoon H, Song Y, Kong J, Ham D (2014) Far-infrared graphene plasmonic crystals for plasmonic band engineering. Nano Lett 14(5):2479–2484. doi: 10.1021/nl500158y CrossRefGoogle Scholar
  31. 31.
    Yeung KY, Chee J, Song Y, Kong J, Ham D (2015) Symmetry Engineering of Graphene Plasmonic Crystals. Nano Lett 15(8):5001–5009. doi: 10.1021/acs.nanolett.5b00970 CrossRefGoogle Scholar
  32. 32.
    Painter O, Srinivasan K, Barclay PE (2003) Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals. Physical Review B 68(3):035214. doi: 10.1103/PhysRevB.68.035214 CrossRefGoogle Scholar
  33. 33.
    Oskooi AF, Roundy D, Ibanescu M, Bermel P, Joannopoulos J, Johnson SG (2010) MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method. Comput Phys Commun 181(3):687–702. doi: 10.1016/j.cpc.2009.11.008 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Juliana Park
    • 1
  • Wonyl Choi
    • 2
  • TaeSun Song
    • 1
  • Wonho Jhe
    • 1
    Email author
  1. 1.Department of Physics & Astronomy and Institute of Applied PhysicsSeoul National UniversityGwanak-ro, Gwanak-guKorea
  2. 2.Department of PhysicsLudwig Maximilian University of MunichMunichGermany

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