The European Physical Journal Special Topics

, Volume 226, Issue 7, pp 1563–1582 | Cite as

Topological boundary states in 1D: An effective Fabry-Perot model

  • E. Levy
  • E. AkkermansEmail author
Open Access
Regular Article
Part of the following topical collections:
  1. From Ill-condensed Matter to Mesoscopic Wave Propagation


We present a general and useful method to predict the existence, frequency, and spatial properties of gap states in photonic (and other) structures with a gapped spectrum. This method is established using the scattering approach. It offers a viewpoint based on a geometrical Fabry-Perot model. We demonstrate the capabilities of this model by predicting the behaviour of topological edge states in quasi-periodic structures.


  1. 1.
    E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987)ADSCrossRefGoogle Scholar
  2. 2.
    P.W. Anderson, Phys. Rev. 109, 1492 (1958)ADSCrossRefGoogle Scholar
  3. 3.
    A.Z. Genack, N. Garcia, Phys. Rev. Lett. 66, 2064 (1991)ADSCrossRefGoogle Scholar
  4. 4.
    M. Störzer, P. Gross, C.M. Aegerter, G. Maret, Phys. Rev. Lett. 96, 063904 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    F. Evers, A.D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008)ADSCrossRefGoogle Scholar
  6. 6.
    G. Maret, T. Sperling, W. Bührer, A. Lubatsch, R. Frank, C.M. Aegerter, Nature Photonics 7, 934 (2013)ADSCrossRefGoogle Scholar
  7. 7.
    L. Bellando, A. Gero, E. Akkermans, R. Kaiser, Phys. Rev. A 90, 063822 (2014)ADSCrossRefGoogle Scholar
  8. 8.
    T. Sperling, L. Schertel, M. Ackermann, G.J. Aubry, C.M. Aegerter, G. Maret, New J. Phys. 18, 013039 (2016)ADSCrossRefGoogle Scholar
  9. 9.
    S.E. Skipetrov, J.H. Page, New J. Phys. 18, 021001 (2016)ADSCrossRefGoogle Scholar
  10. 10.
    M. Kohmoto, B. Sutherland, K. Iguchi, Phys. Rev. Lett. 58, 2436 (1987)ADSCrossRefGoogle Scholar
  11. 11.
    W. Gellermann, M. Kohmoto, B. Sutherland, P.C. Taylor, Phys. Rev. Lett. 72, 633 (1994)ADSCrossRefGoogle Scholar
  12. 12.
    E. Akkermans, E. Gurevich, Europhys. Lett. 103, 30009 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    E. Akkermans, Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics 601, 1 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, Phys. Rev. Lett. 112, 146404 (2014)ADSCrossRefGoogle Scholar
  15. 15.
    J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton university press, 2011)Google Scholar
  16. 16.
    I.E. Tamm, Phys. Z. Sowjetunion 1, 733 (1932)Google Scholar
  17. 17.
    W. Shockley, Phys. Rev. 56, 317 (1939)ADSCrossRefGoogle Scholar
  18. 18.
    T. Goto, A.V. Dorofeenko, A.M. Merzlikin, A.V. Baryshev, A.P. Vinogradov, M. Inoue, A.A. Lisyansky, A.B. Granovsky, Phys. Rev. Lett. 101, 113902 (2008)ADSCrossRefGoogle Scholar
  19. 19.
    E. Abdel-Rahman, A. Shaarawi, J. Mater. Sci. Mater. Electron. 20, 153 (2009)CrossRefGoogle Scholar
  20. 20.
    E.S. Zijlstra, A. Fasolino, T. Janssen, Phys. Rev. B 59, 302 (1999)ADSCrossRefGoogle Scholar
  21. 21.
    X.N. Pang, J.W. Dong, H.Z. Wang, J. Opt. Soc. Am. B 27, 2009 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    M. Hiltunen, L.D. Negro, N.N. Feng, L.C. Kimerling, J. Michel, J. Lightwave Technol. 25, 1841 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    S.V. Zhukovsky, Phys. Rev. A 81, 053808 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    P.A. Kalozoumis, C. Morfonios, N. Palaiodimopoulos, F.K. Diakonos, P. Schmelcher, Phys. Rev. A 88, 033857 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    E. Akkermans, G. Dunne, E. Levy, in Optics of Aperiodic Structures: Fundamentals and Device Applications, edited by L. Dal Negro (Pan Stanford, 2013), Chap. 10Google Scholar
  26. 26.
    E. Levy, A. Barak, A. Fisher, E. Akkermans, arXiv:1509.04028 (2015)
  27. 27.
    Z.Q. Ma, J. Phys. A: Math. General 39, R625 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    M.Z. Hasan, C.L. Kane, Rev. Mod. Phys. 82, 3045 (2010)ADSCrossRefGoogle Scholar
  29. 29.
    E. Witten, arXiv:1510.07698 (2015)
  30. 30.
    Y.E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, O. Zilberberg, Phys. Rev. Lett. 109, 106402 (2012)ADSCrossRefGoogle Scholar
  31. 31.
    A. Dareau, E. Levy, M. Bosch Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, J. Beugnon, arXiv:1607.00901 (2016)
  32. 32.
    F. Baboux, E. Levy, A. Lemaître, C. Gomez, E. Galopin, L.L. Gratiet, I. Sagnes, A. Amo, J. Bloch, E. Akkermans, arXiv:1607.03813 (2016)
  33. 33.
    R. Matloob, H. Falinejad, Phys. Rev. A 64, 042102 (2001)ADSCrossRefGoogle Scholar
  34. 34.
    C. Genet, A. Lambrecht, S. Reynaud, Phys. Rev. A 67, 043811 (2003)ADSCrossRefGoogle Scholar
  35. 35.
    A. Lambrecht, P.A.M. Neto, S. Reynaud, New J. Phys. 8, 243 (2006)ADSCrossRefGoogle Scholar
  36. 36.
    M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014)ADSCrossRefGoogle Scholar

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© The Author(s) 2017

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Rafael research center, Rafael ltd.Haifa 32100Israel
  2. 2.Department of PhysicsTechnion Israel Institute of TechnologyHaifaIsrael

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