OSA's Digital Library

Optics Letters

Optics Letters


  • Vol. 29, Iss. 22 — Nov. 15, 2004
  • pp: 2653–2655

Diffraction and localization in low-dimensional photonic bandgaps

Stefano Longhi and Davide Janner  »View Author Affiliations

Optics Letters, Vol. 29, Issue 22, pp. 2653-2655 (2004)

View Full Text Article

Acrobat PDF (377 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We show that, in low-dimensional photonic bandgaps, wave diffraction resulting from localization in the translational-invariant directions is strongly influenced by the photonic band structure of the periodic crystal, leading to new kinds of wave localization. In particular, for a periodic layered structure we show that, close to a bandgap edge, diffraction is enhanced, with a transition from a parabolic diffraction curve—typical of isotropic media and supporting Gaussian beams—to hyperbolic or elliptic diffraction curves. In the last two cases localization in the form of stationary X-shaped or sinc-shaped waves is possible.

© 2004 Optical Society of America

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(230.1480) Optical devices : Bragg reflectors
(230.4170) Optical devices : Multilayers

Stefano Longhi and Davide Janner, "Diffraction and localization in low-dimensional photonic bandgaps," Opt. Lett. 29, 2653-2655 (2004)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
  2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 85, 1863 (2000).
  3. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, Phys. Rev. Lett. 88, 093901 (2002).
  4. M. Notomi, Opt. Quantum Electron. 34, 133 (2002).
  5. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopolou, and C. M. Soukoulis, Phys. Rev. Lett. 91, 207401 (2003).
  6. See, e.g., I. M. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, Prog. Electron. Res. (PIER) 19, 1 (1998), and references therein.
  7. P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
  8. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, Phys. Rev. Lett. 91, 093904 (2003).
  9. C. Conti and S. Trillo, Phys. Rev. Lett. 92, 120404 (2004).
  10. M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, Opt. Lett. 28, 1090 (2003).
  11. S. Longhi, Opt. Lett. 29, 147 (2004).
  12. S. Longhi, Phys. Rev. E 69, 016606 (2004).
  13. Neglecting the second-order derivative in Eq. 5 corresponds, for an isotropic medium, to the well-known paraxial approximation, which is valid provided that the transverse beam size of, e.g., a Gaussian beam is much larger than its Rayleigh range. As q diverges, the paraxial approximation breaks down, and higher-order derivative terms should be included, marking a continuous transition from the parabolic to the hyperbolic (or elliptic) diffractive regimes.
  14. This is due to the fact that in the monochromatic regime the wave equation in isotropic media is always of the elliptic type, and only the inclusion of temporal degree of freedom may lead to a hyperbolic equation supporting X waves.
  15. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  16. Most usual X waves, such as those considered in Refs. and, actually correspond to a spectral amplitude F(k) ∝ exp(-αk); we consider here a different spectral shape for better comparing the transition from parabolic to hyperbolic localization for a transverse Gaussian field distribution.

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