OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 20 — Oct. 7, 2013
  • pp: 24093–24098

Coherent-form energy conservation relation for the elastic scattering of a guided mode in a symmetric scattering system

Haitao Liu  »View Author Affiliations

Optics Express, Vol. 21, Issue 20, pp. 24093-24098 (2013)

View Full Text Article

Enhanced HTML    Acrobat PDF (836 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We propose a coherent-form energy conservation relation (ECR) that is generally valid for the elastic transmission and reflection of a guided mode in a symmetric scattering system. In contrast with the classical incoherent-form ECR, |τ|2 + |ρ|2≤1 with τ and ρ denoting the elastic transmission and reflection coefficients of a guided mode, the coherent-form ECR is expressed as |τ + ρ|≤1, which imposes a constraint on a coherent superposition of the transmitted and reflected modes. The coherent-form ECR is rigorously demonstrated and is numerically tested by considering different types of modes in various scattering systems. Further discussions with the scattering matrix formalism indicate that two coherent-form ECRs, |τ + ρ|≤1 and |τρ|≤1, along with the classical ECR |τ|2 + |ρ|2≤1 constitute a complete description of the energy conservation for the elastic scattering of a guided mode in a symmetric scattering system. The coherent-form ECR provides a common tool in terms of energy transfer for understanding and analyzing the scattering dynamics in currently interested scattering systems.

© 2013 OSA

OCIS Codes
(130.2790) Integrated optics : Guided waves
(260.2160) Physical optics : Energy transfer
(290.5825) Scattering : Scattering theory

ToC Category:

Original Manuscript: August 5, 2013
Revised Manuscript: September 13, 2013
Manuscript Accepted: September 16, 2013
Published: October 1, 2013

Haitao Liu, "Coherent-form energy conservation relation for the elastic scattering of a guided mode in a symmetric scattering system," Opt. Express 21, 24093-24098 (2013)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).
  3. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature391(6668), 667–669 (1998). [CrossRef]
  4. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455(7211), 376–379 (2008). [CrossRef] [PubMed]
  5. P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005). [CrossRef] [PubMed]
  6. L. M. Tong, F. Zi, X. Guo, and J. Y. Lou, “Optical microfibers and nanofibers:A tutorial,” Opt. Commun.285(23), 4641–4647 (2012). [CrossRef]
  7. H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature452(7188), 728–731 (2008). [CrossRef] [PubMed]
  8. J. Yang, C. Sauvan, H. T. Liu, and P. Lalanne, “Theory of fishnet negative-index optical metamaterials,” Phys. Rev. Lett.107(4), 043903 (2011). [CrossRef] [PubMed]
  9. K. J. Huang, S. Y. Yang, and L. M. Tong, “Modeling of evanescent coupling between two parallel optical nanowires,” Appl. Opt.46(9), 1429–1434 (2007). [CrossRef] [PubMed]
  10. X. P. Huang and M. L. Brongersma, “Rapid computation of light scattering from aperiodic plasmonic structures,” Phys. Rev. B84(24), 245120 (2011). [CrossRef]
  11. G. Y. Li, F. Xiao, L. Cai, K. Alameh, and A. S. Xu, “Theory of the scattering of light and surface plasmon polaritons by finite-size subwavelength metallic defects via field decomposition,” New J. Phys.13(7), 073045 (2011). [CrossRef]
  12. Here the guided mode is defined as a propagative waveguide mode, which obeys an exponential propagation rule exp(ikz) with the propagation constant k being real or approximately real.
  13. For instance, the derivation of the resonance condition in [7] and [8] requires |τ + ρ|≈1 for the elastic scattering of SPPs, which is just the coherent-form ECR under the energy conservative condition.
  14. J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett.83(14), 2845–2848 (1999). [CrossRef]
  15. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A22(9), 1844–1849 (2005). [CrossRef] [PubMed]
  16. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  17. Z. H. Han and S. I. Bozhevolnyi, “Plasmon-induced transparency with detuned ultracompact Fabry-Perot resonators in integrated plasmonic devices,” Opt. Express19(4), 3251–3257 (2011). [CrossRef] [PubMed]
  18. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Theory of surface plasmon generation at nanoslit apertures,” Phys. Rev. Lett.95(26), 263902 (2005). [CrossRef] [PubMed]
  19. L. F. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A14(10), 2758–2767 (1997). [CrossRef]
  20. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express15(18), 11042–11060 (2007). [CrossRef] [PubMed]

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.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited