OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 2 — Jan. 18, 2010
  • pp: 1151–1158

Generalized phase matching condition for 
lossy periodic photonic structures

Xuhuai Zhang and Stephen R. Forrest  »View Author Affiliations

Optics Express, Vol. 18, Issue 2, pp. 1151-1158 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (4456 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The phase matching condition relating the real transverse wave vectors across a periodic boundary has been generalized to the case of complex transverse wave vectors. Based on this generalization, we describe diffraction of a complex Bloch wave propagating within a composite prism, and show that the resulting light in free space is an inhomogeneous plane wave in the presence of losses within the prism.

© 2010 OSA

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(160.3918) Materials : Metamaterials
(050.5298) Diffraction and gratings : Photonic crystals
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Photonic Crystals

Original Manuscript: December 4, 2009
Manuscript Accepted: January 4, 2010
Published: January 8, 2010

Xuhuai Zhang and Stephen R. Forrest, "Generalized phase matching condition for lossy periodic photonic structures," Opt. Express 18, 1151-1158 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62(16), 10696–10705 (2000). [CrossRef]
  2. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002). [CrossRef]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton and Oxford, 2008), pp. 221–222.
  4. A. Damascelli, Z. Hussain, and Z. X. Shen, “Angle-resolved photoemission studies of the cuprate superconductors,” Rev. Mod. Phys. 75(2), 473–541 (2003). [CrossRef]
  5. J. B. Pendry, “Photonic Band Structures,” J. Mod. Opt. 41(2), 209–229 (1994). [CrossRef]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
  7. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell’s law,” Phys. Rev. Lett. 90(10), 107401 (2003). [CrossRef] [PubMed]
  8. R. B. Greegor, C. G. Parazzoli, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental determination and numerical simulation of the properties of negative index of refraction materials,” Opt. Express 11(7), 688–695 (2003). [CrossRef] [PubMed]
  9. X. H. Zhang, M. Davanço, Y. Urzhumov, G. Shvets, and S. R. Forrest, “From scattering parameters to Snell’s law: a subwavelength near-infrared negative-index metamaterial,” Phys. Rev. Lett. 101(26), 267401 (2008). [CrossRef] [PubMed]
  10. N. Garcia and M. Nieto-Vesperinas, “Is there an experimental verification of a negative index of refraction yet?” Opt. Lett. 27(11), 885–887 (2002). [CrossRef]
  11. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). [CrossRef]
  12. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6 Pt 2), 065602 (2003). [CrossRef]
  13. D. de Ceglia, M. A. Vincenti, M. G. Cappeddu, M. Centini, N. Akozbek, A. D'Orazio, J. W. Haus, M. J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77(3), 033848 (2008). [CrossRef]
  14. D. Seetharamdoo, R. Sauleau, K. Mahdjoubi, and A. C. Tarot, “Effective parameters of resonant negative refractive index metamaterials: Interpretation and validity,” J. Appl. Phys. 98(6), 063505 (2005). [CrossRef]
  15. C. Rockstuhl, C. Menzel, T. Paul, T. Pertsch, and F. Lederer, “Light propagation in a fishnet metamaterial,” Phys. Rev. B 78(15), 155102 (2008). [CrossRef]
  16. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78(4), 489–491 (2001). [CrossRef]
  17. K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001), pp. 30–32.
  18. D. R. Smith, P. M. Rye, J. J. Mock, D. C. Vier, and A. F. Starr, “Enhanced diffraction from a grating on the surface of a negative-index metamaterial,” Phys. Rev. Lett. 93(13), 137405 (2004). [CrossRef] [PubMed]
  19. N. W. Ashcroft, and N. D. Mermin, Solid State Physics, Solid State Physics (Holt, Rinehart and Winston, New York, 1976), pp. 368–369.
  20. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon Press, Oxford and New York, 1966), pp. 13–14.
  21. N. F. Declercq, J. Degrieck, and O. Leroy, “The Laplace transform to describe bounded inhomogeneous waves,” J. Acoust. Soc. Am. 116(1), 51–60 (2004). [CrossRef]
  22. W. Huang, R. Briers, S. I. Rokhlin, and O. Leroy, “Experimental-Study of Inhomogeneous Wave Reflection from a Solid-Air Periodically Rough Boundary Using Leaky Rayleigh-Waves,” J. Acoust. Soc. Am. 96(1), 363–369 (1994). [CrossRef]
  23. R. Briers, O. Leroy, O. Poncelet, and M. Deschamps, “Experimental verification of the calculated diffraction field generated by inhomogeneous waves obliquely incident on a periodically rough liquid-solid boundary,” J. Acoust. Soc. Am. 106(2), 682–687 (1999). [CrossRef]
  24. N. F. Declercq, R. Briers, J. Degrieck, and O. Leroy, “The history and properties of ultrasonic inhomogeneous waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 52(5), 776–791 (2005). [CrossRef] [PubMed]
  25. A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell’s law,” Phys. Rev. Lett. 90(13), 137401 (2003). [CrossRef] [PubMed]
  26. 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,” Nature 455(7211), 376–379 (2008). [CrossRef] [PubMed]
  27. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
  28. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
  29. S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B 67(23), 235107 (2003). [CrossRef]
  30. X. H. Zhang, M. Davanco, Y. Urzhumov, G. Shvets, and S. R. Forrest, “A Subwavelength Near-Infrared Negative-Index Material,” Appl. Phys. Lett. 94(13), 131107 (2009). [CrossRef]
  31. I. Tsukerman, “Negative refraction and the minimum lattice cell size,” J. Opt. Soc. Am. B 25(6), 927–936 (2008). [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.


Fig. 1 Fig. 2 Fig. 3

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited