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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 24 — Nov. 21, 2011
  • pp: 24344–24353

Relevance of the light line in planar photonic crystal waveguides with weak vertical confinement

P. Kaspar, R. Kappeler, D. Erni, and H. Jäckel  »View Author Affiliations

Optics Express, Vol. 19, Issue 24, pp. 24344-24353 (2011)

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The concept of the so-called light line is a useful tool to distinguish between guided and non-guided modes in dielectric slab waveguides. Also for more complicated structures with 2D mode confinement, the light lines can often be used to divide a dispersion diagram into a region of a non-guided continuum of modes, a region of discrete guided modes and a forbidden region, where no propagating modes can exist. However, whether or not the light line is a concept of practical relevance depends on the geometry of the structure. This fact is sometimes ignored. For instance, in the literature on photonic crystal waveguides, it is often argued that substrate-type photonic crystal waveguides with a weak vertical confinement are inherently lossy, since the entire bandgap including the line defect modes is typically located above the light line of the substrate. The purpose of this article is to illustrate that this argument is inaccurate and to provide guidelines on how an improved light line concept can be constructed.

© 2011 OSA

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.7370) Optical devices : Waveguides
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Photonic Crystals

Original Manuscript: August 18, 2011
Revised Manuscript: November 4, 2011
Manuscript Accepted: November 4, 2011
Published: November 14, 2011

P. Kaspar, R. Kappeler, D. Erni, and H. Jäckel, "Relevance of the light line in planar photonic crystal waveguides with weak vertical confinement," Opt. Express 19, 24344-24353 (2011)

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  1. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature397, 594–598 (1999). [CrossRef]
  2. S. Mahnkopf, M. Kamp, A. Forchel, and R. März, “Tunable distributed feedback laser with photonic crystal mirrors,” Appl. Phys. Lett. 82, 2942–2944 (2003). [CrossRef]
  3. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]
  4. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics 3, 206–210 (2009). [CrossRef]
  5. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003). [CrossRef] [PubMed]
  6. T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics 1, 49–52 (2007). [CrossRef]
  7. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005). [CrossRef] [PubMed]
  8. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005). [CrossRef] [PubMed]
  9. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: Role of fabrication disorder and photon group velocity,” Phys. Rev. Lett. 94, 033903 (2005). [CrossRef] [PubMed]
  10. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, “Loss engineered slow light waveguides,” Opt. Express 18, 27627–27638 (2010). [CrossRef]
  11. Let σy denote a reflection in the x–z plane, i.e., σyx̂ = x̂, σyŷ = −ŷ, σyẑ = ẑ for the unit vectors x̂, ŷ, and ẑ, respectively. E transforms like a vector, whereas H transforms like a pseudovector under an orientation-reversing map. A mode of even parity is characterized by E(σyr) = σyE(r) and H(σyr) = −σyH(r), a mode of odd parity by E(σyr) = −σyE(r) and H(σyr) = σyH(r). Note that different conventions of parity might be used in other contexts.
  12. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002). [PubMed]
  13. R. März, Integrated Optics: Design and Modeling (Artech House, Norwood, 1995).
  14. It might be interesting to note that the condition of Eq. (1) for guided modes can be proven by a rigorous mathematical analysis, as long as there is a radius R around the core, for which n(r) = n∞ holds for all r ∈ ℝ2 with |r| > R [A.-S. Bonnet-Ben Dhia and P. Joly, “Mathematical analysis and numerical approximation of optical waveguides,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds. (Siam, Philadelphia, 2001), pp. 273–324, Frontiers in Applied Mathematics]. In other words, Eq. (1) holds for all guided modes, if outside of a circle of radius R the background is made up of a homogeneous medium of refractive index n∞. This condition is not fulfilled by the structure of Fig. 1 if we let hbot → ∞. [CrossRef]
  15. D. Marcuse, Theory of Dielectric Optical Waveguides, Quantum Electronics: Principles and Applications (Academic Press, Boston, 1991), 2nd ed.
  16. T. A. Birks, P. J. Roberts, P. S. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995). [CrossRef]
  17. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]
  18. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
  19. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]
  20. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B 22, 1179–1190 (2005). [CrossRef]
  21. W. Kuang and J. D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004). [CrossRef] [PubMed]

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