Advertisement

Optical and Quantum Electronics

, Volume 41, Issue 4, pp 267–283 | Cite as

Effective index approximations of photonic crystal slabs: a 2-to-1-D assessment

  • Manfred Hammer
  • Olena V. Ivanova
Open Access
Article

Abstract

The optical properties of slab-like photonic crystals are often discussed on the basis of effective index (EI) approximations, where a 2-D effective refractive index profile replaces the actual 3-D structure. Our aim is to assess this approximation by analogous steps that reduce finite 2-D waveguide Bragg-gratings (to be seen as sections through 3-D PC slabs and membranes) to 1-D problems, which are tractable by common transfer matrix methods. Application of the EI method is disputable in particular in cases where locally no guided modes are supported, as in the holes of a PC membrane. A variational procedure permits to derive suitable effective permittivities even in these cases. Depending on the structural properties, these values can well turn out to be lower than one, or even be negative. Both the “standard” and the variational procedures are compared with reference data, generated by a rigorous 2-D Helmholtz solver, for a series of example structures.

Keywords

Integrated optics Numerical modeling Photonic crystal slabs Effective index approximation 

Notes

Acknowledgments

This work has been supported by the Dutch Technology foundation (BSIK / NanoNed project TOE.7143). The authors thank Brenny van Groesen, Hugo Hoekstra, and Remco Stoffer for many fruitful discussions.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. Benson T.M., Bozeat R.J., Kendall P.C.: Rigorous effective index method for semiconductor rib waveguides. IEE Proc. J. 139(1), 67–70 (1992)Google Scholar
  2. Benisty H., Weisbuch C., Labilloy D., Rattier M., Smith C.J.M., Krauss T.F., De La Rue R.M., Houdré R., Oesterle U., Jouanin C., Cassagne D.: Optical and confinement properties of two-dimensional photonic crystals. J. Lightwave Technol. 17(11), 2063–2077 (1999)CrossRefADSGoogle Scholar
  3. Benisty H., Labilloy D., Weisbuch C., Smith C.J.M., Krauss T.F., Cassagne D., Béraud A., Jouanin C.: Radiation losses of waveguide-based two-dimensional photonic crystals: positive role of the substrate. Appl. Phys. Lett. 76(5), 532–534 (2000)CrossRefADSGoogle Scholar
  4. Bienstman P., Assefa S., Johnson S.G., Joannopoulos J.D., Petrich G.S., Kolodziejski L.A.: Taper structures for coupling into photonic crystal slab waveguides. J. Opt. Soc. Am. B 20(9), 1817–1821 (2003)CrossRefADSGoogle Scholar
  5. Blair S., Goeckeritz J.: Effect of vertical mode matching on defect resonances in one-dimensional photonic crystal slabs. J. Lightwave Technol. 24(3), 1456–1461 (2006)CrossRefADSGoogle Scholar
  6. Chiang K.S.: Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides. IEEE Trans. Microw. Theory Technol. 44(5), 692–700 (1996)CrossRefADSGoogle Scholar
  7. Chiang K.S.: Analysis of optical fibers by the effective-index method. Appl. Opt. 25(3), 348–354 (1986)CrossRefADSGoogle Scholar
  8. Čtyroký J.: Photonic bandgap structures in planar waveguides. J. Opt. Soc. Am. A 18(2), 435–441 (2001)CrossRefADSGoogle Scholar
  9. Dems M., Nakwaski W.: The modelling of high-contrast photonic crystal slabs using the novel extension of the effective index method. Opt. Appl. 36(1), 51–56 (2006)Google Scholar
  10. Gao D., Zhou Z.: Nonlinear equation method for band structure calculations of photonic crystal slabs. Appl. Phys. Lett. 88, 163105 (1–3) (2007)ADSGoogle Scholar
  11. Gnan M., Bellanca G., Chong H.M.H., Bassi P., De la Rue R.M.: Modelling of photonic wire bragg gratings. Opt. Quantum Electron. 38, 133–148 (2006)CrossRefGoogle Scholar
  12. Hadley G.R.: Out-of-plane losses of line-defect photonic crystal waveguides. IEEE Photon. Technol. Lett. 14(5), 642–644 (2002)CrossRefADSGoogle Scholar
  13. Hammer M.: Hybrid analytical/numerical coupled-mode modeling of guided wave devices. J. Lightwave Technol. 25(9), 2287–2298 (2007)CrossRefADSGoogle Scholar
  14. Hammer, M.: METRIC—Mode expansion tools for 2D rectangular integrated optical circuits. http://www.math.utwente.nl/~hammerm/Metric/ (2009)
  15. Hammer, M., Ivanova, O.V.: On effective index approximations of photonic crystal slabs. IEEE/LEOS Benelux Chapter, Proceedings of the 13th Annual Symposium, pp. 203–206. Enschede, The Netherlands (2008)Google Scholar
  16. Ivanova O.V., Hammer M., Stoffer R., van Groesen E.: A variational mode expansion mode solver. Opt. Quantum Electron. 39(10–11), 849–864 (2007)CrossRefGoogle Scholar
  17. Ivanova, O.V., Stoffer, R., Hammer, M., van Groesen, E.: A vectorial variational mode solver and its application to piecewise constant and diffused waveguides. In: 12th International Conference on Mathematical Methods in Electromagnetic Theory MMET08, Odessa, Ukraine, Proceedings, pp. 495–497 (2008a)Google Scholar
  18. Ivanova, O.V., Stoffer, R., Hammer, M.: A dimensionality reduction technique for scattering problems in photonics. 1st International Workshop on Theoretical and Computational Nano-Photonics TaCoNa- Photonics, Conference Proceedings, p. 47 (2008b)Google Scholar
  19. Ivanova, O.V., Stoffer, R., Hammer, M.: Variational effective index method for 3D vectorial scattering problems in photonics: TE polarization. In: Proceedings of the Progress in Electromagnetics Research Symposium PIERS 2009, Moscow, pp. 1038–1042 (2009a)Google Scholar
  20. Ivanova, O.V., Stoffer, R., Hammer, M.: A variational mode solver for optical waveguides based on quasi-analytical vectorial slab mode expansion. Opt. Commun. (2009b) (submitted)Google Scholar
  21. Kok A., Geluk E.J., Docter B., van der Tol J., Nötzel R., Smit M.: Transmission of pillar-based photonic crystal waveguides in InP technology. Appl. Phys. Lett. 91, 201109 (1–3) (2007)CrossRefADSGoogle Scholar
  22. Kok, A.A.M.: Pillar photonic crystals in integrated circuits. Ph.D. Thesis, Technical University of Eindhoven, Eindhoven, The Netherlands (2008)Google Scholar
  23. Krauss T.F., De La Rue R., Brand S.: Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths. Nature 383, 699–702 (1996)CrossRefADSGoogle Scholar
  24. Liu, T., Panepucci, R.R.: Fast estimation of total quality factor of photonic crystal slab cavities. University/Government/Industry Micro/Nano Symposium UGIM 2008, 17th Biennial, Proceedings, pp. 233–235 (2008)Google Scholar
  25. Lohmeyer, M.: Guided waves in rectangular integrated magnetooptic devices. Cuvillier Verlag, Göttingen, Dissertation, Universität Osnabrück (1999)Google Scholar
  26. Lohmeyer M., Stoffer R.: Integrated optical cross strip polarizer concept. Opt. Quantum Electron. 33(4/5), 413–431 (2001)CrossRefGoogle Scholar
  27. Lohmeyer M., Wilkens L., Zhuromskyy O., Dötsch H., Hertel P.: Integrated magnetooptic cross strip isolator. Opt. Commun. 189(4–6), 251–259 (2001)CrossRefADSGoogle Scholar
  28. März R.: Integrated Optics—Design and Modeling. Artech House Boston, London (1994)Google Scholar
  29. Okamoto K.: Fundamentals of Optical Waveguides. Academic Press, San Diego (2000)Google Scholar
  30. Prather D.W., Shi S., Murakowski J., Schneider G.J., Sharkawy A., Chen C., Miao B.: Photonic crystal structures and applications: perspective, overview, and development. IEEE J. Sel. Top. Quantum Electron. 12(6), 1416–1437 (2006)CrossRefGoogle Scholar
  31. Qiu M.: Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals. Appl. Phys. Lett. 81(7), 1163–1165 (2002)CrossRefADSGoogle Scholar
  32. Qiu M., Azizi K., Karlsson A., Swillo M., Jaskorzynska B.: Numerical studies of mode gaps and coupling efficiency for line-defect waveguides in two dimensional photonic crystals. Phys. Rev. B 64, 155113 (1–5) (2001)ADSGoogle Scholar
  33. Qiu M., Jaskorsynska B., Swillo M., Benisty H.: Time domain 2-D modeling of slab-waveguide-based photonic crystal devices in the presence of radiation losses. Microw. Opt. Technol. Lett. 34(5), 387–393 (2002)CrossRefGoogle Scholar
  34. Qiu M., Mulot M., Swillo M., Anand S., Jaskorzynska B., Karlsson A.: Photonic crystal optical filter based on contra-directional waveguide coupling. Appl. Phys. Lett. 83(25), 5121–5123 (2003)CrossRefADSGoogle Scholar
  35. Shi S., Chen C., Prather D.W.: Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs. Appl. Phys. Lett. 86, 043104 (1–3) (2005)ADSGoogle Scholar
  36. Sopaheluwakan, A.: Characterization and Simulation of Localized States in Optical Structures. Ph.D. Thesis, University of Twente, Enschede, The Netherlands (2006)Google Scholar
  37. Taflove A., Hagness S.C.: Computational Electrodynamis: The Finite Difference Time Domain Method, 2nd edn. Artech House, Norwood (2000)zbMATHGoogle Scholar
  38. van de Velde K., Thienpont H., van Geen R.: Extending the effective index method for arbitrarily shaped inhomogeneous optical waveguides. J. Lightwave Technol. 6(6), 1153–1159 (1988)CrossRefADSGoogle Scholar
  39. van Groesen, E.: Variational modelling for integrated optical devices. In: Proceedings of the 4th IMACS- Symposium on Mathematical Modelling, pp. 5–7. Vienna, Feb (2003)Google Scholar
  40. van Groesen E.W.C., Molenaar J.: Continuum Modeling in the Physical Sciences. SIAM, Philadelphia (2007)zbMATHGoogle Scholar
  41. Vassallo C.: Optical Waveguide Concepts. Elsevier, Amsterdam (1991)Google Scholar
  42. Witzens J., Lončar M., Scherer A.: Self-collimation in planar photonic crystals. IEEE J. Sel. Top. Quantum Electron. 8(6), 1246–1257 (2002)CrossRefGoogle Scholar
  43. Yang L., Motohisa J., Fukui T.: Suggested procedure for the use of the effective-index method for high-index-contrast photonic crystal slabs. Opt. Eng. 44(7), 078002 (1–7) (2005)ADSGoogle Scholar
  44. Zhou W., Qiang Z., Chen L.: Photonic crystal defect mode cavity modelling: a phenomenological dimensional reduction approach. J. Phys. D: Appl. Phys. 40, 2615–2623 (2007)CrossRefADSGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.MESA+ Institute for NanotechnologyUniversity of TwenteEnschedeThe Netherlands
  2. 2.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

Personalised recommendations