Effective index approximations of photonic crystal slabs: a 2-to-1-D assessment
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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.
KeywordsIntegrated optics Numerical modeling Photonic crystal slabs Effective index approximation
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.
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.
- 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
- 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
- Hammer, M.: METRIC—Mode expansion tools for 2D rectangular integrated optical circuits. http://www.math.utwente.nl/~hammerm/Metric/ (2009)
- 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
- 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
- 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
- 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
- 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
- Kok, A.A.M.: Pillar photonic crystals in integrated circuits. Ph.D. Thesis, Technical University of Eindhoven, Eindhoven, The Netherlands (2008)Google Scholar
- 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
- Lohmeyer, M.: Guided waves in rectangular integrated magnetooptic devices. Cuvillier Verlag, Göttingen, Dissertation, Universität Osnabrück (1999)Google Scholar
- März R.: Integrated Optics—Design and Modeling. Artech House Boston, London (1994)Google Scholar
- Okamoto K.: Fundamentals of Optical Waveguides. Academic Press, San Diego (2000)Google Scholar
- Sopaheluwakan, A.: Characterization and Simulation of Localized States in Optical Structures. Ph.D. Thesis, University of Twente, Enschede, The Netherlands (2006)Google Scholar
- 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
- Vassallo C.: Optical Waveguide Concepts. Elsevier, Amsterdam (1991)Google Scholar