Abstract
The paper deals with band-structure calculations of photonic crystals with hexagonal lattice where the unit cell has the shape of a parallelogram. The boundary of this parallelogram, which is identical to the computational window, is modeled with a staircase approximation. In this way Cartesian coordinates can be used to compute the electromagnetic fields in the structure. When using this approach one must relate fields outside the computational window to those inside this area. This is done by utilizing the periodicity of the fields in photonic crystals. The procedure was used to compute the band structures of hexagonal photonic crystals with the Method of Lines and with a Finite Difference Method. The determined results agree very well with those from the literature.
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References
Benson T.M., Sewell P., Sujecki S., Kendall P.C.: Structure related beam propagation. Opt. Quantum Electron. 31, 689–703 (1999)
Bienstman P., Baets R.: Optical modelling of photonic crystals and VCSELs using eigenmode expansion and perfectly matched layers. Opt. Quantum Electron. 33, 327–341 (2001)
Bogaerts W., Bienstman P., Taillaert D., Baets R., De-Zutter D.: Out-of-plane scattering in 1-D photonic crystal slabs. Opt. Quantum Electron. 34, 195–203 (2002)
Cao Q., Lalanne P., Hugonin J.P.: Stable and efficient Bloch-mode computational method for one–dimensional grating waveguides. J. Opt. Soc. Am. A Opt. Image Sci. 19, 335–338 (2002)
Collin R.E.: Field Theory of Guided Waves, 2nd edn, Series of Electromagnetic Waves. IEEE press, New York (1991)
Cryan M.J., Wong D.C.L., Craddock I.J., Yu S., Rorison J., Railton C.J.: Electromagnetic analysis of photonic crystals waveguide operating above the light cone. IEEE Photonics Technol. Lett. 17, 58–60 (2005)
Edelmann A.G., Helfert S.F.: Three-dimensional analysis of hexagonal structured photonic crystals using oblique coordinates. Opt. Quantum Electron. 41, 243–254 (2010)
Guo S., Albin S.: Numerical technique for excitation and analysis of defect modes in photonic crystals. Opt. Expr. 11, 1080–1089 (2003)
Helfert S.F.: Numerical stable determination of Floquet-modes and the application to the computation of band structures. Opt. Quantum Electron. 36, 87–107 (2004)
Helfert S.F.: Determination of Floquet-modes in asymmetric periodic structures. Opt. Quantum Electron. 37, 185–197 (2005)
Helfert S.F.: Applying oblique coordinates to the method of lines. Prog. Electron. Res. PIER 61, 271–278 (2006)
Helfert, S.F.: Introducing oblique coordinates in numerical methods, applied to the computation of band structures. In: ICTON Conference, Rome, Italy, 2:80–83 (2007)
Helfert S.F., Pregla R.: Efficient analysis of periodic structures. J. Lightwave Technol. 16, 1694–1702 (1998)
Joannopoulus J.D., Meade R.D., Winn J.N.: Photonic Crystals—Molding the Flow of Light. Princeton University Press, Princeton (1995)
Joannopoulus J.D., Johnson S.G., Meade R.D., Winn J.N.: Photonic Crystals—Molding the Flow of Light. Princeton University Press, Princeton (2008)
John S.: Strong localisation of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 5, 2486–2489 (1987)
Johnson S.G., Joannopoulos J.D.: Photonic Crystals—the Road from Theory to Practice. Springer, New York (2001)
Pregla R.: Efficient modeling of periodic structures. Int. J. Electron. Commun. (AEÜ) 57, 185–189 (2003)
Pregla R.: Analysis of Electromagnetic Fields and Waves—the Method of Lines. Wiley, Chichester (2008)
Pregla R., Pascher W.: The method of lines. In: Itoh, T. (eds) Numerical Techniques for Microwave and Millimeter Wave Passive Structures., pp. 381–446. Wiley, New York, USA (1989)
Sewell P., Benson T.M., Sujecki S., Kendall P.C.: The dispersion characteristics of oblique coordinate beam propagation algorithms. J. Lightwave Technol. 17, 514–518 (1999)
Villeneuve P.R., Fan S., Johnson S.G., Joannopoulos J.D.: Three-dimensional photon confinement in photonic crystals of low–dimensional periodicity. IEE Proc. Optoelectron. 145, 384–390 (1998)
Wu N., Javanmard M., Momeni B., Soltani M., Adibi A., Xu Y., Lee R.K.: General methods for designing single-mode planar photonic crystal waveguides in hexagonal lattice structures. Opt. Expr. 11, 1371–1377 (2003)
Yablonovitch E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987)
Yamauchi, J., Shibayama, J., Nakano, H.: Propagating beam analysis based on the implicit finite–difference method using the oblique coordinate system. In: OSA Integr. Photo. Res. Tech. Dig., San Francisco, USA, 19–21 (1994)
Ziolkowski R.W., Tanaka M.: FDTD analysis of PBG waveguides, power splitters and switches. Opt. Quantum Electron. 31, 843–855 (1999)
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Helfert, S.F. Staircase approximation of oblique boundaries to compute band structures of photonic crystals. Opt Quant Electron 42, 447–461 (2011). https://doi.org/10.1007/s11082-010-9432-9
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DOI: https://doi.org/10.1007/s11082-010-9432-9