volume | PIER Journals

Abstract

We apply the method of the Broadband Green's Functions with Low wavenumber extraction (BBGFL) to calculate band diagrams in periodic structures. We consider 2D impenetrable objects placed in a 2D periodic lattice. The low wavenumber extraction is applied to the 2D periodic Green's function for the lattice which is used to formulate the surface integral equation. The low wavenumber extraction accelerates the convergence of the Floquet modes expansion. Using the BBGFL to the surface integral equation and the Method of Moments gives a linear eigenvalue equation that gives the broadband (multi-band) solutions for a given point in the first Brillouin zone. The method only requires the calculation of the periodic Green's function at a single low wavenumber. Numerical results are illustrated to show the computational efficiency and accuracy of the method. Because of the acceleration of convergence, an eigenvalue problem with dimensions 49 plane wave Floquet modes are sufficient to give the multi-band solutions that are in excellent agreement with results of the Korringa Kohn Rostoker (KKR) method. The multiband solutions for the band problem and the complementary band problem are also discussed.

1. Ho, K. M., C. T. Chan, and C. M. Soukoulis, "Existence of a photonic gap in periodic dielectric structures," Physical Review Letters, Vol. 65, 3152-3155, 1990.
doi:10.1103/PhysRevLett.65.3152

2. Leung, K. M. and Y. F. Liu, "Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media," Physical Review Letters, Vol. 65, 2646-2649, 1990.
doi:10.1103/PhysRevLett.65.2646

3. Plihal, M. and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B, Vol. 44, No. 16, 8565-8571, 1991.
doi:10.1103/PhysRevB.44.8565

4. Mead, R. D., K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Existence of a photonic bandgap in two dimensions," Applied Physics Letters, Vol. 61, 495-497, 1992.
doi:10.1063/1.107868

5. Kafesaki, M. and C. M. Soukoulis, "Historical perspective and review of fundamental principles in modelling three-dimensional periodic structures with emphasis on volumetric EBGs," Metamaterials, N. Engheta and R. W. Ziolkowski (eds)., Chapter 8, John Wiley and Sons, 2006.

6. Joannopoulos, J. D., S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 2011.

7. Korringa, J., "On the calculation of the energy of a Bloch wave in a metal," Physica, Vol. 13, No. 6, 392-400, 1947.
doi:10.1016/0031-8914(47)90013-X

8. Kohn, W. and N. Rostoker, "Solution of the Schrödinger equation in periodic lattices with an application to metallic lithium," Phys Rev., Vol. 94, 1111-1120, 1954.
doi:10.1103/PhysRev.94.1111

9. Leung, K. M. and Y. Qiu, "Multiple-scattering calculation of the two-dimensional photonic band structure," Physical Review B, Vol. 48, No. 11, 7767-7771, 1993.
doi:10.1103/PhysRevB.48.7767

10. Liu, Z., C. T. Chan, P. Sheng, A. L. Goertzen, and J. H. Page, "Elastic wave scattering by periodic structures of spherical objects: Theory and experiment," Physical Review B, Vol. 62, 2446-2457, 2000.
doi:10.1103/PhysRevB.62.2446

11. Taflove, A. and S. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method, Artech House, 2000.

12. Fan, S., P. R. Villeneuve, and J. D. Joannopoulos, "Large omnidirectional band gaps in metallodielectric photonic crystals," Physical Review B, Vol. 54, 11245-11251, 1996.
doi:10.1103/PhysRevB.54.11245

13. Ziolkowski, R. W. and M. Tanaka, "FDTD analysis of PBG waveguides, power splitters and switches," Optical and Quantum Electronics, Vol. 31, 843-855, 1999.
doi:10.1023/A:1006964830895

14. Hiett, B. P., J. M. Generowicz, S. J. Cox, M. Molinari, D. H. Beckett, and K. S. Thomas, "Application of finite element methods to photonic crystal modelling," IEE Proc-SciMeasurment Technology, Vol. 149, 293-296, 2002.
doi:10.1049/ip-smt:20020642

15. Jin, J.-M. and D. J. Riley, Finite Element Analysis of Antennas and Arrays, Hoboken, Wiley, 2009.

16. Jin, J.-M., The Finite Element Method in Electromagnetics, John Wiley & Sons, 2014.

17. Luo, M., Q. H. Liu, and Z. Li, "Spectral element method for band structures of two-dimensional anisotropic photonic crystals," Physical Review E, Vol. 79, 026705, 2009.
doi:10.1103/PhysRevE.79.026705

18. Bozzi, M., S. Germani, L. Minelli, L. Perregrini, and P. de Maagt, "Efficient calculation of the dispersion diagram of planar electromagnetic band-gap structures by the MoM/BI-RME method," IEEE Trans. on Antennas and Propagation, Vol. 53, No. 1, 29-35, Jan. 2005.
doi:10.1109/TAP.2004.840522

19. Marini, S., A. Coves, V. E.Boria, and B. Gimeno, "Efficient modal analysis of periodic structures loaded with arbitrarily shaped waveguides," IEEE Trans. on Microwave Theory and Tech., Vol. 58, No. 3, 529-536, 2010.
doi:10.1109/TMTT.2010.2040407

20. Tsang, L. and S. Huang, "Full wave modeling and simulations of the waveguide behavior of printed circuit boards using a broadband Green’s function technique,", Provisional U.S. Patent No. 62/152.702, Apr. 24, 2015.

21. Huang, S., "Broadband Green's function and applications to fast electromagnetic analysis of high-speed interconnects,", Ph.D. Dissertation, Dept. Elect. Eng., Univ. Washington, Seattle, WA, Jun. 2015.

22. Huang, S. and L. Tsang, "Broadband Green's function and applications to fast electromagnetic modeling of high speed interconnects," IEEE International Symposium on Antennas and Propagation, Vancouver, BC, Canada, Jul. 2015.

23. Tsang, L. and S. Huang, "Broadband Green's function with low wavenumber extraction for arbitrary shaped waveguide and applications to modeling of vias in finite power/ground plane," Progress of Electromagnetic Research, Vol. 152, 105-125, 2015.
doi:10.2528/PIER15072605

24. Tsang, L., J. A. Kong, K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves, Vol. 2: Numerical Simulations, 705 pages, Wiley Interscience, 2001.

Professor Leung Tsang