Vol. 142

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

Hp-FEM and PML Analysis of Plasmonic Particles in Layered Media

By Mengyu Wang, Kersten Schmidt, Aytac Alparslan, and Christian V. Hafner
Progress In Electromagnetics Research, Vol. 142, 523-544, 2013


In this paper, we introduce a high order finite element (FEM) implementation using perfectly matched layer (PML) for the scattering by plasmonic structures inside layered media. The PML is proven to be very accurate and efficient by a comparative analysis with a commercial FEM software and the Multiple Multipole Program (MMP). A convergence analysis using hp-adaptive refinement inside the PML layer shows that adaptive mesh refinement inside the PML layer is most efficient. Based on this convergence analysis an hp-strategy is proposed, which shows a remarkable error reduction for small additional computational costs.


Mengyu Wang, Kersten Schmidt, Aytac Alparslan, and Christian V. Hafner, "Hp-FEM and PML Analysis of Plasmonic Particles in Layered Media," Progress In Electromagnetics Research, Vol. 142, 523-544, 2013.


    1. Bharadwaj, P., B. Deutsch, and L. Novotny, "Optical antennas," Advances in Optics and Photonics, Vol. 1, No. 3, 438-483, 2009.

    2. Novotny, L. and B. Hecht, Principles of Nano-optics, Cambridge University Press, 2012.

    3. Smajic, J., C. Hafner, and D. Erni, "Design and optimization of an achromatic photonic crystal bend," Opt. Express, Vol. 11, No. 12, 1378-1384, 2003.

    4. Stewart, M. E., C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, "Nanostructured plasmonic sensors," Chemical Reviews, Vol. 108, No. 2, 494-521, 2008.

    5. Sannomiya, T., C. Hafner, and J. Voros, "In situ sensing of single binding events by localized surface plasmon resonance," Nano Letters, Vol. 8, No. 10, 3450-3455, 2008.

    6. Sannomiya, T., C. Hafner, and J. Voros, "Plasmonic nanoparticle based biosensing: Experiments and simulations," Proc. SPIE Plasmonics: Nanoimaging, Nanofabrication, and Their Applications V, Vol. 7395, 73950M, 2009.

    7. Kong, J. A., Electromagnetic Wave Theory, Wiley, New York, 1986.

    8. Ihlenburg, F., Finite Element Analysis of Acoustic Scattering, Springer, Berlin & Heidelberg, Germany, 1998.

    9. Givoli, D., Numerical Methods for Problems in Infinite Domains, Elsevier, Amsterdam and New York, 1992.

    10. Bonnet-BenDhia, A.-S., G. Dakhia, C. Hazard, and L. Chorfi, "Diffraction by a defect in an open waveguide: A mathematical analysis based on a modal radiation condition," SIAM J. Appl. Math., Vol. 70, No. 3, 677-693, Jul. 2009.

    11. Ciraolo, G. and R. Magnanini, "A radiation condition for uniqueness in a wave propagation problem for 2-D open waveguides," Math. Meth. Appl. Sci., Vol. 32, No. 10, 1183-1206, 2009.

    12. Bonnet-BenDhia, A.-S., B. Goursaud, and C. Hazard, "Mathematical analysis of the junction of two acoustic open waveguides," SIAM J. Appl. Math., Vol. 71, 2048-2071, 2011.

    13. Jeresz-Hanckes, C. and J.-C. Nedelec, "Asymptotics for Helmoltz and Maxwell solutions in 3-D open waveguides," Commun. Comput. Phys., Vol. 11, No. 2, 629-646, Feb. 2012.

    14. Schmidt, F., "A new approach to coupled interior-exterior Helmholtz-type problems: Theory and algorithms,", Habilitation Thesis, Free University Berlin, Germany, 2002.

    15. Aksun, M. I. and G. Dural, "Clarification of issues on the closed-form Green's functions in stratified media," IEEE Transactions on Antennas and Propagation, Vol. 53, No. 11, 3644-3653, 2005.

    16. Sauter, S. and C. Schwab, "Boundary Element Methods," Springer-Verlag, Heidelberg, 2011.

    17. Alparslan, A., M. I. Aksun, and K. A. Michalski, "Closed-form Green's functions in planar layered media for all ranges and materials," IEEE Transactions on Microwave Theory and Techniques, Vol. 58, No. 3, 602-613, 2010.

    18. Alparslan, A. and C. Hafner, "Using layered geometry Green's functions in the multiple multipole program," Journal of Computational and Theoretical Nanoscience, Vol. 8, No. 8, 1600-1608, 2011.

    19. Berenger, J.-P., "A perfectly matched layer for the absorption of electromagnetic waves," Journal of Computational Physics, Vol. 114, No. 2, 185-200, 1994.

    20. Jin, J.-M. and W. C. Chew, "Combining PML and ABC for the finite-element analysis of scattering problems," Microwave and Optical Technology Letters, Vol. 12, No. 4, 192-197, 1996.

    21. Chew, W. C., W. H. Weedon, and A. Sezginer, "A 3-D perfectly matched medium by coordinate stretching and its absorption of static fields," Applied Computational Electromagnetics Symposium Digest, Vol. 1, 482-489, Citeseer, 1995.

    22. Bermudez, A., L. Hervella-Nieto, and A. Prieto, "An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems," Journal of Computational Physics, Vol. 223, No. 2, 469-488, 2007.

    23. Collino, F. and P. Monk, "The perfectly matched layer in curvilinear coordinat," SIAM Journal on Scientific Computing, Vol. 19, No. 6, 2061-2090, 1998.

    24. Zschiedrich, L., R. Klose, A. SchÄadle, and F. Schmidt, "A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions," Journal of Computational and Applied Mathematics, Vol. 188, No. 1, 12-32, 2006.

    25. Chen, Z. and H. Wu, "An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures," SIAM J. Numer. Anal., Vol. 41, No. 3, 799-826, 2003.

    26. Bao, G., Z. Chen, and H. Wu, "Adaptive finite-element method for diffraction gratings," JOSA A, Vol. 22, No. 6, 1106-1114, 2005.

    27. Michler, C., L. Demkowicz, J. Kurtz, and D. Pardo, "Improving the performance of perfectly matched layers by means of hp-adaptivity," Numerical Methods for Partial Differential Equations, Vol. 23, No. 4, 832-858, 2007.

    28. Zschiedrich, L., "Transparent boundary conditions for Maxwell's equations,", Ph.D. Thesis, FU Berlin, Berlin, Germany, Nov. 2009.

    29. Nannen, L. and A. Schadle, "Hardy space infinite elements for Helmholtz-type problems with unbounded inhomogeneities," Wave Motion, Vol. 48, No. 2, 116-129, 2011.

    30. Kettner, B. and F. Schmidt, "The pole condition as transparent boundary condition for resonance problems: Detection of spurious modes," Proc. SPIE, Vol. 7933, 79331B-1-79331B-11, 2011.

    31. Kettner, B., "Detection of spurious modes in resonance mode computations --- Pole condition method,", Ph.D. Thesis, FU Berlin, Berlin, Germany, Jul. 2012.

    32. Schwab, C., p- and hp-finite Element Methods: Theory and Applications in Solid and Fluid Mechanisms, Oxford University Press, Oxford, UK, 1998.

    33. Ainsworth, M., "Discrete dispersion relation for hp-version finite element approximation at high wave number," SIAM J. Numer. Anal., Vol. 42, No. 2, 553-575, 2005.

    34. Melenk, J. M. and S. Sauter, "Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichletto-Neumann boundary conditions," Math. Comp., Vol. 79, No. 272, 1871-1914, 2010.

    35. Melenk, J. M. and S. Sauter, "Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation," SIAM J. Numer. Anal., Vol. 49, No. 3, 1210-1243, 2011.

    36. Babuska, I. and B. Q. Guo, "Approximation properties of the h-p version of the finite element method," Computer Methods in Appl. Mechanics Engineering, Vol. 133, 319-346, 1996.

    37. Schmidt, K. and P. Kauf, "Computation of the band structure of two-dimensional photonic crystals with hp finite elements," Computer Methods in Appl. Mechanics Engineering, Vol. 198, 1249-1259, Mar. 2009.

    38. Babushka, I. and W. Rheinbolt, "A posteriori analysis for adaptive finite element computations," SIAM J. Numer. Anal., Vol. 15, 736-754, 1978.

    39. Ainsworth, M. and J. T. Oden, "A posteriori error estimation in finite element analysis," Computer Methods in Appl. Mechanics Engineering, Vol. 142, No. 1-2, 1-88, 1997.

    40. Ainsworth, M. and B. Senior, "An adaptive refinement strategy for hp-finite element computations," Appl. Numerical Mathematics, Vol. 26, 165-178, 1998.

    41. Becker, R. and R. Rannacher, "An optimal control approach to a posteriori error estimation in finite element methods," Acta Numerica, Vol. 10, No. 1, 1-102, 2001.

    42. Demkowicz, L., Computing with hp-adaptive Finite Elements: One and Two Dimensional Elliptic and Maxwell Problems,, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science, 2006.

    43. Schnepp, S. M. and T. Weiland, "Efficient large scale electromagnetic simulations using dynamically adapted meshes with the discontinuous Galerkin method," Journal of Computational and Applied Mathematics, Vol. 236, No. 18, 4909-4924, 2011.

    44. Wihler, T. P., "An hp-adaptive strategy based on continuous Sobolev embeddings," Journal of Computational and Applied Mathematics, Vol. 235, No. 8, 2731-2739, 2011.

    45. BÄurg, M. and W. Dofler, "Convergence of an adaptive hp finite element strategy in higher space-dimensions," Applied Numerical Mathematics, Vol. 61, No. 11, 1132-1146, 2011.

    46. Jackson, J. D., Classical Electrodynamics, 3rd Ed., John Wiley & Sons, 1999.

    47. Fang, Y., N.-H. Seong, and D. D. Dlott, "Measurement of the distribution of site enhancements in surface-enhanced Raman scattering," Science, Vol. 321, No. 5887, 388-392, 2008.

    48. Park, S. J. and R. E. Palmer, "Acoustic plasmon on the Au (111) surface," Physical Review Letters, Vol. 105, No. 1, 016801, 2010.

    49. Pohl, K., B. Diaconescu, G. Vercelli, L. Vattuone, V. M. Silkin, E. V. Chulkov, P. M. Echenique, and M. Rocca, "Acoustic surface plasmon on Cu (111)," EPL (Europhysics Letters), Vol. 90, No. 5, 57006, 2010.

    50. Vattuone, L., M. Smerieri, T. Langer, C. Tegenkamp, H. Pfnur, V. M. Silkin, E. V. Chulkov, P. M. Echenique, and M. Rocca, "Correlated motion of electrons on the Au (111) surface: Anomalous acoustic surface-plasmon dispersion and single-particle excitations," Physical Review Letters, Vol. 110, No. 12, 127405, 2013.

    51. Vattuone, L., G. Vercelli, M. Smerieri, L. Savio, and M. Rocca, "Acoustic surface plasmon dispersion on nanostructured Cu (111)," Plasmonics, Vol. 7, No. 2, 323-329, 2012.

    52. Politano, A., G. Chiarello, V. Formoso, R. G. Agostino, and E. Colavita, "Plasmon of shockley surface states in Cu (111): A high-resolution electron energy loss spectroscopy study," Physical Review B, Vol. 74, No. 8, 081401, 2006.

    53. Politano, A., "Low-energy collective electronic mode at a noble metal interface," Plasmonics, Vol. 8, No. 2, 357-360, 2013.

    54. Schmidt, K. and R. Kappeler, "Efficient computation of photonic crystal waveguide modes with dispersive material," Optics Express, Vol. 18, No. 7, 7307-7322, 2010.

    55. Frauenfelder, P. and C. Lage, "Concepts --- An object-oriented software package for partial differential equations," ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 36, No. 05, 937-951, 2002.

    56. Ramos, J. S. and A. Huerta, "Efficient unstructured quadrilateral mesh generation," International Journal for Numerical Methods in Engineering, Vol. 49, 1327-1350, 2010.

    57. EZ4U, Mesh Generation Environment, , www.lacan.upc.edu/ez4u.htm.

    58. Hafner, C., MaX-1: A Visual Electromagnetics Platform for PCs, John Wiley & Sons, Chichester, UK, 1999.

    59. Hafner, C., Post-modern Electromagnetics: Using Intelligent Maxwell Solvers, Wiley, 1999.

    60. Alparslan, A. and C. Hafner, "Analysis of photonic structures by the multiple multipole program with complex origin layered geometry Green's functions," Journal of Computational and Theoretical Nanoscience, Vol. 9, No. 3, 479-485, 2012.

    61., , COMSOL Multiphysics, http://www.comsol.com/.