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.
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. doi:10.1364/OE.11.001378
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. doi:10.1021/cr068126n
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. doi:10.1021/nl802317d
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. doi:10.1117/12.824683
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. doi:10.1007/b98828
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. doi:10.1002/mma.1084
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. doi:10.1109/TAP.2005.858571
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. doi:10.1109/TMTT.2010.2040354
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. doi:10.1166/jctn.2011.1854
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. doi:10.1006/jcph.1994.1159
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. doi:10.1016/j.jcp.2006.09.018
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. doi:10.1137/S1064827596301406
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. doi:10.1016/j.cam.2005.03.047
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. doi:10.1137/S0036142902400901
26. Bao, G., Z. Chen, and H. Wu, "Adaptive finite-element method for diffraction gratings," JOSA A, Vol. 22, No. 6, 1106-1114, 2005. doi:10.1364/JOSAA.22.001106
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. doi:10.1002/num.20252
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. doi:10.1016/j.wavemoti.2010.09.004
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. doi:10.1137/S0036142903423460
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. doi:10.1090/S0025-5718-10-02362-8
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. doi:10.1137/090776202
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. doi:10.1016/0045-7825(95)00946-9
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. doi:10.1016/j.cma.2008.06.009
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. doi:10.1016/S0045-7825(96)01107-3
40. Ainsworth, M. and B. Senior, "An adaptive refinement strategy for hp-finite element computations," Appl. Numerical Mathematics, Vol. 26, 165-178, 1998. doi:10.1016/S0168-9274(97)00083-4
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. doi:10.1201/9781420011685
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. doi:10.1016/j.cam.2011.12.005
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. doi:10.1016/j.cam.2010.11.023
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. doi:10.1016/j.apnum.2011.07.008
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. doi:10.1126/science.1159499
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. doi:10.1209/0295-5075/90/57006
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. doi:10.1103/PhysRevLett.110.127405
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. doi:10.1007/s11468-011-9310-8
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. doi:10.1103/PhysRevB.74.081401
53. Politano, A., "Low-energy collective electronic mode at a noble metal interface," Plasmonics, Vol. 8, No. 2, 357-360, 2013. doi:10.1007/s11468-012-9397-6
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. doi:10.1364/OE.18.007307
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. doi:10.1051/m2an:2002036
56. Ramos, J. S. and A. Huerta, "Efficient unstructured quadrilateral mesh generation," International Journal for Numerical Methods in Engineering, Vol. 49, 1327-1350, 2010.
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. doi:10.1166/jctn.2012.2049