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On high-order FEM applied to canonical scattering problems in plasmonics

On high-order FEM applied to canonical scattering problems in plasmonics. Mengyu Wang 1 , Christian Engströ m 1,2 , Kersten Schmidt 3 , and Christian Hafner 1 6th Workshop on Numerical Methods for Optical Nano Structures July 6th, 2009. IFH, ETH Zurich, Switzerland

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On high-order FEM applied to canonical scattering problems in plasmonics

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  1. On high-order FEM applied to canonicalscattering problems in plasmonics Mengyu Wang1, Christian Engström1,2, Kersten Schmidt3, and Christian Hafner1 6th Workshop on Numerical Methods for Optical Nano Structures July 6th, 2009 • IFH, ETH Zurich, Switzerland • SAM, ETH Zurich, Switzerland • Group POEMS, INRIA, France

  2. Outline • Scattering problem. • Absorbing boundary conditions(BGT) and formular derivation. • Implementation in CONCEPTS • Numerical results & discussion • ABC vs PML • Acknowledgement • Conclusion IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  3. Scattering problem • Scattering problem is important in the research of nano particles and nano antennas. • The metal behaves as the plasma in optical frequency, yet can have Surface Plasmon effect(SPs). • Some structures, e.g. nano particle pairs, have strong local field enhancement. Difficulties • The simulation of these structures are numerically difficult mainly because of rapid field variation. • The narrow gap is quite demanding for the generation of the mesh. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  4. Absorbing boundary condition(ABC) • Truncation of the domain, we need to put absorption layer(PML) or absorption boundary(ABC). • In this talk, we mostly study ABC, and in the end, we will show some comparison between PML. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  5. The idea of ABC comes from Sommerfeld boundary condition. It’s precise, however, it cannot be implemented numerically. So we replace the Sommerfeld condition at infinity with a boundary condition on the boundary of a truncated domain at radius R. Which leads us to Bayliss-Gunzburger-Turkel(BGT) boundary conditions[1], [1] A.Bayliss, M.Gunzburger, and E.Turkel, “Boundary conditions for the numerical solution of elliptic equations in exterior domains.” SIAM J. Appl. Math., vol. 42, no. 2, pp. 430-451, 1982 IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  6. Derivation(TE) Let u denote the total magnetic field, then the Helmholtz equation is, Multiply by test function v, then integrate by parts, formulating the edge integral by BGT condition, we obtain, Finally we get the variational formulation, IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  7. Implementation in CONCEPTS • CONCETPS is a numerical C++ class library [2] • hp-adaptive FEM on tensor product elements (quadrilateral) • CONCEPTS uses curved elements CONCEPTS is currently under the development of K. Schmidt(INRIA), H. Brandsmeier(SAM ETHZ), R. Kapeler(ITET ETHZ), M. Wang(ITET ETHZ) and several students [2] CONCEPTS http://www.concepts.math.ethz.ch IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  8. CONCEPTS provides classes for bilinear form on 2D space and 1D trace space, which will be assembled as stiffness and mass matrix. In the matrix form, the problem becomes, Here is a piece of code that assembles the stiffness matrix IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  9. Results & Discussions • The following simulation is based on the comparison of ref.[3] • 2D silver circle with radius of 400nm, under wavelength 413 nm • A pair of circles with gap 20nm, under wavelength 413nm Dimensional normalization, R = 1, so k = 6.085… Take the permittivity of silver at 413nm, using Drude model permittivity = -4.995…+i0.2190… [3] J. Smajic, C. Hafner, L. Raguin, K. Tavzarashvili, and M. Mishrikey, “Comparison of Numerical Methods for the Analysis of Plasmonic Structures”, J. Comput. Theor. Nanosci., Vol. 6, pp. 763–774, 2009. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  10. Single disk at 413nm wavelength CONCEPTS results Results from [3] IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  11. Comparison with analytical solution IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  12. Comparison along the boundary of the scatter In [3], Comsol result compared with MMP, with d.o.f. 40273 Comsol mesh from [3] IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  13. CONCEPTS results along the boundary of the scatter CONCEPTS mesh CONCEPTS results, with d.o.f. 2956, polynomial degree 15, computing in 9.7 seconds. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  14. Convergence analysis R1 is the radius of ABC, R0 now is the radius of the disk. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  15. Discussion • We observe that • There is a limit of accuracy for each size of ABC. • When ABC is closer, it converges earlier and faster, but finally reaches lower accuracy. • When ABC is further, it converges later and slower, but finally reaches high accuracy. • We are interested further in • Different k • Different mesh • Different polarization IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  16. Different k = 1 Reaches higher accuracy. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  17. Finer mesh, one more step h-refinement The convergence seems not change, but one thing changes: computation time! IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  18. D.o.f.-time relationship Left, no h-refine; right, 1 step h-refine IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  19. Time-error relationship Left, no h-refine; right, 1 step h-refine How to explain? IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  20. TM polarization Seems not change IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  21. 2 disks case 400nm disks, with gap 20nm, left, CONCEPTS results, right, results from [3] IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  22. Discussion • High local enhancement is observed in the gap. • The results fit quite well. • Adaptivity is highly demanded. • e.g. in the very thin skin depth region, the field decays rapidly, then we need to apply fine mesh but low polynomial degree. Inside the disk, we can use very rough mesh(1 element), but high polynomial degree. • Adaptivity will be the first priority in future work. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  23. Comparison between ABC(BGT type) and PML • It’s normally considered that PML has better performance than ABC. And it’s interesting to compare. • PML is joint work with Holger Brandsmeier. According to [4], we implemented high order radia PML in CONCEPTS, which has very good performance. Test problem k = 1, TE, permittivity = 4.0. [4] F. COLLINO AND P. MONK, The Perfectly Matched Layer in Curvilinear Coordinates, SIAM J. Sci. Comput. 19 (6) (1998) 2061-2090. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  24. ABC results A quite fine ABC result with d.o.f. 27601, takes 320 seconds. Relative L2 error 4.129e-7 IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  25. PML results A quite fine PML solution with d.o.f. 10065, takes 34.6 seconds. Relative L2 error 1.35e-12! IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  26. Discussion • Both ABC and PML and reach high accuracy with high polynomial degrees. • PML reaches 1e-12 accuracy, which is comparable with MAX. • Though the comparison is not very strict, we can have the feeling that PML can beat BGT type ABC. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  27. Acknowledgements The authors express their gratitude to: • SNSF Project no. 119813. • HolgerBrandsmeier, for joint work on PML in CONCEPTS. • JasminSmajic, for providing figures from Ref.[3]. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  28. Conclusions • Scattering problem is important and numerically difficult for plasmonic nano device. • BGT boundary conditions are derived and studied. • They are implemented in CONCEPTS. • The convergence of BGT conditions are studied. • Both one disk and two disks case are studied for certain frequency. • The computation efficiency of high-order polynomial FEM is high. • hp-adaptivity is highly demanded in the research of nano devices. • A comparison between ABC and PML has been done. PML seems to beat ABC. • CONCEPTS is a C++ numerical class library, which has hp-adaptivity and curved tensor product elements. CONCEPTS has good performance in computation. • MAX generates good and reliable reference solution, which helps a lot in the verification of new results. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

  29. Thank you! IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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