1. On high-order FEM applied to canonical scattering 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. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
Outline
Scattering problem.
Absorbing boundary conditions(BGT) and formular derivation.
Implementation in CONCEPTS
Numerical results & discussion
ABC vs PML
Acknowledgement
Conclusion
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

3. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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.
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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.
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

5. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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 , 1982
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

6. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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,
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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
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

8. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

9. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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 = …+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.
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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

11. Comparison with analytical solution
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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
Comsol mesh from [3]
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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.
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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

15. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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

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

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

22. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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.
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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)
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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

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

26. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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.
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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

28. IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures
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.
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IFH/ETHZ 6th Workshop on Numerical Methods for Optical Nano Structures

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