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

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, 20091. IFH, ETH Zurich, Switzerland2. SAM, ETH Zurich, Switzerland3. Group POEMS, INRIA, France

1. IFH, ETH Zurich, Switzerland2. SAM, ETH Zurich, Switzerland3. Group POEMS, INRIA, France

7/6/2010 2IFH/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

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.

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.


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

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,


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

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


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.

Single disk at 413nm wavelength


CONCEPTS results Results from [3]

Comparison with analytical solution


Comparison along the boundary of the scatter


Comsol mesh from [3] In [3], Comsol result compared with MMP, with d.o.f. 40273

CONCEPTS results along the boundary of the scatter


CONCEPTS results, with d.o.f. 2956, polynomial degree 15, computing in 9.7 seconds.

CONCEPTS mesh

Convergence analysis


R1 is the radius of ABC, R0 now is the radius of the disk.

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


Different k = 1


Reaches higher accuracy.

Finer mesh, one more step h-refinement


The convergence seems not change, but one thing changes: computation time!

D.o.f.-time relationship


Left, no h-refine; right, 1 step h-refine

Time-error relationship


Left, no h-refine; right, 1 step h-refineHow to explain?

TM polarization


Seems not change

2 disks case


400nm disks, with gap 20nm, left, CONCEPTS results, right, results from [3]

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.


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.

ABC results


A quite fine ABC result with d.o.f. 27601, takes 320 seconds. Relative L2 error 4.129e-7

PML results


A quite fine PML solution with d.o.f. 10065, takes 34.6 seconds. Relative L2 error 1.35e-12!

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.


Acknowledgements

The authors express their gratitude to:SNSF Project no. 119813.Holger Brandsmeier, for joint work on PML in CONCEPTS.Jasmin Smajic, for providing figures from Ref.[3].


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.


Thank you!


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

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