Finite Element Method for Plasmonics

Plasmonic Structures

Plasmonic structures are made of a positive material (dielectrics) and a negative material (metals at optical frequencies, metamaterials). Surface electromagnetic waves called surface plasmons can appear at the interface.

Plasmonic structure

Applications

Guiding and confining such particular waves in nanophotonic devices reveal a great interest to overcome the diffraction limit, in nanophotonic sensing and related applications.

Challenges

  • Multiple scales
  • Surface plasmons are very sensitive to the geometry (corners)
  • Inaccurate predictions of the near field
  • Hyper-oscillating singularities, called black-hole waves, appear at the corners
  • Standard FEM fail due to spurious reflections

Novel Numerical Method using FEM

Mesh requirements to ensure FEM optimal convergence via the T-coercivity

Standard mesh T-conforming mesh

An hexagonal cavity with an hexagonal negative material inclusion. Left: standard mesh. Right: T-conforming mesh.

Use of Perfectly Matched Layers at the corners to capture the black-hole waves

Scattering problem by a plane wave of a metallic inclusion. Top: standard FEM, spurious reflections at the corners (steady wave). Bottom: multi-scale method, the plasmons propagate towards the corners.

References

  • Mesh requirements for the finite element approximation of problems with sign-changing coefficients A.-S. Bonnet-Ben Dhia, C. Carvalho, P. Ciarlet Jr. Numerische Mathematik, pp 1-38, 2018.

  • Eigenvalue problems with sign-changing coefficients C. Carvalho, L. Chesnel, P. Ciarlet Jr. Compte Rendus Mathématiques, 355 (6), pp 671-675, 2017.

  • On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients A.-S. Bonnet-Ben Dhia, C. Carvalho, L. Chesnel, P. Ciarlet Jr. Journal of Computational Physics, 322, pp 224-247, 2016.

Ongoing Projects

  • T-conforming mesh generator
  • Limiting Amplitude Principle