Finite Element Method for problems in plasmonic structures

Plasmonic structures

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

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 back-holes waves, appear at the corners

Standard FEM fail due to spurious reflexions

Novel numerical method using FEM:

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

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. Left: standard FEM, spurious reflxions at the corners (steady wave). Right: multi-scale method, the plasmons propagates 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