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Project: Efficient and reliable computations of scattering resonances

Scattering resonance computations in nano-optics are important in the development of photothermal cancer therapy, optical antennas, and surface-enhanced Raman scattering. In this project, we use advanced material models for the optical properties of metal-dielectric structures. Our main objectives are to develop robust high order finite element methods and solvers for nonlinear eigenvalue problems to be used in scientific computing and engineering.

The image represents the defect mode of a finite crystal consisting of 5x5 dielectric rods and a point defect obtained by removing the rod at the center (Araujo-Cabarcas, J.C., 2020).

Project information

Project manager at Linnaeus University
Christian Engström
Other project members
Juan-Carlos Araujo-Cabarcas, Umeå University, Sweden; Elias Jarlebring, KTH Royal Institute of Technology, Sweden; Carmen Campos and Jose Roman, Polytechnic University of Valencia, Spain
Participating organizations
Linnaeus University, Umeå University and KTH Royal Institute of Technology, Sweden; Polytechnic University of Valencia, Spain
The Swedish Research Council, 2013–2018
Mathematics (Department of Mathematics, Faculty of Technology)

More about the project

Scattering resonances and quasinormal modes are used to characterize open resonators. Those modes are important in the analysis of a broad range of applications including gravitational-wave signals, musical instruments, and optical resonators.

Scattering resonances replace stationary states for systems in which energy can scatter to infinity. A common technique to terminate the computational domain is a perfectly matched layer (PML). This leads for non-dispersive materials to a linear but highly non-normal eigenvalue problem. An alternative formulation is based on a Dirichlet-to-Neumann (DtN) map, which accounts for the unbounded domain. However, this leads to a nonlinear eigenvalue problem also for non-dispersive materials.

A major challenge in numerical approximation of resonances is the presence of eigenvalues that are unrelated to the scattering resonances of the original operator. These eigenvalues are called spurious eigenvalues and they are in particular common when low-order finite element methods are used with a PML.

In the project, we develop:

  1. High order finite element methods (hp-FEM) for scattering resonance computations with PML/DtN.
  2. Marking strategies for potentially spurious eigenvalues.
  3. Specialized algorithms for nonlinear eigenvalue problems.


Araujo-Cabarcas, J.C., Engström, C. (2020), On spurious solutions encountered in Helmholtz scattering resonance computations in R^d with applications to nano-photonics and acoustics, arXiv:1904.08812.

Araujo-Cabarcas, J.C., Campos, C., Engström, C., Roman, J. (2020). Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures. Journal of Computational Physics. 407. 1-24.

Araujo-Cabarcas, J.C., Engström, C., Jarlebring, E. (2018). Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map. Journal of Computational and Applied Mathematics. 330. 177-192.

Araujo-Cabarcas, J.C., Engström, C. (2017). On spurious solutions in finite element approximations of resonances in open systems. Computers and Mathematics with Applications. 74. 2385-2402.