Project B10 • Numerical methods for nonlinear optics in plasmonic nanogaps

Principal investigators

Project summary

In this project we aim at the construction and the analysis of innovative numerical methods to study nonlinear optics in plasmonic nanogaps which contain a 2D material in their interior. The mathematical modeling consists of (nonlinear) Maxwell equations coupled to a Schrödinger equation via an interface condition, which accounts for the 2D material. Since the optical properties of 2D materials, such as graphene and transition-metal dichalcogenides, can be studied via their interaction with electromagnetic waves, the exploration of the Maxwell–Schrödinger system not only yields a deeper insight into the physics of the 2D materials but also leads to interesting and challenging problems in analysis as well as in numerics. Because of its interdisciplinary aspects, this project will be conducted together with the Theoretical Optics & Photonics group of K. Busch at HU Berlin.

While the electromagnetic waves are governed by 3D Maxwell equations, the dynamics of the Schrödinger equation only takes place in a 2D hypersurface. This situation has not been considered in the mathematical literature so far. To the best of our knowledge, there exists only literature on the wellposedness and numerical solution of the Maxwell–Schrödinger system if both equations are posed on the same spatial domain. We want to construct discretizations in space and time for the fully coupled system and provide rigorous error bounds for these schemes. In addition, together with our partner K. Busch, we will simulate realistic physical applications and thus push forward research in theoretical optics.

Publications

1. B. Dörich and P. Henning. Error bounds for discrete minimizers of the Ginzburg–Landau energy in the high-$\kappa$ regime. SIAM J. Numer. Anal., 62(3):1313–1343, June 2024. URL https://doi.org/10.1137/23M1560938. [preprint] [files] [bibtex]

2. B. Dörich and K. Zerulla. Wellposedness and regularity for linear Maxwell equations with surface current. Z. Angew. Math. Phys., 74(4):131, August 2023. URL https://doi.org/10.1007/s00033-023-02021-w. [preprint] [bibtex]

Preprints

3. B. Dörich. Approximation of minimizers of the Ginzburg–Landau energy in non-convex domains. CRC 1173 Preprint 2025/44, Karlsruhe Institute of Technology, August 2025. [bibtex]

4. C. Döding, B. Dörich, and P. Henning. A multiscale approach to the stationary Ginzburg–Landau equations of superconductivity. CRC 1173 Preprint 2024/21, Karlsruhe Institute of Technology, September 2024. [files] [bibtex]

5. B. Dörich, J. Dörner, and M. Hochbruck. Error analysis of DGTD for linear Maxwell equations with inhomogeneous interface conditions. CRC 1173 Preprint 2024/16, Karlsruhe Institute of Technology, August 2024. Revised version of September 2025. To be published in Mathematics of Computation. [bibtex]