# Mathematics in Optical Technologies

The major challenges in our modern world are health, environment, energy, production, and security. They all are driven essentially by generation and manipulation of photons. That is why photonics is one of the key technologies of the 21st century. Major strategic plans of the U.S. government (Harnessing Light II – Photonics for 21st Century Competitiveness), of the European Commission (Strategic Research Agenda -- Lighting the way ahead), of the European Technology Platform Photonics21, and of the BMBF (Agenda Photonik 2020) acknowledge this trend. The Agenda Photonik 2020 confirms that optical technologies are Germany's most important future technologies and are even more important than the pharmaceutical industry.

To support the complex and rapidly developing optical technologies in
Berlin, ECMath supports research projects in the Innovation Area Mathematics in Optical
Technologies (MOT).
The mathematical research will broaden the research of the Matheon application area Electronic and Photonic devices.

The methods in Innovation Area MOT will span a whole range of mathematical disciplines, from mathematical physics as the modelling language, theory and numerical simulation of partial differential equations to solve high dimensional and multiscale problems to applied stochastics.

## Projects

Prof. Dr. Alexander Mielke

Prof. Dr. Thomas Surowiec

Dr. Marita Thomas

Dr. Dirk Peschka

**Duration:**01.06.2014 - 31.05.2017

**Status:**running

**Located at:**Humboldt Universität Berlin (HU Berlin) , Weierstraß-Institut (WIAS Berlin)

### Description

The goal of the project Mathematical Modeling, Analysis, and Optimization of Strained Germanium-Microbridges is to optimize the design of a strained Germanium microbridge with respect to the light emission. It is a joint project with the Humboldt-University Berlin (M. Hintermüller, T. Surowiec) and the Weierstrass Institute (A. Mielke, M. Thomas), that also involves the close collaboration with the Department for Materials Research at IHP (Leibniz-Institute for Innovative High Performance Microelectronics, Frankfurt Oder).### Website

X Close projectDr. M. Wolfrum

**Duration:**01.06.2014 - 31.05.2017

**Status:**running

**Located at:**Weierstraß-Institut (WIAS Berlin)

### Description

Many modern photonic devices show complex dynamical features in space and time resulting from optical nonlinearities in active, often nanostructured materials. The project is focussed specifically on high-dimensional dynamical regimes in optoelectronic systems. Such a complex spatio-temporal behavior, in which nearly all modes are excited, is characterized by the fact that, in contrast to e.g. solitons or pulsations, it cannot be reduced to a low-dimensional description in terms of classical bifurcation theory. This so-called optical turbulence can be observed both in a Hamiltonian and in a dissipative context. A mathematical treatment of the resulting multi-scale and multi-physics problems presents major challenges for modelling, numerical, and analytical investigations. A simulation of the mostly 2+1 dimensional PDE-systems requires efficient parallelization strategies, instability mechanisms can be described only in terms of amplitude equations, and multi-scale effects in complex device structures can lead to singularly perturbed dynamical problems.### Website

X Close project**Duration:**01.06.2014 - 31.05.2017

**Status:**running

**Located at:**Technische Universität Berlin (TU Berlin)

### Description

Photonic crystals are periodic materials that affect the propagation of electromagnetic waves. They occur in nature (e.g. on butterfly wings), but they can also be manufactured. They possess certain properties affecting the propagation of electromagnetic waves in the visible spectrum, hence the name photonic crystals. The most interesting (and useful) property of such periodic structures is that for certain geometric and material configurations they have the so-called bandgaps, i.e., intervals of wavelengths that cannot propagate in the periodic structure. Therefore, finding materials and geometries with wide bandgaps is an active research area. Mathematically, finding such bandgaps for different configurations of materials and geometries can be modelled as a PDE eigenvalue problem with the frequency (or wavelength) of the electromagnetic field as the eigenvalue. These eigenvalue problems depend on various parameters describing the material of the structure and typically involve nonlinear functions of the searched frequency. The configuration of the periodic geometry may also be modified and can be considered a parameter. Finally, through the mathematical treatment of the PDE eigenvalue problem another parameter, the quasimomentum, is introduced in order to reduce the problem from an infinite domain to a family of problems, parametrised by the quasimomentum, on a finite domain. These are more accurately solvable. In order to solve the problem of finding a material and geometric structure with an especially wide bandgap, one needs to solve many nonlinear eigenvalue problems during each step of the optimization process. Therefore, the main goal of this project is to find efficient nonlinear eigensolvers. It is well-known that an efficient way of discretizing PDE eigenvalue problems on geometrically complicated domains is an adaptive Finite Element method (AFEM). To investigate the performance of AFEM for the described problems reliable and efficient error estimators for nonlinear parameter dependent eigenvalue problems are needed. Solving the finite dimensional nonlinear problem resulting from the AFEM discretization in general cannot be done directly, as the systems are usually large, and thus produce another error to be considered in the error analysis. Another goal in this research project is therefore to equilibrate the errors and computational work between the discretization and approximation errors of the AFEM and the errors in the solution of the resulting finite dimensional nonlinear eigenvalue problems.### Website

X Close project**Duration:**01.06.2014 - 31.05.2017

**Status:**running

**Located at:**Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

### Description

A typical trend in nanotechnology is to extend technology from basically 2D structures to 3D structures, from simple 2D layouts to complex 3D layouts. This has mainly two reasons: (i) There are fundamental physical effects bound to 3D structures, e.g., manifold properties in reciprocal space, and (ii) economic reasons as in semiconductor industry which enforce denser packaging and ever more complex functionalities.The automatic optimization of nano-photonic device geometries is becoming increasingly important and, due to enhanced complexity, increasingly difficult. Typical one-way simulations become unfeasible in many-query and real-time contexts. Model reduction techniques could be a way out. Potentially they offer online speed ups in the order of magnitudes. The reported success, however, is often linked to relatively simple structured objects. Slightly more complex examples fail immediately due to geometric and mesh constrains. To show the potential in real-world examples, however, complex 3D objects including comprehensive parametrizations have to be assembled.

The project aims to establish a link from 3D solid models obtained by CAD techniques, including full parametrizations, to reduced basis models. Establishing this critical link would facilitate systematic device geometry optimizations to be carried out using rigorous 3D electromagnetic field simulations. The main question is, how we can realize a large scale parametrization maintaining topologically equivalent meshes.