Optical Technologies

The major challenges of our modern world lie in the fields of health, environment, energy, production, and security. Scientific progress and innovation in these areas are essentially driven by the generation and manipulation of photons – e.g. in photonic devices, data transmission, sensors, high-resolution microscopy, or the manipulation of biological and other materials, or even the the manipulation of light by light.

That is why photonics is one of the key technologies of the 21st century. Major strategic plans of the German Federal Ministry of Education and Research BMBF (Agenda Photonik 2020), of the European Commission (Strategic Research Agenda - Lighting the way ahead), of the European Technology Platform Photonics21, and of the U.S. government (Harnessing Light II – Photonics for 21st Century Competitiveness) acknowledge this trend. The BMBF Agenda Photonik 2020 confirms that optical technologies are Germany's most important future technologies, even more important than the pharmaceutical industry.

Mathematics plays a key role in optical technologies: It serves as the most precise microscope one can think of. The simulation of photonic structures helps creating more efficient devices, and mathematical modeling and simulation leads to a better understanding of light-matter interaction, including engineering of open quantum systems.

The methods in this application field 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.



Topics

  • Photonic structures (Waveguides, Photonic Crystals)
  • Semiconductor lasers
  • Nonlinear Wave Equations, Solitons
  • Complex nonlinear spatio-temporal dynamics
  • Numerical Methods for Maxwell Equations
  • Semiconductor Transport for Devices with Nanostructures
  • Open Quantum Systems


Running projects

Financed by ECMath

  • OT1

    Mathematical modeling, analysis, and optimization of strained Germanium-microbridges

    Prof. Dr. Michael Hintermüller / Prof. Dr. Alexander Mielke / Prof. Dr. Thomas Surowiec / Dr. Marita Thomas

    Project heads: Prof. Dr. Michael Hintermüller / Prof. Dr. Alexander Mielke / Prof. Dr. Thomas Surowiec / Dr. Marita Thomas
    Project members: Lukas Adam / Dr. Dirk Peschka
    Duration: 01.06.2014 - 31.05.2017
    Status: running
    Located at: Humboldt Universität Berlin / Weierstraß-Institut

    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).

    http://www.wias-berlin.de/projects/ECMath-OT1/
  • OT2

    Turbulence and extreme events in non-linear optics

    PD Dr. Uwe Bandelow / Dr. M. Wolfrum

    Project heads: PD Dr. Uwe Bandelow / Dr. M. Wolfrum
    Project members: Dr Shalva Amiranashvili
    Duration: 01.06.2014 - 31.05.2017
    Status: running
    Located at: Weierstraß-Institut

    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.

    http://www.wias-berlin.de/projects/ECMath-OT2/project_OT2.jsp
  • OT3

    Adaptive finite element methods for nonlinear parameter-dependent eigenvalue problems in photonic crystals

    Prof. Dr. Volker Mehrmann / Dr. A. Miedlar

    Project heads: Prof. Dr. Volker Mehrmann / Dr. A. Miedlar
    Project members: Robert Altmann
    Duration: 01.06.2014 - 31.05.2017
    Status: running
    Located at: Technische Universität 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.

    http://www3.math.tu-berlin.de/numerik/NumMat/ECMath/OT3/

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Financed by others

  • OT-AP1

    Multi-Dimensional Modeling and Simulation of Electrically Pumped Semiconductor-Based Emitters

    PD Dr. Uwe Bandelow / Dr. Thomas Koprucki / Prof. Dr. Alexander Mielke / Prof. Dr. Frank Schmidt

    Project heads: PD Dr. Uwe Bandelow / Dr. Thomas Koprucki / Prof. Dr. Alexander Mielke / Prof. Dr. Frank Schmidt
    Project members: -
    Duration: 01.01.2008 - 31.12.2019
    Status: running
    Located at: Weierstraß-Institut / Konrad-Zuse-Zentrum für Informationstechnik Berlin

    Description

    The aim of this joint project of WIAS and ZIB is the comprehensive and self-consistent optoelectronic modeling and simulation of electrically pumped semiconductor-based light emitters with spatially complex 3D device structure and quantum dot active regions. The required models and methods for an accurate representation of devices, such as VCSELs and single photon emitters, featuring open cavities, strong interactions between optical fields and carriers, quantum effects, as well as heating will be developed and implemented, resulting in a set of tools, that will be provided for our partners in the CRC 787.

    http://www.zib.de/projects/multi-dimensional-modeling-and-simulation-vertical-cavity-surface-emitting-lasers-vcsels / http://wias-berlin.de/projects/sfb787-b4/
  • OT-AP10

    Analysis of discretization methods for nonlinear evolution equations

    Prof. Dr. Etienne Emmrich

    Project heads: Prof. Dr. Etienne Emmrich
    Project members: -
    Duration: 01.09.2012 - 31.12.2018
    Status: running
    Located at: Technische Universität Berlin

    Description

    Nonlinear evolution equations are the mathematical models for time-dependent processes in science and engineering. Relying upon the theory of monotone operators and compactness arguments, we study existence of solutions, convergence of discretization methods, and feedback control for equations of dissipative type. We focus on nonlocality in time (distributed delay, memory effects) and interpret time-delayed feedback control as a nonlocal-in-time coupling. Applications arise in soft matter and dynamics of complex fluids such as liquid crystals.

    http://www.itp.tu-berlin.de/collaborative_research_center_910/sonderforschungsbereich_910/project_groups/a_theoretical_methods/tp_a8/

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