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The Coolest Path Problem
(2009)
We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph.
The introduction of the near-field quantity of optical chirality has emerged in various numerical and few experimental studies of local chirality enhancement due to its relation to the excitation rate of chiral molecules. This time-even pseudoscalar has been dismissed as being a higher-order version of helicity. Nevertheless, we revisit the derivation of the underlying conservation law and define optical chirality in media similar to. We identify the mechanism of chirality conversion by either inhomogeneous or anisotropic space to complement the conservation of optical chirality.
The conservation law of optical chirality in arbitrary space enables the extension of the concept of polarization to the near-field where no distiniguished propagation direction of light is present. We show that the connection of electromagnetic energy and optical chirality provide the ability to define a circular polarization basis in time-harmonic near-field analysis.
In order to illustrate our theory, we present electromagnetic field simulations of simple as well as more complex nanostructures. Results using the well-known far-field polarization concept are readily reproduced and extended from the point of view of chirality conversion.
Reconstruction of photonic crystal geometries using a reduced basis method for nonlinear outputs
(2016)
Maxwell solvers based on the hp-adaptive finite element method allow for accurate geometrical modeling and high numerical accuracy. These features are indispensable for the optimization of optical properties or reconstruction of parameters through inverse processes. High computational complexity prohibits the evaluation of the solution for many parameters. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem allowing to compute solutions for a parameter configuration orders of magnitude faster. The RBM allows to evaluate linear and nonlinear outputs of interest like Fourier transform or the enhancement of the electromagnetic field in milliseconds. We apply the RBM to compute light-scattering off two dimensional photonic crystal structures made of silicon and reconstruct geometrical parameters.
Model order reduction for the time-harmonic Maxwell equation applied to complex nanostructures
(2016)
Fields such as optical metrology and computational lithography require fast and efficient methods for solving the time-harmonic Maxwell's equation. Highly accurate geometrical modelling and numerical accuracy at low computational costs are a prerequisite for any simulation study of complex nano-structured photonic devices. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem based on the hp-adaptive finite element solver JCMsuite capable of handling geometric and non-geometric parameter dependencies allowing for online evaluations in milliseconds. We apply the RBM to compute light-scattering at optical wavelengths of periodic arrays of fin field-effect transistors (FinFETs) where geometrical properties such as the width and height of the fin and gate can vary in a large range.
Optical 3D simulations in many-query and real-time contexts require new solution strategies. We study an adaptive, error controlled reduced
basis method for solving parametrized time-harmonic optical scattering problems. Application fields are, among others, design and optimization problems
of nano-optical devices as well as inverse problems for parameter reconstructions occurring e. g. in optical metrology. The reduced basis method pre-
sented here relies on a finite element modeling of the scattering problem with
parametrization of materials, geometries and sources.
Adaptive sampling strategies for efficient parameter scans in nano-photonic device simulations
(2014)
Rigorous optical simulations are an important tool in optimizing scattering properties of nano-photonic devices and are used, for example, in solar cell optimization. The finite element method (FEM) yields rigorous, time-harmonic, high accuracy solutions of the full 3D vectorial Maxwell's equations [1] and furthermore allows for great flexibility and accuracy in the geometrical modeling of these often complex shaped 3D nano-structures. A major drawback of frequency domain methods is the limitation of single frequency evaluations. For example the accurate computation of the short circuit current density of an amorphous silicon / micro-crystalline multi-junction thin film solar cell may require the solution of Maxwell's equations for over a hundred different wavelengths if an equidistant sampling strategy is employed. Also in optical metrology, wavelength scans are frequently used to reconstruct unknown geometrical and material properties of optical systems numerically from measured
scatterometric data.
In our contribution we present several adaptive numerical integration and sampling routines and study their efficiency in the context of the determination of generation rate profiles of solar cells. We show that these strategies lead to a reduction in the computational effort without loss of accuracy. We discuss the employment of tangential information in a Hermite interpolation scheme to achieve similar accuracy on coarser grids. We explore the usability of these strategies for scatterometry and solar cell simulations.
In many experimentally realized applications, e.g. photonic crystals, solar cells and light-emitting diodes, nano-photonic systems are coupled to a thick substrate layer, which in certain cases has to be included as a part of the optical system. The finite element method (FEM) yields rigorous, high accuracy solutions of full 3D vectorial Maxwell's equations [1] and allows for great flexibility and accuracy in the geometrical modelling. Time-harmonic FEM solvers have been combined with Fourier methods in domain decomposition algorithms to compute coherent solutions of these coupled system. [2,3] The basic idea of a domain decomposition approach lies in a decomposition of the domain into smaller subdomains, separate calculations of the solutions and coupling of these solutions on adjacent subdomains.
In experiments light sources are often not perfectly monochromatic and hence a comparision to simulation results might only be justified if the simulation results, which include interference patterns in the substrate, are spectrally averaged.
In this contribution we present a scattering matrix domain decomposition algorithm for Maxwell's equations based on FEM. We study its convergence and advantages in the context of optical simulations of silicon thin film multi-junction solar cells. This allows for substrate light-trapping to be included in optical simulations and leads to a more realistic estimation of light path enhancement factors in thin-film devices near the band edge.
The Coolest Path Problem
(2010)
Hp-finite-elements for simulating electromagnetic fields in optical devices with rough textures
(2015)
Adaptive sampling strategies for effcient parameter scans in nano-photonic device simulations
(2014)
Rigorous optical simulations of 3-dimensional nano-photonic structures are an important tool in the analysis and optimization of scattering properties of nano-photonic devices or parameter reconstruction. To construct geometrically accurate models of complex structured nano-photonic devices the finite element method (FEM) is ideally suited due to its flexibility in the geometrical modeling and superior convergence properties. Reduced order models such as the reduced basis method (RBM) allow to construct self-adaptive, error-controlled, very low dimensional approximations for input-output relationships which can be evaluated orders of magnitude faster than the full model. This is advantageous in applications requiring the solution of Maxwell's equations for multiple parameters or a single parameter but in real time. We present a reduced basis method for 3D Maxwell's equations based on the finite element method which allows variations of geometric as well as material and frequency parameters. We demonstrate accuracy and efficiency of the method for a light scattering problem exhibiting a resonance in the electric field.
Determining 2D photonic crystal geometries from reflectance spectra with a reduced basis method
(2016)
Solving PDEs on unstructured grids is a cornerstone of engineering and scientific computing. Heterogeneous parallel platforms, including CPUs, GPUs, and FPGAs, enable energy-efficient and computationally demanding simulations.
In this article, we introduce the HPM C++-embedded DSL that bridges the abstraction gap between the mathematical formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different programming models on the other hand.
Thus, the HPM DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HPM DSL, and demonstrate its usage with three examples. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters, is presented.
A code generator and a matching back end allow the acceleration of HPM code with GPUs. Finally, the achievable performance and scalability are demonstrated for different example problems.
Model order reduction for the time-harmonic Maxwell equation applied to complex nanostructures
(2016)
Fields such as optical metrology and computational lithography require fast and efficient methods for solving
the time-harmonic Maxwell’s equation. Highly accurate geometrical modeling and numerical accuracy atcomputational costs are a prerequisite for any simulation study of complex nano-structured photonic devices.
We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem based
on the hp-adaptive finite element solver JCMsuite capable of handling geometric and non-geometric parameter
dependencies allowing for online evaluations in milliseconds. We apply the RBM to compute light-scatteringoptical wavelengths off periodic arrays of fin field-effect transistors (FinFETs) where geometrical properties such
as the width and height of the fin and gate can vary in a large range.
Reconstruction of photonic crystal geometries using a reduced basis method for nonlinear outputs
(2016)
Maxwell solvers based on the hp-adaptive finite element method allow for accurate geometrical modeling and high numerical accuracy. These features are indispensable for the optimization of optical properties or reconstruction of parameters through inverse processes. High computational complexity prohibits the evaluation of the solution for many parameters. We present a reduced basis method (RBM) for the time-harmonic electromagnetic scattering problem allowing to compute solutions for a parameter configuration orders of magnitude faster. The RBM allows to evaluate linear and nonlinear outputs of interest like Fourier transform or the enhancement of the electromagnetic field in milliseconds. We apply the RBM to compute light-scattering off two dimensional photonic crystal structures made of silicon and reconstruct geometrical parameters.
Optical 3D simulations in many-query and real-time contexts require new solution strategies. We study an adaptive, error controlled reduced basis method for solving parametrized time-harmonic optical scattering problems. Application fields are, among others, design and optimization problems of nano-optical devices as well as inverse problems for parameter reconstructions occuring e. g. in optical metrology. The reduced basis method presented here relies on a finite element modeling of the scattering problem with parametrization of materials, geometries and sources.
Solving partial differential equations on unstructured grids is a cornerstone of engineering and scientific computing. Nowadays, heterogeneous parallel platforms with CPUs, GPUs, and FPGAs enable energy-efficient and computationally demanding simulations. We developed the HighPerMeshes C++-embedded Domain-Specific Language (DSL) for bridging the abstraction gap between the mathematical and algorithmic formulation of mesh-based algorithms for PDE problems on the one hand and an increasing number of heterogeneous platforms with their different parallel programming and runtime models on the other hand. Thus, the HighPerMeshes DSL aims at higher productivity in the code development process for multiple target platforms. We introduce the concepts as well as the basic structure of the HighPer-Meshes DSL, and demonstrate its usage with three examples, a Poisson and monodomain problem, respectively, solved by the continuous finite element method, and the discontinuous Galerkin method for Maxwell’s equation. The mapping of the abstract algorithmic description onto parallel hardware, including distributed memory compute clusters is presented. Finally, the achievable performance and scalability are demonstrated for a typical example problem on a multi-core CPU cluster.