## 78-XX OPTICS, ELECTROMAGNETIC THEORY (For quantum optics, see 81V80)

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We present a Newton-like method to solve inverse problems and to quantify parameter uncertainties. We apply the method to parameter reconstruction in optical scatterometry, where we take into account a priori information and measurement uncertainties using a Bayesian approach. Further, we discuss the influence of numerical accuracy on the reconstruction result.

Gaining insights into the working principles of photocatalysts on an atomic scale is a challenging task. The obviously high complexity of the reaction mechanism involving photo-excited electrons and holes is one reason. Another complicating aspect is that the electromagnetic field, driving photocatalysis, is not homogeneous on a nanoscale level for particle based catalysts as it is influenced by the particle’s shape and size.
We present a simple model, inspired by the CO2 reduction on titania anatase, which addresses the impact of these heterogeneities on the photocatalytic kinetics by combining kinetic Monte Carlo with electromagnetic wave simulations. We find that average activity and especially efficiency might differ significantly between different particles. Moreover, we find sizable variation of the catalytic activity on a single facet of a nanocrystal. Besides this quantitative heterogeneity, the coverage situation in general changes laterally on this facet and we observe a concomitant change of the rate-determining steps.
This heterogeneity on all levels of photocatalytic activity is masked in experimental studies, where only the spatially averaged activity can be addressed. Microkinetic models based on experimental findings might therefore not represent the true micro- scopic behavior, and mechanistic conclusion drawn from these need to be handled with care.

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.

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.

Nano-optical scattering problems play an important role in our modern, technologically driven society. Computers, smartphones and all kinds of electronic devices are manufactured by the semiconductor industry which relies on production using photomasks as well as optical process control. The digital world, e.g. the world wide web, is based on optical interconnects and so-called quantum computers based on optics are supposed to be next generation computers. Moreover, global economic progress demands new and sustainable energy resources and one option is to make use of the power stored in optical radiation from the sun. Additionally, understanding fundamental physics such as the optical properties of asymmetric, or chiral, structures could promote future innovations in engineering. In order to understand and manipulate these kinds of processes, physics provides a well established model: the so-called Maxwell’s equations. Stated by James Clerk Maxwell in 1862, this description of the interaction of light and matter still provides a profound basis for the analysis of electromagnetic phenomena. However, real world problems cannot be calculated using simple mathematics. Rather, computer simulations are needed to obtain solutions of the physical model. Finding suitable methods to solve these problems opens up a wide variety of possibilities. On the one hand, there are methods which require long computing times. On the other hand, some algorithms depend on high memory usage. That is why the field of numerics deals with the question which method is optimally suited for specific problems. The aim of this work is to investigate the applicability of the so-called Fourier Modal Method (FMM) to nano-optical scattering problems in general. Since simple analytical solutions are non-existent for most recent physical problems, we use the Finite Element Method (FEM) to double-check performance of the FMM. Mathematics provide reliable procedures to control the errors of numerics using the FEM. Yet up to now it has not been possible to rigorously classify the quality of the Fourier Modal Method’s results. It is not fully understood whether the process of investing more and more computing resources yields more accurate results. So, we have to ask ourselves: does the numerical method invariably converge? In spite of this uncertainty when using the FMM, it is a well established method dating back to the 1980s. This numerical method has recently been used to optimize performance of solar cells [19] as well as to improve the optical properties of so-called single-photon sources [41] which are essential for quantum cryptography. The latter is a promising candidate to increase digital security and revolutionise cryptography techniques. Furthermore, with the help of the Fourier Modal Method an important issue in optics has been partly resolved: angular filtering of light was made possible by using a mirror which becomes transparent at a certain viewing angle [77]. In addition, an improved numerical technique to design so-called Photonic Crystal waveguides based on the FMM was developed recently [15]. Photonic Crystals are used in the fields of optical bio-sensing and for the construction of novel semiconductor devices. Moreover, approaches to link the FMM and the FEM try to combine advantages of both methods to obtain fast and accurate results [81]. These ideas are closely linked to the well-known concept of Domain Decomposition within the FEM [88]. Here, one possibility to couple domains is to use the scattering matrix formalism as it is done in the FMM. In the scope of this convergence study, we state Maxwell’s equations, particularly for periodic geometries. We describe two physical phenomena of nano-optics, namely chirality and opto-electrical coupling, and define the errors of our simulations. Afterwards, the two investigated methods are analysed with respect to their general properties and a way to unify modelling physics when using both algorithms is presented. With the help of various numerical experiments, we explore convergence characteristics of the FMM and draw conclusions about the ability of this approach to provide accurate results and, consequently, its potential for research on technological innovations.

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.

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.