Convergence Study of the Fourier Modal Method for Nano-optical Scattering Problems in Comparison with the Finite Element Method

Please always quote using this URN: urn:nbn:de:0297-zib-56084
  • 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.
Author:Philipp Gutsche
Document Type:Master's Thesis
Tag:Finite Element Method (FEM); Fourier Modal Method (FMM); Nano-optics; Rigorous Coupled Wave Analysis (RCWA)
MSC-Classification:78-XX OPTICS, ELECTROMAGNETIC THEORY (For quantum optics, see 81V80)
CCS-Classification:G. Mathematics of Computing
Granting Institution:Eberhard Karls Universität Tübingen
Advisor:Thomas Judd, Frank Schmidt
Date of final exam:2014/11/01
Date of first Publication:2014/11/01
Page Number:87
Licence (German):License LogoCreative Commons - Namensnennung