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Vector Wave Propagation Method Ein Beitrag Zum Elektromagnetischen Optikrechnen

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Vector Wave Propagation Method Ein Beitrag Zum Elektromagnetischen Optikrechnen Vector Wave Propagation Method Ein Beitrag zum elektromagnetischen Optikrechnen Inauguraldissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der Universitat¨ Mannheim vorgelegt von Dipl.-Inf. Matthias Wilhelm Fertig aus Mannheim Mannheim, 2011 Dekan: Prof. Dr.-Ing. Wolfgang Effelsberg Universitat¨ Mannheim Referent: Prof. Dr. rer. nat. Karl-Heinz Brenner Universitat¨ Heidelberg Korreferent: Prof. Dr.-Ing. Elmar Griese Universitat¨ Siegen Tag der mundlichen¨ Prufung:¨ 18. Marz¨ 2011 Abstract Based on the Rayleigh-Sommerfeld diffraction integral and the scalar Wave Propagation Method (WPM), the Vector Wave Propagation Method (VWPM) is introduced in the thesis. It provides a full vectorial and three-dimensional treatment of electromagnetic fields over the full range of spatial frequen- cies. A model for evanescent modes from [1] is utilized and eligible config- urations of the complex propagation vector are identified to calculate total internal reflection, evanescent coupling and to maintain the conservation law. The unidirectional VWPM is extended to bidirectional propagation of vectorial three-dimensional electromagnetic fields. Totally internal re- flected waves and evanescent waves are derived from complex Fresnel coefficients and the complex propagation vector. Due to the superposition of locally deformed plane waves, the runtime of the WPM is higher than the runtime of the BPM and therefore an efficient parallel algorithm is de- sirable. A parallel algorithm with a time-complexity that scales linear with the number of threads is presented. The parallel algorithm contains a mini- mum sequence of non-parallel code which possesses a time complexity of the one- or two-dimensional Fast Fourier Transformation. The VWPM and the multithreaded VWPM utilize the vectorial version of the Plane Wave Decomposition (PWD) in homogeneous medium without loss of accuracy to further increase the simulation speed. The analysis of sampling-induced deviations in the amplitude shows that the sampling in the aperture needs to meet the Whitaker-Kotelnikov-Shannon (WKS) sampling theorem to provide an accurate reproduction of the elec- tromagnetic vector field. Furthermore, the accuracy of the amplitude de- pends on the sampling in the axis of propagation. A criterion for the samp- ling in the axis of propagation, considering the simulated wavelength, is proposed to minimize the deviation in the amplitude to the results from ii theory. The analysis of the zero order diffraction efficiency in a discretized oblique interface shows the relationship of the amplitude deviations to the discretization in the axis of propagation. The normalized sampling rate is introduced. It provides a sampling criterion to achieve a minimum devi- ation of the simulated amplitude to the results from theory. A plurality of simulations of an oblique layer with different sampling rates are performed to evaluate the theory. The stability of the algorithm and the maintenance of the conservation law is analyzed by investigating the power flux. The re- sults from the VWPM are compared to the results from theory for different polarizations, propagation angles and representative interface configura- tions. In agreement to the observations in several cited publications, the generation and treatment of evanescent modes is identified being critical for the stability of the algorithm and the maintenance of the conservation law. Several techniques of stabilization are presented and it is shown that the model of evanescent waves provides an elegant way to ensure stability and extend the VWPM to evanescent modes. The VWPM is utilized to investigate the effect of micrometer gratings on the local absorption in thin film solar cells and to find a solution for an enhanced absorption over a wide range of wavelengths and spatial fre- quencies. In a simulation model, the angle of incidence of the electro- magnetic vector wave, the period and the duty cycle of one- and two- dimensional micrometer gratings are varied to identify an optimized ge- ometry and analyze its influence on the local absorption. The use of one- or two-dimensional gratings is part of various other research activities as published in [80, 81, 82, 83]. iii Zusammenfassung Auf Basis des Rayleigh-Sommerfeld Beugungsintegrals und der skalaren Wave Propagation Method (WPM) prasentiert¨ diese Arbeit die Vector Wave Propagation Method (VWPM). Die VWPM ist ein propagatives Simula- tionsverfahren zur Berechnung dreidimensionaler Vektorfelder ohne parax- iale Einschrankung¨ und uber¨ den gesamten spatialen Frequenzbereich. Evaneszente Fourier-Moden sind nach der Theorie in [1] modelliert. Die Erweiterung der unidirektionalen zur bidirektionalen VWPM berechnet re- flektierte dreidimensionale Vektorfelder uber¨ die komplexen Propagations- vektoren und die komplexen Fresnel-Koeffizienten. Spezielle Konfigura- tionen der komplexen Propagationsvektoren werden fur¨ die Berechnung von Totalreflexion, evaneszenter Kopplung (optischer Tunneleffekt) und zur Energieerhaltung eingesetzt (reduzierter Propagator). Die Laufzeit der VWPM ist aufgrund der Uberlagerung¨ lokal deformierter Planwellen hoher¨ als die der Beam Propagation Method (BPM). Der prasentierte¨ par- allele Algorithmus skaliert linear mit der Anzahl der Prozesse (Threads). Der nicht-parallelen Anteil Programmcode ist minimal und hat die Zeitkom- plexitat¨ der ein- oder zweidimensionalen Fast-Fourier-Transformation. Fur¨ die Ausbreitung im homogenen Medium wird zur weiteren Reduktion der Simulationsdauer die Plane Wave Decomposition ohne Verlust an Rechen- genauigkeit eingesetzt. Die Analyse der Rechengenauigkeit und deren Abhangigkeit¨ zu den Ab- tastraten in der Apertur und entlang der Propagationsachse zeigt, dass zur exakten Reproduktion des elektromagnetischen Feldes in der Apertur das Whitaker-Kotelnikov-Shannon (WKS) Abtasttheorems zu berucksichtigen¨ ist. Im Zusammenhang mit der Abtastung entlang der Propagationsachse wird die normierte Abtastrate mit Abtastbedingung eingefuhrt,¨ um den Am- plitudenfehler bei diskretisierten schragen¨ Grenzflachen¨ zu minimieren. iv Die Analyse der nullten Beugungsordnung des Phasenelements diskre- tisierter schrager¨ Grenzflache¨ zeigt die Beziehung der Rechengenauigkeit zur Abtastrate. Eine Reihe von Simulationen veranschaulichen die Theo- rie und liefern die Uberpr¨ ufung¨ am Beispiel. Der Energiefluss durch Gren- zflachen¨ ist Grundlage fur¨ die Beurteilung der Stabilitat¨ des Algorithmus und der Energieerhaltung. Die Simulationsergebnisse fur¨ unterschiedliche Polarisationen und Ausbreitungswinkel in reprasentativen¨ Systemkonfigu- rationen werden mit den Ergebnissen der Theorie verglichen. Die vor- liegende Arbeit zeigt in Ubereinstimmung¨ mit der Literatur, dass die Be- handlung der evaneszenten (quergedampften)¨ Fourier-Moden problema- tisch fur¨ die Stabilitat¨ des Algorithmus und die Energieerhaltungseigen- schaften ist. Die Arbeit prasentiert¨ unterschiedliche Ansatze¨ zur Stabil- isierung und verifiziert diese qualitativ. Die Verwendung des reduzierten Propagators erlaubt die Berechnung evaneszenter Fourier-Moden und sta- bilisiert den Algorithmus. Die Anwendbarkeit der VWPM wird in einem praktischen Optimierungs- problem nachgewiesen. Ziel ist die Maximierung der lokalen Absorption in Dunnschicht-Solarzellen¨ uber¨ einen maximales Winkelspektrum durch Verwendung von optimierten resonanten Phasengittern. Anhand eines Simulationsmodells wird durch Parametervariationen eine optimierte Geo- metrie ermittelt. Daruberhinaus¨ ist die Verwendung von ein- oder zweidi- mensionalen resonanten Phasengittern Bestandteil weiterer Forschungsak- tivitaten¨ [80, 81, 82, 83]. v Vorwort Die vorliegende Arbeit entstand in den Jahren 2007 bis 2011 neben meiner Tatigkeit¨ als Entwicklungsingenieur in der Mikroprozessorentwicklung bei der IBM Forschung und Entwicklung GmbH in Boblingen.¨ Die Untersuchun- gen zur optischen Taktverteilung in den Jahren 2005/2006 und der daraus resultierende Kontakt zum IBM Zurich¨ Research Laboratory in Ruschlikon,¨ sowie die Kontakte aus meiner Studienzeit an den Lehrstuhl fur¨ Rechner- architektur und den Lehrstuhl fur¨ Optoelektronik der Universitat¨ Heidelberg waren Ausgangspunkt fur¨ die Arbeiten zur Silizium Photonik in Boblingen¨ und fur¨ diese Dissertation. Die Forschungsarbeit zu optischen on-chip Detektoren und die fur¨ die Detektoroptimierung notigen¨ Simulationen der Ausbreitung elektromagnetischer Wellen lieferte die Motivation fur¨ die vor- liegende Dissertation. Altdorf und Mannheim Matthias Fertig Marz¨ 2011 vi Danksagung Fur¨ die Unterstutzung¨ danke ich Thomas Pfluger,¨ Dieter Wendel, Dr. Cordt Starke, Stefan Wald, Dr. Reinhard Ernst, Dr. Andre Hertwig und Dr. Peter Roth von der IBM Forschung und Entwicklung GmbH in Boblingen;¨ sowie Dr. Thomas Morf, Dr. Nikolaj Moll, Dr. Thilo Stoferle,¨ Dr. Jonas Weiss, Dr. Martin Schmatz, Dr. Bert Offrein und Dr. Thomas Toifl vom IBM Zurich¨ Research Laboratory in der Schweiz fur¨ die hervorragende Zusammenar- beit. Ebenso danke ich Dr. Bastian Trauter, Max Auer und Herrn Professor Dr. Brenner vom Lehrstuhl fur¨ Optoelektronik des Zentralen Instituts fur¨ Technische Informatik der Universitat¨ Heidelberg fur¨ die hervorragende Zusammenarbeit. Mein besonderer Dank gilt Herrn Professor Dr. Brenner fur¨ die Moglichkeit¨ diese Dissertation zu schreiben; fur¨ die Betreuung und die zahlreichen Ideen. Ebenso gilt mein Dank Herrn Professor Dr. Elmar Griese vom Lehrstuhl fur¨ Theoretische Elektrotechnik und Photonik der Universitat¨ Siegen fur¨ die Erstellung
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