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Scattering from Finite Structures : an Extended Fourier Modal Method Scattering from finite structures : an extended Fourier modal method Citation for published version (APA): Pisarenco, M. (2011). Scattering from finite structures : an extended Fourier modal method. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR716275 DOI: 10.6100/IR716275 Document status and date: Published: 01/01/2011 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 09. Oct. 2021 Tot het bijwonen van de openbare verdediging van mijn proefschrift Scattering from Finite Structures: An Extended Fourier Modal Method Opwoensdag 7 september 2011 om 16:00 uur in het Auditorium van de Technische Universiteit Eindhoven. Maxim Pisarenco [email protected] Sint Gerlachstraat 22, 5643NM Einhoven tel: +31 651802573 ± 140 pag. = 9mm rug MAT-GLANS laminaat? Oplage: 200 stuks Tot het bijwonen van de openbare verdediging van mijn proefschrift Scattering from Finite Structures: An Extended Fourier Modal Method Opwoensdag 7 september 2011 om 16:00 uur in het Auditorium van de Technische Universiteit Eindhoven. Maxim Pisarenco [email protected] Sint Gerlachstraat 22, 5643NM Einhoven tel: +31 651802573 ± 140 pag. = 9mm rug MAT-GLANS laminaat? Oplage: 200 stuks Scattering from Finite Structures: An Extended Fourier Modal Method Copyright c 2011 by Maxim Pisarenco, Eindhoven, The Netherlands. All rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author. Cover design by Paul Verspaget and Maxim Pisarenco. A catalogue record is available from the Eindhoven University of Technology Library ISBN 978-90-386-2549-2 NUR 919 Subject headings: boundary value problems; differential operators; eigenvalue problems / electromagnetic waves; diffraction / electromagnetic scattering; numerical methods / inverse problems The work described in this thesis has been carried out under the auspices of - Veldhoven, The Netherlands. Scattering from Finite Structures: An Extended Fourier Modal Method PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 7 september 2011 om 16.00 uur door Maxim Pisarenco geboren te Chişinˇau,Moldavië Dit proefschrift is goedgekeurd door de promotor: prof.dr. R.M.M. Mattheij Copromotor: dr. J.M.L. Maubach Contents Preface ix 1 Introduction1 1.1 Lithography................................... 1 1.2 Problem description.............................. 3 1.3 Objectives and main results.......................... 5 1.4 Outline of the thesis.............................. 6 2 Mathematical modeling9 2.1 Maxwell equations............................... 9 2.2 Interface conditions............................... 11 2.3 Time-harmonic Maxwell equations...................... 13 2.4 Problem geometry and incident field..................... 15 2.5 Boundary conditions.............................. 18 2.5.1 Pseudo-periodic boundary condition................. 18 2.5.2 Radiation boundary condition..................... 19 2.6 Numerical methods for Maxwell equations.................. 20 2.6.1 Finite-difference time-domain method................ 21 2.6.2 Finite element method......................... 22 2.6.3 Fourier modal method......................... 23 2.6.4 Integral equation methods....................... 24 3 Extension of the Fourier modal method for a model problem 27 3.1 Introduction................................... 27 3.2 Standard Fourier modal method....................... 29 3.3 Artificial periodization with PMLs...................... 31 3.4 The contrast-field formulation of the FMM................. 36 3.4.1 Contrast/background decomposition................. 36 3.4.2 Background field solution....................... 37 3.4.3 Contrast field solution......................... 39 3.5 Numerical results................................ 42 vi Contents 4 Generalization to arbitrary shapes and illumination 45 4.1 Standard Fourier modal method....................... 45 4.1.1 TE-polarization............................. 48 4.1.2 TM-polarization............................ 50 4.1.3 Conical incidence............................ 51 4.2 Aperiodic Fourier modal method....................... 54 4.2.1 TE-polarization............................. 59 4.2.2 TM-polarization............................ 61 4.2.3 Conical incidence............................ 63 4.3 Final remarks.................................. 64 5 Stable solution of the coupled linear systems 67 5.1 Introduction................................... 67 5.2 Homogeneous T-matrix and S-matrix algorithms.............. 68 5.3 Non-homogeneous S-matrix algorithm.................... 71 5.4 Numerical results................................ 75 6 Aperiodic Fourier modal method with alternative discretization 79 6.1 Introduction................................... 79 6.2 Two equivalent problems............................ 81 6.3 Discretization of the background field.................... 83 6.4 Recursive linear systems for the vertical problem.............. 85 6.5 Non-homogeneous S-matrix algorithm for repeating slices......... 88 6.5.1 Redheffer notation........................... 88 6.5.2 Non-homogeneous Redheffer star product.............. 89 6.5.3 Global stack description........................ 90 6.5.4 Intermediary coefficients........................ 92 6.6 Summary of computational costs....................... 93 6.7 Far-field recovery................................ 94 6.8 Numerical results................................ 94 7 Conclusions and suggestions for future work 103 A Vector calculus identities 107 B Derivations 109 B.1 Derivation of the convolution term...................... 109 B.2 Discretization using the Galerkin approach................. 109 Bibliography 113 Index 121 Contents vii Summary 123 Samenvatting 125 Curriculum vitae 127 Preface The work described in this thesis has been carried out in the framework of a continuing collaboration between the Centre for Analysis, Scientific computing and Applications (CASA) at Eindhoven University of Technology, and ASML, world leader in the manu- facturing of photolithography systems. It was a delight to work on a topic which posed challenging questions of a theoretical nature and in the same time had a direct practical application. Working on this thesis was a new experience for me that I will remem- ber for life and these lines are being written with genuine satisfaction for finishing the manuscript. I would like to take a moment now to express my gratitude to all people who contributed in one way or another to this thesis. I am indebted to Bob Mattheij for offering me the opportunity to work on a topic which looks as exciting to me now as it did four years ago. I would like to thank him and my copromotor Jos Maubach for their guidance and trust. I consider myself lucky to have Irwan Setija from ASML as a third supervisor. His enthusiasm has been extremely motivating in periods of productive research as well as in moments of stagnation. Our traditional Friday afternoon meetings usually lasted much longer than planned, and were often ended by Irwan’s ”I am very curious about the new results...“. Besides my three supervisors, my defense committee is completed by prof. dr. Gerard Granet, prof. dr. Paul Urbach, prof. dr. Wim Coene and prof. dr. Mark Peletier. I would like to thank them all for the invested time and their valuable comments. I am grateful to several other people who have had a direct contribution to this work. Nico and Mark, former PhD students who have been part of the CASA-ASML collaboration, performed excellent research which resulted in well-written dissertations
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