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Transformation Optics for Controlling DC Magnetic Field KTH Electrical Engineering Transformation Optics for Controlling DC Magnetic Field Fei Sun Doctoral Thesis in Electrical Systems School of Electrical Engineering KTH Royal Institute of Technology Stockholm, Sweden 2014 TRITA-EE 2014:051 KTH School of Electrical Engineering ISSN: 1653-5146 Teknikringen 33 ISRN:KTH/EE--14/051--SE SE-100 44 Stockholm ISBN: 978-91-7595-328-1 Sweden Akademisk avhandling som med tillstånd av Kungl Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredag den 01 December 2014 klockan 10:00 i sal F3, Kungl Tekniska Högskolan, Stockholm. © Fei Sun, December 2014 Tryck: Universitetsservice US AB ii Abstract Static magnetic fields play an important role in many technologies like magnetic resonance imaging, and magnetic sensing. In order to future develop such applications, it is necessary to study some new technologies for enhancing the static magnetic field. Transformation optics provides a new way to design novel devices for static magnetic field enhancement. The purpose of this work is to design some novel DC magnetic devices for static magnetic field enhancement and try to verify their performance experimentally and numerically for future applications. This work provides a new method to control static magnetic fields and pave the way for future potential applications in which high-static magnetic fields are required. The theory of transformation optics is derived in the beginning of the thesis. The proof of the form-invariance of Maxwell’s equations is given in terms of differential geometry. The basic formulae (the transformation rules of fields and materials) are derived by material interpretation. We also extend the theory of transformation optics to the static magnetic field case: we can control the static magnetic field or even manipulate magnets (e.g. ferro- magnets or electric magnets) by the method of coordinate transformation. Some novel devices which can transform magnetic fields produced by magnets (e.g. rescale magnets or cancel magnets) are designed. One main topic of this thesis is to design novel passive DC magnetic concentrators which can amplify the background field and achieve an enhanced DC magnetic field with high uniformity in a large free space region. We propose two methods to achieve this: one way is to use the space compression transformation and the other one is to use the space folding transformation. The performance of the proposed devices is analyzed and verified by numerical simulation based on the finite element method. We also design and realize a concentrator based on the space compression transformation (experimental data is given). This kind of concentrator has many potential applications, e.g., help the magnetic resonance imaging to iii acquire a better spatial resolution and to improve the sensitivity of a magnetic sensor. The other main topic of this thesis is to design a passive compressor/lens that can focus the incident DC magnetic field, and achieve an enhanced DC magnetic field with high gradient in a free space region. We use a finite embedded transformation to design such a compressor and introduce the idea of transforming inside a null-space region to simplify the material requirement of the compressor as well as enhance the focusing performance of the compressor. Numerical simulations are provided to verify the performance of the proposed compressor. This kind of compressor will also have a number of potential applications, e.g., provide better control of magnetic nano-particles in future medical treatment and improve the technology of magnetic separation. Keywords: Transformation optics, DC magnetic field, magnetic field enhancement, finite embedded transformation, high gradient DC magnetic field, magnetic concentrator, null-space medium, transformation magneto- statics, illusions for magnets, Maxwell’s equations. iv Abstrakt I många vetenskapsområden och tekniska tillämpningar spelar statiska magnetiska fält en viktig roll. För utvecklandet av möjliga framtida tillämpningar är det därför angeläget att studera nya metoder för att kunna styra och öka styrkan hos statiska magnetfält. En sådan ny metod för att designa komponenter för styrning av magnetfält är transformationsoptik. Syftet med detta arbete är att designa nya magnetostatiska komponenter för att, inför framtida tillämpningar, erhålla starkare statiska magnetfält, och kontrollera resultaten genom experiment och numeriska simuleringar. Det här arbetet leder till nya sätt att kontrollera statiska magnetfält och banar vägen för framtida tillämpningar vilka kräver starka magnetostatiska fält. Inledningsvis härleds i avhandlingen teorin för transformationsoptik och forminvariansen hos Maxwells ekvationer bevisas genom att använda differentialgeometri. De grundläggande uttrycken (transformationsreglerna för fält och material) härleds utifrån tolkningen av materialet i kroklinjiga koordinater. Vi utvidgar också transformationsoptiken till att inkludera magnetostatiska fält: vi kan kontrollera fältet och även utföra en del trick med magneter (t ex ferromagneter eller elektromagneter) genom koordinattransformationer. En del genuina komponenter som kan transformera magneter (t ex skalning av magneter eller upphävande av deras inverkan) har designats. Det ena huvudämnet för avhandlingen är att designa genuina passiva koncentratorer, för statiska magnetfält, vilka kan förstärka bakgrundsfältet och uppnå förstärkta fält med god homogenitet över stora områden. För detta föreslår vi två metoder: den ena är att använda en rumskomprimerande transformation och den andra är att använda en transformation som viker rummet. Prestandan hos de föreslagna komponenterna har analyserats och verifierats genom numeriska simuleringar baserade på finita elementmetoden. Vi har också utformat ett experiment för att realisera en koncentrator baserad på den rumskomprimerande transformationen. Sådana koncentratorer har v många möjliga tillämpningar: t ex uppnå bättre rumsupplösning i magnetisk resonanstomografi eller öka känsligheten hos magnetiska givare. Det andra huvudämnet för avhandlingen är att designa en passiv kompressor/lins som kan fokusera ett magnetiskt bakgrundsfält, förstärka både fältet och dess gradient, och i fri rymd uppnå förstärkta statiska fält med starka gradienter. För designen använder vi en i rummet begränsad och inbäddad transformation och introducerar konceptet att transformera inom en nollrumsregion för att förenkla kraven på materialet som utgör kompressorn och förbättra dess prestanda. Numeriska simuleringar har gjorts för att verifiera den föreslagna kompressorn. Den här typen av kompressor har också många möjliga tillämpningar: t ex för att uppnå bättre kontroll av magnetiska nanopartiklar i framtida medicinska behandlingsmetoder eller för att förbättra magnetiska separationsmetoder. Nyckelord: omvandling optik, DC magnetfält, magnetfält förbättring, ändlig inbäddade transformation, höggradient DC magnetfält, magnetisk koncentrator, nollrummet medium. vi Acknowledgements I would like to acknowledge people who gave me help during my PhD study. There are many names I should list here. This thesis cannot be finished without their help. At first I would like to express my greatest thanks to my main supervisor Prof. Sailing He. Since the first day I joined his research group, he has given me a lot of helps. Besides his excellent guidance for my scientific research, he also gave me many suggestions and helps for my daily life. I also wish to give my great thanks to my supervisor Prof. Sailing He for his language polishing for this thesis. I also would like to thank Prof. Lars Jonsson. I learned a lot from his course on “the electromagnetic wave propagation”. I want to express my great thanks for his comments on the present work and his careful revision of this thesis. I would like to thank Prof. Martin Norgren for his co-supervision when I studied at KTH. I also want to express my great thanks to him for his translation of the abstract and the summary in this thesis. I want to send my appreciation to Prof. Yungui Ma for his guidance on microwave experiment and discussion on my scientific research in Zhejiang University. I learned many methods and skills on experiments from him which will be a valuable experience in my future career. I want to express my great thanks to him. Dr. Shuai Zhang and my colleague Mr. Kun Zhao also gave me many helps during my PhD life. I would like to express my great thanks to them. I would like to give my great thank to Dr. Pu Zhang for his guiding on the theory of transformation optics and numerical simulation. He gave me a lot of help when I first began my PhD study. I should also express my gratitude to Dr. Yingran He. He also gave me some help on numerical simulation and basic physics. Our Financial Administrator Ms. Carin Norberg and System Administrator Mr. Peter LÖnn always gave me very good administrative and technical supports when I studied at KTH. I would like to express my great thanks to them. I need also express my thanks to Prof. Rajeev Thottappillil, Prof. Peter Fuks, Assistant Prof. Quevedo Teruel Oscar for their help when I studied at KTH. In the Department of Electromagnetic Engineering, there are still many people I want to give my thanks: Mengni Long, Xiaolei Wang, Helin Zhou, Shuang Zhao, Hui Zhang, Lebing Jin, Bing Li, Lipeng Liu, Mariana Dalarsson, Christos Kolitsidas, Elena Kubyshkina, Mauricio Aljure Rey. I need also gave my thanks to Mr. Ziyang Li, Dr. Wenhua Guo, and Dr. Xichen Li for their
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