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KTH Electrical

Transformation for Controlling DC

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

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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 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- , illusions for magnets, Maxwell’s equations.

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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.

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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 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 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 . 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 help when I lived in Stockholm. Prof. Yi Jin, Prof. Jianqi Shen, Prof. Yanxia Cui, Dr. Liu Yang, Prof. Yuan Zhang, Dr. Rui Hu, Dr. Bowen Wang, Dr. Yuqian Ye, Dr. Xuan Li, Dr. Min Yao, gave me many helps and suggestions when I studied at Zhejiang University. I would like to give my thanks to them. I also want to thank my colleagues and friends in Zhejiang University: Xiaochen Ge, Jianwei Tang, Shuomin Zhong, Jing Gu, Hao Zhou, Tuo

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Chen, Fei Ding, Lei Mo, Jiao Lin, Kexin Liu, Yichao Liu, Lu Lan, Zhaofeng Ma, Jianfeng Jiang, Wei Jiang, Borui Li, Tiantian Wu. I hope the name I missed here can accept my appreciation. Thank you for all your helps. I thank the China Scholarship Council (CSC) No. 201206320083 for the financial support of my study at KTH. In the end, I would like to give my sincerely thanks to my parents for their supports and understanding all the time. Thanks to all of you!

Fei Sun Stockholm, October 2014

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List of Publications

List of publications included in this thesis: Journal papers:

I. Fei Sun and Sailing He, “Create a uniform static magnetic field over 50T in a large free space region,” Progress In Electromagnetics Research, Vol. 137, pp. 149-157, 2013. II. Fei Sun and Sailing He, “Static magnetic field concentration and enhancement using magnetic materials with positive permeability,” Progress In Electromagnetics Research, Vol 142, pp. 579-590, 2013. III. Fei Sun and Sailing He, “DC magnetic concentrator and omnidirectional cascaded cloak by using only one or two homogeneous anisotropic materials of positive permeability,” Progress In Electromagnetics Research, Vol. 142, pp. 683-699, 2013. IV. Fei Sun, and Sailing He, “Transformation magneto-statics and illusions for magnets,” Scientific Reports 4, 6593; DOI:10.1038/srep06593, 2014. V. Fei Sun, and Sailing He, “Transformation inside a Null-space Region and a DC Magnetic Funnel for Achieving an Enhanced with a Large Gradient,” Progress In Electromagnetics Research, Vol. 146, pp. 143-153, 2014. VI. Kexin Liu, Wei Jiang, Fei Sun, and Sailing He, “Experimental realization of strong DC magnetic enhancement with transformation optics,” Progress In Electromagnetics Research, Vol. 146, pp. 187-194, 2014. (Invited Paper).

List of publications not included in the thesis:

Journal papers:

VII. Yungui Ma, Fei Sun, Yuan Zhang, Yi Jin, and C. Ong, “Approaches to achieve broadband optical transformation devices with transmuted singularity,” J. Opt. Soc. Am. A, Vol. 29 Issue 1, pp. 124-129, 2012.

VIII. Fei Sun, Yun Gui Ma, Xiaochen Ge, and Sailing He, “Super-thin Mikaelian’s lens of small index as a beam compressor with an extremely high compression ratio,” Optics Express, Vol. 21, No. 6, pp. 7328-7336, 2013.

IX. Lu Lan, Fei Sun, Yichao Liu, C. K. Ong and Yungui Ma, “Experimentally demonstrated an unidirectional electromagnetic cloak designed by topology optimization,” . Letter, Vol. 103, No.12, 121113, 2013.

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X. Yungui Ma, Lu Lan, Wei Jiang, Fei Sun, Sailing He, “A transient thermal cloak experimentally realized through a rescaled diffusion equation with anisotropic thermal diffusivity,” NPG Asia Materials, Vol. 5, pp. e73, 2013.

XI. Fei Sun, and Sailing He, “Extending the scanning angle of a phased array antenna by using a null-space medium,” Scientific Reports 4, 6832; DOI: 10.1038/srep06832, 2014.

XII. Fei Sun, and Sailing He, “A third way to cloak an object: cover-up with a background object,” Progress In Electromagnetics Research, Vol. 149, 173-182, 2014. (Invited Paper).

The reason why papers I to VI are in this thesis is that they have the same main topic “transformation optics for controlling DC magnetic field”. Other papers are other applications of transformation optics (e.g. paper X-XII) or other topics and applications (e.g. paper VIII and IX).

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Acronyms

2D Two-dimensional

3D Three-dimensional

TO Transformation Optics

TE Transverse Electric

TM Transverse Magnetic

MM

DC

HPFM High-permeability Ferromagnetic Material

HTSM High-temperature Superconductor Material

FET Finite Embedded Transformation

FEM Finite Element Method

MRI Magnetic Resonance Imaging

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Contents

1 Introduction ...... 1 1.1 Background and Motivation ...... 1 1.2 Thesis Outline ...... 4

2 Brief Introduction of Differential Geometry ...... 7 2.1 Coordinate Transformations and the Metric Tensor ...... 7 2.2 Covariant and Contra-variant Vectors...... 10 2.3 Scalar Product and Vector Product ...... 12 2.4 The Covariant Differentiation ...... 14 2.5 The Divergence and the Curl ...... 16

3 Theory of Transformation Optics and Transformation Magneto-statics ...... 19 3.1 The Form Invariance of Maxwell’s Equations and Transformation Optics19 3.2 TO in an Orthogonal ...... 24 3.3 Extending the Transformation Optics to Transformation Magneto-statics.27 3.4 Some Examples Designed by Transformation Magneto-statics...... 29

4 Enhancing DC Magnetic Field with High Uniformity ...... 37 4.1 A Theoretical Method for Designing a Passive DC Magnetic Concentrator via Space Compression Transformation ...... 38 4.2 Experimental Realizations of the DC Magnetic Concentrator via Space Compression Transformation ...... 45 4.3 Design a DC Magnetic Concentrator via Space Folding Transformation .. 48

5 Magnetic lenses which Both Amplify the Background DC Magnetic Field and the Gradient of the Field ...... 53 5.1 Theoretical Method to Design a DC Magnetic Lens via Finite Embedded Transformation ...... 53 5.2 Parameters Reduction by Introducing Null-space Medium ...... 57

6 Conclusion and Future Work ...... 65

References ...... 69

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1. Introduction

1.1 Background and Motivation

Static magnetic fields play an essential role in many modern technologies, including particle accelerators, mass spectrometers, metal detection, magnetic sensing [1] (such as magnetoresistance and Hall sensors), magnetic separation processes [2], controlling magnetic nano-particles for gene and drug delivery [3, 4], and magnetic resonance imaging (MRI) [5]. Achieving a higher static magnetic field can provide a breakthrough in various areas. We can classify the enhancement of the static magnetic field into two classes according to its applications: one is to obtain an enhanced static magnetic field with high uniformity (e.g. the MRI needs a high static magnetic field with very high uniformity to achieve a better spatial resolution) and the other is to obtain an enhanced static magnetic field with high gradient (e.g. a high magnetic field with a sharp gradient can provide more accurate control of drug delivery based on magnetic nano-particles in a deeper location in the human body).

Traditionally, people can get an enhanced DC magnetic field mainly by two different methods: one way is to use active magnets with high power supply [6] and the other is to use some passive magnetic lenses composed of super- conductors or ferromagnetic materials to focus/concentrate the incident field [7-11]. Active magnets consume a large amount of electric and also encounter many other limitations (e.g. maximally rated current and maximally rated of the coil, heat-dissipation problem, etc.). Traditional passive magnetic lenses are mainly achieved by super-conductors with various geometrical shapes or different materials [7-11]. These magnetic lenses are passive, meaning that they can achieve additional magnetic flux without further electric power within a background magnetic field. However, magnetic lenses based on simple super-conductors also suffer many limitations: it is hard to achieve much enhancement in a relative large free space region (e.g. superconducting multilayered sheets can only achieve a 2.5 time

1 magnification [11]), they need refrigeration, and the performance is often influenced by quenching and other factors.

In future practical applications (e.g. MRI with better spatial resolution, a more accurate control of drug delivery based on magnetic nano-particles, higher gradient magnetic force for cell manipulations or mineral separation, etc.), there is a higher demand on static magnetic fields: we need an enhanced static magnetic field in a larger free space region with a larger amplification factor, and we need an enhanced static magnetic field with higher uniformity or higher gradient. Traditional magnetic lenses [7-11] cannot achieve these goals. Some novel DC magnetic lenses are needed to meet the demands of future practical applications.

In recent years, transformation optics (TO) has become a very popular research topic [12-18], and has been applied to design many novel optical devices, including optical cloaking [12, 19-24], perfect imaging devices [25- 29], electromagnetic wave concentrators [30-32], rotators [33, 34], optical illusion devices [35-39], super-scatters and super-absorbers [40-45], optical black holes [46, 47], optical [48, 49], beam shifters and splitters [50], polarization controllers [51, 52], optical lenses [53-58], electromagnetic cavities [59-61], special [62-66], source transforming devices [67-71], carpet cloaking [72-76], etc. TO has also been utilized to remove the singularity from optical devices [77-82] and design novel plasmonic nanostructures with broadband response and super-focusing ability [83-86]. The theoretical basis of TO is the form-invariance of Maxwell’s equations under coordinate transformations [30, 87]. TO provides a relation between two spaces: one is a virtual space (referred to as the reference space) and the other is the real space. By applying a special coordinate transformation or choosing a suitable reference space (e.g. a non-Euclidean space [20]), we can obtain some special medium (referred to as the transformation medium) that transforms electromagnetic waves from the reference space to the real space, and thus control the light path following our wishes in the real space. This means that we can largely control the propagation of the electromagnetic wave and view the inverse scattering

2 problem from another perspective: the transformation of a coordinate space. TO is providing a new way to control electromagnetic waves in an unprecedented manner.

However the transformation medium designed by TO is often very complex, which means that it is difficult to realize the device designed by TO with natural materials. For example, invisible cloaking designed by TO [12, 19] requires an inhomogeneous anisotropic impedance matched medium ( equates permeability) and also the existence of some singularities where permittivity and permeability approach zero or infinity. Due to the development of (MMs) [88-92], people can achieve many electromagnetic materials that cannot be found in nature (e.g. those with negative refraction index). A MM is a new type of electromagnetic material composed of artificial units, and has novel electromagnetic properties dependent on these artificial units but not their chemical composition.

In recent years, MMs that work for static magnetic fields have been explored and studied (referred to as DC MM in this thesis) [93-97]. Along with the rapid development of DC MM, many novel devices for the static magnetic field have been designed and even experimentally verified [98-112]. The DC magnetic cloak, which can exclude the background DC magnetic field from the inner hidden region without influencing the outside field, has been designed by various methods [98-102]. This DC magnetic cloak is relevant to a wide range of important applications (e.g. we can prevent metals from being detected by a magnetic metal detector, which means that patients with cardiac pacemakers can undergo magnetic resonance imaging by setting a DC magnet cloak around the cardiac pacemaker). A DC magnetic hose, which performs like a DC magnetic , has been designed with TO for transferring the DC magnetic field an arbitrary distance without loss [103]. Such a DC magnetic hose has been verified experimentally with the help of DC MM and will certainly have many potential applications in the future (e.g. energy transmission and quantum information processing). Other novel devices have also been designed by TO and DC MMs [104-106]. TO and DC

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MM provide a new way to design devices that work for static magnetic fields and will revolutionize magneto-static technologies.

Novel concentrators and lenses that can amplify the static magnetic field passively can also be designed by means of TO and DC MM [106-112]. These devices, which have better performance compared with traditional magnetic lenses, will greatly improve the technologies where high static magnetic fields with high uniformity or high gradient are needed. The main goal of this thesis is to design some novel high-quality magnetic concentrators and lenses by TO and DC MM.

1.2 Thesis Outline

The present thesis is based on six research papers that have been published in international peer-reviewed journals. The present thesis is divided into three parts:

In the first part (Sections 2 and 3), we briefly introduce TO and necessary concepts of differential geometry. Then the fundamental formula of TO for the static magnetic field is derived in detail, and some simple examples (e.g. rescaling magnets and canceling magnets) are given.

In the second part (Section 4), we will focus on how to achieve an enhanced static magnetic field with high uniformity by TO and DC MM. The space folding transformation and the space compression transformation are chosen to design novel magnetic concentrators that can amplify the background static magnetic field passively and achieve an enhanced high static magnetic field with high uniformity in a relative large free space region. The DC MM composed of alternating high-permeability ferromagnetic materials (HPFMs) and high-temperature superconductor materials (HTSMs) is chosen to experimentally realize the magnetic concentrator based on the space compression transformation.

In the third part (Section 5), we focus on how to design a novel magnetic lens/compressor that can passively amplify both the background field and the

4 gradient of the field. We will first use the finite embedded transformation to design such a lens and then simplify its parameters by introducing a null- space medium slab in the reference space. The proposed magnetic lens can also be realized with DC MM, and its performance has been verified by numerical simulations.

At the end of this thesis, the conclusion and future work are presented.

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2. Brief Introduction of Differential Geometry

In this chapter, we introduce some necessary concepts and formulae in differential geometry which will be used to derive the basic formulae of TO in the next chapter. We will only provide a brief introduction and some necessary definitions; the detailed introduction and derivation can be found in the references [15, 17]. The main purpose of this chapter is to derive the divergence and the curl of a vector in an arbitrary coordinate system. We will use them to show the form-invariance of Maxwell’s equations and derive the basic formulae in TO in Chapter 3.

2.1 Coordinate Transformations and the Metric Tensor

First we will introduce the coordinate system and the coordinate transformation. Two coordinate systems {xi, i=1, 2, 3} and {xi’, i’=1, 2, 3} can be related by a coordinate transformation:

xi''= xx ii( ). (2.1)

We have used the Einstein range convention in the above expression: a free index is understood to range over all possible values of the index (e.g. Ai={Ai, i=1, 2, 3}). We will also introduce the Einstein summation convention for convenience of expression: a summation is implied over repeated indices (e.g. i 1 2 3 AiB =A1B +A2B +A3B ). These two conventions are applied throughout this thesis (some exceptions will be indicated).

The natural basis vectors in two coordinate systems are given as {ei, i=1, 2, 3} and {ei’, i’=1, 2, 3}, respectively. The position vectors can be given as i i’ r=x ei and r’=x ei’, respectively. We should note that we use upper index and lower index to distinguish quantities transformed by different rules (this can be seen from Eq. (2.9) later) that are consistent with other literatures [15, 17, and 18]. According to the chain rule, we can get the relation between two coordinate systems:

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i'' ii i ii' dx=Λ=Λii dx;, dx' dx (2.2)

i’ i where Λ i and Λ i’ are defined as:

∂∂xxii' Λ=ii' ;. Λ= (2.3) ii∂∂xxii' '

i i’ Λ and Λ’ are matrices formed by Λ i’ and Λ i, respectively:

ΛΛΛ1 1 1 ΛΛΛ1' 1' 1' 1' 2' 3' 1 2 3 Λ=ΛΛΛΛ=ΛΛΛ2 2 2 2' 2' 2' 1' 2' 3' ;' 1 2 3 . (2.4) ΛΛΛ3 3 3 ΛΛΛ3' 3' 3' 1' 2' 3' 1 2 3

One thing we should mention is that Λ’ and Λ are different matrices (also called transformation matrices). The matrices Λ’ and Λ are the inverse of each i’ i’ i i’ i j’ i’ j’ i’ i i’ i’ other: dx =Λ idx =Λ iΛ j’dx =δ j’ dx ’, thus Λ iΛ j’=δ j’. The symbol δ j’ is the Kronecker delta, which corresponds to the identity matrix [15].

Next we will introduce the metric tensor of the coordinate system, which describes the rule of measurements in the coordinate system. In an arbitrary coordinate system, the line element can be given as:

2 i j ij ij ds=⋅= dr dr( dx ei) ⋅( dx e j) =⋅ e ij e dx dx = gij dx dx , (2.5) where gij=ei•ej . (2.6) is defined as the metric tensor of the coordinate system, which is used to calculate the length in an arbitrary coordinate system. In the Cartesian coordinate system, the natural basic vectors are orthogonal normalized vectors, and thus

gij= d ij in the Cartesian coordinate system (2.7)

Any quantities (e.g. a line element, a surface element, a volume element, etc.) with physical meaning will not change when we change the coordinate

8 system. Therefore, we expected that a line element in an arbitrary coordinate system and in the Cartesian coordinate system should be the same:

2 ij i'' j ds=dij dx dx = gi'' j dx dx (2.2),(2.5) ij ij'' ij '' ⇒dij ΛΛ i'' j dx dx = gi'' j dx dx

ij ⇒gi'' j =ΛΛ i ' j 'δ ij . (2.8)

Eq. (2.8) shows that the metric tensors in an arbitrary coordinate system can be calculated using the coordinate transformation from the Cartesian coordinate system. We can also prove this in another way, which will require introduction of the rule of transformation for the natural basic vector. Any vectors with physical meaning (e.g. intensity, magnetic flux density, etc.) will not change when we change the coordinate system. This can be expressed as:

ii' V= Veii = V e' (2.2) ii'' i ⇒Λi'Veii = Ve' ii''i ⇒V( Λ−=i' eeii' ) 0, for arbitrary V

i ⇒=Λeeii' i' . (2.9)

i’ Similarly we can obtain: ei=Λ iei’. We should note that the components of the vector field Vi and Vi’ will transform like the coordinate displacement dxi and dxi’ in the above derivation. The reason for this is that at each point in the coordinate system {xi, i=1, 2, 3}, the vector is defined from point xi to xi+dxi, and thus the coordinate displacements dxi are the components Vi of a vector field [15]. Therefore they transform based on the same rule. We can summarize the transformation rule of a vector field as:

i'' ii i ii' V=Λ=Λii VV;;' V (2.10)

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ii' ei''=Λ=Λii' ee ii;. e i (2.11)

i As we can see from Eq. (2.10) and (2.11), V and ei transform inversely to ensure that vector V itself is unchanged under the coordinate transformation. From the definition of a metric tensor (2.6) and the transformation rules of a basis vector (2.11), we can also drive the transformation rule of metric tensors from one arbitrary coordinate system to another arbitrary coordinate system:

(2.6) (2.11) ij ij gij''= ei'' ⋅ e j =ΛΛij'' eeij ⋅ =ΛΛijij'' g. (2.12)

If the coordinate system before the transformation is a Cartesian coordinate system, we have gij=δij in (2.12), and then Eq. (2.12) reduces to Eq. (2.8). From the definition of the metric tensor, we can conclude that the metric tensor is a symmetric tensor. The volume element in an arbitrary coordinate system can be expressed with the aid of the metric tensor [15]:

dV= gdx123 dx dx , (2.13) where g=det(G). G is the matrix formed by gij:

ggg11 12 13 =  Gg21 g 22 g 23 . (2.14)

ggg31 32 33

Similarly G’ is the matrix formed by gi’j’:

ggg1'1' 1'2' 1'3' =  Gg'.2'1' g 2'2' g 2'3' (2.15)

ggg3'1' 3'2' 3'3'

2.2 Covariant and Contra-variant Vectors

We can classify vectors into two kinds: covariant vectors (also called one- form [15]) and contra-variant vectors. For example, electric intensity E and

10 magnetic intensity H are covariant vectors in 3D space; density D and magnetic flux density B are contra-variant vectors in 3D space. We will see this fact again from the general form of Maxwell’s equations in an arbitrary coordinate system in Chapter 3.

First we will introduce some necessary definitions. The inverse basis vector can be defined as [15, 17, and 18]:

i i ee⋅=j δ j . (2.16)

The inverse metric tensor is defined as:

ij gij = ee ⋅ . (2.17)

The contra-variant vector is expressed by basis vectors:

i V= Vei , (2.18)

i where V represents components of a contra-variant vector V and ei represents natural basis vectors. The covariant vector (or one-form) is expressed by inverse basis vectors:

i U= Uei , (2.19)

i where Ui are components of a one-form U and e are inverse basis vectors of the coordinate system. We should note that from the physical sense, covariant and contra-variant vectors are different and they will not change under coordinate transformation. However for a given coordinate system, we can either expand a contra-variant vector by inverse basis vectors or a one-form by basis vectors, according to the following transformation rules [15, 17]:

ijij egeege=ji;; = ij (2.20)

i ij j V= gVj;; V i = gV ij (2.21)

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ij i ggjk= δ k . (2.22)

Note a metric tensor and an inverse metric tensor can be used to raise or lower the index. For a general tensor, it can be treated as a tensor product of multiple vectors (e.g. a third order mixed tensor can be expressed as: k i j k i j T=Tij e e ek=AiBjC e e ek). A scalar is a zero-order tensor and a vector is a first- order tensor. Any physical laws expressed by equations should be unchanged under coordinated transformations. Physical equations expressed in the form of tensors naturally satisfy this condition, and we will express Maxwell’s equations by a general tensor-form in Chapter 3 using the language of differential geometry introduced in this chapter.

The rule of transformation of inverse basis vectors, the inverse metric tensor, and components of a contra-variant can be proved with the fact that a vector or scalar product of vectors does not change under coordinate transformations [15, 17]. We provide the results here:

i VVi''= Λ ii; (2.23)

ii' i' ee= Λ i ; (2.24)

i'' j i ' j ' ij gg=ΛΛij. (2.25)

Another thing we should mention is that when we say that some quantity transforms like a covariant vector, it means that this quantity transforms like the component (but not the basis vector) of a covariant vector (see (2.19) and (2.23)). Similarly, if a quantity transforms like a contra-variant vector, this quantity transforms according to the rule of the component (but not the basis vector) of a contra-variant vector (see (2.18) and (2.10)).

2.3 Scalar Product and Vector Product

The scalar product of two vectors in an arbitrary coordinate system can be given as:

12

i j ij ij V⋅= U( V ei) ⋅( U e j) = VU eij ⋅= e gVUij . (2.26)

The scalar product of two vectors should be coordinate-independent, which can be proved by:

(2.10),(2.12) ij ij'' ijij'' V⋅ U = gVUij =( ΛΛijijijg'')( ΛΛ ' 'V U )

ii'' j j ij'' ij '' =ΛΛΛΛi i' j j 'g ij '' VU = gij'' VU

ij i'' j ⇒⋅=V U gVUij = gi'' j V U . (2.27)

The vector product of two vectors in an arbitrary coordinate system can be expressed as [15]:

1 V×=± U[] ijk Vjk U ei . (2.28) g

We choose + in a right-hand coordinate system and – in a left-hand coordinate system. The permutation symbol [ijk] is defined as:

+1, when ijk is an even permutation of 123  [ijk ]= − 1, when ijk is an odd permutation of 123 . (2.29)   0 , otherwise

The entirety ±[ijk]/ g is also called as the Levi-Civita tensor (an anti- symmetric tensor). Next we will show that the vector product of two vectors is still coordinate-independent:

13

11jk'' i ' V×=± U[] ijk Vj U k eii=± [] ijk Λjj V'' Λ k U k Λ i e ' gg

11ijk'' ' i' =± ΛΛi j Λ k[]ijk V jk'' U eii''=±det( Λi )[i ' j ' k '] Vjk'' U e gg 1 = ± [i ' jk ' '] VUjk'' ei' g '

11 ⇒×=±V U[] ijk Vjk U eii=± [ i ' j ' k '] Vj'' U k e ' . (2.30) gg'

2.4 The Covariant Differentiation

Covariant differentiation in differential geometry is different from the partial differentiation in calculus. For a scalar whose covariant differentiation equates to its partial derivation [15, 17]:

∂φ φφ(xxii+∆ ) − () x i φφ= = =∂= φ ;,iiii i lim . (2.31) ∂∆xx∆→xi 0

We use a comma to indicate the partial differentiation and a semicolon to indicate the covariant differentiation. ϕ,i transforms like a covariant vector (see (2.23)). However, the situation of a vector V is different. As we know, the differentiation of a vector requires the subtraction of two vectors located at nearby spatial points: both the vector basis ei and the component under this basis Vi will change when we move the vector from one point to anther nearby i point. According to the Leibniz rule, we should differentiate ei and V separately [15, 17]:

j ∂∂ ∂Ve ∂j =jj = + iiV( Vejj) ei V i ∂∂xx ∂ x ∂ x j jk =∂eVjki +Γ Vji e (2.32) =∂+Γj kj iVVki ej ,

14

k where Γ ij is called the Christoffel symbols, which can be used to calculate the partial differentiation of basis vectors in an arbitrary coordinate system (the result of the partial differentiation of a basis vector can be expressed as the linear combination of all basis vectors in this coordinate system). The Christoffel symbols do not constitute a tensor, as they do not obey the transformation rules of a tensor [15, 17]. The Christoffel symbols are determined by the coordinate system and can be calculated with use of a metric tensor [15]:

1 Γ=i ggil ( + g − g ). (2.33) jk 2 lj,, k lk j jk , l

We can define the covariant differentiation from the Eq. (2.32):

Vj=∂+Γ= VV j kj V j +Γ V kj. (2.34) ;,i i ki i ki .

Similarly we have [15]:

k VVji;,= ji −Γ ij V k. (2.35)

Now the differentiation of a vector can be written as:

∂ j VVe= j . (2.36) ∂xi ;i

j As we can see from Eq. (2.36), V ;i includes both differentiation of the k j vector basis V Γ ki and differentiation of the scalar component under this basis j j V ,i. The most important point is that V ;i transforms like a tensor [15]:

j'' jij VV;'i=ΛΛ ji'; i. (2.37)

j The tensor with components V ;i is called the covariant derivative of vector V. The “covariant derivative” means “correct derivative”, which includes variation of both bases and components. We will use this covariant differentiation in the next section to calculate the divergence and the curl of a vector in an arbitrary coordinate system.

15

2.5 The Divergence and the Curl

After introducing the concept of the covariant differentiation, the divergence and the curl of a vector in an arbitrary coordinate system can be given. The divergence of a vector can be written as [15]:

(2.34) i i ij ∇⋅VV =;,i = Vi +Γ ji V (2.33) 1 =Vi + gg il () +− g g Vj ,i 2 lj ,, i li i ji , l 11 =+−Vi () ggli ggil V j + gg il V j ,i 22jl ,, i ji l li , i 1 =Vi + gg il V j ,,i 2 li i

ij1 =V,,ij + () gV g

1 i = ()gV ,i g

1 i ⇒∇⋅V = ( gV ).,i (2.38) g

And the curl of a vector in an arbitrary coordinate system can be given as [15] (note that we only consider the right hand coordinate system in this thesis):

16

(2.28)(2.34) 1 ∇×V = []ijk Vkj; ei g (2.35) 1 m = []ijk( Vk, j−Γ kj V m ) ei g

(2.33) 1 11ml =[]ijk Vk, j eii − [] ijk g ( glk,, j+− g lj k g kj , l) V m e gg2 1 = []ijk Vkj, ei g

1 ⇒∇×V = [] ijk Vkj, ei . (2.39) g

In the last step, j and k are symmetric in glk,j, glj,k, and gkj,l, and are anti- symmetric in [ijk], thus the result of the summation equates to zero. Now the divergence and the curl of a vector in an arbitrary coordinate system have been given, and will be utilized to rewrite Maxwell’s equations in an arbitrary coordinate system in the next chapter. We should note that the divergence and the curl of a vector in an arbitrary coordinate system can also be derived by other methods (e.g. multi-variable calculus [16]).

17

18

3. Theory of Transformation Optics and Transformation Magneto-statics

In this chapter, we will derive the basic formulas in TO based on the form- invariant of Maxwell’s equations and material interpretation. Then we will extend TO to the DC magnetic field case so that we can design both magnetic materials and magnets by the method of coordinate transformations. Finally we will give some examples (e.g. magnetic illusion devices) designed by the TO derived in this chapter. 3.1 The Form Invariance of Maxwell’s Equations and Transformation Optics

Maxwell’s equations are the foundation of classical electrodynamics and classical optics. First we write Maxwell’s equations in a Cartesian coordinate system (a flat coordinate space) [15]:

 (Di ) = ρ  ,i  (Bi ) = 0  ,i   ∂Bi . (3.1) [ijk] E = −  kj, ∂  t  ∂Di [ijk] H= + J i  kj, ∂t

We can rewrite Maxwell’s equations in an arbitrary coordinate system (e.g. a curved coordinate space) with the aid of the divergence and the curl derived in Chapter 2 (Eqs. (2.38) and (2.39)):

19

 ( gD'i' ) = g ''ρ  ,'i  i'  ( gB'0) =  ,'i  ∂Bi' . (3.2)  [i''' jk] Ekj', ' = − g '  ∂t  ∂Di' [i''' jk] H= g ' + gJ 'i'  kj', ' ∂t

Two sets of Maxwell’s equations can be linked by a coordinate transformation as we have explained in Chapter 2:

i'' ii DD= Λ i  i'' ii BB= Λ i  i'' ii JJ= Λ i . (3.3)  = Λi EEi'' ii  = Λi HHi'' ii

The main trick to TO is the material interpretation [12]: we can assume that the electromagnetic media and the curved coordinates have an equivalent effect on the electromagnetic wave [17], and treat Eq. (3.2) as Maxwell’s equations in a Cartesian coordinate system of a flat space filled with some special medium (called the transformation medium). If we want to treat Eq. (3.2) as Maxwell’s equations in a Cartesian coordinate system, we should make the following definitions:

20

Di' = gD' i'  Bi' = gB' i'  Ji' = gJ' i'  . (3.4) EEi' = i' HH=  i' i'  ρρ'= g ''

The underlined quantities with primes in Eq. (3.4) indicate that the quantities are in the real/physical space; the quantities with primes and without the underline in Eq. (3.3) and (3.4) are ones in the transition space (a curved space in an arbitrary coordinate system); the quantities without primes and without underline in Eq. (3.3) indicate the ones in the reference space (a flat virtual space in the Cartesian coordinate system). In the real space, the space is flat while filled with some transformation medium, which will be deduced later. By combining Eq. (3.3) and (3.4), we can obtain the relationship of quantities in the reference space and the real space:

DgDgDi' =''i'' = Λ ii  i i' i'' ii BgBgB='' = Λ i  JgJgJi' =''i'' = Λ ii  i . (3.5) i EEi' =i'' = Λ ii E  = = Λi HHi' i'' ii H  ρρρ''''=gg =

Considering the metric tensor in a curved coordinate space can be obtained by transforming the metric tensor from a flat Cartesian coordinate space:

(2.11),(2.14),(2.15) gG'= det( ') =det( ΛT GG Λ= ) det( Λ ) det( )

the reference space is a flat Cartesian coordinate space 1 = det(Λ )det()I = det( Λ= ) . det(Λ ')

21

Note that matrices Λ, Λ’, G, and G’ are defined in Eqs. (2.3), (2.14) and (2.15). Therefore we can rewrite Eq. (3.5) as:

 i' 1 DD= Λii'  Λ i  det( ')  i' 1 BB= Λii'  Λ i  det( ')  i' 1  = Λii' JJi . (3.6)  det(Λ ')  i EEi' = Λ ii'  i HHi' = Λ ii'  1 ρρ' =  det(Λ ')

We consider the constitutive relation:

i' ij '' i ij D=εε EDj ' ,; = Ej (3.7)

i' ij '' i ij B=µµ HBj ' ,. = Hj (3.8)

By combing Eq. (3.6), (3.7) and (3.8), we can obtain the transformation medium:

ij'' 11ij'' e = ΛΛi'' je ij ,.µ = ΛΛi'' jµ ij (3.9) det(ΛΛ ') ij det( ') ij

The derivation of Eq. (3.9) is given as below:

22

(3.6) i' ij '' 1 ij'' DE=ee ⇒ Λ=Λii' D j E j ' det(Λ ') i jj' i ijij'' ⇒DE =det( ΛΛΛ ') ij''e j (3.7) ij i j ij'' ⇒ee =det( ΛΛΛ ') ij''

ij'' 1 ⇒ee = ΛΛi'' j ij det(Λ ') ij

Note that the quantities in the transition space do not appear in our final equations. We often drop the underline for quantities in the real space, and rewrite Eq. (3.6) and (3.9) as:

 1 DDi''= Λii  Λ i  det( ')  1 BBi''= Λii  Λ i  det( ')  1 JJi''= Λii  Λ i  det( ') EE= Λi i'' ii . (3.10)  i HHi''= Λ ii  1 ρρ' =  det(Λ ')  i'' j 1 i'' j ij ee= ΛΛij  det(Λ ')  i'' j 1 i'' j ij µµ= ΛΛij  det(Λ ')

Eq. (3.10) is the final system of relationships in TO: the quantities with primes are the ones in the real space and those without primes are the ones in the reference space. This relation can also be derived by other methods (e.g. multi-variable calculus [16]).

23

3.2 TO in an Orthogonal Coordinate System

If the reference space is not in a Cartesian coordinate system but is instead in some other arbitrary coordinate system, we can also obtain the same form of the final relationship (Eq. 3.10) by following a similar process with the aid of the material interpretation (treat the real space as having the same coordinate system as the reference space and filled with some special medium). Note that in other literatures (e.g. [15, 17]), det(Λ’) is expressed as g '/ g , which are essentially the same thing (γ is the determinant of the matrix G in the reference space). However if quantities in Eq. (3.10) are not in the Cartesian coordinate system, we have to transform all quantities to the ones expanded on a normalized basis (in an arbitrary coordinate system, all quantities in Eq. (3.10) correspond to components expanded in the natural basis but not in the normalized basis [15]).

Here we only provide one example of an orthogonal coordinate system. The i relationship between the natural basis (e.g. e and ei ) and the normalized i orthogonal basis (e.g. e and e i ) in an orthogonal coordinate system can be given as:

ii i i eeii e e = = =he, e = ; (3.11) i ii i e g hi

ei ee ii ei= = =, eii = hei . (3.12) ei gii hi where hi is the Lame coefficient or Scale factor in the orthogonal coordinate system. We should note that we do not use the Einstein summation convention in Eqs. (3.11) and (3.12). The components in Eq. (3.10) are expanded on the natural basis. We use quantities with upper “*” to express the components expanded in the normalized orthogonal basis, and their relationship can be given as:

24

i  (3.11)  * i e  i E= Eeii = E = Ei e  hi  (3.12) * iii D= Dei = Dhe ii = D e  i i  (3.11)  *  i e  i H= Heii = H = Hi e (3.13)  h .  i (3.12) *  iii B= Bei = Bhei ii = B e  (3.12) * iii J= J ei = J hei ii = J e  * ρρ=

Combining Eq. (3.13) and (3.10), we can obtain the transformation relationship in an orthogonal coordinate system:

* i *  Ei''Λ ii Eh i i EEii' = =∑∑ = Λ i'  hhhi'ii ii ''  * * hi i HHii' =∑ Λ i'  i hi'  ** i'' i 11ii' hi' i' i D= Dhi'' =∑∑hii Λ= D ΛiD  iidet(ΛΛ ') det( ') h i . (3.14)  **  i''1 hi' ii BB= ∑ Λ i  det(Λ ') h  i i **  i''1 hi' ii JJ= ∑ Λ i  det(Λ ') h  i i  **1 ρρ' =  det(Λ ')

To avoid the confusion, we add the summary symbol in Eq. (3.14). We define the transformation matrix in an orthogonal coordinate system as:

25

h hu∂ i' ii''=ii'' Λ= T iii . (3.15) hii hu∂

ui’ and ui are coordinate variables in orthogonal coordinate systems. Note that we do not use the Einstein summation convention in Eq (3.15). We can rewrite Eq. (3.14) with the help of Eq. (3.15):

 * * = i Eii' TEi' * *  i Hii' = THi'  **1 Di''= TDii  det(T ') i  **  i''1 ii. (3.16) B= TB  det(T ') i   **1 i''= ii J TJi  det(T ')  **1 ρρ' =  det(T ')

Note that the Einstein summation convention is applied in Eq. (3.16). T’ is i’ the matrix formed by T i. Next we explain why det(T’)=det(Λ’):

(3.15) hu∂∂ii'' hu∂∂xxii' T i' =ii'' = i hu∂i ∂∂∂ x ii' xhu i ii hu∂ i' ∂xi ⇒=i' Λ =Λ det(T ') det([ii' ])det( ')det([ ]) det( ') ∂∂x hui

i’ i’ i i Matrices [hi’∂u /∂x ] and [∂x /(hi∂u )] are coordinate rotation matrices between the normalized orthogonal bases in the orthogonal coordinate system and these in the Cartesian coordinate system. The determinants of them are 1. The constitutive relation in an orthogonal coordinate system can be written as

**** ** i ij i ij D=εµ EBjj,. = H (3.17)

26

By combining Eq. (3.16) and (3.17), we can obtain the transformation medium expanded on a normalized orthogonal basis in an orthogonal coordinate system:

**  i'' j 1 i'' j ij ee= TTij  det(T ') . (3.18)  **  i'' j 1 i'' j ij µµ= TTij  det(T ')

3.3 Extending the Transformation Optics to Transformation Magneto- statics

As the DC field is also governed by Maxwell’s equations, we can still use the TO relationship in Eq. (3.10) to design a transformation medium that can control the DC magnetic field by coordinate transformation. However if there are some magnets in the reference space, we cannot use Eq. (3.10), as no magnets appear in this equation. The reason why we cannot use Eq. (3.10) to design magnets is that a linear material relationship (see Eq. (3.8)) has been assumed during the derivation of Eq. (3.10).

In this section, we will derive a general formula that contains the transformation of magnets. Our starting point is from Eq. (3.6), and we rewrite two equations containing the magnetic field in Eq. (3.6) again:

 i' 1 ii' BB= Λ i  det(Λ ') . (3.19)  i HHi' = Λ ii'

If there are some magnets in the reference space, the constitutive relationship should be given as:

Bi'=µµ ij '' HM + µ i'  00j ' , (3.20) i=µµ ij + µ i  B00 HMj

27 where M and M’ correspond to the intensity of magnets in the reference and the real space, respectively. By combining Eq. (3.19) and (3.20), we can obtain:

11ij'' i' ΛΛi'' jµµ ij H + Λi' M i ≡ HM + . (3.21) det(ΛΛ ') ij jj''det( ') i

Eq. (3.21) holds for any magnetic field H’, and therefore we can obtain:

 ij'' 1 µµ= ΛΛi'' j ij  ij  det(Λ ')  . (3.22)  i' 1 ii' MM= Λ i  det(Λ ')

Now we obtain the transformation medium and transformed magnets for the DC magnetic field’s case. We can summarize the above equations (Eqs. (3.19) and (3.22)) for the DC magnetic field (as we have done in Section 3.1, we drop the underline in our final following equations and use quantities with or without primes to indicate the ones in the real space or the reference space, respectively):

 1 BBi''= Λii  Λ i  det( ') HH= Λi  i'' ii  1 . (3.23) µµi'' j = ΛΛi'' j ij  Λ ij  det( ')  i''1 ii MM= Λ i  det(Λ ')

Eq. (3.23) is the key formula in this thesis, and designs in later chapters are based on this one. Note in Eq. (3.23), the Jacobian transformation matrix is expressed as Λ=∂' (',x y ',')/(,,) z ∂ xyz. If the reference space and the real space are not in a Cartesian coordinate system, but in an orthogonal

28 coordinate system, we can do the same process as we did in Section 3.2 to obtain:

**  i''1 ii B= TBi  det(T ')  ** H= THi  i'' ii  **. (3.24) i'' j 1 i'' j ij µµ= TTij  det(T ')  ** i''1 ii M= TMi  det(T ')

i’ In this case T i is expressed in Eq. (3.15), and quantities with upper “*” corresponds to the ones expanded in the normalized orthogonal basis. In all following chapters of this thesis, we use quantities with or without primes to indicate quantities in the real or reference space, respectively.

3.4 Some Examples Designed by Transformation Magneto-statics

In this section, we will design some novel devices working for DC magnetic fields based on the formulae derived in Section 3.3. First we will present an example that gives an illusion of rescaling magnets: a small magnet wrapped by some transformation medium will perform like a big magnet to the outside viewer. Fig. 3.1(a) shows the basic idea of this illusion. We choose a 2D circular magnet with radius R2 in the reference space as an example (see the left picture of Fig. 3.1 (b)): the relative permeability and magnetization intensity in the reference space can be given, respectively, as:

 m , rR∈ [0, ]  m02 m =1, r ∈ [ RR23 , ] ; (3.25)  1, rR∈∞ [3 , ) and

29

 Br0  ,rR∈ [0,2 ] µ  0  M=0, r ∈ [ RR23 , ]. (3.26) 0, rR∈∞ [ , )  3 

Br0 and μm0 are the residual induction and the relative permeability of the magnet, respectively, in the reference space. In the real space (see the right picture of Fig. 3(b)), a small magnet with a radius R1 (R1R3. The transformation relation between the reference space and the real space can be given by:

 Rr12/ R ,r∈ [0, R2 ]  (3.27) r' =( RRrRR31 −) / ( 32 −+) RRR 312( −) /( RRrRR 32 −) , ∈ [ 23 , ];θθ ' = ; zz ' = .   r ,rR∈∞ [3 ,)

Eq. (3.27) shows that the magnet (0R3 in the real space is still air as we apply an identical transformation on the region r>R3 in the reference space. The relative permeability and magnetization intensity in the real space can be calculated with Eqs. (3.24) and (3.27):

 2 RR22 R 2 Br0  diag( , , ) ,r '∈ [0, R1 ]  RR11 R 1µ 0  M'=  0, r '∈ [ RR13 , ]; (3.28) 0, rR '[∈∞ ,)  3   and

30

2  m diag(1,1,( R / R ) )  m0 21 ,rR '∈ [0, ] (3.29) − − +− 1  (R32− Rr )'( +− R 213 RR ) ( R 32− Rr )' ()()'()RR32[ RRrRRR 32 213] m '(= diag , , 2 ),r '∈ [ R13 , R ]. (R32− Rr )' ( R32− Rr )'( +− R 213 RR ) (RRr− ) '  31 ,'rR∈∞ [ , )  3  I

The symbol “diag” indicates a diagonal matrix. I is the identity matrix. Note that the quantities in Eq. (3.28) and (3.29) are expressed in a 2D cylindrical coordinate system. For simplicity, we only consider a 2D case: the magnetic field is in the plane z=0. In this case, μm0 and Br0 only contain the r- component and θ-component, and thus Eq. (3.28) and (3.29) can be reduced to:

 R2 Br0  ,rR '∈ [0,1 ] µ  R10  M'=  0, r '∈ [ RR13 , ], (3.30) 0, rR '∈∞ [ , )  3  and

 mm0 I  ,rR '∈ [0,1 ] − +− −  (R32 Rr )'( R 213 RR ) ( R 32 Rr )' (3.31) m '(= diag , ),r '∈ [ RR13 , ].  (R32− Rr )' ( R32− Rr )'( +− R 213 RR ) ,'rR∈∞ [3 , )  I

Note that for a 2D case, M’ and μ’ also only contain the r’-component and θ’-component in Eqs. (3.30) and (3.31).

31

Figure 3.1: (a) The rescaling magnet illusion. (b) The reference space (left): the circle is the magnet with radius R2 within a white air region. The real space (right): the circle is the magnet with radius R1, the region is the rescaling medium, and the remaining white region is air. (c) The magnetic flux distribution in the region r’>R3 produced by a single magnet with radius R2 = 0.25 m. (d) The magnetic flux distribution in the region r’>R3 = 0.3 m produced by a magnet with radius R1 = 0.1 m warped by the rescaling medium in the region R1

32

The finite element method (FEM) has been used to verify the performance of the device (see Fig. 3.1 (c) and (d)). All simulations based on FEM in this thesis are conducted by the commercial software COMSOL Multiphysics. The

DC magnetic field outside the circular region of radius R3 is entirely unchanged for two cases: we only have a magnet of radius R2 or we have a magnet of a smaller radius R1 (R1< R2) and some rescaling medium. The relative permeabilities of the scaling medium (both along the radial and tangential directions) are larger than zero (see Fig. 3.1 (e) and (f)), which can be realized by the current DC MM [93-101].

The second example is a device that can cancel the DC magnetic field from a magnet. The basic scheme of this illusion is shown in Fig. 3.2 (a) and (b). If we put a magnet in free space, it will produce a non-zero DC magnetic field around it. If we add some complementary magnet (explained in Fig. 3.2 and described by Eq. (3.37)) and transformation medium around this magnet, the total DC magnet field will become zero for the region outside a certain boundary. A specific geometrical structure with h = d = 0.2 m is chosen to demonstrate this illusion (see Fig. 3(c)). The regions with different colors are explained in the caption of Fig. 3(c). The coordinate transformation is given by:

For the and red regions:

x x'=−== ; yyzz ' ; ' ; (3.32) 2

For the purple region:

x'=[ uy ( +−− d ) uy ( d )(/43/4)] x + d +−−++ uy ( d )3/4[ y x /43/2 d] +−− u ( y d )3/4[ y ++ x /43/2 d ]  (3.33)  yy' =   zz' =

And for the other regions (white and green regions):

xxyyzz'= ; ' = ; ' = ; (3.34)

33 where function u is defined as follows:

1,ξ ≥ 0 u()ξ =  . (3.35) 0,ξ < 0

Substituting Eqs. (3.32), (3.33) and (3.34) into Eq. (3.23), we can obtain the transformation medium in each region:

The yellow region (complementary medium):

1 µ '=diag ( − ,2,2) −− ; (3.36) yellow 2

The red region (the complementary magnet):

 1  µµ'red= greendiag( − ,2,2) −−  2 ; (3.37)  = −− M'red diag (1,2,2) M 'green

And the purple region (restoring medium):

 2.5 3 0  m =   'purple  3 4 0 ,in the lower purple triangle region  0 04   m 'purple = diag (0.25,4,4),in the middle purple rectangular region (3.38)  −  2.5 3 0   m '= − 3 4 0 , in the upper purple triangle region  purple  0 04  

Note that the subscript ‘green’ in Eq. (3.37) means the relative permeability or magnetization intensity in the green region of Fig. 3.2 (c), which corresponds to the original magnet to be canceled. Eq. (3.37) shows that the complementary magnet (relative permeability or magnetization intensity) in the red region is related to the magnet to be cancelled filled in the green

34 region. Due to the identical transformation (Eq. (3.34)), the material in the white region is free space.

The FEM simulation is shown in Fig. 3.2 (d) and (e), which verify the performance of this magnet cancelling effect: the DC magnetic field is nonzero outside the region enclosed by the dotted line (see Fig. 3. 2(d)) when there is only one green magnet (the green part in Fig. 3. 2(c)) in the whole space. If the complementary magnet (red region), complementary medium (yellow region), and restoring medium (purple region) are all introduced around the original magnet (green region), the DC magnetic field can be greatly reduced outside the region enclosed by the orange dotted line (see Fig. 3. 2(e)).

To realize the proposed device, we need some negative permeability material. It is still challenging to realize a negative permeability for the DC magnetic field with current MMs. Some MM working in an extremely low may be one possible solution [92]. The applications and detailed explanations for these illusion devices for magnets can be found in our recent paper [105].

35

Figure 3.2: (a) and (b) the scheme of cancelling magnets. (c) A specific design: the green region is the original magnet to be canceled; the red region is the complementary magnet; the yellow region is the complementary medium; and the purple region is the restoring medium. (d) and (e): FEM simulation results. The amplitude of the total DC magnetic flux density distribution outside the whole structure: (d) only original magnet (the green region in (c) with relative permeability 1000 and residual induction 1 T along the x’- direction) is used. (e) all components including the original magnet, the complementary magnet, the complementary medium and the restoring medium are used.

In this chapter, we have derived the basic formulae in TO, extended TO to the DC magnetic field, and designed some novel devices which can transform the magnets. Some more interesting devices can be designed by our method; we will show them in detail in later chapters.

36

4. Enhancing DC Magnetic Field with High Uniformity

Some practical applications require a high uniform static magnetic field in a large free space region. For example, magnetic resonance imaging (MRI) can be used to distinguish pathologic tissue from normal tissue [5]. However, there are still some problems in improving spatial resolution if we want to image some details in deep brain areas. Such an MRI scan requires a static magnetic field with two properties in a large free space region: a uniform field density and high field strength. A higher static magnetic field with high uniformity will provide a better spatial resolution in MRI, which means that we can find diseased cells earlier and more precisely locate lesion areas.

Traditionally there have been two ways to obtain an enhanced DC magnetic field. One way is to use an active magnet (e.g. a coil), which requires a high power supply. In addition, the largest DC magnetic field produced by the magnet is often limited. For example, superconducting magnets can help us achieve a high static magnetic field (e.g. 21T [6]). However, the maximum magnetic field achievable in a superconducting magnet is limited by the field

(“critical field‟, Hc) and the current (“critical current‟, Ic) at which the winding material ceases to be superconducting. The other way is magnetic flux concentration by passive magnetic lenses. With the help of these DC magnetic lenses, we can passively obtain an additional magnetic field enhancement when a weak background DC magnetic field is applied onto these lenses. However traditional magnetic lenses can only obtain an enhanced DC magnetic field in a small region (e.g. a circular region with a diameter of 22mm) with very low enhancement factor (e.g. magnification factor is about 2.5 and 3.2 with superconducting multilayered sheets [113] and HTS bulk cylinders [114], respectively). Traditional methods cannot achieve a greatly enhanced DC magnetic field with high uniformity in a large free space region. Some novel DC magnetic lenses need to be designed for these specific applications and future potential applications.

37

In this chapter, we will use TO to design some novel DC magnetic concentrators/lenses that can achieve an enhanced DC magnetic field with high uniformity in a large free space region. The enhancement factor can be tuned by choosing different geometrical and material parameters of the device.

4.1 A Theoretical Method for Designing a Passive DC Magnetic Concentrator via Space Compression Transformation

The reference space and the real space are shown in Fig. 4.1. The underlying idea here is the space compression transformation: a big red region in the reference space is compressed into a small red region in the real space, and the magnetic field in the space is also compressed (i.e. enhanced). We divide the whole transformed region into many small triangles and number each triangle region (e.g. I, II, III and etc. in Fig. 4.1) so that we can apply a linear transformation on each triangle region and obtain a homogeneous transformation medium in each triangle region [108, 115]. The transformation in each region can be given as:

For the yellow region:

 dd− =++ 10 x' KxKysignx12 ( ') d2 dd20−   y'= Ky3 , (4.1)  zz' =   where

dd21− d2dd10− h1 K12=; K=−= sign ( x ') sign ( y ') ;K3 . (4.2) dd20−−hdd020 h 0

For the green region:

38

 xx' =   hh10− y' =++ M12 x M y sign( y ') h2 , (4.3)  hh20−  zz' = where

hh10− h2 hh 21− M12=−= sign( x ') sign ( y ') ; M . (4.4) hhd202−− hh 20

For the red region (also to be divided into four sub-triangle regions in four quadrants, the following transformation is given for each triangle region):

x'= Nx12 ;' y = N yz ;' = z ; (4.5) where

dh11 NN12=;. = (4.6) dh00

The sign function is introduced by:

 1,x '> 0 sign( x ') =  ; (4.7) −<1,x ' 0

 1,y '> 0 sign( y ') =  . (4.8) −<1,y ' 0

39

Figure 4.1: The reference space (left) and the real space (right). Regions I, II,… VIII are correspondingly transformed from the reference space to the real space. The white region maintains an identical transformation.

With the help of TO (Eq. (3.22), note that there is no magnet in the reference space this time), we can obtain the transformation media in each region:

For the yellow region:

KK2 K 12+ 20 K3 KK 13 K 1  K2 K3 µµ'yellow = 0 0. (4.9) KK11 1 00 KK13

For the green region:

40

1 M 1 0 MM 22 22 MMM11+ 2 µµ'green = 0 0. (4.10) MM22 1 00 M 2

For the red region:

NN121 µµ'red = 0diag ( , , ). (4.11) N2 N 1 NN 12

For a 2D device, Eqs. (4.9) to (4.11) can be reduced to:

KK2 K 12+ 2 K KK K µµ';= 3 13 1 (4.12) yellow 0 K K 2 3 KK11

1 M1 MM µµ';= 22 (4.13) green 0 MMM22+ 11 2 MM22

NN12 µµ'red = 0diag ( , ). (4.14) NN21

We consider a symmetric structure and introduce following geometrical parameters:

dh11= =αα d 0 = h 0  . (4.15) dh22= =ββ d 0 = h 0

41

Substituting Eq. (4.15) into Eqs. (4.12)-(4.14), we can obtain:

22 (βα−+) βα2 ( −1) α −1 −sign( x ') sign ( y ')β αβ( −−1)( βα) βα− µµ';=  (4.16) yellow 0  αβ−−11 −sign( x ') sign ( y ')βα βα−− βα

βα−−11 −sign( x ') sign ( y ') βα−−βα  µµ';green = 0 22 (4.17) α −1 (α−+−1) ( βα) −sign( x ') sign ( y ') βα−( β −−1)( βα)

µµ'red = 0diag (1,1). (4.18)

Note that in this 2D case, the field enhanced region (red region) is free space, which is what we expected. The transformation medium in each region is a homogenous anisotropic medium, which is symmetric in the x’-y’ coordinate system. We can diagonalize the permeability tensor in its principal axis coordinate system (e.g. ξ-χ coordinate system):

mmcoordinate system rotates xx'' xy '' → diag(mm , ), (4.18) mm xx cc xy'' yy '' where

 22 µxx =++cos θµxx'' sin θµyy'' 2sincos θ θµxy''  , (4.19) µ=+−22 θµ θµ θ θµ  cc sinxx'' cosyy'' 2sincos xy'' and

1 2µ θ = arctan xy'' . (4.20) µµ− 2 xx'' yy ''

42

θ (the principal axis angle) is the rotation angle between the x’-y’ system and the ξ-χ system (see the inset of Fig. 4.2 (b)). Substituting Eqs. (4.16) and (4.17) into Eqs. (4.19) and (4.20) respectively, we can obtain the permeability tensors in a diagonal form and the principal axis angle in the yellow and green regions. To further simplify the material requirement of the device, we can take the limit β→∞ in Eqs. (4.16) and (4.17), and obtain:

2 11+−(α ) β →∞ −−α µµ= sign( x ') sign ( y ')( 1) ';yellow 0 α (4.21) −−sign( x ') sign ( y ')(αα 1)

β →∞

µµ' green = 0diag(1,1). (4.22)

As we can see from Eq. (4.21) and (4.22), if β→∞ (which implies d2→∞), the green region approximately becomes air (we need only the yellow region to realize the whole device). We use FEM to verify the performance of the reduced device described by Eqs. (4.21) and (4.22). We choose α=0.4, β=105 and d1=0.2 m and get about a 2.36 field enhancement factor in the center air region (see Fig. 4.2(a)). In theory, the shape of the yellow region should be a rectangle with two infinitely long sides parallel to the x’ axis. Nevertheless, we have to use a device with a finite size (2.8 m long in the x’ direction) to simulate it practically, which is the reason for the degradation of the performance of the device (the uniformity and the enhancement factor are degraded).

We can use the layered ferromagnetic materials and diamagnetic materials to realize such a concentrator in Fig. 4.2 (a). Two principal values of the materials in the yellow region in Fig. 4.2 (a) can be calculated by Eq. (4.19):

μξξ=3.5155 and μχχ=0.2845; the principal axis angle can be calculated with Eq. (4.20): θ=10.9 o in quadrants I and III and θ=-10.9o in quadrants II and IV. We can use two isotropic materials with space ratio 1:1 layer by layer to realize the effective anisotropic medium in the yellow region. Assuming the two

43 materials are in parallel in the χ direction and in series in the ξ direction, we can obtain the required permeabilities by solving the following equations:

0.5 0.5 1 +=  µµµ  12χχ (4.23) µµ+ 12= µ  2 ξξ

We can obtain μ1=0.145 and μ2=6.854, which can be realized by DC MM [93-101] and ferromagnetic materials, respectively. The performance of this layered structure is shown in Fig. 4.2 (b).

Figure 4.2: The absolute value of the total magnetic flux density distribution. A uniform background DC magnetic field of amplitude 1T is imposed onto the devices from the top.

(a) The device is described by Eqs. (4.18), (4.21), and (4.22) with d1=0.2m, α=0.4 and

44

β=105. The green region is air (as β is extremely large in this case, see Eq. (4.22)) and the yellow region reduces to a rectangle infinitely long in the x’ direction (we choose a finite length of 2.8 m in the simulation) and bordered by parallel bounds in the y’ direction. (b)

Two layered isotropic materials (μ1=0.145 and μ2=6.854) are used to realize an effective DC magnetic concentrator.

4.2 Experimental Realizations of the DC Magnetic Concentrator via Space Compression Transformation

In this section, we will show an experimental realization of the DC magnetic concentrator based on the design in Section 4.1. Assuming that α→0 corresponding to an enhancement factor is as high as possible, we can make a diagonalization of Eq. (4.21) by Eqs. (4.19) and (4.20):

2 11+−(a ) β →∞ −−sign( x ') sign ( y ')(a 1) µµ'yellow = 0 a (4.21)  −−sign( x ') sign ( y ')(aa 1)

diagnoliazation 122 11 22 1 (4.24) →µ0diag(α +−+ 1 ( α − 1) + ( − 1) , α +−− 1 ( α − 1) + ( − 1) ) α αα α a →0 2 a → diag( , ); a 2

112µ a →0 θ= arctan xy'' ≈ arctan(a) . (4.25) µµ− 22xx'' yy ''

We can use alternated high-permeability ferromagnetic materials (HPFMs) and high-temperature superconductor materials (HTSMs) to realize the required extremely anisotropic materials in Eq. (4.24). A HPFM has a very high permeability and a HTSM has a very low permeability (nearly zero). The HPFM is an alloy of Fe and Ni with thickness 0.15mm from the Magnetic Shield Corporation. The HTSM is a copper laminated wire (12mm wide and 0.2mm thick) containing ceramic films of yttrium-barium-copper-oxide from the American Superconductor Corporation. The structure of the device in the experiment is shown in Fig. 4.3.

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Figure 4.3: The structure of the device in the experiment. (a) Front view and (b) 3D view: Sketches (left) and photos (right). The HPFM pieces (blue line) with 0.15mm thickness and HTSM pieces (grey line) with 0.2mm thickness are alternately placed. We choose

R1=9.4mm, R2=75mm, and the length of the device is L=70mm along the z’ direction. Ba is the uniform background field. (c) Experimental setup. Liquid nitrogen inside the green tank is used for the refrigeration. The magnetic concentrator is in the center of the Helmholtz coil that is used to generate a uniform background DC magnetic field. A Hall sensor is set inside the concentrator to detect the enhanced DC magnetic field.

46

Although theory provides a suggested rotation angle (see Eq. (4.25)), we have to optimize it due to the finite size of the device along the x’ and z’ direction and the non-ideal permeability of HPFMs and HTSMs (they cannot achieve ideal infinity large or zero in practice). The rotation angle θ has been optimized by the FEM simulation: 5 degrees is chosen to achieve the best enhancement. The FEM simulation results are shown in Fig. 4.4 (a)-(b) and the experimental results are shown in Fig. 4.4 (c). FEM simulation results show that the uniformity of the field is very high in the center air region of our DC magnetic concentrator (see Fig. 4.4(a) and (b)).

We should note that we choose β→∞ and α→0 to simplify the material requirement of the device, which means that the device should be infinitely long along the x’ direction (see Fig. 4.4) and has an infinite enhancement factor in theory. However in experiment, the performance (the enhancement factor and the uniformity) is greatly reduced due to the following reasons: (1) the size of the device is finite along the x’ direction (we cannot achieve infinity in practice); (2) the permeability of HPFM and HTSM are not ideal infinite large or zero; and (3) the device is not infinite along the z’ direction (note that in theory, we design a 2D device which means it should have an infinite length in the z’ direction), hence there are some magnetic flux leakages from the z’ direction.

We have experimentally demonstrated a novel DC magnetic concentrator designed by the space compression transformation. This device can achieve a good enhancement factor (4.74 times) with high uniformity in a relative large air space region (a square with the diagonal length 2R1=18.8mm), which will have many applications in future (e.g. MRI, magnetic sensors, and etc.). Note that the geometrical size of our device is limited by the size of the Helmholtz coils and tank containing liquid nitrogen in this experiment. However the size of our device has no limitation in theory, which means that we can achieve a better enhancement factor in a larger space by designing a larger device if the background field exists in a larger space region. Our research demonstrates a new method for achieving an enhanced DC magnetic field with high uniformity in a relative large air space.

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Figure 4.4: FEM simulation results (a) and (b), and experimental results (c). (a) and (b): the enhancement factor (the ratio of the total field to the background field B/Ba) is shown in the IN and OUT plane (illustrated in the inset in Fig. 4.3(b)). (a) The enhancement factor of 4.81 is achieved in the center region when both HPFM and HTCM work (i.e.

TTc, no refrigeration). (c) Experimental results: the relation between the measured field B at the center of the device and the applied background field Ba for the case at 77K (refrigeration is used: red square) and the case at 300K (no refrigeration: blue circle). The enhancement factor is obtained from the linear fit (dashed lines): 4.74 for 77K (red) and 3.84 for 300K (blue), respectively, which consists well with simulation results.

4.3 Design a DC Magnetic Concentrator via Space Folding Transformation

As we have mentioned in the last section, the DC magnetic concentrator designed by the space compression transformation is often limited by the size of the device in practice; if we want to achieve a higher enhancement, we need a larger device. Sacrificing space to achieve a better enhancement factor is not always feasible in practice. For example, the background field in MRI is produced by an active magnet; the size of the concentrator should be smaller than the magnet and larger than the human body. We can introduce a space folding transformation to tackle this problem. The space folding

48 transformation has been used to design super-scatterers and super-absorbers for electromagnetic waves [40-45]. A small object embedded in some special transformation medium will have a bigger effective scattering or absorption cross-section than the actual size of the whole system. The basic principle of the space folding transformation is shown in Fig. 4.5 (c), and we can write this transformation in the cylindrical coordinate system as:

 R 1 ρρ, '∈ [0,R )  R 1  3

−RR21 RR13− ρ' =+ ρR2 , ρ ' ∈== [RR12 , ]; θθ ' ; z ' z . (4.26)  RR23−− RR 23  ρρ,'(∈∞R ,)  2 

Figure 4.5: The difference between the space compression transformation and the space folding transformation: (a) the space compression transformation, (b) the transition state from the compression to folding transformation, and (c) the space folding transformation.

A circular region with radius R3 in the reference space is compressed or folded into a circular region with radius R1 in the real space; hence the magnetic field in this region is squeezed. The enhancement factor is R3/R1. For the space compression transformation, R3 is smaller than the outside boundary of the device R2; for the space folding transformation, R3 is larger than the outside boundary of the device R2.

By combining Eq. (3.22) with Eq. (4.26), we can obtain the transformation medium in each region:

49

 2 R3  diag(1,1, ),ρ '∈ [0,R1 )  R1  −−  RR13 RR13 ρρ''−−RR222 (4.27)  RR23−ρ ' RR23 −− RR23 µρ'(= diag , , ), '∈ [RR12 , ]. RR−  ρρ''13RR21− ρ '− R2  RR23−  ρ ∈∞  1, ' (R2 , )  

For a 2D device (the DC magnetic field is in the x’-y’ plane, hence only μ’r and μ’θ effect the magnetic field), we can rewrite Eq. (4.27) as:

 1,ρ '∈ [0,R1 )  RR−  ρ '− R 13  2 RR− ρ ' µρ'(= diag 23, ), '∈ [RR , ]. (4.28)  − 12  ρ ' RR13 ρ '− R2  RR23−   1,ρ '∈∞ (R2 , )

The magnetic concentrator is in the region ρ’ ∊ [R1, R2], and the other regions are all air. The key point of the space folding transformation is that the enhancement factor R3/R1 is not only dependent on the geometrical parameter of the device R1, but also on another parameter R3 (we choose R3>R2> R1 and see Fig. 4.5). This is the main advantage of the concentrator based on the space folding transformation: we can use a compact device (i.e. the region of the device R1<ρ’< R2 can be small) to achieve an extremely large enhancement factor (i.e. R3/ R1 is very large) in a relatively large free space region (i.e. R1 is large).

The FEM simulation is used to verify the performance of the proposed device: as shown in Fig. 4.6(a), a uniform background DC magnetic field of amplitude 1T is imposed onto our concentrator described by Eq. (4.28) with

R1=0.2 m, R2=0.3 m, and R3=10 m. The enhancement factor in the center air region (ρ’

50 any other current method, and can reveal a new method to design compact magnetic lenses with high quality (extremely high enhancement factor and high uniformity). The permeability distribution of this device is shown in Fig. 4.6(b) and (c): we should note that it requires negative permeability to realize such a compact concentrator. It is not easy to achieve negative permeability for DC magnetic fields with current MM. A broadband stacked fishnet structure maybe one possible way to realize negative permeability for DC magnetic fields [92].

Figure 4.6: (a) The absolute value of the total magnetic flux distribution simulated by

FEM. (b) and (c) are the permeabilities of the concentrator in the tangential direction μ’θ and in the radial direction μ’ρ, respectively.

51

52

5. Magnetic Lenses which both amplify the Background DC Magnetic Field and the Gradient of the Field

We need a high-gradient DC magnetic field in many applications, e.g., microchip technology for cell separations and manipulations [116-118], magnetic separation [119, 120], drug delivery by nano-magnetic particles [121, 122], etc. People can obtain a high gradient magnetic field simply by using an active magnet, which is an active device and is limited by the heat effect. However if we want to further develop these technologies and applications (e.g., have more accurate control of drug delivery in a deeper location of the human body), a higher magnetic field with higher gradient is needed. We can use some passive magnetic lenses to achieve this. In this chapter, we will use finite embedded transformation (FET) [50] to design a novel DC magnetic lenses that can amplify both the background DC magnetic field and also the gradient of the field. Hence (BB⋅∇) , which the magnetic force is proportional to, is amplified passively. This novel magnetic lens will improve the recent technologies based on the gradient magnetic force [116-122]. 5.1 Theoretical Method to Design a DC Magnetic Lens via Finite Embedded Transformation

The transformations in previous chapters are continuous transformations, and thus the properties of the incident background DC magnetic field are changed only inside an enclosed region and does not change outside the whole device (i.e. the whole devices are invisible to observers outside of the transformed region due to the identical transformation on the region outside the whole device). FET is different from traditional continuous transformations, which is often continuous at the input surface of the device and discontinuous at the output surface of the device. Some reflections can be produced due to the discontinuous transformations at the output surface. The main advantage of using FET is that FET can produce a desired field outside the whole device (e.g. a focusing effect)

53

Now we use FET to design a compressor/lens that can compress the incident DC magnetic field and obtain an enhanced DC magnetic field with high gradient at the output surface of the device. We use (x’, y’, z’) and (x, y, z) to express the coordinate systems in the real space and the reference space, respectively. The reference space here is free space. We assume the input surface of the device is at x’=0 and the output surface of the device is at x’=d. The coordinate transformation can be given by:

xx' =   ((M− 1) xd / += 1) y :α y , x ∈ [0, d ] y '= , (5.1)   yx,∈ ( −∞ ,0) ∪ ( d , ∞ ) zz' = where d is the thickness of the device (the device is in region x’ ∊ [0, d]) and M (0

(M −1) 1/α y ' 0 dα 2 (MM−−11) ( ) µµ= 2 +α (5.2) ' 0 yy'2 ' 0. ddαα 0 0 1/α  

For simplicity we only consider a 2D device: only μ’xx, μ’xy and μ’yy affect the magnetic field. We can rewrite Eq. (5.2) as:

54

(M −1) 1/α y ' dα µµ''xx xy µµ'= :.= (5.3) 0 2 µµ (MM−−11) ( ) ''xy yy yy''2 +α ddαα2

Note that the transformation medium in Eq. (5.3) is symmetric, which means that we can diagonalize it (just as we have done in Section 4.1). By a coordinate rotation, we can rewrite Eq. (5.3) as:

µµ''++ µµ '' µ'(=diag xx yy −κ, xx yy += κ) :diag ( µµ ' , ' ), (5.4) 22 12 where

µ'22+− µ ' 2' µµ ' + 4' µ2 κ = xx xy xx yy xy . (5.5) 2

The FEM simulations have been provided to verify the performance of our device (see Fig. 5.1), and the incident DC magnetic field is gradually compressed as it goes through the device. We have a uniform background DC magnetic field Bb=1 T at the input surface of the device (left side of the device), and a concentrated DC magnetic field with high gradient at the output surface of the device (right side of the device). The higher the value for M, the better the concentration of the DC magnetic field at the output surface will be (see Fig. (a), (d), and (g)). Note that the permeability of the device is always positive, which can be constructed with current DC MMs (e.g. layered diamagnetic substances and paramagnetic materials). If we want a larger degree of the field concentration, M >0 should be as small as possible.

However, in this case (M approaches zero) the smallest value of μ’1 approaches zero and the largest value of μ’2 approaches infinity, which means that the device is more difficult to realize.

An inhomogeneous and highly anisotropic medium is required to realize such a DC magnetic compressor (see Eq. (5.2)), which means that it is hard to

55 realize it experimentally. In the next section, we will introduce the idea of transformation inside a null-space medium to solve this problem and design a novel DC magnetic compressor with better performance and simpler material requirement.

Figure 5.1: 2D FEM simulation results. (a), (d), and (g): the distribution of the absolute value of the total magnetic flux density (a uniform background static magnetic field Bb=1 T is imposed onto the device from the left). The black rectangle shows the profile of the

56 compressor described by Eq. (5.2), and all the other regions are free space. The height of the device is 0.8 m and the thickness of the device is d=0.5 m. We choose M=0.1 (a), M=0.35 (d) and M=0.01 (g). The distribution of the principal values of the permeability

(μ’1 and μ’2) inside the whole compressor: (b) and (c) for M=0.1; (e) and (f) for M=0.35; (h) and (i) for M=0.01.

5.2 Parameters Reduction by Introducing Null-space Medium

In Section 5.1, we choose the free space as the reference space, which leads to the highly anisotropic inhomogeneous permeability inside our compressor (see Eq. (5.2)). To solve this problem, we now use a slab with thickness d filled with the material (called the “null-space” medium) embedded in the free space as the reference space. Next we make a space compression FET again onto the reference space to achieve a novel DC magnetic compressor, with better performance and lower requirement on materials than our previous design.

The null space medium is a special transformation medium that transforms a surface to a volume. In a Cartesian coordinate system, such a null-space medium can perform like a special waveguide for the DC magnetic field (called magnetic hose [103]), and can guide the magnetic flux to an arbitrary distance without loss. In a cylindrical coordinate system, such a null-space medium can be used as a hyper-lens with magnification for super-resolution imaging [123]. We will first design a null-space slab using TO: the reference space is the free space and the coordinate transformation can be given as:

 xx,0≤  x'= xd / ∆ ,0 ≤ x ≤∆ ;' y = y ,' z = z . (5.6)  x−∆+ dx, >∆

Note that here we use (x’, y’, z’) and (x, y, z) as the coordinate systems in the real space and the reference space, respectively. The transformation medium can be calculated with the help of TO (Eq. (3.22)):

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 d ∆∆when ∆→0 diag( , , )→∞diag ( ,0,0), xd'∈ [0, ] µ '.=  ∆ dd (5.7) xd'∈ ( −∞ ,0) ∪ ( , ∞ )  1,

Note that the slab region 0

xx''= '    yx' , ''< 0  y''= fxy ( ', ') , 0 ≤≤ x '' d , (5.8) y' , xd '' >   zz''= ' where f (x’, y’) can be an arbitrary continuous function satisfying the boundary condition that f (0, y’)=y’ and f (d, y’)=My’ to ensure a space compression transformation with 0 d are free space as we use an identical transformation in these regions. The compressor is filled in the region 0

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 10f x' 2 d 2 2 ∆ µ '' = ffxx''(( ) + ( f y ') ) 0 , (5.9) ∆fdy'  2 ∆ 00 d where

∂∂ff ff:= ,: = . (5.10) xy''∂∂xy''

For a 2D case, the DC magnetic field is within the plane x’’-y’’: we can rewrite Eq. (5.9) as:

1 fx' d 2 µ '' = 2 2 ∆ . (5.11) ∆  f y' ffxx''()( ) + ( f y ')  d

Good performance of the proposed device is verified by FEM in Fig. 5.2. When other parameters are the same, a smaller M results in a larger compression (see Fig. 5.2 (c) and (d)). In terms of TO, this can be easily understood: considering the transformation relationship y’’=My’ at the output surface of the device, if other parameters are kept the same, a smaller M means a stronger compression.

If we don’t introduce the null-space slab as the reference space (∆=d, and see Fig. 5.2 (a)), the effect of DC field compression is lower than the case where the null-space slab is introduced (∆<

59 thus means a faster compression (under the same compression degree M in the vertical direction) along the horizontal direction. A faster compression inside the device leads to achieve a higher gradient after the compressor. A faster compression also leads to a better enhanced DC magnetic field at the output surface of our compressor: the free space’s diverging effect on the DC magnetic flux is the same once the magnetic flux passes through the compressor; more rapid compression inside the compression will lead to a better convergence effect on the magnetic flux at the output surface of the device. When ∆ becomes smaller, the converging effect is enhanced and the diverging effect is still the same, and hence the effect of enhancement is stronger.

Figure 5.2: The absolute value of the total magnetic flux density distribution. A uniform background DC magnetic field with amplitude 1T is imposed from the left onto our magnetic compressor with height H=20 cm, thickness d=10 cm. A linear transformation

60 function f (x’, y’)=[(M-1)x’/d+1]y’ is chosen. The maximum enhancement degree in the center region after the compressor is 4.8, 7.6, 7.9 and 15.5 for (a), (b), (c) and (d) respectively. M=0.1, 0.1, 0.1, and 0.05 from (a) to (d), and ∆=10cm, 1cm, 0.1cm, and 0.1cm from (a) to (d), respectively. The white regions indicate where the magnetic field is beyond the color bar. When ∆ decreases from d to 0 while the other parameters remain constant, which means that the reference space is changing from a free space to a null- space slab inserted in a free space, the concentration effect increases (comparing the results of Fig. 3(a), (b) and (c)).

Figure 5.3: FEM simulation results. (a), (c), (e) and (g) show the amplitude of the gradient distribution along the x’’ direction, and correspond to the device in Fig. 5.2 (a), (b), (c) and (d), respectively. Arrows show the location of the back surface of the device. (b), (d), (f), and (h) show the amplitude of the gradient distribution along the y’’ direction 5mm behind the back surface of the device, which correspond to the device in Fig. 5.2(a), (b), (c), and (d), respectively. Arrows show the location of the center symmetrical axis of the device. The x’’ and y’’ directions are shown in Fig. 5.2.

If we choose Δ<

61

d 1 fx' µ '' ≈ 2 . (5.12) ∆f y' ffxx''

We can also diagonalize this symmetrical tensor:

coordinate system rotation ∆→0 dd1 fx' 1 2 m '' ≈2 → diag( (1+ fx' ),0) →∞ diag ( ,0). (5.13) ∆∆ffyy''ffxx''

As shown in Eq. (5.13), if Δ approaches zero, we can realize this DC magnetic compressor simply by layered superconductors (μ→0) and ferromagnetic materials (μ→∞) [106, 110]. The orientation of these layers is indicated by the principal axis angle, which can be determined with Eq. (4.20):

2µ '' 2 f ∂∂fxy( ', ') y ' tan(2θθ ) =xy =x' ⇒==tan f =. (5.14) µµ− −2 x' ∂∂ ''xx '' yy 1 fx' xx''

Note that here θ is the principal axis angle of the anisotropic medium but not the coordinate variable in a cylindrical coordinate system. Eq. (5.14) reveals that θ also corresponds to the slope of the compression function f (x’, y’). Considering that the shape of function f (x’, y’) can be arbitrary provided that it is a continuous function and satisfies the boundary condition (f (0, y’)=y’ and f (d, y’)=My’), the slope of this function (and also the orientation of layered media θ) can be arbitrary provided that it changes continuously.

From the above analysis, we can greatly simplify the material requirement of our DC magnetic compressor, provided that Δ approaches zero. We can simply utilize layered magnetic materials (one type with very high relative permeability, e.g., ferromagnetic materials, and the other type with very low relative permeability, e.g., super-conductors) to realize our compressor. The boundaries of this layered medium, whose slopes correspond to the slope of the compression function f (x, y), can be arbitrary provided that it changes continuously and satisfies the boundary condition yB=MyA with 0

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Figure 5.4: (a) and (b) show the absolute value of the total magnetic flux distribution when a uniform background DC magnetic field with amplitude 1 T is imposed onto our reduced compressor from the left. (a) A 2D reduced compressor. (b) A 3D reduced compressor. The insert provides the 3D geometrical shape of this funneled compressor. The maximum field enhancements are about 9 and 14.1 times in (a) and (b), respectively. (c) The experimental sample of this magnetic compressor (composed by layered HPFMs and HTSMs).

The FEM simulation has been utilized to verify the performance of our reduced compressor (see Fig. 5.4). A 2D reduced compressor with height H=20 cm and thickness d=10 cm, composed of 10 superconductor layers with

63 relative permeability μ=10-6 and 10 ferromagnetic material layers with relative permeability μ=106, shows a very good DC magnetic field compression effect (see Fig. 5.4(a)). A 3D compressor can be easily constructed by rotating a 2D one along its symmetrical axis (x direction). We also simulate a 3D reduced compressor (a 3D funnel structure), which also shows a very good field compression effect (see Fig. 5.4(b)).

The experiment of this funnel magnetic compressor is in progress during the writing of this thesis see Fig. 5.4(c) for the experimental realization of the proposed compressor. Experimental measurement results will be reported in upcoming articles.

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6. Conclusion and Future Work

In this thesis, we have extended transformation optics to the DC magnetic field case and designed many novel devices for the DC magnetic field. Based on differential geometrical theory and Maxwell’s equations, we derive the basic formulae for transformation optics. We extend transformation optics to the case in which there are magnets in the reference space, and design some special devices for transforming magnets (e.g. rescaling magnets and cancelling magnets).

We propose two different DC magnetic concentrators based on the space compression transformation and the space folding transformation, which can both amplify the background DC magnetic field to achieve an enhanced field with high uniformity. The concentrator based on the space compression transformation is not complex, and has been verified by both numerical simulation and experiment. The advantages of this concentrator are that it is easily made, has high uniformity and the enhancement factor can be tuned by the geometrical size of the device. The concentrator based on the space folding transformation can provide a very good field enhancement by a compact device. The advantages of this concentrator include: a very high enhancement factor (not limited by its size), compactness, and the property that the enhanced DC magnetic field can be in a large free space region with high uniformity. The performance of this concentrator has been verified by numerical simulations.

We have also proposed a DC magnetic field compressor, which can amplify both the background field and the gradient of the incident field. We have introduced the idea of transforming inside a null-space medium to simplify the material requirement of such a compressor. We have designed a funnel structure composed of layered ferromagnetic materials and super-conductors to realize it. Numerical simulations have shown the good performance of the proposed compressor. The advantages of this compressor include: (1) it can passively achieve a higher degree of field enhancement; (2) it can passively

65 achieve a higher field gradient; and (3) the requirement of the materials’ permeability is very simple. The function of this novel DC magnetic compressor is as a funnel that concentrates DC magnetic flux, which may present many potential practical applications in the future (e.g. cell manipulations by gradient magnetic field, DC magnetic field focusing, high- gradient magnetic separation, high magnetic force generation, drug delivery by magnetic nano-particles in deeper tissues, etc.).

There is still much follow-up work to do in the future: (1) the study of how to achieve negative permeability with the current material for DC magnetic fields. We need to find a new way to extend the meta-material with negative permeability from the electromagnetic wave to the DC magnetic field. Applications for such a work include cancelling magnets and a concentrator based on the space folding transformation; (2) to experimentally demonstrate a device that can rescale the magnet and the 3D funnel compressor for amplifying the gradient of the field; and (3) further novel devices by TO is also a long-term goal.

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Sammandrag I den här avhandlingen har vi utökat transformationsoptiken till att inkludera statiska magnetiska fält och designat många nya komponenter som baseras på statiska magnetfält. Utifrån teorin för differentialgeometri och Maxwells ekvationer, härleder vi de grundläggande uttrycken för transformationsoptik. Vi utvidgar transformationsoptiken till fallet när det finns magneter i referensrummet och designar några särskilda metoder för att omvandla magneter (t ex skalning av magneter eller upphävande av deras inverkan).

Vi föreslår två olika magnetostatiska koncentratorer baserade på en rumskomprimerande transformation samt en rumsvikande transformation, vilka båda kan förstärka ett magnetostatiskt bakgrundsfält för att åstadkomma ett förbättrat fält av god homogenitet.

Koncentratorn baserad på rumskomprimering kan göras enkel, vilket har verifierats genom både numeriska simuleringar och experiment. Fördelarna med denna koncentrator är bl a: lätt att uppnå hög grad av homogenitet och förstärkningsfaktorn kan stämmas av utifrån den geometriska storleken på koncentratorn. Koncentratorn baserad på den rumsvikande transformationen kan göras kompakt. Fördelarna med denna koncentrator är: mycket hög förstärkningsfaktor (inte begränsad av sin storlek), kompakt utförande, förstärkt magnetostatiskt fält med god homogenitet i ett stort område utanför magneten. Designen av denna koncentrator har verifierats genom numeriska simuleringar.

Vi föreslår också en magnetostatisk kompressor, som kan både förstärka bakgrundsfältet och dess gradient. Vi introducerar idén att transformera inuti en nollrumsregion, för att förenkla de materiella kraven. Vi designar en trattstruktur består av skiktade ferromagnetiska material och supraledare, för att förverkliga en sådan kompressor. Numeriska simuleringar uppvisar goda resultat för den föreslagna kompressorn. Fördelarna med denna passiva kompressor är: 1) den kan uppnå en högre grad av fältförstärkning 2) den kan uppnå en högre fältgradient 3) den ställer inga höga krav på materialets permeabilitet. Funktionen hos denna nya magnetostatiska kompressor är som

67 en tratt som koncentrerar det magnetiska flödet. Den kan leda till många potentiella praktiska tillämpningar i framtiden t ex manipulera celler genom magnetfältsgradienter, magnetostatisk fältfokusering, starka fältgradienter för magnetisk separation, skapa starka magnetiska krafter, injicera medicin i djupt belägen vävnad genom transport av magnetiska nanopartiklar, etc.

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