Fin 2006 apparaissait dans la revue Science la première cape d'invisibilité électromagnétique. Réalisée par l'équipe de D. R. Smith à l'université de Duke aux Etats-Unis et conçue théoriquement par J. B. Pendry ce nouveau dispositif électromagnétique éveilla l'intérêt de la communauté scientifique. Quelques mois auparavant c'était une innovante et fascinante technique mathématique qui avait été dévoilée sous le nom « optique de transformation » ou plus communément « transformation d'espace ». Cet outil extrêmement puissant, basé sur la réinterprétation des équations de Maxwell, offre un énorme potentiel pour la conception de dispositifs aux propriétés extraordinaires. De plus, le couplage avec l'ingénierie des résonateurs à métamatériaux permet la réalisation des milieux générés par ces transformations. En effet, des recherches récentes ont établi la possibilité de fabriquer des structures dans les domaines microondes et optiue, permettant de contrôler la lumière d'une manière inconcevable avec les matériaux naturels. A partir d'éléments sub-longueur d'onde que sont les résonateurs à métamatériaux, il est possible d'obtenir les distributions spatiales bien spécifiques de la permittivité  et de la perméabilité , offrant ainsi le potentiel pour guider et contrôler le flux de l'énergie par une ingénierie d'espace. Ces métamatériaux ont ouvert la porte à de nombreuses applications qui, jusque-là, avaient été considérées comme impossibles. Ceux-ci permettent en effet de s'abstraire des réponses électromagnétiques des matériaux naturels ainsi que de leur composition chimique, grâce au contrôle de la forme et des dimensions de leurs cellules unités, ces dernières pouvant être vus comme des atomes artificiels homogènes puisque leurs dimensions sont beaucoup plus petites que la longueur d'onde. L'apparition de la méthode de la transformation d'espace en 2006 a donc apporté une nouvelle vision sur les fondations de l'optique. Analogue à la relativité générale où temps et espace ne font plus qu'un et sont modifiés par l'énergie et la masse, la transformation d'espace montre qu'il -vi-particularly in the technical realization of several prototypes. Moreover, I also want to thank my colleagues, Alexandre Sellier, Natalia Dubrovina, Quynh Le Van, Simon Marcellin, Anna Niang, Yulong Fan, Eleana Bochkova and Badreddine Ratni for their concern and accompany.

Finally, I would like to

-ii-est possible de déformer l'espace à volonté afin d'imposer la trajectoire des rayons lumineux.

Cette ingénierie d'espace fournit un contrôle du flux lumineux jusqu'à une précision nanométrique. Ainsi, la relativité générale trouve une application dans de nouveaux systèmes optiques basés sur la transformation d'espace où la lumière est guidée par les métamatériaux d'une manière contrôlée et prédéfinie. Ce point de vue n'est pas totalement nouveau puisque déjà dans les années 1920 W. Gordon avait pu remarquer que des milieux isotropes en mouvement pouvaient apparaitre comme une géométrie d'espace-temps pour le champ électromagnétique.

L'objectif de cette thèse est donc de comprendre la théorie de l'optique de transformation et d'appliquer ce concept pour concevoir de nouveaux systèmes électromagnétiques liés aux antennes. J'expose dans un premier temps le principe de l'optique de transformation ainsi que les bases et concepts mathématiques nécessaires à l'utilisation d'un tel outil. Dans cette thèse, je me consacre donc au concept de l'optique de transformation qui permet de contrôler le trajet des ondes électromagnétiques à volonté en appliquant une variation spatiale judicieusement définie dans les paramètres constitutifs. Il est exploré pour concevoir des nouveaux types d'antennes et de dispositifs dans le domaine des micro-ondes.

Deux méthodes sont utilisées pour concevoir les dispositifs : la transformée de coordonnées et la transformation spatiale. La transformation de coordonnées est faite en redéfinissant les paramètres électromagnétiques. Suivant cette approche, les équations de Maxwell sont écrites de telle façon qu'elles sont invariantes dans les transformations. En fait les transformations de coordonnées fournissent les tenseurs de perméabilité et permittivité appropriés dus à l'invariance des équations de Maxwell dans ces transformations, ce qui rend possible le contrôle de la lumière à toute échelle, du macroscopique jusqu'au microscopique. En créant les distributions de permittivité et perméabilité, l'espace peut être déformé à volonté, rendant ainsi possible par exemple la création de trajectoire "fantaisistes" avec des indices pouvant être négatifs.

Il en résulte ainsi la conception de deux systèmes électromagnétiques proposés dans cette thèse ; -iii-un dispositif d'illusion et un adaptateur de guide d'ondes. Le dispositif d'illusion capable de modifier l'apparence d'une émission électromagnétique et de la délocaliser, est validé par le biais de simulations numériques. En effet, en comprimant l'espace contenant un élément rayonnant, nous montrons qu'il est capable de modifier son diagramme de rayonnement et aussi de faire en sorte que la source d'émission apparaisse à l'extérieur de cet espace. Nous décrivons la transformation de l'espace métrique et le calcul des paramètres du matériau. Des modèles continus et discrets avec les valeurs des paramètres électromagnétiques calculés sont proposés.

Une réduction des paramètres électromagnétiques suivant une polarisation bien fixe pour la source est également proposée afin de penser à une possible fabrication du dispositif en utilisant des valeurs réalisables de la permittivité et la perméabilité de métamatériaux existants.

Nous concevons par la suite un adaptateur entre deux guides d'ondes de sections différentes.

Après une étude de plusieurs techniques de transformation différentes, nous en proposons une basé sur un système de double adaptateur afin d'assurer une transmission entre deux guides avec un rapport de 11,4 entre les sections. Les milieux provenant de cette méthode présentent une perméabilité et une permittivité anisotrope. Cependant, nous montrons qu'une transformation de type 'inverse proportionnelle' conduit à la conception d'un système adaptateur à partir d'un matériau avec les paramètres du matériau physiquement réalisables. Les paramètres complexes sont simplifiés et discrétisés pour une réalisation du dispositif. Le prototype fabriqué à partir de résonateurs à métamatériaux électriques et magnétiques de cet adaptateur de guide d'ondes est ensuite présenté. Une validation du concept est réalisée en comparant les distributions de champs simulé et mesuré.

Dans une seconde partie, la transformation spatiale basée sur l'équation de Laplace et quasiconforme est étudiée pour concevoir des lentilles large bande pour les applications antennaires.

Cette méthode de conception fait appel à un ensemble de conditions aux limites qui définissent l'effet du dispositif, tout en limitant la transformation à l'intérieur du matériau. En particulier, les angles entre les lignes de coordonnées sont quasiment conservés et il y a une anisotropie limitée -iv-des coordonnées « carrées », autrement dit, elles sont toujours sensiblement carrées, plutôt que rectangulaires. Contrairement à la transformation de coordonnées, cette technique permet au module conforme des deux domaines d'être différent que dans une certaine mesure, ce qui constitue une caractéristique cruciale pour une conception fonctionnant sur une large bande de fréquence et donc à partir de matériaux non résonants. Pour une transformation en deux dimensions, cela conduit à la prescription d'un matériau d'anisotropie très limitée qui peut être bien approximée par une réponse diélectrique uniquement.

Cette transformation spatiale quasi-conforme est donc utilisée pour concevoir deux lentilles ; une lentille de focalisation capable de restaurer les émissions en phase d'un réseau conforme d'éléments rayonnants et une lentille à dépointage qui permet de dévier la direction du faisceau rayonné d'une antenne. La lentille de focalisation est appliquée à un réseau d'éléments rayonnants conformé sur une surface cylindrique. Elle permet de compenser les déphasages introduits par la localisation des éléments sur la surface cylindrique et ainsi annuler l'effet de défocalisation pour produire une émission en phase. La deuxième lentille est conçue pour produire un dépointage de faisceau à partir d'une source avec un rayonnement dans la direction normale au plan de la source. Ces deux lentilles sont réalisées à partir d'un matériau isotrope tout-diélectrique grâce à la technologie d'impression en trois dimensions (3D). Les prototypes permettent de valider expérimentalement la fonctionnalité des lentilles sur une large bande de fréquence.

Les travaux présentés dans cette thèse montrent donc toute la richesse du concept de l'optique de transformation pour concevoir de nouveaux systèmes électromagnétiques.

ACKNOWLEDGEMENTS

As is often the case of overseas students, I have found the experience during my thesis fascinating but full of challenges at the same time. Today when I finish this thesis and retrospect, I find that all the challenges came together with rewards. During these three years, I appreciate and honor all the people that have participated in my life and offered help generously. I would like to spend my ink on the words of gratitude for them.

I would like to begin with thanking my thesis supervisor, Prof. André de Lustrac. Prof. De Lustrac welcomed me into his group when I was at the crossroad of working or continuing my academic career. The subject of the thesis he offered to me is brand new and interesting to me.

He has shown me nothing but patience and encouragement even at the lows of my research work.

I want to send my appreciation to my co-supervisor Prof. Shah Nawaz Burokur for his guidance.

From project modelling to experimental setups, even on the manuscription of scientific papers, his generous tutoring has permeated into all the details in my work. He also helped me out of difficulties in personal life, such as accommodation problems and so on. He has made my work and life here in Paris filled of warmth and fun. I would like to give my great thanks to Dr. Paul-Henri Tichit. My thesis started by continuing the coordinate transformation based designs of Dr. Tichit's thesis. Dr. Tichit has taught me everything he could give on this subject. As to a freshman in face of transformation optics, the knowledge he imparted to me make the rudiments of the subject much easier to me.

In addition I need also express my gratitude to the opportunity for discourse and collaboration with many colleagues. Dr. Anotole Lupu, Dr. Aloyse Degiron and Dr. Tatiana Teperik all have offered me help in physic problems and give me instructions and ideas. I also thank Dr. G.P.

Piau from AIRBUS Innovation Works for his scientific support along this thesis, and proposed. An all-dielectric prototype is realized by 3D printing and validated experimentally by near-field and far-field measurements. The radiation emitted by the conformal lens antenna system is consistent with that of a planar array of radiators. In Chapter 5, the design procedure of a lens antenna, which steers the radiated beam of an array of similarly fed patch antennas, is presented. Two prototypes are fabricated for this device, one by using electric LC metamaterial resonators and the other by all-dielectric 3D printing. The dielectric prototype allows validating the beam steering functionality over a broad frequency range spanning from 8 to 12 GHz.

Finally, we conclude on the two different design concepts and validation technologies. The principle of metamaterials relies on sub-wavelength structures engineering. By this engineering the metamaterial has the functionality as electric and magnetic particles that behave in the same fashion as atoms and molecules of continuous media. It is found out that for metamaterials with certain particle structures, such as resonant objects [3,4] or cut wires [5,6],

the effective  and  of the equivalent medium can achieve both positive and negative values across certain frequency bands. Although the concept of negative  has been accepted and understood for a long time, negative permeability is a less familiar phenomenon typically not found generally in natural materials, except for ferrite materials.

In naturally occurring materials with positive index of refraction, solutions of the Maxwell Equations yield propagating electromagnetic waves in which the phase and energy velocities are in the same direction. To explain why this is true, imagine a uniform plane wave propagating in an unbounded medium so that no boundary conditions are applied to the wave. Taking the direction of propagation of this plane wave with angular frequency  to be in the +z direction, we can express the electric and magnetic fields as

x e E t z E kz t j ) , ( ) ( 0    (1-1) y e E t z H kz t j ) , ( ) ( 0     (1-2)
where the spatial wave number k = kr + jki is in general complex, with ki contributing to attenuation along the direction of propagation. The power flow in such a medium is given by the time averaged Poynting vector (1-3)

As presented in Figure 1-2, in most natural materials,  and  are positive, and this forces kz to assume a continuum of real values. Thus, S maintains a non-zero value, and energy is free to propagate in the +z direction.

Not classically considered is the case when both r < 0 and r < 0. When both  and  are simultaneously negative, the material will have a refractive index n < 0. The wave number becomes negative, while the power flow S flows in the +z direction. Thus, the power flow becomes anti-parallel with the phase velocity, and because of this property, negative index materials are referred to as "left-handed" materials since a simultaneously negative  and  makes the vector triplet [k, E, H] left handed. 'Left-Handed' materials are so-called because Figure 1-3 shows the difference when an incident ray propagates into a positive index and a negative index. In a positive index medium, waves with a larger wave-vector than in the free space are evanescent: rather than oscillating in free space, they decay exponentially. In a negative index medium, however, the amplitudes of these evanescent waves grow exponentially-allowing for the transmission of information about features on a much finer scale than is available using conventional optical elements.

A further significant advantage of the manual patterning of a material is that it can be made to be inhomogeneous-the properties of the material can change as a function of position. This allows for the curving of light ray trajectories throughout the bulk of a material and can lead to a significantly increased phase space in which to design optical devices.

Lots of applications based on metamaterials have been proposed since J. Pendry introduced Split Ring Resonators (SRR) and wire structures to realize respectively negative permeability and permittivity medium (ref). Then they were used to create the first negative index material [8]. In my thesis, metamaterials are implemented in microwave device applications, designed by transformation optics. So here we present several metamaterial applications that we consider interesting for the thesis and for the antenna domain. In Figure 1234, a low index metamaterial for directive emission is introduced [9]. This material has an index lower than 1. It's shown that the specific properties of this metallic composite Section 1.1: State of art on metamaterials and metamaterial-based devices -8-material can strongly modify the emission of an embedded source. In this application, under proper conditions the energy radiated by a source embedded in a slab of metamaterial will be concentrated in a narrow cone in the surrounding media. In Figure 12345D. Werner together with colleagues from the Pennsylvania State University and Lockheed Martin Commercial Space Systems have succeeded enabling specific device performance over usable bandwidths by employing dispersion engineering of metamaterial properties [10]. In particular, they have designed metamaterials that considerably improve performances of conventional horn antennas over greater than an octave bandwidth with negligible loss and advance the state of the art in the process.

A new type of compact flexible anisotropic metamaterial (MM) coating is proposed in Figure 1-6, which greatly enhances the impedance bandwidth of a quarter-wave monopole to over an octave [11]. The MM coating has a high effective permittivity for the tensor component oriented along the direction of the monopole. By properly choosing the radius and tensor parameter of the MM coating, another resonance at a higher frequency can be efficiently excited without affecting the fundamental mode of the monopole. Additionally, the similar current distributions on the monopole at both resonances make stable radiation patterns possible over the entire band. T. Driscoll and colleagues have presented a radial gradient-index lens with an index of refraction ranging from -2.67 (edge) to -0.97 (center), shown in Figure 123456. They showed that the lens can produce field intensities at the focus that are greater than that of the incident plane wave.

A biplanar gradient lens with negative gradient is constructed which can produce field intensities at the focus that are greater than that of the incident plane wave. by T. Driscoll et al. [12] as shown in Figure 1234567.

In the CRIME group at IEF, other studies related to metamaterial surfaces for Fabry-Perot type directive antennas and lenses are conducted [13][14][15][16][17][18].

These applications are either based on the antenna impedance matching or the dispersion of the MMs. In my thesis, we focus on the application of metamaterials for transformation optics -10-designs. The concept of transformation optics (TO) will be discussed in the next section.

Transformation optics

Transformation optics (TO) is one of the most powerful conceptual techniques for the design of devices which leverage complex materials. TO enables to alter the field distribution by choosing material parameters which cause light to 'behave' as if it was in a transformed set of coordinates [18,19] and novel applications have been explored rapidly and quantitively, including: In order to provide the reader with the background information necessary to understand the rest of the dissertation, this section would be a full introduction to different kinds of transformations.

Transformation optics relies on what is termed the 'form-invariance' of Maxwell's equations [START_REF] Post | Formal Structure of Electromagnetics[END_REF], which means that transformation optics provide a method to modify the field by changing the variables in the Maxwell's equations. This chapter begins with a derivation of this property beginning with the familiar form of Maxwell's equations. And then it is followed by a description of the consistent manner in which transformation optical calculations may be performed for arbitrary geometries. Transformations based on Laplace's equations will also be introduced. Some applications will act as examples to help understanding the principles.

Coordinate transformations

In this section, the transformation from Cartesian coordinates to other orthogonal coordinate systems is introduced by the statement of the media equations. Several applications of such coordinate transformations will also be introduced later. We start from the Maxwell's equations to explain the principles and then focus to write out the material parameters in a very explicit way for future studies by means of numerical finite element methods. Coordinate transformation is achieved by redefining the constituent material parameters. The Maxwell's equations are written such that the differential form of the equations is invariant under coordinate transforms:

. 0 , , ,

                  H E t H E J t E H     
(1-4)

These equations are rewritten in covariant notation in Cartesian coordinates. The invariance form is completely a property of the differential equations. The covariant notation is used for compactness.

. 0

,               j ij i j ij i j ij k j ijk i j ij k j ijk H E t H E J t E H        (1-5)
i, j and k are the indices that take value from 1 to 3. We note that a particular coordinate is identified by x i , for example (x, y, z). And i

 is short for i x   /
. The completely antisymmetric Levi-Cevita tensor is expressed by ijk  .

We consider that the properties of these equations are under a coordinate transform between x and x i ', which is the transformed coordinate (x', y', z'). E and H are transformed as

' ' ' i i i i i i i i E A E E A E   (1-6)
where

' i i
A is the Jacobian matrix and is given by

i i i i x x A    ' ' (1-7)
So after several steps of derivations and simplifications, the transformed equations can be written as:

. 0 , ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '               j ij j j i i i j ij j j i i i j ij j j i i k j k j i i i i j ij j j i i k j k j i H A A A A E A A A t H A A A E J A A t E A A A H        (1-8)
When we compare the Maxwell's equation before and after a coordinate transformation, we can find that there exist some transformation rules of the material parameters, current and charge that will make the transformed equations to be in the identical form of the equations before transformation. With a proper choice of materials and sources, one can arrive at an identical set of equations to those which are found under an arbitrary coordinate transformation. In another word, if we choose a certain material whose properties are as below:

A J A A J A A A A A A i i i ij j j i i j i ij j j i i j i           ' ' ' ' ' ' ' ' ' ' ' (1-9)
Then the transformed Maxwell's equations can be rewritten as

              j j i i j j i i j j i k j k j i i j j i k j k j i H E t H E J t E H        (1-10)
Now we continue with the transformation between two Cartesian coordinate systems. For the transformation between any two kinds of orthogonal coordinate systems, such as between cylindrical coordinate system and spherical coordinate system and etc., we define the original coordinate system as (u, v, w) and the transformed coordinate system as (u', v', w'), to distinguish with the typical Cartesian coordinate expression (x, y, z). The mapping

) det( / ) det( / ' ' ' ' ' ' ' ' ' ' i i ij C j j i i j i C i i ij C j j i i j i C A A A A A A       (1-11)
) ' , ' , ' ( ) , , ( w v u w v u  can be written as         ) , , ( ' ' ) , , ( ' ' ) , , ( ' ' w v u w w w v u v v w v u u u (1-12)
The transformation media equations in the orthogonal coordinate can be expressed as

) det( / ) det( / ' ' ' ' ' ' ' ' ' ' k k kl O l l k k l k O k k kl O l l k k l k O T T T T T T       (1-13) with ) /( ) ( ' ' ' k u k u k k u h u h T k k    . k u
h is defined in the line element expression in the orthogonal coordinate in the original space. In Cartesian coordinate, the line element is the original space is

² ² ² ² dz dy dx ds   
. While in the orthogonal coordinate in the original space, the line element is

² ² ² ² ² ² ² )² , , ( ² )² , , ( ² ) , , ( ² 2 dw h dv h du h dw w v u h dv w v u h du w v u h ds w v u w v u       (1-

Cartesian coordinate Cylindrical coordinate Spherical coordinate

² ² ² ² dz dy dx ds    ² ² ² ² ² dz d d ds       ² ² sin ² ² ² ² ²    d r d r dr ds            1 1 1 w v u h h h         1 1 w v u h h h           sin 1 r h r h h w v u
So the Jacobian transformation matrix in the orthogonal coordinate is

                                     w w h h v w h h u w h h w v h h v v h h u v h h w u h h v u h h u u h h T w w v w u w w v v v u v w u v u u u ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' (1-15)
For example, if the transformation takes place between two cylindrical coordinate systems

) ' , ' , ' ( ) , , ( z z     
, the Jacobian transformation matrix will take the form:

                                     z z z z z z T ' ' 1 ' ' ' ' ' ' ' ' ' 1 '                   (1-16)
Normally, the parameters we obtain above will be assigned to the transformed media as properties and simulated by the numerical finite element methods, for instance the COMSOL Multiphysics finite element-based electromagnetics solver. So the required parameters should be expressed in the form of

) ' , ' , ' ( z y x . ) det( / ) det( / ' ' ' ' ' ' ' ' R R R R R R T l k O j i C T l k O j i C       (1-17)
where the rotation matrix R is given as

                                     ' ' 1 ' ' 1 ' ' 1 ' ' 1 ' ' 1 ' ' 1 ' ' 1 ' ' 1 ' ' 1 ' ' ' ' ' ' ' ' ' w z h v z h u z h w y h v y h u y h w x h v x h u x h R w v u w v u w v u (1-18)
In cylindrical and spherical coordinate systems, the expressions of matrix R are presented in

Cylindrical coordinate

Spherical coordinate

            1 0 0 0 ' cos ' sin 0 ' sin ' cos     R               0 ' sin ' cos ' cos ' sin ' cos ' sin ' sin ' sin ' cos ' cos ' cos ' sin            

R

Coordinate transformation has been applied to design plenty of devices. In my thesis, there are two projects based on coordinate transformation that will be presented. In the following, we take an example for discovering how this concept works in designs. In 2008, D. R. Smith proposed a device which can bend the beam by certain angle [START_REF] Rahm | [END_REF]. This beam bending device is based on coordinate transformation. The sketch of the principle for the device in presented in As we can see from the formulation, the transformation is between an original Cartesian coordinate and a transformed cylindrical coordinate. To be more in details, the x-axis is transformed to the radial distance r', and the y-axis is transformed to a scaled azimuth axis.

Following the recipe we have discussed above, one obtains the permittivity and permeability tensors of the beam bend expressed in cylindrical coordinate system

            r r r j i j i / / / ' ' ' '      (1-19) with ) /( 1     k
. Note that r' was substituted by r for possible simulations in Comsol.

The polarization of the electric field was chosen to lie in the z-direction normal to the (x-y) plane of the wave propagation. The frequency considered in the simulations was 8.5 GHz.

However it should be noted that the transformation-optical approach is valid for any arbitrary frequency. The dark grey lines represent the direction of the power flow. Figure 123456789shows the spatial distribution of the electric field (color map) in the example of a π/2-bend. In

Space transformation based on Laplace's equation

The specific coordinate transformation is used in turn to determine the distribution of constitutive parameters which is discussed in the last section. These determined parameters are assigned to some TO device to obtain functional physical field distributions. However, the field distribution within the volume of the device is of no consequence in most instances: only the fields on the boundaries of the device are relevant, since the function of most optical devices is to relate a set of output fields on one port or aperture to a set of input fields on another port or aperture. From the TO point of view, device functionality is determined by the properties of the coordinate transformation at the boundaries of the domain. Since there is an infinite number of transformations that have identical behaviour on the boundary, there is considerable freedom to find a transformation that is close to optimal in the sense that it maximizes a desired quantity, such as isotropy.  and permeability '

 in the transformed space can be expressed as

) det( / ' ' ' A A A T       
. A is the Jacobian transformation tensor, which characterizes the geometrical variations between the original space Ω and the transformed space Ω'. The determination of the matrix A is the crucial point for designing transformation mediums. A can be obtained from the mapping point to point defined by coordinated transformation formulations. A can also be obtained from the mapping between two domains by specifying proper boundary conditions.

Here we take an arbitrary shaped cloak as example to introduce the transformation based on To ensure a cloak without reflections, the internal deformation field of the cloak layer must be continuous and wave impedance must be constant along the structure. The deformation tensor is calculated by the partial derivative of displacement with respect to the original coordinate, so the displacement fields must be smooth enough. It is known that the Laplace's equations with Dirichlet boundary conditions will always give rise to harmonic solutions [START_REF] Courant | Methods of Mathematical Physics[END_REF]. This suggests that the displacement field inside of the cloak layer can be calculated by solving Laplace's equations ΔU ′ = 0 with the boundary conditions U'(a) = a' and U'(b) = b'. In order to eliminate the singular solution of the Laplace's equations, we can use the inverse form of the Laplace's equations as

3 , 2 , 1 , 0 ² ² ² ² ² ² ' 3 ' 2 ' 1                   i U x x x i (1-20)
where Ui denotes the original coordinates in the original space.

After solving this equation by a PDE solver provided by the commercial software such as COMSOL Multiphysics, we are able to obtain

i j x x ' /  
, which means we can obtain the matrix A -1 . Then A can be calculated and so does the permittivity and permeability.

A two-dimensional arbitrary cloak is then proposed and verified numerically as presented in Figure 1-11. This concept of transformation provides generally complex parameters. Often these parameters are too high to achieve. But from the field distribution, the cloaking device performs a good functionality.

Conformal mapping

An invertible mapping is called conformal mapping if it preserves angles. We start discussing the conformal mapping by a review of the 2D mapping between (x, y) and (x', y') systems of the

form   ) , ( ' ), , ( ' ' y x y y x x X 
. The tensors can be written as:

              1 0 0 0 ' ' ' ' ' ' 0 ' ' ' ' ' ' 2 2 2 2 y x y y x x y y x x y x y y y x y x y x y x x x (1-21)
Ideally, the constitutive tensors have only diagonal components, this implies that

y y x x y x y x ' ' ' '   . Since yy xx   
, we can find that

x y y x y x y x ' ' ' '    (1-22)
And these equations are the well-known Cauchy-Riemann equations that define conformal maps. Now we can find out that the coordinates X' satisfy the vector form of Laplace's equation

0 ' 2   x (1-23)
This leads to the simple material parameters, permittivity and permeability that are given below:

            1 0 0 0 1 0 0 0 1 A   (1-24)
TO media that are of the form of Equation I-24 are often described as "isotropic" and "dielectric-only". The full transformation could be obtained with an isotropic dielectric if the electric field is polarized to be parallel to the z-axis, (TM z polarization). It seems to be a challenge to find conformal mappings analytically while explicitly solving the Cauchy-Riemann differential equations, for example, the analysis of mapping between z = x + iy and z' = x' + iy' is very complex However, the limitations of conformal maps arise when we truncate our transformation domain. That is, once boundary conditions are applied to the governing differential equations, it is not always possible to achieve conformal modules M, the aspect ratio of the differential rectangle corresponding to a set of orthogonal coordinates, equal to unity, and hence the Cauchy-Riemann equations do not apply universally to all geometries.

Quasi-conformal transformation optics (QCTO)

As we discussed previously, much of the power of the TO is determined by the transformation at the boundary of the domain. For instance, we might require that our transformation does not introduce reflections or change the direction of a wave entering or exiting our transformed domain. These conditions introduce additional restrictions to our transformation. The most straightforward way to satisfy these conditions is to stipulate that the coordinates are the same as free space on the boundary of our transformed domain, for instance Dirichlet boundary conditions. This extra constraint severely limits the scope of conformally-equivalent domains. Specifically, Chapter 1: Transformation Optics: Definition and related tools -23-let's take the mapping between two quadrilateral domains for instance. Once the sides of the quadrilateral domain have been specified, the region can only be mapped to another quadrilateral that shares the same conformal module. If the domain is rectangular, then M is the aspect ratio of the entire domain. Another concern relates to the boundary conditions directly.

While Dirichlet boundaries are ideal for most purposes, they may be incompatible with our requirement of orthogonality at all points in the mapped domain. If we simultaneously specify

) ( ' x x
and M at the boundary, the problem becomes over-determined and we are not guaranteed that the mapping will be orthogonal at the boundary [START_REF] Tang | [END_REF]. Instead, we require a combination of

  1 1 ) ( , ,    c T A M M M Diag A AA (1-25)
The conformal module provides an immediate measure of the anisotropy of the TO medium, as well as its required magnetic response, since

y x    M . We have x y y x y Mx y Mx ' ' ' '    (1-26)
The inverse form of this equation is We find that we recover the Laplacian vector for the inverse problem:

0 2   x (1-28)
The following steps are the same as we have discussed previously in the part of the transformation based on Laplace's equation with Dirichlet boundary conditions.

We see that the cost of the quasi conformal (QC) map is immediately clear: the in-plane material tensors elements are no longer equal to each other. However, it is generally the case that small deformations of space create small perturbations to the conformal module of the physical domain. Li and Pendry [START_REF] Post | Formal Structure of Electromagnetics[END_REF] suggested that the "small" anisotropy can be ignored in this case. The geometric average of these quantities is simply

2 / 1 ) (    c z A M n  (1-29)
Since the virtual and physical domains are no longer of the same size, the resulting distributions of material parameters are only approximately correct, and will introduce a number of aberrations. Despite its limitations, the QC method has found many applications. QC method may be applied to flatten conventional dielectric lens-antennas and parabolic reflector-antennas without significant loss in performances [START_REF] Tang | [END_REF][93][94]. The quasi-isotropic properties can then further be implemented by all-dielectric devices, which benefits a wide frequency band of operation.

Numerical calculation softwares

In this section, I present the different softwares used as tools to design transformation optics devices.

Mathematica

Wolfram Mathematica is a computational software program used in many scientific, engineering, mathematical and computing fields, based on symbolic mathematics. In my thesis, Wolfram Mathematica is mainly used as a tool for parameter calculations of coordinate transformation designs. By coding the formulation of the coordinate transformation designs, Mathematica can provide the simplified formulations of the tensors, such as , . Applying a polarization of the field or other techniques, Mathematica can complete the parameter reduction.

After assigning values to the variables, a parameter distribution or even discretization can be achieved.

COMSOL Multiphysics

COMSOL Multiphysics® is a general-purpose software platform, based on advanced numerical methods, for modeling and simulating physics-based problems. In the thesis, COMSOL is used for 2-D near field and far field calculations. In COMSOL, continuous formulations can be assigned to material properties. This is very convenient for transformation optic designs. In addition to conventional physics-based user interfaces, COMSOL Multiphysics also allows for entering coupled systems of partial differential equations (PDEs). The PDEs can be entered directly or using the so-called weak form. The PDE solver can be used for the calculation of the space transformations. The designs using the method of transformation based on Laplace's equation can be calculated by the PDE solvers. Then the parameters from this calculation can be To calculate the effective parameters of the resonators shown Figure 1-15, a homogenization procedure is briefly described in the following part of this section. The homogenization procedure is based on the use the coefficients of transmittance and reflectance obtained by simulations and measures. Parameters such as the index n and the impedance Z can be calculated depending on the transmission and reflection from the following equations [21]:

2 2 2 2 ) 1 ( ) 1 ( t r t r Z       (1-30)   ) ( 1 2 1 ) cos( 2 2 t r t nkd    (1-31)
Thus, we can reduce parameters such as the effective permittivity and the effective permeability Two causal conditions for passive media must, however, be followed:

0 ) Re(  Z and 0 ) Im(  Z (1-32)
Indeed, this method involves reversing the coefficients of reflection and transmitting a uniform metamaterial sample. Since the 1970s, Nicholson and Ross [START_REF] Nicolson | Instrumentation and Measurement[END_REF] were interested in this method. Smith and colleagues retraced the method by applying it successfully to basic bricks metamaterials and adding conditions branches selection conditions when multifaceted functions such as the argument of a complex number [96]. Note that the index of a medium is set over the entire frequency band in which material is excited by a single electromagnetic mode. However defining an impedance requires continuity of material or less, the assumption of long wavelength compared to the size of the heterogeneities of so that diffraction phenomena are negligible.

Experimental setups

Two experimental setups have been used during my thesis to validate the proposed concepts from fabricated prototypes. One is based on near-field measurements in order to be able to show the wavefronts and the other one is based on far-field measurements to show antennas radiation patterns.

Near-field measurement setup

A first experimental system aiming to scan the electric near-field microwave radiation is used.

The electric field is scanned by a field-sensing monopole probe connected to one port of a vector network analyser by a coaxial cable. The probe is mounted on two orthogonal linear computer-controlled translation stages as presented in Figure 1-16, so that the probe can be moved within the radiation region of the system under test.

By stepping the field sensor in small increments and recording the field amplitude and phase at every step, a full 2D spatial field map of the microwave near-field pattern is obtained either in free-space or in parallel-plate TEM waveguide configuration. The total scanning area can cover a surface area of 400 x 400 mm2. Microwave absorbers are applied around the measurement

Far-field measurement setup

The second experiment system consists of measuring the far-field radiation patterns of the antenna in an anechoic chamber. Figure 1-17 shows the far-field measurement system.

In such an emission-reception setup, the fabricated transformation based deviece is excited by a designed source which can be in the form of a patch antenna or patch array or even monopole or dipole. A dual-polarized wideband horn antenna is used as the receiver to measure the radiated power level of the emitter. The measurements are done for computer-controlled elevation angle varying from -90° to +90°. The microwave source is a vector network analyser (Agilent 8722

ES) that we also use for detection. The feeding port is connected to the metamaterial antenna by means of a coaxial cable, and the detecting port is connected to the horn antenna also by means of a coaxial cable.

Validation technology

In the previous sections, we have introduced the transformation optics concept and the softwares used for the numerical calculations and also the types of measurements performed. In my thesis, the implementation of microwave devices designed by transformation optics concept requires the realization of permittivity and permeability materials or cells. We implement both resonant metamaterial resonators and all dielectric materials. The metamaterial resonators will provide a wide range of permittivity or permeability values ranging from negative to positive.

High permittivity values around 15 can also be produced. But they will suffer from the narrow frequency band limitation. These resonators are normally fabricated by printed circuit board technology.

On the contrary, the all-dielectric materials can only provide a limit range of positive permittivity values larger than 1. But the advantage is the wide frequency band and an easy adaptation to THz and optics domain. The all-dielectric prototypes presented in this thesis are fabricated by 3D printing and therefore we are able to realize air holes with extremely small radius.

In the following parts, I present the metamaterial and dielectric cells used to produce permittivity and permeability values requires in my proposed designs.

Electric LC (ELC) resonators

An electromagnetic metamaterial is a man-made composite material comprising a periodic array of subwavelength inclusions. Typically, a single metallic metamaterial inclusion can be considered as an LC resonant circuit with its inductance and capacitance influenced by its shape and dimensions.

The ELC resonator is presented and illustrated in To examine the performance of the ELC resonator, the electromagnetic properties of the structure are analysed numerically by applying the appropriate periodic conditions on a unit cell.

The simulation environment of a resonator is given in Figure 12345678910111213141516171819. Two Floquet ports are used along the direction of propagation. These ports are placed at a sufficiently large distance so that the incident wave is planar at the surface of the resonators. A de-embedding allows to cancel phase delay due to the distance travelled by the wave in vacuum from the exciting ports.

Split ring resonator (SRR)

Split ring resonators (SRRs) consist of a pair of subwavelength concentric metallic rings, etched on a dielectric substrate, with gaps on opposite sides. We use the SRR structure for its ability to present a magnetic resonance and hence the possibility of producing negative permeability values.

The electromagnetic properties of SRRs have been analysed in lots of works [98] and [99]. A SRR behaves as an LC resonator that can be excited by an external magnetic flux, exhibiting a strong diamagnetism above their first resonance. SRRs can also exhibit cross-polarization effects (magneto-electric coupling) [99] so that excitation by a properly polarized time-varying external electric field is also possible. Figure 1-20 shows the basic topology of the SRR, as well as the equivalent-circuit model proposed in [98].

The numerical analysis of the SRR resonator is similar to the simulation we have introduced for the ELC resonators. The only difference is the polarization of the incident wave is according to the one shown in Figure 1234567891011121314151617181920.

All-dielectric material

To realize the transformation designs, positive permittivity values above 1 can be realized from non-resonant cells to benefit a broad frequency range. Air holes in a dielectric host medium of relative permittivity h = 2.8 is therefore considered in my thesis. Suppose that air and the dielectric are mixed together, the effective parameter can be calculated by:

h h a a e f f      (1-33)
where a = 1 is the relative permittivity of air and fa and fh are the volume fraction of the air holes and the host material, respectively. By adjusting the volume fraction of the air holes in the dielectric host medium, the effective permittivity of the cell can then be engineered.

Conclusion

In this chapter, the principle of metamaterials is presented, and several antenna applications The interests in invariance of Maxwell equations to coordinate transformations have been revived in 2006 when U. Leonhardt [21] and J. B. Pendry et al. [101] showed how the paths of electromagnetic waves travelling within a material can be controlled at will by applying a judiciously defined spatial variation in the constitutive parameters. The concept of coordinate transformation electromagnetics, presented in the introduction chapter, appears to be a convenient tool to design devices or components with special properties difficult to obtain with naturally occurring materials and has been successfully utilized to experimentally validate the emblematic electromagnetic cloak [79] [102]. Other conceptual and functional electromagnetic devices have then proposed using this method. To name a few, field concentrators [79], beam bends and expanders [44][56] [60][103], waveguide tapers [104] and field rotators [67] have been designed. For microwave antenna applications, focusing lens antennas have been theoretically designed [28] [105]. The performances of an omnidirectional retroreflector [106] based on the transmutation of singularities and a Luneberg lens [107] have been experimentally demonstrated.

Furthermore, techniques of source transformation [35][71] [74,75][ [108][109][110][111], where source distribution is included in the transformed space, have offered new opportunities for the design of active devices. In this Part I, we will discuss two devices that have been designed based on coordinate transformation optics in the framework of this thesis. One of the two has been fabricated by making use of resonant metamaterials, typically electric-LC (ELC) resonators, introduced by Schurig et al. for its electric resonance [97].

Part I consists of three chapters, where two designs of microwave devices are introduced in Chapter 2 and Chapter 3 respectively. The concepts of these two designs are both based on coordinate transformation optics. In Chapter 2, we introduce a coordinate transformation optics approach to delocalize sources inside a metamaterial assigned shell to appear outside the shell.

The design we propose consists in a novel radiating device that can change the position of an apparent emitting source. By applying radial transformations that compress space, and then that match it to the surrounding environment, we show how a source placed in the core region can be apparently displaced at a distance outside the second matching region corresponding to a -39-space folding. Moreover, the radiation pattern of the virtual source can be changed and we show that an omnidirectional radiation can be changed into a directive one. We describe transformation of the metric space and the calculation of the material parameters. In Chapter 3, following the transformation optics approach still, we design a taper between two waveguides of different cross sections. Three different transformation techniques are presented so as to achieve a low-reflection taper between the two waveguides. The media obtained from these three methods presents complex anisotropic permittivity and permeability. The taper is finally designed with two compression parts and two matching parts. This structure allows us to succeed in the tapering function with a factor of 11 between the two different cross sections of the waveguides. However we show that using a reciprocal transformation leads to the design of a taper from a material with physically achievable material parameters. The complex parameters are simplified and discretized for a physical realization. The fabricated prototype of this waveguide taper is then presented. Full wave simulations are performed at different frequencies to validate the proposed two devices. The tapering function is also further verified by experimental demonstrations showing the scanned near field of the device. Complex electromagnetic structures can be designed by using the powerful concept of transformation electromagnetics as it has been introduced previously [128]. In this chapter, we define a spatial coordinate transformation that shows the possibility of designing a device capable of producing an illusion on an antenna radiation pattern as shown in Figure 2-1. Indeed, by compressing the space containing a radiating element, we show that it is able to change the radiation pattern and to make the radiation location appear outside the latter space. Both continuous and discretized models with calculated electromagnetic parameter values are presented. A reduction of the electromagnetic material parameters is also proposed for a possible physical fabrication of the device with achievable values of permittivity and permeability that can be obtained from existing well-known metamaterials. Following that, the design of the proposed antenna using a layered metamaterial is presented. Full wave numerical simulations using Finite Element Method (FEM) are performed to demonstrate the performances of such a device.

Introduction

As well known to all, transformation electromagnetics concept has motivated a series of studies on illusion optics devices [28, 29] [112-115], which can create optical illusions. For example, an object can be made to appear like another one by transforming the scattered light outside a virtual boundary into that of the object chosen for the illusion [28]. A shrinking device able to transform an object virtually into another one of smaller size with different material parameters has further been experimentally demonstrated [114]. A "ghost" illusion device, which is capable of creating multiple virtual ghost images of the original object's position has also been realized

and validated [115]. These three designs are illustrated in Figure 2-2. The concept of this step is more less the same as the shrinking device we presented in the previous paragraph. The two dielectric ghosts are created by increasing the permittivity value at the region of the 'ghost'. Actually these ghosts don't really come from the transformation. One can also fabricate this device by just implanting the shrinking device in two dielectric blocks.

Here we bring up a real challenge of creating the illusion only by applying a coordinate transformation. The question we ask ourselves is whether we can delocalize the emission of radiating element contained in a material shell such that we have the impression that the emission appears like coming from a virtual source at another location? With such manipulation, we will therefore be able to generate an electromagnetic radiation from a location where no radiator is physically present. In this chapter, we use transformation electromagnetics to change the appearance of a source and virtually delocalize the emission outside its core material shell.

Such an illusion device can be potentially used to fool radar detectors. We will show how from a coordinate transformation that compresses space and then expands it, it is not only possible to transform the electromagnetic radiation of a small antenna into one equivalent to that of a large one, but also to change its spatial location. We discuss the space parameters generated from our study and validate the concept through numerical simulations. We propose a structure in transverse magnetic (TM) polarization with simplified reduced parameters. Finally a layered structure of the metamaterial is presented in order to make a first step towards the physical realization process of this illusion device.

Spatial coordinate transformation formulation

The basic idea of our concept is to virtually modify the spatial location of a radiating electromagnetic source outside the material shell in which it is placed. An observer will then have the impression the emitter is elsewhere instead of its real physical location, as illustrated in lens can be viewed as a medium where space is totally suppressed and folded outside. Such manipulation has been referred to as embedded transformation [61]. Other cases of perfect lenses having a cylindrical geometry have been utilized as magnifying devices through linear transformations [51][116]. For the proposed delocalization of electromagnetic radiation, different transformation methods can be applied. For example, a radial transformation using a sinusoidal function is firstly presented. Such transformation has been introduced in our recent work [117].

The concept of the sinusoidal transformation

The concept of the sinusoidal transformation is presented in Figure 2 The angular part of the transformation is not changed and we assume a z-independence of the transformation for possible parameter reduction. Thus, any source placed at a certain distance in material will have 3 images source as represented with the projection on the r' axis and at different positions.

We consider (r, θ, z) to be the coordinates in the initial fictitious cylindrical space and (r', θ', z')

to be the coordinates in the final real physical transformed space. In the transformation, we assume θ and z independence. Mathematically this transformation is expressed as:

        z z' θ θ' f(r) r' (2-1)
where f(r) corresponds to a perfect radial transformation and takes the form f(r) =

(arcsin(r/A))/B. In order to calculate the parameters we need to determine the Jacobian matrix

(2-2) in the cylindrical coordinates:

                                                      1 0 0 0 1 0 0 0 ) ( ' ' ' ' ' ' ' ' ' r r f z z z r z z r z r r r r J cyl       (2-2)
This Jacobian matrix does not take into account the equivalence with the Cartesian coordinates which allows to create the material. We therefore need to introduce the component of the metric tensors into the final Jacobian matrix J, given by: As we can see here, this sinusoidal transformation is mainly between the r axis and r' axis in cylindrical coordinate systems. This delocalization function is achieved by one step of transformation. The field is firstly compressed when the sinusoidal function is increasing, then it is extended when the function is decreasing, and finally it is matched when the curve of the function cross the line r = r' at the point r' = R. The transformation used to achieved our material engineering device is defined as g(r') = Asin(Br') where A = R/sin(BR). This condition assures the perfect impedance matching outside between the air and the material and keeps the blue line of the Figure 2-3(c) unchanged. The value of variable B is considered equal to 61.2 in order to delocalize the source outside the shell.

The value of the permeability and permittivity components are very strong at the maximum point of g(r') when r'=R0 and we have plotted in the Figure 2-6 the components in a fixed interval. We can remark that there two zones: a red one which corresponds to the core zone with positive values creating the stretching and the delocalisation of the space and a blue one which corresponds to negatives values and to the space folding and a matching zone for the impedance matching. This second zone corresponds to a space where the propagation of the waves are backward and opposite to the core and the air area.

If we take a closer look at the distribution of the permittivity, it is obvious that all the four parameters have both positive and negative values. This is not surprising, since the transformation has two regions. One region is the compression region and the other region is the

The concept of the linear transformation

However in such sinusoidal transformation, the material parameters values can be very high and even after the parameter reduction process, they can be still difficult to achieve with common metamaterial structures. So we have to search for another transformation; one which can provide reasonable material parameter values and less complex permittivity distribution. In fact, many kinds of transformations can create this folding and extension concept. As long as the function increases with a differential coefficient larger than 1, then decreases with a differential coefficient smaller than -1, and finally matches at the boundary of the shell, it will create this illusion. At last, we propose another transformation which is linear. This linear transformation is simple to calculate and the material properties can be possibly achieved. The principle of the proposed transformation is depicted in Figure 234567. The space is first compressed by a linear transformation with a slope lager than 1 and then extended by another linear transformation whose slope is negative. In this transformation, a part of the space is folded near the boundaries of the material shell.

Such folding of space corresponds to the zone between r = R1 and r = R2, presenting a negative derivative on the transformation in Figure 2-7 and corresponding also to the use of negative material parameters. The space transformation is decomposed into two linear parts: a first part where the value of r increases linearly, corresponding to a space compression (derivative > 1) and a second part where r decreases linearly (derivative < 1) where the space is extended in order to assure a perfect impedance matching with vacuum. Any source placed at a certain distance in the material shell will produce three images at different positions as represented by the projections RA, RB and RC on the r' axis.

Here in the transformation, the function f(r) of Equation (1-1) is given as:

        2 1 1 0 R r q R q R r b r a q r r f               (2-6) With variables q > 1 , q R R R R a     1 2 1 2 and   q R R R R q b      1 2 1 2 1
. This transformation consists in two steps. In the domain where

q R r    1 0
, the field is compressed and in the domain

2 1 R r q R   
, the field is released to match at the boundary. Even though this transformation is not continuous, it provides much simpler material properties.

The transformation used to achieve the engineered material device is defined as:

        2 1 1 ' ' 0 ' ' ' R r R R r a b r r q r g r              (2-7)
This condition assures a perfect impedance matching between the material and vacuum and leaves the boundary at r = R2 unchanged. The radius R2 of the whole material shell is considered here to be equal to 5cm and the radius R1 of the compression zone is taken as 2cm. The ratio q in the compression zone is chosen to be 4.5 which can lead to a maximum delocalization As before, the three components of the permittivity take values varying from negative to positive. The extremum values depend on the value of q. If we increase q, the real source in the shell will be delocalize to a further location. But meanwhile, the maximum positive value and the minimum negative value will increase. For this reason, we have choose q = 4.5, so that the maximum positive value 20 is possible to achieve.

Comparison between sinusoidal and linear transformations

Actually, there could exist many kinds of transformation which can create this delocalization. In this thesis, we only take two of them as examples. The sinusoidal transformation can achieve this functionality by only one step, while the linear transformation consists in two steps. The distribution of the permittivity and permeability of the sinusoidal transformation is continuous in most cases and there are singular points in some parts. The distribution of the permittivity and permeability of the linear transformation is separated into two parts due to the two steps of transformation. One part is purely positive and constant, the other part is negative. The parameter values given by the linear transformation are achievable because they are relatively moderate. The positive part can be fabricated with certain kind of dielectric material, such as ceramics whose permittivity can achieve high values at room temperature. The negative part can be realized by resonant metamaterials.

NUMERICAL SIMULATIONS AND RESULTS

In this section, FEM based numerical simulations with Comsol Multiphysics [118] are used to design and characterize the proposed illusion device at microwave frequencies (300 MHz-300 GHz). The validation of the material is performed in a two-dimensional configuration in a transverse magnetic (TMz) mode. Continuity and matching conditions are respectively applied to the internal and external boundaries. The line current source of length l = 5 mm along the zaxis is placed at an offset distance d = 1.9 cm from the centre. The frequency is 5 GHz, with a wavelength of 6 cm. The source isis an electric dipole with a length of 5 mm.

Parametric study of q

The corresponding calculated z-component of the electric field (Ez) is depicted in Figure 23456789.

The radiation emitted by the source in free space is illustrated in Figure 23456789). An isotropic antenna diagram is obtained due to the small width of the source compared to the operating wavelength. In Figure 2-9(b), the material shell defined for the linear transformation with q = 4.5 as represented by the parameters calculated in Figure 2345678, is applied around the source. Such a configuration leads to two virtual sources; one appearing in the zone between R1 and R2 and another one outside R2 at r = 8.55 cm, with therefore a delocalization distance D = 6.65 cm.

Using q = 6 and the appropriate electromagnetic parameters leads to a virtual source outside R2 As addressed in [START_REF] Chen | [END_REF], surface waves supported by the negative electromagnetic parameters propagate on the material shell and the delocalized virtual source is created by the interference of the surface waves since there is no space for the evanescent field to decay exponentially. The absence of space is due to the fact that the negative index medium cancels optically some of the surrounding positive index material. This phenomenon is similar to what occurs in a perfect lens from a negative index slab.

In fact, the location of the virtual source r = dq where d is the distance between the real source and the center of the material, as presented in Figure 234567. If the position of the real source d is fixed, d = 1.9 cm in our case, the location of the virtual source is a proportional function of q.

And then the distance D between the real source and the virtual source is D = d(q-1). In another word, if we increase the value of q, the virtual source will appear at a further place from the real source. Here we conclude that when the virtual source is fixed, by changing the value of q, we can control the position of the virtual source as we wish. Based on this result, we start to wonder what if we fix the value of q, and place the real source at different positions considering that we are limited by the size of the material. This question will be discussed in the next section.

Parametric study of the position of the real source

The two virtual sources observed in Figure 2 When the source is placed in the region 0 ≤ r ≤ R1/q as in Figure 2-11(a), the virtual emission will be located in the compression zone. However the real source placed in the region R1/q ≤ r ≤ R2/q produces a virtual emission in the matching zone (Figure 2-11(b)). Finally, when the source is placed in the region R2/q ≤ r ≤ R2 as in Figure 2-11(c), the virtual emission is produced in vacuum outside the material shell.

In

Parametric study of the dimensions of the shell

In this study, we fix the scale factor q = 4.5. The position of the real source will always be at the same location as we mentioned just before. First we make a parametric study on the dimensions R1 and R2 of the shell, while keeping the ratio R1/R2 always equals to 2/5.

The results are presented in Figure 2-12. We increase the dimension R2 of the shell from 5 cm to 10 cm, and as shown in the figure, the distribution of the electric field Ez inside the shell is scaled. But the distribution of the electric field outside the shell remains the same. No matter how large the shell is, the virtual source always appears at r = 8.55 cm. Of course this is only true in the situation that the real source remains at the same location and in the compression region. So, we can conclude that as long as the ratio of the thickness of the two regions remains the same, the location of the virtual source doesn't depend on the dimension of the shell.

LAYERED MATERIAL DESIGN

The simulations in the previous section have shown the possibility of achieving a delocalization of electromagnetic radiation. A possible realization is considered at a frequency of 10 GHz. In order to realize the material described in Figure 2-8, the shell is discretized into seven zones;

one zone in the central region with radius R1 = 2 cm, where space is compressed and six others for the matching zone corresponding to negative permittivity and permeability values.

If we consider a metamaterial unit cell with dimension 5 mm at the operating frequency of 10

GHz, four cells will then be used in the first zone and one cell only is needed in each of the other six zones. Choosing a polarization along z for the magnetic field direction, we are able to The thickness of the layers in the matching zone are set to d = 5 mm in order to design the metamaterial cells and this model will later be tested under the frequency 10 GHz with the wavelength of 3 cm. Using a compression ration q = 4.5 at such frequency, it will be possible to

show the change in nature of the electromagnetic radiation. We assume the value in the centre of each zone to be the value of the whole zone. The material parameter values, namelyzz, rr and , are summarized in Table 2-I.

Split Ring Resonators (SRRs) [120], known to produce a magnetic resonance where the permeability ranges from negative to positive values, can be efficiently used to produce the zz values. Cut wires [121] or electric-LC (ELC) [97] resonators are good candidates to produce the rr and  values as we have introduced in Chapter 1. As stated in the introduction chapter, metamaterial resonators such as SRRs, ELCs and cut wires can be considered to produce the desired material parameters. The previous achievements using these metamaterial resonators suggest that the material parameters can be realized through the use of metamaterial structures and that the proposed device is feasible. Full wave simulations are performed with the layered material having electromagnetic parameters of Table I and material shell and the discretized shell. Even if the reconstruction of the wavefronts is not perfect due to the discontinuity of the discretized material, the field distributions are in good qualitative agreement with the simulation performed using the continuous material parameters.

We can clearly observe that the radiation pattern of the source in the transformed material is directive. This is due to the compression of the space in which the source is placed. A higher q at 5 GHz leads to similar change in radiation pattern as shown in Figure 2-16.

Figure 2-16:

Directive radiation pattern at 5 GHz when a higher q is applied.

Finally, we should mention that the use of metamaterials implies the presence of losses due to their resonant nature. Since a negative index medium is needed in the device to amplify evanescent waves so as to obtain an image as clear as the source, the device will be very sensitive to losses. High losses in the negative index medium will definitely deteriorate the image formed and no delocalization will be observed, presented in Figure 2-17.

CONCLUSION

In this chapter, we have introduced a delocalization device designed using two kinds of coordinate transformations. The device can create an image of the real source in the radiated field by folding the space. A sinusoidal transformation is first applied. This transformation can achieve the illusion functionality by only one step of transformation, but the parameters values are too high to achieve. Then a linear transformation has been used to design the material that is able of producing a virtual emission outside the latter material with achievable values of the metamaterial parameters. When the real source remains at the same location and in the compression region, the location of the virtual source doesn't depend on the dimension of the shell. If the real source in the compression zone is close to the boundary or just inside the extension zone, two other images will be created. Here we try to place the real source in the compressing region inside the core shell, in order to create a ghost target outside the shell to protect the real source from radar detectors or other potential usage. For a practical fabrication of the device with common metamaterial structures, a layered material has also been proposed

-1

and tested at 10 GHz. But this discretization is still not perfect and must be optimized. It has been shown that the material shell creates the illusion that an electromagnetic radiation comes from outside the shell while the real source is placed in the latter. But for the moment, due to the complex permittivity and permeability distributions, it is still difficult for us to achieve simultaneously the three material parameters of the device. 3.1 Introduction. Basically there are two main reasons that limit the development of this subject. The first reason is that the value of the permittivity and permeability relies on the difference of the width of the two waveguide. The larger this difference is, the higher permittivity and permeability value one can obtain. When the input waveguide is 4 or 5 times wider than the output waveguide, the parameters can reach about 25, which is already the maximum value for metamaterial resonators to realize. The second reason is the complex material properties. In most cases, there exist four parameters to realize. This is also a big challenge to overcome.

But in this thesis, these two problems are solved. In the model I proposed, the width for the input waveguide is 11.4 times of the width of the output waveguide. We manage to achieve this huge difference by achievable permittivity and permeability. Moreover, by several successive steps of parameter reduction, we simplify the four parameters into to two. And a prototype is fabricated and experimentally verified.

In P.-H. Tichit's work, three different kinds of transformation were introduced: linear, parabolic and exponential [67]. The sketches and formulations of these three kinds of transformation are In Tichit's study, the size difference between the input and output waveguides is only 5 times, which is limited by the unachievable large parameter values. My contribution in this field is to develop this tapering device by increasing the size difference between the input and output waveguide to 11.4 times. My final goal is to realize a prototype for an experimental demonstration of the tapering device.

Spatial coordinate transformation formulation

In order to study the performance of different kinds of coordinate transformations and the material properties they provide, we introduce four other types of transformations. These are: cosinusoidal, parabolic, logarithmic and reciprocal transformations. We compare their permittivity and permeability distributions and the electric field distributions in the device respectively. For all the models of the four new transformations in this chapter, the width of the input waveguide is set to a = 28.5 cm and the width of the output waveguide is set to b2 = 2.5 cm. The length of the taper l is fixed to 18 cm. The following designs will be tested at 8 GHz in simulations.

Cosinusoidal transformation

Firstly a cosinusoidal transformation is introduced. At the beginning of the taper, when y = a/2, the cosinusoidal function starts from its maximum, while at the end of the taper, the function stops at its minimum. The formulation of the transformation approach is presented in the Equation below:

Parabolic transformation

The parabolic function introduced in this section is a concave upward function. The minimum of this function appears at the end of the taper when y = b1/2. The formulation of the transformation approach is presented in the Equation (3-2): The electric field and norm of electric field for such configuration are presented in Figure 345678respectively. The wavefronts show a clear curve of a parabolic function. In the output waveguide, the field is stronger than the one in the cosinusoidal transformation, but a reflection can be observed from the norm of electric field in Figure 3-8(b).

(3-2)

Logarithmic transformation

We continue our study on the logarithmic transformation. The logarithm function has a negative coefficient. So the function decreases from a/2 when x = 0 to b/2 when x = l. The formulation of the transformation approach is presented in the Equation (3-3):

            z z mx a a y y x x ' )) 1 ln( 2 ( 2 ' ' (3-3)
where Even though the transmission from the reciprocal transformation is much better than any other transformations we have introduced, there is still a parameter that we cannot realize for the moment. So we have to explore another method which can taper the cross section of two waveguides of width ratio 11.4 by achievable material properties.

Reciprocal transformation

Two tapers and two adapters system

In this section, we introduce a two tapers and two adapters system in order to be able to propose a realistic solution for prototyping. The top view of the structure of the system is presented in

               z z l b x b a l b y x x ' y ' ' 1 1 1 1 1 (3-5)
Formulation of the transformation in the second taper part:

                 z z l b l l x b b l b y x x ' - y ' ' 3 2 2 1 2 1 3 2 (3-6)
The electromagnetic properties of the transformation-optical medium can now be readily obtained by calculation. We then can obtain the permittivity and permeability tensors of the two steps of the waveguide tapers.

                    ' , ' 0 0 0 ' , ' ' , ' 0 ' , ' ' , ' y x y x y x y x y x zz yy xy xy xx       (3-7)
The components values of  tensor are given in Table 3-I where a non-diagonal term ( xy  ) appears. This non-diagonal term is necessary to guide electromagnetic waves in the x-y plane like it is the case for this taper.

    ' , ' ' , ' y x y x zz xx     ' , ' y x xy   ' , ' y x yy  First taper 1 1 ' ) ( bl x b a bl   1 ' ) ( bl y a b  ' ) ( ' ) ( 1 2 1 2 2 2 2 1 2 x b a bl l b y b a l b     Second taper 3 1 2 1 1 3 1 ) ' )( ( l b l l x b b l b     3 1 1 ' ) ( l b y b b  ) ' )( ( ' ) ( 2 1 1 3 1 2 3 2 1 2 3 2 1 2 2 1 l l x b b l b l b l b y b b      
The components for the two taper parts have different expressions. But the principle of the two taper parts are the same, it is just the variables and locations in the coordinate system that are different. As we can see in the table, there exist four parameters for both of the two taper parts.

In order to realize this design, a further parameter reduction needs to be performed.

The different geometrical dimensions of the initial and transformed spaces are, a = 28.5 cm, b1 = 10.5 cm, b2 = 2.5 cm, l1 = 7.5 cm, l2 = l4 = 4.5 cm and l3 = 6 cm respectively. For the first taper part, the maximum value is 2.74 and the minimum value is -1.32. And for the second taper part, the maximum value is 7.25 and the minimum value is -2.5. These values can be obtained from electric LC resonators or split rings resonators. An input wave port is placed on the left boundary while an output wave port is placed on the right boundary of the output waveguide. The electric field is compressed two times in the two taper parts respectively. And the field is adapted by the two adapter parts to minimize the reflection on the boundaries between each two parts. In Figure 3-15(b), the field intensity is light before the tapering, and it becomes extremely high which means highly concentrated after the compression. The transmission of this system can go up to -0.2 dB, i.e. 98%.

Parametric study on the two adapters

The material in the two adapters are isotropic. So the material properties in the adapters are variations of index. In the first adapter, the index starts from n1 on the left boundary and decreases to 1 on the right boundary. In the second adapter, the index starts from n2 on the left boundary and decreases to 1 on the right boundary. At first, we set n2 = 3.3, and make a parametric study on n1 from 1 to 4. The total transmission of the system is presented in Figure 3-16. As shown in the figure, when n1 = 2.1, the transmission is closest to 0 dB and the reflection is about -25 dB which is the minimum value throughout the whole study. So we select this value as the starting index value of the first adapter.

For the second adapter. We set n1 = 2.1, and the same parameter study for n2 is performed from 1 to 4. The result is presented in Figure 3-17. In the curve, we can see that, when n2 = 3.3, the transmission is the highest and the reflection is the lowest. From both of the two parametric studies on the index, we find out the starting index for the two adapters which can ensure the best transmission for the system.

2D discrete model

We discretized the continuous model in Figure 3-15 and full wave simulations have been done to verify the discrete model. We choose the size of the unit cells to be 15 mm × 15 mm for the two taper parts. The discrete permittivity or permeability of each cell is constant and equals to the permittivity or permeability value at the center of each cell. In this case, the first taper is composed of 95 (19×5) unit cells, while the second taper is composed of 28 (7×4) unit cells.

Parameter reduction

Since there are four parameters to realize for the moment, we have to find a way to simplify them by keeping only two of them. In order to do this, we separate the simplification into two steps as shown in Figure 345678910111213141516171819. The first step is to calculate the eigenvalues and eigenvectors of each unit cell. For each unit cell, the eigenvectors are different from others. One of the eigenvectors is the same as z-axis. The vector is along the direction of the magnetic field and the other vector is orthogonal to the direction of magnetic field. The permittivity and permeability distributions after the first step are presented in Figure 3

Experimental validation using ELC and split rings resonator

We consider here a source that has a z-polarization of electric field. The parameter  stands for a permeability component.

The structure of the ELC resonator used to engineer the permittivity parameters of the transformed tapering system is presented in Figure 3-20(a). The 35 µm thick copper resonator is printed on a low loss RT/Duroid ® 5880 substrate having a dielectric permittivity 2.2, a low loss tangent 0.001 and a thickness t = 0.51 mm. The properties of the resonator are characterized numerically using Finite Element Method (FEM) simulations done with ANSYS commercial code HFSS [START_REF]High Frequency Structure Simulator[END_REF]. Using reflection and transmission responses from the sample, effective parameters can be extracted using the retrieval procedure described in Chapter 1 "General Definition". This is possible since the structure period along the propagation direction is very small compared to the working wavelength.

In performing the retrieval, we assume along propagation direction a z-direction size pz = 5 mm corresponding to a cubic unit cell. Extracted permittivity and permeability are also shown in GHz, so that it will provide the permeability we need for the tapering system by verifying the dimension variables: b, b1, g1, g2. This tapering system requires a directive planar wave input, which is 28.5 cm long. The wave vector has to be orthogonal to the input boundary of the system. But unfortunately in our lab, we cannot provide such a long and directive source at 8 GHz. We have to use the source which is Section 3.4: Experimental validation using ELC and split rings resonator -86-2.5 cm long. We put this source 25 cm far from the taper in order to have an input that close to plane wave. We simulated this source and the electric field distribution of the source is presented in Figure 3 The electric field is scanned by our near-field measurement system described previously between two parallel metallic plates (TEM waveguide). The electric field of the source and the tapering system are presented in Figure 3-22(c) and (d) respectively. As shown in the figure, a quite overall good qualitative agreement is noted between the simulated and measured characteristics.

Conclusion

In this chapter, firstly we have firstly presented a review on the former works about the waveguide taper by our group and other research teams. Then four new kinds of coordinate transformations have been introduced with the main goal to fabricate a prototype of this tapering system. These four transformations are compared and discussed both on the provided material properties and the field distributions. Then we considered a tapering system which consists two tapers and two adapters. Such solution allows to work with material parameters that can be obtained from resonant metamaterials. Parametric study are done for the starting index of the two adapters to optimize the transmission. Then the 2D model we designed is discretized and the electric field distribution is calculated for this discrete model. A parameter reduction has been performed to allow working with only two material parameters. One is permittivity and the other one is permeability. ELC and Split Ring Resonators are used to provide permittivity and permeability values respectively at 8 GHz. Finally a 3D model is designed and fabricated.

Numerical simulations and measurements of the electric field for both the source and the tapering system are presented and compared. A quite overall good qualitative agreement is noted between the simulated and measured characteristics. These structures designed using coordinate transformation concept technique generally exhibit anisotropy and spatial inhomogeneity and sometimes present a challenge for practical implementations, leaving lots of devices unrealized experimentally. Metamaterial structures like split ring resonators (SRR) [30] and electric LC (ELC) [31] resonators, to name the most common, have widely participated in the implementation of TO based devices. However, their resonant nature limits the frequency bandwidth and performances of the devices. To overcome such limitations, quasi-conformal transformation optics (QCTO) has been proposed. QCTO helps to minimize the anisotropy of the constitutive materials, allowing the possibility of alldielectric implementations. As a result, nearly-isotropic gradient index (GRIN) materials with broad bandwidth and low losses can be employed, opening the way to broadband antennas and devices.

In this part, two lenses that perform different functionalities are presented. Both of these two lenses are designed by the method of transformation optics based on Laplace's equation. This part is composed of three chapters; Chapter 4 to Chapter 5. In Chapter 4, a functional lens, which is capable of restoring in-phase emissions over a broad frequency range, is designed, fabricated and validated experimentally. In Chapter 5, we introduce a lens capable of changing the direction of propagation of an electromagnetic radiated beam. This beam steering lens is also designed using transformation optics based on Laplace's equation. Two different prototypes are proposed for this design. One makes use of electric LC resonators and is fabricated by Printed Circuit Board (PCB) technology, and the other uses an all-dielectric material and is fabricated through 3D printing technology. Microwave antennas conformed on non-planar surfaces are of great interest to the scientific community and to the large public because they can potentially provide communication in any direction and more importantly, they can be easily mounted on masts or embedded in the skin of vehicles and therefore be drown in the landscape such that they become less visually intrusive.

RESTORING IN-PHASE EMISSIONS FROM NON-PLANAR RADIATING ELEMENTS USING WAVE-MATTER INTERACTION

distorted due to the different phase shifts caused by the varying path lengths of the electromagnetic (EM) waves due to the location of the individual antennas on the curved surface.

Therefore to avoid the degraded antenna performances, complex systems such as phase shifters are essential to compensate for the introduced phase shifts. But today, with the advent of metamaterials and space transformation concept, novel low cost devices can be proposed to replace the complex phase shifting systems. Recently, theoretical studies have shown the possibility of enhancing the directivity through the use of a QCTO-based lens conformed on a non-linear array of radiating elements that follow a cylindrical or other arbitrary shape [START_REF] Oliveri | [END_REF]124]. The lens is capable of restoring in-phase emissions from a conformal array of antennas so as to obtain performances similar to that of a linear one, as presented in Figure 4-2. The lens totally changes the emission of the cylindrical antenna array and transforms a diffused emission into a directive one. However, the lens in these works is quite large compared to the antenna array and has been evaluated only theoretically.

In this chapter, we present the theoretical formulation to transform the radiation emitted by a cylindrical array into that of a planar one. We then synthesize the material to fabricate the conformal lens and we perform the experimental demonstration of in-phase restoration of the emission by the proposed lens over a wide frequency range spanning from 8 to 12 GHz.

Laplace's equation is used to determine the constitutive electromagnetic parameters of the transformation medium. The conformed cylindrical antenna array used as excitation source radiates through the conformal lens corresponding to the transformed medium, which restores radiation patterns similar to those of a planar antenna array. Full wave simulations based on finite element method is used to validate the design method. An all-dielectric prototype is fabricated through easily available inexpensive 3D printing technology. Measurements are performed and reported experimental near-field distributions and far-field radiation patterns demonstrate clearly broadband focusing properties of the lens and in-phase directive emissions.

Design of the directivity enhancement lens 4.2.1 Concept and analytical design

The design of the conformal lens is presented in Figure 4-3. In the x-y plane, we suppose that the coordinate transformation between the physical space (x', y') and the virtual space (x, y) is x = f(x', y') and y = g(x', y'). The mathematical equivalence of this mapping is expressed by a Jacobian matrix J whose elements are defined by

    ' , ' , y x y x J   
. By solving Laplace's equations in the virtual space with respect to specific boundary conditions, J can be obtained: The physical space performs an inverse function of the virtual space. Thus the Jacobian matrix of this inverse transformation from (x, y) to (x', y') can be represented by J -1 . Here we assume that the conformal module of the virtual space is 1 while the conformal module of the physical space is M. Once J -1 is known, the properties of the intermediate medium can be calculated. In

         A D C B D C B A B A A D D C C B y n y y y x n x x     (4-2)
where nˆ is the outward normal to the surface boundaries.

It can be observed that the effective property tensors obtained from Laplace's equation are not isotropic in the x-y plane anymore. But if the conformal module M of the physical space is not quite different with the conformal module of the virtual space, which is 1 in this case, Li and Pendry suggested that the small anisotropy can be ignored in this case [79]. The near-isotropy resulting from the quasi-conformal mapping leads to the approximation. For simplicity, the transformation deals with a two-dimensional (2D) model with incident TE-polarized wave having only a z-directed component. Considering the polarization of the excitation, the properties of the intermediate medium can be further simplified as:

1 , ) det( 1       J r (4-3)
where r is the permittivity of free space. The determination of the mapping is introduced by solving Laplace's equations subject to predefined boundary conditions by Comsol Multiphysics Partial Differential Equation (PDE) solver [117].

Finite element method based 2D numerical simulations with COMSOL Multiphysics are firstly used to numerically validate the calculated transformed conformal lens. Scattering boundary conditions are set around the computational domain and an array of eight dipole antennas is used as source. The different dipoles are excited with equal amplitude and phase so as to obtain a directive beam at boresight direction. The electric field of the exciting sources is polarized along the z direction. The lens is considered to be 2 cm thick and is parameterized with R = 29 cm and r = 27 cm. Our main aim in this chapter is to recover a directive beam in the main lobe of a conformal array of radiators, similar to that of a planar one. For this, we consider the simplified lens with only isotropic permittivity which can be fabricated from all dielectric materials. Moreover such materials provide wide-band operation compared to the use of resonant metamaterials. However, Chapter 4: Restoring in-phase emissions from non-planar radiating elements using wave-matter interaction -99-such simplified dielectric lens shows radiation leakage on its lateral sides.

Dimension study of the lens

For the dimension of the lens, we intend to find a smaller one in size which is easy for possible fabrication. This small lens also should provide achievable parameter distribution. Of course, the quality of the functionality is also supposed to be guaranteed. For these reasons, we launch a parametric study on the dimension of the lens.

Parametric study of the thickness of the lens

In this study, we set r = 27 cm and d = R-r. We vary d in the following discussion and evaluate the permittivity distribution inside the lens region. As shown in Figure 4-8, d is set to 2 cm, 4 cm and 6 cm respectively. In Figure 4-8(a), when d = 6 cm, the range of the permittivity starts from -0.84 to 1.89. The highest value is located at the bottom center of lens. And the lowest value mostly distributes on the two sides of the lens. In Normally you study a conformal lens for a given curvature radius. This radius does not change in the study. Then this parametric study seems inappropriate? But you could make a parametric study concerning the lens length.

In order to study the grade of the conformation (how much the lens is conformed), we continue the study of the lens by evaluating the radius of the inner factor of the lens. In this study, we keep the straight-line distance AB in Figure 4-3(a) the same. Moreover, the thickness of the lens

d = R -r = 2 cm.
The inner radius r is chosen to be 22.5 cm, 27.5 cm and 32.5 cm respectively.

As we can see in Figure 4-10, the range of the permittivity distribution decreases as r increases.

The maximum value of zz reduces from 3.26 to 2.45. And the minimum value of zz increases from -1 to -0.31. Of course, if r is large, the bottom boundary is more close to the planar ground plane. In another word, the conformal grade is quite small. The radiated E-field is presented in Figure 4-11. The first observation we can make is that in Figure 4-11(a) and (d), is the fact that despite the lens is able to focus the radiated beam, there is a leakage on both sides of the lens. In this case r = 22.5 cm, and the thickness of the lens is 2 cm, the lens region is too small compared to the region that we conform. We can also state here that the conformal module in too large for such case. In this situation, when we ignore the anisotropy and keep only zz, the lens doesn't perform well any more. On the other side for the other two lenses, the focusing phenomenon is very clear and no parasitic lobes are observed.

2D discretization and full-wave simulations

A 2D discrete lens is designed for further realistic full-wave numerical simulations. The dimensions of the lens are set as r = 27 cm, d = 2 cm. The permittivity profile of the designed lens is shown in Figure 4-12(a). According to effective medium theory, if the operating wavelength is large enough with respect to the size, the composite material can be considered isotropic and homogenous. We therefore propose a discrete lens model, as presented in Figure The conformal array presents a distorted diagram with a lower radiation level than the planar array. The use of the lens above the conformal array allows restoring the in-phase emission to create a radiation pattern with a clear directive main beam. The transformed lens is capable of retaining similar performances to the planar microstrip patch array. Although the second lobe is still a little bit high, the difference between the first lobe and second lobe is about 12 dB.

We further continue to study this lens at several other frequencies. The same simulation as in The far field patterns at 8 GHz and 12 GHz are presented in Figure 4-16. The results here at both of the two frequencies present similarities in the sense that the lens is able to increase the level of the first lobe of the conformal patch array. Although the pattern of the conformal patch and lens system is not identical to the pattern of the planar array, the lens can still enhance the level between of the main beam (how much?). Now we take a closer look at the red curves in reduced with a much thicker and larger lens. This is because that when we ignore the anisotropy of the material parameters, the index on the two sides of the lens is changed.

Dielectric material analyse and 3D prototype proposal

Since only zz varies from 1.5 to 2.6 in the lens design, we are able to consider using it over a broad frequency range. The lens is thus realized from non-resonant cells. Air holes in a dielectric host medium of relative permittivity h = 2.8 is therefore considered. Suppose that two materials are mixed together, the effective parameter can be calculated by: By adjusting the volume fraction of the air holes in the dielectric host medium, the effective permittivity of the cell can then be adjusted. Figure 4

Full-wave simulations on 3D discrete model

Full-wave simulations using Ansys HFSS [START_REF]High Frequency Structure Simulator[END_REF] are performed to numerically verify the functionality of the lens.

A microstrip patch antenna array composed of eight linear radiating elements is used as primary source and the simulation results from 6 GHz to 14 GHz are presented in Figure 4-19. It must be noted that for each tested frequency, a different microstrip patch array is used. Indeed such patch radiators present a narrow band frequency response. Therefore to cover the 8 GHz to 12 GHz frequency band, three different arrays have been used. However, it is supposed that in real In Figure 4-20, for the planar array, a main lobe with a directivity reaching 15.3 dB is observed.

It should be noted that here a sectorial beam, i.e. a wide beam in one plane and a narrow beam in the other plane is obtained since we are using a linear array of radiating elements. For the conformal array, a lower directivity is obtained for the primary radiated lobe (12.3 dB) and a slight distortion can also be observed in the main beam. However in presence of the designed dielectric lens, the distortion disappears and a directivity of 15.7 is obtained, which agrees quantitatively with the planar array. A slightly higher directivity is even observed, confirming the fact that the lens is able to create a constructive interference between the emissions of the radiating elements of the conformal array and hence produce an overall in-phase emission. The color scale in Figure 4-20 is in decibels on the right. In Figure 4-23, the S11 of the patch array antenna and the dielectric lens antenna system are presented. The measurement is carried out at 8 GHz, 10 GHz and 12 GHz respectively in order to cover the whole X-band. The blue curves represent the S11 of the patch array and the red curves represent the S11 of the patch and lens system. It's clear that the dielectric lens keeps the same bandwidth as the patch array antenna alone. But the center operation frequency is shifted down a little.

A first experimental setup is set to scan the near-field microwave radiation. The electric field is scanned by a field-sensing monopole probe connected to one port of an Agilent 8722ES vector network analyzer by a coaxial cable.

The probe is mounted on two orthogonal linear translation stages (computer-controlled Newport MM4006), so that the probe can be translated with respect to the radiation region of the system under test. By stepping the field sensor in small increments and recording the field amplitude and phase at every step, a full 2D spatial field map of the microwave near-field pattern is acquired in free-space. The total scanning area covers 400 x 400 mm 2 with a step resolution of 2 mm. Microwave absorbers are applied around the measurement stage in order to suppress undesired scattered radiations at the boundaries. A quite overall good qualitative agreement is noted between the simulated and measured characteristics. From the different plots, a clear highly directive radiation lobe is observed at all tested frequencies for the planar array. As predicted in the 3D simulations and confirmed in the Chapter 4: Restoring in-phase emissions from non-planar radiating elements using wave-matter interaction -115-electric field cartographies, the conformal array presents a distorted diagram with a lower radiation level than the planar array. The use of the lens above the conformal array allows restoring the in-phase emission to create a radiation pattern with a clear directive main beam.

The transformed lens is capable of retaining similar performances to the planar microstrip patch array. However, some imperfections exist due to the assumption of the isotropy of the material property.

Conclusions

In summary, we have presented the experimental validation of a compact all-dielectric conformal lens operating on a wide frequency range and capable of restoring in-phase emissions from a conformal array of radiators. The lens has been designed by making use of Laplace's equations and has been tested over a wide frequency band ranging from 8 GHz to 12 GHz. Such a lens is able to compensate the out of phase distribution occurring on a conformal array of radiating elements and to allow a control of the emission in order to produce a main radiated lobe boresight direction. Furthermore, calculated and measured near-field distributions and farfield antenna patterns show directive emissions from the conformal array. The proposed method is low-cost and presents potential airborne and trainborne applications in communication systems and environments where antennas are conformed on non-planar surfaces. The method is general and can be freely applied to any 2D and 3D shapes. In this chapter, based on transformation electromagnetics, the design procedure of a lens antenna, which steers the radiated beam of a patch array, is presented. The demonstrated metamaterial lens placed above a linear array of similarly excited radiating elements deflects the radiated beam to an off-normal direction, as presented in Figure 5-1. The space transformation modifies the space above the array of radiators and thereby alters its radiating pattern. Laplace's equation is adopted to construct the mapping between the virtual space and the physical space.

To validate the design methodology, two prototypes have been fabricated and tested. The first one is a metamaterial-based lens prototype, designed using electric LC resonators. The second one is a broadband all-dielectric compact low-cost lens prototype, fabricated through threedimensional (3D) polyjet printing technology. An array of radiators composed of four planar microstrip antennas realized using standard lithography techniques is used as excitation source for the lens. Both the far-field radiation patterns and the near-field distributions are measured and reported. Measurements agree quantitatively and qualitatively with numerical full-wave simulations and confirm the corresponding steering properties. It is observed that by placing the lens on a feeding source instead of phase shifters, we are able to steer the radiation emitted by the latter source with a lens that can operate over a large frequency band.

The experimental validation of the lens paves the way to a wide range of potential applications in electromagnetics and daily communications employing lenses and antennas. This concept of wave control can be applied in other scientific engineering domains such as energy harvesting or radiation illusion. More generally, the ability to modify the space around an object so as to transform its radiating behavior or its electromagnetic response can lead to other unprecedented applications in the fields of wave propagation.

Designs of the beam steering lens and 2D analysis

The scalar two-dimensional Helmholtz equation is form invariant with respect to coordinate transformations which are equivalent to a conformal mapping. Thus QCTO is an approximate solution of minimizing the anisotropy for general boundary conditions. Here, we use QCTO to propose the design of a lens capable of deflecting electromagnetic waves. To transform the angle between the wave vector k and y-axis from 0° to 45°, we first study two different designs based on Laplace's equation in this section.

Beam steering lens design 1

In this first example we want design a lens which steers the beam of a directive antenna at 45°.

                      A D D C C B B A A D D C C B y n W x y y y x n x x  (5-2)
where nˆ is the outward normal to the surface boundaries. The designed lens is fully parameterized with W = 1 m and H = 0.5 m too have a super directive source at 10 GHz. The properties of lens design 1 are shown in Figure 5-3.

It can be observed that the effective property tensors obtained from Laplace's equation are not isotropic in the x-y plane. But if the conformal module M of the physical space is not quite different with the conformal module of the virtual space, which is 1 in this case, Li and Pendry suggested that the small anisotropy can be ignored in this case [79]. Considering the polarization of the excitation, the properties of the intermediate medium can be further simplified as:

    ' , ' , y x y x J    (5-1) 0 ' ' , 0 ' '

Numerical validation of full parametric lens (design 1)

In this section, finite element method based numerical simulations are used to design and characterize the proposed transformed beam steering lens. In the simulation model using Comsol Multiphysics, scattering boundary conditions are set around the computational domain.

Study on the height dimension of full parametric lens

For the proposed dimension, lens design 1 has shown consistent behavior as in the virtual space.

Then we begin to wonder what will happen if we change the dimension of the lens. So we perform a parametric study on the height of the lens H. In the original design, H was set to 0.5 m. In the following study, we will test one model whose height is reduced to 0.2 m and another model whose height is increased to 0.8 m. The length of the source Ls remains the same as before.

The material properties of lens design 1 analysed from COMSOL Multiphysics are shown in the material parameter will be higher. In this study, we keep the ratio W/H = 2. We first try the check the field distribution by reducing the size of the lens from 1m to 30 cm, then two other dimensions are studied (20 cm and 10 cm). It is clear that even though the sizes of the lens we present in Figure 5-11 are different, the field distributions are the same. It is important at this stage to note that the deflection angle depends mostly on the material parameters of the lens and very little on the size of the lens.

Frequency study on the lens design 1 after parameter reduction

Lens design 1 is supposed to be a wideband lens which is not limited by frequency. So we tested this design model at several different frequencies.

As we have shown in the dimension study, the size of the lens can be shrunk. In this study, the dimension of the lens is W = 10 cm and H = 5 cm. The length of the source Ls = 5 cm, presented as the red segment in Figure 5-12. As shown in Figure 5-12 below, the field distributions at 8

W=30 cm W=20 cm W=10 cm H=15 cm H=10 cm H=5 cm (a) (b) (c) 1 -1                           E H G F H G F E E H H G G F y n W x y y y x n x x  (5-4)
-132-These parameters are anisotropic before the parameter reduction. The radiated beam is deflected by 45° after propagating through the lens. From this simulation, we can note that the functionality of the lens design 2 is capable of the beam steering functionality.

Parameter reduction

The parameter we assign to the lens in Figure 5 The conformal module of the lens design 2 is much smaller than the conformal module of lens design 1. In another word, the anisotropy of this design is small enough to be neglected. So when we keep only on permittivity component, the lens does not lose the steering functionality. Since the electric field of the source is polarized along z direction, we further simplify the design by keeping only the properties along z direction, which is zz in this case. -135-

The discretization has caused reflection between the unit cells and on the boundary of the lens. Therefore, instead of 28°, the discrete lens is able to steer the beam by 22.5°.

Experimental validation using electric LC resonators

We fabricate the transformed lens by using electric LC (ELC) resonators and we experimentally validate the proposed lens in the microwave domain around 10 GHz. As known to all, the metamaterial suffers the narrow bandwidth limitation. But due to our fabrication resources, where it is impossible to realize permittivity values above 2.8 from all-dielectric material as in the previous chapter, the only way to achieve such values is by using metamaterials. Here, we make use of ELC resonators. However, such solution can be quite complex due to the large number of different needed values. Therefore, to demonstrate experimentally the mechanism of beam deflection, a prototype is fabricated using low cost printed circuit board (PCB) technology, as shown by the photography in Figure 5 A 3D discrete lens is designed for further realistic numerical simulations. The permittivity profile of the designed lens is shown in Figure 5-18. Full-wave simulations using Ansys HFSS 33 have been performed on the 3D discrete lens to verify the beam steering functionality. The microstrip patch antenna array, which operates at 10 GHz as we introduced just before, is used as primary source and the corresponding radiated emissions at 10 GHz are presented in A quite overall good qualitative agreement is noted between the numerical and experimental performances. The characteristics obtained from the lens are further compared to the excitation source alone so as to show the influence of the lens. As it can be clearly observed from the

Experimental validation using all-dielectric material

Besides the ELC metamaterial lens, an all-dielectric lens prototype presenting a graded permittivity profile is also fabricated for potential wide frequency band through threedimensional (3D) polyjet printing technology. The dielectric material used for this fabrication is the same as the one we have mentioned in Chapter 4 for the in-phase emissions restoring lens.

As known, the permittivity of this material in our polyjet printing fabrication facilities is 2.8.

We decide to fabricate the part of the beam steering lens that has the permittivity value below 2.8. Therefore the part between 2.8 to 4 is discarded.

We therefore propose a discrete lens model, as presented in As we can see here, although we have neglected part of the lens that contained high permittivity value, the lens performs almost the same functionality with the original one. The emitted beam is steered by an angle around 22.5° at all these three frequencies.

A 3D discrete lens is designed for further realistic numerical simulations. The permittivity profile of the designed lens is shown in Figure 5-29(a). The profile is divided into 170 cubic cells. In order to be able to use the lens over a broad frequency range, the lens is realized from non-resonant cells. Air holes in a dielectric host medium of relative permittivity h = 2.8 is therefore considered. Full-wave simulations using Ansys HFSS have been performed on the 3D discrete lens to verify the beam steering functionality. The microstrip patch antenna array composed of four equally fed linear radiating elements is used as primary source and the corresponding radiated emissions at 8 GHz, 10 GHz and 12 GHz are presented in Figure 5-32. Therefore to cover the 8 GHz to 12

GHz frequency band, three different arrays have been used. However, it is supposed that in real applications, a primary source presenting a wideband frequency response will be used. Here a sectorial beam, i.e. a wide beam in one plane and a narrow beam in the other plane is obtained since we are using a linear array of radiating elements. As it can be observed in Figure 5-32(c), in presence of the designed 3D dielectric lens, the radiated wavefronts undergo a deflection of 28°, confirming the 2D simulation results and the fact that the lens is able to modify the direction of wave propagation. Moreover, the lens is able to enhance the directivity of the radiated beam from 16 to more than 31.

In Figure 5-33, the S11 of the patch array antenna and the dielectric lens antenna system are presented. The measurement is carried out at 8 GHz, 10 GHz and 12 GHz respectively in order to cover the whole X-band. The blue curves represent the S11 of the patch array and the red curves represent the S11 of the patch and lens system. It's clear that the dielectric lens keeps the same bandwidth as the patch array antenna alone. But the center operation frequency is shifted down a little.

Conclusion

To conclude, in this chapter, we have proposed a concept to manipulate electromagnetic waves and design a beam steering lens. The latter manipulation is enabled by using quasi-conformal transformation and Laplace's equation is utilized to construct the mapping between the virtual space and the physical space. Numerical verifications have been performed on two different designs; one where the conformal module of the mapping is close to 1 and another one were the conformal module is far from 1. The radiation of a current line source has been steered off the normal in both designs. It has also been observed that when ignoring material anisotropy in the design where the conformal module is far from 1, the beam steering functionality is weakened and hence, beam steering angle is less compared to the case where anisotropic parameters are used. However, when the conformal module is close to 1, material anisotropy can be ignored without deterioration of the steering performances.

We have also presented the experimental validation of a metamaterial based beam steering lens when the conformal module is far from 1. ELC resonators have first been used to tailor the material parameters of the lens by a judicious engineering of the geometrical dimensions. The lens has been tested experimentally on a frequency band spanning from 9.3 GHz to 10.7 GHz.

Another experimental realization of a compact all-dielectric beam steering lens operating on a wider frequency range has then presented. The lens has been tested over a broad frequency band spanning from 8 GHz to 12 GHz. Such lenses are able to both enhance the directivity and modify the direction of propagation, allowing a control of the emission. The concept has been validated through calculated and measured near-field distributions and far-field antenna patterns.

The proposed method is ease of fabrication, low-cost and presents potential airborne and trainborne applications in communication systems and environments where radiation direction needs to be controlled in a passive way. The proposed designs open the way to a wide range of potential applications in electromagnetics and daily communications employing lenses and antennas.

Conclusion -151-

CONCLUSION

Transformation optics is an efficient tool to manipulate the electromagnetic field distributions.

The form-invariance of Maxwell's equations is exploited to design complex electromagnetic media with desired and particular properties. The device that is assigned by the designed electromagnetic media can perform a certain functionality that have never been demonstrated before or that are very difficult to realize with naturally common materials.

Coordinate transformation is able to compress, rotate or fold the space at will. It is a useful and powerful concept that can really control the field at one's will. It has been demonstrated in this thesis that, by applying a two-step linear transformation, an illusion device can be designed based on coordinate transformation. This illusion device is able to change the radiation pattern and to make the radiation location appear outside the latter space. In addition, a tapering device can be designed by several kinds of transformations. However, the material properties designed from coordinate transformation are always complex or singular. In most case, the parameter tensor consists of non-diagonal components whose value are either negative or too high to be achieved. Therefore, although the designs based on coordinate transformation are usually fascinating, very few of them can be realized. Even they can be realized, normally resonant metamaterials are required, which is limited by narrow frequency band.

Transformation optics based on Laplace's equation is another method to reshape the electromagnetic field. Other than mapping the virtual and physical space from point to point as in coordinate transformation, the mapping can also be established by solving Laplace's equation with specified boundary conditions. A beam focusing lens and a beam steering lens are designed by this concept and validated experimentally. Transformation based on Laplace's equation provides a much easier way for the device designing. Indeed for quasi-conformal transformation optics, an in-plane isotropic material can be applied to the designs which can further be realized by non-resonant all-dielectric materials, allowing a broad band operations. However, this concept requires the transformed virtual and physical space to be quite similar with each other, where no subversive mapping can be established.

Transformation optics therefore requires the use of engineered values of permittivity and permeability. Metamaterials which are complex artificial materials can be used to fabricate structures which mimic a material response that has no natural equivalent. This capability has led to an explosion of interests in the physical properties of these materials and the ways in which they can be used. Two structures of resonant metamaterial resonators are implemented in this thesis to realize near zero permeability values and high permittivity values. But these resonators present the drawback of having narrow-band resonances. Thus, frequency independent designs from transformation optics are limited by the resonators. All-dielectric materials are generally utilized for QCTO designs. Such materials can be for example engineered by dielectric 3D printing as it has been done in this thesis. These prototypes operate on a wide frequency to fabricate the isotropic designs. Keywords : transformation optics, metamaterials, lens, antenna, gradient index and 3D printing Abstract : The concept of transformation optics which can control electromagnetic waves at will by applying a judiciously defined spatial variation in the constitutive parameters is explored to design novel type antennas and microwave devices. In a first part, based on coordinate transformation, an illusion device able to modify the appearance of an electromagnetic emission and then delocalize it is validated through numerical simulations. A waveguide tapering device allowing to assure quasi-total transmission between two waveguides of different crosssections is also designed and validated experimentally through the use of metamaterial resonators.

In a second part, space transformation based on Laplace's equation is studied to design lenses for antenna applications. A beam focusing lens able to restore in-phase emissions from a conformal array of radiators and a beam steering lens that allows deflecting the radiated bema of an antenna are designed. Prototypes fabricated by an alldielectric isotropic material through threedimensional (3D) polyjet printing technology allow to experimentally validate the functionality of the lenses on a wide frequency range.

  I: HU, HV, AND HW EXPRESSIONS IN DIFFERENT COORDINATE SYSTEMS. ........................ 15TABLE 1-II: EXPRESSIONS OF ROTATION MATRIX R IN DIFFERENT COORDINATE SYSTEMS. ...... 16 TABLE 2-I: DISCRETIZATION OF THE PARAMETERS FOR THE LAYERED MATERIAL SHELL. ........ 61 TABLE 3-I: COMPONENTS VALUES OF  TENSOR FOR THE TRANSFORMATION OF TWO TAPER PARTS. ................................................................................................................................. 77 List of Figures -xiii-

FIGURE 1 - 1 : 4 FIGURE 1 - 2 : 6 FIGURE 1 - 3 : 6 FIGURE 1 7 FIGURE 1 - 5 : 8 FIGURE 1 - 6 : 9 FIGURE 1 - 7 : 10 FIGURE 1 - 8 :

 11412613617158169171018 FIGURE 1-1: (A) METAMATERIAL IMPLEMENTATION-CLOAKING DEVICE [1]. (B) A MODEL OF 3D

FIGURE 1 - 9 : 18 FIGURE 1 - 19 FIGURE 1 - 21 FIGURE 1 - 23 FIGURE 1 - 26 FIGURE 1 - 27 FIGURE 1 - 28 FIGURE 1 - 32 FIGURE 1 - 33 FIGURE 1 - 34 FIGURE 1 - 35 FIGURE 2 - 41 FIGURE 2 - 2 : 43 FIGURE 2 - 3 :

 1918119121123126127128132133134135241224323 FIGURE 1-9: DISTRIBUTION OF THE ELECTRIC FIELD COMPONENT NORMAL TO THE PLANE OF

45 FIGURE 2 - 4 : 46 FIGURE 2 - 5 : 50 FIGURE 2 - 7 : 51 FIGURE 2 - 8 : 53 FIGURE 2 - 9 : 55 FIGURE 2 - 10 : 57 FIGURE 2 - 58 FIGURE 2 - 59 FIGURE 2 - 60 FIGURE 2 - 61 FIGURE 2 - 62 FIGURE 2 - 63 FIGURE 3 - 65 FIGURE 3 - 2 : 67 FIGURE 3 - 3 :

 452446255027512853295521057258259260261262263365326733 FIGURE 2-4: A NEGATIVE REFRACTIVE INDEX MEDIUM BENDS LIGHT TO A NEGATIVE ANGLE

68 FIGURE 3 - 4 : 70 FIGURE 3 - 5 : 71 FIGURE 3 - 6 : 71 FIGURE 3 - 7 : 72 FIGURE 3 - 9 : 73 FIGURE 3 - 74 FIGURE 3 - 75 FIGURE 3 - 75 FIGURE 3 - 76 FIGURE 3 - 79 FIGURE 3 - 80 FIGURE 3 - 80 FIGURE 3 - 81 FIGURE 3 - 82 FIGURE 3 - 84 FIGURE 3 - 85 FIGURE 3 - 86 FIGURE 4 - 91 FIGURE 4 - 2 : 93 FIGURE 4 - 95 FIGURE 4 - 4 : 97 FIGURE 4 - 5 : 98 FIGURE 4 - 6 : 98 FIGURE 4 - 7 : 99 FIGURE 4 - 8 :FIGURE 4 - 9 :

 6834703571367137723973374375375376379380380381382384385386491429349544974598469847994849 FIGURE 3-4: SKETCH DIAGRAM OF FOUR KINDS OF TRANSFORMATION. (A) COSINUSOIDAL, (B)

102 FIGURE 4 - 103 FIGURE 4 - 104 FIGURE 4 - 105 FIGURE 4 - 105 FIGURE 4 - 106 FIGURE 4 - 16 : 107 FIGURE 4 - 108 FIGURE 4 - 109 FIGURE 4 - 110 FIGURE 4 - 111 FIGURE 4 - 112 FIGURE 4 - 112 FIGURE 4 - 113 FIGURE 4 - 114 FIGURE 4 - 115 FIGURE 5 - 1 : 117 FIGURE 5 - 2 : 119 FIGURE 5 - 3 : 122 FIGURE 5 - 5 123 FIGURE 5 - 6 : 124 FIGURE 5 - 7 :

 10241034104410541054106416107410841094110411141124112411341144115511175211953122551235612457 FIGURE 4-10: PERMITTIVITY DISTRIBUTION OF DIFFERENT RADIUS. (A) R = 22.5 CM. (B) R = 27.5

126 FIGURE 5 - 9 : 127 FIGURE 5 - 127 FIGURE 5 - 129 FIGURE 5 - 129 FIGURE 5 -

 126591275127512951295 FIGURE 5-9: NORMALIZED ANTENNA RADIATION PATTERNS SHOWING THE BEAM STEERING

FIGURE 5 - 24 :

 524 FIGURE 5-19: (A) REAL PART OF ELECTRIC FIELD AND (B) NORM OF THE ELECTRIC FIELD

  Figure 1-1(a) shows the most well-known metamaterial implementation-cloaking at microwave frequencies [1]. And Figure 1-1(b) shows a model of 3D metamaterial presented by the U.S. Department of Energy's Ames Laboratory [2].

Figure 1 - 1 :

 11 Figure 1-1: (a) metamaterial implementation-cloaking device [1]. (b) A model of 3D

  is antiparallel to the Poynting vector-seemingly violating the right-hand rule.

Figure 1 - 2 :

 12 Figure 1-2: Effective medium parameter diagram showing the four quadrants of material

Figure 1 - 3 :

 13 Figure 1-3: (left) Comparison of refraction of an incident ray propagating into a positive

Figure 1 - 4 :

 14 Figure 1-4: (a) geometrical interpretation of the emission of a source inside a slab of a low

Figure 1 - 5 :

 15 Figure 1-5: Metamaterial liner geometry, properties and incorporation into a rectangular

Figure 1 - 6 :

 16 Figure 1-6: Configuration of (a) the quarter-wave monopole antenna and (b) the same

Figure 1 - 7 :

 17 Figure 1-7: Diagrams showing (a) actual picture of a lens disk, (b) blow-up illustrating

Chapter 1 :

 1 Transformation Optics: Definition and related tools -13-

'



  are the components of the permittivity and permeability tensors in the original Cartesian coordinate.



  are the components of the permittivity and permeability tensors of the transformed media in the new Cartesian coordinate. The subscript C stands for Cartesian coordinate system. And the subscript O in the following equations stands for other orthogonal coordinate systems.

Figure 1 - 8 .

 18 Figure 1-8.The formulation of this transformation is also presented in the figure. The variable k is a scale factor and  is the angle of the bend.

Figure 1 - 8 :

 18 Figure 1-8: Illustration of the embedded coordinate transformation for  = π/5 [81].

Figure 1 -

 1 9(a) the material properties were calculated for γ = 0.02 whereas in Figure1-9(b) the tensors were obtained for γ = 0.3. As can be seen from Figure1-9, each material realization bends the incoming light by α = π/2.-18-

Figure 1 - 9 :

 19 Figure 1-9: Distribution of the electric field component normal to the plane of propagation

Chapter 1 :

 1 Transformation Optics: Definition and related tools -19-1.2.2.1 Transformations based on Laplace's equation with Dirichlet boundary conditions As we have mentioned in Eq. 1-11, the permittivity '

  Laplace's equation with only Dirichlet boundary conditions[82,83] [101].

Figure 1 - 10 :

 110 Figure 1-10: The scheme of constructing an arbitrary cloak [82].

Figure 1 -

 1 Figure 1-10 shows the scheme for constructing an arbitrary shaped cloak. Suppose an original

Figure 1 - 11 :

 111 Figure 1-11: Material parameters (a) zz, (b) xx, (c) xy and (d) yy. Electric field

  Dirichlet and Neumann boundaries to simultaneously fix the geometry of the transformed domain and maintain orthogonality on the boundary.

Figure 1 - 12 :

 112 Figure 1-12: Depiction of the quasi-conformal mapping using intermediate

Figure 1 -

 1 Figure 1-13 shows an example of coordinate transformation design in COMSOL. In the area of

Figure 1 - 13 :

 113 Figure 1-13: Interface of COMSOL Multiphysics.

Chapter 1 :

 1 Transformation Optics: Definition and related tools -27-assigned as the material properties of the transformed zone.1.3.3 ANSYS HFSSANSYS HFSS software is a commercial software for simulating 3-D full-wave electromagnetic devices. It is one of several industrial tools used for antenna design, and the design of complex RF electronic circuit elements including filters, transmission lines, and packaging.In HFSS, with the help of the drawing tools we are able to simulate a 3-D model that is quite similar to a realistic model. In my thesis, the first utilization of HFSS is to design antennas that can feed the lenses we proposed by transformation optics. Then together with the sources, 3-D models of the lens-antennas can be simulated as in real case.As show by the example inFigure 1-14, the patch antenna is feed by a circular waveport and conformed on to cylindrical surface. The dielectric material of the lens is assigned the same properties of the material for fabrication. Air holes of different radius made in the lens as in 2-D designs.

Figure 1 - 14 :

 114 Figure 1-14: Sketch of simulation setup of antennas.

Figure 1 - 15 :

 115 Figure 1-15: Sketch of simulation setup of unit cells.

Section 1 . 4 :

 14 Experimental setups -30-stage in order to suppress undesired scattered radiations.

Figure 1 - 16 :

 116 Figure 1-16: Near field scanning experimental setup.

Figure 1 - 17 :

 117 Figure 1-17: Far field scanning experimental setup.

Figure 1 -

 1 18(a). This resonator consists of an inductive loop connected to a capacitive element. Only the capacitive element formed by the central gap couples strongly with the field electrical (Figure1-18(b)) to produce an electric response.

Figure 1 - 18 :

 118 Figure 1-18: Electric-LC resonator [97]. (a) A typical ELC resonator is composed of a

Chapter 1 :

 1 Transformation Optics: Definition and related tools -33-

Figure 1 - 19 :

 119 Figure 1-19: Sketch of simulation setup of ELC resonators.

Figure 1 - 20 :

 120 Figure 1-20: Split ring resonator [98]. A typical SRR is composed of two rings with gaps

Figure 1 - 21 :

 121 Figure 1-21: Photograph of Objet Eden260VS 3D printer and its product.

  Section 1.6: Conclusion -36-based on metamaterials are introduced at first. Then the design concepttransformation opticsis introduced. Then coordinate transformation and transformations based on Laplace's equation, including transformation with only Dirichlet boundary conditions, conformal mapping and quasi-conformal transformation optics, are introduced. The numerical calculation softwares, near field and far field experimental setups we use for the thesis are presented. For the realistic validation (simulated and experimental) of the designs, electric and magnetic metamaterial resonators and dielectric materials engineering are also presented. All the design concepts, numerical calculation tools, validation technologies and experimental setups will be used for the designs of the microwave devices we propose in the thesis. -37-PART I DESIGNS OF MICROWAVE DEVICES BASED ON COORDINATE TRANSFORMATION -38-

Chapter 2 :Figure 2 -

 22 Figure 2-1:Diagrammatic sketch of the function of the Delocalization device.

Figure 2 -

 2 Figure 2-2(a)-(c) show the concept of the optical transformation of an object into another object.

Figure 2 - 2 :

 22 Figure 2-2: Simulated electric field distributions of (a)-(c) the optical transformation of an

Figure 2 -

 2 Figure 2-2(d)-(f) represent the electric field distribution of the shrinking device of an arbitrary

Figure 2 -

 2 Figure 2-2(g)-(i) represent the field distribution of a point source transmitting through a ghost

Figure 2

 2 Figure 2-3.

Figure 2 - 3 :

 23 Figure 2-3: Principle of the transformation: A detector will observe the electromagnetic

Figure 2 - 4 :

 24 Figure 2-4: A negative refractive index medium bends light to a negative angle with the

- 5 .

 5 Such transformation consists in stretching a part of the space contained in a radius outside the shell and compressing it in the radius r = R of the material, as illustrated in Figure 2-3. In Figure 2-3(a), an observer detects the source at the location of the long red line. But this source is a virtual source created by the illusion. The real source is located at the short red line inside the material shell, shown in blue. The properties of the material in the shell are related to the sinusoidal transformation Chapter 2: Illusion optics: Optically transforming the nature and the location of electromagnetic emissions -47-presented in Figure 2-3(b). By doing continuous transformation, a part of a space is folded near the boundaries of the material shell. This continuous transformation (not embedded) is represented in Figure 2-3(c) where the material shell is represented in blue line, which has not been transformed to assure perfect space metric matching at the boundary of the shell.

Figure 2 - 5 :

 25 Figure 2-5: Principle of the sinusoidal transformation: An observer outside the region

3 ) 4 )

 34 The permittivity and permeability tensors ψ are then calculated as ) of such calculation are given by (1-4) in the cylindrical base: This final components in the Cartesian coordinates are given by the expression (2-5) and are calculated using passage matrix into the 2 different coordinate system:

  extension region. These two parts are opposite. Negative values can be possibly achieved. But the extremum values of zz, -11 and 53, are almost impossible to realize.

Figure 2 - 6 :

 26 Figure 2-6: Representation of the components of the permittivity and permeability tensor

Figure 2 - 7 :

 27 Figure 2-7: The transformation used to create such device is achieved with two steps of

Chapter 2 :

 2 Illusion optics: Optically transforming the nature and the location of electromagnetic emissions -53-distance D = 0.07 m at r maximum (or r' = R1).

Figure 2 -

 2 Figure 2-8 shows the variation in the permittivity tensor components in the transformed space.

Figure 2 - 8 :

 28 Figure 2-8: Variation of the material parameter tensor components ψrr, ψθθ and ψzz with q

Chapter 2 :

 2 Illusion optics: Optically transforming the nature and the location of electromagnetic emissions -55-at r = 11.4 cm and a delocalization distance D = 9.5 cm, as shown in Figure 2-9(c). When q = 7.5, this transformation created a virtual source outside the material shell at r = 14.25 cm and a delocalization distance D = 12.35 cm, as shown in Figure 2-9(d).

Figure 2 - 9 :

 29 Figure 2-9: Electric field (Ez) distribution of a point source placed at the position (x, y) =

  -9(b) and (c) are in good agreement with the schematic of Figure 2-7(b) where it is obvious that three emissions points (the real and the two virtual sources) are present. They are defined by the projection of the source on the r axis directly on the r' axis; there are 2 intersections with the linear transformation curve and one intersection with the curve r = r'.

.

Figure 2 - 10 : 1 Chapter 2 :

 21012 Figure 2-10: Electric field (Ez) distribution in the case of a linear transformation with q =

Figure 2 - 11 :

 211 Figure 2-11: Electric field distribution in the case of a linear transformation with q = 4.5

Figure 2 -

 2 11(d) -(f), we continue another study of the positions of the real source in the extension region. When the real source moves further from the centre of the material shell, the position of the virtual source moves back closer to the centre of the shell. This can be explained by the principle of this linear transformation as shown in Figure 2-7 in the extension zone.

Figure 2 - 12 : 1 Chapter 2 :

 21212 Figure 2-12: Electric field (Ez) distribution of a point source placed at the position (x, y) =

Figure 2 - 13 :

 213 Figure 2-13: Electric field (Ez) distribution of a point source placed at the position (x, y) =

  three effective parameters, which are zz, rr and . The discretized zones are illustrated in Figure 2-14(a) and (b).

Figure 2 - 14 :

 214 Figure 2-14: Representation of the linear transformation and the layered material for a

  the performances of the device are presented in Figure 2-15.

Figure 2 - 15 :

 215 Figure 2-15: Magnetic field distribution for TM wave polarization at 10 GHz. (a) Source

Figure 2 -

 2 15(b) and (c) show the field distribution when the source is placed at 1.9 cm from the centre in respectively the continuous

Figure 2 - 17 :

 217 Figure 2-17: High loss in the negative index deteriorates the delocalization.

Figure 3 -

 3 Figure 3-1:Sketch diagram of the waveguide tapering functionality.

Figure 3 - 2 :

 32 Figure 3-2: Transformed taper (green lines) between two waveguides (black lines) with

Figure 3 - 3 :

 33 Figure 3-3: Representation of components of the permittivity and permeability tensors 

Figure 3 - 4 :

 34 Figure 3-4: Sketch diagram of four kinds of transformation. (a) cosinusoidal, (b) parabolic,

Figure 3 -Figure 3 - 5 :

 335 Figure 3-5 shows the permittivity and permeability distributions of this transformation. For xx

Figure 3 - 6 :

 36 Figure 3-6: (a) Real part of electric field and (b) norm of electric field distribution for the

Figure 3 - 7 :

 37 Figure 3-7: Representation of components of the permittivity and permeability tensors θ

Figure 3 -

 3 Figure 3-7 shows the permittivity and permeability distributions of this transformation. For xx

Figure 3 - 8 :

 38 Figure 3-8: (a) Real part of electric field and (b) norm of electric field distribution for the

Figure 3 - 9 :

 39 Figure 3-9: Representation of components of the permittivity and permeability tensors θ

Figure 3 -

 3 10 presents the electric field and norm of electric field distributions. The wavefronts show a clear curve of a logarithmic function. In the output waveguide, the field is stronger than the one in the cosinusoidal transformation and almost similar to the one in the parabolic transformation.

Figure 3 -

 3 Figure 3-10: (a) Real part of electric field and (b) norm of electric field distribution for the

Finally a reciprocal

  Figure 3-11. For xx, zz and xy, the range of the value is achievable. But the positive value 69

Figure 3 -

 3 Figure 3-12 shows the electric field amplitude and norm distributions. In the output waveguide,

Figure 3 - 11 :

 311 Figure 3-11: Representation of components of the permittivity and permeability tensors θ

Figure 3 -

 3 Figure 3-12: (a) Real part of electric field and (b) norm of electric field distribution for the

Figure 3 - 13 .

 313 Figure 3-13. The input and output waveguides are shown as the dark red zones in free space

Figure 3 - 13 :

 313 Figure 3-13: Sketch of two transformed tapers (orange zones) and two adapters (yellow

Figure 3 -

 3 Figure 3-14 shows the variation in the permittivity and permeability tensor components

Figure 3 - 14 :

 314 Figure 3-14: Variation of the material parameter tensor components ψxx, ψxy, ψyy and ψzz

Figure 3 -

 3 Figure 3-15: (a) Real part of electric field and (b) norm of electric field distribution for the

Figure 3 - 16 :

 316 Figure 3-16: Parametric study on the starting index of the first adapter n1.

Figure 3 - 17 :

 317 Figure 3-17: Parametric study on the starting index of the second adapter n2.

Figure 3 -

 3 Figure 3-18: (a) Real part of electric field and (b) norm of electric field distribution for 2D

Figure 3 -

 3 Figure 3-19: (a) Sketch of the principle of the parameter reduction. (b) Permittivity and

  -19(b). As we can see, four parameters are reduced to three. And the maximum value is 12.5, which is achievable with metamaterial resonators. Choosing a polarization along z for the electric field, and adjusting the dispersion equation in polar coordinates in each unit cell, a further parameter reduction can be achieved. At last only two parameters are kept. The final parameter distribution after parameter reduction is then presented in Figure 3-19(c). Only two parameters are needed and the maximum value of the permittivity is 25.8.

Section 3 . 4 :

 34 Experimental validation using ELC and split rings resonator -84-

Figure 3 -

 3 Figure 3-20(a). As shown in the figure, the ELC resonator has a resonance at about 8.5 GHz.

Figure 3 -

 3 Figure 3-20(b) represents the structure of the split ring resonator used to engineer the magnetic

Figure 3 -

 3 Figure 3-20: (a) Typical ELC unit cell structure and the extracted permittivity and

Figure 3 -

 3 Figure 3-21: (a) 3D discrete model design and (b) the fabricated prototype.

  -22(a). As shown in the figure, the wavefronts are not quite planar. And the electric field of the tapering system with this source as input is also numerically calculated and presented in Figure3-22(b). The field is distorted compared to the simulations with a long wave port as input in Figure3-18. But we can still observe some compression of the fields inside the taper.

Figure 3 - 22 :

 322 Figure 3-22: Numerically calculated real part of electric field distribution of (a) 2.5 cm

  TO) has enabled to control EM fields in unprecedented and unbelievable ways through the use of judiciously engineered materials with parameters that vary spatially. The designs we introduced based on coordinate transformation have been discussed in Part I.

Figure 4 - 1 :

 41 Figure 4-1 : Diagrammatic sketch of the function of the conformal lens and the

Figure 4 - 2 :

 42 Figure 4-2: Coordinate systems for lens designs for 2-D dipole arrays on a conducting

Figure 4 -

 4 2(a) and (b) represents the conformal mapping of this principle. The planar antenna array is conformed into a cylindrical one. The electric field distribution of this lens is shown in Figure 4-2(c) and compared to the antenna array alone presented in Figure 4-2(d).

Figure 4 -

 4 3(a) represents the virtual space which a sector of a cylindrical surface. The physical space presented in Figure 4-3(b) is formed by a 54° angular sector of the annular region between the circles of radius R and r. Segments AB and  ' 'B A are perfect electric conductor boundaries, corresponding to the ground plane of the antenna array represented by the black line elements are the patches The segments DA and CB have the same length and are equal to R-r. The transformation established between the physical and virtual spaces aims to transform the conformed patch array into a linear planar one.

Figure 4 -

 4 Figure 4-3:Illustration showing the space mapping from the virtual space to the physical

Figure 4 - 4 :

 44 Figure 4-4: Permittivity distribution (a) xx., (b) xy, (c) yy, (d) zz.

Figure 4 - 5 :

 45 Figure 4-5: Real part of electric field distribution for three different configuration of the

Figure 4 - 6 :

 46 Figure 4-6: 2D far field radiation pattern at 10 GHz for three different configurations.

Figure 4 - 7 :Figure 4 - 7

 4747 Figure 4-7: Real part of electric field distribution for three different configuration of the

Figure 4 - 8 :

 48 Figure 4-8: Permittivity (zz) distribution of different thickness of the lens, (a) d = 6 cm. (b)

Figure 4 -

 4 Figure 4-8(b), when d = 4 cm, the maximum value of the permittivity has increased from 1.89 to

Figure 4 - 9 :

 49 Figure 4-9: Real part of electric field distribution emitted from the continuous lens. (a) d =

2

 2 Parametric study of the radius of the sector:

Figure 4 - 10 :

 410 Figure 4-10: Permittivity distribution of different radius. (a) r = 22.5 cm. (b) r = 27.5 cm. (c)

Figure 4 - 11 :

 411 Figure 4-11: Real part of electric field distribution emitted from the continuous lens. (a) r

Section 4 . 4 :

 44 2D discretization and full-wave simulations -104-4-12(b), which is composed of 92 unit cells (13 of the row that consists 4 cells, 12 of the row that consists 3 cells and 2 of the row that consists 2 cells). The respective permittivity of each cell is considered to be constant and equals to the permittivity value at the center of each cell. In the discrete approximation, the two edges with permittivity values close to 1 are deleted from the continuous lens such that permittivity ranges from 1.5 and 2.6 in the discrete lens.

Figure 4 - 12 :

 412 Figure 4-12: Design of 2D discrete model composed of 92 cubic cells.

Figure 4 -

 4 Figure 4-13 Real part of electric field and electric field norm distribution emitted from the

Figure 4 - 14 :Figure 4 - 15 :

 414415 Figure 4-14: 2D far field radiation pattern at 10 GHz for three different configurations.

Figure 4 -

 4 Figure 4-13 is performed at 8 GHz and 12 GHz. The field distributions, shown in Figure 4-15,

Figure 4 - 16 :

 416 Figure 4-16: 2D far field radiation pattern for three different configurations. (a) 8GHz. (b)

Figure 4 -

 4 Figure 4-14 and Figure 4-16. At 0° and 180°, the level of the pattern is much higher than the

  where a = 1 and fa and fh = 1-fa are respectively the volume fraction of the air holes and the host material, respectively.

Figure 4 - 17 :

 417 Figure 4-17: Effective permittivity of the cell composed of air hole in a dielectric host

  -17 shows the typical values of effective permittivity that can be obtained for a cubic cell in the different regions of the discrete lens. A 3D discrete lens is designed for further realistic full-wave numerical simulations. The permittivity profile of the designed lens is shown in Figure 4-12. The profile is divided into five different regions composed of 92 cubic cells.

Figure 4 - 18 :

 418 Figure 4-18: Design of the 3D discrete lens composed of five regions.

Section 4 . 6 :

 46 Full-wave simulations on 3D discrete model -110-applications, a primary source presenting a wideband frequency response will be used. The structures of the patch arrays are presented in Figure 4-19(a)-(c). And the return loss is plotted in Figure 4-19(d).

Figure 4 - 19 :

 419 Figure 4-19: The conformal microstrip patch array. (a) Resonance at 8 GHz. (b)

Figure 4 - 20 :

 420 Figure 4-20: 3D simulated radiation patterns at 10 GHz. (a) Planar array with a

Figure 4 - 21 :

 421 Figure 4-21: Simulated radiation patterns in the focusing plane (x-y plane). (a) 8 GHz. (b)

Figure 4 - 22 :

 422 Figure 4-22: Photography of the fabricated prototype.

Figure 4 - 23 :

 423 Figure 4-23: Measured S11 distribution at (a) 8 GHz, (b) 10 GHz and (c) 12 GHz, where

Figure 4 - 24 :

 424 Figure 4-24: Real part of electric field distribution at 10 GHz. (a) Planar array. (b)

Figure 4 - 25 :

 425 Figure 4-25: Measured real part of electric field distribution for the conformal array in

Figure 4 - 26 :

 426 Figure 4-26: Measured radiation patterns in the focusing plane (x-y plane). (a) 8 GHz. (b)

Chapter 5 :

 5 Coherent beam control with a transformation optics based lens -117-5 COHERENT BEAM CONTROL WITH A TRANSFORMATION OPTICS BASED LENS

Figure 5 - 1 :

 51 Figure 5-1: Diagrammatic sketch of the function of the beam steering lens.

Figure 5 - 3 :

 53 Figure 5-3: Anisotropic material parameter values for design 1 of the beam steering lens,

Section 5 . 1 :

 51 Designs of the beam steering lens and 2D analysis -122-

Figure 5 - 4 :

 54 Figure 5-4: (a) Electric field and (b) norm of the electric field distribution at 10 GHz

Figure 5 -

 5 Figure 5-5.Figure 5-5(a) represents the properties when H = 0.2 m. Although this lens area is

Figure 5 - 1 Chapter 5 :

 515 Figure 5-5.Figure 5-5(a) represents the properties when H = 0.2 m. Although this lens area is

Figure 5 - 5 Figure 5 - 6 :

 5556 Figure 5-5 Material parameter values for design 1 of the beam steering lens. (a) H = 0.2 m.

  material[125] or by drilling holes in a high constant dielectric[126,127].5.1.1.5 Study on the height dimension of lens design 1 after parameter reduction.The same dimension study is performed on this model after parameter reduction. H is set to be 0.2 m and 0.8 m respectively. The electric field distribution and the norm of the electric field are presented in Figure5-8.

Figure 5 - 8 :Figure 5 - 9 :

 5859 Figure 5-8: (a) and (c) Real part of electric field and (b) and (d) norm of the electric field

Figure 5 - 10 :

 510 Figure 5-10: Beam splitting device. (a) Real part of electric field distribution at 10 GHz

  distribution and the norm of the electric field show clearly the transmission of four splitted beams through the lenses. The angle of deflection of the beams with respect to the normal is 28° as for the lens design 1. 5.1.1.7 Dimension study on both height and width of the lens after parameter reduction. A parametric study on the size of the isotropic lens allows showing that a reduction in the size of the lens does not alter the deflection angle, as presented in the different parts of Figure 5-11.

Figure 5 - 11 :

 511 Figure 5-11: Parametric study performed on the size of the lens. When reducing the

  -15 is anisotropic. Here we continue to test the functionality of the lens design 2 by ignoring the anisotropy by keep only zz. The electric field distribution and the norm of the electric field are presented in Figure 5-16. Different from the simulation result for lens design 1, the isotropic lens design 2 can still deflect the beam by 45°.

Figure 5 -

 5 Figure 5-16: (a) Real part of electric field and (b) norm of the electric field distribution at

Figure 5 - 18 :

 518 Figure 5-18: Discretization of the proposed lens design 1. The permittivity (zz) distribution

Figure 5 -

 5 Figure 5-19: (a) Real part of electric field and (b) norm of the electric field distribution at

  -20.

Figure 5 - 20 :

 520 Figure 5-20: (a) Photography of the metamaterial based fabricated prototype. (b) Top

Figure 5 -= 4 . 8

 548 Figure 5-22: (a) Typical ELC unit cell structure. (b) Resonance of a parameterized unit

Figure 5 -

 5 23(a). Indeed such patch radiators present a narrow band frequency response. Here a sectorial beam, i.e. a wide beam in one plane and a narrow beam in the other plane is obtained since we are using a linear array of radiating elements. As it can be observed, in presence of the designed 3D ELC-based lens, the radiated wavefronts undergo a deflection of 13°. The color scale is in dB.

Figure 5 - 23 :Figure 5 - 24 .

 523524 Figure 5-23: Simulated 3D radiation patterns in linear scale. (a) Linear array of patch

Figure 5 - 24 :

 524 Figure 5-24: Measured S11 distribution at 10 GHz, where the blue curve represents the

Chapter 5 :

 5 Coherent beam control with a transformation optics based lens -141-dotted blue and continuous green traces on the different plots, representing the simulated and measured radiation patterns respectively, the primary source alone produces a radiated beam at boresight direction ( = 0°). The designed lens allows to steer the radiated beam by 13° at 10 GHz, as depicted by the dashed violet and continuous red traces in Figure 5-25(b), corresponding to respectively simulations and measurements on the lens-antenna system.

Figure 5 - 25 :

 525 Figure 5-25: Simulated and measured far-field radiation patterns in the deflection lane (x-

Figure 5 - 26 :

 526 Figure 5-26: Measured real part of electric near-field distributions at (a) 9.3 GHz, (b) 10

Figure 5 - 27 :Chapter 5 :

 5275 Figure 5-27: Measured real part of electric near-field distributions at (a) 9 GHz, (b) 11

Figure 5 - 28 :

 528 Figure 5-28: Frequency study on beam steering angle variation.

Figure 5 -

 5 29(a), which is composed of only 170 unit cells. The respective permittivity of each cell is considered to be constant across the cell and is equal to the average permittivity within the cell. As illustrated by the colour plot in Figure5-29(a), the permittivity zz values range from 1 to 2.8 in the discrete approximation.

Figure 5 - 29 :

 529 Figure 5-29: Beam steering lens realization. (a) Calculated discrete lens with 170 values of

Figure 5 - 30 :

 530 Figure 5-30: Return loss of the array of equally fed microstrip patch antenna used as wave

Figure 5 -

 5 Figure 5-31: (a)-(c) 2D simulated electric field distribution at 8 GHz, 10 GHz and 12 GHz,

Chapter 5 :

 5 Coherent beam control with a transformation optics based lens -147-

Figure 5 - 32 :

 532 Figure 5-32: Simulated 3D radiation patterns in linear scale. (a) Linear array of patch

Figure 5 - 33 :

 533 Figure 5-33: Measured S11 distribution at (a) 8 GHz, (b) 10 GHz and (c) 12 GHz, where

Figure 5 - 34 :Chapter 5 :

 5345 Figure 5-34: Measured real part of electric near-field distribution at (a) 8 GHz, (b) 10

Figure 5 - 35 :

 535 Figure 5-35: Comparison of the normalized simulated and measured radiation patterns in

Université

  Paris-Saclay Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France Titre : Optique de transformation : Application aux antennes et aux dispositifs micro-ondes Mots clés : optique de transformation, métamatériau, lentille, antenne, gradient d'indice et impression 3D Résumé : Le concept de l'optique de transformation qui permet de contrôler le trajet des ondes électromagnétiques à volonté en appliquant une variation spatiale judicieusement définie dans les paramètres constitutifs, est exploré pour concevoir des nouveaux types d'antennes et de dispositifs micro-ondes. Dans une première partie, basée sur la transformation de coordonnées, un dispositif d'illusion capable de modifier l'apparence d'une émission électromagnétique et de la délocaliser, est validé par le biais de simulations numériques. Un dispositif de transition de guide d'ondes, permettant d'assurer une transmission quasi totale entre deux guides d'ondes de sections transversales différentes est également conçu et validé expérimentalement par l'utilisation des résonateurs à métamatériau. Dans une seconde partie, la transformation de l'espace basée sur l'équation de Laplace est étudiée pour concevoir des lentilles pour les applications antennaires. Une lentille de focalisation capable de restaurer les émissions en phase d'un réseau conforme d'éléments rayonnants et une lentille à dépointage qui permet de dévier la direction du faisceau rayonné d'une antenne sont conçues. Des prototypes fabriqués à partir d'un matériau isotrope tout-diélectrique grâce à la technologie d'impression en trois dimensions (3D) permettent de valider expérimentalement la fonctionnalité des lentilles sur une large bande de fréquence. Title : Transformation optics: Application for antennas and microwave devices

2D Two-dimensional 3D Three-dimensional TO Transformation Optics QCTO Quasi-conformal Transformation Optics TE Transverse Electric TM Transverse Magnetic TEM Transverse Electromagnetic MM Metamaterial ELC Electric LC SRR Split Ring Resonator PDE Partial Differential Equation FEM Finite Element Method HFSS High Frequency Structural Simulator

  WHERE THE BLUE CURVES REPRESENT THE S11 OF THE PATCH ARRAY AND THE RED CURVES REPRESENT THE S11 OF THE PATCH AND LENS SYSTEM. ..................................... 148 NORMAL DIRECTION IS OBSERVED. ............................................................................ 148

	List of Figures List of Abbreviations and Acronyms
	LIST OF ABBREVIATIONS AND ACRONYMS quasi-conformal transformation optics, are introduced. The definition of the material properties
	for these three kinds of transformations are provided by PDE solvers from Comsol Multiphysics
	FIGURE 5-34: MEASURED REAL PART OF ELECTRIC NEAR-FIELD DISTRIBUTION AT (A) 8 GHZ, (B) numerical simulation software. The numerical calculation softwares, near field and far field
	10 GHZ AND (C) 12 GHZ, WHERE AN ARRAY OF PATCH RADIATORS IS USED AS EXCITATION experimental setups we use for the thesis are presented. For the realistic validation (simulated
	FEED FOR THE LENS. A DIRECTIVE BEAM SHOWING PLANAR WAVEFRONTS ORIENTED IN AN and experimental) of the designs, electric and magnetic metamaterial resonators and dielectric
	materials engineering are also presented. OFF-FIGURE 5-35: COMPARISON OF THE NORMALIZED SIMULATED AND MEASURED RADIATION PATTERNS IN THE FOCUSING PLANE (X-Y PLANE). (A) 8 GHZ. (B) 10 GHZ. (C) 12 GHZ. .. 149 The manuscript is then decomposed into two parts. Part I consists of two chapters related to two GENERAL INTRODUCTION microwave device designs based on coordinate transformations. In Chapter 2, an illusion device
	which is able to change the radiation pattern and location of a source is presented. Numerical
	calculations are performed and a discrete model is proposed. Chapter 3 introduces a scheme to
	Transformation optics, as a powerful designing tool based on space transformation, opens up taper the electric field from a wide waveguide into an output waveguide that is 11.4 times
	many further ways to control electromagnetic (EM) waves, which can be employed to design smaller than the input. We develop the method and show how to reduce the parameters' value
	applications based on engineered materials like metamaterials and dielectrics. This thesis aims and also the number of the parameters needed for the design. A prototype is fabricated by
	to apply the transformation optics concept for microwave devices designs. Following the resonant metamaterials and is experimentally validated by near-field measurements.
	previous works done in the framework of Paul-Henri Tichit's thesis in CRIME group at IEF, my In Part II, two designs of transformations based on Laplace's equation are presented in two
	thesis works started by continuing with the design concepts based on classical coordinate different chapters. Such type of transformations allows realizations from materials with
	transformations. Then I focused on the transformation based on Laplace's equation, and two dielectric-only response and thus broadband operations. In Chapter 4, a functional lens which
	lens antennas are designed by this concept. My thesis is financially supported by the French restores in-phase emissions from a non-planar array of radiators over a broad frequency range is
	Ministry of Higher Education and Research. Fabrication and measurements have been made
	with the help of the AIRBUS Group Innovations.
	In the first chapter, the state of art of metamaterials is discussed firstly. The principle of
	metamaterials is presented, and several antenna applications based on metamaterials are
	introduced. Then the design concept -transformation optics is introduced. Coordinate
	transformation as the most well-known methodology is presented firstly. An example of a beam
	bending is studied helping for a better understanding. Then transformations based on Laplace's
	equation, including solutions with only Dirichlet boundary conditions, conformal mapping and
	-xxvi--2-	-xxvii-

14 )

 14 For different coordinate systems, of course, hu, hv, and hw have different expressions. For

	Cartesian coordinate	( x	, y	, z	)	, cylindrical coordinate	(		, 	, z	)	and spherical coordinate
	( r	, 	, 	)	these expressions are presented in Table 1-I.

Table 1 -I: hu, hv, and hw expressions in different coordinate systems.

 1

Table 1 -

 1 II.

Table 1 -II: Expressions of rotation matrix R in different coordinate systems.

 1

Table 2 -I: Discretization of the parameters for the layered material shell.

 2

	Zone	zz	rr	
	1	20.3	1	1
	2	-6.6	-2.2	-0.3
	3	-5	-1.7	-0.4
	4	-3.9	-1.3	-0.5
	5	-3.1	-1	-0.6
	6	-2.4	-0.8	-0.7
	7	-2	-0.7	-0.9

Table 3 -I: Components values of  tensor for the transformation of two taper parts.

 3

Acknowledgements

-v-Acknowledgements each domain. The thick black line represents a PEC boundary in each domain.

We now start to discuss the mapping between two domains that don't share the same conformal module. As presented in Figure 1-12, the physical domain is a rectangle domain with a hump on the bottom boundary whose conformal module is M. We intend to map this physical domain to Section 1.2: Transformation optics -24-

the rectangle virtual domain whose conformal module is 1 on the left of the figure.

To compensate for the mismatch in conformal modules, we first map our virtual domain to an intermediate domain having the same conformal module as the physical domain. And the Jacobian matrix of the conformal transformation Ac is assumed to be obtained. The simplest way to do this is with a uniform dilation of the form y' = My. We can then conformally map this intermediate domain onto the physical domain with a conformal transformation.

The combined dilation and conformal mapping produce material tensors of the form:

-153-BIBLIOGRAPHY

However, when several radiating elements are conformed on a non-planar surface, a radiation presenting multiple beams and defocusing effect, is emitted if no phase shifters are used to compensate for the different phase shifts introduced due to the location of the individual radiators. The broadband directive in-phase emission from an array of sources conformed cylindrically is numerically and experimentally reported in this chapter. By using the space transformation theory, a functional lens, which is capable of restoring in-phase emissions over a broad frequency range, is proposed and realized. The operating principle is presented of Figure 4-1. An all-dielectric, compact and very low-cost lens prototype presenting a graded refractive index is realized through three-dimensional (3D) polyjet printing. An array consisting of eight planar microstrip antenna elements, fabricated using standard lithography technique, is used as Transverse Electric (TE) polarized wave launcher to feed the lens. To validate the proposed lens, we experimentally demonstrate the broadband focusing properties and in-phase directive emissions. Both the far-field radiation patterns and the near-field distributions are measured and reported. It is observed that measurements agree quantitatively and qualitatively with numerical full-wave simulations and confirm the corresponding physical characteristics.

Introduction

Conformal antennas have received a lot of attention. Modern communication systems, aerospace application, high-speed vehicles and missiles require antennas conformed to their surfaces, with the main goal to reduce aerodynamic drag. Such conformal antennas are able to offer various advantages such as a potential of 360° coverage so as to secure continuous communications, a minimal level of unwanted radiations, low weight, and the possibility of obtaining significant gains. When the curved surface has a large radius compared to the operating wavelength, the influence of curvature on the performances is low. However, when the curvature radius is small compared to wavelength, the antenna's radiation patterns are terms of fields' equivalence with the virtual space upon the outer boundaries, Neumann and Dirichlet boundary conditions are set at the edges of the lens. The boundary conditions for the conformal lens are:

Concept of the lens design 1

For this first design, the virtual space which is free space filled with air and the physical space which is the transformed medium lens are respectively presented in Figure 5-2(a) and (b). The points B', C' and D' in the physical space share the same location as B, C and D in the virtual space. We consider the length of the segment CD to be equal to W and that of BC to be H. In Figure 5-2, a color plot of the contour of the design is shown. For simplicity, the transformation deals with a 2D model with incident Transverse Electric (TE) polarized wave. In this case, the electric field only has the z-directed component. Suppose that the coordinate transformation between the physical space (x', y') and the virtual space (x, y) is x = f(x', y'), y = g(x', y'). The mathematical equivalence of this mapping can be expressed by a Jacobian matrix J whose elements are defined by

. By solving Laplace's equations in the virtual space with respect to specific boundary conditions, the Jacobian matrix J of the mapping can be obtained:

The physical space performs an inverse function of the virtual space. Thus the Jacobian matrix of this inverse transformation from (x', y') to (x, y) can be represented by J -1 . Here we assume that the conformal module of the virtual space (vacuum) is 1 while the conformal module of the physical space is M. Once J -1 is known, the properties of the intermediate medium can be calculated. In terms of fields' equivalence with the virtual space upon the outer boundaries, Neumann and Dirichlet boundary conditions are set at the edges of the lens. For the lens design For further fabrication, we intend to have a designed lens that is small in size. When H = 0.2 m, the maximum permittivity value is 8.85, which is a big challenge for all dielectric fabrication. In this study, we fix the dimension of the lens as W = 1 m and H = 0.5 m. In the lens area, we assign an isotropic material property after the reduction from the design we introduced before. As shown in Figure 567, the lenses are now assigned only a permittivity parameter along zdirection, while the permeability is isotropic. It is clear that, since the physical space of lens design 1 is distorted and the conformal module is far from 1, the lens still steers the beam but by an angle of 28°, smaller than the expected 45°. This is due to the fact that the conformal module is much larger than 1. So the anisotropy cannot be ignored if a deflection of 45° is still desired.

Therefore, the beam steering functionality is weakened when the anisotropy is ignored. If we consider the zz distribution for lens design 1, it can be noted that zz value ranges from 1 to 6.4.

Such values can be achieved using for example, nano-and micro-sized titanates dispersed in a The rectangle EFGH is mapped from the quadrilateral E'F'G'H', where EF = W, FG = H. In the quadrilateral E'F'G'H', the different dimensions are:

. The determination of the mapping is introduced by solving Laplace's equations subject to predefined boundary conditions by Comsol Multiphysics Partial Differential Equation (PDE) solver.

For the lens design 2, the boundary conditions are:

It is clear that for the design 2 the conformal module difference between the virtual space and physical space is smaller than in design 1. In summary, the anisotropy in design 2 is smaller than in design 1.

The designed lenses is fully parameterized with W = 1 m and L = 0.5 m. The properties of the lens design 2 shown in Figure 5-14 share similar parameter variation as lens design 1, but with different range of values. For the component zz, the range was from 1 to 6.4 in design 1, while in design 2, the range is from 0.4 to 5.8.

Numerical validation of lens design 2

In this section, finite element method based numerical simulations are used to design and characterize the proposed transformed beam steering lens design 2. As before in the simulation model using Comsol Multiphysics, scattering boundary conditions are set around the computational domain. The same configuration as for the simulation of the lens design 1 is used.

A current line source of length Ls = 0.1 cm is used and the electric field of the source is polarized along the z-direction. The operation frequency is set to 10 GHz.

Discretization of the lens design 1

In the distribution of zz for design 2, the value between 0.4 and 1 distributes in most part of the lens area. This is impossible to realize by the 3D all dielectric printing. In this section, we select the lenses based on design 1 with parameter reduction when H = 5 cm for discretization and further implementation. The properties of the transformed medium are shown in Figure 5-18.

According to the effective medium theory, if the operating wavelength is large enough (relative to the actual size of unit cell size), the composite material can be considered isotropic and homogenous. We choose the size of the unit cells to be 5 mm × 5 mm. In this case, the whole lens is composed of 200 (20×10) unit cells. The discrete permittivity of each cell is constant and equals to the permittivity value at the centre of each cell.

The discrete model of the proposed lens design is presented. After the discretization, the range of the permittivity has decreased, which is shown in Figure 5 simulations done with ANSYS commercial code HFSS. 40 For an electromagnetic wave incident with a wave vector and field polarization of Fig. 2(a), the ELC resonator will exhibit an electric resonance. Using reflection and transmission responses from the sample, effective parameters can be extracted using the retrieval procedure reported in chapter 1. This is possible since the structure period along the propagation direction is very small compared to the operating wavelength. In performing the retrieval, we assume along propagation direction a cell size pz = 5 mm corresponding to a cubic unit cell. Extracted permittivity and permeability are shown in Figure

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