Keywords: Metasurface antenna, leaky-waves, periodic surface, aperture field implementation. iii

In this thesis, a method for the implementation of arbitrary aperture field distributions using tensorial metasurfaces is introduced. Sinusoidally modulated metasurfaces are used in order to generate leaky waves with control on both phase and amplitude. The desired aperture phase distribution is obtained using a new local holography formulation. On the other hand, the amplitude distribution is controlled by varying modulation indices and average impedance depending on the position. A separate control of the aperture field components is achieved by modulating the impedance tensor elements independently. The theoretical formulation of the method is presented in details by taking into account the implementation method and antenna adaptation issues. The method is applied to design a wide range of radiation patterns examples both for farfield and near-field applications. The design procedure was first validated with simulations results for a working frequency of 20GHz giving a good agreement with the theoretical results. Several metasurfaces were then manufactured and measured for working frequencies of 10 GHz, 12.25 GHz and 20GHz. The consistency of the measurements and the simulation results proves that a good control of the aperture field phase and amplitude distributions is achieved using the proposed method.

List of Tables

 Table 3.
List of Figures

Introduction

In recent years, a great interest was given to satellite communication systems as a possible solution to the deployment of high speed agile communication systems and their spreading to non-military applications. This increased the need for low-profile highperformance antennas. In addition, these antennas have to be light, low-cost and easy to manufacture in order to be compatible with mass market production.

In parallel, metamaterials have become an appealing subject of research as they exhibit exotic physical properties that cannot be found in nature such as single or double negative materials [START_REF] Alù | Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency[END_REF]- [START_REF] Pendry | Negative refraction makes a perfect lens[END_REF]. In the case of 2D structures, the equivalents of metamaterials are called metasurfaces [START_REF] Holloway | An Overview of the Theory and Applications of Metasurfaces: The Two-Dimensional Equivalents of Metamaterials[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF]. They have been used recently in various applications such as frequency selective surfaces [START_REF] O'nians | Antenna Feed System Utilizing Polarization Independent Frequency Selective Intermediate Reflector[END_REF]- [START_REF] Munk | Frequency Selective Surfaces: Theory and Design[END_REF], the control surface-wave wave-front [START_REF] Maci | Metasurfing: Addressing Waves on Impenetrable Metasurfaces[END_REF]- [START_REF] Bosiljevac | Nonuniform metasurface Luneburg lens antenna design[END_REF], the control of field transmission [START_REF] Pfeiffer | Metamaterial Huygens' Surfaces: Tailoring Wave Fronts with Reflectionless Sheets[END_REF]- [START_REF] Yu | Flat Optics: Controlling Wavefronts With Optical Antenna Metasurfaces[END_REF] and in leaky-wave antennas [START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF].

Leaky-wave antennas are based on the propagation properties of surface-waves on sinusoidally modulated metasurfaces. By acting on the modulation parameters, it is possible to control the propagation of the surface waves or to generate radiating modes (leaky-waves) in order to realize antennas [START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF].

In previous work, sub-wavelength printed patches over a grounded dielectric slab have been used to implement scalar [START_REF] Patel | A printed leaky-wave antenna based on a sinusoidallymodulated reactance surface[END_REF]- [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] and tensor metasurface antennas [START_REF] Minatti | A Circularly-Polarized Isoflux Antenna Based on Anisotropic Metasurface[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF]. In the scalar case, the direction of the field above the metasurface is dictated by the source [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF]. This severely limits the design possibilities. However, a control of the radiated beam polarization, has been achieved in [START_REF] Minatti | Spiral Leaky-Wave Antennas Based on Modulated Surface Impedance[END_REF]- [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF].

Tensor metasurfaces allow higher degrees of freedom for the antenna design, they have been used to generate various types of radiation patterns such as circularly polarized (CP) waves [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF], [START_REF] Minatti | Modulated Metasurface Antennas for Space: Synthesis, Analysis and Realizations[END_REF] or isoflux shaped beam antennas for space applications [START_REF] Minatti | A Circularly-Polarized Isoflux Antenna Based on Anisotropic Metasurface[END_REF]- [START_REF] Minatti | Modulated Metasurface Antennas for Space: Synthesis, Analysis and Realizations[END_REF].

Recently, metasurface antennas have also been used in order to generate radiation pattern with control over amplitude, phase and polarization [START_REF] Minatti | Flat Optics for Leaky-Waves on Modulated Metasurfaces: Adiabatic Floquet-Wave Analysis[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF].

In this thesis report, we investigate the performances of tensor metasurface antennas as a possible candidate for space applications. We introduce a new approach for the generation of radiating aperture field distributions using modulated tensorial metasurfaces as a generalization of the solution proposed in [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] for the tensorial case. Our objective is to propose a systematic procedure for the implementation of general aperture field distributions with control of amplitude, phase and polarization [START_REF] Teniou | Implementation of radiating aperture field distribution using tensorial metasurface[END_REF]- [START_REF] Teniou | Tensorial Metasurface Antennas Radiating Polarized Beams Based on Aperture Field Implementation[END_REF]. A separate control of the aperture field components is achieved by modulating the impedance tensor elements independently. A new exact formulation for the first order holography principle is introduced in order to generate the desired aperture phase distribution. On the other hand, the amplitude distribution is controlled using modulation indices and average impedances that vary depending on the position on the metasurface. The proposed method is general and can be applied to any arbitrary aperture field distribution.

This rapport is structured as follows:

Chapter 1. Context and state of the art

In this section, we present the context of this PhD thesis and the corresponding state of the art. In order to do that, we develop each term of the thesis title: "Metasurface antennas for space applications". First, the fundamental parameters used to describe the performance of an antenna are defined. Then, the potential needs to which we try to respond in the space communication systems are introduced. Finally, a state of the art about metasurface antennas is presented.

Antenna

Figure 1: Antenna as a transition device [START_REF] Balanis | Antenna theory[END_REF].

An antenna is an electrical device that converts electrical power into radiated electromagnetic waves (and vice versa). In other words, the antenna is a transitional structure between free-space and a guiding device as shown in Figure 1 [START_REF] Balanis | Antenna theory[END_REF].

Source Transmission line Antenna Radiated free-space wave

In order to describe antenna performances, several parameters have been defined. In this section we will present the parameters that are used in this work.

Radiation pattern

The radiation properties of an antenna depend on the distance between the observation point and the antenna. This led to the definition of two main radiation regions. The first region is the near-field region, where the angular distribution of the field is dependent upon the distance from the antenna. Then, the far-field region is the region where the field angular distribution becomes independent of this distance. In general, the far-field region is considered for distances greater than An antenna radiation pattern is a graphical representation of the far-field radiation properties of the antenna as a function of space coordinates. It can either represent the intensity of the radiated field, the power density, the directivity, the phase, the polarization or other electromagnetic properties of the radiated field. In general, it is traced in the spherical coordinates and in a logarithmic scale in decibels dB (Figure 2). This logarithmic scale is more used because it accentuates the variations of the radiation pattern in the regions with low values.

 

Side lobes

Local maxima present in an antenna radiation pattern are called lobes. In the case of an antenna designed to radiate in one particular direction, the lobe in that direction is larger than the others and is called the "main lobe". The remaining lobes are called side lobes.

They usually represent radiation in undesired directions and need to be minimized.

Beamwidth

The beamwidth is an antenna parameter related to the radiation pattern. It describes the width of the main lobe in the case of single beam antenna or the width of the desired lobes in the case of multi-beam antenna. The beamwidth is an angular separation between two directions with the same radiation intensity on the opposite sides of the main lobe direction.

There are several definitions of the antenna beamwidth in the literature. The most widely used are  Half Power Beamwidth (HPBW): It is defined as the angle separating the two directions in which the radiation intensity is equal to half the intensity in the maximum direction.  First Null Beamwidth (FNBW): It is defined as the angle separating the two directions where there is a first null in the radiation intensity.

Other definitions exist in the literature. However, the term beamwidth usually refers to the Half Power Beamwidth.

The beamwidth is a key parameter of the antenna radiation pattern. It is often used as a trade-off between it and the side lobe level (SLL). In other words, the beamwidth decreases as the side lobe level increases and vice versa [START_REF] Balanis | Antenna theory[END_REF].

Directivity

The directivity of an antenna is the ratio of the radiation intensity in a given direction to the average radiation intensity in all direction. This average radiation intensity is equal to the total power radiated by the antenna divided by 4 (equation ( 1)). ).

For an aperture antenna of surface a S the directivity is defined as:

where a E is the aperture field distribution.

Gain

Another important antenna parameter is the gain. The gain of an antenna is closely related to the directivity. This parameter describes the losses of the antenna as well as its directional properties (Figure 3). On the other hand, the directivity is only affected by the radiation pattern of the antenna.

The gain is defined as the ratio of the radiation intensity at a certain direction to the radiation intensity obtained if the input power accepted by the antenna was radiated isotropically (equation ( 3)). As in the case of directivity, the direction of maximum radiation is considered when no particular direction is specified.

Figure 3: Directivity Vs Gain.

Antenna efficiency

Several efficiencies can be defined for antennas depending on the considered losses. In general, the total efficiency 0 e can be written as [START_REF] Balanis | Antenna theory[END_REF]:

In the case of aperture antennas, an additional important efficiency is defined as follows [START_REF] Balanis | Antenna theory[END_REF]:

e ap pu A D e AD  ( 6
)
where p A is the physical area of the aperture e A is the effective area of the aperture. It corresponds to the area with uniform distribution that gives the same directivity D as the considered aperture.

u D is the directivity given by an aperture of area p A with a uniform distribution.

The aperture efficiency is then introduced in the total efficiency, leading to: 0 r cd ap e e e e  (7)

Polarization

The polarization of an antenna in a certain direction is defined as the polarization of the wave transmitted by the antenna [START_REF] Balanis | Antenna theory[END_REF]. As the variation of the components of the radiated field may vary differently with the direction from the antenna, different parts of the radiation pattern may have different polarizations.

The polarization of electromagnetic waves describes the variation of the direction of the electric-field vector. It can either be linear, circular or elliptical.

Linear polarization: an electromagnetic wave is linearly polarized if the electric field vector is always oriented along the same line. This is accomplished either when the electric field has only one component or two orthogonal linear components that are in time phase or 180° out of phase.

Antenna 11

Circular polarization: an electromagnetic wave is circularly polarized if the electric field vector traces a circle as a function of time. This is accomplished when electric field has only two orthogonal linear components, with the same magnitude and that have phase difference that is an odd multiple of 90°. If the rotation of the vector is clockwise (resp counterclockwise), the wave is right-hand (resp left-hand).

Elliptical polarization: an electromagnetic wave is elliptically polarized if the electric field vector traces an ellipse as a function of time. Despite being the general case of electromagnetic wave polarization, it is defined in practice as the case where the polarization of the wave is neither linear nor circular. The same rule as in the circular case is used in order to define the orientation of the elliptical polarization (right-hand or lefthand).

Axial ratio

As stated in the previous subsection, the electric field vector traces an ellipse as a function of time for elliptically polarized antennas. The ratio between the major axis and the minor axis of this ellipse defines the axial ratio of the antenna. The axial ratio is always bigger than one and gives the polarization of an antenna as follows:

 Linear polarization: For linearly polarized waves, there is only a major axis in the curve traced by the electric field vector. The minor axis is then equal to 0. As a result, the axial ratio is equal to infinity.  Circular polarization: For circularly polarized waves, the major axis and the minor axis are equal. The axial ratio is then equal to 1 (or 0dB). The more the axial ratio is close to 0dB, the more the polarization is close to a circular polarization.  Elliptical polarization: As in the previous subsection, the elliptical polarization is defined as the case were the polarization is neither linear nor circular. As a result, the polarization is elliptical when the axial ratio is neither equal to 1 (0db) nor to infinity.

Bandwidth

The bandwidth of an antenna is the range of frequencies within which a certain antenna characteristic conforms to some objective performances. As these antenna characteristics doesn't necessarily have the same frequency dependence, there are multiple charac-terizations of the bandwidth. Among the possible definitions of the bandwidth we can distinguish two different groups:

Pattern bandwith: This bandwidth can be defined with respect to several aspects of the radiation pattern of the antenna depending on the application (side lobe level, beamwidth, direction of the main lobe, axial ratio …). As an example, we can consider the frequency band where:

 The amplitude of main lobe does not decrease lower than -3dB from its value at the central frequency.  The axial ratio is smaller than 3dB. This definition is useful for circularly polarized beams.  The side lobe level is smaller than a certain value (for example -10 or -20dB).

 The radiation characteristics of the antenna are preserved. As for example: an omnidirectional radiation pattern.

It should be noted that there is no clear definition of the pattern bandwidth. Its definition will greatly depend on the needs of the considered application.

Impedance bandwidth: In general case, the impedance bandwidth is defined with respect to the scattering parameter 11 S . In this case, the bandwidth of the antenna is the region were the 11 S is smaller than -10dB. This definition is widely used in various antenna applications.

Antennas for space applications:

Satellite communications and satellite broadcasting have always stimulated antenna development by providing challenging constraints to the researchers. Depending on the applications, several types of performances are expected from satellite antenna systems.

For example, one may need an antenna pointing at broadside or with a tilted beam, radiating one beam or multiple beams, with circular or linear polarization or even with a different polarization for each beam.

Antennas for space applications: 13

In recent years, the need for higher speed data links with greatly increased mobility is expanding significantly. This gave interest to the use of Satellite-On-The-Move (SOTM) systems in the Ka frequency band . SOTM systems are systems able to establish communication with a satellite and maintain it while moving. These systems need agile beam steering antenna solutions. Among them, the mechanical beam steering solutions are currently the most attractive due to their low cost. However, their large size limits their application to restricted areas such as military systems and large transports.

The need for SOTM systems for the mass market such as plane transportation, trains, or even personal vehicles motivates the need for antenna systems with the following properties:

 Light weight  Small dimensions  Low cost  High performances

 Simplicity of manufacturing

Another challenging problem in satellite communication systems is the radiation of contoured beams. Contoured beams are beams were the radiated power fits a desired coverage that corresponds for example to a country or to a certain territory. In order to obtain this type of radiation patterns, two main approaches involving parabolic reflector antennas are generally used (see Figure 4). A first approach is to produce the shaped contour using a parabolic reflector fed by an array of feeders where each feeder is responsible for a part of the radiated field [START_REF] Parini | Design aspects and tolerances in array fed contour beam antennas of INTELSAT VI type[END_REF]. Using this type of antennas requires complex feeding networks and may increase the power consumption and the weight of the antenna.

The second approach is to shape the surface of the reflector in order to act on the width of the radiated beam to obtain the desired radiation pattern [START_REF] Cherrette | A method for producing a shaped contour radiation pattern using a single shaped reflector and a single feed[END_REF]. However, the design of these shaped reflectors can be time-consuming and sometimes very costly. The performances of these methods were further improved in the literature. As an example, the feeder position was optimized in [START_REF] Abd Rahman | Design of a satellite antenna for Malaysia beams by ray tracing method[END_REF]. In [START_REF] Mahajan | A Method of Generating Simultaneous Contoured and Pencil Beams From Single Shaped Reflector Antenna[END_REF], a single shaped reflector in conjunction with multiple feeders was used in order to generate a simultaneous contoured and pencil beam. However, the methods are still costly and time consuming.

In general antennas systems for SOTM systems and Contoured beams systems have the following limitations:  They are bulky and/or heavy.

 They are complex to design and are not suited for a large scale production.

 They are costly (especially for contoured beams systems).

In this work, metasurface antennas performances are investigated as a possible candidate in order to overcome these limitations. Several antennas designs are studied: single beam, multiple beams, beam shaping and near field focusing.

Metasurface antennas

Before introducing metasurfaces, we start by introducing metamaterials. So what are metamaterials?

Metamaterial

The term metamaterial comes from the Greek word meta (to go beyond) meaning "to go beyond the material". They consist of materials engineered to have properties that are not found in nature. They are generally composed of repeating patterns with subwavelength scale (Figure 5). Their exotic properties come from the arrangement of the repeating pattern (shape, geometry, size, orientation). Two dimensional metamaterials can be realized by arranging electrically small scatterers or holes into a two-dimensional pattern on a surface or interface. These surface versions of metamaterials are called metasurfaces (Figure 7).

Air Conventional Material

Interface Air Negative index material Interface

Metasurfaces

Metasurfaces are 2D equivalents of metamaterials. As they are 2D structures, this interesting type of metamaterials are thinner than 3D metamaterials. As a result, they offer the possibility of less bulky structures.

As for metamaterials, the word metasurface is not defined clearly, making the term "metasurface" ambiguous. Thus, it can be found in the literature for various types of applications widely discussed in [START_REF] Holloway | An Overview of the Theory and Applications of Metasurfaces: The Two-Dimensional Equivalents of Metamaterials[END_REF]. The article cites papers that use metasurfaces for the achievement of smart antennas, novel wave-guiding structures and absorbers. However, the emphasis of this article focuses on non-modulated metasurfaces which are not the objective of our study.

Metasurfaces offer a lot of possibilities. Among them, we can cite the following classes:

 Frequency selective surface metamaterials [7]-[8]  Metasurfaces for the control of surface-wave wave-front [9]-[11]  Metasurfaces screens for the control of field transmission [12]-[14]

 Metasurfaces for aperture antennas [START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF] In our work, we are interested in the use of modulated metasurfaces for the design of aperture antennas. This type of metasurfaces is based on leaky wave generation from a surface wave. In what follows, the term "metasurface" will be used to describe this third type only.

Metasurfaces for aperture antennas (Leak-wave antennas)

This type of antennas is based on the use of modulated metasurfaces in order to generate leaky waves. The periodic modulation of metasurfaces produces an infinity of discrete surface wave modes. When the periodicity of the modulation is increased, some of these modes may become propagating. These waves will then propagate away from the metasurface. In [START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF], Oliner presents a rigorous treatment of propagation on sinusoidally modulated surface impedances. In his paper, the surface under study is a reactance that varies sinusoidally in the direction of propagation. The article discusses the properties of the guided and radiating modes resulting from the impedance modulation. In order to avoid the superposition of multiple radiating modes, the periodicity of the metasurface has to be chosen in order to have only the 1  mode as a propagative mode.

The basic idea of metasurface antenna is then to control the impedance modulation above the metasurface in order to transform a launched surface wave into a local -1 leaky mode able to radiate a desired radiation pattern.

The electromagnetic properties of metasurfaces can be described in terms of surface impedance (analogous to material parameters for volumetric metamaterials). The surface impedance tensor s Z is defined as the ratio of the transverse electric field t E and the transverse magnetic field t H at the surface S :

   t s t s E Z n H Z J ( 8
)
where n is the unit vector normal to the surface.

 t J n H is the surface current.

In other words, s Z describes the electromagnetic behavior of the complex medium be- low the surface. It is important to stress the fact that the surface impedance strongly depends on the considered wave-mode and wavenumber (spatial dispersion). Properties of the impedance tensor are further discussed in the next chapter.

Depending on the nature of s Z (scalar or tensor), metasurface antennas can be divid- ed into two groups: scalar metasurfaces and tensor metasurfaces.

Scalar metasurfaces

Scalar metasurfaces are metasurfaces that have scalar surface impedance. In this case equation ( 8) reduces to:

ŝ Z  tt E
n H . The main challenge in this kind of metasurface is the control of the generated leaky wave in order to produce the desired radiation pattern.

In [START_REF] Patel | A printed leaky-wave antenna based on a sinusoidallymodulated reactance surface[END_REF], a design procedure of a sinusoidally modulated scalar metasurface is presented. The designed metasurface is used in order to generate a tilted beam radiation pattern. It is implemented using printed metallic strips over dielectric substrate (Figure pattern using scalar metasurfaces. The proposed solution is based on the holography principle. In this approach, an objective phase wave obj  and an incident phase wave surf  are introduced. The idea is to produce a term proportional to * obj surf  so that the interaction between the metasurface and the incident wave produces the objective field distribution (see equation ( 9)).

2 * ()

obj surf surf obj surf       (9)
In the case of a tilted beam radiation pattern, the resulting metasurface impedance pattern is shown in figure 9. (a). The metasurface is implemented using metallic patches printed over dielectric substrate. The corresponding radiation pattern is illustrated figure 9. (b). The holographic principal was also extended to tensor case and it will be discussed in the next subsection. The main issue of scalar metasurfaces is the limited control of the radiated leaky wave polarization. The direction of the electric field above the metasurface is defined by the type of excitation used in order to generate the surface wave. It is therfore difficult to generate a circularly polarized beam using this kind of antennas. Some solutions has been proposed in order to overcome this limitation.

(a) (b) In [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] a new method for the aperture field generation (instead of beam generation) has been presented by our lab. By controlling the modulation parameters of the surface impedance, the leaky wave mode is used in order to generate a desired aperture field distribution. This aperture distribution can correspond to any objective far-field or nearfield radiation pattern. This approach significantly broadens the design possibilities offered by scalar metasurfaces (except for polarization control limitations). A spiral shape is tilted in order to generate a tilted beam with circular polarization as illustrated in figure 12. The advantage of this solution with respect the previous designs is that the whole metasurface participate in the radiation pattern instead of having parts of the metasurface for each polarization component. In addition, using this method, a nearfield metasurface antenna radiating a Bessel beam was designed. The designed antenna has a good focusing effect within the non-diffractive region of the Bessel beam.

Another interesting approach for near-field control was introduced in [START_REF] Martinez-Ros | Non-Uniform Sinusoidally Modulated Half-Mode Leaky-Wave Lines for Near-Field Focusing Pattern Synthesis[END_REF]. In this paper, eight mono-directional leaky wave lines are used in order to produce a near-field focus- Scalar metasurfaces offers an interesting range of possibilities and are relatively easy to implement. However, their performances are limited when it comes to polarization control. Moreover, the aperture efficiency is pretty limited (lower than 20%) in this type of metasurfaces [START_REF] Minatti | Spiral Leaky-Wave Antennas Based on Modulated Surface Impedance[END_REF]- [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF]. In addition, some more complex radiation patterns with complex polarization configurations would be difficult and maybe impossible to obtain using this kind of metasurfaces. This motivates the need for investigating tensor metasurfaces. Although this kind of metasurface is more difficult to control and design, they offer a broader set of possibilities. This type of metasurfaces will be discussed in the next subsection.

Tensor metasurfaces

In contrast with scalar metasurfaces, the surface impedance (defined in equation ( 8)) of this group of metasurfaces is a tensor. The ratio between the tangential electric field and the tangential magnetic field will have four components instead of one. For example if we write the surface impedance tensor s Z in the cylindrical coordinates, we obtain:

ZZ ZZ         s Z ( 10
) (a) (b)
The possibility of modulating each element of the surface impedance results in an increase of the degrees of freedom of the metasurface design. This significantly widens the possibilities offered by this type of antennas in term of radiation pattern, especially regarding polarization control.

In [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF], the holography principle introduced in the previous sub-section is generalized for tensor metasurfaces. The principle is then used in order to design a tilted beam circularly polarized metasurface antenna using sliced gap unit cell topology. The same antenna is reintroduced in [START_REF] Minatti | Modulated Metasurface Antennas for Space: Synthesis, Analysis and Realizations[END_REF] alongside various tensor metasurfaces antenna designs (broadside CP, tilted beam CP, sector isoflux …etc). Some examples of the achieved radiation patterns are illustrated in Figure .16. The article summarizes the different steps in the design of tensor metasurface antennas and shows some of the designs realized so far.

In some recently published articles [START_REF] Minatti | Flat Optics for Leaky-Waves on Modulated Metasurfaces: Adiabatic Floquet-Wave Analysis[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF], metasurface antennas are used in order to generate radiation pattern with control over amplitude, phase and polarization. In these papers, a modulation index variating in space is used in order to control the leakage parameter of the field over the metasurface. By controlling this leakage parameter and thus the power converted into leaky-wave by the metasurface, a tapering of the amplitude distribution over the metasurface is achieved. This method is used in order to improve the performances of previously designed metasurface by the same group. Figure 17 presents a circularly polarized broadside beam metasurface antenna with a tapered amplitude distribution.

Goal of the thesis

The goal of this thesis is to introduce a new approach for the design of tensorial metasurface antennas generalizing the procedure presented in [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] to the tensorial case.

In this approach, we focus on the control of the aperture field above the metasurface in order to generate the desired radiation pattern. Using the proposed method, arbitrary aperture distribution can be generated with control on amplitude, phase and polarization. The objectives of this thesis are summarized below:

1-Full control of the generated leaky wave. By treating the general case of a hybrid surface wave excited on the metasurface, the objective is to have an independent control over the generated leaky wave components. 2-Introduce a general approach for the generation of arbitrary aperture field distribution. The objective is to propose a method for generating arbitrary phase and amplitude distribution by acting on the modulation parameters of the impedance tensor. 3-Implementation of the obtained metasurface using printed patches over dielectric substrate. The method is treated for the widely used cylindrical wave excitation case. A design algorithm treating the particularities of the cylindrical excitation case is presented. 4-Validation of the design procedure with numerical simulations. The method is validated for wide range of objective radiation patterns treating the cases of both far-field and near field radiation control. The simulation results are obtained mostly using the commercial software Ansys Designer and compared with the radiation pattern given using ideal aperture distribution as well as equivalent magnetic currents over the metasurface. 5-Validation of the design procedure with measurements. Several prototypes are manufactured and measured for the frequencies of 10GHz, 12.25GHz and 20GHz. Far-field and near-field metasurfaces antennas are designed. Some manufacturing challenges as well as antennas adaptation are treated.

Chapter 2. Formulation and theoretical background

In this section, the leaky wave generation mechanism is presented in the case of scalar and tensorial metasurfaces. In addition, a new method for the generation of aperture field based on the control of leaky waves is introduced. In the presented method, an independent control of the phase and amplitude of each component of the leaky wave aperture field distribution is achieved allowing implementing any arbitrary aperture field distribution.

Theoretical background

This section presents some theoretical background about leaky wave metasurface antennas. The leaky wave generation mechanism is presented for the scalar case and then extended to the tensor case.

Scalar leaky wave metasurfaces

A surface wave propagating through a scalar metasurface is described by the equation ( 8) (where the surface impedance s Z is a scalar). Let's consider the case of a mono- directional TM surface wave (SW) propagating along the x direction through reactive surface of impedance

s jX  s Z
(Figure 18). The magnetic field above the metasurface has the following form:

ŝw t jk x Ie  t H (x) y (11)
where

sw t
k is the wavenumber of the propagating surface wave. k as the free-space wavenumber, we can write the z component of the wave vector as

  2 2 0 sw sw zt k k k  (12)
In the case of a surface wave, sw t k is greater than the free-space wavenumber 0 k . As a result, sw z k is purely imaginary and the wave does not propagate in the z direction.

Let's now introduce a sinusoidal modulation of the reactance surface along the propagation direction as described by equation ( 13) :

2 ( ) 1 cos ss Z x jX M x a         (13)
where s X is the average reactance.

M is the modulation index.

a is the periodicity of the modulation.

The periodicity of the surface impedance leads to the generation of an infinite number of Floquet modes that propagates through the metasurface [START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF]. As a result, the electric field and the magnetic field over the metasurface can be expressed in the follow-

ing form: , , ŝw tn sw tn jk x n n jk x n n Ie Ve                t t H (x) y E (x) x (14)
where n I and n V are the current and voltage modal coefficients of the Floquet's modes.

, ,0

2

sw sw t n t n k k n a    
is the corresponding wavenumber of each mode.

,0 sw t k is the wavenumber of the mode 0.

For small values of M, the tangential electric field on the metasurface can be approximated at the first order by writing the sinusoidal modulation of equation ( 13) in the exponential form [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF]. This leads to:

22 22 sw sw sw tt t j k x j k x jk x aa s MM jIX e e e                      t E (x) x (15)
The tangential electric field is then composed of three modes with different corresponding wavenumbers:

 The mode 0 of wavenumber The leaky wave direction of propagation is related to the modulation periodicity by the following equation [START_REF] Patel | A printed leaky-wave antenna based on a sinusoidallymodulated reactance surface[END_REF]:

1 0 2 sin( ) sw t k a k      (16)
where 1   is the propagation direction of the mode -1.

Using equation ( 16), the modulation periodicity needed in order to obtain a leaky wave radiating at the direction 1   can be easily calculated. However, some important points have to be considered in the metasurface design:

 Higher order modes can also be propagating modes if 0 2 sw t n kk a   for 1 n  . This aspect has to be checked in order to avoid undesired radiated beams. A possible solution is to use a modulation index M small enough to neglect higher order modes (In our case M<0.5).  When the metasurface is modulated, the wavenumber of the incident surface wave becomes complex. This latter has been calculated as in [START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF] and used instead of sw t k in order to have a more accurate design procedure.

Non-modulated tensorial metasurfaces

Before extending the leaky wave generation principle to the tensorial case, the properties of non-modulated tensorial metasurfaces are presented. In particular, this section discusses the properties of the surface impedance and the propagating modes traveling through the metasurface.

Surface impedance tensor

As stated in the previous chapter, the surface impedance of these metasurfaces is described by a tensor. Equation ( 8) is then written as follows:

        tt S S         sw ss ρ ρ E ρ Z k n H ρ Z J ρ (17)
where

ˆxy     ρ
x y is a point on the antenna surface.

n is a unit vector normal to the surface S.

    ˆt S   r J r n H r is the equivalent surface current.
The impedance

  sw s

Zk

tensor is defined for a discrete set of wavenumbers, namely:

          ˆˆ, , , , cos sin sw sw sw t t t k n k n          sw k k x y (18)
where sw t k is the wavenumber associated with the n-th mode propagating along the direction ˆsw t k defined by the angle  (Figure 20) at the angular velocity  .

We will focus our attention on the dominant mode (n=0) propagating on a lossless reciprocal metasurface. For a lossless metasurface, the impedance tensor is anti-Hermitian ( †  ss ZZ ). If in addition, the metasurface is reciprocal, s Z is a purely imaginary tensor [START_REF] Hoppe | Impedance boundary conditions in electromagnetics[END_REF].

Thus, on a general 2D orthogonal system, placed on the surface S, equation ( 17) be-comes:

1 11 12 2 11 12 1 2 12 22 1 12 22 2 E X X H X X J jj E X X H X X J                                 (19)
In what follows, we consider that a surface wave propagating at a direction ˆsw t k is excited on a metasurface placed in the x-y plane (Figure 20).

Propagating modes

In order to study the surface wave modes propagating on a non-modulated tensorial metasurface, a procedure similar to the one presented in [START_REF] Bilow | Guided waves on a planar tensor impedance surface[END_REF] is introduced.

For a tensorial metasurface, the dominant mode is a hybrid EH surface mode. In cylindrical coordinates, it is described by the following equation:

    (2) 1 (2) 1 ˆH ˆˆH sw t TM TE t sw t TM TE t V V k I I k                   E ρ H ρ   (20)
where:

LW ŷ n x ˆ( )  h ρ  ρ ˆ( ) sw t  k ρ  (2) 1
H is the first order Hankel Function of the second kind.

The wavenumber sw t k of the surface wave mode can be found using transverse resonance technique as follows:

 The impedance boundary condition (equation [START_REF] Pandi | Design of Scalar Impedance Holographic Metasurfaces for Antenna Beam Formation With Desired Polarization[END_REF]) is written between the modal coefficients leading to:

mm me TM TM TM me ee TE TE TE XX V I I j XX V I I                       s Z (21)
where ,, mm me ee X X X are the TM and TE components of the impedance tensor.

 By introducing the quantity  defined as:

2 0 1 sw t k k       (22)
the dyadic free-space impedance is written as :

0 0 0 0 0 TM TM TM TE TE TE j V I I j V I I                           Z ( 23
)
where 0

 is the free space impedance  By applying the transverse resonance technique condition (

0 det 0     s ZZ
) we obtain a second order polynomial equation of un-

known  :     2 2 2 0 0 0 0 ee mm ee me mm X X X X X            (24)
 Solving the second order polynomial equation leads to the expression of  :

    2 2 2 2 2 2 0 0 0 0 4 2 mm ee me mm ee me ee mm ee X X X X X X X X X            (25)
 The wavenumber sw t k of the surface wave mode is finally obtained as:

2 0 1 sw t kk    (26)
 For the solution of equation ( 25), the modal coefficients of equation [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] must satisfy :

  0 0 mm TE TM me me TE TM ee X II X X II X                    (27)
As for the scalar case, the surface wave wavenumber sw t k have to be greater than the free-space wavenumber. As a result, the quantity  has to be a real positive number. Depending on the values of the impedance tensor components, we can distinguish four cases [START_REF] Bilow | Guided waves on a planar tensor impedance surface[END_REF]- [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF]:

 Case 1: 0 mm ee XX .
In this case, equation ( 24) has two solutions different in sign. That means that a single surface wave type, corresponding to the positive solution of  , propagates on the metasurface. In order to have only one surface wave excited on the metasurface, we place ourselves in the first case of study.

 Case 2:   2 2 2 2 00 40 mm ee me ee mm X X X X X       . In this case, equation (24)

Wave-modes generated by modulated tensorial metasurfaces

After studying the surface wave modes propagating on non-modulated tensorial metasurfaces, the leaky wave mechanism that was presented for scalar metasurfaces is extended to the tensorial case.

In this study, we place ourselves in the case of a sinusoidal modulation of each component of the impedance tensor along the direction of propagation of the form:

  2 1 cos ij ij ij ij X x X M x p           (28)
where ij X is a component of the impedance tensor.  If the periodicities ij p are multiple of the same periodicity p , then the global problem is periodic. The dispersion problem can then be treated using a discrete infinite number of independent transmission lines [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF]. For tensorial metasurfaces, lines with different polarizations (TE/TM) are coupled together.  If the periodicities ij p are different, the global problem is aperiodic. As a result, the dispersion problem cannot be treated using conventional techniques based on Floquet-mode decomposition. In [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF], a circuital approach for the analysis of the propagation characteristics of waves guided by scalar or tensorial metasurfaces modulated by functions with discrete spectrum is proposed. This paper presents a solution to the dispersion problem of this case of study (modulation with different periodicities).

As for the scalar case, the infinite number of Floquet-modes can be replaced by a first order approximation of the electric field for small values of the modulation indices ij M . This approximation is obtained by writing the sinusoidal terms of equation [START_REF] Teniou | Complex aperture field generation using tensorial metasurfaces[END_REF] in the exponential form and replacing them in [START_REF] Pandi | Design of Scalar Impedance Holographic Metasurfaces for Antenna Beam Formation With Desired Polarization[END_REF]. As a result, we obtain: In fact, as for scalar metasurfaces, the wavenumber of the mode 0 propagating on a modulated metasurface is different from the one on a non-modulated metasurface.

) 22 j x j x pp j x j x pp j x j x pp j x j x pp MM E j X J X J X J e X J e MM X J e X J e MM E j X J X J X J e X J e MM X J e X J e                              (
This wavenumber becomes complex and can be obtained using the procedure presented in [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF].

If
It should be noted that the components 12 X and 21 X are distinguished in equations ( 29) and [START_REF] Teniou | Tensorial Metasurface Antennas Radiating Polarized Beams Based on Aperture Field Implementation[END_REF]. This is done in order to later check the anti-Hermitian property of the proposed formulation.

Goal of the proposed procedure

In this work, we propose to control the leaky wave described by equation ( 30) in order to generate an objective aperture distribution of the form: It can be seen from equation [START_REF] Teniou | Tensorial Metasurface Antennas Radiating Polarized Beams Based on Aperture Field Implementation[END_REF] that the phase of the leaky wave is affected by the periodicity of the modulation while the amplitude is controlled by the modulation index and the average impedance.

The working principle of these metasurfaces can then be summarized as follows: "The use of modulated tensorial metasurfaces to generate leaky waves. These leaky waves are then used in order to obtain an objective aperture field giving the desired radiation pattern (Figure 21)".

Aperture field generation

The generation of aperture field distribution using modulated tensorial metasurfaces can be decomposed into two main aspects:

 Generation of the phase distribution.  Generation of the amplitude distribution.

Phase distribution generation

Phase matching between the leaky wave and the objective aperture distribution is obtained using holography principle. This principle is adapted to surface wave to leaky wave conversion. The impedance variation law results from the interaction of the incident wave with the objective aperture field.

In principle, two possible approaches may be considered to obtain the objective phase distribution:

 First approach: For each component of the tangential field, equating the phase of each individual term of [START_REF] Teniou | Tensorial Metasurface Antennas Radiating Polarized Beams Based on Aperture Field Implementation[END_REF] with the phase of (31).  Second approach: For each component, equating the phase of the sum of the terms of ( 30) with the phase of (31).

First approach

The objective phase distribution is imposed on each term of the generated leaky wave (described by equation ( 30)). This condition is summarized by the following set of equations:

                                            (32)
These equations lead to the following conditions for the modulation periodicities:

                11 12
pp J E J E pp J E J E                 (33)
where

    sgn 1 2 ij ij X     and 12 21    .
By defining the incident vector phase wave inc Ψ as the phase of the current J and the objective vector phase wave obj Ψ as the phase terms of the objective electric field [START_REF] Balanis | Antenna theory[END_REF] , we obtain:

                (35)
By replacing the imaginary parts of ( 35) in [START_REF] Teniou | Complex aperture field generation using tensorial metasurfaces[END_REF], the impedance tensor s Z can be cast in the following compact form:

          **
                      s Z X X M ( 36
)
where X is the average reactance tensor, M is the modulation index matrice,  and are the outer and the Hadamard products respectively.

As stated in a previous sub-section, the impedance tensor s Z is anti-Hermitian (equa- tion ( 19)). Equation [START_REF] Soukoulis | Photonic crystals and light localization in the 21st century[END_REF] does not guarantee that as the off diagonal terms depends on field components that cannot be controlled at priory (incident field and objective field). A possible solution is to decompose the impedance tensor into the Hermitian part H and the anti-Hermiatian one A defined by: † † , 22

ss   ss Z Z Z Z HA [START_REF] Hoppe | Impedance boundary conditions in electromagnetics[END_REF] and then use A as the impedance tensor. This solution leads to the formulation presented in [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF]. However, this technique neglect a significant part of the impedance tensor (Hermitian-part) that guarantee the exact objective phase distribution.

Second approach

The objective phase distribution is now imposed on the sum of the terms of the leaky wave. This can be summarized as follows:

                    (38)
It can be seen from equation [START_REF] Bilow | Guided waves on a planar tensor impedance surface[END_REF] that obtaining the desired phase distribution is not a trivial task as a significant number of elements needs to be controlled simultaneously. , pp in order to obtain the desired phase distribution. Moreover, an anti-Hermitian impedance tensor can then be obtained by imposing that [START_REF] Pfeiffer | Metamaterial Huygens' Surfaces: Tailoring Wave Fronts with Reflectionless Sheets[END_REF]

21

XX  without neglecting terms of the impedance tensor.

In this work, this approach is adopted as no impedance term is neglected. To simplify the calculations a new local framework formulation is introduced. To the best of our knowledge, no work has been published on this important aspect. This completely novel formulation is described in the next section.

Local framework formulation

In this approach, we take advantage of the fact that physical properties (energy con-

                    1 loc tt loc tt loc ss                 E ρ R ρ E ρ H ρ R ρ H ρ Z ρ R ρ Z ρ R ρ (40)
In equation [START_REF] Orfanidis | Electromagnetic Waves and Antennas[END_REF] and what follows, the superscript loc indicates that the considered quantity is written in the local framework. In this local framework, equation ( 19) simplifies as :

loc loc loc loc loc E X X X X HJ jj E X X X X                                 (41)
It can be seen from equation [START_REF] Iliopoulos | Micro-wave multi-beam generation using an anisotropic metasurface[END_REF]

X X M X X M                       (42)
The anti-Hermitian property of the impedance tensor is then guaranteed by imposing [START_REF] Pfeiffer | Metamaterial Huygens' Surfaces: Tailoring Wave Fronts with Reflectionless Sheets[END_REF] 21 loc loc XX  as this component does not affect the electric field. In addition, 22 loc X is a free quantity that can be used for other design purposes. However, as the surface wave wavenumber sw t k depends on the impedance components, 22 loc X can not be selected arbitrarily.

Finally, the impedance tensor that allows an exact phase match between the leaky wave and the objective wave is written in the global framework as follows:

             

Amplitude distribution generation

As explained in the previous sub-sections, the first order approximation of the amplitude of the generated leaky wave is proportional to the product ij ij XM . Thus, the idea is to change the average impedance and the modulation index along the metasurface in order to obtain the desired amplitude distribution above the metasurface. This amplitude control includes two aspects :

 The compensation of the spreading factor of the surface wave.

 The generation of the desired amplitude distribution.

It should be noted that the proposed amplitude generation technique is developed in the local framework that was developed in the previous sub-section. We decompose the procedure into two steps :

2.3.2.1
Step 1: Compensation of the surface wave spreading factor

The primary step of the amplitude control is the compensation of the attenuation due to the propagation of the surface wave on the metasurface. The first objective is then to obtain uniform amplitude of the generated leaky wave.

This objective is achieved using a modulation index and average impedance that vary depending on the position on the metasurface leading to the following condition:

            11 11 2 12 12 2 loc loc loc loc loc loc X M H c X M H c              ρ ρ ρ ρ ρ ρ ( 44
)
where c is a constant that depends on the possibilities offered by the implementation method. This constant is maximized in order to have the maximum amplitude of the generated leaky wave.

Equation ( 44) imposes a condition on the product ij ij XM but not on their individual values. However, since the dispersion constant sw t k is affected mainly by the average impedance values, smooth continuous variations of these have to be imposed to avoid undesired reflections. In the following, a linear variation of loc ij X along the direction of propagation is supposed. However, other variation laws could be used. Figure 24 presents an example of spreading factor compensation. The example treats a single component case on a distance of 6 . It can be seen from the example that using a product XM varying depending on the position, the amplitude of the generated leaky wave is constant (Figure 24.(e)).

It should be noted that the variation of the average impedance leads to a variation of the dispersion constant sw t k . This may affect the phase of the incident wave along the metasurface and may lead to a detuning in the phase matching of the leaky wave. To solve this, the average impedances variation laws are imposed before the phase x  x 

x 

x 

x  matching. Therefore, the detuning due to this variation is automatically corrected by the local holographic procedure.

2.3.2.2

Step 2: Generation of the desired amplitude distribution

After compensating the spreading factor of the surface wave, the product ij ij XM is modified in order to obtain arbitrary amplitude distributions of the generated leaky wave. This is obtained by multiplying the modulation indices obtained in the previous step by the objective amplitudes as described by the following equation :

            ,, 11 11 1
,, 12 [START_REF] Pfeiffer | Metamaterial Huygens' Surfaces: Tailoring Wave Fronts with Reflectionless Sheets[END_REF] 2 Equation ( 44) and (45) can then be combined in a single equation giving the amplitude matching condition described as follows:

loc loc u loc obj loc loc u loc obj M M E M M E              ρ ρ ρ ρ ρ ρ (
                , 11 11 2 1
X M H c E X M H c E                ρ ρ ρ ρ ρ ρ ρ ρ (46)

Conclusion of the chapter

In this chapter, a method for the generation of arbitrary aperture field distributions using tensor metasurfaces is presented. The working principle of these metasurface antennas is based on the conversion of excited surfaces waves to leaky waves propagating away from the metasurface.

By introducing a sinusoidal modulation of the metasurface impedance components, an infinity of hybrid modes are generated. The modulation parameters are chosen in such a way that only modes of order -1 are non-negligible leaky waves. These modes are then controlled using the modulation parameters in order to obtain the desired aperture field distribution above the metasurface. As the modulation parameters are controlled independently, the metasurface is not periodical; Even though each individual component is periodical.

x  x  x  x  x  (
On one hand, the phase of the generated leaky wave is controlled using modulation of the periodicities of the individual components. The objective phase distribution is obtained using the holographic principle in the tensorial case. In order to obtain the desired phase matching and at the same time ensuring the physical feasibility of the metasurface (anti-Hermitian impedance tensor), a local framework is introduced at each point of the metasurface. The holography principle is then applied in this local framework (equation ( 42)) to obtain the desired phase distribution above the metasurface.

On the other hand, for each component, the amplitude of the generated leaky wave is proportional to the product of the average impedance with the modulation index. The desired amplitude distribution is then obtained by varying these parameters depending on the position on the metasurface. The two parameters are used in order to compensate the spreading factor of the excited surface wave while at the same time imposing the desired amplitude distribution above the metasurface (equation ( 46)).

Equations ( 42) and ( 46) are the main blocks of the proposed formulation describing idealized continuous metasurfaces. In practice, these metasurfaces are implemented by printing sub-wavelength asymmetric patches over a dielectric substrate as already proposed in [START_REF] Patel | A printed leaky-wave antenna based on a sinusoidallymodulated reactance surface[END_REF]- [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] and [START_REF] Minatti | A Circularly-Polarized Isoflux Antenna Based on Anisotropic Metasurface[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF]. In the next chapter, the implementation method of this type of metasurfaces as well as the excitation of the surface wave is presented.

Chapter 3. Implementation of the metasurfaces

Introduction

In the previous chapter, leaky wave generation using modulated tensorial metasurfaces was addressed. By controlling the generated leaky waves, arbitrary aperture field distribution can be generated with control on both phase and amplitude distributions.

The metasurfaces treated in the previous chapter are idealized continuous metasurfaces with arbitrary excitation of the surface wave. In our work, we focused on the most common case of a cylindrical surface wave excitation obtained using a coaxial probe placed at the center of the metasurface. In addition, the metasurfaces are implemented by printing subwavelength metallic patches over a grounded dielectric substrate resulting in a square grid of printed patches discrete metasurface.

In the present chapter, we start by developing the previous formulation for the cylindrical surface wave excitation case. Practical considerations such as field continuity, surface wave wave-front and incident wave estimation are treated. Then, the implementation of the discrete metasurface on a grounded dielectric slab is addressed. A new unit-cell having multiple degrees of freedom is presented. Finally, a step by step design algorithm starting from the objective aperture field to the corresponding metasurface design is presented.

Cylindrical wave excitation

In most practical cases, a cylindrical surface wave is excited using a coaxial probe placed at the center of the metasurface [START_REF] Patel | A printed leaky-wave antenna based on a sinusoidallymodulated reactance surface[END_REF]- [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] and [START_REF] Minatti | A Circularly-Polarized Isoflux Antenna Based on Anisotropic Metasurface[END_REF]- [START_REF] Faenzi | Realization and Measurement of Broadside Beam Modulated Metasurface Antennas[END_REF] (Figure 26). For this widely used geometry, the cylindrical coordinates system represents a natural global framework. The impedance boundary condition will then have the following form:

                E X X H j E X X H                                    ρ ρ ρ ρ ρ ρ ρ ρ (47)
In the previously developed formulation, a linear variation of the average impedance Chapter 3. Implementation of the metasurfaces 52 was imposed in order to compensate for the spreading factor of the surface wave. Imposing a radial variation of the average impedance introduces radial symmetry (invariance with respect to the azimuthal angle  ) in In addition, we assume that under a small modu- lation index, the impedance modulation does not affect the radial propagation of the surface wave.

    ij ij XX    ρ , thus   0 ij X      .

Local framework derivation

The local framework at each point of the metasurface is derived by replacing equation ( 27) into [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] to obtain the expression of the tangential electric field   t  E ρ and magnetic field

  t  H ρ , yielding:                             0 (2) 0 1 0 (2) 1 ˆH ˆH t TM sw t TM sw X I j k X X I k X                                     
              0 2 1 ˆl oc X X            ρρ x ρρ ρ ρ ρ  ( 49
)
where the quantity

          2 2 0 1 XX            ρ ρ ρ ρ is used for normalisation.
The second unit vector of the local framwork is then defined as

                0 12 1 loc loc X X               ρρ x ρ x ρ z ρ ρ ρ ρ  (50)
The transformation matrix    R ρ from the global framework to the local framework is then directly given by

                      0 0 1 1 X X X X                             ρρ ρ ρ ρ R ρ ρρ ρ ρ ρ (51)
The magnetic field and the current density in the local framework are given by:

                          (
                    H ρ R ρ H ρ ρ ρ x ρ J ρ n H ρ ρ ρ x ρ (52)
The transformation matrix    R ρ has to be evaluated at each point of the metasurface in order to write the different electromagnetic quantities in the local framework.

Incident wave amplitude and phase estimation

A key step of the metasurface design is the estimation of the incident field defined by   t  H ρ . In the case of cylindrical surface wave, the term   TM I  ρ needed in (52) is the only unknown. This term can be found numerically using the geometrical optic based procedure introduced in [START_REF] Casaletti | Polarized Beams Using Scalar Metasurfaces[END_REF] and extending it to the tensorial case.

2 1 cos l ij ij ij ij X X M p                      Associated local tangent problem   sw t k  ρ Dispersion analysis    s Z ρ     2 1 H sw k  s Z Source     2 1 H sw k       Tensor impedance

Local modulation Quasi-Cylindrical wavefronts

The impedance variation of the sinusoidally modulated metasurface can be interpreted as a two scales problem :

 A component with rapid variation rate, corresponding to sinusoidal variation of the impedance and governed by the periodicities Under optic approximation, the metasurface can be described by an equivalent impedance s X obtained using the local periodic problem. The metasurface can then be considered as a slowly-variating equivalent surface impedance

  s X  ρ .
It should be noted that even though the metasurface is loss-less, the equivalent impedance has a real part describing the energy loss due to leaky wave radiation [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF].

The propagation problem can be solved numerically with the following steps:  The metasurface is divided into sectorial areas where the propagation can be considered as mono-directional. For each direction  , an associated local tan- gent problem with radial direction of propagation is defined (figure 28). Each local problem is defined by the impedance tensor described by equation (43).  Each sectorial area is then divided into N sub-sections, along the propagation direction, in which the modulation parameters ( ij M , ij X , and ij p ) are assumed to be constant (figure 28).  In each sub-domain, the dispersion problem is solved analytically using the procedure developed in our lab [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF]. Using the proposed method, the complex wavenumber  The local framework transformation matrix    R ρ defined by the equations (51) and ( 52) is calculated at each individual sub-domain.  The incident magnetic field in the n-th section is then approximated in the local framework as

          (2) , 1 , , 2 Ĥ loc TM sw loc t n t n Ik         H ρ ρ ρ x ρ (53)
The complex magnitude   , TM n I   ρ is obtained by imposing the continuity of the magnetic field across two adjacent sectors as described by the following equa-tion:

        1 (2) 1, 1 , 1, , (2) 1 ,
              (54)
As the continuity of magnetic field is enforced by the equation ( 54) and the variation of the average impedance is smooth and continuous, we can assume a quasiconservation of power at the different interfaces.

After estimating the incident wave on the metasurface and establishing the impedance variation using equations ( 42) and ( 46), the next step is to physically implement the metasurface structure.

Physical implementation of the metasurface

The impedance variation law found with the proposed formulation is independent from the approach used to implement the metasurface. Indeed, the surface impedance s Z takes into account all of the interaction effects between the wave, the metasurface and the media below the 0 z  plane.

Metasurface structure

In this work, the metasurface structure is implemented using sub-wavelength metallic elements printed in a square lattice over a grounded dielectric substrate (Figure 29). The proposed unit cell is a circular patch with a variable v-shaped slot (Figure 31). It presents an axis of symmetry corresponding to the middle of the v-shaped slot. The It should be noted that the rotation of the incident angle  is not equivalent to the rotation  of the unit cell in the opposite direction (see figure 32). The two structures are not equivalent due to the periodicity of the structure in a square lattice. It is therefore important to consider these two rotation parameters separately. Another important aspect is the symmetry of the metasurface. In the case of a symmetrical metasurface, the impedance tensor is diagonal when the incident wave propagates in the direction of the symmetry axis. However, it should be noted that symmetry of the unit cell is not equivalent to the symmetry of the metasurface. In the example of figure 33.(a), the unit cell has two axis of symmetry while the metasurface structure is not symmetrical. This is due to the square lattice of the metasurface. A symmetrical unit cell gives a symmetrical metasurface when the axes of symmetry of the cell are parallel to the axes of the square periodicity of the structure (Figure 33.(b)).

Unit cell parameters

Surface impedance calculation

In order to estimate the impedance tensor corresponding to a given unit cell, the metasurface is assumed to be locally periodical (in terms of spatial periodicity and not modulation periodicity). As a result, the impedance tensor is calculated for an infinite periodical metasurface composed of same unit cell (the unit cell of interest).

axes of symmetry of the cell axes of symmetry of the cell

Zk

is then obtained by adding the contribution of the grounded dielectric slab at the resonant wavenumber sw k .

The metasurface design procedure presented in the chapter 2 is independent of the implementation method used in order to obtain the surface impedance distribution.

The MoM model will then be modified or replaced if another implementation method is used (multi-layer substrate…). As stated in the two previous sub-sections, the impedance tensor is affected by the unit cell parameters, the surface wave wavenumber sw k and the structure of the metasurface (printed patches over a grounded dielectric slab).   and 0   , a working frequency of 20GHz and a substrate Rogers TMM6 of relative permittivity 6 and thickness 1.27mm.

Spatial discretization of the metasurface

In order to implement the metasurface with a square grid of printed patches, the impedance variation law    s Z ρ needs to be discretized with respect to the position on the metasurface. In this context, the metasurface is discretized into a square grid of discretization step xy    . The average impedance, the modulation index and the periodicity of each impedance component are assumed to be constant inside the unit cells. Their value is set to be equal to the central value of the impedance modulation parameters. The latter variation is only "apparent" in the sense that, since the impedance elements are small compared to the wavelength, the real physical boundary condition "feel" by the wave is a sort of average impedance between adjacent different values. This effect recreates a continuous impedance condition (homogenization).

Implementation algorithm

In this section, the algorithm for the design and implementation of modulated tensorial metasurfaces for aperture field generation is summarized and detailed. Figure 38 shows the algorithm flowchart describing each step needed for the implementation of an arbitrary aperture field distribution () obj t  E ρ . The algorithm described in this chapter was implemented in a MATLAB code. The algorithm is composed by the following steps:

Preliminary steps

These steps are independent of the objective aperture field distribution. They are carried out as follows :

2 s sw 0 0 X k k 1                          loc loc
X X 1 M Im H ( ) X X 1 M Im H ( )                              ρ ρ ρ ρ ρ ρ   (2) , , 1 H l TM n t n sw I k     H     11 11
,

loc loc X M   ρ ρ     12
X M H E X M H E                ρ ρ ρ ρ ρ ρ ρ ρ

Objective Field (input)

Impose the variation of and

Start

 

X   ρ   X   ρ  Step 1.1 :
The working frequency is fixed

 Step 1.2 :
The dielectric substrate and the unit cell design are selected. It should be noted that these two elements are chosen so that the achievable impedance variation range is the largest possible.  Step 1.3 : Generation of the impedance database using the MoM model for a fixed discretization step.  Step 1.4 : The average impedance variation laws and the maximum values of the modulation indices are fixed. These values are obtained by ensuring that the impedance variation stays always inside the impedance database.  Step 1.5 : The metasurface is decomposed into subdomains, along the direction of propagation, in which the surface modulation is assumed to be constant. For a cylindrical surface wave excitation, the metasurface is divided into angular subdomains and then each subdomain is divided into N elementary sections.

These important steps are further described in the table 3.1.

Step

Input

Step

  ij X  ρ ,   max ij M 1.5
-Divide the metasurface into sectors and sections. -Table 3.1: Preliminary steps of the design algorithm.

Continuous impedance determination

After performing the preliminary steps, the objective aperture field distribution is de- The steps for the calculation of the continuous impedance variation are further described in table 3.2.

Step

Input

Step description Eq Output

For each sector (loop on the sectors)

Starting from source and following the propagation path

2.1   ij X  ρ ,   max ij M
Estimate the propagation constant by solving the local dispersion problem [START_REF] Casaletti | Guided Waves on Periodically Modulated Scalar and Tensorial Reactance Surfaces: a Circuital Approach[END_REF].

  t  H ρ 2.3   ij X  ρ
Define the local framework transfor-

mation matrix    R ρ . (51) ,(52)    R ρ ,   loc t  H ρ , , ( )
loc obj t  E ρ 2.4   loc t  H ρ , , ( )
loc obj t  E ρ ,   ij X  ρ Calculate the modulation indices   11 loc M  ρ and   12 loc M  ρ . (46)   11 loc M  ρ ,   12 loc M  ρ 2.5   loc t  H ρ , , ( )
loc obj t  E ρ ,   ij X  ρ ,   11 loc M  ρ ,   12 loc M  ρ Calculate the impedance variation law    s Z ρ . (43)    s Z ρ
Table 3.2: Continous impedance determination algorithm.

Metasurface design generation

After calculating the impedance varation law, the metasurface design is generated with the following steps:

 Step 3.1 :
The metasurface is discretized into a rectangular grid with a discretization step equal to the unit cell spatial periodicity (

Feeder design 69

The steps corresponding to the metasurface design generation are described in Table 3.3.

Step

Input

Step description Output

x y d    

Discretize the metasurface into rectangular grid.

 

,

kl xy 3.2   ij X  ρ
Define the discrete impedance as the value at the center.

 

,

d ij k l X x y 3.3 DB ij X ,   , kl xy
Find the unit cell giving the desired impedance in the impedance database. DXF file Table 3.3: Final metasurface design generation.

 

Feeder design

The last step of the metasurface antenna implementation is the design of the feeding structure. The geometry shown in figure 39 is used to excite a cylindrical surface wave on the metasurface. The feeding structure consists of a circular patch of outer radius 2

r with an annular slot of inner radius 1 r and width e . The inner circle of the feeder is connected to the core of a coaxial cable from below the metasurface. For a fixed substrate thickness and relative permittivity, the metasurface antenna is adapted by acting on the parameters 1 r , e and 2 r of the feeder. The effects of these pa- rameters on the antenna adaptation are presented in the next chapter.

Conclusion of the chapter

In this chapter, practical considerations in the metasurface implementation are presented.

First, the widely used cylindrical surface wave excitation was presented. The local framework definition and the impedance variation laws were described in this special case. In addition, a method for the estimation of the incident wave amplitude and phase variation was presented.

Then, the different aspects of the implementation using printed patches over a grounded dielectric substrate were presented. A new unit cell design offering an additional degree of freedom was introduced. In addition, considerations about the spatial discretization of the metasurface and the surface impedance calculation were addressed.

After that, the algorithm used for the generation of the metasurface design was sum-

h substrat Ground plane coax 1 r 2 r e Top view
Side view marized and detailed with a flowchart and different explanation tables. The algorithm started with preliminary steps independent of the objective aperture distribution.

Then, the procedure for the calculation of the impedance variation was presented. In the final step of the algorithm, the DXF of the metasurface structure is generated.

In the last part of the chapter, the feeding structure used in order to generate a cylindrical surface wave was presented. The structure consists of a circular patch with an annular slot in it. The parameters of the feeder are used in order to control the antenna adaptation.

The implementation procedure was presented in the case of cylindrical wave excitation. However, it can be generalized to other types of excitations by estimating the incident wave analytically or numerically. In addition, other structures such as multilayered substrate should be investigated.

In the next chapter, simulation results are presented. The effects of the parameters of the feeder on the antenna adaptation are studied. Then several metasurface designs with various radiation patterns are simulated with the software ANSYS Designer. The simulation results are compared with the theoretical ideal aperture field radiation pattern.

Chapter 4. Numerical Results

Introduction

In this chapter, a numerical validation of the proposed procedure through commercial software analysis is presented. In the first part of the chapter, the effects due to the variation of the feeder's geometrical parameters on the antenna impedance matching are studied by parametric analysis on HFSS.

In the second part, several antenna metasurface designs are introduced and analyzed using the software ANSYS Designer. Designs treating far-field as well as near-field radiation pattern are presented. Simulation results are compared with radiation pattern radiated by the ideal aperture field distribution and by the estimated equivalent magnetic currents above the metasurface.

Study of the parameters of the feeder:

In order to take into account the effect of the coaxial cable, the feeding structure described in the previous chapter (Figure 39) is imported in the simulation software HFSS. Figure 40 illustrates the metasurface, the feeding structure and the coaxial cable designs in HFSS.

In order to study the effect of the parameters 1 r , e and 2 r of the feeder (Figure 39), a parameter-sweep simulation is performed. Due to the complexity of the structure, the analysis is performed on the central portion of the metasurface surrounded by absorbing boundary conditions. The dimensions of the portion are chosen in such a way that increasing it would not affect the scattering parameter variation. In our study, each parameter of the feeder is studied separately in order to understand its effect on the antenna matching. The simulations were performed for a Rogers TMM6 substrate of thickness h=1.27mm and relative permittivity εr=6.

In figure 41 In figure 42, the variations of the scattering parameter S 11 in dB with respect to the frequency are presented for different values of the width e of the annular slot. The simulations were performed for 2 3.8mm r  and 1 0.8mm r  . As for 1 r , the parameter e mainly affects the level of S 11 at the resonance. However, it has almost no effect on the resonant frequency.

Finally, figure 43 presents the variation of the scattering parameter S 11 in dB for different values of the outer radius 2 r of the patch. The parameter sweep was performed for 0.25mm e  and 1 0.8mm r  . Unlike the two previous parameters, a variation of 2 r produces a shift of the resonant frequency. The step by step adaptation procedure is described in Table 4.1.

Metasurface designs

In order to validate the proposed design procedure, several metasurface antennas examples have been generated and simulated using commercial software. The obtained results are compared with :

 The theoretical far field resulting from the ideal continuous aperture field distribution. It is labelled "Ideal Aperture" in the different plots of the radiation patterns and represented in blue.  The free-space radiation pattern of the equivalent magnetic current ( ) 2 t   J ρ n E . This radiation pattern is obtained using the Equivalence Theorem. The magnetic currents are estimated by following the propagation path of the surface wave using the impedance values found in the database. It is labelled "Equiv Current" in the different plots of the radiation patterns and represented in red.

All the examples presented in this chapter are designed on a Rogers TMM6 substrate of relative permittivity εr=6 and thickness h=1.

Tilted beam linearly polarized antenna

The first design example is an antenna radiating a single beam at the direction )

obj E  (c) obj E  (d) arg( ) obj E  .
As expected, the radiation pattern presents a tilted beam pointing at the desired direction with the desired linear polarization. The simulated beamwidth is equal to 15°. phase ( )

E  phase ( ) E 
The cross-polarization levels differ between the two software (-17dB in FEKO and -24dB in Designer). This discrepancy is probably due to the varying-quality mesh that was used in the two simulations. The differences between the simulation and the theoretical radiation pattern may have several causes. For instance:

 The dielectric substrate is supposed infinite in the simulation software.  The feeder does not convert all the input power into surface wave. As a result, part of this energy is radiated directly.  The surface wave that was not converted into leaky waves may be radiated or reflected when reaching the end of the metasurface.  The surface wave reflected at the end of the metasurface can be converted into undesired leaky waves radiating in unpredicted directions.  The mesh quality greatly affects the value of the off-diagonal reactance com-  as the frequency changes. In fact, a frequency variation leads to a differ- ent value of propagation constant sw t k . This change will affect the phase of the incident wave producing a phase error (proportional to the frequency shift) in result of the holography. In addition, the variation of the propagation constant also affects the values of the impedance tensor elements.

The results of figures 47 and 48 validate the design procedure in the case of tilted beam LP radiation pattern. In order to reduce the undesired reflections at the edges of the metasurface, larger metasurfaces will be considered. Therefore, only results obtained with Ansys Designer will be presented in the upcoming examples.

Broadside circularly polarized antenna

The second example is a broadside circularly polarized antenna producing the pattern shown in figure 49. A circular polarization (CP) can be obtained by superposing two linearly polarized beams with orthogonal polarization and 2

 phase shift. Thus, the relative aperture field distribution is given by Figure 52 represents the far field radiation pattern. The RHCP and the LHCP components in the 0   cut-plane are represented in figure 52.(a) and (b), respectively. The 3D radiation pattern is represented in figure 52.(c). It can be seen from the figure that the radiation pattern corresponds the predicted broadside RHCP radiation. In addition, a good agreement is obtained between the simulation and the theoretical results. The simulated beamwidth is equal to 8° with a side lobe level of -12dB. A better agreement can be obtained by further increasing the antenna dimensions to diminish the undesired edges reflections. The results given in figure 52 validate the metasurface design procedure in the case of broadside RHCP radiation pattern.

    ˆ( ) 1 2 obj t j   E ρ x y (57)

Multi-beam metasurface antenna:

As a third example, we consider a metasurface antenna radiating multiple beams. In order to obtain a multi-beam radiation pattern, we propose to superpose the aperture field distribution corresponding to each beam. This leads to [START_REF] Teniou | Implementation of radiating aperture field distribution using tensorial metasurface[END_REF]- [START_REF] Teniou | Tensorial Metasurface Antennas Radiating Polarized Beams Based on Aperture Field Implementation[END_REF]:

1 1 ( ) ( ) beams N obj k t t k beams N      E ρ E ρ (58)
where beams N is the number of beams.

()

k t  E ρ is the objective field corresponding to each individual beam.
In this report, two examples of multi-beam metasurface antennas are presented. The first antenna radiates two beams with different polarizations. The second antenna has a more complex radiation pattern of four beams radiating in different directions with different types of polarizations. For each antenna, the objective aperture distribution is calculated using equation (58) by variating the number of beams and the nature of their polarization.

Two beams antenna

The first multi-beam antenna example that we considered is a metasurface radiating two beams (Figure 53) with the following characteristics:

RHCP Phi

sin cos sin sin

u v         1 1 , 30 , 0          2 2
, 45 ,135

     û v (a) (b) (c) (d)

Four beams antenna

As a further validation of the multi-beam design approach, a more complex objective radiation pattern is considered (Figure 57). In this example, the considered metasurface radiates four beams with the following characteristics [START_REF] Teniou | Implementation of radiating aperture field distribution using tensorial metasurface[END_REF]: -Linear


 -Linear  RHCP LHCP     2 2
, 45 ,180

         3 3
, 30 ,90 The radiation pattern was normalized with respect to the maximum beam amplitude.

         1 1 , 30 , 0        
For this complex radiation pattern, the metasurface was simulated for different frequencies. Figure 61 shows the frequency behavior of the two linearly polarized beams presents in the 0 In view of the diversity of the realized radiation pattern and the complexity of the achieved multi-beam radiation patterns, these results confirm that the proposed method is able to implement general radiating aperture field distributions.

  cut-plane.

Flat top circularly polarized antenna

In this section, we implement a flat-top RHCP radiation pattern. For this kind of antennas, often used in satellite applications, the normalized radiation pattern n F is relatively constant within a desired angle range while it is below a certain threshold outside of this region.

The desired radiation pattern can then be described by the following equation

  1 if if f n f F                ( 59
)
where  is the maximum oscillation with respect to the desired level. 1   is the threshold.   . In addition, the simulation results are in close agreement with the theoretical results. As a result, we can consider that beam shaping is achievable using the proposed design procedure.

Near field multi-beam focusing antenna

After validation of the design procedure for far field patterns, the control of the near field is investigated. As our design procedure allows the control of the aperture field distributions, it can be used for near-field applications using appropriate objective aperture distributions. In this section, an example of near field multi-beam focusing antenna is presented. It can be seen from the figures that the field is focused in the two planes when propagating away from the metasurface. In addition, we notice high level of the field near the metasurface, especially at its center. These elevated levels of the field are probably due near field radiated by the feeder. This is due to the fact that not all the energy injected in the feeder is converted into surface waves.

Feeder effect cancellation

In order to investigate the effect of the feeder on the radiation pattern, a nonmodulated metasurface is designed and imported on Designer. As the metasurface is not modulated, no leaky wave component is radiated. Thus, the radiated field above the metasurface is mainly due to the effect of the feeder in presence of the metasurface. The near field corresponding to this non-modulated metasurface is simulated and then subtracted from the field of the near field multi-beam focusing metasurface.

Even though this approach is not exact, it can give an idea about the radiation resulting from the leaky wave and the perturbation caused by the feeder radiation.

In figure 70

Conclusion of the chapter

In this chapter, several numerical results are presented in order to validate the proposed design procedure.

In the first part of the chapter, we studied in HFSS the effects of the geometrical parameters of the feeder on the antenna adaptation by varying the latter in different simulations on HFSS. We concluded from the parameter study that the resonant frequency was mainly affected by the outer radius parameter 2 r while the two remaining parameters [START_REF] Alù | Pairing an epsilon-negative slab with a mu-negative slab: Resonance, tunneling and transparency[END_REF] r and e mainly affected the level of the scattering parameter at the resonant frequency. Finally, a step by step algorithm for the adaptation of a metasurface antenna using the considered feeding structure was presented.

After that, several far field radiation patterns were implemented and validated with numerical software. At first a 3 radius metasurface radiating a linearly polarized tilted beam was designed and simulated with FEKO and DESIGNER. The results were in close agreement with the theoretical results and corresponded to the objective radiation pattern. Then, several 5 radius metasurfaces were designed. Using these distributions could be implemented using the proposed method.

Finally, an example of near field control using metasurfaces is presented. In the considered example, a metasurface focusing four beams at a desired height is designed in collaboration with the IETR laboratory of the University of Rennes1. The simulation of the metasurface presented satisfactory results of near field beam focusing.

The study of last example led to the conclusion that the feeder significantly affects the near field radiation pattern of metasurface antennas. In fact, subtracting the contribution of the feeder from the simulated near field led to a significant improvement of the results. A future study should be performed in order to reduce this effect and maximize the surface wave energy conversion achieved by the feeder.

The simulation software HFSS is good at describing the feeding structure. However, it cannot simulate large size antennas. On the other hand, larger metasurfaces can be simulated using Designer. However, it does not give a good analysis of the feeding structure. As a result, HFSS is used in order to simulate the scattering parameter while Designer is used for the radiation pattern simulation. In addition, both Commercial software simulation cannot separate the different mechanism governing metasurface radiation. It can be very useful to develop an in-house code for the analysis of these structures as it allows the separation of the different radiation components such as: leaky waves, surface waves, radiation of the feeder, radiation at the edge of the metasurface…

In the next chapter, several metasurface antennas are manufactured and measured as a further validation of the proposed design procedure.

Chapter 5. Experimental results

Introduction

In this chapter, measurement results are presented as a validation of the proposed metasurface design procedure. At first, the manufacturing mechanical limitations are presented leading into a condition on the impedance database generation. Then a first prototype is manufactured on a substrate FR4 in order to check the precision obtained with the considered manufacturing process. The designed prototype radiation pattern is then measured to asses the validity of the simulation results. In the last part of this chapter, several examples of metasurface antennas are manufactured and measured.

The masurements of the scattering parameter are compared with the results of HFSS while the radiation patterns are compared with the results of Designer.

Manufacturing limitations

As the metasurface consists of small patches printed over a grounded dielectric substrate, the main manufacturing limitation is the gap dimension between two conducting elements. Depending on the manufacturing technique (chemical, laser etching,...) this gap cannot be smaller than a certain minimum value. In the following, we denote this minimum gap by the variable min g .

Figure 72 shows two adjacent unit cells of the metasurface with identical shape. The figure is used to show how the typical geometry of our metasurface looks like.

Judging from the figure, two type of gaps can be distingished:

 The gap g due to the presence of a v-shaped slot in the unit cell design.  The gap p g present between two adjacent patches. This gap results from the spacial periodicity.

By imposing the limitation condition on these two gaps, we will derive a condition on the variation range of the filling factor defined as % d a d   .

Figure 72: Description of the two gaps present in the structure: The gap g due to the v-shaped slot and the gap p g due to the periodicity of the structure.

 Condition on the slot gap :

In our study, we considered unit cells with a v-shaped slot of width equal to 20% of

 Condition on the distance between two adjacent cells :

The distance between two adjacent patches is minimal when the considered patches dimension is maximal. By considering two adjacent patches with the same dimension we can express the gap p g as a function of the filling factor % a as follows:

    % 2 2 1 2 1 p d g d d d a d d              (
Depending on the manufacturing process (the minimum achievable gap min g ), equation ( 67) is used in order to limit the variation range of the filling factor % a when generating the impedance database. Using this condition, we can ensure that the final metasurface geometry can be manufactured. For example, if we consider a practical case of a mimimum achievable gap min 150µm g  and a periodicity ' 2.5mm d  , we obtain that the filling factor has to satisfy: % 0.3 0.97 a  .

It can be seen from the Figure 35 (section 3.3.4) that the filling factor affects the variation range of the impedance components (in particular for the cross polarization elements). As a result, we must check the variation range of the impedance tensor elements when equation (67) is satisfied.

Another important limitation of the manufacturing process is related to the opening angle  of the v-shapet slot of the unit cell. When this angle is too small, the triangular part of the unit cell is so small that it can detache during the manufacturing process. To avoid this problem, we only considered slot opening angles  that are greater than 60°.

Mechanical validation test

High performance dielectric substrates used for space application are generally expensive and may have a long delivery time. In order to test the precision of the manufacturing process and avoid wasting expensive substrates, a metasurface prototype has been printed on a commun cheaper substrate (FR4 in our case).

The metasurface prototype have the following specifications: Figure 75 shows the feeding structure with and without a soldered coaxial cable. The feeding structure was also checked in the microscope and did not present undesired defaults. In light of these results, we decided to proceed with the manufacturing of practical metasurface antennas on an adequate substrate as a validation of the proposed metasurface design procedure.

Measurement of metasurface antennas

In this section, the measurement results for the different manufactured metasurface antennas are presented.

Measurement of the test prototype

Before measuring functional metasurfaces, the previously manufactured test prototype is measured. The measurement results are then compared with simulations in order to check the validity of our approach and explore the limitations of the simulation tools. The first step was the measurement of the antenna reflection parameter (S 11 ) of the antenna prototype. Figure 76 shows the comparison between measurements and HFSS simulations. It can be seen from the figure that the antenna is matched (S 11 <-10dB) in the frequency band [11.5-13]GHz. Therefore, the antenna is properly working at the desired frequency of 12.25GHz. In addition, a close agreement is obtained between the simulation and the measurement.

As second step, we measured the radiation pattern of the prototype at the GeePs Laboratory in France. Two orthogonal linear components were measured using the described setup in order to obtain the right hand and the left hand circular components of the radiated field.

Figure 79 shows the measurement results (in black) of the far field radiation pattern in the 0   cut-plane at 12.25GHz. The measurements are compared with the simula- tion results (in red) obtained using Ansys Designer. The RHCP components are represented by solid lines while the LHCP components are represented with dashed lines.

The measured radiation pattern corresponds to the expected broadside RHCP beam.

The measured cross-polarization level is equal to -20 dB and the side lobe level is equal to -27 dB. The beamwidth is equal to 5.3°. A close agreement is obtained be-tween the simulation results and the measurements in the region of the main lobe. The discrepancy outside of this region is probably due to the fact the dielectric substrate is considered infinite in the simulation. There is therefore a discontinuity in the end of the metasurface that can lead to undesired radiations. In addition, due to the complexity of the structure, the mesh quality was significantly limited by the available computer memory. The measured gain is equal to 18dBi. The same gain value was obtained in the simulation. The low value of the gain is due to the high dissipation factor of the FR4 substrate ( tan 0.017

 

).

Figure 79: Measurement (in black) and simulation (in red) of the circular components of the far field radiation pattern for the 0   cut-plane [30]. Solid lines represent RHCP components and dashed lines represent LHCP components. The working frequency is 12.25GHz.

Broadside RHCP Antenna

After measuring the radiation pattern of the test prototype, a similar antenna was manufactured on a substrate more suitable for space applications. The considered antenna has the following specifications:  Amplitude distribution law: in order to minimize the side lobe levels, a Hamming amplitude distribution was used [START_REF] Harris | On the use of windows for harmonic analysis with the discrete Fourier transform[END_REF].

As for the first prototype, the metasurface was manufactured by the company INOVEOS and is represented in figure 80. Also this metasurface exhibits the spiral shape associated to the circular polarization. In figure 81, measurement and the simulations of the scattering parameter S 11 with respect to frequency are presented. The solid line corresponds to measurements while the dashed lines are simulations results obtained using HFSS. The effective relative permittivity of the substrate for the considered frequency (equal to 6.3) is different from the value used for the metasurface design. As a result, the simulation results are presented in the case of ϵ r =6 (in black) and ϵ r =6.3 (in red). As it can be seen from the figure, measurements are closer to the simulations obtained with ϵ r =6.3 (value given by Rogers). In addition, we noticed undesired oscillations in the measured scattering parameter. In our opinion, these oscillations are caused by the reflection of the surface wave at the edge of the metasurface. The reflected surface wave generates the oscillations when it returns to the feeder. This effect is less visible in the case of the FR4 because the propagating surface wave is attenuated by the high level of dielectric losses ( tan 0.017

 

). The level of the undesired oscillations can be reduced by increasing the metasurface dimensions. Thus, the propagating surface wave will either be converted into leaky waves or attenuated through propagation. As the scattering parameter curve does not correspond to the simulation, we considered the frequency of maximum broadside gain as the working frequency. In order to measure the gain, the radiation pattern of the metasurface is compared to the radiation pattern of a standard gain horn antenna. Figure 83 gives the variations of the total gain at broadside with respect to the frequency. The maximum gain value is equal to 26dBi at the frequency of 9.5GHz. For this antenna size, this corresponds to a total efficiency 0 30% e  . This value is not far from the theoretical aperture efficiency for a hamming distribution and for this dimension ( 36% ap e 

). This two values leads to a product 83.33% r cd ee 

. This product is high due to the low level of the dielectric losses introduced by the TMM6 ( tan 0.0023

 

).

- In figure 84, the measurement and the simulation results for the broadside RHCP metasurface radiation pattern are presented for the 0   cut-plane. The antenna ra- diates the expected RHCP beam at broadside with cross-polarization smaller than -18 dB and side lobe levels smaller than -30dB. The measured beamwidth is equal to 4.6°.

At the working frequency of 9.5GHz, the simulation results are pretty similar to the measurement results. Figure 85 shows the variation of the radiation pattern of the manufactured metasurface with respect to the frequency. The figure gives the circular components of the radiated field at the 0   cut-plane for the frequencies 9 GHz (in red), 9.5GHz (in black) and 9.9GHz (in blue). We can see from the figure that the cross-polarization level and the side-lobe level are relatively stable in the considered bandwidth. In addition, by calculating the variation of the axial ratio in dB with respect to the frequency (see Figure 86), we can see that the axial ratio is smaller than 3dB for the bandwidth between 9GHz and 9.9 GHz. In light of the figures 84-86 we can see that, despite the frequency shift of the working frequency, the performances of the manufactured metasurface are quite promising and represent a first validation of the design procedure.

Four-beams metasurface antenna

The second antenna to be manufactured is the four-beam metasurface. This antenna is similar to the one presented in the previous chapter (Numerical Results). However, it was designed for a working frequency of 10GHz and manufactured on substrate ROGERS TMM6 of relative permittivity ϵ r =6 and thickness h=2.54mm. The antenna parameters are the same as those of the manufactured broadside RHCP metasurface.

A Hamming amplitude distribution was also applied in order to reduce the side lobes level. The metasurface was manufactured by INOVEOS and is represented in figure 87. The corresponding radiation pattern will be measured soon at the GeePs laboratory. The near field was then measured at the IETR. Figure 90 gives the measurement results for the horizontal cut at a height of 5 z   . The figures 90.(a) and (b) show the linear components of the electric field while the figure 90.(c) shows the total field. It can be seen from the figure that the radiated near field exhibits the desired near field focusing behavior. The obtained beams have the expected polarizations (two beams polarized along the x axis and two beams along the y axis). The beamwidth is equal to 0.6λ with side-lobe levels about -12dB. In light of the results obtained in the previous chapter, we assume that they are mainly caused by the spatial radiation of the feeder. Indeed, this effect should be more predominant in near-field applications.

Conclusion of the chapter

In this chapter, an experimental validation of the metasurface design procedure is presented. In the first part, the limitations of the metasurfaces manufacturing were presented. The main limitation is the minimum distance between the conductors composing the structure. This limitation emerges in two parameters of the structure :

(1) The gap g due to the v-shaped slot present in the unit cell. (2) The gap p g between two adjacent unit cells that is due to the spatial periodicity of the structure. This constraint led to a condition on the filling factor ( % a ) variation range. By satisfying this condition during the impedance database generation, one can insure that the designed metasurface respects that manufacturing limitations.

Then, a metasurface prototype design was manufactured on a substrate FR4 to assess the manufacturing precision of the metasurface geometry. This step was performed in order to check if the unit cells shapes were manufactured with a satisfying level of precision. The prototype was designed for a metasurface radiating a broadside RHCP beam at the frequency of 12.25GHz. The structure was manufactured by the company INOVEOS based in Brive-la-Gaillarde in France with a chemical process giving a very good level of precision. The prototype radiation pattern was then measured at the GeePs laboratory giving results in good agreement with the simulation.

After that, a broadside RHCP metasurface was manufactured on a substrate ROGERS TMM6 that is more suitable for space applications. The metasurface is similar to the one presented in the simulation chapter but with an objective working frequency of 10GHz. However, a frequency shift of the adaptation bandwidth as well as undesired oscillations were noticed when measuring the scattering parameter S 11 . In our opinion, this is probably due to undesired reflections at the edge of the metasurface and to a difference in the relative permittivity of the substrate. The far field radiation pattern of the metasurface antenna was measured in the GeePs laboratory giving the desired broadside RHCP beam with a cross polarization level of -18 dB and side lobe level smaller than of -30 dB. In addition, the antenna parameters (cross polarization level, side lobe level and axial ration) are relatively stable over a bandwidth between 9GHz and 9.9GHz. The maximum gain is equal to 26dB at a frequency of 9.5GHz. This corresponds to total efficiency 0 30% e  . A four-beam metasurface was also manufactured for an objective working frequency of 10GHz and is ready to be measured at the same laboratory. This is probably caused by the spatial radiation of the feeder. Indeed, as stated in the previous chapter, not all the energy injected in the feeder is converted into surface wave. This leads to undesired radiations by the feeding structure.

Despite the undesired oscillations in the scattering parameters, the manufactured metasurfaces presented the desired radiation pattern in all the presented examples. In addition, the TMM6 antennas (broadside RHCP and near-field focusing) presented good performances and they represent a first experimental validation of the proposed metasurface design procedure. In future work, additional sets of metasurfaces will be manufactured in order by improve their performances such as : side lobe levels reduction, efficiency enhancement, edge reflections reductions...

6 .

 6 Conclusions and Prospective ......................................................................... 6.1 Summary of the contribution .............................................................................. 6.2 Future work ......................................................................................................... 6.3 List of publications ............................................................................................. REFERENCES ............................................................................................................... vii

1 :

 1 Preliminary steps of the design algorithm. .................................................Table 3.2: Continou UMR5213 s impedance determination algorithm. ...................... Table 3.3: Final metasurface design generation. ......................................................... Table 4.1: Adaptation steps of the feeder. ................................................................... viii

Figure 1 : 5 Figure 2 : 6 Figure 3 : 9 Figure 4 : 14 Figure 5 : 15 Figure 6 : 16 Figure 7 :

 1526394145156167 Figure 1: Antenna as a transition device [31]. ............................................................... 5 Figure 2: Radiation pattern example in a logarithmic scale. ......................................... 6

Figure 8 : 19 Figure 9 : 20 Figure 10 : 21 Figure 11 : 21 Figure 12 : 22 Figure 13 : 23 Figure 14 : 24 Figure 15 : 25 Figure 16 : 25 Figure 17 : 26 Figure 18 : 29 Figure 19 : 32 Figure 20 : 34 Figure 21 : 39 Figure 22 :Figure 23 :Figure 24 :Figure 25 :Figure 26 :Figure 27 :Figure 28 :Figure 29 :Figure 30 :Figure 31 :Figure 32 :Figure 33 :Figure 34 :Figure 35 :Figure 36 :Figure 37 :Figure 38 :Figure 39 :Figure 40 :Figure 41 :Figure 42 :Figure 43 :Figure 44 :Figure 45 :Figure 46 :Figure 50 :Figure 51 :Figure 52 :Figure 53 :Figure 54 :Figure 59 :Figure 61 :Figure 62 :Figure 63 :Figure 65 :Figure 67 :Figure 70 :Figure 74 :Figure 75 :Figure 76 :

 81992010211121122213231424152516251726182919322034213922232425262728293031323334353637383940414243444546505152535459616263656770747576 Figure 8: Metasurface antenna designed in [16]. (a) Photograph of fabricated antenna (b) Measured co-polarization and cross-polarization E-plane radiation ...................... 19

Figure 84 :Figure 86 :Figure 87 :Figure 88 :Figure 89 :

 8486878889 Figure 84: Measurement (in black) and simulation (in red) of the circular components of the far field radiation pattern of the broadside RHCP metasurface for the 0   cut-plane. Solid lines represent RHCP components and dashed lines represent LHCP components. The working frequency for the simulations and the measurements is 9.5GHz. ...................................................................................................................... 121

Chapter 1 -Chapter 2 -Chapter 3 -Chapter 4 -Chapter 5 -Chapter 6 -

 123456 Context and state of the art: In this chapter, the general context if this thesis is presented. In addition, key antennas parameters needed in this study are presented. Finally, an overview of different metasurface examples present in the literature is given. Formulation and theoretical background: Theoretical background about scalar and tensor metasurface properties is given. In addition, the formulation of the proposed method is detailed. Our approach is based on a local holographic principle and variable impedance modulation parameters. Implementation of the metasurfaces: This chapter gives metasurface implementation considerations such as surface-wave excitation and feeding structure. A step by step algorithm for the implementation of metasurfaces is given, starting from the objective aperture field distribution to the final metasurface structure. Numerical Results: In this chapter, numerical results obtained using commercial software are presented. First, the feeding structure parameters are studied in order to obtain the adaptation of the metasurface. Then, examples of metasurface designs with various objective radiation patterns are presented. Experimental Results: This chapter gives experimental results obtained from the measurement of manufactured metasurfaces. In addition, constraints related to the manufacturing method (laser etching technique) are presented. Conclusions and Prospective: Conclusions and future work are presented

Figure 2 :

 2 Figure 2: Radiation pattern example in a logarithmic scale.

U

  is the radiation intensity rad P is the total radiated power If no direction is specified, the directivity is calculated with respect to the direction of the maximum radiation intensity. The directivity indicates the antenna ability to concentrate power in a desired direction. The bigger is the directivity, the more power is radiated in the maximum direction. It is generally calculated in dB (

is the conduction efficiency d e is the dielectric efficiency 11 S

 11 is the input port voltage reflection coefficient The efficiencies c e and d e describe the Ohmic losses introduced by the conductors and the dielectric respectively. They are usually determined experimentally but cannot be Losses relation between the gain and the directivity as follows cd G e D 

Figure 4 :

 4 Figure 4: Contoured beams methods: (a) Using an array of feeders. (b) Using shaped reflectors [33].

Figure 5 :Figure 6 :

 56 Figure 5: Example of a 3D metamaterial structure [36].

Figure 7 :

 7 Figure 7: An example of a metasurface consisting of 2D arranged scatterers [6].

8 .

 8 (a)). Using the relation between the direction of the radiated beam and the modulation periodicity, a metasurface radiating a tilted beam was implemented. The corresponding radiation pattern is represented in Figure.8.(b).

Figure 8 :

 8 Figure 8: Metasurface antenna designed in [16]. (a) Photograph of fabricated antenna (b) Measured co-polarization and cross-polarization E-plane radiation

Figure 9 :

 9 Figure 9: Tilted beam metasurface antenna [17] (a) Scalar impedance pattern formed from holographic interference (b) Measured radiation pattern

Figure 10 :

 10 Figure 10: Spiral shaped metasurface antenna [18]. (a) Spiral shape (b) Radiation pattern

Figure 11 :Figure 12 :

 1112 Figure 11: Tilted beam circularly polarized antenna [19]. (a) Modified metasurface to radiate circular polarized waves (b) Radiation pattern

  ing pattern antenna. The eight lines are disposed symmetrically in order to produce the desired focusing pattern (Figure.13).

Figure 13 :

 13 Figure 13: Near field focusing pattern antenna [21]. (a) Principle (b) Simulated near field.

  Figure.14 Illustrates the obtained metasurface and the measured radiation pattern.

Figure 14 :

 14 Figure 14: Tilted beam circularly polarized metasurface antenna [17]. (a) A portion of the tensor holographic impedance surface (b) Radiation pattern with the RHCP in black and the LHCP in gray

Figure 15 :

 15 Figure 15: Circularly polarized isoflux antenna [22]. (a) Photograph of the realized prototype. (b) Gain measurements at 8.6 GHz (up: right hand component. down: left hand component).

Figure 16 :

 16 Figure 16: Example of metasurface antenna designs presented in [23]. (a) Sector isoflux antenna (b) Tilted beam circularly polarized antenna.

Figure 17 :

 17 Figure 17: Radiation pattern measurement of a broadside circularly polarized metasurface antenna with tapered amplitude distribution [26]: (a)

Figure 18 :

 18 Figure 18: A surface wave propagating through a mono-directional reactance surface.

Figure 19

 19 Figure 19 illustrates the effect of the modulation periodicity on leaky wave generation. It presents the electric field on the half space above the metasurface with respect to the term 2 sw t k a   . The figure was obtained by calculating, on a MATLAB code,the total electric field corresponding to the tangential field described by equation[START_REF] Oliner | Guided waves on sinusoidally-modulated reactance surfaces. Antennas and Propagation[END_REF]. It can be seen from the figure that only a surface wave is present when the metasurface is not modulated. By increasing the modulation periodicity, we notice a first case where

Figure 19 :

 19 Figure 19: Effect of the modulation periodicity on leaky wave generation at a frequency of 20GHz .

Figure 20 :

 20 Figure 20: Metasurface geometry and leaky wave generation.

I

  are the voltage and current modal coefficients of the surface wave,

  ij X is the average impedance of each component. ij M is the modulation index of each component. ij p is the periodicity of modulation for each component. Depending on the values of the individual periodicities ij p , two main cases can be distinguished when introducing the sinusoidal modulation of the impedance components:

  [START_REF] Teniou | Modulated Tensorial Metasurfaces for Aperture Field Generation[END_REF] Each component of the electric field , ij E is composed of six terms depending on the surface current components k J . These terms can be decomposed into three main groups depending on their corresponding wavenumber:  Two terms of the form ij k XJ : these terms correspond to the mode 0 excited on the metasurfaces. They are surface waves of wavenumber 0

Figure 21 :

 21 Figure 21: Leaky wave control for aperture field generation.

1 obj

 1 For example, if we consider the first component of the objective field

  servation and reciprocity) are conserved under orthogonal transformations. At each point of the metasurface we define a local framework such that the magnetic field   t  H ρ is directed along a local axis (Figure22).

Figure 22 :

 22 Figure 22: Local framework definition at each point of the metasurface.

Figure 23 Figure 23 :

 2323 Figure 23 presents an example of the generation of an objective phase distribution. The figure illustrate for one component, the incident wave (Figure 23.(a)), the objective wave (Figure 23.(b)) and the corresponding impedance variation obtained using equation (43) (Figure 23.(c)). Then, figure 23.(d) presents the phase of the generated leaky wave. It can be seen from the figure that the phase of the generated leaky wave corresponds to the objective phase distribution.

Figure 24 :

 24 Figure 24: Compensation of the surface wave spreading factor example. (a) Objective amplitude distribution. (b) Amplitude of the incident wave. (c) Variation of the modulation index M . (d) Variation of the average impedance X . (e) Amplitude of the gen- erated leaky wave (normalized).

  modulation indices giving the desired amplitude distribution.

Figure 25

 25 Figure 25 presents an example of amplitude generation using variating modulation index along the propagation direction. The figure illustrates the objective amplitude variation, the amplitude of the incident wave, the variation of the modulation index and the average impedance and the amplitude of the generated leaky wave. The objective amplitude presented in this example is a tapered amplitude distribution (Figure 25.(a)). It can be seen from the figure that the amplitude of the generated leaky wave (Figure 25.(e)) corresponds to the objective amplitude variation.

Figure 25 :

 25 Figure 25: Generation of arbitrary field amplitude distribution. (a) Objective amplitude distribution. (b) Amplitude of the incident wave. (c) Variation of the modulation index M . (d) Variation of the average impedance X . (e) Amplitude of the generated leaky wave (normalized).

Figure 26 :

 26 Figure 26: Cylindrical wave surface wave excitation.

Figure 27 :k

 27 Figure 27: Radial variation of the effective impedance over the metasurface.

Figure 28 :

 28 Figure 28: Radial variation of the effective impedance over the metasurface.





  A component with slow rate given by the variation of the modulation parameters (

Figure 29 : 3 .Figure 30 :

 29330 Figure 29: Metasurface structure composed by periodic metallic elements printed over a grounded substrate [39].

Figure 31 :

 31 Figure 31: Unit cell design: circular patch with a v-shaped slot.



  described by the following parameters: The lattice periodicity d . This quantity corresponds to the unit cell dimension and it should not be confused with the modulation periodicity of the impedance component ij p .  The patch dimension d . For the considered unit cell topology, it corresponds to the diameter of the circular patch.  The filling factor % d a d   . It is defined as the ratio of the unit cell dimension over the structure periodicity. It is expressed in %.  The slot size g . It gives the width of the v-shaped slot. In our case, it is expressed as a fraction of the filling factor.  The opening angle of the v-shaped slot. It is expressed by the parameter  .  The incident angle of the electromagnetic wave. It gives the angle between the direction of propagation of the incident wave and the axes of the square lattice. It is expressed by the parameter  .  The cell rotation. It gives the angle between the axe of symmetry of the unit cell and the square lattice of the structure. It is expressed by the parameter  .

Figure 32 :Figure 33 :

 3233 Figure 32: Comparison between the rotation of the unit cell and the incident angle of the surface wave. (a) Rotation of the incident angle (     , 0 , 30      ). (b) Rotation

Figure 34 :

 34 Figure 34: Printed metasurface equivalent TM and TE transmission lines.

Figure 35 :

 35 Figure 35: Reactance levels corresponding to the unit cells of slot size 0.2 g  and angles 60   and

Figure 35

 35 Figure 35 presents the variations of the impedance parameters written in the cylindrical coordinates with respect to the filling factor (horizontal axis) and the incident angle (vertical axis). These values have been calculated using the MoM model for a unit cell for parameters 0.2 g  , 60

Figure 36 :

 36 Figure 36: Tensorial reactance database generation. Component X  of the impedance tensor for a substrate Rogers TMM6 of relative permittivity 6 and thickness 1.27mm. The working frequency is 20GHz.

Figure 37

 37 Figure 37 presents the variation of the reactance X  of a given metasurface for different values of the discretization step ,, 3 6 12 xy

Figure 37 :

 37 Figure 37: Effect of the spatial discretization on the reactance variation. Effect of the discretization on the X  component for different discretization steps : (a) Continuous impedance. (b) 12 xy      . (c) 6 xy      . (d) 3 xy      . (e) Impedance scale.

Figure 38 :

 38 Figure 38: Flowchart of the metasurface implementation algorithm.

  After that, starting from the source and following the propagation path, the impedance calculated on each sector by performing the following steps : Step 2.1 : The propagation constant ,,sw tn k  is found by solving the local dispersion problem [39].  Step 2.2 : The magnetic field amplitude and phase are found using equation (54) by imposing the continuity of the magnetic field at the sections interface.  Step 2.3 : The transformation matrix    R ρ is found and used in order to write the incident and the objective fields in the local framework.  Step 2.4 : The modulation indices using the amplitude matching condition described by equation (46).  Step 2.5 : The impedance variation obtained using equation (43) (local holography formulation) and equation (46).

  are defined as the value of the continuous impedance at the center of the squares.  Step 3.3 : At each discrete point, the unit cell giving the desired impedance is found in the impedance database. The unit cell is described by the orientation angle  , the slot opening  , and the filling factor. It should be noted that the impedance database is scanned for a fixed incident angle corresponding to the position   , kl xy.  Step 3.4 : A DXF (Drawing eXchange Format) file corresponding to the impedance design is generated.

Figure 39 :

 39 Figure 39: Structure of the cylindrical wave excitation feeder.

Figure 40 :

 40 Figure 40: Simulation of the metasurface in HFSS. (a) Metasurface structure. (b) Structure of the feeder. (c) Coaxial cable modelization.

Figure 41 :

 41 Figure 41: Variation of the scattering parameter S 11 with respect to frequency for different values of 1 r . The remaining parameters have fixed values 0.25mm e  and 2 3.7mm r  .

Figure 42 :

 42 Figure 42: Variation of the scattering parameter S 11 with respect to frequency for different values of e . The remaining parameters have fixed values 1 0.8mm r  and 2 3.8mm r  .

Figure 43 :value of 2 r

 432 Figure 43: Variation of the scattering parameter with S 11 respect to frequency for different values of 2 r . The remaining parameters have fixed values 0.25mm e  and 1 0.8mm r  .

12 

 12 Figure 43 : S 11 = -24dB).

Figure 44 :

 44 Figure 44: Objective far field radiation pattern: linearly polarized beam along the phi axis radiating in the direction     00 , 30 , 0     .

Figure 47 Figure 45 :

 4745 Figure 47 presents the far field results obtained using ANSYS Designer and FEKO. The phi and theta components of the electric field in the 0   cut-plane are repre- sented in figure 47(a) and (b) respectively. In figure 47(c) a 3D representation of the far field radiation pattern obtained using Designer is represented.

Figure 46 :

 46 Figure 46: Generation of the metasurface design using the MATLAB Algorithm: (a) Variations of 11 () loc X  ρ (Ω) (b) Variations of 12 () loc X  ρ (Ω) (c) Structure of the metasurface imported in ANSYS Desinger.

Figure 47 :

 47 Figure 47: Far field radiation pattern (normalized) results for the linealry polarized

Figure 48 :

 48 Figure 48: Phi component of the far field radiation pattern at the 0   cut plane for different frequencies [27].

Fig. 48

 48 Fig.48 represents the phi component of the radiated field in the frequency range   19, 21 GHz . As for any LW antennas, the beam direction deviates from the design direction 0

Figure 49 :Figure 50 :Figure 51 :Figure 52 :

 49505152 Figure 49: Objective far field radiation pattern: right hand circularly polarized beam radiating at broadside     00 , 0 , 0     .

  In figure 51.(a)-(b), the corresponding reactance tensor components are represented in the local framework. The DXF of the metasurface was then generated and imported in ANSYS Designer (figure 51.(c)) in order to simulate the structure. The final antenna layout is composed of 10732 patches.

Figure 53 :

 53 Figure 53: Objective far field radiation pattern: two beams radiation pattern.

Figure 54 :Figure 55 :

 5455 Figure 54: Aperture field distribution giving a two beams radiation pattern. a) obj E  ; (b) obj E  (c) arg( ) obj E  ; (d) arg( ) obj E  .

Figure 56 :

 56 Figure 56: Far field radiation pattern (normalized) results for the two beams metasurface [29]-[30]: (a) Linear components in the 0   cut-plane. The theta components are represented in solid lines while the phi components are represented in solid lines (b) Circular components in the 135  



  Beam 1: A LP beam along the phi axis radiating in the direction Beam 2: A LP beam along the theta axis radiating in the direction Beam 3: A RHCP beam radiating in the direction   Beam 4: A LHCP beam radiating in the direction   choice is motivated by the fact that the four beams have all the possible polarization states. In addition, the different directions of the beams add up to the complexity of the problem.

Figure 57 :

 57 Figure 57: Objective far field radiation pattern: four beams radiation pattern with different directions and polarizations.

Figure 58 :

 58 Figure 58: Aperture field distribution giving a four beams radiation pattern. a) obj E  ; (b) obj E  (c) arg( ) obj E  ; (d) arg( ) obj E  .

Figure 59 :

 59 Figure 59: Generation of the four beams metasurface using the MATLAB Algorithm: (a) Variations of 11 () loc X  ρ (Ω) (b) Variations of 12 () loc X  ρ (Ω) (c) Structure of the metasurface imported in ANSYS Desinger.

Figure 60 presents the

  Figure 60 presents the results of the far field radiation pattern provided by Designer. The figure 60.(a) presents the linear components in the 0   cut-plane. The theta components are represented in dashed lines while the phi components are represented in solid lines. Figure 60.(b) gives the circular components in the 90   cut-plane. The RHCP components are represented in solid lines while the LHCP components are represented in dashed lines. The 3D representation of the total field is given in figure 60.(c).

Figure 60 :

 60 Figure 60: Far field radiation pattern (normalized) results for the four beam metasurface [27]-[28]: (a) Linear components in the 0   cut-plane. The theta components are represented in dashed lines while the phi components are represented in solid lines (b) Circular components in the 90   cut-plane. The RHCP components are repre- sented in solid lines while the LHCP components are represented in dashed lines (c) 3D representation of the radiation pattern.

25 Figure 61 :Figure 62 :

 256162 Figure 61: Variations of the multi-beam radiation pattern with respect to frequency for the 0   cut-plane [27]. (a) Theta components. (b) Phi components. The considered frequencies are: 19.5 GHz (in red), 20GHz (in black) and 20.5GHz (in blue). The radiation pattern was normalized with respect to the maximum beam amplitude.

  Figure 62 gives the frequency behavior of the two circularly polarized beams presents in the 90   cut-plane. The radiation patterns were simulated for the frequencies 19.5 GHz (in red), 20GHz (in black) and 20.5 GHz (in blue). It can be seen from the figure that the radiation performances (side lobe levels and cross polarization levels) are stable in the considered frequency band.

f 1 JFor a flat top of angle 20 f

 120 defines the angular region of interest. To generate the radiation pattern of the form (59) with RHCP, we proposed a new aperture distribution consisting of multiplying the bi-dimensional inverse Fourier transform of characteristic function of a circular disc [40] by the term   ˆî  xy . The final objective aperture field distribution is given by [27]: is the first order Bessel function of the first kind.  , the metasurface design algorithm produces the reac- tance component variation shown in figure 63.(a)-(b) (local framework). The corresponding geometry is represented in figure 63.(c). The far field radiation pattern was simulated using Designer and the results are represented in figure 64. Figures 64.(a) and (b) show the far field radiation pattern for the RHCP and LHCP components in the 0   cut-plane, where figure 64.(c) presents the 3D radiation pattern. It can be seen from the figure 64 that, for angles f   , a RHCP radiation pattern is obtained with a relatively constant amplitude of oscillations 1dB

Figure 63 :Figure 64 :

 6364 Figure 63: Generation of the two beam metasurface using the MATLAB Algorithm: (a) Variations of 11 () loc X  ρ (Ω) (b) Variations of 12 () loc X  ρ (Ω) (c) Structure of the metasurface imported in ANSYS Desinger.

4. 3 . 5 . 1 Figure 65 :

 35165 Figure 65: Objective near field radiation pattern at the height 5 z   . (a) Ex compo- nent. (b) Ey component. (c) Total field Et.

Figure 66 :

 66 Figure 66: Aperture field distribution giving the four beams near field focusing [41] -[42]. a) obj x E ; (b) obj y E

EFigure 67 :

 67 Figure 67: Generation of the four-beams near field focusing metasurface using the MATLAB Algorithm: (a) Variations of 11 () loc X  ρ (Ω) (b) Variations of 12 () loc X  ρ (Ω) (c) Structure of the metasurface imported in ANSYS Desinger.

Figure 68 :

 68 Figure 68: Simulated near field horizontal cut-plane in ANSYS Designer at the height 5 z   . (a) Ex component. (b) Ey component. (c) Total field Et.

Figure 69 :

 69 Figure 69: Simulated near field vertical cut-plane in ANSYS Designer. (a) Ex component in the XZ plane. (b) Ey component in the YZ plane.

Figure 70 :

 70 Figure 70: Simulated near field horizontal cut-plane in ANSYS Designer at the height 5 z   with feeder effect cancellation. (a) Ex component. (b) Ey component. (c) Total field Et.

Figure 71 :

 71 Figure 71: Simulated near field vertical cut-plane in ANSYS Designer with feeder effect cancellation. (a) Ex component in the XZ plane. (b) Ey component in the YZ plane.

  large variety of far field radiation patterns were achieved, ranging from multi-beam antennas to a flat top circularly polarized antenna. The simulations were in close agreement with the theoretical results. In light of the variety and complexity of the achieved radiation patterns, we concluded that arbitrary aperture field

Figure 73 :

 73 Figure 73: Manufactured prototype for mechanical validation. (a) Whole structure. (b) Zoom on the metasurface.

Figure 74 :Figure 75 :

 7475 Figure 74: Microscopic view of the unit cell.

Figure 76 :

 76 Figure 76: Variations of the scattering parameter S 11 (in dB) of the metasurface prototype with respect to frequency (in GHz). The solid line gives the measurement results while the dashed line shows the simulation results obtained using HFSS.

Figure 77 :

 77 Figure 77: Measurement of the far field radiation pattern in the GeePs Laboratory. (a) Anechoic chamber. (b) Standard Gain Horn Antenna (reference). (c) Measured antenna.

  Figure 77 shows the measurement setup for the far field radiation pattern. It consists of an anechoic chamber containing a reference antenna (figure 77.(b)) and the Antenna Under Test (AUT, figure 77.(c)) placed at a distance of 7m from each other. Before proceeding with the measurement, the AUT is aligned with the reference antenna using laser alignment as shown in figure 78.

Figure 78 :

 78 Figure 78: Laser alignment of the measurement antenna with the reference antenna.



  Substrate : Rogers TMM6 of thickness h=2.54mm and relative permittivity ϵ r =6. The dissipation factor of this substrate is tan 0.0023   .  Working frequency : 10GHz.  Dimension : circular metasurface of diameter 38cm.  Unit cell dimension : ' 2.5mm d  . Objective radiation pattern : broadside RHCP.

Figure 80 :

 80 Figure 80: Manufactured broadside RHCP metasurface (printed on a substrate ROGERS TMM6).

Figure 81 :

 81 Figure 81: Variations of the scattering parameter S 11 (in dB) of the manufactured broadside RHCP metasurface with respect to frequency (in GHz). The solid line gives the measurement results while the dashed line shows the simulation results obtained using HFSS.

3 Figure 82 :

 382 Figure 82: Variations of the RHCP component (normalized) in the 0  cut-plane with respect to the frequency and the incident angle  .

Figure 83 :

 83 Figure 83: Variations of the gain at broadside in dBi with respect to the frequency.

Figure 84 :

 84 Figure 84: Measurement (in black) and simulation (in red) of the circular components of the far field radiation pattern of the broadside RHCP metasurface for the 0   cut-plane. Solid lines represent RHCP components and dashed lines represent LHCP components. The working frequency for the simulations and the measurements is 9.5GHz.

Figure 85 :

 85 Figure 85: Variations of the measured radiation pattern with respect to frequency at the 0   cut-plane for the broadside RHCP antenna. (a) RHCP components. (b) LHCP components. The considered frequencies are: 19.5 GHz (in red), 20GHz (in black) and 20.5GHz (in blue). The radiation pattern was normalized with respect to the maximum beam amplitude.

5 . 4 .

 54 Measurement of metasurface antennas 123

Figure 86 :

 86 Figure 86: Variations of the axial ratio at broadside in dB with respect to the frequency.

Frequency

Figure 87 : 5 . 4 . 4 . 5 . 4 .

 8754454 Figure 87: Manufactured four-beams metasurface (printed on a substrate ROGERS TMM6).5.4.4 Near field multi-beam focusing metasurface antenna

Figure 88 :

 88 Figure 88: Manufactured near-field multi-beam focusing antenna [41] -[42].

Figure 89 :Figure 90 :

 8990 Figure 89: Variations of the scattering parameter S 11 (in dB) of the near-field multibeam focusing antenna [41]-[42]. The solid line gives the measurement results while the dashed line shows the simulation results obtained using HFSS.

  Finally, a near-field multi-beam focusing metasurface was manufactured in collaboration the IETR laboratory of the University of Rennes1. The metasurface of working frequency 20GHz was measured at the IETR. A horizontal cut of the Cartesian components of the electric field for a height of 5 z   was presented. The measured near field comported the desired four focused beams with the expected polarization. The side-lobe levels of the manufactured metasurface are equal to -12dB.

  have two solutions with the same sign as

								X . That means that there
								ee
	are either no SW or two SW propagating on the metasurface with
	different wavenumbers ,1 sw t k and ,2 . sw t k
	 Case 3: 	mm ee X X		 00 4 2 2 2 me X    	2	ee mm X X	. After calculations, this
	condition leads to	0 X  and me	mm ee XX	2  . In this case, the term  0 
	either has no positive solution or a single positive solution corresponding
	to a TE and a TM surface wave modes, propagating independently with
	the same wavenumber sw k .
					t
	 Case 4: 	mm ee X X		 00 4 2 2 2 me X    	2	ee mm X X	. In this case,  is com-

plex. That does not correspond to any physical surface wave.

  that only the first column of loc

	yields:
	11 loc	11 loc	1	11 loc		loc obj	,1	* loc ,1 obj	
	21 loc	21 loc	1	21 loc		loc obj	,2	* loc ,1 obj	

s Z affects the local electric field. The second approach described by equation

[START_REF] Bilow | Guided waves on a planar tensor impedance surface[END_REF]

reduces to the use of the holographic principle on the first collumn of the local impedance tensor. This

Table 4 .

 4 1: Adaptation steps of the feeder.

in this chapter.

2 D  , where D is the largest dimen- sion of the antenna and  is the wavelength.

Chapter 6. Conclusions and Prospective

Summary of the contribution

In this thesis, a method for the generation of arbitrary aperture field distribution using modulated tensorial metasurfaces is presented. Chapter 2 gives the formulation of the method as well as some theoretical background about scalar and tensorial metasurfaces. When a surface wave propagates through a sinusoidally modulated metasurface, leaky waves that propagates away from the surface are generated. The phase of these waves can be controlled by acting on the periodicity of the modulation using a holographic principle. On the other hand, the amplitude is controlled using modulation indices and average impedances that vary depending on the position on the metasurface. A new local framework formulation is introduced in order to simplify the different calculations. The idea is to control these leaky waves in order to generate the desired aperture distribution. By modulating the different impedance elements separately, the aperture field components can be controlled independently.

In chapter 3, the different aspects of the metasurface implementation are presented.

The previous formulation is written in the case of a cylindrical surface wave excitation. The metasurface is implemented using sub-wavelength patches printed over a grounded dielectric substrate. The chapter also describes the unit cell design as well as the impedance database generation. Finally, the different steps of the metasurface design algorithm are described, from the specification of the objective aperture field to the generation of the antenna DXF file. In chapter 4, numerical results obtained using simulation software are presented. At first, the effects of the parameters of the feeder on the adaptation of the antenna are studied. Each parameter was investigated separately leading to the development of a step-by-step adaptation algorithm. After that, several metasurface designs for far field radiation pattern control were simulated: broadside RHCP metasurface, tilted beam linearly polarized metasurface, multi-beam metasurface and flat top RHCP metasurface. For this wide range of examples, the simulated radiation pattern was in close agreement with the field radiated by the ideal aperture field distribution and by the estimated magnetic field currents above the metasurface. Finally, a near-field multi-130 beam focusing design was generated in collaboration with the IETR laboratory of the University of Rennes1. The theoretical aperture field distribution was calculated by our partners and was then injected in our algorithm in order to generate the metasurface structure. The simulated near field corresponded to the objective radiation but was significantly affected by the radiation of the feeder.

Chapter 5 was dedicated to the presentation of the manufacturing process as well as some initial experimental results. A metasurface prototype as well as two metasurfaces structures (broadside RHCP and four-beams) were manufactured by the company INOVEOS that is based in Brive-la-Gaillarde in France. The prototype was manufactured on a FR4 substrate for mechanical validation of the manufacturing process. On the other hand, the two other metasurfaces were manufactured on a ROGERS TMM6 substrate that is more suitable for space applications. The prototype and the broadside RHCP metasurfaces were measured in the GeePs laboratory. The measured radiation pattern was in good agreement with the simulation results. However, a frequency shift was present in the TMM6 metasurface. In the final part of the chapter, a near field multi-beam focusing metasurface was manufactured and measured in the IETR laboratory. The measured near field exhibits the expected four beams with the desired polarizations.

Future work

The obtained simulation and the experimental results are very promising and prove that a good control of the aperture field distribution is achieved with the proposed procedure. However, several aspects need to be considered in order to improve the performance of these metasurfaces:  The feeder affects the field radiated by the metasurface, especially in the case of near field application. As a result, it is important to optimize the feeding structure in order to maximize the energy that is converted into surface wave (and therefore minimize the energy that is radiated by the feeder).  The considered aperture field distributions were used as a first validation of the method. They have to be optimized in order to improve the antenna performances such as side lobe levels and directivity. Appendix 131  Undesired edges reflections should be reduced by increasing the antenna dimension and maximizing the surface wave to leaky wave conversion.  Metasurface antennas are narrow band. It is then important to improve their bandwidth and find a compromise between the bandwidth and the efficiency.

A possible solution is to use multi-layered substrate.  The reconfigurability of the metasurface should also be investigated.

 Commercial simulation software are pretty limited when it comes to analyzing metasurfaces. It can be very useful to develop an in-house code for the analysis of these structures as it allows the separation of the different radiation mechanisms such as: leaky waves, surface waves, radiation of the feeder, radiation at the edge of the metasurface…

List of publications

The work in this thesis has been published in the following peer-reviewed journals and conference proceedings.

Peer