Contribution to the Study and Design of Antennas for the Generation of Radio Waves Bearing Orbital Angular Momentum

for directing this thesis. His scientific intuition, knowledge, guidance and patience have contributed significantly to this work and inspired me a lot. I would also like to express my deep gratitude to Dr. Christian Brousseau for his cosupervision and the great support he has given me throughout this work, and to Prof. Olivier Emile for valuable discussions and suggestions. Their contributions make this thesis possible.

In addition, I would like to thank Prof. Ala Sharaiha for supporting me to attend the EuCAP conference and for helping present the paper at the LAPC conference.

Many thanks to Laurent Le Coq, Xavier Morvan, Jean-Christophe Le Cun, Eric Moricet, Christophe Guiton and David Levalois for technical advices and willingness to help me realize the antennas and perform the measurements. I also wish to express my gratitude to all the staff at IETR for passing through a beautiful and unforgettable time together and for the help they have given me during these years.

Especially I thank the PhD student Abdullah Haskou for his enthusiasm and the secretary Noëlle Le Ber for her kindness. I would like to thank my parents for all their support during my study. I thank my wife Wenlin Kuai for her encouragement and company and for giving birth to our baby which is the biggest gift for me till now in my life.

I would also like to acknowledge the China Scholarship Council (CSC) for the scholarship support during my PhD study.

I Résumé en Français

I. Introduction et objectifs

Le fait que les ondes électromagnétiques (EM) peuvent porter du moment angulaire est connu depuis assez longtemps, mais c'est seulement en 1992 qu'il a été démontré théoriquement par Allen et al. [1] que certains types de faisceaux pouvaient porter un moment angulaire orbital (OAM en anglais) de lħ par photon, où l est un nombre entier appelé "charge topologique" ou l'ordre du mode OAM. Alors que la phase d'une onde plane habituelle est constante sur le front d'onde, la phase  des ondes OAM possède une variation linéaire le long de la coordonnée angulaire  (angle de roulis):  = l. Si l'on décrit le champ dans le front d'onde en coordonnées polaires, il varie comme e -jl .fl(r).e -jkz , où k est le vecteur d'onde, z, l'axe de propagation et fl(r), la variation du champ le long de la coordonnée radiale.

Les propriétés de l'OAM ont suscité beaucoup d'intérêt dans de nombreux domaines. Les premières études concernant l'interaction de l'OAM avec la matière ont cité comme exemples d'applications potentielles, les pinces optiques [2], le lecteur optique de micromachines [3], le piégeage et guidage des atomes [4], etc. Dans le domaine radiofréquence, l'OAM a été proposé pour améliorer l'efficacité spectrale [5][START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Wang | Terabit free-space data transmission employing orbital angular momentum multiplexing[END_REF][START_REF] Yan | Highcapacity millimetre-wave communications with orbital angular momentum multiplexing[END_REF][START_REF] Hui | Multiplexed millimeter wave communication with dual orbital angular momentum (OAM) mode antennas[END_REF] dans les communications sans fil, en créant plusieurs sous-canaux de propagation correspondant aux différents modes l du moment angulaire orbital.

Il existe également des applications possibles dans le domaine de l'identification d'objet [START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio domain[END_REF] ou celui des radars [START_REF] Edfors | Is orbital angular momentum (OAM) based radio communication an unexploited area?[END_REF][START_REF] Tamagone | Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Tamburini | Reply to Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF].

Depuis 1992, plusieurs méthodes ont été proposées pour générer des ondes portant de l'OAM. Dans le champ optique, il existe principalement trois techniques: la lame de phase spirale [START_REF] Liu | Detecting orbital angular momentum through division-of-amplitude interference with a circular plasmonic lens[END_REF][START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF][START_REF] Niemiec | Characterization of an OAM flat plate antenna in the millimeter frequency band[END_REF], les modes Laguerre-Gauss [1,[START_REF] Bai | Generation of orbital angular momentum (OAM) radio beams with phased patch array[END_REF][START_REF] Bai | Experimental circular phased array for generating OAM radio beams[END_REF] et le réseau diffractant [START_REF] Padgett | An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes[END_REF]. Dans le domaine de la radio, deux grandes familles d'antennes ont été proposées pour générer des ondes OAM. La première est basée sur la transformation de l'onde plane en une onde OAM. Pour cela on peut citer: la lame de phase spirale [START_REF] Turnbull | The generation of free-space Laguerre-Gaussian modes at millimeter-wave frequencies by use of a spiral phaseplate[END_REF][START_REF] Schemmel | A modular spiral phase plate design for orbital angular momentum generation at millimetre wavelengths[END_REF], la lame de phase plate [START_REF] Sueda | Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses[END_REF][START_REF] Khonina | The phase rotor filter[END_REF], le réseau diffractant [START_REF] Harris | Optical helices and spiral interference fringes[END_REF] et le réflecteur en spirale [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Gabor | A new microscopic principle[END_REF]. La deuxième famille est basée sur l'emploi de réseaux circulaires de dipôle à phase contrôlée [5,[START_REF] Siegman | Lasers[END_REF].

La deuxième méthode est en réalité une étude purement théorique qui n'a pas été suivie de réalisation et d'expérimentation à cause de complexités d'ordre pratique. L'objectif de cette thèse est donc de concevoir et de réaliser de nouveaux types d'antenne dans la gamme des Résumé en Français II radiofréquences pour répondre aux besoins actuels et futurs des applications nécessitant des ondes électromagnétiques ayant un moment angulaire orbital.

II. Génération d'OAM avec déphaseur circulaire et réseau d'antennes patch

Nous présentons d'abord un réseau circulaire à commande de phase à l'aide d'un déphaseur original pour générer le mode l = 1 du moment angulaire orbital. Ce générateur d'ondes OAM utilise quatre antennes « patch » alimentées par une ligne de transmission circulaire originale fonctionnant à la fréquence 2.5 GHz.

Conception d'antenne

Pour la création d'un mode OAM l = 1, nous avons utilisé N = 4 antennes avec un écart de phase entre élément successifs de 90° (2πl/N = π/2 radians). A cet effet, nous avons conçu un déphaseur circulaire qui non seulement fournit l'écart de phase de 90° mais aussi assure l'adaptation d'impédance à chaque discontinuité en utilisant des transformateur quart-d'onde (Fig. 1a). Le déphaseur fonctionne à 2.5 GHz et dispose de 5 ports: le port1 est l'entrée et est adaptée à 50 Ω, tandis que les autres ports sont adaptés à 100Ω et utilisés comme sorties pour alimenter les antennes patch situées sur un cercle. Le logiciel HFSS a été utilisé pour optimiser la longueur et la largeur des arcs micro-rubans et permettre l'adaptation de l'antenne, ainsi que l'obtention d'une amplitude uniforme et un écart de phase de 90° au niveau des ports de sortie.

Résultats de simulation et de mesure

Fig. 3: Caractéristiques simulées du déphaseur: les déphasages (à gauche) et les coefficients de transmission (à droite) à 2.5 GHz. Le déphasage des S21, S31, S41, S51 est de -168 ° -261 ° -350 ° -439°, respectivement; et leur amplitude, de -6.4 dB, -6,6 dB, -6,9 dB, -7 dB.

Les résultats simulés du déphaseur sont présentés à la Fig. 3. Nous pouvons observer qu'à 2.5 GHz (ligne verticale noire), entre deux ports voisins, l'erreur de phase est de 4° et l'erreur d'amplitude de 0.6 dB. Ainsi, le déphaseur a un très bon accord avec les spécifications requises.

La Fig. 4 montre la cartographie 2D de l'amplitude et la phase de la composante Ex du champ électrique dans un plan perpendiculaire parallèle à la surface de l'antenne et situé à 60

IV mm au-dessus de celle-ci. On peut voir qu'au centre, l'amplitude est beaucoup plus faible qu'en périphérie, ce qui montre la présence du vortex caractéristique des ondes à OAM. La phase tourne autour du centre avec un déphasage de 2π en un tour. Ceci démontre en effet que l'amplitude et la phase correspondent à une onde ayant un OAM de l = 1. La Fig. 5 présente les diagrammes de rayonnement et de phase en 3D de l'antenne OAM pour la polarisation principale Ex (voir Fig. 1b). On retrouve sur le diagramme 3D de l'amplitude, le trou central caractéristique des ondes OAM. La forme en spirale du diagramme de phase, tracée en coordonnées polaires, correspondant à la variation linéaire de la phase en fonction de l'angle de roulis (, sur la Fig. 5).

III. Conception d'une antenne directive OAM en utilisant la cavité de Fabry-Perot

L'antenne précédente est simple, compact et facile à réaliser, mais sa directivité est relativement faible. Dans ce chapitre, nous allons l'intégrons dans une cavité Fabry-Perot (FP) pour améliorer sa directivité.

Structure de base de la cavité Fabry-Perot

La cavité FP est très sélective en fréquence et a été utilisée à l'origine, comme un filtre de fréquence en optique. Dans les applications antennaires, elle est souvent utilisée en tant que filtre spatial pour améliorer les performances de l'antenne telles que la directivité [START_REF] Cheng | Generation of electromagnetic waves with arbitrary Orbital Angular Momentum modes[END_REF][START_REF] Mahmouli | 4-Gbps uncompressed video transmission over a 60 GHz Orbital Angular Momentum wireless channel[END_REF][START_REF] Mahmouli | Orbital angular momentum generation in a 60GHz wireless radio channel[END_REF].

où c est la vitesse de la lumière à l'intérieur de la cavité (espace libre), f0, la fréquence de travail, n, un nombre entier correspondant au numéro de mode de cavité, , la phase du coefficient de réflexion du PRS et 0, l'angle de dépointage du faisceau sortant. Dans notre étude, n est choisi à 0 pour utiliser le premier mode et donc l'épaisseur D minimale.

Conception d'antenne directive à base de cavité FP

Nous avons étudiée 2 structures différentes pour la surface réfléchissante PRS : une avec des lignes micro-rubans métallisés déposés sur un substrat FR4 et l'autre composée de tubes métalliques. Ici nous allons présenter la deuxième qui a été retenue pour la réalisation. La géométrie de cette antenne FP OAM est représentée à la Fig. 9. Étant donné que les ondes OAM générées par le réseau circulaire sont plus directives lorsque le rayon du réseau devient plus grand, nous choisissons un rayon de réseau de 110 mm à la fréquence de 2.5 GHz. La directivité de l'antenne dépend de la réflectivité de la surface PRS (c'est-à-dire du diamètre et de la périodicité des tubes), de l'épaisseur D de la cavité et de la superficie du PRS. Pour obtenir une meilleure directivité de l'antenne OAM, ces paramètres ont été optimisés à l'aide du logiciel IX qui est caractéristique des ondes OAM. En outre, avec l'utilisation de la cavité FP, la directivité de l'antenne est augmentée et les lobes secondaires sont sensiblement atténués.

Pour donner une meilleure vue de l'influence de la cavité FP sur la directivité de l'antenne OAM, nous faisons une comparaison des diagrammes de rayonnement E-plan à la Fig. 11. On peut voir qu'avec l'utilisation de la cavité FP, la directivité de l'antenne passe de 7 à 16.6 dB.

La directivité maximale est obtenue pour un angle de 10 °.

IV. Génération d'onde guidée à OAM dans guide circulaire

La lame de phase spiral (SPP -Spiral Phase Plate) a été proposée pour transformer une onde plane en une onde transportant du moment angulaire orbital [START_REF] Liu | Detecting orbital angular momentum through division-of-amplitude interference with a circular plasmonic lens[END_REF][START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF][START_REF] Niemiec | Characterization of an OAM flat plate antenna in the millimeter frequency band[END_REF]. Elle a été utilisée en optique et en radiofréquence pour générer des ondes à OAM dans l'espace libre, mais très rarement dans un guide d'onde, sauf dans [START_REF] Rodes | Nouveaux concepts d'antenne à base de matériaux BIE métalliques[END_REF]. Nous présentons dans ce chapitre, deux antennes

V. Génération d'OAM dans un guide d'onde circulaire en combinant des modes guidés

Il a été démontré en optique que les modes de Laguerre-Gauss (LG) peuvent être produits en combinant deux modes Hermite-Gauss (HG). Suivant cette idée, un guide d'onde rectangulaire en silicium a été utilisé pour générer des faisceaux OAM optiques en combinant les modes E x 21 et E x 12 [START_REF] Vu | Antennes à bande interdite électromagnétique directive: contribution à l'élargissement de la bande passante et à l'évaluation de l'impédance[END_REF]. De la même manière, en combinant certains modes propres d'un guide d'onde métallique, on peut créer des ondes OAM dans les bandes de fréquences radio. Dans cette partie, nous proposons deux méthodes pour obtenir le mode OAM l = 1: i) la combinaison de deux types de modes TE21; ii) la combinaison des modes TM01 et TE21.

Combinaison de deux types de modes TE21

La Fig. 22 montre la distribution du champ électrique, ainsi que celle de l'amplitude et de la phase de la composante Ex du premier type de mode TE21 et la Fig. 23, celles du deuxième type de mode TE21. On peut voir que les diagrammes du champ E des deux types de mode TE21 sont assez similaires. La différence est une rotation de π/2 entre les deux diagrammes aussi bien en champ qu'en amplitude et en phase. Il en va de même pour la composante Ey.

Combinaison de modes TM01 et TE21

De la même manière que précédemment, on peut créer une onde OAM en combinant le mode TM01 et le premier type de mode TE21 avec une différence de phase de π/2 en entre les deux.

Pour

VI. Conclusions et perspectives

General Introduction

Context and objectives of the study

It is well known from Maxwell's theory that electromagnetic (EM) radiation carries both linear momentum (energy) and angular momentum. The latter has two parts: Spin Angular Momentum (SAM) which corresponds to the polarization of an EM wave and Orbital Angular Momentum (OAM) which is associated with the spatial distribution of an EM wave.

The SAM of an EM wave carries ±ħ per photon and it has been recognized for a long time. However, the OAM has not attracted the attention of the researchers until 1992 when Allen et al. [1] recognized that some types of beams possess an orbital angular momentum of lħ per photon, where l is an integer called the "topological charge" or the order of the OAM mode. Whereas the phase of a usual plane wave is constant on the wavefront, the phase  of OAM waves undergoes a linear variation along the angular coordinate  (roll angle):  = l.

The field hence varies as e -jl .fl(r).e -jkz , where k is the wave vector, z is the propagation axis and fl(r) represents the field variation along the radial coordinate.

The special characteristics of OAM have attracted a great deal of interests from many domains. The early studies concerning the interaction of OAM with matter evoked the applications of optical tweezers [2], optical drive of micro-machines [3], atoms trapping and guiding [4]. In radio domain, OAM has been firstly proposed to improve spectral efficiency [5][START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Wang | Terabit free-space data transmission employing orbital angular momentum multiplexing[END_REF][START_REF] Yan | Highcapacity millimetre-wave communications with orbital angular momentum multiplexing[END_REF][START_REF] Hui | Multiplexed millimeter wave communication with dual orbital angular momentum (OAM) mode antennas[END_REF] in radio communications, by creating multiple sub-channels of propagation corresponding to the twisting degree of the EM wave. Despite this point is still subject to intense debates [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio domain[END_REF][START_REF] Edfors | Is orbital angular momentum (OAM) based radio communication an unexploited area?[END_REF][START_REF] Tamagone | Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Tamburini | Reply to Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF], there are also possible applications of OAM for object identification [START_REF] Uribe-Patarroyo | Object identification using correlated orbital angular momentum states[END_REF] and radars [START_REF] Liu | Orbital-Angular-Momentumbased electromagnetic vortex imaging[END_REF][START_REF] Lin | Improved OAM-based radar targets detection using uniform concentric circular arrays[END_REF][START_REF] Lin | Super-resolution orbital angular momentum based radar targets detection[END_REF].

The OAM beams can be generated easily in optical domain, especially when the programmable spatial light modulator (SLM) is available [START_REF] Coomber | Optically addressed spatial light modulators for replaying computer-generated holograms[END_REF]. The existing optical devices can also provide mature schemes to manipulate the OAM beams such as combining, splitting, collimation, concentrating, and detecting [START_REF] Cai | Integrated Compact Optical Vortex Beam Emitters[END_REF][START_REF] Genevet | Holographic detection of the orbital angular momentum of light with plasmonic photodiodes[END_REF][START_REF] Rui | Demonstration of beam steering via dipole-coupled plasmonic spiral antenna[END_REF][START_REF] Liu | Detecting orbital angular momentum through division-of-amplitude interference with a circular plasmonic lens[END_REF]. Hence, the optical OAM beams can be deftly applied to many fields. However, when it comes to the radio domain, as the wavelength is much longer than the optics, it is difficult to manipulate the radio OAM beams, such as beam combining and splitting, so that the coaxially transmitting cannot be easily ensured. Therefore, among the applications of OAM at radio frequencies, the generation of OAM is very important and sometimes decides the performance of the whole system.

Up to now, in the radio domain, two main families of antennas have been proposed to generate OAM waves. In the first one, the OAM wave is generated by a transformation of a plane wave. This can be achieved by using the spiral phase plate [START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF], flat drilled phase plate [START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF][START_REF] Niemiec | Characterization of an OAM flat plate antenna in the millimeter frequency band[END_REF][START_REF] Bennis | Flat plate for OAM generation in the millimeter band[END_REF] and spiral reflectors [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Tamburini | Experimental verification of photon angular momentum and vorticity with radio techniques[END_REF]. In the second family, the OAM wave is directly generated using some circular phased arrays [5,[START_REF] Tennant | Generation of OAM radio waves using circular time-switched array antenna[END_REF][START_REF] Bai | Generation of orbital angular momentum (OAM) radio beams with phased patch array[END_REF][START_REF] Bai | Experimental circular phased array for generating OAM radio beams[END_REF][START_REF] Bai | Generation of OAM radio waves with three polarizations using circular horn antenna array[END_REF].

The main objective of this thesis is to design several new antennas for the generation of radio OAM waves, which is necessary for any existing and future application.

Outline of the thesis

This thesis consists of 4 major chapters:

In the first chapter we introduce the properties of EM waves bearing OAM and review the typical methods for generating these waves in both optical and radio domains.

In the second chapter an OAM antenna using 4 patches and an original circular phase shifter-power divider is developed. This antenna works at 2.5 GHz and generates an OAM wave with l = 1. This chapter consists of two major parts. The first part is the parametric study of an ideal phased patch array, which is necessary for the design of a practical OAM antenna; the second part presents the design procedure, the simulation data and the measurements of this novel OAM antenna.

In the third chapter we present a method to enhance the directivity of the previously developed OAM antenna by using the Fabry-Perot (FP) cavity. A FP cavity is generally composed of a ground plane, an air cavity and a partially reflecting surface (PRS). It is highly frequency selective and is often used to improve the antenna directivity in antenna applications.

We embed the previous OAM antenna inside the FP cavity with two different PRS. The first one is made of periodic metallic strips printed on a FR4 substrate, and the second, made of periodic metallic tubes. Each FP cavity is optimized associated with the phased patch array to obtain a maximum antenna directivity. The design procedure and the simulation data of the two FP OAM antennas are given, as well as the measurements of the second antenna.

In the fourth and last chapter, we present some horn antennas and a waveguide antenna for the generation of radio OAM waves. These structures are all based on a circular metallic waveguide. Actually, we have developed two major methods to obtain the OAM mode: i) transformation of the guided modes of a circular waveguide by using the spiral phase plate; ii) combination of the guided modes. It is well known from Maxwell's theory that electromagnetic (EM) radiation carries both energy and momentum. The momentum may have two parts of contributions: linear momentum and angular momentum [1]; the latter can be decomposed into two distinct components: spin and orbital. The spin angular momentum (SAM) corresponds to the polarization of an EM wave and the orbital angular momentum (OAM) is associated with the spatial distribution of an EM wave.

Since the concept of momentum is not specific to the EM waves, it is also related to the classical mechanics, we will firstly give a brief presentation of this concept in this area before turning to the EM waves. Thereafter, a vector is represented by a bold letter (e.g. E represents the electric field vector).

I.1.1 Momentum in classical mechanics

The second law of Newton which was published in 1687 states that for a body with a constant mass m, the acceleration suffered by the body is proportional to the net force applied and inversely proportional to its mass m. A more general form is to say that the force is equal to the change of momentum per unit time:

𝑭 = 𝑑𝒑 𝑑𝑡 (I-1)
where F is the force applied on the object, and 𝒑 = 𝑚𝒗 is the momentum equal to the product of its mass m and its speed v.

The angular momentum of a point M is the moment of the momentum p with respect to an original point O (𝑂𝑀 ⃗⃗⃗⃗⃗⃗ × 𝒑). Therefore it plays a role similar to the linear momentum, but in the case of rotation. We will find again these two conceptions concerning the EM waves.

I.1.2 Momentum of electromagnetic waves

I.1.2.1 Linear momentum: Poynting vector

The Poynting vector is a vector whose direction indicates the direction of propagation of an EM wave in the absence of angular momentum, and whose intensity is equal to the wave power density. It is expressed in W/m² and is defined by:

𝑺 = 𝑬 × 𝑯 (I-2)
where E and H represent respectively the electric and magnetic fields.

The momentum p of a photon is then expressed as:

𝒑 = ħ𝑘 (I-3)
where k represents the wave number and ħ the reduced Plank constant.

I.1.2.2 Angular momentum: SAM and OAM

In 1909, Poynting made the hypothesis that a circularly polarized EM wave carries angular momentum [START_REF] Poynting | The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light[END_REF]. His reasoning is based on an analogy between a mechanical modela uniformly revolving shaft of circular section, and a beam of circularly polarized light. He concluded that the angular momentum delivered to unit area per second is:

𝑃𝜆 2𝜋

where P is the radiation pressure exerted on a surface by a beam of wavelength λ. In current notation, this amounts to writing that the angular momentum is equal to σħ, with 𝜎 = ±1

corresponding respectively to the left and right circular polarization.

This component of the angular momentum is called the spin angular momentum (SAM), because it is associated with the spin of the photon (see Figure I-1a) which can be visualized by rotating the photon itself around its axis. Consequently, it can exist two values of σ, of which each corresponds to one direction of rotation [START_REF] Tannoudji | Photons et atomes -Introduction à l'électrodynamique quantique[END_REF]. Since it is not necessary for the photon to have an external reference, the SAM is also called the intrinsic angular momentum. There exists a second type of angular momentum which is defined in relation to an external reference: it is therefore an extrinsic angular momentum. It can be represented by an orbit of the photon around a point (see Figure I-1b) and that is why it is called the "orbital angular momentum (OAM)". We find the equivalence between the sign ("+" or "-") and the direction of rotation, but this time the value is related to the speed of rotation, i.e. the number of jump of 2π per period. Therefore, it can exist infinitely in theory.

In 1992, a team from the University of Leiden [1] studied again on this subject, especially on the transformation of orbital angular momentum through a Gaussian beam. In particular, they were interested in the Laguerre-Gaussian (LG) modes which are defined by the polynomials of the same name and possess a varying term of exp(-il) in their expression. Here l is an integer number called the "topological charge" or the order of the OAM mode and  is the roll angle.

In this case, l is directly involved in the expression of the OAM carried by the beam: more exactly, the latter would be of lħ per photon to match with the σħ of the SAM and the ħk of the Several years later, this phenomenon is observed experimentally by the measurements of the distributions of the intensity and phase [START_REF] Padgett | An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes[END_REF]. An EM wave carrying OAM can be decomposed into (like any EM wave, within the scope of paraxial approximation) a sum of LG beams, noted LGpl and given by [START_REF] Allen | The orbital angular momentum of light[END_REF][START_REF] Siegman | Lasers[END_REF][START_REF] Kolobov | Quantum Imaging[END_REF]:

𝐿𝐺 𝑝𝑙 = √ 2𝑝! 𝜋(𝑝 + |𝑙|)! 1 𝑤(𝑧) [ 𝑟√2 𝑤(𝑧) ] |𝑙| 𝑒𝑥𝑝 [ -𝑟 2 𝑤 2 (𝑧) ] 𝐿 𝑝 |𝑙| [ 2𝑟 2 𝑤 2 (𝑧) ] 𝑒𝑥𝑝(𝑖𝑙𝜑) 𝑒𝑥𝑝 [ 𝑖𝑘 0 𝑟 2 𝑧 2(𝑧 2 +𝑧 𝑅 2 ) ] 𝑒𝑥𝑝 [-𝑖(2𝑝 + |𝑙| + 1) tan -1 ( 𝑧 𝑧 𝑅 )] (I-4)
where the 1/e radius of the Gaussian term is given by 𝑤(𝑧) = 𝑤(0)[(𝑧 2 + 𝑧 𝑅 2 )/𝑧 𝑅 2 ] 1/2 with w(0) being the beam waist, zR the Rayleigh range, and (2𝑝 + |𝑙| + 1) tan -1 (𝑧/𝑧 𝑅 ) the Gouy phase. 𝐿 𝑝 |𝑙| (𝑥) is an associated Laguerre polynomial, obtained from the more familiar Laguerre polynomials by:

𝐿 𝑝 |𝑙| (𝑥) = (-1) |𝑙| 𝑑 |𝑙| 𝑑𝑥 |𝑙| 𝐿 (𝑝+|𝑙|) (𝑥) (I-5)
l is the order of the OAM mode and p is the number of radial nodes in the intensity distribution.

For the order l = 0, we find a Gaussian distribution. For any other value of l, the intensity is distributed in the form of a ring (donut) centered on the beam axis. As an example, the intensity However, since OAM is a physical phenomenon throughout the EM spectrum, each frequency band can also take advantage of the benefits of other ones. In the following parts, we will introduce the methods which are often used at either optics or radio frequency bands.

Because the OAM was firstly studied in the optics field, we will at first introduce the techniques used in this field. In optics, two main families of tools are proposed for generating OAM beams: transformation optics that transforms the fundamental beam to an LG beam using "optics" and diffraction methods that use diffraction.

I.2.1 Spiral phase plate

A spiral phase plate (SPP) is a tool which enables the transformation of a plane wave carrying no orbital angular momentum to a wave bearing topological charge. It has been used

in optics for many years [START_REF] Beijersbergen | Helicalwavefront laser beams produced with a spiral phaseplate[END_REF][START_REF] Turnbull | The generation of free-space Laguerre-Gaussian modes at millimeter-wave frequencies by use of a spiral phaseplate[END_REF][START_REF] Schemmel | A modular spiral phase plate design for orbital angular momentum generation at millimetre wavelengths[END_REF], and can be realized by different techniques, such as vapor deposition [START_REF] Sueda | Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses[END_REF], lithography [START_REF] Khonina | The phase rotor filter[END_REF], etc.

To better explain how the SPP works, let's firstly take a brief look at the concept of wave propagation. When a wave passes through a medium with a refractive index of n1 and a height of L, its phase 𝜱 is increased by a quantity expressed by:

𝛷 = 2𝜋 𝜆 × 𝑛 1 × 𝐿 (I-6)
where λ is the wavelength.

However, the phase variation of an OAM wave on the wavefront is 2πl in one turn. To transform a plane wave or a wave with a plane wavefront, we can use a plate with a variable thickness. A spiral phase plate is made of the material with a constant refractive index and a varying thickness along the roll angle φ (see Figure I-5).

The phase shift δ𝜱 introduced by the SPP is then proportional to the roll angle and is expressed by:

𝛿𝛷 = 2𝜋×(𝑛 1 -𝑛 2 )×𝐿 𝑚𝑎𝑥 𝜆 (I-7)
where Lmax corresponds to the height difference for φ = 2π and n2, the refractive index of air.

Thus, to obtain a 2πl phase variation on the wavefront, the height difference should be equal to: passing through a SPP. Usually, the higher the topological charge is, the bigger the height difference is. It is also possible to maintain the height difference identical to that of the order l = 1 for the higher order modes by using a SPP with several summits [START_REF] Biener | Formation of helical beams by use of Pancharatnam-Berry phase optical elements[END_REF].

𝐿 𝑚𝑎𝑥 = 𝑙×𝜆 𝑛 1 -𝑛 2 (I-8)

I.2.2 Laguerre-Gaussian mode

The experiment investigated by Allen in 1992 shows that the Laguerre-Gaussian (LG)

modes have rotational symmetry along their propagation axis and carry an orbital angular momentum of lħ per photon [1]. The LG mode can be directly generated [START_REF] Harris | Optical helices and spiral interference fringes[END_REF], transformed from a Hermite-Gaussian (HG) mode using a cylindrical lens mode converter [START_REF] Beijersbergen | Astigmatic laser mode converters and transfer of orbital angular momentum[END_REF], or produced by properly combining two HG modes [1]. For instance, a LG01 mode can be obtained by

I.2.3 Diffraction gratings

First of all, we can remind that an idealized diffraction grating is an optical component generally made up of a set of narrow parallel lines with a spacing of d, which must be wider than the wavelength of interest. It may be of the reflection or the transmission type. For a reflection grating, the above features are represented by the reflective strips and for a transmission one, by the slits. In the following part, only the case of the transmission grating will be presented, because the conclusions are similar in both cases.

When a diffraction grating is illuminated by a light beam, the slits will split or diffract the light into several beams travelling in different directions. The result is a regular alternation of constructive and destructive interferences, which creates multiple beams whose directions can be determined by the following relationship:

𝑑(sin 𝜃 𝑖 ± sin 𝜃 𝑚 ) = 𝑚λ (I-11)
where θi and θm are respectively the incidence and the diffraction angles of the light beam, λ the wavelength, d the spacing between two neighboring slits and m an integer called the diffraction order.

When the light is normally incident on the gratings (θi = 0), the diffraction angle can be obtained by the following formula:

𝜃 𝑚 = arcsin (±𝑚 𝜆 𝑑 ) (I-12)
Thus, for a given frequency and incidence angle, there exists several diffracted waves propagating in different directions.

Another way to see a diffraction grating is to define it as the pattern that would be obtained by the interference between the incident and the diffracted beams. In this way, it is possible to record the interference pattern between the two beams and then to use it to generate one beam from the other. This is the technique called the "holography" proposed by D. Gabor [START_REF] Gabor | A new microscopic principle[END_REF]. The interference pattern is called a hologram.

In the case of a hologram for generating OAM beams, the reference beam is a Gaussian beam, and the interference pattern is characterized by l-fold dislocation at its center (see Figure I-8). Due to their special shape, this type of hologram is often called a "forked diffraction grating" [START_REF] Bazhenov | Screw dislocations in light wavefronts[END_REF][START_REF] Heckenberg | Laser beams with phase singularities[END_REF]. The appropriate hologram can be generated by computer, or it can be formed by recording onto photographic film, the interference pattern between a plane wave and the beam one seeks to produce. What makes the holographic approach particularly appealing is the commercial availability of spatial light modulators (SLMs) [START_REF] Coomber | Optically addressed spatial light modulators for replaying computer-generated holograms[END_REF]. These pixelated liquid crystal devices can be programmed through the video interface of a computer to act as holograms and therefore, they have taken the place of the photographic film. Furthermore, numerically calculated holographic patterns can be displayed on an SLM. These devices produce reconfigurable, computer controlled holograms that allow a simple laser beam to be converted into an exotic beam with almost any desired phase and amplitude structure. And the beam pattern can be changed many times per second to meet experimental requirements. power ends up in the first order diffracted spot. This can be partly overcome by blazing the grating [START_REF] Heckenberg | Laser beams with phase singularities[END_REF]. When mode purity is of particular importance, it is also possible to create rather more sophisticated holograms where the contrast of the pattern is varied as a function of radius such that the diffracted beam has the required radial profile [START_REF] Arlt | Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam[END_REF].

As shown in

I.3 Methods for generating OAM waves at radio frequencies

The first radio OAM simulations were performed by Thidé et al. in 2007 [10]. Since then, the study of EM waves carrying OAM in the radio frequency domain is growing. In addition to the application of such waves is still subject to intense debates [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio domain[END_REF][START_REF] Edfors | Is orbital angular momentum (OAM) based radio communication an unexploited area?[END_REF][START_REF] Tamagone | Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Tamburini | Reply to Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF], there are also problems of generating these waves. In fact, even though the OAM is a naturally physical property in spite of the frequency, the methods available for generating OAM waves in the radio domain are not totally the same as that in optics due to the difference in the frequency scale.

Up to now, in radio frequency bands, two main families of antennas have been proposed

to generate OAM waves: plane wave transformers and circular phased arrays. Both families will be presented in this section.

I.3.1 Plane wave transformers

In the first family, the OAM wave is generated by a transformation of a plane wave.

Several antennas have been proposed and studied to realize this transformation.

I.3.1.1 Spiral phase plate

The first antenna presented here is brought from the optics. As presented in the last section, the Spiral Phase Plate can be used to generate optical beams carrying OAM. In the same manner, a SPP has been realized in the 30 GHz frequency band for generating radio OAM waves [START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF]. Whereas in the optical field, a SPP is illuminated by a Gaussian beam that can be regarded as a plane wave, in the radio field a conventional horn antenna that is commercially To improve the reflectivity of the SPP, an ultralow reflectivity SPP was proposed in [START_REF] Hui | Ultralow reflectivity spiral phase plate for generation of millimeter-wave OAM beam[END_REF].

Unlike the previous SPP which has a smooth and continuous surface, this SPP is composed of unit cells whose equivalent permittivity and thickness are designed to satisfy the impedance matching condition based on transmission line theory. To ensure a good mechanical integrity, the phase plate must have as few holes as possible:

the maximum ratio of the air can be selected as 33% [START_REF] Bennis | Flat plate for OAM generation in the millimeter band[END_REF], which corresponds to a minimum

I.3.1.4 Reflector antennas

The generation of OAM waves with a reflecting system is based on a principle similar to that of the Spiral Phase Plate: the phase variation of the propagated wave has a dependency on the roll angle. In the case of the SPP, this variation is realized in the process of transmission, while in the case studied here, it is done during the reflection on a surface. Therefore, the appearances of these antennas are also close: they all have a height difference Hmax along the roll angle of the reflector which is defined by:

𝐻 𝑚𝑎𝑥 = 1 2 × 𝑙 × 𝜆 (1-16)
where λ is the wavelength at the working frequency, l is the order of the OAM mode.

To our knowledge, only two different reflectors have been proposed to generate OAM waves. The first one is made from styrofoam blocks, covered with an electrically conductive and reflective surface [START_REF] Tamburini | Experimental verification of photon angular momentum and vorticity with radio techniques[END_REF]. To realize a total 2π phase shift in one turn, the spiral reflector is Following this experiment, a second one was done in the real environment [START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF]. The authors used an 80-cm twisted parabolic reflector dish, working in the 2.

I.3.2 Circular phased arrays

The proposal to generate waves bearing OAM in the radio field was made for the first time by using a circular phased antenna array [START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio domain[END_REF]. It is generally composed of N identical short dipoles (see Figure I-23), where each one is fed by the same signal but with an incremental phase shift compared to its neighbor. To realize a 2πl phase shift in one turn, the phase shift δϕ between two successive dipoles is expressed by [5]: The number of the array elements N has an additional influence on the OAM antenna: it determines the largest OAM mode lmax that can be generated [5], where -N/2 < lmax < N/2.

𝛿𝜙 = 2𝜋𝑙 𝑁 (1-17)
This method has two main advantages. The first one is that each of the antennas making up the system can be controlled independently of the others; and as a result, the second is that the phase of the generated wave can be well known and controlled. Its disadvantage is the increase of the system complexity.

Following this, several other studies showing the variations of this method have been published. The first proposal is to use a circular TSA (Time-Switched Array) to generate OAM waves at lower cost [START_REF] Tennant | Generation of OAM radio waves using circular time-switched array antenna[END_REF]. The elements of the array are excited with unit amplitude and uniform phase at a fundamental working frequency, but are energized sequentially such that each element is only switched on for a time period. Later, in another study the same group presented the design of a circular phased array of 8 patch antennas [START_REF] Bai | Generation of orbital angular momentum (OAM) radio beams with phased patch array[END_REF]. Again, all the patch antennas are fed using the same signal but with an incremental phase shift between each patch. Besides the ideal model, the authors also give the practical design [START_REF] Bai | Experimental circular phased array for generating OAM radio beams[END_REF]. As shown in

I.4 Conclusions

In this chapter we introduced the characteristics of the EM waves bearing OAM and reviewed the typical methods for generating such waves.

Different from a usual plane wave, the OAM waves have a helical phase front which makes it a potential candidate for radio communications [5][START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Wang | Terabit free-space data transmission employing orbital angular momentum multiplexing[END_REF][START_REF] Yan | Highcapacity millimetre-wave communications with orbital angular momentum multiplexing[END_REF][START_REF] Hui | Multiplexed millimeter wave communication with dual orbital angular momentum (OAM) mode antennas[END_REF] and radar applications [START_REF] Liu | Orbital-Angular-Momentumbased electromagnetic vortex imaging[END_REF][START_REF] Lin | Improved OAM-based radar targets detection using uniform concentric circular arrays[END_REF][START_REF] Lin | Super-resolution orbital angular momentum based radar targets detection[END_REF].

Thus, it is necessary to investigate the methods for generating these waves.

In optical field, three main techniques were used to generate OAM beams: spiral phase plate, Laguerre-Gaussian mode and diffraction grating. In radio field, two main families of antennas were proposed to generate OAM waves: plane wave transformers (spiral phase plate, flat phase plat, diffraction grating and spiral reflector) and circular phased arrays.

Chapter II Generation of Radio OAM Waves with Circular Phase Shifter and Array of Patch Antennas

II.1 Introduction

In the previous chapter, we mentioned that the circular phased antenna array is a common way to generate OAM waves in the radio frequency band. However, apart [START_REF] Bai | Experimental circular phased array for generating OAM radio beams[END_REF], the existing structures are remained mainly at simulation level.

In this chapter we present a circular phased array using an original combination of a phase shifter and a power divider to generate a wave bearing an OAM mode l = 1. Besides the design procedure and the simulation data, the full experimental characteristics of the antenna are given.

This OAM-wave generator works at a frequency of 2.5 GHz. It uses four patches which are fed by a single and original transmission line. The structure is simple, compact and easy to realize.

It can be used in many domains like radio communications and radar applications.

This chapter is organized as follows. Firstly, we give a brief introduction of the structure of a single patch antenna and a patch antenna array; then we investigate the influences of the array parameters on the OAM antenna performances, which is necessary for the design of a practical OAM antenna; finally, we design, realize and characterize this novel OAM antenna.

II.2 Patch antenna structure

II.2.1 Single patch antenna

A typical patch antenna is composed of a very thin metallic strip (patch), a ground plane and a substrate that separates them. The radiating patch is usually etched on the substrate by chemical method, and it may be square, rectangular, circular, elliptical or any other geometry.

There are many configurations that can be used to feed the patch antennas. The four most popular are the microstrip line, coaxial probe, aperture coupling and proximity coupling. width W and the approximate length L of a patch antenna can be easily found in an antenna handbook, and they are expressed as following [START_REF] Balanis | Antenna theory: analysis and design[END_REF]:

𝑊 = 𝑐 2𝑓 𝑟 √ 2 𝜀 𝑟 +1 (II-1) 𝐿 ≈ 𝑐 2𝑓 𝑟√ 𝜀 𝑟 (II-2)
where c is the speed of light in free space, fr the resonant frequency of the patch antenna, and εr the relative permittivity of the substrate. Thus, when the patch antenna resonates at 2.5

GHz, the calculated width and length are respectively 36.5 and 28.6 mm. However, since some of the waves radiated by the patch travels in the substrate, some in the air, εr needs to be replaced by an effective dielectric constant εreff. Besides, due to the fringing effect, the patch looks greater than its actual dimension with an extension in length on each end by a distance of ∆L.

Consequently, equation (II-2) is modified as following:

𝐿 = 𝑐 2𝑓 𝑟√ 𝜀 𝑟𝑒𝑓𝑓 -2∆𝐿 (II-3)
where

𝜀 𝑟𝑒𝑓𝑓 = 𝜀 𝑟 +1 2 + 𝜀 𝑟 -1 2 (1 + 12 ℎ 𝑊 ) - 1 2
(II-4)

∆𝐿 = 0.412 × (𝜀 𝑟𝑒𝑓𝑓 +0.3)( 𝑊 ℎ +0.264) (𝜀 𝑟𝑒𝑓𝑓 -0.258)( 𝑊 ℎ +0.8) × ℎ (II-5)
where h is the thickness of the substrate.

The recalculated length of the patch antenna using the new equation is equal to 28.2mm, which is the initial value used for simulation. According to the simulations, to resonate at 2.5

GHz, the length of single patch antenna is 27.4 mm. Moreover, the input impedance of the patch antenna has a dependency with the position of the feeding point. To realize a 100 Ω impedance matching, the position of the probe is optimized at 3.7 mm from the bottom side of the patch antenna.

II.3 Parametric study of the circular phased patch array

In Thidé's work [START_REF] Thidé | Utilization of photon orbital angular momentum in the low-frequency radio domain[END_REF], the circular phased array for generating OAM waves is composed of N short dipoles, where each one is fed using the same signal but with an incremental phase shift compared to its neighbor. However, this antenna is only at simulation level. Here we intend to use patch antennas for the purpose of realization. Compared to the dipoles, the patches have two major advantages: 1) the radiation of the dipole can be considered as omnidirectional, i.e. it radiates in both the front and back directions in the plane perpendicular to the dipole. With the patch, the back radiation is considerably decreased and therefore, the radiated power is doubled in the forward direction; 2) the dipole needs a symmetric power supply, typically with 2 parallel wires. However, the patch can be directly supplied by a coaxial cable and therefore, needs no "balun" which further complicates the antenna realization. In this part we make a detailed parametric study of the ideal phased patch array, which is the first step to design a practical OAM antenna.

II.3.1 Number of array elements

We have firstly varied the number of the array elements to study its influences on the quality of the generated OAM wave at 2.5 GHz. The array radius is kept at 60 mm (0.5×λ). To According to [5], we can know that to create an OAM wave with a circular phased array of N antennas, the theoretically largest OAM mode lmax that can be generated is predicted by the following equation: -N/2 < lmax < N/2. Therefore, we use N = 8 patch antennas (see Figure II-6(c)) to generate different OAM modes at 2.5 GHz. When the elements are all in phase, no OAM wave is generated (i.e. l = 0); when they are fed using the same signal but with an incremental phase shift of 2πl/8 between each element, the corresponding OAM mode is produced.

II.3.2 Array radius

The array radius R is another important parameter of the circular phased array. To study its influences on the properties of the generated OAM wave, we use an array of 4 patch antennas

II.4 OAM antenna design

According to the above analysis, it can be concluded that N = 4 is the smallest number of array elements for generating an OAM wave with a quite good quality. In this case, the array is smaller and easier to realize. Therefore, we developed a new antenna using four patches and an original and simple phase shifter-power divider to generate an OAM wave with l = 1. The full design procedure of this "OAM antenna" will be presented in this part.

II.4.1 Design of phase shifter-power divider

For creating an OAM mode l = 1 with N = 4 antennas, the phase step between each element should be 2πl/N = π/2 radians = 90°. To this purpose, an original circular phase shifterpower divider is designed which supplies the required 90° phase difference between two neighboring antennas, as shown in -350°, -439°, respectively; and the corresponding amplitudes are -6.4 dB, -6.6 dB, -6.9 dB, -7 dB.

Finally, it can be remarked that using a mirror effect, the phase shifter can be used to feed an antenna bearing an OAM mode l = +1 or -1.

II.4.2 Design of OAM antenna

II.4.2.1 Simulation model

The practical design of the OAM antenna is shown in The 2D magnitude and phase patterns of the Ex component (see (θ = 0°, when the OAM antenna faces to the transmitting antenna) with a step of 1°. The OAM antenna is characterized for both horizontal and vertical polarizations. From these measurements, the field of the main polarization Ex is determined using this express:

𝐸 𝑥 = 𝐸 𝑉 cos 𝜑 + 𝐸 𝐻 cos 𝜃 sin 𝜑 (II-6)
where EH and EV are respectively the E-fields of the horizontal and vertical polarizations.

Support

Horn antenna θ between theoretical and experimental results is very weak for small  angle, but increases with . Consequently, the ideal OAM behavior is respected in the vicinity of the vortex and the agreement remains still good until  = 30°, where the beam amplitude is maximum. -40

E-plane (xOz cut) E-plane (yOz cut) (a)

θ = 10° θ = 30° θ = 50° (b)
-20 0 0 30 60 90 120 150 180 -150 -120 -90 -60 -30 Measurement Simulation -20 -10 0 0 30 60 90 120 150 180 -150 -120 -90 -60 -30 1 2 3 410 2 0 30 60 90 120 150 180 -150 -120 -90 -60 -30 Theory (=) Simulation Measurement 1 2 3 410 2 0 30 60 90 120 150 180 -150 -120 -90 -60 -30 1 2 3 410 2 0 30 60 90 120 150 180 -150 -120 -90 -60

II.5 Conclusions

In this chapter we developed an antenna using four patches and an original and simple phase shifter-power divider to generate an l = 1 OAM wave at 2.5 GHz.

We firstly investigated the influences of the array parameters (number of array elements (N) and array radius (R)) on the performance of an ideal phased patch array and made the following conclusions: i) as N increases, the OAM modes become purer and of better quality, but the array becomes larger and more complex to realize; ii) N = 4 the smallest number of array elements that can produce the mode l = 1 with quite good quality; iii) as R increases, the OAM beam becomes more directive, but naturally some side lobes appear and the antenna geometry becomes larger.

Then, we presented the design of a practical OAM antenna which is based on a circular phase shifter-power divider and an array of 4 patches. The array radius was set at 50 mm. Appropriate weights (phase and amplitude) were obtained for the array elements. Both the 3D radiation & phase patterns and the 2D magnitude & phase patterns of the electrical field confirm the generation of an OAM bearing wave.

Finally, a prototype was realized and characterized. Good agreement is obtained between theoretical results, simulation data and measurements.

Chapter III

Design of a Directive OAM Antenna by

Using Fabry-Perot Cavity

III.1 Introduction

In the previous chapter, we developed a novel antenna to generate an l = 1 OAM wave at 2.5 GHz. This OAM-wave generator is based on a 4-element circular phased array of patch antennas. It is simple, compact and easy to realize, but its directivity is relatively low. To overcome this drawback, in this chapter, we embed this OAM antenna inside a Fabry-Perot (FP) cavity to enhance its directivity. The FP cavity is composed of a ground plane, an air cavity and a partially reflecting surface (PRS).

This chapter is organized as follows. First of all, we simply analyze the model of a FP cavity and study its influences on the directivity of an antenna; then the developed OAM antenna in Chapter II is embedded inside two different FP cavities and each FP cavity is optimized for obtaining a maximum directivity of the OAM antenna; finally, a prototype is realized and characterized.

III.2 Fabry-Perot cavity

A Fabry-Perot (FP) cavity is highly frequency selective and is originally used as frequency filters in optics. In antenna applications, it is often employed as space filter to improve the antenna performances such as directivity [START_REF] Akalin | A highly directive dipole antenna embedded in a Fabry-Perot type cavity[END_REF][START_REF] Boutayeb | Directivity of an antenna embedded inside a Fabry-Perot cavity: Analysis and Design[END_REF][START_REF] Gardelli | Array thinning by using antennas in a Fabry-Perot cavity for gain enhancement[END_REF].

III.2.1 Structure of the FP cavity

The structure of the FP cavity is shown in Figure III-1. It is generally composed of a ground plane which eliminates the back radiation, a primary source and a partially reflecting surface (PRS). The ground plane and the PRS are separated by the cavity thickness D. This cavity is also defined by its lateral dimension L. Different methods have been employed to model a FP antenna. The first approach is based on the theory of transmission lines [START_REF] Talleb | Characterization of electronically controlled electromagnetic band gap structures[END_REF], the second utilizes the dispersion diagram of a metamaterial [START_REF] Enoch | A metamaterial for directive emission[END_REF], the third takes advantage of the theory of the leakage waves [START_REF] Oliner | Leaky-wave antennas[END_REF][START_REF] Jackson | Leaky-wave propagation and radiation for a narrow-beam multiple-layer dielectric structure[END_REF], and the last utilizes the method of the Fabry-Perot interferometer which is one of the most used for extracting the antenna parameters of the FP antennas [START_REF] Trentini | Partially reflecting sheet arrays[END_REF][START_REF] Boutayeb | Antenna inside PBG and Fabry-Perot cavities[END_REF][START_REF] Thevenot | New advancements to exploit the potentialities of the EBG resonator antennas[END_REF][START_REF] Mahdjoubi | An overview on the design and properties of EBG antennas[END_REF].

III.2.2 Analysis model of the FP cavity

The analysis method based on the FP interferometer has been detailed in many papers and PhD theses. To summarize, consider the 2D FP structure consisting of a PRS and a ground plane with infinite lengths. The PRS is characterized by its reflection and transmission coefficients (r and t) and the ground plane by its reflection coefficient (rg). Because the FP cavity has cylindrical symmetry around the z axis and has no dependencies on the azimuth coordinate φ, the φ variations are already included in these coefficients. of the first transmitted wave is "1×t". In the same manner, the second transmitted ray has an amplitude of "t×r×rg", because it is first reflected on the PRS, then on the ground plane and finally transmitted from PRS. The phase lag of the second ray compared to the first one is

( 𝐷 𝑐𝑜𝑠 𝜃 + 𝐷 cos 𝜃 × cos 2𝜃) × 𝑘 = 2𝑘𝐷 cos 𝜃,
where k is the propagation constant inside the cavity and θ the incidence angle between -90° and 90°. The amplitude of the n th transmitted ray is "t×(r×rg) n-1 " and the phase lag compared to the first one is 2×(n-1)×k×D×cos(θ). The total external field of the FP cavity can be determined by the sum of all the transmitted fields, and therefore, the transmission coefficient of the FP structure is given by:

𝑇 𝐹𝑃 = 𝑡[1 + (𝑟 • 𝑟 𝑔 )𝑒 -𝑗2𝑘𝐷 cos 𝜃 + ⋯ + (𝑟 • 𝑟 𝑔 ) 𝑛 (𝑒 -𝑗2𝑘𝐷 cos 𝜃 ) 𝑛 + ⋯ ] (III-1)
The resultant series constitute in fact a geometric series whose expression has the following simple and compact form:

𝑇 𝐹𝑃 = 𝑡 1-𝑟•𝑟 𝑔• 𝑒 -𝑗2𝑘𝐷𝑐𝑜𝑠𝜃 (III-2)
When the ground plane is a perfect electric conductor (PEC) (rg = -1) and the phase of the PRS reflection coefficient is α, the cavity transmission coefficient can be expressed as:

𝑇 𝐹𝑃 = 𝑡 1+|𝑟|•𝑒 𝑗(𝛼-2𝑘𝐷𝑐𝑜𝑠𝜃) (III-3)
On the other hand, the cavity resonance is defined as the case where all the transmitted rays are in phase and make a constructive interference. In this case, the radiation pattern is maximum in the θ direction. According to (III-3), TFP is maximum as a function of θ if its denominator is minimum. This happens if the term |𝑟| • 𝑒 𝑗(𝛼-2𝑘𝐷𝑐𝑜𝑠𝜃) is a negative real number:

1 + |𝑟| • 𝑒 𝑗(𝛼-2𝑘𝐷𝑐𝑜𝑠𝜃) = 1 -|𝑟|, 𝛼 -2𝑘𝐷𝑐𝑜𝑠𝜃 = (2𝑛 + 1)𝜋 𝐷 = 𝑐 4𝜋𝑓 0 cos 𝜃 [𝛼 + (2𝑛 + 1)𝜋], n = 0, 1, 2, 3…… (III-4)
where c is the speed of the light inside the cavity (free space), f0, the working frequency, n, an integer corresponding to the cavity mode number. In our study, n is set to 0 to use the first cavity mode.

The antenna directivity depends highly on α and | r |. The higher is | r |, the smaller is the denominator and the higher is the TFP maximum. Moreover, | r | is highly dependent on the PRS chosen to form the cavity. In the next sections we embed the OAM antenna inside the FP cavity with the PRS made of metallic strips or tubes to enhance its directivity.

III.2.3 Influences of the dimension of the FP cavity

The cavity must have a finite dimensions to be easily realizable. Various studies have shown the influences of the lateral dimension (L) of the cavity on the antenna directivity. The directivity increases with the lateral dimension of the cavity. However, it does not increase infinitely with the cavity surface. This means that from a certain dimension, the electric field is totally radiated and increasing the surface does not improve anymore the directivity. For small dimensions, the directivity of the FP antenna decreases significantly since part of the energy is diffracted at the edges of the cavity, which results in the occurrence of the side lobes. To limit this loss of energy, metallic walls can be added to close the cavity. We can see from Figure III-3 that it is possible to obtain a directivity higher than 18 dB for L > 5×λ without metallic walls and for L > 3×λ with metallic walls with a reflectivity of 0.96 of the PRS [START_REF] Ronciere | Antennes à bande interdite électromagnétique et à cavité Fabry-Pérot reconfigurables[END_REF].

Figure III-3: Influences of the lateral dimension of the cavity on the directivity [68]

The study carried out on this problem [START_REF] Ronciere | Antennes à bande interdite électromagnétique et à cavité Fabry-Pérot reconfigurables[END_REF] has highlighted the influences of the small lateral dimension on the resonance condition. In fact, when the lateral dimension decreases, the resonance frequency increases. Thus, we should use the following resonance conditions:

For a cuboid cavity with a square PRS, 𝑓 = where r is the radius of the cavity.

Therefore, in the following study it is necessary to adjust the cavity thickness to be close to the resonance condition to maximize the directivity. The bandwidth is a critical parameter of the FP cavity because it depends heavily on the reflectivity of the PRS. Normally the higher is the reflectivity, the narrower is the bandwidth.

Various studies have addressed this problem [START_REF] Rodes | Nouveaux concepts d'antenne à base de matériaux BIE métalliques[END_REF][START_REF] Vu | Antennes à bande interdite électromagnétique directive: contribution à l'élargissement de la bande passante et à l'évaluation de l'impédance[END_REF]. To obtain a maximum bandwidth, the variation of the phase versus the frequency should be slowest.

In summary, to obtain a maximum directivity of a FP cavity, it is appropriate to impose a strong reflectivity at the PRS; to obtain a possibly maximum bandwidth, a slow variation of the phase of the PRS refection coefficient versus the frequency is necessary. Due to this fact, a PRS made of metallic strips with a filling ratio (b/λ) larger than 20% should satisfy these conditions [START_REF] Vu | Antennes à bande interdite électromagnétique directive: contribution à l'élargissement de la bande passante et à l'évaluation de l'impédance[END_REF].

III.3.2 Influences of the array radius on the directivity of the phased patch array

In the previous chapter, we have already made a detailed study on the circular phased patch array. Here we briefly show the influences of the array radius on the antenna directivity.

The configuration of the OAM antenna is shown in Figure III-6. The patch array is manufactured on a FR4 substrate with a thickness of 1.6 mm and a relative permittivity of 4.4 and works at 2.5 GHz. All the patch antennas are probe fed using the same signal but with an incremental phase shift between each patch. For creating an OAM mode l = 1 with N = 4 antennas, the phase step between two neighboring elements should be 2πl/N = π/2 rad = 90°. radius increases, but naturally some side lobes appear. When the array radius keeps increasing, the side lobe becomes even comparable with the main lobe, which means a strong loss of energy.

Figure III-7: E-plane radiation patterns with different array radiuses R at the frequency of 2.5 GHz

III.3.3 Optimization of the FP cavity associated with the phased patch array

It has been demonstrated that the FP cavity is able to enhance the directivity of an antenna [START_REF] Akalin | A highly directive dipole antenna embedded in a Fabry-Perot type cavity[END_REF][START_REF] Boutayeb | Directivity of an antenna embedded inside a Fabry-Perot cavity: Analysis and Design[END_REF][START_REF] Gardelli | Array thinning by using antennas in a Fabry-Perot cavity for gain enhancement[END_REF]. Along this way, we would like to embed the above OAM antenna inside the FP cavity to enhance its directivity of the main lobe, as shown in Figure III-8. Because the OAM waves generated by the patch array are more directive with a bigger array radius, here we choose an array radius of 110 mm.

R

To obtain a best directivity of the OAM antenna, the width and period of the metallic strips, the cavity thickness, as well as the lateral dimension of the cavity will all be optimized.

III.3.3.1 Optimization of the width and period of the strips

The optimization is divided into three steps. The first step is to optimize the width and period of the metallic strips, where the cavity aperture has a constant and big enough dimension of 5λ × 5λ.

The aim of this optimization is to obtain a maximum directivity of the OAM antenna by choosing a best reflectivity of the PRS. To simplify the process, we maintain the period (b) of the strips at 20 mm and vary the width (a) of the strips from 1 to 6 mm with a period of 1 mm.

We simulate again a unit cell of the PRS, the resulting magnitudes and phases of the PRS reflection coefficient are shown in Table III-1. As expected, the magnitude (| r |) and phase (α) both increase with the width of the strips. In addition, according to (III-4), the cavity thickness depends on the working frequency, the phase of the reflection coefficient of the PRS and the incidence angle of the wave rays.

Consequently, the cavity thickness will also vary with the width of the strips. Besides, from

Figure III-6 we can see that when the array radius is 110 mm, the angle value corresponds to the maximum directivity is at ±15° for the main lobe, which means the value of the incidence angle here is 15°. Thus, the initial thickness of the cavity (D) is equal to

𝜆 4π×cos(15°) × (𝛼 + 𝜋)
for the cavity mode n = 0, where λ is the wavelength. For example, choosing a = 1 mm, the corresponding | r | and α are respectively 0.75 and 0.77π, and the calculated cavity thickness is 55 mm with λ = 120 mm. The whole results are also shown in Table III-1.We can see that the cavity thickness increases with the width of the metallic strips as well.

The variation of the E-plane maximum directivity of the OAM antenna as a function of the width of the strips is shown in Figure III-9. We can observe that the OAM antenna achieves a maximum directivity of 15.6 dB with a width of 4 mm (τ = 20% for a period of 20 mm).

Usually, the directivity of the FP cavity should increase with the magnitude of the reflection coefficient of the PRS. However, in our case it is noted that the behavior is not as expected. magnitude of the reflection coefficient of the PRS. Besides, due to this fact, an appropriate width of the strips is necessary to reduce the side lobe level.

Figure III-10: E-plane radiation patterns with different widths of strips at the frequency of 2.5 GHz (period of strips: 20 mm)

In summary, to obtain a maximum directivity of the FP OAM antenna, it is suitable to impose a strong reflectivity at the PRS; on the other hand, to obtain a lower side lobe level, the reflectivity should not be too high. Based on these two points, the strips with a width of 4 mm and a period of 20 mm will be utilized.

III.3.3.2 Optimization of the cavity thickness

The second step is to optimize the thickness of the FP cavity. During this step, we maintain the width and period of the strips at 4 and 20 mm and still employ a cavity aperture with a dimension of 5λ × 5λ.

When the phased patch array is embedded inside the FP cavity, the angle value corresponding to the maximum directivity is at ±11° instead of ±15°, so the previously calculated cavity thickness may not be sufficiently precise. Due to this fact, we have varied the cavity thickness (D) a little to further optimize the antenna directivity, the results are shown in To better show the influences of the cavity thickness on the antenna directivity, Figure

III-11 presents the E-plane radiation patterns with the calculated and optimized cavity thicknesses. We can observe that as presented above, the OAM antenna achieves a higher directivity with the optimized cavity thickness. Moreover, the side lobe level is significantly decreased at the same time, which is another advantage of this thickness. means the OAM waves become more directive. This indicates that to obtain a higher directivity, it is appropriate to increase the dimension of the cavity aperture. However, the directivity does not increase infinitely as it approaches the limit of operation and a big dimension will make the cavity very difficult to realize in practice, a trade-off between the antenna directivity and the dimension of the cavity aperture is therefore necessary. Finally, we choose the cavity aperture with a dimension of 600 mm × 600 mm for obtaining an antenna directivity of 16.2 dB for the angle of 11°.

III.3.3.4 Performance of the optimized OAM antenna

According to the above optimization results, to obtain a maximum directivity of the OAM antenna at 2.5 GHz, a good condition is: The maximum directivity is obtained for an angle of 11°.



III.3.4 Design of the phase shifter-power divider

As what is done in the previous chapter, here we re-design the circular phase shifterpower divider with a larger array radius of 110 mm to supply the required 90° phase difference between two neighboring antennas. The combined phase shifter and power divider is manufactured on a FR4 substrate with a thickness of 1.6 mm and relative permittivity of 4.4. It error = ±0.6dB). Thus, the phase shifter-power divider has quite a good agreement with the required specifications.

Figure III-15: Simulated parameters of the phase shifter-power divider: phase shifts (left) and transmission coefficients (right). At 2.5 GHz, the phase shifts of S21, S31, S41, S51 are -182°, -273°, -363°, -452°, respectively; and the corresponding amplitudes are -7.5 dB, -7.7 dB, -7.9 dB, -8.8 dB.

III.3.5 Characterization of the FP OAM antenna with the phase shifter-power divider

The configuration of the FP OAM antenna with the phase shifter-power divider is shown in Figure III-16. The OAM antenna contains 3 FR4 substrates which all have a dimension of 300 mm × 300 mm × 1.6 mm and a relative permittivity of 4.4. The phase shifter-power divider is manufactured on the bottom substrate and the patch array on the top one. The big ground plane is between the middle and the top substrates. All the patch antennas are connected with the phase shifter-power divider through probes and fed by a 50 Ω coaxial cable from the back side of the bottom substrate. To obtain a good radiation pattern and satisfy the impedance matching, the length and width of single patch antenna are optimized at respectively 26.6 mm and 36.5 mm and the position of the feeding point is 5 mm from the bottom side of the patch antenna. Compared to the ideal FP OAM antenna, the maximum directivity decreases slightly by 0.2 dB.

The 2D radiation patterns correspond to the xOz and yOz cuts of the 3D plot. It can be seen that the E and H planes both obtain a maximum directivity at the angle of 11°. Besides, the side lobe level of H-plane is higher than E-plane. We present the phase patterns (Figure III-18c) for three values of the spherical angle (aperture)  : 5°, 15° and 25°. The spiral form of the phase variation versus the roll angle  is due to the polar representation of the linear phase variation. We can observe that the phase variation in one turn is 2π and the quality of the phase pattern is good at small values of  angle, but degrades as  increases. According to the simulation results, the directivity of the OAM antenna is significantly enhanced by using the FP cavity. However, the big dimension of the substrate makes it difficult

-150 -120 -90 -60 -30 E-plane H-plane 2 4 6 0 30 60 90 120 150 180 -150 -120 -90 -60 -30  = 5° = 15° = 25°
to realize this antenna in practice. To solve this problem, in the next part we propose another design method.

III.4 Design of the FP OAM antenna with the PRS made of metallic tubes

In this part, the metallic strips are replaced by metallic tubes. In this way, the big substrate is no more needed which makes it easier to realize the FP cavity.

III.4.1 Characterization of the PRS

The geometry of the PRS made of periodic metallic tubes is shown in vice versa, when e/λ decreases, the magnitude tends to be 1 (no transmission) and the corresponding phase tends to be π. For a constant e/λ, the higher is δ, the bigger is the magnitude of the reflection coefficient and the closer to π is the phase. In this new design, the diameter of the tubes is set at 4 mm. To obtain a maximum directivity of the OAM antenna, the reflection coefficient of the new PRS should be close to the previous one, i.e. the magnitude and phase of the reflection coefficient of the new PRS should be near 0.94 and 0.89π, respectively. According to the simulations, this corresponds to a period of the tubes of 26 mm.

III.4.2 Influences of different kinds of walls on the OAM antenna directivity

As shown in Figure III-22, the FP cavity is closed on the sides using 4 walls, of which two are used to support the metallic tubes. The circular patch array is still manufactured on a FR4 substrate with a thickness of 1.6 mm and a relative permittivity of 4.4 and works at 2.5

GHz. All the array elements are probe fed using the same signal but with a 90° phase shift between each element. Again the array radius is set at 110 mm and the cavity aperture has a dimension of 600 mm × 600 mm. To obtain an optimal directivity of the OAM antenna, in this section the influences of different kinds of walls on the antenna directivity will be investigated. a maximum directivity for the angle of 10°, and the H-plane has a higher side lobe level than E-plane. Besides, as the thickness of the PVC wall increases, the side lobe level becomes higher in E-plane, but remains almost the same in H-plane. Therefore, it is appropriate to use the PVC wall with a small enough thickness to obtain a low side lobe level of the E-plane. However, to decrease the side lobe level of H-plane, a new material of the wall is needed. PEC walls in E-plane and 2 PVC walls in H-plane. We can observe that for both E and H planes, the side lobe level is low enough compared to the main lobe. Therefore, the results are good enough.

E-plane H-plane

Figure III-26: E and H plane radiation patterns of the FP OAM antenna with 2 PEC walls in E-plane and 2 PVC walls in H-plane

III.4.3 Characterization of the FP OAM antenna with the phase shifter-power divider

III.4.3.1 Simulation model

The configuration of the practical design of the second FP OAM antenna is shown in Figure III-27. The structure of the OAM antenna itself is the same as the previous one. The patch array is connected with the same phase shifter-power divider to obtain a 90° phase difference between two neighboring elements and a uniform amplitude for all the elements. The main difference between the two FP antennas is the PRS that makes up the FP cavity. For the first one, it is made of metallic strips which are printed on a FR4 substrate; for the latter one, it is made of metallic tubes. Besides, to the purpose of realization, the latter FP cavity is closed by 2 PEC walls in E-plane and 2 PVC walls in H-plane.

-40

-20 0 20 0 30 60 90 120 150 180 -150 -120 -90 -60 -30 E-plane H-plane

III.5 Conclusions

In this chapter we presented a method to enhance the directivity of an OAM antenna by embedding it inside a FP cavity. This OAM antenna is based on a 4-element circular phased patch array with an array radius of 110 mm and works at 2.5 GHz.

We used the FP cavity with two different PRS. The first one is made of a set of periodic metallic strips printed on a FR4 substrate, and the second, made of metallic tubes. We chose the strips with a width of 4 mm and the tubes with a diameter of 4 mm. The influences of the parameters of the FP cavity on the antenna directivity were studied. To obtain a maximum directivity, the periods of the strips and the tubes were optimized at 20 and 26 mm, the cavity thickness at 58 mm and the dimension of the cavity aperture at 600 mm × 600 mm. In addition, the second PRS is closed on the sides using 4 walls: 2 PVC walls in H-plane for supporting the A circular phase shifter-power divider was used to supply the required phases and amplitudes for the array elements. The design procedure and the simulation data of the two practical FP OAM antennas were given, as well as the measurements of the second antenna.

IV.1 Introduction

In the previous chapters, the OAM waves are generated by a circular phased array of patch antennas. An ideal circular array is flexible to generate different OAM modes, but with a low directivity. To improve this, we have embedded it inside a Fabry-Perot cavity. However, with the use of FP cavity, the overall dimension of the system is increased, which makes the antenna difficult to realize.

In this chapter, we present a new and simple way to create middle directivity OAM antenna using a horn structure. The OAM waves are generated by two main methods: i) transformation of the guided modes of a circular waveguide with a spiral phase plate (SPP); ii) combination of the guided modes. For the first method, we present three structures based on TE11, TM01 and TE21 modes; for the second one, the OAM waves are created either by combing the two kinds of TE21 modes or TM01 with the1 st kind of TE21 modes. This chapter is organized as follows. Firstly, we simply introduce the structure of a circular waveguide; then, we present the first method to generate OAM waves; finally, we present the second method to generate OAM waves.

IV.2 Circular waveguide

A waveguide is a structure that guides waves. At radio frequencies, it is usually a hollow metallic pipe. It allows the EM waves to pass through its interior, and its metallic walls provide excellent shielding between the exterior and interior. It is often utilized in conjunction with a horn antenna to transmit the EM waves to the free space. Although a waveguide may assume The circular waveguide is occasionally used as an alternative to the rectangular waveguide. Its round cross section makes it easy to machine, and it is often used to feed the conical horns. Like other waveguides, a circular waveguide can support both transverse electric (TE) and transverse magnetic (TM) modes. These modes have a cut-off frequency, below which the EM energy is severely attenuated. The equation for calculating the cut-off frequency fc of a circular metallic waveguide can be easily found in a waveguide handbook, and it is expressed as following [START_REF] Elliott | An introduction to guided waves and microwave circuits[END_REF]:

𝑓 𝑐 = 𝑐𝜂 2𝜋𝑟 (IV-1)
where c is the speed of light in free space, η a constant depending on the guided mode, and r the radius of the waveguide.

For a circular waveguide, the first three modes are TE11, TM01 and TE21 modes, and the corresponding values of η are respectively 1.841, 2.405 and 3.054. Therefore, when the diameter of the waveguide is set as 4.4 cm (the value used in the next part), the calculated cutoff frequencies of these three modes are 4, 5.2 and 6.6 GHz.

IV.3 Generation of OAM waves by using spiral phase plate

In the first chapter, we mentioned that a Spiral Phase Plate (SPP) is able to transform a plane wave to a wave bearing an OAM topical charge, and it has already been used in both optical and radio domains. However, the technique of SPP is usually used to generate OAM waves in free space, it is rarely used in the waveguide except in [START_REF] Kristensen | Angular momentum and spinorbit coupling for microwave photons[END_REF]. In this part, we present three original methods to generate the OAM mode in a waveguide by combining the classical modes of a circular waveguide with the SPP. The first one is based on TE11 mode, the second one on TM01 mode, and the last one on TE21 mode.

IV.3.1 TE11 mode and a single SPP

IV.3.1.1 SPP in waveguide

The first structure to generate OAM wave is presented in Figure IV-3. The SPP is made of Teflon (εr ≈ 2.1) and is placed inside the waveguide with the same diameter. The height step h of the SPP has been chosen to generate an OAM mode l = 1. The TE11 mode is excited ideally by a waveport in HFSS software. The E-field is polarized along x axis. The other end of the waveguide is covered by a Perfectly Matched Layer (PML) to make sure that the reflected wave is totally vanished. The phase of the Ex component of the TE11 mode is constant on the wavefront. Thus, to create an OAM mode l = 1, the SPP should realize a phase shift of 2π in one turn (along  angle) and the required step height h is given by [START_REF] Beijersbergen | Helicalwavefront laser beams produced with a spiral phaseplate[END_REF]:

ℎ = 2𝜋 𝑘 1 -𝑘 2 (IV-2)
However, in waveguide, k1 and k2 should be replaced by kg1 and kg2 which are the guided propagation constants for waveguide filled respectively with Teflon and air:

𝑘 𝑔𝑖 = √𝑘 0 2 𝜀 𝑖 -𝑘 𝑐 2 = √ ( 2𝜋𝑓 𝑐 ) 2 𝜀 𝑖 -( 𝜂 𝑟 ) 2 (IV-3)
where f is the working frequency, c the speed of light in free space, εi the dielectric permittivity of the material inside the waveguide (i = 1 for Teflon and i = 2 for air), r the radius of the waveguide, and η a constant which depends on the guided mode. For TE11 mode, η is equal to 1.841.

Here, we set the diameter of the empty waveguide as 4.4 cm. The corresponding cut-off frequency of the TE11 mode is 4 GHz. Since the wavelength of the working frequency needs to be quite small compared to the waveguide diameter, we choose a working frequency of 40

GHz, and therefore, the height step of the SPP becomes: h = 1.7 cm.

The magnitude and phase patterns of the Ex component for the wave passing through the SPP are plotted in Figure IV-5. We can note that the magnitude at the center is much smaller than the surroundings and the phase variation in one turn is equal to 2π. This shows indeed that both the amplitude and the phase correspond to a wave bearing an OAM mode with l = 1. We can also observe that even at the frequency of 40 GHz, the magnitude and phase of the Ex field are not quite satisfactory. To improve the results, one solution would be to increase again the frequency. In the next section we propose another solution by using a horn structure. The waveguide has a diameter of 4.4 cm, and the cut-off frequency of the TM01 mode is 5.2 GHz. The working frequency can be chosen to be much lower than that of the first structure; in fact, it just needs to be higher than the cut-off frequency of the TM01 mode. We choose a working frequency of 7.4 GHz, and therefore, the height step of each half-turn SPP becomes: h

IV.3.2.2 Horn antenna design

We also design a practical conical horn antenna based on TM01 mode to make the OAM waves more directive, as shown in Figure IV-16. As discussed above, for TM01 mode, the working frequency can be quite near to the cut-off frequency, so no high-order modes will exist.

Besides, the SPPs can be kept inside the waveguide. The diameters of the waveguide and the horn aperture are respectively 4. We can also observe that the quality of the phase pattern degrades as  increases. The magnitude and phase patterns of the propagated wave, plotted on a circular area with a radius of 7 cm, lying 1 cm above the horn aperture, are presented in Figure IV-19. We can observe that the magnitude at the center is much smaller than the surroundings and the phase rotates around the center with a 2π phase shift in one turn. This shows indeed that both the amplitude and the phase correspond to a wave carrying an OAM mode with l = 1. Besides, compared to the previous horn antenna, the magnitude of Ex field is improved. The diameter of the empty waveguide is 4.4 cm, and the corresponding cut-off frequency of the TE21 mode is 6.6 GHz. Similar to the TM01 mode, the working frequency just needs to be higher than the cut-off frequency of the TE21 mode. We choose a working frequency of 10 GHz, and therefore, the height step of each half-turn SPP becomes: h = 2.7 cm.

The magnitude and phase patterns of the Ex component for the wave at the end of the SPPs are shown in Figure IV-22. We can note that the magnitude pattern has a null at the center and the phase pattern has a linear variation along the roll angle  with a 2π phase shift in one turn. This confirms the generation of an OAM bearing wave with l = 1. We can also observe that the results are as good as those of the second structure.

IV.4 Generation of OAM waves by combining guided modes

In the first chapter, we showed that in optics the Laguerre-Gaussian (LG) modes have rotational symmetry along their propagation axis and carry an OAM of lħ per photon. They can be produced by combing two Hermite-Gaussian (HG) modes. As an example, a helically phased

LG01 mode (l = 1 or -1) is obtained by combining two HG modes of HG01 and HG10 with a π/2 phase difference between them [1]. Along this way, a rectangular silicon waveguide has already been used to generate optical OAM beams by combining the E x 21 and E x 12 modes [START_REF] Zhang | Generating in-plane optical orbital angular momentum beams with silicon waveguides[END_REF]. In the same manner, by combining some proper modes of a metallic waveguide we can create OAM waves in radio frequency bands. In this part, we propose two methods to obtain the OAM mode of l = 1: i) combination of two kinds of TE21 modes; ii) combination of TM01 and TE21 modes.

IV.4.1 Combination of two kinds of TE21 modes

The first method to generate OAM waves in a circular metallic waveguide is based on the combination of two kinds of TE21 modes. Because it is difficult to excite a pure TE21 mode in practice, we only study the ideal situation where the two TE21 modes are excited by a waveport in HFSS software. In the next section we propose another method by combining the TM01 and TE21 modes, as well as the design of an "OAM antenna" based on this method.

IV.4.2 Combination of TM01 and TE21 modes

The second method to generate OAM waves in a circular metallic waveguide is similar

IV.5 Conclusions

In this chapter, we have presented two main methods to generate radio OAM waves in a circular waveguide. The first method is to transform the classical guided modes of a circular waveguide to an OAM wave by using SPP. A horn structure is used to create a directive OAM antenna. The OAM mode of l = 1 is generated for the Ex component, based on TE11 mode with a single SPP, or TM01 mode with two half-turn SPPs, or TE21 mode with two half-turn SPPs.

The resulting magnitude and phase patterns of the electrical field, and the radiation and phase patterns all confirm the generation of an OAM bearing wave. There are some differences between the three horn antennas: i) to enable the working frequency to be near cut-off, for TE11 mode, the SPP should be placed inside the horn, while for TM01 and TE21 modes, it can be maintained inside the waveguide; ii) the magnitude pattern of Ex field of the horn antenna based on TM01 mode is better than those of the other two antennas; iii) the horn antenna based on TM01 mode has also a smoother and more regular radiation pattern with lower side lobes.

The second method is by a combination of the classical guided modes. Actually, we have developed two different ways to obtain the OAM mode: i) combination of two kinds of TE21 modes; ii) combination of TM01 and 1 st kind of TE21 modes. The OAM mode of l = 1 is generated for both the Ex and Ey components. The resulting magnitude and phase patterns of the electrical field confirm the generation of an OAM bearing wave. In addition, we designed an antenna using two printed monopoles and a single transmission line to simultaneously create the TM01 and TE21 modes and hence, to generate an OAM wave with l = 1. Both the electrical field patterns and the radiation patterns of this antenna evidence the presence of an OAM carrying wave.

Conclusions and Perspectives

In this thesis we have presented several new antennas for generating radio waves bearing Orbital Angular Momentum (OAM), which is necessary for any existing and future application.

In the first chapter we introduced the properties of waves bearing OAM and reviewed the typical methods for generating these waves. According to the working frequency, these methods can be used either in optics or in radio domain. In optics, three main techniques were used to generate OAM beams: spiral phase plate, Laguerre-Gaussian mode and diffraction grating. In radio domain, two main families of antennas were proposed to generate OAM waves: plane wave transformers (spiral phase plate, flat phase plat, diffraction grating and spiral reflector)

and circular phased arrays.

In the second chapter, we developed an OAM antenna using 4 patches and an original circular phase shifter-power divider to generate an l = 1 OAM wave at 2.5 GHz. We firstly investigated the influences of the array parameters (number of array elements (N) and array radius (R)) on the performance of an ideal phased patch array and made the following conclusions: i) as N increases, the OAM modes become purer and of better quality, but the array becomes larger and more complex to realize; ii) N = 4 the smallest number of array elements that can produce the mode l = 1 with quite good quality; iii) as R increases, the OAM beam becomes more directive, but naturally some side lobes appear and the antenna geometry becomes larger.

On the other hand, we designed and realized an OAM antenna based on these conclusions.

This antenna consists of 4 elements with an array radius of 50 mm. Appropriate weights (phase and amplitude) were obtained for the array elements with the use of the phase shifter-power divider. Both the 3D radiation & phase patterns and the 2D magnitude & phase patterns of the electrical field confirm the generation of an OAM bearing wave with l = 1. Besides, good agreement is obtained between theoretical results, simulation data and measurements.

In the third chapter we present a method to enhance the directivity of the previous developed OAM antenna by using a Fabry-Perot (FP) cavity. A FP cavity is generally composed of a ground plane, an air cavity and a partially reflecting surface (PRS). It is highly frequency selective and is often used to improve the antenna directivity in antenna applications.

We embed the OAM antenna inside the FP cavity with two different PRS. The first one is made of a set of periodic metallic strips printed on a FR4 substrate, and the second, made of metallic tubes. We chose the strips with a width of 4 mm and the tubes with a diameter of mm. The influences of the parameters of the FP cavity on the antenna directivity were studied.

To obtain a maximum directivity, the periods of the strips and the tubes were optimized at 20 and 26 mm, the cavity thickness at 58 mm and the dimension of the cavity aperture at 600 mm × 600 mm. In addition, the second PRS is closed on the sides using 4 walls: 2 PVC walls in Hplane for supporting the tubes and 2 PEC walls in E-plane for mechanical balance and decreasing the side lobe level of the radiation pattern. The FP OAM antenna finally achieved a directivity of 16.2 dB with an enhancement of 9.2 dB in E and H planes.

In the practical design, a circular phase shifter-power divider was used to supply the required phases and amplitudes for the array elements. The design procedure and the simulation data of the two practical FP OAM antennas were given, as well as the measurements of the second antenna.

In the fourth and last chapter, we presented several horn antennas and a waveguide antenna for generating radio OAM waves. These structures are all based on a circular metallic waveguide. Actually, we have developed two major methods to obtain the OAM mode: i) transformation of the guided modes of a circular waveguide by using the spiral phase plate; ii) combination of the guided modes.

For the first method, the OAM mode of l = 1 is generated for the Ex component, based on TE11 mode with a single SPP, or TM01 mode with two half-turn SPPs, or TE21 mode with two half-turn SPPs. The resulting magnitude and phase patterns of the electrical field, and the radiation and phase patterns all confirm the generation of an OAM bearing wave. There are some differences between the three proposed horn antennas: i) to enable the working frequency to be near cut-off, for TE11 mode, the SPP should be placed inside the horn, while for TM01 and TE21 modes, it can be maintained inside the waveguide; ii) the magnitude pattern of Ex field of the horn antenna based on TM01 mode is better than those of the other two antennas; iii) the horn antenna based on TM01 mode has also a smoother and more regular radiation pattern with lower side lobes.

For the second method, the OAM mode of l = 1 is generated for both the Ex and Ey components, by combining the two kinds of TE21 modes or TM01 and 1 st kind of TE21 modes.

The resulting magnitude and phase patterns of the electrical field confirm the generation of an OAM bearing wave. In addition, we designed an antenna using two printed monopoles and a single transmission line to simultaneously create the TM01 and TE21 modes and hence, to generate an OAM wave with l = 1. Both the electrical field patterns and the radiation patterns of this antenna evidence the presence of an OAM carrying wave.

In continuation of this thesis, the following perspectives are considered:

 Realize the horn antennas and the waveguide antenna for validation of the simulation results.

 Use the proposed OAM horn antennas as a "primary source" for simple parabolic reflectors to obtain high directivities at a low cost.

 Use the proposed OAM horn antennas as a "primary source" for twisted parabolic reflectors to produce a higher order OAM while maintain the high directivity.

 Apply the proposed OAM antennas to radio communications [5][START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Wang | Terabit free-space data transmission employing orbital angular momentum multiplexing[END_REF][START_REF] Yan | Highcapacity millimetre-wave communications with orbital angular momentum multiplexing[END_REF][START_REF] Hui | Multiplexed millimeter wave communication with dual orbital angular momentum (OAM) mode antennas[END_REF] and radars [START_REF] Edfors | Is orbital angular momentum (OAM) based radio communication an unexploited area?[END_REF][START_REF] Tamagone | Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF][START_REF] Tamburini | Reply to Comment on 'Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF].

Fig. 1 :

 1 Fig. 1: Les modèles de simulation : (a) Géométrie du déphaseur circulaire; (b) Configuration de l'antenne OAM

Fig. 2 :

 2 Fig. 2: Prototypes du déphaseur (à gauche) et du réseau d'antennes à onde OAM (à droite)

Fig. 4 :Fig. 5 :

 45 Fig. 4: Cartographie 2D en amplitude (à gauche) et en phase (à droit) du composant Ex pour l'antenne OAM à 2.5 GHz

Fig. 6 :Fig. 7 :

 67 Fig. 6: Coefficient de réflexion S11 simulé et mesuré de l'antenne OAM

Fig. 8 :

 8 Fig. 8: Structure de la cavité FP

HFSS et nous obtenons les valeurs suivantes :Fig. 9 :

 9 Fig. 9: Géométrie de l'antenne à cavité FP

Fig. 10

 10 Fig. 10 Les diagrammes de rayonnement 3D du réseau d'antenne OAM : (a) sans cavité FP; (b) avec cavité FP

Fig. 11 :Fig. 12 :

 1112 Fig. 11: Comparaison des diagrammes de rayonnement dans le plan E de l'antenne OAM avec et sans cavité FP(rayon de réseau égal à 110mm)

Fig. 13 :

 13 Fig. 13: Cartographie 2D en amplitude (à gauche) et en phase (à droit) du composant Ex pour l'antenne à cavité FP à 2.5 GHz

Fig. 14 :Fig. 15 :

 1415 Fig. 14: Mode TE11 circulaire classique. (a) Distribution du champ électrique; (b) Amplitude normalisée de la composante Ex ; (c) Phase de la composante Ex

Fig. 16 :Fig. 17 :Fig. 18 :Fig. 19 :

 16171819 Fig. 16: Diagramme de rayonnement 3D de l'antenne cornet OAM à la fréquence de 20 GHz avec un seul SPP placé à l'intérieur du cornet

Fig. 20 : 4 GHzFig. 21 :

 20421 Fig. 20 : Diagramme de rayonnement 3D de l'antenne cornet OAM avec deux demi-SPPs à la fréquence de 7.4 GHz

Fig. 22 :Fig. 23 :

 2223 Fig. 22: Mode guidé classique TE21 (premier type) le guide d'onde circulaire : (a) Distribution de champ électrique; (b) Amplitude normalisée de la composante Ex; (c) Phase de la composante Ex

Fig. 24 :l = 1 Fig. 25 :

 24125 Fig. 24: Evolutions des diagrammes en amplitude (en haut) et en phase (en bas) de la composante Ex pour la génération d'une onde OAM, en combinant deux types de mode de TE21 avec une différence de phase de π/2entre les deux

Fig. 26 : 1 Fig. 27 :

 26127 Fig. 26 : Distributions du champ électrique pour les modes TM01 et TE21 : (a) TM01 (b) TE21 de type 1

Fig. 28 :

 28 Fig. 28: Diagrammes en amplitude (à gauche) et en phase (à droite) de l'onde propagée: (a) Composante Ex; (b) Composante Ey

Fig. 29 :

 29 Fig. 29 : Diagrammes de rayonnement (à gauche) et de phase (à droite) 3D de l'antenne OAM à 8.3 GHz : (a) Composant Ex; (b) Composante Ey

Figure I- 1 :

 1 Figure I-1: (a) The SAM of an EM wave; (b) the OAM of an EM wave[START_REF]Angular momentum of light[END_REF]

  momentum. The Poynting vectors associated with a linearly polarized LG beam are shown (in the near field and without taking into account the beam divergence) in Figure I-2, with l being related to the number of spirals. We can see that the vectors are not in the beam axis, but turn around.

Figure I- 2 :

 2 Figure I-2: The spiraling curve represents the Poynting vectors of a linearly polarized LG beam of radius w(z) [1]

  and phase plots of LG modes of LG01, LG11 and LG21 are plotted in Figure I-3. It can be seen that in terms of its intensity on a plane perpendicular to the propagation axis, an LG mode comprises p + 1 concentric rings with a zero on-axis intensity. In addition, the phase distribution varies from 0 to 2π along the roll angle for each ring.

Figure I- 3 :I. 2

 32 Figure I-3: Normalized intensity (top) and phase (bottom) plots of LG modes: LG01, LG11, and LG21(left to right) showing the p + 1 concentric rings and the effect on the phase pattern[START_REF] Yao | Orbital angular momentum: origins, behavior and applications[END_REF]

Figure I- 4 :

 4 Figure I-4: Plane phase front for (a) l = 0 and helical phase fronts for (b) l = 1; (c) l = 2; (d) l = 3[START_REF] Yao | Orbital angular momentum: origins, behavior and applications[END_REF]

Figure I- 5 :

 5 Figure I-5: Geometry of the spiral phase plate with a constant refractive index and a varying thickness along the roll angle φ

Figure I- 6 :

 6 Figure I-6:A helically phased wave with l = 2 generated from a plane wave using a SPP[START_REF] Yao | Orbital angular momentum: origins, behavior and applications[END_REF]

  combining two HG modes of HG01 and HG10 with a π/2 phase difference between them, as shown in Figure I-7.

Figure I- 7 :

 7 Figure I-7: Combination of two HG modes of HG01 and HG10 to produce a helically phased LG01mode: normalized intensity plots (top) and corresponding phase profiles (bottom)[START_REF] Yao | Orbital angular momentum: origins, behavior and applications[END_REF]

Figure I- 8 :

 8 Figure I-8: View of the holograms for different topological charges: (a) l = 1; (b) l = 3.

Figure I- 9 ,

 9 when the reference wave output of a conventional laser passes through a 3-fold dislocation hologram illuminated at normal incidence, three main beams are observed at the outputs[START_REF] Padgett | Light's orbital angular momentum[END_REF]. Two of them are identical but propagate with opposite diffraction angles and topological charges of l = ±3. The third beam, not shown here, propagates in the extension of the reference wave and carries no OAM. The efficiency of this transformation is related to the level of sophistication in hologram design. For a binary hologram (see FigureI-8) which comprises only black and white areas with no grey scale, very little of the incident

Figure I- 9 :

 9 Figure I-9:A pair of helically phased waves with l = ±3 transformed from a plane wave using a hologram[START_REF] Padgett | Light's orbital angular momentum[END_REF]

  available and suitable for the working frequency of the SPP is more often used as the incident wave source (see Figure I-10). The simulated results of this OAM horn antenna are presented in Figure I-11.

Figure I- 10 :

 10 Figure I-10: Structure of the OAM horn antenna with spiral phase plate[START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF]

Figure I- 13 :I. 3 . 1 . 2

 13312 Figure I-12: (a) Unit cell of the proposed SPP; (b) Explosive view of the model including the SPP,connector and horn antenna[START_REF] Hui | Ultralow reflectivity spiral phase plate for generation of millimeter-wave OAM beam[END_REF]

Figure I- 14 :

 14 Figure I-14: Side view (left) and top view (right) of the flat drilled phase plate[START_REF] Bennis | Flat plate for OAM generation in the millimeter band[END_REF]

Figure I- 15 :

 15 Figure I-15: Variation of the refractive index as a function of the roll angle[START_REF] Bennis | Flat plate for OAM generation in the millimeter band[END_REF]

Figure I- 16 :

 16 Figure I-16: Magnitude (left) and phase (right) patterns generated by the OAM horn antenna with theflat drilled phase plate[START_REF] Niemiec | Etude des propriétés du moment angulaire orbital des ondes électromagnétiques : développement de capteurs, transfert de moment et applications[END_REF]

Figure I- 17 Figure

 17 Figure I-17: (a) Side view of an open-end waveguide fed flat phase plate antenna for OAM wave generation; (b) View of a unit cell in this design and the phase shift versus hole radius[START_REF] Cheng | Generation of electromagnetic waves with arbitrary Orbital Angular Momentum modes[END_REF]

Figure I- 20

 20 Figure I-20 presents the geometry of the hologram and the corresponding intensity distribution of the far-field diffraction pattern. The hologram is realized by etching the copper on a printed circuit board. As expected, three output beams are observed in Figure I-20b. The middle peak amplitude corresponds to the Gaussian beam with no OAM characteristic and the other two beams on either side of the Gaussian beam have an intensity profile which may correspond to an OAM bearing wave. Unfortunately, it is difficult to conclude the presence or absence of the OAM wave, because no study of the phase was presented, which is essential when one wants to define a wave carrying OAM.

  trimmed into a staircase-like structure with eight steps, where each one corresponds to a π/4 phase shift (see Figure I-21a). The measurements were performed at 2.4 GHz in a big anechoic chamber. The spatial maps (see Figure I-21b) of the magnitude and the phase of the electrical field show the existence of OAM spatial field distribution. Due to the discretization of the phase mask, the measured phase distribution is not good but remains visible.

Figure I- 21 :

 21 Figure I-21: Generation of radio OAM waves with a spiral reflector: (a) view of the realized reflector;(b) measured (top) and simulated (bottom) results[START_REF] Tamburini | Experimental verification of photon angular momentum and vorticity with radio techniques[END_REF]

4

 4 GHz frequency band, to induce a linear phase distribution along the roll angle. The elevations at different values of roll angle are shown in Table. 1. Figure I-22b shows the intensity map of the radio beam vortex at 40 meters in open space in the region harboring the singularity. The intensity distribution in this region presents some fluctuations caused by environmental interference effects. Although they do not show, the authors state that the phase pattern confirms the expected vortex.

Figure I- 22

 22 Figure I-22: (a) View of the twisted parabolic antenna; (b) intensity map in the far field[START_REF] Tamburini | Encoding many channels on the same frequency through radio vorticity: First experimental test[END_REF]

Figure I- 23 :

 23 Figure I-23: Configuration of a circular antenna array (N = 12, dipoles) for generating waves bearing OAM. The phase shift δϕ between two neighboring antennas is defined in (I-17) [5].

  Figure I-24 shows the far-field amplitude and phase patterns of the circular TSA for different OAM modes. It is observed that the amplitude pattern has a null at the center and the phase pattern has a linear variation along the roll angle with a 2πl phase shift in one turn. Moreover, the null size increases with the number of the OAM mode.

Figure I- 24 :

 24 Figure I-24: Far-field amplitude (left) and phase (right) patterns generated by a circular TSA for different OAM modes: (a) l = 1; (b) l = 2 [27]

Figure I- 25 :

 25 Figure I-25: Geometry of OAM circular phased array: (a) ideal model; (b) practical design; (c)prototypes[START_REF] Bai | Experimental circular phased array for generating OAM radio beams[END_REF]

Figure I- 29 :

 29 Figure I-29: Configuration of single horn antenna (left) and geometry of the horn antenna array(right)[START_REF] Bai | Generation of OAM radio waves with three polarizations using circular horn antenna array[END_REF]

Figure II- 1 :

 1 Figure II-1: 3D model of a single patch antenna

  Figure II-3, its 3D radiation pattern. We can see that it resonates at 2.5 GHz and its magnitude pattern maximum is in the boresight direction.

Figure II- 2 :Figure II- 4 :

 24 Figure II-2: Simulated reflection coefficient of a single patch antenna

Figure II- 6 : 8 Figure

 68 Figure II-6: Geometry of the phased patch array with different numbers of array elements: (a) N = 4; (b) N = 6; (c) N = 8

Figure II- 7 :

 7 Figure II-7: 2D phase (left) and magnitude (right) patterns of Ex component for OAM mode l = 1 with different numbers of array elements at 2.5 GHz: (a) N = 4; (b) N = 6; (c) N = 8.

Figure II- 9 Figure II- 9 :Figure

 99 Figure II-9 presents the 2D magnitude and phase patterns of Ex component of the generated wave for different OAM modes of l = 0, 1, 2, 3. The observation window is still a circular area with a radius of 300 mm, lying 60 mm above the patch antenna array. We can note that for l = 0, the phase and magnitude patterns correspond to a plane wave; for l = 1, 2, 3, the phase pattern has a linear variation along the roll angle with a 2πl phase shift in one turn and the magnitude pattern has a null at the center. This shows indeed that an OAM bearing wave is generated. Moreover, as l increases, the null size becomes larger and the generated OAM wave becomes less uniform and more disperse. The 3D radiation patterns (in linear value) are shown in Figure II-10. We can observe the characteristics similar to those of the E-field magnitude pattern.

(Figure II- 11 :

 11 Figure II-11: 3D and 2D radiation patterns for OAM mode l = 1 with different array radiuses at 2.5 GHz: (a) R = 0.4×λ; (b) R = 0.7×λ; (c) R = λ; (d) E-plane radiation pattern.

Figure II- 12 .

 12 The combined phase shifter and power divider is manufactured on a FR4 substrate with a thickness of 1.6 mm and a relative permittivity of 4.4. It works at 2.5 GHz and has 5 ports: port1 is the input and is matched to 50 Ω, while the others are used as outputs to feed the patch antennas located on a circle with a radius of 50 mm.The impedance of each part of the microstrip arcs is marked in Figure II-12. HFSS software is used to optimize the length and the width of the arcs to ensure the antenna matching, a uniform amplitude at the output ports and a 90° phase shift between two successive outputs.

Figure II- 12 :

 12 Figure II-12: Geometry of the phase shifter-power divider

Figure II- 13 :

 13 Figure Simulated parameters of the phase shifter-power divider: phase shifts (left) and transmission coefficients (right). At 2.5 GHz, the phase shifts of S21, S31, S41, S51 are -168°, -261°, -350°, -439°, respectively; and the corresponding amplitudes are -6.4 dB, -6.6 dB, -6.9 dB, -7 dB.

Figure II- 14 .

 14 Figure II-14: Configuration of the OAM antenna: (a) Top view; (b) 3D view

  Figure II-16).

Figure II- 15 :Figure II- 16 :II. 4 . 2 . 2

 1516422 Figure II-15: 2D magnitude (left) and phase (right) patterns of Ex component for the OAM antenna at 2.5 GHz

Figure II- 17 :

 17 Figure II-17: View of the realized phase shifter-power divider (left) and the OAM antenna (right)

Figure II- 18 :

 18 Figure II-18: View of the measurement setup

Figure II- 20 :

 20 Figure II-20: Simulated and measured 2D radiation and phase patterns of the OAM antenna at 2.5GHz: (a) Normalized magnitude patterns; (b) Normalized phase patterns (in degree).

Figure III- 1 :

 1 Figure III-1: Structure of the FP cavity (PRS: Partially Reflecting Surface)

Figure III- 2 :

 2 Figure III-2: Principle of the FP cavity

  L and D are respectively the lateral dimension and the height of the cavity.For a cylindrical cavity,

III. 3 Figure III- 5 :

 35 Figure III-4: (a) Geometry of the PRS made of metallic strips; (b) a unit cell of the PRS

Figure III- 6 :

 6 Figure III-6: Configuration of the 4-element circular phased patch array

Figure III- 8 :

 8 Figure III-8: Configuration of the OAM antenna inside the FP cavity

Figure III- 9 :

 9 Figure III-9: Maximum directivity of the OAM antenna versus the width of the metallic strips at the frequency of 2.5 GHz (period of strips: 20 mm)

Figure III- 11 :III. 3 . 3 . 3

 11333 Figure III-11: E-plane radiation patterns with the calculated and optimized cavity thicknesses at the frequency of 2.5 GHz (width of strips: 4 mm, period of strips: 20 mm)

  Figure III-12 presents the 3D radiation patterns of the patch antenna array (array radius equals to 110 mm) with and without FP cavity. We can observe a null in the center which is one of the characteristics of the OAM waves. In addition, with the use of the FP cavity, the antenna directivity is enhanced and the side lobes are significantly suppressed at the same time.

Figure III- 12 :

 12 Figure III-12: 3D radiation patterns of the OAM antenna (array radius equals to 110mm): (a) with FP cavity; (b) without FP cavity (width of strips: 4 mm, period of strips: 20 mm, cavity thickness: 58 mm, cavity aperture: 600 mm × 600 mm)

Figure III- 13 :

 13 Figure III-13: Comparison of E-plane radiation patterns of the OAM antenna with and without FP cavity (array radius equals to 110mm) (width of strips: 4 mm, period of strips: 20 mm, cavity thickness: 58 mm, cavity aperture: 600 mm × 600 mm)

works at 2 . 5

 25 GHz and has 5 ports: port1 is the input and is matched to 50 Ω, while the others are used as outputs to feed the patch antennas located on a circle. The impedance of each part of the microstrip arcs is marked in Figure III-14. HFSS software is used to optimize the length and the width of the arcs to ensure the antenna matching, a uniform amplitude at the output ports and a 90° phase shift between two successive outputs.

Figure III- 14 :

 14 Figure III-14: Geometry of the phase shifter-power divider, feeding the four patches

Figure III- 16 :

 16 Figure III-16: Configuration of the FP OAM antenna with phase shifter-power divider

Figure III- 18 :

 18 Figure III-18: Radiation and phase patterns of the FP OAM antenna at 2.5 GHz: (a) 3D radiation pattern; (b) 2D radiation patterns in E and H planes; (c) Normalized phase patterns (in radians) at different θ angles

Figure III- 19 :

 19 Figure III-19: Normalized magnitude (left) and phase (right) patterns of Ex component for the FP OAM antenna at 2.5 GHz (circular observation window with a radius of 300 mm, lying 30 mm above the PRS)

Figure III- 20 :

 20 Figure III-20: (a) Geometry of the PRS made of metallic tubes; (b) a unit cell of the PRS

Figure III- 21 :

 21 Figure III-21: Reflection coefficient of the PRS versus the normalized period (e/λ) for different values of δ at 2.5 GHz: (a) Magnitude; (b) Phase.

Figure

  Figure III-22: Configuration of the ideal model of the second FP OAM antenna with the PRS made of metallic tubes

Figure III- 23 :

 23 Figure III-23: Comparison of E and H plane radiation patterns of the FP OAM antenna for three different cases: no walls, 5-mm-thick wall and 15-mm-thick wall

Figure III- 25 :

 25 Figure III-25: Top view of the FP cavity for showing the material of the walls

Figure III- 26

 26 Figure III-26 presents the E and H plane radiation patterns of FP OAM antenna with 2

Figure III- 27 :

 27 Figure III-27: Configuration of the practical design of the second FP OAM antenna

Figure III- 28 Figure III- 28 :

 2828 Figure III-28 shows the 3D radiation pattern of the FP OAM antenna at 2.5 GHz. We can observe a null in the center that is characteristic of OAM waves. The magnitude and phase patterns of the Ex component of the propagated wave, plotted on a circular area with a radius of 300 mm, lying 30 mm above the PRS, are presented in Figure III-29. It can be seen that the magnitude at the center is much smaller than the surroundings and the phase rotates around the center with a 2π phase shift in one turn. This shows indeed that both the amplitude and the phase correspond to an OAM bearing wave with l = 1.

Figure III- 30 :

 30 Figure III-30: Comparison of E-plane radiation patterns of the FP OAM antenna with different array radiuses

Figure III- 31 :

 31 Figure III-31: Realized prototype of the FP OAM antenna (array radius equals to 60 mm): (a) Front view; (b) Back view.

Figure III- 32 :

 32 Figure III-32: Simulated and measured return losses of the FP OAM antenna (array radius equals to 60 mm)

  walls in E-plane for mechanical balance and decreasing the side lobe level of the radiation pattern. The FP OAM antenna finally achieved a directivity of 16.2 dB with an enhancement of 9.2 dB compared to the phased patch array alone.

Figure IV- 1 :

 1 Figure IV-1: Some examples of waveguides[START_REF] Sadiku | Waveguides[END_REF]

  Figure IV-2 presents the simulated propagation constants versus frequency for the first three modes of a circular waveguide. We can see that the simulated cut-off frequencies, where the propagation constant becomes 0, have a good agreement with the theoretical values.

Figure IV- 2 :

 2 Figure IV-2: Simulated propagation constants versus frequency for the first three modes (TE11, TM01 and TE21 modes) of a circular waveguide with a diameter of 4.4 cm

Figure IV- 3 :

 3 Figure IV-3: Geometry of the waveguide for generating OAM wave, using TE11 mode with a single SPP (Spiral Phase Plate) (PML: Perfectly Matched Layer)

Figure IV- 4 :

 4 Figure IV-4: Field distribution of TE11 mode and the corresponding magnitude and phase patterns of Ex component: (a) Field distribution; (b) Normalized magnitude pattern; (c) Phase pattern.

Figure IV- 5 :IV. 3 . 1 . 2

 5312 Figure IV-5: Normalized magnitude (left) and phase (right) patterns of Ex component for the structure with a single SPP at the frequency of 40 GHz

Figure IV- 6 :Figure IV- 7 :

 67 Figure IV-6: Geometry of the OAM horn antenna with a single SPP (SPP inside the waveguide)

Figure IV- 9 :Figure IV- 10 :Figure IV- 11 :

 91011 Figure IV-9: Geometry of the OAM horn antenna with a single SPP (SPP inside the horn): (a) 3D view of the antenna structure; (b) 2D view for showing the probe parameters

Figure IV- 12 :

 12 Figure IV-12: Normalized magnitude (left) and phase (right) patterns of Ex component for the OAM horn antenna with a single SPP at the frequency of 20 GHz (observation window: circular area with a radius of 5 cm, lying 2 cm above the horn aperture)

Figure IV- 13 :

 13 Figure IV-13: Geometry of the waveguide using TM01 mode, with two half-turn SPPs (PML: Perfectly Matched Layer)

Figure IV- 14 :

 14 Figure IV-14: Field distribution of TM01 mode and the corresponding magnitude and phase patterns of Ex component: (a) Field distribution; (b) Normalized magnitude pattern; (c) Phase pattern.

= 3 .

 3 6 cm. The magnitude and phase patterns of the Ex component for the wave at the end of the SPPs are given in Figure IV-15.We can observe that the magnitude pattern has a null at the center and the phase pattern has a linear variation along the roll angle  with a 2π phase shift in one turn. This shows indeed that an OAM bearing wave with l = 1 is generated. We can also note that the results are better than those of the first structure.

Figure IV- 15 :

 15 Figure IV-15: Normalized magnitude (left) and phase (right) patterns of Ex component for the structure with two half-turn SPPs at the frequency of 7.4 GHz (based on TM01 mode)

Figure IV- 16 :

 16 Figure IV-16: Geometry of the OAM horn antenna with two half-turn SPPs, based on TM01 mode: (a) 3D view of the antenna structure; (b) 2D view for showing the probe parameters

Figure IV- 17 :Figure IV- 18 :

 1718 Figure IV-17: Reflection coefficient of the OAM horn antenna with two half-turn SPPs (based on TM01 mode)

Figure IV- 19 :

 19 Figure IV-19: Normalized magnitude (left) and phase (right) patterns of Ex component for the OAM horn antenna with two half-turn SPPs at the frequency of 7.4 GHz (based on TM01 mode) (observation window: circular area with a radius of 7 cm, lying 1 cm above the horn aperture)

Figure IV- 20 :

 20 Figure IV-20: Geometry of the waveguide with two half-turn SPPs, using TE21 mode (PML: PerfectlyMatched Layer)

Figure IV- 21 :

 21 Figure IV-21: Field distribution of the 1 st kind of TE21 mode and the corresponding magnitude and phase patterns of Ex component: (a) Field distribution; (b) Normalized magnitude pattern; (c) Phase pattern.

Figure IV- 22 :

 22 Figure IV-22: Normalized magnitude (left) and phase (right) patterns of Ex component for the structure with two half-turn SPPs at the frequency of 10 GHz ((based on TE21 mode)

Figure IV- 23 :

 23 Figure IV-23: Geometry of the OAM horn antenna with two half-turn SPPs (based on TE21 mode)

Figure IV- 24 :

 24 Figure IV-24: 3D Radiation pattern of the OAM horn antenna with two half-turn SPPs at the frequency of 10 GHz (based on TE21 mode)

  Figure IV-26, the 2 nd kind of TE21 mode. One can see that the E-field patterns of the 2 nd kind of TE21 mode have quite a similar distribution compared to the 1 st kind of TE21 mode. The main difference is just a π/2 rotation of both magnitude and phase patterns. Moreover, due to the symmetrical distribution of the electrical field, for each kind of TE21 mode the magnitude and phase patterns of Ey component have an identical distribution compared to the Ex component, the only difference is that both the magnitude and phase patterns rotate π/2 (see Figures IV-27

  and IV-28).

Figure IV- 26 :

 26 Figure IV-26: Field distribution of the 2 nd kind of TE21 mode and the corresponding magnitude and phase patterns of Ex component: (a) Field distribution; (b) Normalized magnitude pattern; (c) Phase pattern.

Figure IV- 27 :l = 1 Figure IV- 28 :

 27128 Figure IV-27: Evolutions of magnitude (top) and phase (bottom) patterns of Ex component for the generation of an OAM wave, by combining two kinds of TE21 modes with a π/2 phase difference

Figure IV- 29 :

 29 Figure IV-29: Magnitude (left) and phase (right) patterns of the combined TM01 and TE21 modes: (a) Ex component; (b) Ey component

Figure IV- 30 : 12 andFigure IV- 31 :

 301231 Figure IV-30: Configuration of the proposed OAM antenna, using the combination of TM01 and TE21 modes

Figure

  Figure IV-32: Magnitude (left) and phase (right) patterns of the propagated wave: (a) Ex component; (b) Ey component (observation window: circular area with a radius of 3 cm, lying 1 cm above the waveguide aperture)

Figure IV- 33

 33 Figure IV-33 shows the 3D radiation and phase patterns of the OAM antenna. For each component, the radiation patterns in magnitude and phase depict the vortex that is characteristic of an OAM wave. The spiral form of the phase pattern, plotted in polar coordinates, corresponding to the linear variation of the phase against the roll angle (φ, in Figure IV-33b).

Figure IV- 33 :

 33 Figure IV-33: 3D radiation (left) and phase (right) patterns of the OAM antenna: (a) Ex component; (b) Ey component

IV.1 Introduction ............................................................................................................ IV.2 Circular waveguide................................................................................................. IV.3 Generation of OAM waves by using spiral phase plate ......................................

  SPP in waveguide ................................................................................. IV.3.2.2 Horn antenna design ............................................................................. IV.3.3 TE21 mode and two half-turn SPPs ................................................................. IV.3.3.1 SPP in waveguide ................................................................................. IV.3.3.2 Horn antenna design .............................................................................

	ideal model and the practical design of the proposed antennas are presented and characterized
	based on simulations.
	IV.3.1 TE11 mode and a single SPP ...........................................................................
	IV.3.1.1 SPP in waveguide .................................................................................
	IV.3.1.2 Horn antenna design .............................................................................
	IV.3.2 TM01 mode and two half-turn SPPs ...............................................................
	IV.3.2.1
	The field distributions and the corresponding field patterns
	(magnitude and phase) of the utilized guided modes are given and analyzed. Besides, both the

IV.4 Generation of OAM waves by combining guided modes .................................... IV

  .4.1 Combination of two kinds of TE21 modes ...................................................... IV.4.2 Combination of TM01 and TE21 modes ..........................................................

IV.5 Conclusions ............................................................................................................. Conclusions and Perspectives .......................................................................... References ......................................................................................................... Bibliography of the author ..............................................................................

	I
	Chapter I
	Orbital Angular Momentum of
	Electromagnetic Waves

.1 What is the Orbital Angular Momentum?

Table I -

 I

	0 = 2π	1/2	6.25
	π/2	3/8	4.69
	π	1/4	3.12
	3/2 π	1/8	1.56

1: Elevations for given values of the roll angle for deforming the parabolic antenna [6]

Roll angle

Elevation (in units of λ) Elevation (cm)

Table III - 1 :

 III1 Magnitudes and phases of the PRS reflection coefficient and the corresponding cavity thickness for different widths of the metallic strips (period of strips: 20 mm)

	Metallic strips
			FP cavity
	D
	Ground plane
	Width (a) of	Magnitude of S11	Phase of S11 of PRS	Cavity thickness
	metallic strips (mm)	of PRS	(in unit of π)	(mm)
	1	0.75	0.77	55
	2	0.88	0.84	57.1
	3	0.92	0.88	58.4
	4	0.94	0.89	58.7
	5	0.97	0.92	59.6
	6	0.98	0.94	60.3

Table III

 III

	-2. We can observe that the E-plane maximum directivity of the OAM antenna changes
	slightly with the cavity thickness, while the angle value corresponding to the maximum
	directivity maintains at 11°; the OAM antenna achieves a best directivity of	16.2 dB with a

cavity thickness of 58 mm, which is the final value we have used.

Table III -

 III 2: Maximum directivity value and position of the OAM antenna for different cavity thicknesses (width of strips: 4 mm, period of strips: 20 mm)

	Cavity thickness (mm) Max directivity (dB)	Position (°)
	59	15	11
	58.7 (calculated value)	15.6	11
	58.5	15.9	11
	58	16.2	11
	57.5	16	11

Table III -

 III 3: Maximum directivity value and position of the OAM antenna for different dimensions of the cavity aperture (width of strips: 4 mm, period of strips: 20 mm, cavity thickness: 58 mm)

	Dimension of cavity aperture (mm×mm)	Max directivity (dB)	Position (°)
	360 × 360	13.9	15
	480 × 480	14.3	14
	600 × 600	16.2	11
	720 × 720	17	10

Acknowledgements

First and foremost I would like to express my deepest gratitude to my supervisor Prof. Kouroch Mahdjoubi for providing me the chance to study at IETR, University of Rennes 1, and

Abstract

It is well known from Maxwell's theory that electromagnetic (EM) radiation carries both linear momentum (energy) and angular momentum. The latter has two parts: Spin Angular Momentum (SAM) which corresponds to the polarization of an EM wave and Orbital Angular Momentum (OAM) which is associated with the spatial distribution of an EM wave. The SAM has only two states (left and right) and is used to double the channel capacity in telecommunications. On the other hand, the OAM can theoretically have an infinite number of states called the OAM modes. Therefore, the first applications of OAM have been proposed in wireless communications at radio frequencies. However, first of all, it is necessary to develop the antennas for generating such waves.

The objective of this thesis is to design the antennas for the generation of radio waves bearing OAM. The manuscript contains three parts. In the first part, an antenna using 4 patches and an original phase shifter is developed and tested to generate an OAM wave. In the second part, a Fabry-Perot (FP) cavity is used to enhance the directivity of this antenna. The third part is to generate guided OAM waves. Some horn antennas are used to radiate these waves with good directivity.

Key words: orbital angular momentum, patch antenna, Fabry-Perot cavity, horn antenna, waveguide antenna, spiral phase plate.