The present thesis deals with the design of a planar antenna array for the communication between a civil aircraft (Dassault's Falcon) and an Inmarsat satellite. The final goal is to provide high speed Internet access in X/Ku-Band (8-12GHz / 12-18GHz), and possibly even in K/Ka- Band (18-27GHz / 17-40GHz). The study starts with the Ku-Band only, then evolves towards the transmission band and finishes with their Ka-Band counterparts. The main constraint of the thesis is that the final result should be a low profile antenna that will be easy to integrate below the fuselage of an aircraft. At the same time, the antenna has to provide a beam steering mechanism in order to track the position of the targetted satellite while the aircraft is moving. The usual way to achieve it is to implement some dedicated electronic chips below each radiating element of array. Dassault Aviation has its own alternate and more mechanical oriented solutions. Since the communication system is intended for top-notch business jets produced in small series, cost is not here a decisive criterion. This thesis assumes the existence of the beam-steering system and concentrates on a proof of concept and careful design of the antenna elements that will constitute the final array. They have to satisfy tight constraints in terms of size, return loss, mutual coupling with neighbors, efficiency, radiation pattern and dual circular polarization quality. Printed multilayer antennas have been retained as the best candidates, and themain parts of the thesis have been dedicated to their study. The thesis concludes with the presentation of an apparently completely unrelated and independant topic, namely the simulation of nanoscale 2D planar structures based on Graphene. The rationale for this last chapter is to provide a prospective study of the use of Graphene to create a variable capacitance. This component is always needed in the beamforming network of reconfigurable and scanning arrays. Our laboratory is developing a deep knowledge of the electromagnetic properties of Graphene-based devices and these varicaps could lead to very interesting applications in high frequencies, once the Graphene technology has definitely progressed. Thanks to electrostatics Green's functions, a variable capacitor based on Graphene has been simulated, and a Quantumeffect -the Quantum capacitance of Graphene- has been successfully tackled from a numerical point of view.