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This thesis is a study of a new kind of Fabry-Perot long-baseline optical resonator proposed for gravitational waves interferometric detectors: the Mesa beam profile Fabry-Perot cavity. A detailed experimental work has been performed on a prototype built in the LIGO laboratories of the California Institute of Technology. The aim of this experiment is to explore all of the main properties of such an optical cavity including: the reliability of its control; beam distortions due to surface imperfections and misalignments; the efficiency of the coupling to a gaussian input beam. The advantage to use a non-spherical optics resonator consists in the possibility to significantly reduce test masses thermal noise which is expected to be the fundamental limit for advanced gravitational waves interferometric detectors, such as Advanced LIGO. It is possible to generate a flat-top, wider laser beam probe of such a detector just reshaping the profile of its mirrors. The characteristic Mexican hat' mirror profile has been designed to support a flat-top beam, the so-called Mesa beam. Three test mirrors have been manufactured by the LMA laboratory (Lyon, France) in order to study the behavior of this new family of laser beams in our Fabry Perot prototype. A brief introduction about the problem of estimating mirror thermal noises, their physical sources and how to treat them formally, using, for example, Levin's direct approach is presented. After that, this thesis deals with all the physical characteristic of finite size optical resonators and the related experimental issues like the lock acquisition techniques. A overview of the Mesa beam design and related theoretical issues is included. The second part of this thesis treats all the experimental issues of our experiment: a Fabry-Perot, folded, suspended optical resonator 7 meters long conceived to store Nd:YAG laser light with optics sizes scaled down from the Advanced LIGO baseline parameters. It is placed inside a vacuum pipe and the spacing between the mirrors is determined by three INVAR rods. A cavity finesse of about 100 is achieved by tuning the reflectivity of the input (flat) mirror. The other two mirrors, the folding mirror, and the end mirror, which can be either a spherical or a Mexican hat mirror, have high reflectivity coatings. During this year of operations, the cavity was always operated in air. The stability of the mechanics has been tested with a spherical mirror, 8 meters radius of curvature. The input and output optics layout has been developed to match and study this preliminary cavity configuration. The control electronics, necessary to keep the cavity locked on a resonance, has been made and assembled so that it is possible to use either a side-lock feedback on the laser frequency or the cavity length dithering technique. A high voltage driver circuit has been designed and assembled to drive the mirror piezo actuators. A preliminary study of the cavity beam profile was performed for the spherical optics configuration: several beam profile samples of the fundamental mode and higher order modes were compared to the theoretical predictions. The last part of this work was the characterization of the cavity behavior with the first Mexican hat mirror. Since spherical symmetry is lost for such a resonator, the resonant beam depends on the particular mirrors and input beam alignment. Beam profiles recorded were compared with realistic simulations based on the Fast Fourier Transform implementation of the beam propagation and using the actual Mexican hat mirror map. Finally, the first Mesa beam fundamental mode was acquired and analyzed.