Demonstrated by the James-Webb Space Telescope launch, the scientific community shows more and more interest in mid-infrared observations. The diffraction limit of an instrument is inversely proportional to the diameter of the telescope: the larger the primary mirror, the higher the resolution. Therefore, the instrument METIS (Mid-Infrared ELT-based Imager and Spectrograph), based on the 37-meter aperture Extremely Large Telescope, will take advantage of the giant mirror to achieve an unrivaled resolution in mid-infrared. However, to reach this limit, the perturbations created by the temperature fluctuations in the atmosphere need to be removed by Adaptive Optics (AO). METIS will already have an AO system, called Single Conjugate Adaptive Optics (SCAO). However, this system will require a Natural Guide Star (NGS) with a K-band magnitude ofmK Æ 11. This limits considerably the sky coverage where the atmospheric compensation (thus high-resolution observations) could be done. The idea is to use an artificial guide star, called Laser Guide Star (LGS), to get rid of this constraint. This is performed by shooting a laser at the 90-km sodium layer, tuned so that it excites the atoms there that re-emits toward the ground. For a telescope with the size of the ELT, several lasers are recommended in a configuration called Laser Tomography AO or Multi-Conjugate AO. The plurality of the lasers allows removing errors such as the focus anisoplanatism. However, because of the complexity and cost of such a solution, the only available answer for the ELT/METIS is a Single Laser AO system (SLAO). This will be less complex and expensive at the expense of a decrease in the performances. The question is: what are the performances of such a SLAO system, and how could it be made? The Yorick Adaptive Optics (YAO) simulator has been used to estimate the performances of the SLAO system for the ELT/METIS. The elongation of the LGS beam, created by the finite thickness of the sodiumlayer, plays a crucial role in the process and has direct consequences on the performances. A parameter analysis has been performed to define the most suitable Field of View, number of sub-apertures, pixels size to tackle the problem. An algorithmhas also been developed inMATLAB, coupledwith the results fromthe YAOalgorithm. The goal of this algorithm was to estimate the correction that could be brought by the so-called truth sensing on the features left by the LGS correction because of the beam elongation. This sensing will require a natural guide star coupled with the existing SCAO sensor of the METIS instrument and will work in addition to the SLAO sensor. Once the parameters settled, it has been possible to create an optical design with ZEMAX. Because the distance between the telescope and the sodium layer varies relatively to the zenith angle, the focal point of the LGS beamchanges accordingly: the optical system needs to tackle this. It was also required to find a solution to separate the LGS beam from the science beam: two solutions have been described, with their advantages and limitations detailed. Finally, a mechanical implementation of the systemhas been made to explain how such a system could be hanged. The work performed has allowed estimating the Strehl ratio obtained for the SLAOsystemwith the ELT/METIS, which would be around 60%. These results have been confirmed by the European Southern Observatory and their AO experts. The study made on the truth sensing of the system has allowed putting constraints on the natural guide star brightness required (mK Ç 16.5), and the sky coverage of the SLAO system which would be almost 100%. The optical design made with ZEMAX has shown great performances so that the perturbations induced by the system are small enough to be able to measure correctly the ones created by the atmosphere. The solution proposed with the mechanical design could pave theway for further study in the domain. The Adaptive Optics with a single laser guide star is complicated, especially for a telescope with the size of the ELT. Using a laser guide star instead of a natural one creates additional complications that need to be solved to achieve good performances, especially if only one LGS can be used. The solution proposed would allow to considerably increase the sky coverage of the system, which is currently limited to few percents with the SCAO system. However, further studies need to be done, such as the evaluation of the real impact of the NGS magnitude on the system performances, or the design of the complete and realistic mechanical design.