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Guidelines are provided for designing multishell Luneburg lenses (LL) whatever the primary source is. Three different optimization problems that minimize three different norms of the discrepancy between the ideal relative permittivity and the reconstructed one with respect to the thickness and permittivity of each shell are detailed. The performances of the three so-optimized LL are compared by simulations with those of previously proposed approaches. The highest directivity is obtained for the minmax norm. An investigation of the influence of the number of shells on the lens antenna performances, directivity, sidelobe levels and aperture efficiencies for various lens diameters, is also proposed in order to better know how to choose these parameters. Finally, measurements done with a multishell Luneburg lens fed by a waveguide are compared with simulations to show, through excellent agreements between theory and measurements, the validity of this study.

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