Abstract: Annular gain media offer various advantages to the design of lasers. The advantages comprise efficient laser excitation by internal pumping sources for solid state lasers and large discharge surfaces for diffusion-cooling of high power CO$-2$/-lasers. To benefit from these advantages it is necessary to find a suitable resonator. The presented solution to the resonator problem consists of a hybrid resonator which is stable in radial and unstable in azimuthal direction. Basically, it can be considered as the annular equivalent of the well-known stable-unstable slab resonator. A geometrical model of the resonator indicates that for proper resonator function the mirror shapes must respect certain constraints. A diffraction model of the loaded resonator gives detailed information on the beam properties and extraction efficiencies. To reduce the computational effort for beam propagation an approximate diffraction kernel for annular beams has been used. Experimental data on a 1 kW diffusion-cooled CO$-2$/-laser are reported. !21
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