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We present a new scheme for an optical resonator for production of Bessel and Bessel-Gauss light beams. The resonator with Bessel modes is composed of two plane mirrors with an axicon placed close to one of them. If this mirror is concave, the modes are Bessel-Gauss light beams. Analytical expressions relating parameters of the resonator and characteristics of its modes are obtained and analyzed. The results are verified with the Fox-Li algorithm. The resonator scheme was implemented in an experiment to confirm the possibility of the generation of zero-order Bessel beams. It was found that multipass modes can also oscillate in the resonator if its apertures are large enough.

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