We present a detailed study of the unstable Bessel resonator. The cavity of this laser consists of a reflective axicon and a convex spherical output coupler. A matrix method analysis for the bare resonator is employed to extract the eigenmodes and eigenvalues of the cavity, which allows us to obtain the fundamental and higher-order modes. Diffractive losses and relative phase shift behavior, in terms of both the varying radius of curvature of the output coupler and the cavity length, are studied with the matrix method and the Fox-Li algorithm. Direct comparison of the transverse mode profiles with a similar resonator employing a concave output coupler is performed, showing excellent agreement for large values of the radius of curvature. We also considered the effects of varying the aperture of the output coupler and the wedge angle of the axicon on the transverse mode profiles and diffractive losses.