We develop a round-trip matrix diagonalization method for quantitative description of selection of radially or azimuthally polarized beams by birefringence-induced bifocusing in a simple laser resonator. We employ different focusing between radially and tangentially polarized light in thermally stressed laser rods to obtain low-loss stable oscillation in a radially polarized Laguerre–Gaussian, \mathrm{LG}\left(0,1\right)* , mode. We derive a free-space propagator for the radially and azimuthally polarized \mathrm{LG}\left(0,1\right)* modes and explain basic principles of mode selection by use of a round-trip matrix diagonalization method. Within this method we calculate round-trip diffraction losses and intensity distributions for the lowest-loss transverse modes. We show that, for the considered laser configuration, the round-trip loss obtained for the radially polarized \mathrm{LG}\left(0,1\right)* mode is significantly smaller than that of the azimuthally polarized mode. Our experimental results, obtained with a diode side-pumped Nd:YAG rod in a flat–convex resonator, confirm the theoretical predictions. We achieved a pure radially polarized \mathrm{LG}\left(0,1\right)* beam with M^2=2.5 and tens of watts of output power.