An analysis of the vector wave equation was conducted. A class of self-similar solutions with inhomogeneous polarization corresponding to the resonator modes is deduced. The modes with inhomogeneous polarization can be selected by a diffraction element with polarization selectivity used as one of the resonator mirrors. Diffraction elements with high polarization selectivity of about 100% are necessary for generating `pure' radially polarized modes. The radially polarized beam provides a higher energy efficiency (the product of the depth of the cut by the cutting velocity) for laser cutting metals than the circularly polarized main mode does under the same conditions. The two limiting cases of resonance absorption on the spherical plasma target could be realized using axially polarized beams: the resonance absorption is maximum in the case of radial polarization and does not occur in the case of azimuthal polarization.