Generation of polarisation-nonuniform modes in a high-power $\rm{CO}_2$ laser

A method describing radially and azimuthally polarised laser modes is presented, which is devoid of intrinsic contradictions and unjustified restrictions. The solutions of the wave equation found in the paper satisfy Maxwell's equation $\nabla E=0$. This allowed the calculation of all the components of the fields of such modes, including the longitudinal one. Expressions for tight focusing of radially and azimuthally polarised modes are obtained in the Debye approximation. The methods of intracavity generation of such modes in a high-power industrial $CO_2$ laser are experimentally realised and methods of mutual transformation of these modes outside the cavity are developed. The intracavity generation of azimuthally polarised modes was achieved by using as a rear highly reflecting mirror either $V$-shaped axicons or metal diffraction mirrors with the relief period comparable to the wavelength. Radially polarised radiation was generated by using diffraction mirrors with the relief period of the order of two wavelengths. The polarisation-nonuniform $TEM_{01}^*$- and $TEM_{11}^*$ modes of power up to 1.8 kW are generated for the first time. In this case, the degree of polarisation was close to 100% within the entire radiation power range both in the near- and far-field zones. By using two half-wave phase shifters, the azimuthally polarised radiation is transformed to radially polarised radiation and vice versa, the energy efficiency of this transformation being 92%. These results open up the possibility for industrial applications of such radiation.