The mathematical modeling of the laser beam propagation in the unstable resonator with the mirrors of the arbitrary form in non-orthogonal coordinate systems

The mathematical modeling of the laser beam propagation in the unstable resonator with the mirrors of the arbitrary form is represented in the article. The method considered is based on using the non-orthogonal coordinate systems, which allows the adequate reduction of full wave boundary problem to the system of the evolution Shrödinger's equations. The features of the method described are neglecting from the Siegman-Siclash transformations. As a result it doesn't need to interpolate the field near the back mirror on the detailed grid, that is necessary generally for the solution of the equation, describing the radiation propagation after the field reflection from the back mirror. The spectral methods and iterations have been used for the numerical solution of the boundary problems. The telescope resonator with the cylindrical mirrors has demonstrated the features of the unstable resonator calculations when the active medium has been saturated.