M.A. Ogar, V.A. Feofilaktov

The article describes principles of the three-dimensional mathematical model of optical resonators with the possibility of accounting "slanting" beams, which has a big importance in the problem of calculating the multipass schemes currently used in high-flow carbon monoxide lasers. We analyzed a lot of articles in this problem. In many of these articles there were proposed the methods of calculating in three-dimensional approximation. However, not all of them provide the possibility of accounting "slanting" beams. Several articles have proposed the method of lens-transformation to account the "slanting" beams. But for calculating some schemes with this method, the step along optical axis must be sufficiently small, which leads to a significant increase in computation time and the required computer memory. At the beginning of the article, the method of solution of wave equation is described. Because of the rather large wave front curvature of the spherical wave in the unstable resonator, different coordinate systems are chosen to define the plane and spherical waves. For a plane wave it was chosen the rectangular coordinate system, and for a spherical wave - the modified spherical coordinate system. In both coordinate systems proportional three-dimensional grids are built. On the basis of these grids the difference scheme is built. The distributions of complex amplitudes are set at evenly spaced reference surfaces. To define the active medium the volumetric distributions of saturated gain coefficient, refractive index and total power density (represented by cubic matrices), the saturation intensity, and the coefficient of non-selective losses are used. In-between local parameters of the active medium are calculated using trilinear interpolation. Then the method of calculation is expounded and the program block-scheme is shortly described. The main differences between this program and comparable ones are described, (for example, the implementation of two independent methods for increasing the speed of calculation convergence). Finally, the example of the calculation of multipass unstable resonator with illustrations is shown. Present program have great opportunities, as it allows to simulate and to study the operation of the resonators with complicated configurations, including various schemes «control oscillator + amplifier». It allows us to find the amplitude and the phase distribution of the electromagnetic field with a good accuracy in the cavity of multipass unstable resonators. The results of these calculations are used for determination of the power, the far-field distribution of complex amplitude, the angular divergence and the radiation intensity of the output.