Abstract: In this paper, the analytical expressions of selfreproducing modes in general stable and unstable spherical mirror resonators are derived by Fresnel-Kirchhoff diffraction integral method. The law of self-reproducing modes changing from stable resonator to unstable resonator is revealed. Furthermore, the self-reproducing mode in the unstable resonator is also obtained under the condition of geometric optics approximation. By comparing with the diffraction integral results, it is found that the spatial distributions of the lowest-order even symmetric mode (i.e., fundamental mode) and the lowest-order odd symmetric mode of the unstable resonator obtained based on geometric optics approximation are consistent with the diffraction integral results. For higher-order self-reproducing modes, when the unstable resonator is close to the stable resonator, there is a large difference between the results from the geometric optics approximation and the diffraction integral. The results in this paper can be used as an important supplement to the teaching content of laser principle.

Key words:  laser principle, self-reproduding mode, diffraction integral method, geometrical optics approximation.