| The output beam quality of the laser,which is determined mainly by the laser mode in the resonator,decides the characteristics of the transmission and focusing operation. The calculation of the laser mode shows its importance in the applications of the laser machining and optical measurements. It provides the basis for the design of the resonators, external optical path and beam shaping system. The mode solutions of the typical resonators, such as the plane parallel resonator and the confocal spherical resonator, were given by analytic theories. But the relevant calculations of novel resonators and newly developed optical components show the limit of this method. Numerical methods, including Hermite-Gaussian expansion, Fast Fourier Transform (FFT) method, Finite Difference Method (FDM) and Finite Element Method (FEM), achieve in solving above problems. In particular, the FEM shows its advantages of good flexibility and high efficiency over the other methods. It has been widely used in fiber optics as well as in mode calculation of laser resonator with resonators without media in it so far. A numerical matrix method (NMM) for mode calculation based on FEM is proposed in this paper. It can describe the series of eigen modes in the resonator with good adaption to various mirror shapes. The mode calculations of the spherical confocal resonator is performed by using the NMM. The results show the advantages by compared with the traditional methods. Then the method is also exploited to analyze the confocal unstable resonator without and with media inside. The results are contrasted with that of another article to prove its effectiveness. No one has use this method in mode calculations of resonator with media inside. This shows this method can be used to calculate modes of resonator in practical research and real applications. As the implementation of the algorithms,a software for the resonator mode calculations is developed, which is introduced in the last part of this article. |
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