Fresnel integral equations of the transverse multimode are solved by the numerical integration method in the cylinder coordinate. Under the initial conditions of the uniform and random distribution given respectively for the complex light-field amplitude, TEM_n0 modes of the confocal and stable cavities with circular mirrors of spherical surfaces as well as the plane cavity with circular mirrors are computed and analized. Computational results indicate that the numerical results of single mode for spherical cavity coincide with the analytical results. The results of the multimode not only conform to the behavior of the multimode space distribution, but also show that those of the fundamental mode dominate.

The advantages and disadvantages of the 3D paraxial wave equations in the confocal unstable cavity solved by the difference methods and 3D Fresnel integral equations solved by the numerical integral method and the fast Fourier transform (FFT) method are analyzed. It is assumed that the laser beam in the unstable cavity transfers by the modified Guassian spherical wave. By using a coordinate transformation, the diverging or converging beam is converted into equivalent collimated beam. The paraxial wave equation of the modified function is solved by FFT method. Under the same conditions, the obtained results coincide with those in the internal and external references. The near field and far field results computed in confocal unstable cavity of the COIL experiment are reasonable.

In this paper, we first analyse by the geometric optics method the relation of angular sensitivity and deviation distance sensitivity with the magnification m and tilt angle β respectively. Then the changes of near-field phase and far-field intensity, which were led to by the tilt of a single mirror and two mirrors with different deviated direction respectively, are computed and analysed by C-6 code. The computational results indicate that the tilt of mirrors really lead to the near field phase changing from uniform to nonuniform distribution and the laser beam quality from high to low.