Abstract: 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.

Key words: Guassian spherical wave, paraxial wave equations, confocal unstable cavity