A theoretical analysis of the laser resonator with a rectangular hollow dielectric waveguide and external reflector as well as some numerical results of that are presented in this paper. Special features of this method are as follows: 1. A complicated propagation process has been represented by transformation matrix. Thus

analysis of the lowest order mode of the complicated resonator is reduced to a problem solving for maximum eigenvalue from a complex matrix equation; 2. An approximate multi-mode theory is employed to describe more completely and accurately the mode construction

propagation characteristic of the cavity mode as well as the effects of higher order mode; 3. According to the diffraction principle and the theory of ray optics

a simple equation of the equivalent system is derived to replace the iterative diffraction integral to describe the propagation of the radiation field in free space outside the waveguide

so that the computation is much simplefied.More than ten figures are shown as a result of numerical computation for some typical resonators of CO2 waveguide laser and reflect the relations between the resonator geometry and mode characteristic from various aspects. The loss property of the fundamental mode in a matched resonator is identical with reference [3]

and the characteristic of the cavity mode of an ideally matched resonator approaches that of the fundamental mode

and both are identical with the result from the single-mode theory. Emphasis is laid on the following subjects

such as: the effect of the light spot size at the waveguide entrance on the coupling efficiency; the dependence of mode properties on parameters of the spherical mirror; characteristics of the lowest loss mode composition and the mode selectivity; the behavior of the near field characteristic with the flat mirror system; the influence of the guide geometry and the optical constant of dielectric materials on the loss properties of resonator modes

ect. Some new ideas are suggested and some useful conclusions are drawn which are helpful in the research and development of gas waveguide laser.