We use the residue of the map to study the stability and Gaussian beam dynamics of a general optical resonator. The map is constructed from space domain matrix formalism. For a lossless system we find that the residue is a function of resonator g-parameters' product and new critical stable curves exist within the geometrically stable regions. The dynamic behavior of a dissipative resonator with loss optical element is similar to a damping oscillatory motion and can be determined by the corresponding conservative resonator without those loss elements.
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