Abstract: Generally, the ABCD law for the complex parameter q of the TEM$-00$/ Gaussian beam is not valid for high-order modes. The focussing of a Gaussian beam means imaging of the beam waist. Under this condition, the spot size in the virtual part of the parameter q should be substituted by the Rayleigh range of a certain resonator. Thus, the ABCD law of the parameter q can be used for the high-order modes or their superposition. The calculation shows that when the focus lens moves along the optical axis, the change of focal length and focal depth is not relative to the orders of the Gaussian beam modes for an ideal resonator without distortion. For high power gas lasers or solid lasers, the distortion of the resonator components can change the resonator parameters, and also the Rayleigh range. In this situation, for a certain resonator, its focus characteristics, including the change of focal length and focal depth, are relative to the output power. It's important for the applications of the laser space craft vehicle and flying optical processing.!4
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