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By considering the Gaussian spatial distributions of the intracavity photon density and the initial population-inversion density as well as the influence of the acoustic-optic (AO) Q-switch, we provide the coupled rate equations for a doubly Q-switched laser with both an AO modulator and a Cr 4+ -doped saturable absorber (SA). These coupled rate equations are solved numerically. The key parameters of an optimally coupled doubly Q-switched laser are determined,and a group of general curves are generated for the first time. These key parameters include the optimal normalized coupling parameter, the optimal normalized SA parameters and the normalized parameters of the AO Q-switch, which can maximize the output energy. Meanwhile, the corresponding normalized energy, the normalized peak power and the normalized pulsewidth are given. The curves clearly show the dependence of the optimal key parameters on the parameters of the gain medium, the SA, the AO Q-switch,the resonator and the spatial distributions of the intracavity photon density. Sample calculations for a diode-pumped Nd3+:YVO4 laser with both an AO modulator and a Cr4+ :YAG SA are presented to demonstrate the use of the curves and the related formulas.