To improve the energy extraction efficiency of the high repetition rate and short pulse CO₂ master-oscillator power-amplification system in the light source of laser plasma extreme ultraviolet lithography, it is necessary to study the coupling matching characteristics between the beam intensity cross-section distribution of the seed beam and the gain distribution of the laser amplifier. Based on the Euler-Lagrange formula and Frantz-Nodvik equation, the optimal intensity distribution function of the seed beam under a specific gain distribution is solved by variational method. The effects of the seed beam width, intensity distribution, and gain distribution of the laser amplifier on the power extraction are examined. The results of numerical simulation show that for the laser amplifier with a gain radius of 1 cm, the optimal beam radius of the fundamental mode Gaussian beam with a pulsed seed light power of 500 W is 0.67 cm, and the optimal beam width of the low-power seed light is not equal to the gain width of the amplifier. The optimal beam radius of the higher-order super-Gaussian beam with a pulsed seed power of 2000 W is close to the gain region radius. The optimal beam radius corresponding to the 8-order super-Gaussian beam is 0.9 cm, and the power extraction value is 3409 W. The higher-order gain distribution and the matched super-Gaussian beam can significantly enhance the energy extraction value of the amplifier. The results of this study will provid e theoretical guidance for designing pulsed CO₂ main-oscillation power amplifier systems.