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It is demonstrated theoretically that the quantum well (QW) semiconductor optical amplifiers (SOAs) with high differential gain will show fast gain recovery dynamic characteristics. Based on the calculation of energy band structure, the effects of compressive strain and p-type modulation doping on the gain, the differential gain, and linewidth enhancement factor (a-factor) are investigated. The peak of the differential gain spectrum shifts to longer wavelength, and the peak value is significantly enhanced by increasing compressive strain. Meanwhile the gain is obviously increased by increasing the p-type modulation doping concentration. By comparing the gain and phase recovery dynamics in three different types of SOA samples, it is shown that the QW SOA with highest differential gain has the shortest gain recovery time among the three samples.