The distributions of Amplified Spontaneous Emission (ASE) in the Dye Laser Amplifier (DLA) as well as the excited level and triplet state populations of the dye have been reported. The store energy of dye molecules and the amplification of fs pulses in condition of under saturable energy were calculated by mathematical simulation. According to our analyzing, the decreasing of store energy caused by triplet state absorbing must be taken into account when designing a DLA. Experimentally with Q-switched Nd:YAG synchronouly pumped two-stage dye laser amplifier, we have got the amplification gain of 1000 and 60 for the first and the second stage respectively. To the first stage, our calculation result can be fit well with the experiment.