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Measurements of the switch-off time of an optically switched Fabry-Perot InGaAsP/InP laser amplifier indicate that the device can turn off in less than the spontaneous carrier lifetime. The switching dynamics of the amplifier are numerically modeled using the van der Pol equation with optical injection and the rate equation for the carrier density. The simulations show that the device switches off faster as its injection current is increased toward the threshold level. A qualitative analytical analysis is presented which shows that this fast switch off is due to the very large cavity finesse of the amplifier when biased near its lasing threshold.