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Measuring the turn-on delay of diode lasers provides useful information on carrier recombination dynamics, particularly Auger recombination, essential for their design for high-speed modulation and power-efficient performance. Here we present a rigorous, comprehensive relationship between the time delay and the Auger recombination coefficient. We demonstrate the application of this formulation by extracting this coefficient for AlGaInAs/InP quantum wells incorporated in long-wavelength vertical-cavity surface-emitting lasers.