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Many physical effects can potentially limit the peak achievable output power of single emitter broad area diode lasers under high current, pulse-pumped operation conditions. Although previous studies have shown reliable operation to high pump levels (240 A, 300 ns, and 1 kHz), power was found to saturate. We present here results of a systematic study to unambiguously determine the sources of this power saturation. A combination of detailed measurements and finite element device simulation were used for the diagnosis. We find that the measured power saturation is dominated by electron leakage caused by band bending at high bias due to the low mobility of the p-type waveguide. However, the power saturation is only fully reproduced when longitudinal spatial hole-burning is included. Higher powers are expected to be achieved if higher energy barriers and lower confinement factors are used to mitigate leakage and longitudinal hole-burning, respectively.