We present an experimental investigation on the period-one dynamics of an optically injected InAs/GaAs quantum dot laser as a photonic microwave source. It is shown that the microwave frequency of the quantum dot laser's period-one oscillation is continuously tunable through the adjustment of the frequency detuning. The microwave power is enhanced by increasing the injection strength providing that the operation is away from the Hopf bifurcation, whereas the second-harmonic distortion of the electrical signal is well reduced by increasing the detuning frequency. Both strong optical injection and high detuning frequency are favorable for obtaining a single sideband optical signal. In addition, particular period-one oscillation points of low sensitivity to the frequency detuning are found close to the Hopf bifurcation line.