Fluctuations of the optical power incident on a photodiode can be converted into phase fluctuations of the resulting electronic signal due to nonlinear saturation in the semiconductor. This impacts overall timing stability (phase noise) of microwave signals generated from a photodetected optical pulse train. In this paper, we describe and utilize techniques to characterize this conversion of amplitude noise to phase noise for several high-speed (>10 GHz) InGaAs P-I-N photodiodes operated at 900 nm. We focus on the impact of this effect on the photonic generation of low phase noise 10 GHz microwave signals and show that a combination of low laser amplitude noise, appropriate photodiode design, and optimum average photocurrent is required to achieve phase noise at or below -100 dBc/Hz at 1 Hz offset a 10 GHz carrier. In some photodiodes we find specific photocurrents where the power-to-phase conversion factor is observed to go to zero.