Nonlinear dynamics of semiconductor lasers has found many interesting applications in microwave photonics technology. In particular, a semiconductor laser under optical injection of proper strength and optical frequency detuning can enter into the dynamical period-one (PI) state through Hopf bifurcation. The resulting optical output carries a broadly tunable high-speed microwave modulation without employing any expensive microwave electronics. It is therefore a desirable source for radio-over-fiber (RoF) applications. The PI state can also be adjusted to have a nearly single sideband (SSB) optical spectrum. It is an advantageous property for long distance fiber transmission because it minimizes the microwave power penalty that is induced by chromatic dispersion. In this work, we investigate in detail the properties of the PI state and the effect of fiber dispersion as a function of the injection conditions. Based on a well-established rate equation model, the results show that the generated microwave frequency can be several times higher than the intrinsic relaxation resonance frequency of the laser. With a large injection strength and an injection detuning frequency higher than that required for Hopf bifurcation, the generated microwave power is nearly constant and the optical spectrum is close to SSB. We simulate the effect of fiber chromatic dispersion and the result shows a maximum microwave power penalty of less than 2 dB. The characterization of the PI state is useful in guiding the design of RoF systems based on optically injected semiconductor lasers.