We present a theoretical study into the dynamics and bifurcations of a semiconductor laser subject to delayed optical feedback, as modelled by the Lang-Kobayashi equations. For the case of a short external cavity, of the order of a few centimeters, there is a limited number of external cavity modes (ECMs), which makes it possible to apply advanced techniques from dynamical systems, such as the continuation of ECMs and their bifurcations, and the computation of unstable manifolds. From the physical point of view, a short cavity is characterized by the fact that the delay time in the external cavity is of the same order of magnitude as the period of the relaxation oscillation of the laser. In this regime the optical feedback phase is known to play an important role. We provide a detailed overview of how the dynamics depends on the feedback phase, which is in good agreement with recent experimental measurements.