With the increased interest for interstellar exploration after the discovery of exoplanets and the proposal of the Breakthrough Starshot project to perform a fly-through mission of Alpha Centauri to capture the first images of an Earth-like exoplanet, this paper investigates the optimization of photon-sail trajectories in Alpha Centauri for various mission applications. The prime objective is to find the optimal steering strategy for the photonic sail to get captured in Alpha Centauri after a minimum-time transfer from Earth. By extending the idea of the Breakthrough Starshot project with a deceleration phase at arrival, the mission's scientific yield is increased. In addition to capture into an orbit at the center of the habitable zone of one of the stars, a transfer trajectory to an orbit around the other star and an orbit-raising maneuver to the outer edge of the habitable zone are also investigated and added to the proposed mission scenario. The trajectories are optimized for minimum time of flight with the trajectory optimization tool InTrance, which makes use of evolutionary neurocontrol. The results show that an increase in technological development is required to obtain a feasible time of flight of approximately one century, using a two-sided reflective sail. The optimized time of flight from our Solar System until capture in Alpha Centauri ranges from 20,000 years for current sail technology to less than 80 years for a futuristic graphene-based ultralight sail. The latter allows a maximum injection speed into Alpha Centauri of 5.5\% of the speed of light. The results in this work show an average improvement of 30\% in terms of time of flight for lightness numbers smaller than a graphene-based sail. A sail as proposed by the Breakthrough Starshot project, however, would need over 2000 years to travel from our Solar System until capture about Alpha Centauri B, such that a fly-through mission that arrives within a lifetime is arguably the best option for a near-term mission to Alpha Centauri.