The first mission proposals to visit the Alpha Centauri system use photon-sail acceleration as a mode of propulsion to reach this stellar system closest to our own Solar System. To prepare for a future mission, the photon-sail dynamics in the system is investigated. Planar Lyapunov orbits around the colinear classical Lagrange points are designed to explore the Alpha Centauri system. This has been done before in other systems like the Earth-moon and Sun-Earth systems, but not yet in an elliptical binary star system. Starting with an initial guess in the circular restricted three-body problem without photon-sail acceleration, a Multiple Shooting Differential Correction (MSDC) algorithm changes the trajectory to a periodic orbit. A continuation method increases the eccentricity to match e = 0.5208, which is the eccentricity of the inner binary system of Alpha Centauri. The lightness number of the photon sail is increased to add photon-sail acceleration to the model up to a defined maximum of ?ZNe = 2. A set of five constant steering laws is chosen to investigate its effect. Next to that, the moment at which the periodic orbit starts in terms of the true anomaly is varied as well. This results in a set of 40 families of periodic orbits with increasing lightness numbers. Depending on the orientation, the augmented Lyapunov orbit either shrinks into smaller orbits or expands into larger orbits when increasing the lightness number. If the orbit shrinks, it can either converge into an artificial equilibrium point or the photon-radiation pressure on the sail can become minimal. In that case, the Lyapunov orbit becomes (almost) independent on the lightness number and reaches ?ZNe = 2. If the orbit expands, the maximum velocity will eventually go to infinity. At this vertical asymptote, the maximum lightness number is found. The initial true anomaly of Alpha Centauri 0 has a great effect on the Lyapunov orbits around L2 and L3 in the classical ER3BP. For 0 = 0, the orbit either converges to an AEP or the maximum velocity goes to infinity. For 0 = , a few orientations can reach ?ZNe = 2. To further explore Alpha Centauri, an adaptive differential evolution algorithm is used to design trajectories between the Lyapunov orbits. The performance of the algorithm is expressed as the Euclidean difference between the states at the end of the departure leg and the beginning of the arrival leg. Three different lightness number of ? = 0.1, 0.5 and 2 are used for these trajectories. With a lightness number of 0.1, the dimensionless Euclidean error is in the range of 1E-1 to 1E-3, depending on the Lyapunov orbits. With this lightness number, the stars are also used as a gravity assist. For larger lightness numbers, the Euclidean error becomes negligible in the range 1E-7. With a lightness number of 2, the time of flight during the trajectory is significantly lower. In future research, this can be further decreased using an MSDC algorithm.