In this paper, a trajectory optimization algorithm for multiple unmanned aerial vehicles (UAVs) is investigated for free space optic (FSO) based wireless communication networks, which is composed of multiple UAVs and multiple ground terminals (GTs). Considering the FSO channel model and line of sight (LoS) probability, the altitude of the UAVs for FSO connection between each GT and UAV is decided by the predetermined radius of the communication area of the GT. A multi-UAV trajectory optimization (MUTO) scheme is proposed to maximize the service time. In the MUTO scheme, first, the network is divided into multiple sectors using graph partitioning, and assigned to each UAV, and then the traveling salesman problem (TSP) is used to determine the order of the GTs that the UAV passes through. The number of UAVs that maximizes the time for the UAV to stay within the communication area of the GT is derived. The UAV trajectories are determined by sequentially minimizing the energy consumed when moving between GTs and maximizing the service time by applying successive convex approximation. In addition, to assist operations that use multiple UAVs, two collision avoidance schemes are applied to the MUTO scheme. First is the additional constraints (AC) method, which uses a geographic formula to control the distance between the UAVs, and second is the initial delay (ID) method, which deliberately adds a delay to the UAV trajectories to provide sufficient UAV spacing. The simulation results show that under low density operation conditions, MUTO-ID and MUTO-AC are sufficient to avoid all UAV collisions, and in overly dense UAV operations with very close UAV trajectories, MUTO-ID can reduce collisions by approximately 80%, MUTO-AC can reduce collisions by approximately 85%, and MUTO-ACID can reduce collisions by approximately 95%, when compared to the MUTO scheme with no collision avoidance support applied.