In this work, we investigate an unmanned aerial vehicle (UAV)-enabled wireless power transfer (WPT) network with multiple ground sensor nodes (SNs). A UAV is operated at a fixed altitude with a directional antenna array and is designed to wirelessly transfer energy to the SNs. We consider a non-linear energy harvesting (EH) model and a directional antenna structure of uniform linear array (ULA) where we apply an analog directional beamforming scheme. Taking the fairness issue into account, we consider a problem aiming at maximizing the minimum harvested energy among all SNs during a fixed time period by jointly optimizing the UAV trajectory and the orientation of the directional antenna on the UAV. However, the complex antenna pattern expression of analog directional beamforming and the implicit non-linear function in the EH model introduce significant difficulties in handling the non-convex problem of the joint design. To tackle these difficulties, we propose and adopt a modified approximate antenna pattern model, i.e., a modified cosine antenna pattern, and reformulate the original problem via quantizing the UAV trajectory in the time domain. Later, by employing a convex property in the EH model and a proved lemma, we successfully construct a tight convex approximation for the reformulated problem, based on which the problem can be solved via a proposed iterative algorithm and the objective converges to an efficient suboptimal solution. Finally, we provide numerical results to confirm the convergence of the proposed algorithm, examine the approximation error and evaluate the system performance. The results show the performance advantage of the directional antenna in UAV-enabled WPT networks than the omni-directional antenna case, and illustrate how the directional antenna of the UAV overcomes its coverage limitation