## 1 Introduction
Energy beamforming has been recognized as a promising technology for significantly increasing the energy transfer efficiency of RF-based WET systems, particularly compared to single-antenna omni-directional transmission [2; 3; 4]. An energy transmitter with multiple antennas can simultaneously focus energy beams toward devices in the desired directions, thereby overcoming high propagation path loss without increasing transmit power or bandwidth. In this regard, various energy beamforming techniques have been proposed for use in terrestrial wireless networks with perfect channel state information (CSI) as well as with imperfect CSI [3; 4; 5; 6; 7]. Moreover, due to the advantage of having LoS channels, beamforming techniques combined with resource allocation and optimization have been widely considered to enhance power transfer performance in various UAV-enabled systems [14; 25]. Beamforming combined with optimization of placement and resource allocation has been studied to maximize energy efficiency in wireless-powered UAV communication systems with non-orthogonal multiple access (NOMA) [26]. Additionally, hybrid beamforming with resource allocation has been explored in UAV-enabled wireless-powered mobile edge computing networks [27]. Furthermore, the joint optimization of beamforming, transmit power, power-splitting ratio, and UAV trajectory was proposed to enhance communication performance in UAV-enabled relay networks with wireless power transfer [28]. For UAV-enabled wireless-powered communication networks (WPCNs), beamforming techniques have also been explored using a backscattering scheme [29] and reconfigurable intelligent surfaces (RIS) [30], by jointly optimizing time allocation. Various beamforming techniques for UAV-enabled systems have been proposed in conjunction with resource allocation and optimization. However, existing studies have primarily focused on WPCNs or simultaneous wireless information and power transfer (SWIPT) systems, with an emphasis on optimizing communication performance. To the best of our knowledge, despite its significant utility, an energy beamforming technique that focuses solely on increasing energy transfer efficiency to simultaneously charge multiple devices, rather than on communication performance, has not been fully studied to realize its potential gains in UAV-enabled WET systems. Motivated by the aforementioned observations, we focus on energy beamforming design to optimize energy transfer efficiency for UAV-enabled WET systems. In practical WET networks, multiple devices have different energy requirements, and their charging times vary based on the amount of energy needed. Therefore, UAV altitude and energy beamforming must be optimized accordingly. To this end, this paper investigates a joint design of UAV altitude and energy beamforming to minimize the overall charging time required for all energy-harvesting devices (EHDs) to meet their energy requirements. Thus, a large number of EHDs are efficiently served simultaneously while avoiding unnecessary energy transfer from the UAV. Our main contributions are summarized as follows:  
* We propose the joint design of UAV altitude- and channel statistics-based energy beamforming, where the EHDs' energy demands are considered in efficiently and simultaneously serving the EHDs while reducing the additional resources and costs associated with obtaining perfect channel state information. In contrast to existing works on UAV-enabled WET, which consider only the LoS dominant channel without small-scale fading, we adopt the more general air-to-ground (A2G) Rician fading channel while also taking into account the characteristic nature of the aerial channel in practical UAV scenarios, where the channel statistics depend on the altitude of the UAV. * Due to the highly non-convex and non-linear nature of our design problem, we first jointly optimize UAV altitude and energy beamforming in a single-EHD scenario to draw insights.