## 1 Introduction  
In recent years, unmanned aerial vehicles (UAVs) have garnered considerable attention due to their ability to perform tasks that are challenging or hazardous for humans, including reconnaissance, surveillance, and search and rescue [1; 2; 3]. Compared to a single UAV, multiple cooperative UAVs can provide broader coverage, greater flexibility, and robustness through versatility [4; 5; 6]. Therefore, the flying ad hoc network (FANET) has become a hot topic as it provides an effective real-time communication solution for UAVs' collaboration. However, the UAVs in FANETs usually encounter several challenges, such as limited communication resources, intensive contention from neighboring UAVs, and even external interference, which can hinder meeting network capacity and delay communication requirements. In order to increase network capacity, the usage of multiple channels [7; 8] and directional antennas [8; 9; 10; 11] are essential for FANETs. Due to their advantages in terms of high gain and focused beamforming capabilities, directional antennas can cover a larger communication range compared to omnidirectional antennas [12] and are well-suited for jamming environments [13]. At the same time, UAVs equipped with narrow beam angle directional antennas pose design difficulties for the media access control (MAC) protocol. Neighbor discovery for a MAC protocol must fulfill the requirement that any pair of neighbor UAVs steers their main lobe toward each other to complete main lobe rendezvous and operate in different modes (reception and transmission mode) simultaneously within a limited time. Additionally, the MAC protocol must guarantee that main lobe rendezvous interruption and data transmission interruption are avoided due to the high mobility of UAVs. Last but not least, the design of the MAC protocol needs to complete simultaneous channel rendezvous [14] and main lobe rendezvous [15] in a multichannel FANET with directional antennas. Many studies have investigated neighbor discovery in directional networks. Neighbor discovery for directional FANETs can be generally classified into two types: one is to achieve neighbor discovery with the help of omnidirectional antennas [16; 17; 18; 19], while the other adopts directional antennas directly to achieve neighbor discovery [15; 20; 21; 22; 23; 24; 25; 26]. The communication range of omnidirectional antennas is commonly smaller than that of directional antennas. Therefore, the neighbor set discovered through the former is a subset of that of the latter. There are two ways to extend the communication range with omnidirectional antennas. One is to exploit multihops [27], whereas the increase of multihops could introduce negative impacts such as large delays, additional interruptions, and a higher bit error rate (BER) [28; 29]. The other method is dual-band wireless networks [30]. According to the Friis transmission equation [31], the high loss of the high-frequency band is offset by the gain of the directional antenna so that the omnidirectional antenna communication range is the same as that of the directional antenna. However, this approach requires coordination between different frequency bands. Therefore, many studies for neighbor discovery focus on using directional antennas directly. In [20], an adaptive directional antenna protocol for terahertz networks (ADAPT) was proposed to achieve high throughput. However, this protocol was designed for the directional networks with centralized architecture, which is not suitable for FANETs. There are two methods of neighbor discovery for FANETs with directional antennas: probabilistic [21; 22; 23] and deterministic protocols [15; 24; 25; 26]. The disadvantage of probabilistic protocols is that the discovery delay cannot be guaranteed, which may lead to extreme delays for two neighbor UAVs to discover each other. Deterministic protocols, on the other hand, achieve neighbor discovery with a bounded delay. In [25; 26], an iterative search was proposed to decrease the delay of neighbor discovery.