## 1 Introduction  
In the current era, there is considerable demand for high-speed communications [1; 2; 3; 4; 5; 6]. With the emergence of microstrip structures and the growing demand for lightweight and compact structures [7; 8], Vivaldi commonly features a gradually narrowing slot design. Initially introduced by [PERSON] in 1979, the Vivaldi was further refined by [PERSON]. The Vivaldi operates over a wide bandwidth, offering high efficiency and a consistent radiation pattern as a travelling-wave [2; 9]. Vivaldi has attracted attention due to its array featuring a wide scanning angle and ultra-wide bandwidth [10]. Recently, the frequency band from 30 to 300 GHz has been considered the key to addressing the data explosion in 5G/6G systems [11; 12]. The advantage of the Vivaldi antenna lies in its extremely wide bandwidth. Successfully designed Vivaldi antennas are widely used in wireless communication systems, radar, telescopes, remote sensing systems, satellite communications, and more. This is due to its stable performance characteristics, wide bandwidth, and high directivity. This allows for it to deliver optimal performance over a wide range of frequencies. It can be adjusted according to specific application requirements [2; 7; 8; 13]. For instance, in wireless communication systems, to examine the composition of soil beneath the surface, ground-penetrating radar must send high-power pulses into the earth. To achieve deeper signal penetration and higher resolution, ground-penetrating radar antennas are required to operate in lower frequency bands, featuring high gain and ultra-wide bandwidth [9]. Vivaldi is a type of continuously scaled antenna, utilised in wideband wireless communication due to its large bandwidth and small lateral aperture among other prominent features [8]. The Vivaldi antenna has a compact footprint and features lower power consumption and complexity [2; 12]. While there are other types of wideband antennas, the Vivaldi antenna has advantages over others in terms of weight, cost, end-fire radiation, and scanning angle capabilities, particularly in the simplicity of integrating the system [8; 14; 12]. However, in systems such as radar, communication, and various electronic devices, space is often at a premium. It is necessary to keep the spacing between antenna units to a minimum. The operational principle of the Vivaldi antenna often leads to significant mutual coupling among neighbouring elements, primarily due to space-wave coupling. It can lead to adverse consequences, such as reduced antenna efficiency and inaccurate radiation patterns [12]. However, to enhance the gain of the antenna, researchers have proposed various methods. Among them, antenna arrays, etched grooves, and techniques such as dielectric lenses have gained widespread use. Typically, antenna arrays can significantly improve gain but require consideration of coupling and power distribution, making the design complex. Additionally, creating slots in the antenna can decrease its operating frequency and enhance its gain, yet this measure reduces the antenna's directivity. Adding a high dielectric constant to the antenna enables it to reliably boost its gain, although the extent of improvement is confined [9]. To achieve high performance, one often overlooks the miniaturisation in systems for communication. Therefore, researching small-sized, wideband, high-gain antennas is extremely important [15, 16, 17]. As mentioned, the Vivaldi antenna has a very wide bandwidth and can be adjusted according to specific application requirements, thereby offering optimal performance across a broad frequency range. Reference [7] employs a Vivaldi antenna array, phase switching, and an IC receiver for wireless broadband reception in 5G, achieving a gain of 8-13 dB, with a power consumption of 22.8 mV at 2 GHz. Reference [7] aims to increase the gain by using a decoupled director in a tapered slot. Around the slot, two symmetric sets of rectangular slits have been introduced to enhance the impedance matching.