## 1 Introduction  
The X-ray has been widely used in medical diagnosis, security checks, and drug testing since [PERSON] discovered it in 1895 [1]. It has made remarkable progress in communication, material analysis, high-resolution imaging, and other fields [2; 3; 4; 5]. X-ray sources can be roughly classified into two types: thermal cathodes and cold cathodes. Traditional thermal cathode X-ray sources exhibit limitations such as low current densities, high operating temperatures, and long response times, which render them unsuitable for miniaturization. Moreover, high-power operations can lead to material degradation and shorten the working lifespans of devices [6; 7; 8]. On the contrary, cold cathode X-ray sources are based on the principle of field emission; when the cathode surface is subjected to a strong electric field, electrons can break through the potential barriers and escape, and electrons bombard the anode metal to produce X-rays. This type of X-ray source exhibits good real-time performance, low power consumption, and easy integration [9; 10; 11; 12; 13; 14]. Many materials have been tried as cold cathodes. Specifically, carbon nanotubes (CNTs) have attracted much attention due to their excellent thermal properties, large aspect ratio, high electrical conductivity, and mechanical stability [15; 16; 17; 18; 19]. Compared to other cathode materials, they can be easily made into various shapes of emitters by patterned growth, making them a popular choice for cathodes in vacuum devices [20; 21; 22; 23; 24]. Some reports have explored the applications of CNTs such as X-ray microscopes [25], portable X-ray systems [26], cathode ray tube type lighting elements, and vacuum fluorescence display panels [27]. The field emission performance of CNTs is relatively stable, and their manufacturing process has gradually developed. It is well known that a gate electrode is necessary during CNT field emission processes to control and modulate the emission current [28; 29; 30; 31]. A change in the gate voltage alters the distribution of the electric field between the gate and the cathode electrodes, causing variations in the field emission current of CNTs. Many studies have been conducted on gate electrodes to improve field emission performance. The effects of concave grids on field emission properties were investigated in [32]. The effects of the grid aperture ratio and the relative position between the cathode and gate on field emission performance were explored in [33]. Achieving digital control of miniature X-ray tubes using a focusing-functional gate was accomplished in [34]. The influence of the alignment and dislocation relationship between the grid and cathode on electron transmittance was studied in [35]. A grid structure combining coarse and fine grids was designed to enhance the cathode current in [36]. The leakage current was reduced and the transmission ratio was improved through the self-aligned structure of the gate and CNTs in [37]. Based on point-like CNTs, the focusing effect of the electron beam spot through indium tin oxides (ITOs) glass was tested in [38]. Our team has previously reported on some papers in the field of CNT field emission, including the production of CNTs of various morphologies, the enhancement of the stability of CNTs, and the application and design of cold cathode X-ray tubes [39; 40; 41]. In the process of designing X-ray tubes based on CNTs, we found that the gate not only affects the electron transmittance but also the divergence of the electron beam, which affects the performance of the X-ray tube. However, previous reports have mainly focused on improving the electron transmittance of the grid, and research on the influence of the grid on the electron beam spot is lacking. Therefore, this paper presents a study on the grid aperture ratio. In actual experiments, different grid aperture ratios can change the electric field distribution on both sides of the grid, impacting the current of each electrode, electron transmittance, and electron beam spot size.