## 1 Introduction  
X-ray radiation has long been used in various fields, such as medical diagnosis, industrial applications, material characterization, and scientific research. A traditional X-ray tube comprises a metal filament (hot cathode) and a metal target (anode), which are enclosed in an evacuated glass envelope. The metal filament is heated to over 1000 \({}^{\circ}\)C to emit electrons, which are accelerated by an applied electric field to bombard the metal target, generating X-rays [1]. These thermionic emission (TE) X-ray tubes have many limitations. Metal filaments must be heated to a high temperature to boil off electrons, so the response time is long. Moreover, the energy distribution of the emitted electrons from the heated cathode is wide, inhibiting their focusing and the X-ray imaging resolution. Also, the high temperature may lead to evaporation of the filament material or its oxidation by residual gases, reducing its lifetime. The typical lifetime of a TE X-ray tube is less than a year and a large portion of their failures thereof is filament-related.  
Field emission (FE) of electrons from the surface of a conductor was proposed by [PERSON] and [PERSON] in 1928 [2]. In FE, electrons are drawn from high-aspect-ratio or low-work-function cathode materials to the vacuum under a high electric field. This phenomenon is used in a variety of applications such as FE display panels [3], backlight units [4], field emission lamps (FELs) [5], and X-ray tubes [6]. FE-type X-ray tubes have many desirable properties. Electrons are extracted from the metal cathode by an applied electric field while the temperature of the emitters is significantly lower than that of the TE filaments. A colder cathode has the benefits of a longer lifetime and a much faster response. The FE current depends on the applied voltage, and the X-ray intensity depends on the FE current. Thus, the X-ray intensity can be conveniently and very rapidly controlled by varying the cathode voltage. Many researchers have worked on the design and fabrication of FE X-ray tubes [7,8].  
The typical FE X-ray tubes may comprise three parts, which are an FE cathode grown with electron emitters, an anode (metal target), and a gate electrode. Also, magnetic lenses or extra biased focusing electrodes can be used to focus the electrons onto the anode. Since nano-materials were discovered, carbon nanotubes (CNTs) [9, 10, 11], carbon nano-coils (CNCs) [12, 13, 14, 15], and CNTs decorated with nanodiamonds [16] have been identified as excellent FE cathode materials because of their high aspect ratio, good thermal stability, and good mechanical strength. The extremely high aspect ratios of CNTs and CNCs give rise to a large field enhancement factor, \(\beta\)[17], which makes them excellent field emitters with low turn-on fields. Researchers have found that CNCs exhibit several advantages over CNTs, including better long-term current emission stability, higher current density, and longer operation lifetime [18,19].  
The resolution of an X-ray image depends on the spot size of the electron beam on the anode. To minimize the spot size, a focusing electrode [20] is commonly used, but its use requires an extra bias circuit and results in a more complicated tube design. In this research, a simple diode-type FE X-ray tube is designed and focusing is achieved without resorting to magnetic lenses or active focusing electrodes.