## 3 Results and Discussion  
### Simulation of the Self-Focusing Cathode Design  
Figure 2 displays the geometry of the X-ray tube for simulation. The red part is the anode, which has a diameter of 15 mm and is biased at an applied voltage of 50 kV; the blue part is the metallic cup for electrostatic focusing which has a diameter of 15 mm with a 12-mm-diameter aperture on the top to focus the emitted electrons. The CNC FE cathode disk (not shown) was situated at the bottom of the metallic cup. The whole cathode (CNC FE disk and cup) was grounded. The distance from the electron emitters to the anode was fixed at 20 mm, and the cup height was varied to minimize the size of the focal spot on the anode.  
Figure 2: Geometry of the X-ray tube in SolidWorks. Red and blue parts are anode and cup-shaped cathode, respectively. CNC FE disk (diameter = 15 mm) is at the bottom ofocusing cup; a 12-mm-diameter aperture is at the top of cup.  
In the design herein, the electron-emitting area was quite large. This was advantageous because it would greatly reduce the average current density if the total FE current was fixed, resulting in a longer lifetime of the CNC emitters.  
The electrostatic potential distribution and, hence, the electric field distribution were then calculated using Comsol Multiphysics. Finally, the trajectories of the emitted electrons accelerated by the Coulomb force were simulated using Simion 8.0. The calculated trajectories are shown in Figure 3.  
Figure 3a presents the simulation results for an X-ray tube without the focusing cup; electrons are dragged by the applied voltage in straight trajectories to bombard the anode. In contrast, in Figure 3b, the trajectories of the electrons emitted from a cup-shaped cathode clearly converged onto a small focal spot. After testing with various heights of the cup-shaped cathode, the optimal cup height was about 5 mm, which yielded the smallest focal spot size of 0.1 mm (diameter). Figure 4 shows the three-dimensional (3D) electron trajectories in the X-ray tube with a 5-mm-high focusing cup.  
Note that the focusing cup is maintained at the same voltage as the CNC FE disk. Thus, no extra bias circuit is needed, and the structure and fabrication of such an X-ray tube are simple.  
Figure 4: Three-dimensional (3D) diagram of electron trajectories in the X-ray tube with a 5-mm-high focusing cup.  
Figure 3: Trajectories of emitted electrons (**a**) without and (**b**) with focusing cup design. Vertical axis in (**b**) represents electric potential \(V\).  
To understand why this simple design offered such a good focusing effect for a large-area cathode, Figure 5 shows the electric field distribution on the CNC FE disk (a) without and (b) with the focusing cup design. Comparing Figure 5a with b, the metallic cup wall clearly has a strong shielding effect: the electric field strength near the disk edge was reduced from 9.5 \(\times\) 10\({}^{5}\) V/m (center) to below 1.2 \(\times\) 10\({}^{5}\) V/m. In contrast, the electric field without the focusing cup was nearly constant everywhere on the CNC FE disk. This shielding effect forced the electric field lines within the cup to converge towards the central axis, so the electrons were guided by this field distribution and focused onto the anode.