# Spatial and Momentum Mapping Modes for Velocity Map Imaging Spectrometer
## 3 Simulation
### High-Resolution Momentum Mapping Mode  
The optimization in the previous paper [11] was for 1000 eV electrons, which is called high-energy mode. However, VMI is mainly used for measuring electrons within tens of eV kinetic energy. So, we add the working mode for the electrons with kinetic energy up to 150 eV, which is called the middle-energy mode. The resolution is sacrificed in high-energy mode due to the upper limit of the applicable high voltage. This sacrifice is unnecessary in the middle-energy mode, so we expect a better resolution. With the electron kinetic energy upper limit decreases, we can reduce all voltages proportionally.  
Figure 4 illustrates the simulated electron-flying trajectory. The source region is defined by a \(3\times 3\) point array in a \(1\times 3\) mm\({}^{2}\) region. The kinetic energy of electrons is 150 eV. The emission angle ranges from 0 to 180\({}^{\circ}\) with an interval of 30\({}^{\circ}\). We calculate the spread of each electron cloud on the detector for every emission angle. The goal is set as the sum of them, and then we reduce this goal as much as possible in the optimization.  
Figure 5a summarizes the simulation from Figure 4. The radius is plotted as a function of TOF for electrons with different kinetic energies. To obtain the simulated relative resolution, we calculated the average radius \(R\) and the maximum radius difference \(\Delta R\) of the nine particles on the detector's surface at an emittance angle of 90\({}^{\circ}\). Then, the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{3}{c}{**Geometry (mm)**} & \multicolumn{6}{c}{**Voltages (V)**} \\ \hline  & \multirow{2}{*}{**DIST**} & \multirow{2}{*}{**TH**} & \multirow{2}{*}{**ID**} & \multicolumn{4}{c}{**Ion**} & \multicolumn{2}{c}{**Electron**} \\ \cline{5-10}  & & & & \(\times\)**1.0** & \(\times\)**2.2** & \(\times\)**2.8** & \(\times\)**5.9** & \(\times\)**7.6** & **  
relative energy resolution is calculated as \(\Delta E/E\cong 2\Delta R/R\). Figure 5b demonstrates the relative energy resolution at electron energies ranging from 0 to 150 eV in high-resolution momentum mapping mode. All the electrons at different kinetic energies and emission angles are nicely focused. \(\Delta E\) depends weakly on the electron energy. The best relative energy resolution of 0.15% is predicted at 150 eV, which is the highest kinetic energy.  
Figure 4: Simulation of the high-resolution momentum mapping mode. The source region is defined as 3 \(\times\) 3 points from a rectangular region of dimensions 1 \(\times\) 3 mm\({}^{2}\). The kinetic energy of electrons is 150 eV. The colors represent the different emission angles of the electrons. The emission angle ranges from 0 to 180\({}^{\circ}\) with an interval of 30\({}^{\circ}\).  
Figure 5: Summary of the simulation results in the high-resolution momentum mapping mode. using SIMION 8.2 from Scientific Instrument Services, Palmer, MA, USA (**a**) Electron kinetic energies from outside to inside are 150, 75, 50, and 25 eV, respectively. (**b**) Simulated electron kinetic energy resolution \(\Delta\)E/E and \(\Delta\)E as a function of electron kinetic energy. The resolution of the high-resolution momentum mapping mode is around 0.22 eV (red line), and we can reach a relative resolution of 0.15% (black line) at 150 eV.