# Spatial and Momentum Mapping Modes for Velocity Map Imaging Spectrometer
## 4 Experiment
### High-Resolution Momentum Mapping Mode  
To evaluate the performance of the high-resolution momentum mapping mode of the VMI spectrometer, we measured the ATI image on Ar [29]. By scaling proportionally the VMI lens setting, the photoelectrons with an energy of up to 50 eV were covered. The measured 2D momentum spectrum is shown on the left side of Figure 7a and the reconstructed 2D-momentum distribution using polar-onion-peeling (POP) inversion [30,31] on the right side, respectively. The POP method was chosen here because it is very robust against low statistics and does not accumulate noise on the axis of the spectrum. Then, in order to show a high-energy structure, the electron energy spectrum was obtained by angular integrating the reconstructed image between the 0\({}^{\circ}\) and 12\({}^{\circ}\) with respect to the polarization axis and shown in Figure 7b. To obtain the instrument's resolution, we fit ATI peaks between 40.7 and 47.9 eV with Gaussian functions, as depicted by the black line in the inset of Figure 7b. As shown by  
Figure 7: ATI measurement on Ar. (**a**) Raw 2D-momentum distribution of electrons is shown on the left side, and the 2D-momentum distribution reconstructed by the polar onion peeling method is shown on the right side. The intensity in the interaction region is \(1.4\times 10^{14}\) W/cm\({}^{2}\). (**b**) Photoelectron spectra of Ar ionized by 800 nm linearly polarized laser pulses. The photoelectron kinetic energy spectrum was obtained by integration of the momentum distribution 0\({}^{\circ}\) to 12\({}^{\circ}\) with reset to the laser polarization axis. The red squares represent the photoelectron spectrum, and the peak at 44.6 eV is selected to represent the resolution, as shown by the blue line in the enlarged injector picture. [PERSON] et al. [32], the width of peaks at the plateau region may change. So, the fitting parameters are the peak width and height. A narrow ATI peak near 44.6 eV is chosen, and its full width half maximum (FWHM) is determined as 0.66 eV in the least square fitting. This width is contributed by the intrinsic width of the ATI peak and the instrument resolution. According to theoretical calculations [32], the intrinsic width of this ATI peak is about 0.3 eV. So, the energy resolution of the spectrometer is estimated as \((\sqrt{0.66^{2}-0.3^{2}})=0.59\) eV, corresponding to 1.3% at 44.6 eV electron kinetic energy. In Table 3, we have listed the performance of several VMI spectrometers designed for low, middle, and high kinetic energy, respectively. The results indicate that our VMI spectrometer works excellently in measuring both high-energy and middle-energy electrons. There are several mechanisms that can deteriorate the energy resolution. Electric field distortions near the wire mesh can change the direction of passing charged particle [26]. The velocity change \(\Delta v_{x}\) in the \(x\) direction reaches a maximum at the wire, which is estimated as  
\[\Delta v_{x}=\frac{q\Delta Ed}{4 mv_{y}}. \tag{2}\]  
Here, \(\Delta E\) is the electric field difference between the two sides of the mesh, \(d\) is the mesh cell size, and \(v_{y}\) is the particle velocity in the \(y\) direction. In the spatial mapping mode with the magnification factor of 6.4, \(\Delta v_{x}\) is estimated as \(4.1\times 10^{-5}\) mm/ns. After the remaining time of flight of 5000 ns, the image distortion is about 0.2 mm. In the momentum mapping mode, \(\Delta v_{x}\) is estimated as 0.005 mm/ns. After the remaining time of flight of 10 ns, the image distortion is about 0.05 mm, which corresponds to a relative energy resolution of 0.33%. The estimated image distortions agree reasonably with the observed images in Figures 6 and 7. In the original VMI paper by [PERSON] [2] that showed the microlens effect, two homogenous electric field regions are separated by a mesh. So, there is no focusing and the measured ion image is by far much worse than the one measured in VMI mode. After that, a mesh is often ascribed to be the principal killer of the resolution.