## 2 Near-Cathode Particle Dynamics
This is in comparison to the one shown in Figure 3b where a flattop transverse laser pulse (truncation ratio of 1) is applied instead. Figure 3c captures a transient comparison of the transverse sliced emittance along the bunch between the transverse flattop case (blue curve) and the truncated-Gaussian case (red curve) at the end of the emission process. The comparison shows a tendency of lower emittance (left axis) around the current peak (right axis) for the truncated-Gaussian case, although the absolute emittance value at this moment apparently cannot yet represent the bunch qualities. Moreover, Figure 3d shows a vector plot of relative displacements (lengths of the vectors) on the transverse plane from the macro-particles in the flattop case (arrow tails) to those in the truncated-Gaussian case (arrow heads). Note that each pair of the macro-particles used in the vectors are born at the same time at the cathode position and are further tracked through the emission process, and that the diameter of the transverse particle distributions is the same at the cathode. This indicated that the development of the beam self-fields by utilizing a transversely optimized truncated-Gaussian distribution at the cathode plane can allow a higher charge density towards the center of the bunch at the end of the photoemission, when it is compared to the case with a flat transverse distribution. Based on the optimization approach, as shown in Figure 2, a transverse truncated-Gaussian laser distribution with an edge-to-center intensity ratio of about 0.65 is chosen and employed for beam dynamics simulations in Section 3. Figure 3: A transient view for the characteristics of near-cathode particle dynamics at the end of the photoemission process. (**a**) on-tracking transverse particle distribution for a truncated-Gaussian (TG) laser distribution with an optimal truncation ratio at the cathode; (**b**) on-tracking transverse particle distribution for a transverse uniform (RU) laser distribution at the cathode; (**c**) comparison of slice emittance along the bunches at the end of the photoemission between bunches (**a**,**b**); (**d**) vector plot of relative displacements on the transverse plane of the macro-particles in bunch (**a**) with respect to bunch (**b**). The arrows point from the macro-particles in (**b**) to those in (**a**), in which these macro-particles paired in the vectors are born at the same time clock at the cathode. The length of the arrows corresponds to the displacements in meter at this moment. Figure 2: Optimization of a transverse Gaussian cathode drive laser distribution by adjusting the truncation ratio, i.e., a defined edge-to-center intensity ratio. (**a**) total extracted bunch charge versus emission time; (**b**) exemplary transverse cathode laser spot and its horizontal cut. Note that this is not to scale and only for illustration purpose; (**c**) computed transverse electric space-charge fields during emission at the observation point A as marked in (**a**,**d**); (**d**) mean bunch position versus emission time; (**e**) exemplary intensity projections of a horizontal cut to the transverse cathode laser spot as sketched in (**b**): different curves correspond to transverse cathode laser distributions with various edge-to-center (truncation) intensity ratios of 1 (uniform, black curve), 0.9 (pink), 0.8 (dark green), 0.6 (red), 0.4 (yellow) and 0.2 (light green), respectively. Note that subplots (**b**,**e**) are for illustration purpose and not to scale; (**f**) computed transverse electric space-charge fields at the end of emission at the observation point B as marked in (**a**,**d**).