When electrons are accelerated by radially polarized ultrashort pulses (RPUP), it is generally believed that tighter focusing can lead to better acceleration effects. However, in the case of tight focusing, due to the diffraction effect, the peak intensity of the pulse will decrease significantly after leaving the focus, which limits the electron acceleration range. The longitudinal peak intensity distribution (LPID) of the pulse is influenced by the degree of focusing, which can be determined by the beam waist size. Therefore, the LPID can also be measured by the beam waist. Clarifying the numerical relationship between the LPID and the beam waist is beneficial for determining the effect of the LPID on electron acceleration. At the same carrier frequency, the spatiotemporal electric field gradients of pulses are higher if the duration of the ultrashort laser pulse is shorter. The ultrashort duration of the pulse allows the electron to approach the peak intensity of the pulse more easily and be accelerated by the peak electric field to obtain high final kinetic energy. The short pulse duration leads to spectral blueshift and a blueshift of the center instantaneous frequency of the pulse. In this paper, the relationships are analyzed between the spectral blueshift, the instantaneous center frequency blueshift, the spatiotemporal electric field gradients, and the beam waist. The influence of these properties on electron acceleration is also studied.

We derive the expressions for focused RPUP by using the sink-source model. We investigate how the spatiotemporal properties of sub-cycle, single-cycle, and few-cycle laser pulses influence electron acceleration. The electron is initially located on the z-axis with an initial velocity of zero. Along the optical axis, only the longitudinal electric field needs to be considered, as the transverse electric field component and magnetic field are zero. At the initial time, the pulse is far from the electron, so the interaction between the electron and the pulse at this time can be ignored. By studying the spectra blueshifts and the blueshifts of the center instantaneous frequency of the pulses for different pulse durations, the relationship is analyzed between the spectral blueshift, the instantaneous center frequency blueshift, and the spatiotemporal electric field gradients. Additionally, the relationship between the LPID and beam waist size is explored by studying the LPID of the focused pulse for different beam waist sizes. After considering the radiation-reaction force on the electron, the modified relativistic Newton-Lorenz equation is used to study electron acceleration. In this study, we examine the effect of the beam waist, pulse duration, initial phase, and electron’s initial position on the acceleration, which in turn helps us understand the role of the spatiotemporal properties of the pulses in electron acceleration.

For the waist spot size w0=0.4μm, the maximum final kinetic energy of the electron is 49.718 MeV for the 0.45-cycle pulse and 34.830 MeV for the 2.60-cycle pulse. The maximum final kinetic energy of the electron for the 0.45-cycle pulse is 43% higher than that for the 2.6-cycle pulse. This is because the spatiotemporal electric field gradients experienced by the electron in the 0.45-cycle pulse are significantly higher than those in the 2.6-cycle pulse during acceleration. The peak intensity of the pulse experienced by the electron in the case of the 0.45-cycle pulse is also higher than that in the case of the 2.6-cycle pulse (Fig. 7). The instantaneous frequencies of the 0.45-cycle pulse experienced by the electron are also significantly higher than those of the 2.6-cycle pulse (Fig. 7). For the waist spot size w0=5μm, the maximum final kinetic energy of the electron is 2.92GeV for the 0.45-cycle pulse, and is 2.04GeV for the 2.6-cycle pulse. The maximum final kinetic energy of the electrons for the 0.45-cycle pulse is also 43% more than that of the 2.6-cycle pulse. The instantaneous frequency of the pulse at the position of the electron and the maximum kinetic energy gain of the electron in the case of the sub-cycle pulse are significantly higher than those in the case of the few-cycle pulse. The larger instantaneous frequency change of the pulse indicates a larger electric field gradient, which means that electrons can approach the peak intensity of the pulse more closely. The maximum kinetic energy of electrons for the waist spot w0=5μm increases by more than one order of magnitude compared to that for the waist spot w0=0.4μm. This indicates that the LPID is the most important factor affecting the electron acceleration. The electric field gradient of the pulse is another important factor affecting the electron acceleration.

The spectra of the sub-cycle, single-cycle, and few-cycle laser pulses are blue-shifted. The instantaneous frequencies of these pulses change with time. The central instantaneous frequency of the pulse is higher if the pulse duration is shorter. The change in the instantaneous frequency over time can be regarded as an indicator of the change in the spatiotemporal electric field gradient. The electric field gradient, the change in the instantaneous frequency, and the central instantaneous frequency of the pulse are all greater if the pulse duration is shorter. The LPID of the focused pulse is the largest at the focus. The range over which the peak intensity of the pulse remains high on both sides of the focus on the optical axis is larger if the beam waist size is larger. The LPID of the pulse and the electric field gradient are important factors affecting electron acceleration. If the peak intensity of the pulse is maintained at a high level over a larger range on both sides of the focus, the exit kinetic energy of the electron can be increased by orders of magnitude. Therefore, the upper limit of the electron kinetic energy gain is determined by the LPID of the pulse if the maximum peak intensity is constant. For the single-cycle and sub-cycle pulses, the significant blue shift in the central instantaneous frequency is a sign that the pulse has high electric field gradients in both time and space. The high spatiotemporal electric field gradients of these pulses also significantly increase the electron’s final kinetic energy, which helps to approach the upper limit of the electron kinetic energy gain.