We propose an all-optical technique to record the time information of relativistic electron beams with sub-optical-cycle duration. The technique is based on the interaction of the electron beam with the ponderomotive potential of an optical traveling wave generated by two counter-propagating circularly polarized optical fields at different frequencies in vacuum. One of the optical pulses is a vortex laser pulse, and the other is a normal Gaussian laser pulse. The time information of the electron beam is mapped into the angular information, which can be converted into a spatial distribution after a drift section. Thus, the temporal profile and arrival time of the electron beam can be retrieved from the spatial distribution of the electron beam. The measurement has a dynamic range comparable to the period of the optical intensity grating formed by two counter-propagating laser pulses. This technique may have wide applications in many research fields that require sub-optical-cycle electron beams.

Ultrashort terahertz (THz) beam is often affected by spatio-temporal distortion due to its long wavelength, and this spatio-temporal coupling will degrade the focusing effect. Here, we develop a numerical model for calculating the vector focusing of ultrashort pulses with arbitrary spatial and temporal distribution, which adopt the vector diffraction integral and coherent spectrum superposition methods. Leveraging it, the focusing of a broadband pulse can be characterized together with its vector characteristics and we explore the influences of multifarious spatio-temporal couplings on the focusing of ultrashort THz pulses. In addition, the possibility of focusing the ultrashort THz radiation from an undulator to 1 GV/m is also discussed. This will provide further guidance for the non-equilibrium state of matter induced by ultrashort strong-field THz pulses.

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