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The latest-generation light sources, such as Free-Electron Lasers and fourth-generation synchrotron radiation facilities, require challenging ultrafast optical measurements. In particular, there is generally a strong need for characterizing relativistic electron bunch shapes with sub-picosecond to femtosecond resolution. A particularly promising method relies on ultrafast measurement techniques based on the use of chirped laser pulses for probing the electric field created by the electrons either in the near field (in the vicinity of the bunches) or in the far field (i.e. looking at the radiation emitted by the bunches). However, developing such electric-field-recording techniques presents important challenges, given the ultrafast timescales, the repetition rates (up to hundreds of MHz), as well as the need for single-shot characterization. We present here a new strategy allowing pulses to be recorded in single shot with picosecond resolution at megahertz acquisition rates. The method is based on a combination of the so-called photonic time-stretch data acquisition, allowing long record lengths and novel electro-optic sampling techniques. We will also present a promising strategy to overcome the current time-resolution limitation of the existing setup, which requires new ideas from the ultrafast measurements community.