Fig. 1. Layout of the Faraday-rotation measurement. The polarizations of the light passing above and below the beam are rotated oppositely.

Fig. 2. Layout of the Michelson-type interferometer. The polarizer only allows the transmission of the polarization-rotated light. The probe beam is amplified $N_1$ times by a pair of lenses and then split into L1 and L2 by a beam splitter. L1 and L2 interfere with each other and the interference fringes are recorded by a CCD camera after being amplified $N_2$ times by another amplification system.

Fig. 3. (a) Magnetic field distribution of the electron beam and (b) the polarization rotation angle of the probe laser using Eq. (1) in the $\mathit{y}\text{-}\mathit{z}$ plane from PIC simulations. The distribution of the polarization rotation angle of a Gaussian magnetic field profile (c) without and (d) with considering the transit time of the probe light.

Fig. 4. Intensity distributions of (a) L2 from Eq. (7) and (b) L1 and L2 from Eq. (10) in the plane $\mathit{x}\text{-}\mathit{y}\text{-}{\mathit{z}}^{\prime }$ with ${\tau }_{\mathrm{pro}}=100\mathrm{fs}$; the normalized integral intensity traces of L2 along the (c) $y$ axis and (d) $x$ axis; (e) the normalized integral intensity profile of the overlapping region along the $x$ axis; (f) the normalized intensity of the retrieved interference fringes in the $x$ axis.

Fig. 5. Intensity distributions of (a) L2 from Eq. (7) and (b) L1 and L2 from Eq. (10) in the plane $\mathit{x}\text{-}\mathit{y}\text{-}{\mathit{z}}^{\prime }$ with ${\tau }_{\mathrm{pro}}=250\mathrm{fs}$; the normalized integral intensity traces of L2 along the (c) $y$ axis and (d) $x$ axis; (e) the normalized integral intensity profile of the overlapping region along the $x$ axis; (f) the normalized intensity of the interference fringes in the $x$ axis.

Fig. 6. (a) Interference fringe interval vs. the incident angle between L1 and L2. (b) The resolution of the measurement as a function of $N_2$ for different angles.