Fig. 1. Schematic of radiative mechanisms arising from electron–matter interactions in a homogeneous material at a single frequency (see Fig. 4 for details of the radiation). The lower half of the image is the normal Smith–Purcell radiation (SPR) in vacuum, in which only negative harmonics electron beams contribute to the evanescent field. In the medium (upper half), normal SPR excited by the −1st harmonic, reverse SPR excited by the 1st harmonic, and Cherenkov (CR) can be present at the same time when the electron velocity is faster than twice the light velocity in the medium. The free electron velocity is 0.6c, with the refractive index n of 4, the grating period of 5 μm, and the radiation frequency of 30 THz.

Fig. 2. Dispersion relations for electron velocities of (a) 0.2c, (b) 0.4c, and (c) 0.6c. The black dotted line is the light line in vacuum, and the solid black line is the light line in the medium with a refractive index of 4. The gray area in (a) is normal SPR; the blue area in (b) is special SPR; the yellow area in (c) is reverse SPR. (d) Relationship between the radiation angle and frequency with the refractive index of medium fixed to 4 and electron velocity between 10 eV and 160 keV. (e) Relationship between the frequency of SPR at 0° and electron velocity between 10 eV and 160 keV. The three types of SPR shown in (e) correspond to the dispersion relations shown in (c).

Fig. 3. Schematic of the SPR wavefront with electron beam velocity of (a) 0.2c, (b) 0.4c, and (c) 0.6c. The gray square in the middle is the grating, and the electrons pass through the lower surface of the grating. The upper part of the grating is the medium with n=4, and the lower part is the vacuum. Time-domain contour of the real part of the electric field propagating along the x axis Ex of SPR with electron beam velocity of (d) 0.2c, (e) 0.4c, and (f) 0.6c. Frequency of SPR as a function of the probe angle in medium and vacuum with electron beam velocity of (g) 0.2c, (h) 0.4c, and (i) 0.6c.

Fig. 4. (a) Frequency of SPR as a function of the angle in a medium and vacuum with electron beam velocity of 0.6c. The yellow triangle is the focal point of the black dotted line at f=30  THz and with different types of radiation. The label of each yellow triangle corresponds to the label in (b)–(e). Theoretically calculated field distribution of (b) normal SPR in vacuum, (c) reverse SPR in medium, (d) CR in medium, (e) and (f) normal SPR in medium at 30 THz.

Fig. 5. Schematics of (a) free electron excitation of a metasurface, (b) XOZ plane with period L and gap g of 5 and 0.5 μm, respectively, and (c) XOY plane. The metasurface is composed of a 45° angle grating. (d) Direct observation of the simulation of (left) forward-propagation reverse SPR (CR) and (right) backward-propagation normal SPR. The real part of the electric field propagating at 30 THz along the z axis is observed at the XOY plane 5 μm above the metasurface.