Fig. 1. (a) Schematic of reverse Smith-Purcell radiation (SPR) generated by free electrons in photonic crystals. The yellow region represents the metal grating, with free electrons skimming along its surface. Normal Smith-Purcell radiation is generated in the vacuum above the grating, while the hexagonally arranged air-column photonic crystals (PhCs) beneath the grating produce reverse Smith-Purcell radiation. (b) One-dimensional dispersion curve of the PhCs along the Γ-M direction. (c) Equal-frequency surfaces (EFS) of the PhCs at 260 GHz. The red dashed line represents the wavevector of free electrons. The inset shows the EFS of the photonic crystals from 255 GHz to 300 GHz.

Fig. 2. (a), (b) Propagation wavevector diagrams for (a) normal Smith-Purcell radiation (N-SPR) in vacuum and (b) reverse Smith-Purcell radiation (R-SPR) in photonic crystals (PhCs). (c) Angle-frequency relationship for N-SPR and R-SPR from simulations and numerical calculations. (d)–(g) Simulated field distributions for frequencies from 250 GHz to 300 GHz, with the grating in the center and free electrons passing through the surface of the grating. The upper (lower) half shows the N-SPR (R-SPR) field distribution.

Fig. 3. (a) Radiation angles for reverse Smith-Purcell radiation (R-SPR) and normal Smith-Purcell radiation (N-SPR) at 300 GHz with grating periods ranging from 108 μm to 120.7 μm. The inset shows the dispersion relation for the photonic crystals at 300 GHz (red line) and for free space (blue line). The free electrons wavevector ke is on either side of 0.5×2π/a for grating periods of 119 μm and 120.7 μm, respectively. (b), (c) Field distributions of N-SPR and R-SPR for grating periods of 119 μm and 120.7 μm. (d) Radiation angle as a function of grating period for varying background dielectric constants ε of the photonic crystal (9 to 13), with a duty cycle of g=0.45 and a period of a=630  μm. (e) Radiation angle as a function of grating period for varying duty cycles of the photonic crystal (g=0.44–0.47), with a background dielectric constant ε of 12.96 and a period of a=630  μm. (f) Radiation angle as a function of grating period for varying photonic crystal periods (a=570–660), with a background dielectric constant ε of 12.96 and a duty cycle of g=0.45. (g) Radiation angle as a function of grating period for varying free electron energies (2.8–3.1 keV), with a background dielectric constant of 12.96 and a duty cycle of g=0.45.

Fig. 4. (a), (b) Propagation wavevector diagrams for reverse Smith-Purcell radiation excited by 2.5 keV and 3.5 keV free electrons in photonic crystals with a=630  μm, g=0.45, and ε=12.96. The waves for x>0 are backward waves, while the waves for x<0 are forward waves. (c), (d) Relationship between the frequency and radiation angle for reverse Smith-Purcell radiation (R-SPR, red line) and normal Smith-Purcell radiation (N-SPR, black line) excited by 2.5 keV and 3.5 keV free electrons in photonic crystals and vacuum.