Free electron radiation [1–5] including Cherenkov radiation [6–8], transition radiation [9,10], and Smith-Purcell radiation has become a versatile platform for studying light-matter interactions [11,12], enabling the development of tunable light sources and enhancing particle detection [13–16]. Recently, reverse Cherenkov radiation, in which radiation is emitted in the direction opposite to the motion of charged particles, resulting in a reversed Cherenkov cone, has been extensively studied [6,17–19]. The reverse Cherenkov radiation enhances our understanding of electron-material interactions and offers a new level of control over radiation emission. Surprisingly, the exploration of reverse Smith-Purcell radiation, also driven by free-electron-material interactions, has been relatively limited.

Smith-Purcell radiation occurs when free electrons interact with a periodic structure such as a grating. Smith-Purcell radiation has attracted significant interest due to its potential applications in generating tunable light sources across microwave, terahertz [20,21], and X-ray frequencies [22–25]. However, Smith-Purcell radiation emits light in all directions, with the frequencies decreasing monotonically as the observation angle increases relative to the direction of the free electrons’ motion, similar to the Doppler effect [26–28]. These characteristics limit the collection efficiency of Smith-Purcell radiation, hindering its application in developing a broadband light source or a high-power radiation source in a specific collection direction. Various methods have been proposed to harness the potential of Smith-Purcell radiation [29,30], including the use of free electron lasers [31,32], superradiance excited by strong nonlinear interactions [33], and resonant cavities [34]. These methods rely on conventional mechanisms to manipulate the radiation characteristics [35–39]. Advancements in artificial electromagnetic materials, particularly photonic crystals, provide a new way to manipulate electromagnetic radiation [40–44]. Photonic crystals have resulted in the observation of many unique phenomena such as negative refraction [45–49], reverse Cherenkov radiation [19,50,51], and reverse Doppler radiation [52,53]. The reverse Doppler effect refers to a phenomenon in which the frequency of light (or other waves) increases when the source is moving away from the observer and decreases when it approaches the observer. This behavior is the opposite of the traditional Doppler effect. Reverse Cherenkov radiation refers to a phenomenon in which radiation is emitted in the opposite direction of a particle’s motion, effectively reversing the direction of the Cherenkov cone. These unique phenomena arise from the precise design of the photonic band structure, allowing for tailored characteristics like zero-refractive-index behavior [54], slow light propagation [55], and enhanced light-matter interactions. By tuning the dispersion properties, photonic crystals can support left-handed characteristics, leading to novel phenomena like backward wave propagation and enhanced control of radiation direction. In this work, we demonstrate a new type of reverse effect, namely, reverse Smith-Purcell radiation. The main feature of reverse Smith-Purcell radiation is the reversal of the relationship between radiation frequency and radiation angle.

Here, we achieve reverse Smith-Purcell radiation in photonic crystals with left-handed properties, where the lower-frequency components propagate forward, while the higher-frequency components propagate backward [schematic is shown in Fig. 1(a)]. The reverse Smith-Purcell radiation confines the radiation to a narrow angular range, approximately within ±40°, which provides a way to obtain broadband light sources in a specific observation angle. Furthermore, by precisely adjusting the grating geometry and the kinetic energy of the free electrons, we can control both the radiation direction and output frequencies. We would like to emphasize that the reverse Smith-Purcell radiation achieved here is fundamentally distinct from the inverse Smith-Purcell effect that refers to the process where charged particles are accelerated by interacting with an incident electromagnetic wave [56]. We would like to point out that a photonic crystal has also been employed to enhance the properties of X-ray scintillation via the Purcell effect by manipulating the local density of optical states through photonic or plasmonic structures [57,58]. Here, we utilize photonic crystals to facilitate reverse Smith-Purcell radiation, allowing for precise control over both the direction and frequency of the emitted radiation. Our results provide a promising platform to study unexplored light-matter interactions, and open avenues to obtain tunable, broadband light sources.

In conventional Smith-Purcell radiation, the output wavelength is given as [59] λ=L−n(1β−cosθ),(1)where L is the grating length, β≡v/c, v is the velocity of the incident free electron, c is the speed of light in vacuum, n is the radiation order, and θ is the observation angle, referring to the angle between the observation direction and the moving direction of the free electron beam. The above equation indicates that the wavelength of Smith-Purcell radiation is determined by the kinetic energy of the incident free electrons, the grating length, and the observation angle θ. We investigate Smith-Purcell radiation at frequencies around 300 GHz, using a grating with a period of L=118μm, height of h=100μm, and length of 500 μm, combined with free electron energy of 3 keV. The periodicity of the grating causes the dispersion relation of the free electron to unfold periodically with a periodicity of 2π/L. In Fig. 1(b), the red dashed lines represent the 0th and −1st order free electron dispersions. The −1st order intersects the free-space radiation region [gray area in Fig. 1(b)] within a frequency range of 247–307 GHz, corresponding to a radiation angle from 180° to 0°.

To achieve left-handed properties in this frequency range, we designed a hexagonal arrangement of photonic crystals. The designed air-cylinder photonic crystals have a lattice period of a=630μm, with a cylinder radius of g×a (g=0.45), and a background dielectric constant of ε=12.96, suitable for materials such as Si or AgGaS. The band structure and equal-frequency surfaces of these photonic crystals [as shown in Figs. 1(b) and 1(c)] were calculated using the plane-wave expansion method [52,60]: ∑G‖(k‖+G‖)·(k‖+G‖′)κ^(G‖−G‖′)A(k‖|G‖′)=ω2c2A(k‖|G‖),(2)where k‖=kxex+kyey, G‖=b1ex+b2ey, κ^(G‖−G‖′) is the Fourier expansion coefficient, and A(k‖|G‖′) is a column vector. k‖ represents the wavevector. The left side of the eigenvalue equation is a symmetric positive-definite matrix along the diagonal, meaning all of its eigenvalues are positive. Therefore, ω2/c2 can be considered as the eigenvalue λ of the matrix, and hence ω=λ·c2. We optimized the overlap between the photonic crystal’s left-handed frequency range and the Smith-Purcell radiation range by adjusting the lattice period and dielectric constant. The lattice period defines the photonic crystal’s band structure, which positions the bandgap to align with the desired radiation frequencies. The dielectric constant influences the refractive index contrast, fine-tuning the bandgap’s width and position, ensuring efficient coupling with the Smith-Purcell radiation range. Specifically, within the frequency range of 245–305 GHz, the frequency at the Γ point (the center of the Brillouin zone) decreases as the Bloch wavevector (k) increases, as depicted in Fig. 1(b). This behavior indicates that the group velocity is opposite to the phase velocity (k·v<0), resulting in left-handed properties (n<0) [45]. Consequently, the left-handed frequency range of the photonic crystals closely aligns with the Smith-Purcell radiation range (247–307 GHz), making these photonic crystals highly promising for further Smith-Purcell radiation studies.

For instance, at 260 GHz, the wavevector of a 3 keV free electron interacting with a grating of 118 μm is given by ke=cos(ω/u)−2π/L, as shown by the red dashed line in Fig. 1(c). The direction of reverse Smith-Purcell radiation propagation (indicated by the black arrow) is determined by the continuity of the tangential wavevector component and the gradient of the equal-frequency surfaces of photonic crystals. At this frequency, reverse Smith-Purcell radiation propagates forward, while normal Smith-Purcell radiation propagates backward in free space. This unique dispersion property of the photonic crystals, combined with Smith-Purcell radiation, provides a novel method to control radiation direction and frequency. The photonic crystals with left-handed properties were introduced on the lower half of the grating [Fig. 1(a)], spatially separated from the free electrons to avoid direct interactions, allowing simultaneous observation of Smith-Purcell radiation in both photonic crystals and vacuum for comparison. The combined double-periodic structure of photonic crystals and gratings effectively avoids a mismatch between the Smith-Purcell radiation range and the left-handed frequency range of photonic crystals.

Smith-Purcell radiation excited by free electrons in vacuum shows a monotonic red shift as the frequency decreases. The emission propagation diagram of Smith-Purcell radiation in free space is circular [Fig. 2(a)]. In contrast, in photonic crystals, Smith-Purcell radiation propagation, analyzed using equal-frequency surfaces, exhibits an irregular wave-like pattern [Fig. 2(b)]. For frequencies above 300 GHz, the wavevector of the free electrons exceeds 0.5×2π/a, resulting in forward Smith-Purcell radiation propagation, similar to normal Smith-Purcell radiation, yet with a larger radiation angle, near 90°. For frequencies between 275 GHz and 300 GHz, reverse Smith-Purcell radiation propagates backward, opposite to normal Smith-Purcell radiation, with an increasing angle as the frequency decreases. Normal Smith-Purcell radiation in free space propagates backward between 248 GHz and 275 GHz, whereas it propagates forward with an increasing radiation angle as the frequency decreases in photonic crystals. Normal Smith-Purcell radiation and reverse Smith-Purcell radiation both exhibit backward propagation below 248 GHz, with the free electrons’ wavevector less than 0.5×2π/a.

Reverse Smith-Purcell radiation is confined within the range of 47° to 132° due to the dispersion properties of photonic crystals. As the radiation angle increases from 47° to 90° and exceeds 90°, the reverse Smith-Purcell radiation frequency rises from 247 GHz to 274 GHz; as the angle continues climbing, the frequency ranges from 275 GHz to 308 GHz.

The CST simulation results further illustrate the differences between the following two scenarios. When in free space, the radiation angle decreases from 155° to 34° [Figs. 2(d)–2(g)] with the frequency increasing from 250 GHz to 300 GHz, consistent with the theoretical calculations [Fig. 2(a)]. However, in photonic crystals, the radiation characteristics are significantly modified by the periodic structure: the backwards-propagating radiation at 250 GHz and 260 GHz is transformed into forward propagation [Figs. 2(d) and 2(e)], while the forward-propagating radiation at 280 GHz and 290 GHz changes to backward propagation [Figs. 2(f) and 2(g)]. The radiation angles extracted from the simulation are highly consistent with the theoretical predictions [dashed lines in Fig. 2(c) represent theoretical calculations, and circles represent simulation results].

We observed that reverse Smith-Purcell radiation excited by free electrons tends to switch between forward and backward radiation at the frequency extremes of Smith-Purcell radiation, determined by the relationship between ke and 0.5×2π/a, where higher frequencies correspond to larger ke. To better analyze this directional transition, we fixed the observation frequency at 300 GHz and varied the grating period from 108 to 120.7 μm to tune ke, resulting in a normal Smith-Purcell radiation angle between 10° and 90°. In photonic crystals, the corresponding reverse Smith-Purcell radiation angle ranges from 53° to 92° [Fig. 3(a), red line]. The Smith-Purcell radiation angle at 300 GHz is consistently above 53°, as determined by the relationship between k=ω/c and 2π/a. Notably, ke reaches 0.5×2π/a at 119.5 μm grating period, marking the critical point where the radiation direction switches [shown in the inset of Fig. 3(a)], and the effective refractive index of the photonic crystals transitions from negative to positive. The effective refractive index is negative for L<119.5μm, leading to reverse Smith-Purcell radiation, and positive for L>119.5μm. The simulated field distribution shows excellent agreement with the theoretical calculations [circles in Fig. 3(a)]. Simulation results indicate that when L=119μm, the Smith-Purcell radiation angle shifts from 30° to 92° in vacuum and the photonic crystals, respectively, suggesting backward propagation. Conversely, Smith-Purcell radiation in vacuum (10°) and in the photonic crystals (87°) exhibits forward propagation [solid circles in Figs. 3(b) and 3(c) correspond to Fig. 3(a)] at L=120.7μm. Hence, we confirm that reverse Smith-Purcell radiation in photonic crystals can be realized only when ke<0.5×2π/a.

To understand the robustness of reverse Smith-Purcell radiation, we have investigated the effects of the background dielectric constant, the period of the photonic crystals, and the energy of free electrons on this phenomenon. Figures 3(d) and 3(g) demonstrate that variations in the background dielectric constant and duty cycle of the photonic crystal result in changes in the Smith-Purcell radiation angle with respect to the grating period, while the transition point between normal and reverse Smith-Purcell radiation remains at L=119.5μm. Figures 3(f) and 3(g) illustrate how variations in both the grating period and the energy of free electrons shift the transition point, offering enhanced control over the reverse Smith-Purcell radiation effect. The presence of a small number of point imperfections in asymmetric photonic crystals still enables the achievement of reverse Smith-Purcell radiation. Furthermore, we observed that even with variations in the photonic crystal period (a) and the duty cycles (g) of up to 4%, the reverse Smith-Purcell radiation phenomenon remains detectable, showcasing the robustness of the device against fabrication variations. Grating imperfections with one defect in 10 periods and 1% free electron energy spread do not affect the reverse Smith-Purcell radiation phenomenon. It is noteworthy that the photonic crystals’ sensitivity to the incident angle and frequency is determined by the shape and magnitude of equal-frequency surfaces, respectively, with the equal-frequency surfaces being non-circular in this structure, and a smaller equal-frequency surfaces area corresponding to higher frequencies.

By adjusting the velocity of free electrons, the radiation characteristics of Smith-Purcell radiation within photonic crystals can be precisely controlled. For a free electron kinetic energy of 2.5 keV, the bandgap of photonic crystals plays a crucial role in limiting Smith-Purcell radiation excitation to frequencies above 235 GHz [Fig. 4(a)]. The wavevector diagram shows that most radiation is induced by free electrons with ke>0, producing backward waves, opposite to normal Smith-Purcell radiation. Theoretical calculations show that reverse Smith-Purcell radiation occurs within specific angular ranges [Fig. 4(b)]. As the observation angle increases, the frequency decreases from 250 GHz to 235 GHz. However, due to the bandgap of photonic crystals, angles from 65° to 90° do not support Smith-Purcell radiation excitation. Beyond 90°, the frequency shifts from 278 GHz to 251 GHz. This shift reflects a transition from forward to backward propagation. As the free electrons’ energy decreases, the Smith-Purcell radiation frequency range also decreases. This frequency range can overlap with the photonic crystal’s bandgap, providing a filtering effect that further refines the radiation characteristics by restricting certain frequency components. When ke<0.5×2π/a, the Smith-Purcell radiation distinctly propagates backward. When the kinetic energy of the free electrons is increased to 3.5 keV, the frequency range for Smith-Purcell radiation in free space expands. In this case, radiation is primarily induced by free electrons with ke<0, resulting in backward propagation [Fig. 4(c)]. This leads to similar behavior where, for ke<0.5×2π/a, normal Smith-Purcell radiation shows backward characteristics, and reverse Smith-Purcell radiation is confined to specific angular ranges defined by the dispersion properties of the photonic crystals [Fig. 4(d)]. The ability to precisely control the kinetic energy of the free electrons and the corresponding wavevector enables fine-tuned manipulation of both normal Smith-Purcell radiation and reverse Smith-Purcell radiation, making it a powerful tool for directing and modulating electromagnetic radiation.

In conclusion, we have proposed the concept of the reverse Smith-Purcell radiation effect, where the radiation direction is reversed, using left-handed materials and demonstrated its feasibility within photonic crystals through both theoretical calculations and electromagnetic simulations. Moreover, we analyzed the process of refractive index transition from negative to positive at the Brillouin zone boundary in photonic crystals at approximately 300 GHz. In the designed photonic crystals, not only is the propagation direction of Smith-Purcell radiation reversed, but we also compress the full-angle Smith-Purcell radiation into a narrower angular range. The properties of reverse Smith-Purcell radiation in photonic crystals are dictated by the arrangement of photonic crystals, as seen in its band structure and the equal-frequency surfaces. As a result, the desired reverse Smith-Purcell radiation effect can be achieved by adjusting the structural parameters of the photonic crystals or incorporating additional materials with unique characteristics.

This reverse Smith-Purcell radiation phenomenon is not limited to the microwave range but can potentially extend to visible light by designing suitable photonic crystals or left-handed materials. Bandwidth is determined by free electron parameters, grating period, and photonic crystal design, which can be optimized for the desired bandwidth. Stronger angle control from photonic crystals requires a higher effective dielectric constant, which is typically accompanied by higher losses. These losses can be mitigated by incorporating impedance matching layers. Additionally, experimental conditions, such as material choice and free electron parameters, should be optimized for each frequency range to ensure optimal performance. Higher free electron densities improve interaction efficiency but also introduce additional losses. These effects can be minimized by selecting materials with lower absorption at specific frequencies, optimizing the free electrons, and using impedance matching layers to reduce material losses. The proposal and research of reverse Smith-Purcell radiation reveal a new physical phenomenon that opens up exciting possibilities for controlling radiation direction and frequency with high precision. Furthermore, reverse Smith-Purcell radiation, combined with the enhanced properties of the photonic crystals [61], holds great potential for achieving high-power, tunable Smith-Purcell radiation sources. If an artificial material with a stable refractive index over a specific bandwidth and angle can be designed, we could potentially realize broadband, single-angle Smith-Purcell radiation. This has profound implications for the development of tunable THz Smith-Purcell radiation sources and their applications in advanced technologies such as imaging, communications, and particle detection.