Traditional solid and gas lasers cannot directly generate X-ray or γ-ray. In addition to decay radiation sources, high-energy electronic radiation can effectively generate such high-energy photons. For example, using electrons emitted by electron guns to bombard targets can generate X-ray through bremsstrahlung, or injecting relativistic electrons into undulators can also generate soft/hard X-ray. The former is the oldest X-ray generation plan in history, widely used in medicine, industrial production, and other fields, whereas the latter is free-electron laser (FEL), currently one of the most cutting-edge topics. However, X-rays generated by bremsstrahlung have some defects, such as large divergence angles and poor coherence. Traditional FEL requires a large array of undulatory magnets, and this large volume of strong magnetic system cannot be deployed in such a short period (micrometer level), so it is difficult to establish such a device. If we want to use the FEL system to generate X-ray or even γ-ray, there are two direct ways: (1) we can accelerate the free electrons to extreme relativistic energy, so the shortwave radiation can be generated by these high-energy electrons via relativistic frequency up-conversion effect. (2) We can also build a longer undulator system to generate high-order harmonics and then amplify the output. Both of these plans will make FEL systems larger and more expensive, and such complex precision systems will also reduce their reliability and increase the difficulty of construction and maintenance.1

To solve the above problems, many scientists have taken different approaches and attempted non-magnetic array undulator configurations. Elias proposed the concept of “electromagnetic wave undulator” in 1979, pointing out that electromagnetic waves can interact with free electrons to generate a wave period much smaller than a magnetic array does.2 The most intuitive principle is the inverse Compton scattering (ICS) gamma laser, where electrons interact with high-energy laser pulses and undergo transverse acceleration under the laser electromagnetic field, then stimulating radiation. At this point, the oscillation period of the electron is close to the oscillation period of the laser pulse, which is much shorter than the traditional undulator period. From a quantum perspective, it can be understood that a free electron undergoes ICS with a low-energy photon, generating a high-energy photon. Massachusetts Institute of Technology (MIT) proposed an ICS gamma laser system based on a linear accelerator in 2009.3 In 2012, Ta Phuoc et al. proposed a “fully optical Compton gamma light source,” where the generation, acceleration, and oscillation of free electrons are all achieved by a single light source.4 With the development of laser wake field acceleration, it has been found that when electrons are accelerated inside plasma bubbles, generated by ponderomotive force, they will produce cyclotron betatron radiation, with significantly shorter oscillation periods than solid undulators. Electrons, without extremely high energy, can also produce strong X-rays.5^,6

In addition, the concept of an “on-chip free-electron light source” is also one of the cutting-edge types of research in recent years. There are two technical routes in this plan: (1) surface plasmon polaritons (SPP) amplification and (2) micro/nano-structure modulation of electrons. The former includes on-chip light sources that interact with free electrons and SPPs,7^,8 whereas the latter includes Smith-Purcell radiation generated by the interaction between electrons and micro/nano gratings,9^–11 and periodic near-field generated by micro periodic structures affect electron beams.12^–15 Although on-chip light sources based on Smith-Purcell radiation can also produce X-rays,16 their directionality and monochromaticity are generally inferior to near-field undulators.

With the improvement of micro/nano-processing technology, micro/nano-structures with periodic tens of nanometers can be prepared. Using appropriate external pumping, the near-field can be generated around these periodic micro/nano-structures. When an electron beam passes through the structure, it is subjected to the near-field, generating periodic acceleration and radiation.

In this paper, we propose a model that utilizes the interaction between microwaves or lasers and nano-sized structures to generate a quasi-static periodic near-field, thereby inducing electron oscillations to generate X-ray or even γ-ray and use simulation to obtain the radiation’s characters of the device. Through theoretical derivation and simulation verification, it is found that when the driving electric field intensity is determined, the number of photons radiated by electrons with γ≫1 per unit time and per unit solid angle in the undulator is independent of the electron energy.

One of the best ways to induce small periodic oscillations of free electrons under the influence of an external field is to use electromagnetic near-field modulation. For a dielectric medium in an electric field with no surface charge, if the medium is in a vacuum environment, it is easy to obtain (ε1·E→1−ε0·E→2)·e→n=0 and (E→1−E→2)×e→n=0 from Gaussian theorem, where ε0 and ε1 are the vacuum’s and the dielectric’s dielectric constants, E→1 and E→2 are the electric fields inside and outside the dielectric, and e→n is the unit vector of the interface. It can be seen that if a negative y-axis electric field is applied to the dielectric cylindrical array in a vacuum [see Fig. 1(a)], electric field distortion will occur near the cylinder [see Fig. 1(b)]. This is because the dielectric will generate displacement polarization under the effect of an external electric field, and dipole enhancement will also occur between array elements. The most noteworthy aspect is that the induced electric field of the dielectric column contains an x-component electric field which is not included in the external electric field [red and blue blocks in Figs. 1(b) and 1(c)].

Using micro/nano-processing technology, dielectric nanopillars with a diameter of 40 nm are arranged in two staggered columns along the y-axis with a center spacing of 50 nm, resulting in an x-component periodic staggered near-field channel along the y-axis [see Figs. 1(a) and 1(b)]. This near-field channel will impose a periodic transverse oscillation electric field force on the injected electrons, which can make them oscillate with extremely short periods equal to 50 nm, then radiating short-wavelength photons [see Fig. 1(d)].

To apply an electric field in the y-axis on a dielectric nanopillar array, we cannot simply use a static electric field, because a high-voltage static electric field can easily break down the medium. Thus, we use the radio frequency (RF) microwave standing wave to induce a near-field. By placing the dielectric nanopillars at specific positions in the RF cavity, the polarization of the microwave is parallel to the y-axis, the channel is at the antinode of the electric wave and the nodal position of the magnetic wave generates a near-field which will drive electrons lateral oscillate and minimize the influence of the background magnetic field as well. When an electromagnetic wave passes through an array of nanopillars which orthogonal to its polarization direction, if the nanopillars’ diameter and spacing period are far less than the wavelength, the electromagnetic wave will bypass the array due to diffraction effects and not hinder its propagation. Therefore, it can form a stable standing wave field in this structure. In addition, the nano dielectric materials’ low absorption rate of RF photons makes them less susceptible to being damaged under high-energy microwave heating.

The displacement polarization of bound electrons in a dielectric requires a certain response time, which is generally in the range of femtoseconds to sub-nanoseconds, much shorter than the oscillation period of RF microwaves. Therefore, this near-field is relatively stable. The RF cavity of an electron induction accelerator cannot sustain an electromagnetic field whose strength is higher than the GV/m level.17^,18 We hope that electrons will experience a near-field electric field strength of 10MV/m on the x-axis in the oscillation channel. If the nanopillar array is made of silicon, the standing wave amplitude should be about 80MV/m. This is feasible in engineering and there is no risk of thermal damage in the short term. If dielectric ceramic materials with higher dielectric coefficients and higher thermal damage threshold are used, the strength of the standing wave can be further reduced while the near-field electric field strength remains unchanged, or a stronger near-field can be generated using a stronger driving field to enhance the power of the light source.

It is worth noting that when the direction of the microwave electric field is reversed, the direction of the induced near-field will also be reversed, which will cause the oscillation of electrons a sudden change with the π phase. This is quite unfavorable for the monochromaticity of electron radiation. Therefore, considering that the microwave resonant cavity and microwave source of RF accelerators widely use S-band microwave (with a wavelength of about 10 cm), the length of the near-field channel of the device should be <5cm, so that the direction of the electric field remains unchanged when electrons travel through the channel. In fact, to ensure that the driving near-field intensity is sufficiently high, the near-field channel’s length is preferably <2cm, allowing electrons to be injected into the device with the highest microwave electric field.

Electrons are subjected to both transverse near-field oscillations and longitudinal acceleration driven by the background microwave electric field. For relativistic electrons with a velocity close to the speed of light, the total increment of their longitudinal velocity when passing through the oscillation channel does not exceed the order of 105m/s, which can be ignored compared with the longitudinal velocity close to the speed of light and the corresponding energy gain is not higher than the energy dissipation of the high-quality electron beam.19 Besides, as the electron beam travels through the structure, a series of electromagnetic modes excited by it will interact with the structure to produce a wakefield which will cause the energy loss of subsequent electrons. For example, a 1 fC, 5 MeV electron pulse will produce a deceleration wakefield with a rapidly decaying wake potential of up to 2 kV and the strength of the wakefield is proportional to the total charge in the interaction area. For a short relativistic electron pulse, because of the retarded effect, its wakefield affects the subsequent electron pulses.20 Because the nanostructures are discrete dielectric materials and cannot couple with the induced electromagnetic modes to form surface waves, the tail field will rapidly decay. Another way in which energy is lost by radiation is the Smith-Purcell (S-P) effect. This is essentially a form of Cherenkov diffraction radiation, which is particularly effective when electrons propagate near the conductor grating, but the radiation efficiency decreases significantly when near the dielectric material. However, the gain condition of S-P radiation requires 4πx0/λu∼1/γ, that for electrons with energies higher than 2 MeV, the left side is an order of magnitude higher than the right side, so S-P radiation is significantly suppressed. Thus, by controlling the charge of the electron pulses and the repetition rate of an electron beam, each electron pulse will be affected by a wakefield which is comparable to the magnitude of the longitudinal electric field. Moreover, there is no necessary precise phase-locked condition between electrons and the electric field in this model, such small changes in longitudinal velocity will not have a significant impact on the radiation output, which we will talk about later in Sec. 4.

In addition, due to the long RF wavelength, compared with the length of the near-field channel, the near-field can be regarded as an approximate electrostatic field when the relativistic electrons undergo several oscillation cycles, which is used in the mathematical derivation in the following text.

According to the Lienart-Wiechart potential, the instantaneous electric field strength of a moving electron at the observation point with distance R can be expressed as (Gaussian unit)21E→(x→,t)=e[n→−β→k3γ2R2]ret+ec[n→×(n→−β→)×β→˙k3R]ret,(1)where e is the elementary charge, n→ is the unit vector from the electron to the observation point, β→ is the normalized velocity of the electron, k=1−n→·β→(t′)=dt/dt′ is the time dilation factor, γ=1/1−β2 is the Lorentz factor, [·]ret is the retarded bracket, and c is the light speed in the vacuum. Only the second term can radiate to a distance. It can be seen that when electrons are subjected to external forces, they will emit electromagnetic waves transmitting perpendicular to the direction of the force.

When a relativistic electron travels in a straight channel line under an extremely short period of a transverse reciprocating electric field, it can be seen from Eq. (1) that this kind of electric field, with an extremely short period and not very high intensity, will cause the electron to apply a periodic oscillating electric field (i.e., radiate electromagnetic waves) to the forward observation point, and the electron’s trajectory will hardly produce significant deflection.

In this case, β→ approaches 1 and has a direction similar to n→, i.e., β→∼n→, and k=1−n→·β→(t′)≈1−1+12γ2=12γ2, n→−β→≈n→(1−1+12γ2)=12γ2n→, γ≫1. By substituting the second term of Eq. (1), the electric field of the radiation can be obtained as E→(x→,t)=ec[n→×(12γ2n→×β→˙)(12γ2)3R]ret=ec[n→×(n→×β→˙)R4γ4]ret.(2)

For relativistic electrons, there is a conversion relationship between the electromagnetic wave frequency ν(θ) radiated by them and the electron oscillation frequency ν0 on their oscillation plane (x-y plane):22ν(θ)≈2γ2ν0/(1+γ2θ2), i.e., the relativistic frequency up-conversion relationship (see Fig. 2). Among them, θ is the angle between the radiation direction and the direction of electron motion, which can be approximated as the angle between the radiation direction and the y-axis in this model [see Fig. 1(a)]. Due to the structure of the device and the characteristic of small-amplitude transverse oscillations of the electrons, we can consider that the θ∼0 is a constant, as approximated in the aforementioned derivation.

If electric polarization near-field and photon number are used to describe the power, according to the up-conversion relationship, the number of photons radiated by an electron per second and per unit solid angle in the undulator can be written as dN(t)dΩ=e2γ4πc2|n→×(n→×E→Pγme)|2·12γ2hν0=e22πc2hν0|n→×(n→×E→Pme)|2,(3)where E→P is the polarization field and me is the electron mass. It can be seen that for relativistic electrons with γ≫1, the number of photons radiated per unit time and unit solid angle is independent of the electron energy [see Fig. 1(d), which is a schematic diagram of the radiation generated by this model].

Because the amplitude of the electron’s oscillation is too small, it is also necessary to examine whether its oscillation process has special gain behavior from the perspective of quantum interaction. Taking the parameters mentioned in Sec. 2, when a 5 MeV 0.75 fs electron pulse with a charge of 1 fC oscillates in the undulator, its Pierce parameter ρ∼10−6, when the detuning parameter δ∼500 satisfies the resonance condition, which leads to a 5% resonance frequency shifts to positive values than the classical results. However, the detuning parameter does not satisfy the threshold condition δ<δT≡3/22/3, which means the FEL is “below the threshold” of exponential gain and exhibits a classical small signal gain,23 this light source is based on the synchrotron radiation (SR) of free electrons. Due to the quantum recoil effect, the electron that emits the photon will lose a considerable amount of momentum and leave the gain range, so the number of saturation radiation photons of an electron pulse is equivalent to the number of electrons in it.24 Therefore, in the following, we will calculate the radiation spectrum of the electrons in this device without considering the exponential gain of the FEL.

To generate electromagnetic waves with shorter wavelengths, the energy of electrons should be high enough to generate strong relativistic frequency up-conversion effects. However, the emissivity of high-energy electron beams is high, making it difficult to pass through channels with a width of only about 30 nm without colliding with nanorods, which will cause ionization. Therefore, we will mainly discuss low-energy relativistic electrons with energies of ∼1 to 6 MeV and high-energy electrons with energies of ∼10 to 20 MeV. These electrons can be supplied from compact accelerators and RF electron guns, even on-chip electron sources, which can be easily obtained in the laboratory.

The near-field distribution resulting from the interaction between the electromagnetic field and dielectric nanopillar array has been obtained through the simulation software COMSOL^® [see Fig. 1(b)], then using the particle tracking module simulates the motion of electrons and inputs their acceleration into the Lienart-Wiechart potential calculation to obtain the electron radiation intensity. In this simulation, the device is composed of silicon columns with a diameter of 40 nm and generates a near-field of 108V/m under the drive of a 10-cm wavelength RF microwave. In experimental environments, electron beams have a certain degree of energy dissipation, usually 1% or even lower, which can cause spectral broadening. However, for this model, the oscillation period of relativistic electrons is fixed, and their radiation photon frequency ∝γ2 while the energy of relativistic electrons ∝γ. Therefore, the spectral broadening effect caused by electron dispersion should be less than 0.01% and can be ignored.

The photon energy of lower energy relativistic electron radiation is in the range of ∼0.1 to 10 keV (see Fig. 3). The photon energy of 1 MeV electron radiation is about 0.2 keV, belonging to the extreme ultraviolet region [see Fig. 3(b)], whereas the photons of 3 and 6 MeV electron radiation are in the soft X-ray and X-ray region. In Fig. 3(a), it can be observed that the number of photons radiated by 1 MeV electron per unit of time per unit of solid angle is much lower than that of 3, 6 MeV electrons, which deviates from the results of Eq. (3) to a certain extent. This is because the approximation mentioned earlier was made when the Lorentz factor γ≫1, which is not accurate for lower energy electrons.

As observed in Fig. 3, the spectrum of electron radiation in this undulator is highly monochromatic. This is because the oscillation frequency of electrons in this device is entirely determined by the spatial arrangement of the dielectric nanopillar array, exhibiting high robustness and stability. For a nanopillar array with strict periodicity, the oscillation frequency of the electrons is also quasi-monochromatic. However, the radiation spectrum of electrons is not completely monochromatic due to variations in the near-field intensity caused by phase changes in the driving field.

To obtain radiation with shorter wavelengths, we can use higher-energy electron beams, but correspondingly, higher-energy electron beams have higher electron emissivity and it is more difficult to compress the beam diameters. Due to the fact that nanorods can extend along the z-axis, elliptical electron beams compressed along the x direction can be used, but this process is also very difficult for extreme relativistic electrons. To reduce the possibility of ionization caused by electron beams colliding with nanorods, the current intensity must also significantly decrease, which means that the number of photons generated per unit time will also decrease. Therefore, an electron beam of ∼10 to 20 MeV is an appropriate choice for generating hard X-ray.

Figure 4 shows the single electron radiation intensity of 10, 15, and 20 MeV electrons. It can be seen that for electrons exceeding 10 MeV, their radiation spectrum is in the hard X-ray region. Figure 4(a) shows that the number of photons per unit time and per unit solid angle emitted by strongly relativistic electrons with different energies is approximately equal, consistent with that given by Eq. (3).

As mentioned earlier, compared with traditional free electron SR light sources, the transverse displacement and the changes in the velocity direction of electrons in this system are very small. The extremely small transverse oscillation amplitude causes the undulator factor of this device K∼0, resulting in a very small divergence angle of radiation,25 which can be approximated as 1/γ. Therefore, we can say that the higher energy the electron has, the smaller the divergence angle of radiation is. The output area of the device is only on the order of μm2, so the radiation brightness is quite high.

Figure 5 shows the relationship between the radiation intensity generated by electrons in the undulator and the radius of the dielectric nanopillars when the array period is constant. The transverse near-field excited by the driving electromagnetic field decays with the square of the distance near the nanopillars, and the intensity of the dipole enhancement effect between the nanopillars is proportional to the inverse of the spacing between adjacent nanopillars. When the radius of the nanopillar is <30% of the array period, the magnitude and distance of the lateral force acting on the electrons will be greatly attenuated, greatly reducing the radiation intensity.

Although the quasi-static field generated by the microwave near-field undulator has the advantage of not requiring precise phase matching which makes the radiation spectrum highly robust, its radiation intensity is limited by the electric field amplitude of the driving microwave and cannot reach a higher level. To increase its radiation intensity by several orders of magnitude, we naturally thought of using a pulsed laser with a tilted wavefront to produce a similar quasi-static field.

In addition to using microwaves, another way to generate a quasi-static near-field is to use a laser beam with a tilted wavefront. Figure 6 shows that two wavefront tilted laser beams with the same phase are incident from both sides of the medium array to form a standing wave, the phase velocity of which matches the longitudinal motion velocity of the electrons, thereby maintaining a quasi-static near-field around the electron beam; however, when observed from the laboratory reference frame, this near-field is not quasi-static. Although this local quasi-static field that maintains phase matching with the longitudinal motion of electrons has more stringent implementation conditions and lower robustness than the microwave-driven scheme in actual experiments, its higher radiation efficiency makes up for these shortcomings.

Using pulsed lasers, the intensity of the driving field can reach a very high level, and its upper limit is mainly determined by the surface damage threshold of the dielectric material. For an ultrashort laser with a pulse width of 10 fs, the medium surface breakdown strength is near 2J/cm2,26^,27 i.e., ∼50GV/m. Due to the electric dipole enhancement effect, the field strength at the point with the highest energy density on the surface of the dielectric column is about ten times that of the driving electric field. Therefore, the longitudinal driving field strength is up to about 5GV/m; however, the stronger laser field also enhances the electric dipole enhancement effect of the dielectric array, improving the lateral electric field conversion efficiency, resulting in a transverse near-field strength of about 2GV/m, which is two orders of magnitude higher than the near-field generated by the microwave drive scheme. According to the analysis in Secs. 3 and 4, when the transverse-driven near-field intensity increases by two orders of magnitude, the radiation energy flux density generated by a single electron oscillation will increase by four orders of magnitude, that is, the number of photons radiated by the periodic undulation of a single electron increases by four orders of magnitude.

Figure 7 shows that the undulation period of this model is much smaller than the general dielectric-based undulators, and is close to the limit of plasma-based undulators. Although the driving field amplitude of dielectric-based undulators is smaller than that of plasma-based undulators, the former is more robust to the undulation period than the latter and is less likely to be damaged during operation.

In addition, the 10GV/m longitudinal acceleration field enables the electron beam to better resist the deceleration effect of the wakefield, so both the maximum charge allowed per electron pulse and the maximum repetition frequency of the electron pulses can be increased, which further improves the photon generation efficiency of the device.

We propose a mechanism based on the interaction between microwave and nanostructures to generate periodic oscillations of relativistic electrons in the near-field, promoting the generation of extremely short wavelength electromagnetic radiation. To obtain higher radiation efficiency, the higher energy femtosecond pulse laser with a tilted wavefront can be used instead of microwave as the driving source to generate a similar periodic quasi-static field around the electron beam. The melting point of dielectric materials such as monocrystalline silicon and dielectric ceramics is mostly above 1000°C and has a low absorption rate of RF photons, and the damage threshold to fs pulse lasers has also reached 2J/cm2 which makes this configuration less prone to thermal damage during operation with either microwave or laser driving systems. The microwave-driven configuration does not require strict phase matching between electrons and the driving field. It is easy to implement and has high robustness, but has low radiation power. By controlling the charge injected into the structure with an RF photocathode electron gun or a desktop accelerator system, it can be used as a desktop high-energy single-photon radiation source to produce hard X-ray single-photon radiation with good directionality. The pulsed laser-driven configuration requires a relatively strict matching relationship between the longitudinal velocity of electrons and the phase velocity of the driving light field. It has lower robustness, but the radiation power is four orders of magnitude higher than the microwave-driven configuration and allows the injected electron beams to have higher repetition frequencies and higher charge densities, thereby further improving the radiation generation efficiency. Therefore, the stronger near-field generated by the laser will make it possible to expand the width of the oscillation channel while maintaining the force intensity of electrons on the central axis, which means that extreme relativistic electrons with higher emissivity can also be injected into it to generate γ-ray or even hard γ-ray. This implies that, in addition to laser-plasma interaction and laser-solid interaction quantum electrodynamics (QED) effect,29 we may have an additional means to obtain γ-ray. The dielectric nanopillars can be fabricated with electron-transparent materials such as silicon nitride30 and silicon carbide31 and arranged in multiple columns, allowing for the injection of a large spot-size electron beam and simultaneous undulation in each column to enhance the output power.

Moreover, due to the many similarities between this structure and the structure of the dielectric laser accelerator (DLA), both use an external driving electromagnetic wave field to irradiate periodic nanostructures and generate near-field modulation of electron motion. Therefore, after adjusting the period and arrangement of the nanostructure, higher energy laser irradiation can also be used to apply for higher acceleration to electrons in this device, thereby increasing the power of electron oscillation radiation. Furthermore, this structure can be etched on the same substrate as the DLA system, allowing the acceleration, regulation, and oscillation radiation processes of the electron beam to be completed on the same chip, within a distance of a few centimeters.

Biographies of the authors are not available.