In Figure 1, an electron beam with an energy chirp is first generated by Linac 1, and then the electron beam is compressed by the magnetic bunch compressor (BC), where the length of the electron beam can be controlled by adjusting the energy chirp and the intensity of the BC. Generally, the longitudinal structures of the electron beam can be expressed by the following equation: S z , σ z = exp − z 2 2 σ z 2 2 π σ z , (1) where Z is the longitudinal coordinate of the electron beam and σ z is the length of the electron beam (rms). After that, the compressed electron beam passes through another accelerator section (Linac 2) to reach the final beam energy. The beam then produces transition radiation by passing through a metal foil directly, without the modulation and dispersion sections (chicane) in Figure 1, which are treated as drift sections (turning powers off). We use an aluminum (Al) foil as an example since the material of the foil has little effect on THz transition radiation [39, 40].

The transition radiation can be divided into forward radiation and backward radiation, which have different formation length L. Compared with forward radiation, backward radiation has a much shorter formation length [41, 42] and is easier to be collected. If only considering backward radiation, the transition radiation of a single electron can be expressed by the Ginzburg–Frank formula [36]. d 2 P e d Ω d k = r e m c 2 π 2 β 2 sin 2 ⁡ θ 1 − β 2 cos 2 ⁡ θ 2 , (2) where r e is the classical electron radius, m is the rest mass of the electron, Ω is the solid angle, k is the wave number of radiation, β is the relative speed, and θ is the angle measured against the backward radiation. For an electron beam, the form factor of the electron beam F k , σ z = ∫ e i k z S z , σ z d z 2 is the square of the Fourier transform result for the normalized longitudinal particle distribution within the electron beam, which greatly affects the intensity of transition radiation. The form factor of the compressed electron beam can be expressed as follows: F b k , σ z = e − k σ z 2 . (3)

The transition radiation of an electron beam can be expressed by the Nodvick–Saxon formula [43]. d 2 P d Ω d k = d 2 P e d Ω d k N + N N − 1 × F k , σ z , (4) where N is the number of the electrons. According to Eq. 4, the total radiation intensity can be simplified as N d 2 P e d Ω d k for F k , σ z = 0 , which is called incoherent transition radiation. On the contrary, the total radiation intensity can be simplified as N 2 d 2 P e d Ω d k for F k , σ z = 1, which is known as CTR. When substituting Eqs. 2, 3 into Eq. 4 and only considering CTR, the CTR intensity of the compressed electron beam can be obtained using the following equation: d 2 P d Ω d k = N 2 β 2 r e m c 2 π 2 sin 2 ⁡ θ 1 − β 2 cos 2 ⁡ θ 2 × e − k σ z 2 . (5)

Taking d Ω = 2 π sin ⁡ θ d θ and integrating wave number k of radiation, the angular distribution of the CTR can be expressed as follows: d P d θ = N 2 β 2 r e m c 2 π σ z × sin 3 ⁡ θ 1 − β 2 cos 2 ⁡ θ 2 . (6)

According to Eq. 6, the CTR has a conical distribution with the maximum intensity at angle θ = sin − 1 3 / γ 2 − 1 . When γ is much greater than 1, the θ is approximately equal to 3 / γ . In addition, taking d k = − 2 π / λ 2 d λ and integrating solid angle Ω , the energy spectrum of the CTR can be expressed as follows: d P d λ = N 2 r e m c 2 2 ln 2 γ − 1 2 π σ z × Erf 2 π σ z λ , (7) where Erf x = 2 / π ∫ 0 x e − η 2 d η is the error function. From Eq. 7, the CTR is a broad-band THz radiation with a center frequency related to length σ z .

To obtain narrow-band THz radiation, the CTR generated by an electron bunch train is also introduced. In Figure 1, an ultrafast laser is stretched by a parallel grating pair to introduce a linear chirp with chirp parameter α = 1 / σ 0 σ o u t , where σ 0 is the initial laser pulse length (rms) and σ o u t is the stretched laser pulse length. Furthermore, the stretched laser pulse is divided into two laser pulses by a beam splitter to introduce tunable time delay τ by a tunable optical delay line. Finally, the two laser pulses are recombined to obtain a frequency-beating laser pulse with a beating frequency [28, 29] f = α τ π . (8)

According to Eq. 8, the beating frequency can be continuously tuned by adjusting chirp parameter α and time delay τ .

The electron beam at the exit of Linac 2 is sent into a modulation section together with the frequency-beating laser pulse to interact with each other. Due to the electron phase differences in the initial electron beam, some electrons obtain energy from the frequency-beating laser, while the other electrons transfer energy to the frequency-beating laser during the interaction. The energies of the electrons in the electron beam become different, resulting in periodic energy modulation over the wavelength of the frequency-beating laser [44, 45]. Then the modulated beam is sent to the chicane to convert the energy modulation into density modulation and obtain a pulse train structure with a THz period, which can be described as M Gaussian microbunches with a time interval of T. At the same time, the frequency-beating laser pulse is reflected out by an optical mirror. Finally, the electron bunch train passes through the Al foil to generate narrow-band THz radiation. Here, the form factor of the electron bunch train can be expressed as follows [46, 47]: F t k , σ z = e − k σ s 2 × sin M k c T / 2 M s i n k c T / 2 2 , (9) where σ s is the length of microbunch in the electron bunch train (rms). According to Eq. 9, the form factor has peaks at the resonance wavelength and its harmonics, which can be tuned by manipulating the frequency-beating laser pulse. Substituting Eqs 2, 9 into Eq. 4, the CTR intensity of the electron bunch train can be expressed as follows: d 2 P d Ω d k = N 2 β 2 r e m c 2 π 2 sin 2 ⁡ θ 1 − β 2 cos 2 ⁡ θ 2 × sin M k c T / 2 M s i n k c T / 2 2 × e − k σ s 2 . (10)

Taking d k = − 2 π / λ 2 d λ and integrating solid angle Ω , the energy spectrum of the CTR can be expressed as follows: d P d λ = 4 N 2 r e m c 2 − β + 1 + β 2 tanh − 1 ⁡ β M 2 λ 2 β × − 1 + cos 2 π c M T λ − 1 + cos 2 π c T λ × e − 4 π 2 σ s 2 λ 2 . (11)

In this simulation, an electron beam with a beam energy of 125 MeV, a pulse length of 1.7 ps, and a charge of 500 pC is produced at the entrance of the BC, and the bunch length can be easily adjusted with a minimum of 100 fs by tuning the current of the BC. The beam is then accelerated to 1.4 GeV by Linac 2. Finally, the beam passes through the Al foil with a thickness of several μ m to produce the CTR. Here, an ideal longitudinal structure of the electron beam described in Eq. 1 is adopted, and then the form factors with different pulse lengths are shown in Figure 2.

According to Figure 2, the form factor will decrease more slowly and have a relative wider bandwidth with the decrease of the beam length, and the CTR generated by the compressed electron beam will have a broad-band spectrum. Figure 3 shows the angular distributions of the CTR with a beam length of 100 fs and different energies.

According to Figure 3, the angular distribution has a double-lobe distribution, which conforms to the conical distribution of the CTR. In addition, the intensity of the CTR will decrease and the divergence will increase with decrease in beam energy. For the electron beam with a beam energy of 1.4 GeV, the CTR has the maximal intensity at the angle of 6.28 × 10 − 4 rad and has a good orientation, which is easy to be collected using an off-axis parabolic mirror. Figure 4 shows the CTR spectra generated by the electron beams with a beam energy of 1.4 GeV and different beam lengths.

From Figure 4, if f is less than c / 2.35 σ z , the CTR intensity increases rapidly and has a maximum at c / 6 σ z ; otherwise, the enhancement effect disappears. The CTR intensity will increase with the decrease in the beam length and the center frequency is also proportional to the beam length; thus, an electron beam with a relative shorter beam length is beneficial to generate CTR radiation with a higher frequency and wider spectral bandwidth. However, compressing the length of the electron beam to below 100 fs generally is a significant challenge. The SXFEL can compress the electron beam to a final bunch length of 100 fs, and the beam can be used to generate CTR radiation with a pulse energy of 342 µJ and a frequency up to 5 THz. Moreover, the bunch length of the electron beam at SXFEL can easily be adjusted between 100 fs and 1.7 ps so the broad-band THz radiation can be produced from 0.1 to 5 THz.

To obtain THz radiation with a higher frequency and narrow spectral bandwidth, the CTR generated by a frequency-beating laser-modulated electron beam is also introduced. In this simulation, a laser pulse with the central wavelength of 800 nm and the initial pulse duration of 30 fs is adopted. The laser pulse is first stretched to 42 ps by the paralleled grating pair, and then a time delay of 40 ps is introduced to obtain the frequency-beating signal at the frequency of 10 THz (30 μ m ). The longitudinal amplitude distributions and spectrum of the frequency-beating laser pulse are shown in Figures 5A, B, where one can observe that the fundamental wavelength is about 30 μ m .

The beating frequency can be tuned by adjusting chirp parameter α and time delay τ , which is limited by the initial laser power and pulse length. In theory, the beating frequency can be easily adjusted from 0.1 to 30 THz [29], while the number of modulation periods can decrease with the decrease in the beating frequency for limited electron bunch length, which may broaden the spectral bandwidth. In addition, the fundamental Fourier component decreases from 0.23 to 0.15 when the beating frequency is adjusted from 10 to 15 THz according to Figures 5C,D. For a larger beating frequency, a larger time delay τ will further decrease the fundamental Fourier component, which can decrease the final radiation power. To obtain a laser pulse with a larger beating frequency and sufficient fundamental Fourier component, higher initial laser power and shorter initial pulse length are required. Therefore, laser pulses with beating frequencies from 5 to 15 THz and sufficient fundamental Fourier component can be easily obtained by the frequency-beating technique. As an example, the frequency-beating laser pulse with a frequency of 10 THz is used as a seed in the following modulator.

Here, an electron beam with a full pulse length of 2 ps and a beam energy of 1.4 GeV at the exit of Linac 2 is adopted to simulate the performance. The frequency-beating laser pulse and the electron beam are sent into the modulation section to interact with each other and obtain the energy modulation with 20 periods at a frequency of 10 THz (100 fs). The electron beam passes through the modulator and the chicane with R 56 of 13.01 cm to convert the energy modulation into density modulation. The longitudinal phase spaces after the chicane and corresponding current profile of the electron bunch train are shown in Figure 6.

From Figure 6, one can find that the electron bunch train includes 20 microbunches, where the pulse length of each microbunch σ s is 19.2 fs and time interval T is 100 fs. Figure 7 shows the form factors of the electron bunch trains with different numbers of microbunches and the CTR spectrum generated by the electron bunch train with 20 microbunches.

From Figure 7, one can observe that the form factors have an apparent peak at 10 THz with a form factor of 0.23, and the CTR generated by the electron bunch train has a narrow spectral bandwidth. In addition, the spectral bandwidth decreases from 2.69 μ m to 0.66 μ m as the number of the microbunches increases; thus, a relative longer electron beam can also decrease the spectral bandwidth. The pulse energy of the CTR generated by the electron bunch train with 20 microbunches can reach 91 μJ at 10 THz. It is worth mentioning that the longitudinal space charge force has limited influence on the frequency-beating laser-modulated electron beam so the method can be adopted for a higher beam charge to generate THz radiation with higher pulse energy. Furthermore, the radiation frequency can be easily tuned from 5 to 15 THz by adjusting the beating frequency.

To compare the performance of the actual electron bunch train and the Gaussian microbunches, the form factors of the CTR generated by the actual electron bunch train and the 20 Gaussian microbunches are shown in Figure 8.

Figure 8 shows that the form factor of the actual electron bunch train at the fundamental radiation is 0.22, which is close to the form factor of the Gaussian microbunches (0.23). However, the form factors of these two cases are quite different at the second harmonic. The form factor of the Gaussian microbunches is only 0.003 at the second harmonic, while the form factor of the actual electron bunch train is 0.05. Moreover, the form factor of the actual electron bunch train shifts from 10 to 10.6 THz compared with that of the Gaussian microbunches. These differences are caused by several reasons: the actual electron bunch train has a variable time interval, unlike the Gaussian microbunches, and the length of each microbunch of the actual electron bunch train is shorter than that of the Gaussian microbunches. Nevertheless, the CTR of the actual electron bunch train at the fundamental frequency can be estimated using the Gaussian microbunches.