## 2 Method and Principles  
Figure 1 shows the schematic layout of the proposed technique. The electron beam is produced from a photocathode gun with a beam energy of several keV, sequentially the beam energy will be enhanced to tens of MeV by a short LINAC. Then the electron beam is sent into a short undulator (modulator) to interact with a chirped frequency beating laser pulse and get sufficient energy modulation. This energy modulation is converted into density modulation by a small chicane. After that, the bunched electron beam will emit intense coherent THz radiation in the following radiator. In the proposed technique, the critical point is the preparation of the seed laser, which involves tunable THz frequency signal. This process contains the optical heterodyning of two linearly chirped broadband pulses, as presented in Figure 1. To generate the required seed laser, ultra-fast laser pulses are stretched to create large frequency chirps. Here, we consider a Gaussian optical pulse with a central frequency at \(\omega_{0}\), an initial amplitude of \(2A\) and a transform-limited bandwidth of \(\sigma\). The electrical field can be written as \(E_{0}=2 Ae^{-(i\omega_{0}t+\frac{\mu^{2}}{\sigma^{2}})}\). After propagating through a linearly dispersive medium, the laser pulse undergoes a frequency-dependent phase modulation, which can be written as [46]:  
\[\phi(\omega)=\phi(\omega_{0})+\tau_{0}(\omega-\omega_{0})+\frac{(\omega-\omega _{0})^{2}}{2\mu}+\ldots\, \tag{1}\]  
where \(\tau_{0}\) is the group delay at \(\omega_{0}\), \(\mu\) is the carrier frequency sweep rate (linear chirp rate), \(\mu=\frac{d\omega}{dt}\). The chirped laser pulse firstly passes through the optical pulse stretcher (generally based on gratings) to broaden the pulse length of the lasers to \(\sigma_{n}\). when \(\sigma_{n}\ \gg\ \sigma\), \(\sigma_{n}\) can be calculated as \(\sigma_{n}=\sigma\sqrt{(1+4/\mu^{2}\sigma^{4})}\). The optical pulse stretcher can disperse the chirped pulse to different position, thus broadening the pulse length, which can be tuned as required. Here we ignore the high-order chirps, and the electrical field of the laser pulse becomes  
\[E_{f}=A\sqrt{\frac{\sigma}{\sigma_{n}}}e^{-[\frac{\mu^{2}}{\sigma_{n}^{2}}(1-i \frac{\omega_{0}}{\sigma})}\times e^{i[\phi_{0}-\omega_{0}(t+\tau_{0})-\frac{ \pi}{4}]}. \tag{2}\]  
Figure 1: Schematic layout of the proposed technique. The length from photocathode gun to the exit of chicane is about 25 m (include photocathode gun, LINAC, modulator, chicane, and drifts), the total undulator length is about 10–15 m (a undulator length of 4 m is enough for FEL saturation, tapered undulators are required to obtain higher pulse energy). After that, a beam splitter is used to split the initial ultra-short pulse and send them to two branches. One of the branches contains a delay line (for example, a Michelson interferometer arrangement) that is utilized to control the time delay \(\tau\) between these two pulses. Finally, the two laser pulses are recombined and sent into the modulator. Considering that these pulses are linearly chirped, the frequency content of the two pulses at every point in time domain differs by a constant frequency \(f_{0}\), which is determined by the time delay \(\tau\) and initial frequency chirp. The intensity of the combined pulse is then given by  
\[I=\frac{1}{2}\Big{|}E_{f}\Big{(}t+\frac{\tau}{2}\Big{)}+E_{f}\Big{(}t+\frac{ \tau}{2}\Big{)}\Big{|}^{2}=I^{+}(t)+I^{-}(t)+E_{0}{}^{2}\bigg{(}\frac{\sigma}{ \sigma_{n}}\bigg{)}\times e^{(-\frac{t^{2}}{\sigma_{n}^{2}})}e^{(-\frac{t^{2} }{2\sigma_{n}^{2}})}\times\cos\bigg{(}\frac{2t\tau}{\sigma\sigma_{n}}+\omega_{ 0}\ \tau\bigg{)}, \tag{3}\]  
where \(I^{\pm}(t)=E_{0}{}^{2}\Big{(}\frac{\sigma}{2\sigma_{n}}\Big{)}e^{-2(\pm\frac {\pm}{2})^{2}/\sigma_{n}^{2}}\) is the intensity of low-frequency (DC) component.