2 Principle

In order to compress the electron beam, an energy chirp is usually introduced by either the radiofrequency (RF) or an external laser. After passing through a bunch compressor (BC), the high-energy particles move forward while the low-energy ones move backward, indicating the compression or stretching of the electron beam. For a given strength of the BC, the compression factor depends on the energy chirp of the electron beam, which is determined by the amplitude and frequency of the RF wave or laser. In the FEL facilities, the electron beam is usually compressed to several kA by the BC in the linear particle accelerator (linac). The RF used for bunch compression ranges from 1.3 to 12 GHz^[ Reference Akre, Dowell, Emma, Frisch, Gilevich, Hays, Hering, Iverson, Limborg-Deprey, Loos, Miahnahri, Schmerge, Turner, Welch, White and Wu ^{35 ]}. For advanced electron beam manipulation techniques, the energy chirp is usually induced by an external laser with peak power of approximately 10 GW and pulse energy of approximately 10 mJ in a short undulator (modulator). The required pulse energy of the laser is proportional to the beam length and square of the beam energy^[ Reference Emma, Bane, Cornacchia, Huang, Schlarb, Stupakov and Walz ^{21 ,} Reference Huang, Ding, Huang and Qiang ^{22 ]}. The frequency of the laser is about two orders of magnitude higher than that of the RF wave, resulting in a two orders of magnitude larger energy chirp in the electron beam with the same additional energy spread. As a result, the peak current of the electron beam could be enhanced to several tens of kilo-ampere, making the laser-based method much more efficient for generating energy chirp in the electron beam than the RF wave.

In the modulator, the energy modulation is introduced through the laser–electron beam interaction. The energy modulation is converted into density modulation after the beam passes through the magnetic chicane downstream of the modulator. Now we consider that the electron has the initial coordinate $\left(s,p\right)$ , with $s=z-{\overline{v}}_zt$ being the relative coordinate, $z$ the longitudinal coordinate along the electron beam, ${\overline{v}}_z$ the average velocity of the electron beam along the $z$ -axis and $t$ the time. Besides, we define a relative energy $p=\left(\gamma - {\gamma}_0\right)/\gamma$ , where $\gamma$ is the Lorentz factor of the electron and ${\gamma}_0$ denotes the average value for the electron beam. The coordinate of the electron at the end of the magnetic chicane is as follows:

(1) $$\begin{align}{s}^{\prime}=s+{R}_{56}{p}^{\prime },\end{align}$$

(2) $$\begin{align}{p}^{\prime}=p+{p}_0\sin \left(\omega t+{\phi}_0\right),\end{align}$$

where $\omega =2\pi c/\lambda$ is the frequency of the external laser at the wavelength of $\lambda$ , $c$ is the light speed, ${p}_0$ is the energy modulation amplitude in the modulator, ${\phi}_0$ is the initial phase and ${R}_{56}$ is the longitudinal dispersion of the magnetic chicane. In order to maximize the peak current, the optimum dispersion $\rho$ should be as follows:

(3) $$\begin{align}\rho =\frac{\lambda }{2\pi {p}_0},\end{align}$$

where the condition $\mathrm{d}s/ \mathrm{d}p=0$ is satisfied. The longitudinal phase space evolution of the electron beam after passing through the modulator and dispersion section is given in Figure 1 (left).

Figure 1 Longitudinal phase space of the electron beam with normal laser modulation (left), frequency beating with the unchirped laser (middle) and with the chirped laser (right).

Now we consider the frequency beating cases. The electron is transported into the modulator to interact with two unchirped lasers with the central frequencies ${\omega}_1$ , ${\omega}_2$ and energy modulation amplitudes ${p}_1$ , ${p}_2$ , assuming ${\omega}_1>{\omega}_2$ . The coordinates of the electron at the end of the magnetic chicane are as follows:

(4) $$\begin{align}{s}^{\prime}=s+{R}_{56}{p}^{\prime },\end{align}$$

(5) $$\begin{align}{p}^{\prime}=p+{p}_1\sin {\omega}_1t+{p}_2\sin \left({\omega}_2t+\Delta \phi \right),\end{align}$$

where $\Delta \phi$ is the phase difference between two lasers. We take $\Delta \phi =0$ and assume that the two lasers have the same envelope with ${p}_1={p}_2={p}_0$ ; then the energy modulation could be written as follows:

(6) $$\begin{align}{p}^{\prime}=p+2{p}_0\sin \left(\frac{\omega_1+{\omega}_2}{2}t\right)\cos \left(\frac{\omega_1-{\omega}_2}{2}t\right).\end{align}$$

It is obvious that the right-hand side of Equation (6) can be separated as the fast-oscillating term and the slow-oscillating term. The frequency beating node, indicating the slow-oscillating term and equal to the wavelength of the current spike, is dominated by the wave number difference of two lasers. The longitudinal phase space evolution of the electron beam along the modulator and dispersion section is shown in Figure 1 (middle), from which one can see that the current spikes are equispaced. The wavelength of the seed laser is much shorter than the distance between the current spikes.

Furthermore, we consider the case that two linearly-chirped lasers are adopted to interact with the electron beam in the modulator with the central frequencies ${\omega}_1$ and ${\omega}_2$ . An unchirped laser passes through the dispersion component and the phase modulation $\phi \left(\omega \right)$ can be expanded in the Taylor series at ${\omega}_0$ as follows^[ Reference Zhang, Wang, Jiang, Li, He, Zhang, Jia, Wang and He ^{36 ]}:

(7) $$\begin{align}\phi \left(\omega \right)=\sum \limits_{m=0}^{\infty}\frac{{\left(\omega -{\omega}_0\right)}^m}{m!}{\phi}_{{m}},\end{align}$$

where the dispersion coefficient ${\phi}_{{m}}$ evaluates the mth derivative of $\phi \left(\omega \right)$ with respect to $\omega$ . We consider the linear chirp and ignore the third and higher order terms of Equation (7). The coordinate of the electron at the end of the magnetic chicane is as follows:

(8) $$\begin{align}{p}^{\prime}=p+{p}_1\sin \left({\omega}_1t+{\alpha}_1{t}^2+{\phi}_1\right)+{p}_2\sin \left({\omega}_2t+{\alpha}_2{t}^2+{\phi}_2\right),\end{align}$$

where the chirp parameter $\left|{\alpha}_{{i}}\right|\cong 1/{\sigma}_0{\sigma}_{{i}}$ , and pulse duration ${\sigma}_{{i}}={\sigma}_0{\left(1+4{\phi}_{{i}}^2/{\sigma}_0^4\right)}^{1/2}$ . Here, ${\sigma}_0$ is the input transform-limited pulse duration. We assume that the two chirped lasers have the same central wavelength ${\omega}_1={\omega}_2={\omega}_0$ , the same envelope ${p}_1={p}_2={p}_0$ and opposite frequency chirps ${\alpha}_1=-{\alpha}_2=\alpha$ . We take ${\phi}_1={\phi}_2=0$ . The energy modulation can be written as follows:

(9) $$\begin{align}{p}^{\prime}&=p+{p}_0\sin \left({\omega}_0t+\alpha {t}^2\right)+{p}_0\sin \left({\omega}_0t-\alpha {t}^2\right)\nonumber\\ &=p+2{p}_0\sin ({\omega}_0t)\cos (\alpha {t}^2).\end{align}$$

The evolution of the longitudinal phase space of the electrons is shown in Figure 1 (right), from which one can see that the gradually varying spacing current enhancement along the electron beam is achieved, which could be used to generate the isolated attosecond X-ray pulses in the following chicane-undulator segments.

The schematic layout of the proposed scheme is shown in Figure 2. In the radiator, the radiation pulse train, which reflects the longitudinal current distribution of the electron beam, is generated in the first undulator segment. The undulator length should be optimized to make the peak power high enough to suppress the shot noise but far from saturation to prevent the degradation of the electron quality. After the first delay line, the target radiation pulse is shifted forward to the upstream current peak and the microbunching (nm level) in the electron beam is washed out. Due to the unequal interval of the current spikes (several hundred nm), other radiation pulses will miss the current peak and the radiation gain processes are suppressed. The ultra-short target pulse is continuously amplified in the downstream undulator segments by repeating the above process.

Figure 2 Schematic layout of the proposed scheme.

3 Simulation

To show the performance of the proposed technique, 3D start-to-end simulations have been carried out based on the parameters of the SXFEL-HE (under design), which consists of a 3-GeV linac and two undulator lines. A high-quality electron beam with the charge of 500 pC is generated in the photoinjector and then compressed to about 1 kA in the linac with two-stage BC chicanes. The longitudinal phase space of the electron beam at the exit of the linac, simulated by the tracking code ASTRA^{[ 37 ]} in the injector and ELEGANT^[ Reference Borland ^{38 ]} in the linac, is shown in Figure 3 (left). The bunch length of the electron beam is about 400 fs by the end of the linac. The normalized slice emittance is about 1.0 μm⋅rad and the slice energy spread is about 50 keV. These simulation results fit quite well with the measurements at the SXFEL facility^[ Reference Feng, Liu, Chen, Zhou, Zhang, Qi, Gu, Wang, Jiang, Li, Wang, Wang, Zhang, Feng, Li, Lan, Li, Zhang, Deng, Xiang, Liu and Zhao ^{39 ,} Reference Zeng, Feng, Gu, Wang, Zhang, Li and Zhao ^{40 ]}.

Figure 3 Longitudinal phase space of the electron beam at the end of the linac (left) and at the exit of the ESASE section (right). The bunch head is to the right.

In the modulation section, a transform-limited laser pulse (longitudinal Gaussian distribution) with the central wavelength of 800 nm and pulse duration of 100 fs (full width at half maximum (FWHM)) is split and sent into two branches. The separated two optical pulses are stretched by a dispersive medium with opposite dispersions, resulting in two 1.1 ps (FWHM) long laser pulses with opposite frequency chirps. Finally, these two chirped laser pulses are sent through a splitter to recombine to a single laser pulse, as shown in the left-hand part of Figure 2, and sent into the modulator to interact with the electron beam to generate the energy modulation. The peak power of each chirped laser pulse is about 15 GW with laser waist of about 300 μm and pulse energy of about 16 mJ. The energy modulation is converted into density modulation in the downstream dispersion section, generating the gradually varied spacing pulse train. The longitudinal phase space of the electron beam at the exit of the modulation section, simulated by the 3D algorithm^[ Reference Zeng, Feng, Wang, Zhang, Qi and Zhao ^{41 ]}, is shown in Figure 3 (right). One can find that the peak current of about 5 kA is achieved with the current spacing of around 40 fs. The head and the tail parts of the electron beam, containing 5.1% particles in total, are cut during the simulation, which has little contribution to the radiation process but will result in the distortion of the electron beam in the longitudinal direction. Subsequently, the electron pulse train is transported to the downstream radiator.

It is worth pointing out that the longitudinal space charge effect is not negligible due to the high peak current^[ Reference Geloni, Saldin, Schneidmiller and Yurkov ^{42 ,} Reference Ding, Huang, Ratner, Bucksbaum and Merdji ^{43 ]}. The longitudinal space charge field, according to Ref. [Reference Ding, Huang, Ratner, Bucksbaum and Merdji43], could be calculated by the following:

(10) $$\begin{align}Ez\approx -\frac{Z_0{I}^{\prime }(s)}{4\pi {\overline{\gamma}}_z^2}\left(2\ln \frac{{\overline{\gamma}}_z{\sigma}_z}{r_{\mathrm{b}}}+1-\frac{r^2}{r_{\mathrm{b}}^2}\right),\end{align}$$

where ${Z}_0=377\ \Omega$ is the free space impedance, ${I}^{\prime }(s)= \mathrm{d}I/ \mathrm{d}s$ is the derivative of the electron current profile with respect to the longitudinal bunch coordinate $s$ , ${\overline{\gamma}}_z=\gamma /\sqrt{1+{K}^2/2}$ , $K$ is the undulator parameter, ${\sigma}_z$ is the root mean square (rms) bunch length of the current spike, ${r}_{\mathrm{b}}$ is the beam radius of a uniform transverse distribution and $r=\sqrt{x^2+{y}^2}$ . Here, we take $K=2.67$ , $\gamma =5871$ , ${r}_{\mathrm{b}}=100$ μm and ${\sigma}_z=293$ nm. With these parameters, the accumulated energy modulation $\Delta E$ after the 39 m undulator is as given in Figure 4. One can see that the longitudinal space charge produces additional energy chirps in the current spikes with a peak-to-peak energy variation of about 20 MeV. The longitudinal space charge effect can significantly degrade the FEL performance and is taken into account in the simulations based on GENESIS^[ Reference Reiche ^{44 ]}.

Figure 4 Electron beam energy modulation from the longitudinal space charge field from the 39 m undulator.

Thirteen 3-m-long undulators in total are used with the period length of 30 mm. The numbers of undulators used in the first to ninth undulator segments are chosen as 3, 2, 2, 1, 1, 1, 1, 1 and 1, respectively. The simulation results are summarized in Figure 5. In the first undulator segment, an attosecond pulse train, which reflects the comb-like current distribution, is generated with the peak power of about several tens of megawatt. Then the attosecond pulse train is shifted forward in the downstream delay line, so that the target radiation pulse (the last one) can overlap the adjacent electron pulse while the others meet the low-current part of the electron beam because of the unequal interval of the current distribution. Meanwhile, the microbunching accumulated in the first undulator segment is smeared out when the beam passes through the delay line. An X-band deflecting cavity is located at the end of the undulator line with temporal resolution better than 10 fs, which helps one to choose the strength of the delay line during future experimental research. The relative temporal jitter between the laser pulses and the electron beam is only tens of femtoseconds at FEL facilities, for example, approximately 50 fs rms at SXFEL^[ Reference Feng, Liu, Chen, Zhou, Zhang, Qi, Gu, Wang, Jiang, Li, Wang, Wang, Zhang, Feng, Li, Lan, Li, Zhang, Deng, Xiang, Liu and Zhao ^{39 ]}. A stable comb-like structure without obvious shot-to-shot variations had been experimentally demonstrated at FERMI^[ Reference Roussel, Ferrari, Allaria, Penco, Di Mitri, Veronese, Danailov, Gauthier and Giannessi ^{45 ]}, indicating that convenient optimization methods will be feasible for the tuning of the proposed scheme. Besides, one could increase the pulse duration of the seed laser and the electron beam to further increase the tolerance of temporal jitters.

Figure 5 The output radiation evolution and the spectrum along the undulator line at the end of the first, fourth and ninth undulator segments.

In the second undulator segment, the target radiation pulse is shifted and interacts with a ‘fresh’ part of the electron beam, leading to continuous amplification, similar to the direct seeding scheme^[ Reference Lambert, Hara, Garzella, Tanikawa, Labat, Carre, Kitamura, Shintake, Bougeard, Inoue, Tanaka, Salieres, Merdji, Chubar, Gobert, Tahara and Couprie ^{46 ]}. The amplification of the satellite pulses is suppressed, leading to continuous growth of the contrast, which is defined as the percentage of the target radiation pulse energy. By repeating the process of the undulator amplification and radiation delay, an isolated radiation pulse is achieved with a peak power of 330 GW and a pulse duration of about 620 as after nine undulator segments. The contrast of the target radiation pulse is over 99%. The bandwidth of the final output is about 0.5%, which is about 1.1 times that of the Fourier transform limit. The final peak power of the radiation can be further improved by optimizing the parameters of the frequency beating to generate more current spikes and employ more undulator segments. The output pulse duration can be further shortened by using shorter undulators, which helps to decrease the slippage effect.

3D simulations for SASE have also been performed, sharing the same electron beam at the end of the linac and the same undulator parameters. The comparison results are given in Figure 6. It is easy to find that the normal SASE saturates at around 35 m with the peak power of about 20 GW, which is one order of magnitude lower than that of the proposed scheme.

Figure 6 FEL gain curves for the proposed scheme and the normal SASE.