# Toward the Generation of an Isolated TW-Attosecond X-ray Pulse in XFEL
## 3 Methods with Current Modulation
### Single Current Spike I: Recycling Method  
A different approach has also been undertaken. Instead of using multiple current spikes, the possibility of a single current spike for TW-as X-ray generation has been examined. Note that whereas a current of ~10 kA peak was used in the multi current spikes method, a higher peak current such as 30 kA is needed in the single current spike method.  
The first idea introduced was to recycle a single current spike to amplify an X-ray pulse many times [42]. Because an X-ray pulse quickly slips out from a current spike, the current spike still shows low energy spread and good emittance. Hence, the electrons in the current spike may contribute several times to the amplification process. In other words, if an X-ray pulse is delayed again to match the same current spike, the electrons contribute further to the amplification, as shown in Figure 6.  
Figure 5: (**a**) Irregularly-spaced current spikes. The electron bunch has multiple current spikes with different spacing (top). Multiple radiation pulses (green small spikes) are generated from the first undulator unit (second line). The red colored line is now our TRP. The radiation pulses are delayed to match the TRP to the second-last current (third line). In the second undulator unit, only the TRP is matched with a current spike, and is amplified. The electron bunch is then further delayed to match the amplified TRP to the third-last current spike (fourth line). This process is repeated to the end (bottom). (**b**) Simulation results with four different conditions. (i) The original method used in [PERSON] [37]. (ii) Linearly tapered wiggler (the field between each magnet pole is changed linearly along the beam propagation direction) in Figure 3 with a 1.5 cycle laser pulse. (iii) Well-type tapered wiggler n Figure 3 with a 1.5 cycle laser pulse. (iv) Chirped laser method which shows highest radiation power. (v) Laser pulse stacking method (a laser pulse is divided to several pulses and is summed again with different arrival times, such as \(\tau_{1},~{}\tau_{2},\ldots\) as shown in the top part of Figure 5a). See [PERSON] et al. [40] for the details. Reproduced from [PERSON] et al. [40] with permission.  
Simulation results obtained using Elegant [46] and Genesis 1.3 code [47] are shown in Figure 7. The results with three different cases are compared. The Pohang Accelerator Laboratory X-ray Free Electron Laser (PAL-XFEL) parameters were adopted, in which the electron beam energy was 10 GeV. When an optimal tapering to the undulator gap distribution was used, a radiation power up to 1.2 TW with a pulse duration of 100 attoseconds could be obtained.  
In this method, the delay of both the electron and the X-ray is needed. There would be a vibration problem in X-ray mirrors in addition to the power supply stability, as in the case of Tanaka [37]. In Tanaka [37], the X-ray delay unit was used just one time at the first stage. However, in [PERSON] et al. [42], the delay unit was used for every step in the amplification stage. Multiple reflections from X-ray mirrors would decrease the overall radiated power that can be achieved.