5.3 Experiment results
5.3.3 Nonlinear longitudinal phase space manipulation
The multipole magnets impart ann-th order correlation to the transverse phase space (e.g., ∆x′∝xn).
Thus, a series of multipole magnets would provide or eliminate any correlations that can be approxi- mated by a polynomial series. This concept is adopted by the DEEX beamline to correct the nonlinearity in the initial LPS. Here, the first EEX beamline converts all longitudinal properties to horizontal prop- erties. Multipole magnets in the middle section correct the nonlinear correlation. Then, the second EEX beamline converts the linearized phase space back to the LPS. The octupole is able to mitigate harmful double-horn features of the bunch, which appear after a strong compression [21, 42]. We demonstrated the nonlinear correction using a single octupole magnet in the DEEX beamline. From this demonstration, the suppression of double-horn features was also observed.
(a) (b)
(c) (d)
(e) (f)
Figure 31: Third-order correction using an octupole magnet. (a)–(e) show measured longitudinal phase spaces with different octupole magnet strengths. The gradient of the octupole magnet was (182, 0, -182, -364, -546) T/m3. Each value corresponds to a–e, respectively. (f) shows the third-order coefficient of the polynomial fitting and the final bunch length (σz,4) for each of the displayed cases.
The bunch charge was increased to 700 pC to strengthen the space-charge force at low energy; thus, the third-order correlation in the LPS became more evident. Additionally, the bunch compression was performed such that the higher-order correlations were more evident in the LPS, and they affected the current and energy distributions. For the 700 pC charge, the initial rms bunch length at the compres- sor entrance was 0.67±0.03 mm. The DEEX beamline compressed the bunch length to 0.43±0.02 mm, and the current distribution after the compression exhibited a double-horn feature, owing to folded structures that originated from a third-order correlation [see Fig. 31(b)].
Figures 31(a)–(e) show single-shot measurement results of the LPS with different octupole magnet strengths. Figure 31(f) shows the third-order coefficient from the polynomial fitting of the measured LPS and the corresponding rms bunch length. The third-order coefficient of a polynomial fit can represent the strength of a third-order correlation in LPS. To fit the phase space using a polynomial series, the averaged center of slice distribution was used. Spectral slices were used instead of the typical time-slices owing to the S-shapes in LPS. When the longitudinal beam distribution is fl(z,δ), the averaged longitudinal position of each spectral-slice can be calculated as follows:
¯ z(δ) =
R∞
−∞z fl(z,δ)dz R∞
−∞fl(z,δ)dz. (5.10)
Subsequently, ¯z(δ)can be fitted using a polynomial function with a maximum order of three as follows:
¯
z(δ)≈a0+a1δ+a2δ2+a3δ3. (5.11) Here, the coefficienta3represents the third-order correlation in the LPS distribution.
The third-order coefficient of the polynomial fit indicates the strength of the third-order correlation in the LPS distribution. An octupole strength of 182 T/m3 [Fig. 31(a)] provided a stronger third-order compression. Thus, the outer particles on the head and tail of the bunch were pushed further inside, and the third-order coefficient decreased from -17.77 m to -45.43 m. Owing to a strengthened third- order compression, the bunch was further compressed, whereas the double-horn feature became stronger compared with the octupole-off case in Fig. 31(b). When the polarity of the octupole-magnet field was flipped, the third-order compression weakened, and the third-order coefficient increased. When the oc- tupole strength was -364 T/m3 [Fig. 31(d)], the third-order coefficient became 2.43 m, which was the smallest magnitude among the presented cases. The phase space distribution was almost linear in this case, and the double-horn feature disappeared. When the octupole strength was increased further, the LPS distribution exhibited an S-shape in the opposite direction to that in Fig. 31(b). The bunch was significantly lengthened, and the third-order coefficient further increased.
Chapter 6
On-going efforts to utilize DEEX based LPS manipulations
6.1 Sub femtoseconds bunch compression
Figure 32: Layout of single EEX beamline. The beamline consists of two doglegs, a TDC, a fundamental mode cavity and three sextupoles.
Bunch length after the EEX beamline relies on beam size, phase space slope, and emittance to the entrance as shown in Eq. (5.6). The beam size and the slope are controllable variables using quadrupole, while emittance is fixed. Most of cases, electron beams from a photocathode gun have larger longitudi- nal emittance than transverse one. Due to this reason, bunch compression using single EEX is feasible to achieve shorter bunch length than the DEEX beamline or chicane bunch compressor. In this section, we explored feasibility of the sub-fs bunch generation via linear EEX dynamics and numerical simu- lation. The simulation was done with ELEGANT [56] including higher order effect and 1D coherent synchrotron radiation (CSR). In addition, initial beam was assumed 6D Gaussian.
When incoming beam to the EEX beamline satisfy minimum bunch length condition (sx=-R51/R52), the final bunch length can be expressed as
σz,f = R52εx,i
σx,i
, (6.1)
whereσx,i andεx,i are horizontal beam size and emittance to the EEX beamline entrance.R52 is (5,2) elements of the EEX beamline transfer matrix. In this study, we adopted an fundamental mode cavity (FMC) that eliminates thick-cavity effect from the TDC in the EEX beamline as shown in Fig. 32 [38].
Althoughεx,iandR52are, we can varyσz,f by controllingσx,i.
A large initial beam size generates a small bunch length at the exit in the linear regime, but it gener- ates strong higher orders effects. In Fig. 33, we scan the initial beam size and compared corresponding final bunch length from Eq. (6.1) to ELEGANT simulations under different conditions to investigate the limiting effects. Case (a) in Fig. 33 is the simulation result with higher order effects but no CSR.
Case (b) uses sextupole magnets to suppress the higher order effects (see Fig. 32) and there is no CSR yet. Case (c) is the simulation with sextupole magnets and CSR effects. When initialσx,iis less than 0.1 mm, analytical calculation and case (a) shows good agreement while the bunch length above this range rapidly increases due to the strong higher order effects from the beam size. To suppress the higher order effects, we introduced three sextupole magnets to the EEX beamline as shown in Fig. 32. The sextupole setting is optimized for theσx,i of 0.1 mm, and Fig. 33 shows that the bunch length curve for case (b) is clearly lower than the case (a). When CSR is introduced to the simulation, it normally increases the bunch length. However, the bunch length growth is ignorable with the beam and beamline parameters in Table 3 which has a low peak current and a small bending angle. The final bunch length with all physics is 0.9 fs for 0.1 mm ofσx,i , and it can be reduced further if we increaseσx,i little more (0.14 mm provides 0.72 fs).
Figure 33: Final bunch length from different initial horizontal beam size. Blue line: analytical calculation based on Eq. (6.1). Case (a): ELEGANT simulation without sextupoles and CSR effect. Case (b): ELE- GANT simulation with sextupoles, but no CSR effect. Case (c): ELEGANT simulation with sextupoles and CSR effect.
The limiting factors, higher order effects and CSR effects, increase the bunch length and the emit- tance at the same time. At the entrance to the beamline, the horizontal emittance is 0.05 µm and the longitudinal emittance is 0.1 µm. Due to EEX, the final horizontal and longitudinal emittance should be 0.1 µm and 0.05 µm if there is no emittance growth. However, the higher order effect increased the longitudinal emittance to 0.061 µm. This emittance growth from the higher order effect is suppressed when sextupoles are introduced. The final longitudinal emittance is 0.054 µm in this case. In the previous section, we found that the CSR effect is negligible to the final bunch length. However, the emittance is sensitive to the CSR effect. When CSR is applied to the simulation, the longitudinal emittance increases to 0.066 µm. Among all cases, the final horizontal emittance is well preserved.
Beamline parameters Values Beam parameters Values
Bending angle 5◦ Energy 100 MeV
Dipole-Dipole 5 m Charge 2 pC
Dogleg’s dispersion -0.47 m Bunch length 0.167 ps
TDC kick strength 2.13 m−1 Horizontal slope (σxx′,i/σx,i2 ) -0.97 m−1 FMC voltage -718 kV Vertical slope (σyy′,i/σy,i2) 0 m−1 1st sextupole geometry strength 1309.51 m−3 Chirp (σzδ′,i/σz,i2 ) -0.0393 m−1 2nd sextupole geometry strength -1138.23 m−3 Normalized horizontal emittance 0.05 µm 3rd sextupole geometry strength 17.68 m−3 Normalized vertical emittance 0.05 µm Normalized longitudinal emittance 0.1 µm Table 3: ELEGANT input parameters. Left is beamline parameters and right is initial beam conditions.