# X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Accelerator Lattice Overview

## 3. Laser Modulator

## 4. Bunch Compressors

#### 4.1. First Bunch Compressor

#### 4.1.1. BC1 Using a Conventional C-Chicane

#### 4.1.2. BC1 Using a Double C-Chicane

#### 4.1.3. BC1 Comparison of Designs

#### 4.2. Second Bunch Compressor

#### 4.2.1. BC2 Mechanism, Design, and Performance

#### 4.2.2. Determination of LABC Parameters

## 5. Linac Sections

#### 5.1. First Linac Section

#### 5.2. Second Linac Section

## 6. Beam Analysis at X-ray Undulator Entrance

#### 6.1. Beam Properties

#### 6.2. Achieved Mitigation of Beam Instabilities

#### 6.3. Preliminary X-ray Lasing Calculation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of the proposed MaRIE XFEL accelerator lattice using an LABC scheme for the second stage of the bunch compression. The two different configurations of the first bunch compressor (BC1) are shown. The evolution of the beam properties in lattice 1 using a conventional C-chicane as BC1 is shown at the top, and for lattice 2, which uses a double C-chicane as its BC1, is shown at the bottom. The evolution of the energy modulation, given in the variation of the Lorentz factor, is shown at the exits of BC1, the laser modulator (LM), and the second bunch compressor (BC2), explaining the development of the sequence of the high current spikes to be used for lasing in the X-ray undulator at the end of the lattice. In the figure, the instances of the matching optics (MO) are also shown.

**Figure 2.**The evolution of the Twiss parameters, along the beamline for (

**a**) lattice 1 using a conventional C-chicane as BC1 and (

**b**) lattice 2 using a double C-chicane as BC1. An overview of the lattice elements is provided at the top of the respective plot. Note that the profiles of the $\mathsf{\beta}$-functions in lattice 2 inside the section of L2 appeared unaffected by the quadrupoles because only a small focusing strength was needed in this case.

**Figure 3.**Energy modulation contrast ratio of the entire beam downstream of the laser modulator. The central section of the plot that is shaded in yellow is expanded and provided at the top, showing the details of the electron beam energy modulation.

**Figure 4.**The (

**a**) horizontal ($\widehat{x}$) and (

**b**) vertical ($\widehat{y}$) transverse momentum along the beam. (

**a**).

**i**and (

**b**).

**i**show the transverse momentum distribution before the laser modulation. (

**a**).

**ii**and (

**b**).

**ii**show the transverse momentum distribution after the laser modulation. Downstream of the laser modulator, the vertical momentum is slightly perturbed in an uncorrelated manner, but clear modulation is seen on the profile of the horizontal momentum.

**Figure 5.**Performance comparison of BC1 configurations in a conventional C-chicane (lattice 1) and a double C-chicane (lattice 2). The electron beam is divided into 1000 slices (

**a**) at the exit of L1, (

**b**) at the exit of BC1 using a conventional C-chicane, and (

**c**) at the exit of BC1 using a double C-chicane. (

**a**).

**i**, (

**b**).

**i**, and (

**c**).

**i**show the normalized emittances and the currents of each slice along the beam; (

**a**).

**ii**, (

**b**).

**ii**, and (

**c**).

**ii**show the distributions of the transverse centroids; (

**a**).

**iii**, (

**b**).

**iii**, and (

**c**).

**iii**show the distributions of the average transverse momentum shifts.

**Figure 6.**Contour plot of the value of ${\mathsf{\sigma}}_{{\overline{{x}^{\prime}}}_{slice}}$, the current-weighted standard deviation of the slice-average horizontal kick of all the beam slices, varying according to C-1 compression ratio and C-2 bending magnet length ${l}_{d,2}$ at ${B}_{d,2}=0.10$ T. The white marker is placed at the minimum of ${\mathsf{\sigma}}_{{\overline{{x}^{\prime}}}_{slice}}$ on the contour plot. A 2D scan was performed for each ${B}_{d,2}$ value, constituting a 3D parameter scan.

**Figure 7.**Summary plot of the entire 3D parameter scan for the optimization of BC1 using a double C-chicane scheme. The minimum value of the current-weighted standard deviation of the slice-average horizontal kick of all 1000 slices of the beam ${\mathsf{\sigma}}_{{\overline{{x}^{\prime}}}_{slice},min}$ (blue), the horizontal beam size (green) as well as the projected horizontal normalized emittance (red) at the double C-chicane BC1 exit, and the total length of the double C-chicane BC1 lattice (brown) were calculated for each value of the C-2 dipole magnetic field ${B}_{d,2}$ used in the parameter scan, and the results were plotted.

**Figure 8.**Longitudinal phase space of the electron beam plotted in the form of the energy modulation contrast ratio $\mathsf{\Delta}\gamma /{\mathsf{\sigma}}_{\gamma ,0}$ vs. the relative longitudinal coordinate, and beam current plotted within one laser wavelength (1 $\mathsf{\mu}\mathrm{m}$), (

**a**) before and (

**b**) after the bunch compression by BC2. Within each energy modulation period by the laser modulator, BC2 converts the energy modulation into the longitudinal electron density modulation, enhancing the average beam current of 500 A prior to BC2 to a current spike of 3 kA downstream of BC2. Note that the modulation amplitude after BC2 decreased due to the increase in the uncorrelated energy spread by the ISR effect in BC2.

**Figure 9.**Performance of BC2 in lattice 1, where the electron beam had previously been compressed by BC1 of a conventional C-chicane configuration. The electron beam is divided into 20,000 slices at (

**a**) L2 exit and (

**b**) BC2 exit. (

**a**).

**i**and (

**b**).

**i**show the normalized emittances and the currents of each slice along the beam; (

**a**).

**ii**and (

**b**).

**ii**show the distributions of the transverse centroids; and (

**a**).

**iii**and (

**b**).

**iii**the distributions of the average transverse momentum shifts.

**Figure 10.**The evolution of the longitudinal energy distribution within the beam from (

**a**) the photoinjector exit, (

**b**) at the end of the linear accelerator section, and (

**c**) at the end of the linearizer section. The final form of the longitudinal energy distribution within the beam is the chirp that is used in BC1.

**Figure 11.**Illustration of the uncorrelated energy spread distribution along the beam, plotted for the beam at the photoinjector exit; at the L1 exit with IBS calculation included in the Elegant simulation; and at the L1 exit, without considering IBS.

**Figure 12.**Comparison of the longitudinal distributions of the electron beam properties at the entrance of the X-ray undulator, after the beam traversed lattice 1 (

**a.i**–

**f.i**) and lattice 2 (

**a.ii**–

**f.ii**). The beams are divided into 20,000 slices with equal lengths, and the beam properties per slice are shown for (

**a**) beam currents, (

**b**) transverse centroid shifts, (

**c**) transverse average momentum shifts, (

**d**) normalized transverse emittances, (

**e**) rms transverse beam sizes, and (

**f**) rms transverse beam momentum profiles.

**Figure 13.**Comparison of the URWW effect on the electron beam (

**a**) from lattice 1, processed with our LABC method and (

**b**) with the conventional bulk compression method. In each plot, the distributions of the fractional URWW beam energy depression and of the predicted depressed beam energy distribution within the beam are provided. Note that the head of the beam is to the right of the plot.

Parameter | Value | ||
---|---|---|---|

Undulator length | ${L}_{u}$ | 1.50 | m |

Undulator number of periods | ${N}_{u}$ | 10 | |

Undulator period | ${\mathsf{\lambda}}_{u}$ | 0.15 | m |

Undulator peak field | ${B}_{0}$ | 0.53 | T |

Undulator parameter | K | 7.45 | |

$\mathsf{\xi}={K}^{2}/(2+{K}^{2})$ | 0.965 | ||

$\mathcal{J}={J}_{0}(\mathsf{\xi}/2)-{J}_{1}(\mathsf{\xi}/2)$ | 0.708 | ||

Laser wavelength | ${\mathsf{\lambda}}_{L}$ | 1.0 | $\mathsf{\mu}\mathrm{m}$ |

Laser beam waist | ${w}_{0}$ | 2.0 | mm |

Laser beam Rayleigh length | ${z}_{R}$ | 12.6 | m |

Laser peak power | ${P}_{L}$ | 3.4 | GW |

Laser pulse length | ${\mathsf{\tau}}_{L}$ | 1.2 | ps |

LM modulation amplitude | ${A}_{L}$ | 23.6 |

Parameter | Value | ||
---|---|---|---|

${\mathsf{\alpha}}_{x}$ | 16.15 | ||

${\mathsf{\beta}}_{x}$ | 679.6 | m | |

${\mathsf{\alpha}}_{y}$ | 16.15 | ||

${\mathsf{\beta}}_{y}$ | 679.7 | m | |

Horizontal norm. emittance | ${\mathsf{\u03f5}}_{n,x}$ | 65.9 | nm rad |

Vertical norm. emittance | ${\mathsf{\u03f5}}_{n,y}$ | 65.9 | nm rad |

Horizontal rms beam size | ${\mathsf{\sigma}}_{x}$ | 174.6 | $\mathsf{\mu}\mathrm{m}$ |

Vertical rms beam size | ${\mathsf{\sigma}}_{y}$ | 174.6 | $\mathsf{\mu}\mathrm{m}$ |

Unco. frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta},0}$ | $2.91\times {10}^{-6}$ | |

Co. frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta}}$ | $1.12\times {10}^{-2}$ | |

Energy chirp coefficient | h | 24.3 | ${\mathrm{m}}^{-1}$ |

rms beam length | ${\mathsf{\sigma}}_{z}$ | 477.3 | $\mathsf{\mu}\mathrm{m}$ |

Parameter | Value | ||
---|---|---|---|

Compression ratio | ${C}_{1}$ | 25 | |

${R}_{56}$ | −41.2 | mm | |

Dipole magnetic field | ${B}_{d}$ | 0.60 | T |

Dipole length | ${l}_{d}$ | 0.315 | m |

BC1 total length | 8.11 | m |

Parameter | Value | ||
---|---|---|---|

C-1 compression ratio | ${C}_{1,1}$ | 7.50 | |

${R}_{56,1}$ | −37.1 | mm | |

C-1 dipole magnetic field | ${B}_{d,1}$ | 0.24 | T |

C-1 dipole length | ${l}_{d,1}$ | 0.600 | m |

C-2 compression ratio | ${C}_{1,2}$ | 3.33 | |

${R}_{56,2}$ | −4.0 | mm | |

C-2 dipole magnetic field | ${B}_{d,2}$ | 0.10 | T |

C-2 dipole length | ${l}_{d,2}$ | 0.600 | m |

BC1 total length | 21.78 | m |

**Table 5.**Twiss parameters and other beam properties at L2 exit, as in lattice 1. Beam energy 12 GeV.

Parameter | Value | ||
---|---|---|---|

${\mathsf{\alpha}}_{x}$ | −0.4179 | ||

${\mathsf{\beta}}_{x}$ | 426.1 | m | |

${\mathsf{\alpha}}_{y}$ | −0.3318 | ||

${\mathsf{\beta}}_{y}$ | 458.6 | m | |

Horizontal norm. emittance | ${\mathsf{\u03f5}}_{n,x}$ | 93.3 | nm rad |

Vertical norm. emittance | ${\mathsf{\u03f5}}_{n,y}$ | 86.8 | nm rad |

Horizontal rms beam size | ${\mathsf{\sigma}}_{x}$ | 41.4 | $\mathsf{\mu}\mathrm{m}$ |

Vertical rms beam size | ${\mathsf{\sigma}}_{y}$ | 41.2 | $\mathsf{\mu}\mathrm{m}$ |

Unco. frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta},0}$ | $5.04\times {10}^{-6}$ | |

Co. frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta}}$ | $7.76\times {10}^{-5}$ | |

rms beam length | ${\mathsf{\sigma}}_{z}$ | 18.6 | $\mathsf{\mu}\mathrm{m}$ |

Parameter | Value | ||
---|---|---|---|

Compression ratio | ${C}_{2}$ | 6 | |

${R}_{56}$ | −2.0 | mm | |

Dipole magnetic field | ${B}_{d}$ | 0.52 | T |

Dipole length | ${l}_{d}$ | 0.600 | m |

BC2 total length | 34.28 | m |

Parameter | Value | ||
---|---|---|---|

${\mathsf{\alpha}}_{x}$ | −14.78 | ||

${\mathsf{\beta}}_{x}$ | 201.69 | m | |

${\mathsf{\alpha}}_{y}$ | −14.79 | ||

${\mathsf{\beta}}_{y}$ | 201.74 | m | |

Horizontal norm. emittance | ${\mathsf{\u03f5}}_{n,x}$ | 54.5 | nm rad |

Vertical norm. emittance | ${\mathsf{\u03f5}}_{n,y}$ | 54.5 | nm rad |

Horizontal rms beam size | ${\mathsf{\sigma}}_{x}$ | 191.4 | $\mathsf{\mu}\mathrm{m}$ |

Vertical rms beam size | ${\mathsf{\sigma}}_{y}$ | 191.4 | $\mathsf{\mu}\mathrm{m}$ |

Unco. frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta},0}$ | $1.34\times {10}^{-5}$ | |

rms beam length | ${\mathsf{\sigma}}_{z}$ | 477.4 | $\mathsf{\mu}\mathrm{m}$ |

**Table 8.**Lattice 1 electron beam Twiss parameters and other properties at X-ray undulator entrance. Beam energy 12 GeV.

Parameter | Value | ||
---|---|---|---|

${\mathsf{\alpha}}_{x}$ | $5.087\times {10}^{-5}$ | ||

${\mathsf{\beta}}_{x}$ | 1.670 | m | |

${\mathsf{\alpha}}_{y}$ | $-2.145\times {10}^{-4}$ | ||

${\mathsf{\beta}}_{y}$ | 2.006 | m | |

Horizontal norm. emittance | ${\mathsf{\u03f5}}_{n,x}$ | 98.3 | nm rad |

Vertical norm. emittance | ${\mathsf{\u03f5}}_{n,y}$ | 86.8 | nm rad |

Horizontal rms beam size | ${\mathsf{\sigma}}_{x}$ | 2.64 | $\mathsf{\mu}\mathrm{m}$ |

Vertical rms beam size | ${\mathsf{\sigma}}_{y}$ | 2.72 | $\mathsf{\mu}\mathrm{m}$ |

MB frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta},\mathrm{MB}}$ | $7.5\times {10}^{-5}$ | |

MB beam portion | 38.8% | ||

LABC parameters | A | 16.1 | |

B | 0.0927 |

**Table 9.**Lattice 2 electron beam Twiss parameters and other properties at X-ray undulator entrance. Beam energy 12 GeV.

Parameter | Value | ||
---|---|---|---|

${\mathsf{\alpha}}_{x}$ | $1.768\times {10}^{-4}$ | ||

${\mathsf{\beta}}_{x}$ | 2.158 | m | |

${\mathsf{\alpha}}_{y}$ | $1.885\times {10}^{-3}$ | ||

${\mathsf{\beta}}_{y}$ | 1.292 | m | |

Horizontal norm. emittance | ${\mathsf{\u03f5}}_{n,x}$ | 66.1 | nm rad |

Vertical norm. emittance | ${\mathsf{\u03f5}}_{n,y}$ | 98.9 | nm rad |

Horizontal rms beam size | ${\mathsf{\sigma}}_{x}$ | 2.47 | $\mathsf{\mu}\mathrm{m}$ |

Vertical rms beam size | ${\mathsf{\sigma}}_{y}$ | 2.33 | $\mathsf{\mu}\mathrm{m}$ |

MB frac. energy spread | ${\mathsf{\sigma}}_{\mathsf{\delta},\mathrm{MB}}$ | $7.4\times {10}^{-5}$ | |

MB beam portion | 40.4% | ||

LABC parameters | A | 16.1 | |

B | 0.0948 |

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**MDPI and ACS Style**

Xu, H.; Anisimov, P.M.; Carlsten, B.E.; Duffy, L.D.; Marksteiner, Q.R.; Robles, R.R.
X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression. *Appl. Sci.* **2023**, *13*, 2285.
https://doi.org/10.3390/app13042285

**AMA Style**

Xu H, Anisimov PM, Carlsten BE, Duffy LD, Marksteiner QR, Robles RR.
X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression. *Applied Sciences*. 2023; 13(4):2285.
https://doi.org/10.3390/app13042285

**Chicago/Turabian Style**

Xu, Haoran, Petr M. Anisimov, Bruce E. Carlsten, Leanne D. Duffy, Quinn R. Marksteiner, and River R. Robles.
2023. "X-ray Free Electron Laser Accelerator Lattice Design Using Laser-Assisted Bunch Compression" *Applied Sciences* 13, no. 4: 2285.
https://doi.org/10.3390/app13042285