# Analytical Scaling Laws for Radiofrequency-Based Pulse Compression in Ultrafast Electron Diffraction Beamlines

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## Abstract

**:**

## 1. Introduction

## 2. Longitudinal Envelope Equation

#### 2.1. Envelope Equation

#### 2.2. Solution in a Drift

#### 2.3. Single Particle Dynamics and Non-Linear Phase-Space Correlations in the RF Buncher

#### 2.4. Emittance Growth Mechanisms and the Relationship between Different Longitudinal Phase Space Definitions

#### 2.4.1. $(z,{z}^{\prime})$ Trace Space Emittance

#### 2.4.2. $(z,\delta )$ Phase Space

#### 2.5. Bunch Length Limit in Absence of Space-Charge Effects

## 3. Space-Charge Limits to Compression

#### 3.1. An Example of Geometry Factor Calculation: Gaussian Distribution Case

#### 3.2. Effect of the Longitudinal Space-Charge Force on the Minimum Bunch Length

## 4. Bunch Compression Limits for Different Charge Distributions

## 5. X-Band Cavity Compensation

#### 5.1. Analytical Estimates

#### 5.2. Start-to-End Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of RF ballistic bunching scheme. A velocity chirp is imparted on an electron using an RF cavity so that the tail of the beam has a higher energy than the head. During the following drift, the particles in the tail catch up with the particles in the head, resulting in strong longitudinal compression.

**Figure 2.**(

**Left**) Trace spaces of the beam at the exit of the prebuncher after the linear chirp is subtracted from the distribution for high-energy (

**a**) and low-energy (

**c**) cases compared with the analytical predictions from Equation (11) (green curves). (

**Right**) Longitudinal trace spaces at the temporal waist for the high-energy (

**b**) and low-energy (

**d**) cases compared with the predictions from Equation (13) (green curves). The current profiles at the focus are also shown in black.

**Figure 3.**Comparison between analytical estimates (dashed lines) and GPT simulations (solid lines) for an initial bunch length of 0.65 ps. The trace space emittance after the buncher is shown in blue, and emittance growth in $(z,\delta )$ phase space is shown in orange.

**Figure 4.**Final bunch length as a function of the initial bunch length for the (

**a**) high-energy and (

**b**) low-energy cases. The analytical curves are also shown and are found to be in very good agreement with the simulations.

**Figure 5.**(

**a**) Longitudinal field of the bunch compared with the linear field component of the 3D Gaussian field and with the beam-averaged generalized force term that appears in the envelope equation. The total charge in the bunch in this simulation is $Q={10}^{5}e$. (

**b**) Geometry factors for the transverse (blue) and longitudinal (purple) field plotted as a function of the rest frame aspect ratio.

**Figure 6.**The evolution of longitudinal and transverse beam sizes are shown in purple. The space charge envelope equation term is shown in green, and its approximation is shown as a dotted line with aspect ratio fixed, i.e., A = 1.

**Figure 7.**(

**a**) Final waist size plotted versus initial bunch length compared with GPT. The green and black lines show the asymptotic behavior in the regimes where space charge and RF emittance growth dominate the dynamics. The blue line is the analytical expression that considers all effects discussed in this paper (

**b**) The optimum phase space found from the scan in (

**a**) for an input bunch length of 72 μm. The RMS bunch length is 1 μm. Particles are color-coded as a function of their radial coordinate (blue corresponds to on-axis). (

**c**) Evolution of the emittance along the line. The growth observed in GPT is due to the non-linearities of the space-charge field. (

**d**) Final emittance in GPT as a function of initial bunch length. The relative importance of the space-charge contribution to the emittance can be inferred by subtracting the expected emittance growth due to the buncher dynamics non-linearities. The cross-over point (where space charge becomes the dominant effect) can be used to estimate the optimal injection condition and hence the minimum final bunch length achievable for a given setup.

**Figure 8.**(

**a**) Minimum bunch length analytical estimates compared with GPT simulation as the charge in the 4.6 MeV energy beam is varied for Gaussian (purple) and uniformly filled ellipsoidal (blue) beam distributions. (

**b**) Minimum bunch length versus $\gamma $ and N for the Gaussian distribution assuming a constant focal length f = 1.88 m.

**Figure 9.**(

**a**) Illustration of beamline setup for the RF emittance growth compensation. A short x-band cavity is used to compensate for the curvature in $(z,{z}^{\prime})$ space imparted by the S-band linac and gun and linearize the output trace space. (

**b**) GPT simulation of phase space at the exit of the S-band gun, the exit of the X-band linearizer, the exit of the buncher, and at the longitudinal focus.

**Figure 10.**Analytical predictions for the RMS bunch length at the longitudinal waist for 160 fC charge and 4.6 MeV energy beams plotted with respect to initial bunch length for compensated (solid) and uncompensated (dashed) cases. Space-charge effects are taken into account assuming a constant aspect ratio equal to 0.6 for Gaussian (purple) and uniformly filled ellipsoidal (blue) distribution. These results are compared with GPT simulations of the linearizer beamline in compensation mode. The red, purple, and blue squares show the results of start-to-end simulations for varying laser pulse lengths following the beam from the cathode located in an S-band RF gun.

**Figure 11.**Comparison of predicted bunch length dependence on charge for fixed input bunch length. Application to the Gaussian line charge.

**Figure 12.**Final phase spaces for a beam shorter than, equal to and longer than the optimal bunch length, respectively. The phase spaces are color coded with respect to the radial coordinate.

Parameter | High Energy | Low Energy |
---|---|---|

Focal length | 1.88 m | 1 m |

Beam kinetic energy | 4.6 MeV | 150 keV |

Norm. transverse emittance | 100 nm | 8.3 nm |

RMS transverse beam size | 100 μm | 100 μm |

Cavity Frequency | 2.856 GHz | 2.856 GHz |

Relative energy spread | ${10}^{-5}$ | ${10}^{-5}$ |

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

Charge | ${10}^{6}e$ |

Laser Spot Size | 10 μm |

Cathode MTE | 0.5 eV |

Optimal laser pulse length | 0.95 ps (rms) |

Gun Accelerating Gradient | 94.7 MV/m |

Gun Phase | 35.5° |

Linearizer accelerating voltage | 1.8 MV |

Linearizer phase | 173.5° |

Buncher accelerating voltage | 6.75 MV |

Buncher phase | 101° |

Final kinetic energy | 4.5 MeV |

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**MDPI and ACS Style**

Denham, P.; Musumeci, P.
Analytical Scaling Laws for Radiofrequency-Based Pulse Compression in Ultrafast Electron Diffraction Beamlines. *Instruments* **2023**, *7*, 49.
https://doi.org/10.3390/instruments7040049

**AMA Style**

Denham P, Musumeci P.
Analytical Scaling Laws for Radiofrequency-Based Pulse Compression in Ultrafast Electron Diffraction Beamlines. *Instruments*. 2023; 7(4):49.
https://doi.org/10.3390/instruments7040049

**Chicago/Turabian Style**

Denham, Paul, and Pietro Musumeci.
2023. "Analytical Scaling Laws for Radiofrequency-Based Pulse Compression in Ultrafast Electron Diffraction Beamlines" *Instruments* 7, no. 4: 49.
https://doi.org/10.3390/instruments7040049