^{1,a)}

### Abstract

Ultrafast electron diffraction (UED) relies on short, intense pulses of electrons, which because of Coulombic repulsion will expand and change shape as they propagate. While such pulse expansion has been studied in other contexts, efforts to model this effect for typical UED parameters have only arisen fairly recently. These efforts have yielded accurate predictions with very simple models, but have left a number of unexplained results (such as the development of a linear self-similar profile with sharply defined end points). The present work develops a series of models that gradually incorporate more physical principles, allowing a clear determination of which processes control which aspects of the pulse propagation. This will include a complete analytical solution of the one-dimensional problem (including a fundamental limitation on temporal resolution), followed by the gradual inclusion of two-dimensional and inhomogeneous effects. Even very simple models tend to capture the relevant on-axis behavior to within 10% or better. This degree of success can be traced to the manner in which the pulse transitions from one dimensional to two dimensional. We also present methods for determining the most appropriate model for a given situation and suggest paths toward future modeling improvements as the field evolves.

Mike Armstrong provided invaluable comments on the first draft of this manuscript. This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.

INTRODUCTION

BACKGROUND AND OVERVIEW

ANALYSIS

One-dimensional model

2D effects: The mean-field model and variants

2D effects: Building on the mean-field models

Simple models yield accurate , and energy spread

All models that include profile shape evolution show the development of a linear self-similar profile on a time scale dictated by the plasma frequency

Two-dimensional profile evolution results in sharply defined turnaround points

Off-axis and inhomogeneous behaviors are more complex

SUMMARY AND CONCLUSION

### Key Topics

- Phase space methods
- 35.0
- Chirping
- 32.0
- Mean field theory
- 28.0
- Space charge effects
- 24.0
- Carrier density
- 19.0

## Figures

Schematic of a photocathode used for ultrafast electron diffraction (UED). The present work focuses on the free-space propagation of the bunch from the grid to the sample. A separate laser beam (not shown) drives changes in the sample.

Schematic of a photocathode used for ultrafast electron diffraction (UED). The present work focuses on the free-space propagation of the bunch from the grid to the sample. A separate laser beam (not shown) drives changes in the sample.

Universal self-expansion curves for a Gaussian pulse in the 1D limit. Only half of the symmetric pulse is shown, in the reference frame of the pulse. The initial density is given by , while the time scale is set by . All axes are nondimensionalized as indicated. Times are , , , , , , and . (a) Charge density vs longitudinal position. (b) Velocity vs longitudinal position. The asymptotic linear chirp line is included for . The flattopped central part of the pulse approaches this line.

Universal self-expansion curves for a Gaussian pulse in the 1D limit. Only half of the symmetric pulse is shown, in the reference frame of the pulse. The initial density is given by , while the time scale is set by . All axes are nondimensionalized as indicated. Times are , , , , , , and . (a) Charge density vs longitudinal position. (b) Velocity vs longitudinal position. The asymptotic linear chirp line is included for . The flattopped central part of the pulse approaches this line.

Conceptual sketch for the mean-field approach. (a) A rotationally symmetric electron bunch in its center-of-mass frame, with half maximum widths and . (b) Coulomb’s law yields an electric field, the component of which governs the longitudinal expansion. (c) The electric field will be roughly linear out to a “typical” distance which we take as . Under the linear approximation the pulse will expand self-similarly in the direction. Deviations from linearity will change the shape of the pulse as time passes. A similar construction can model the transverse expansion.

Conceptual sketch for the mean-field approach. (a) A rotationally symmetric electron bunch in its center-of-mass frame, with half maximum widths and . (b) Coulomb’s law yields an electric field, the component of which governs the longitudinal expansion. (c) The electric field will be roughly linear out to a “typical” distance which we take as . Under the linear approximation the pulse will expand self-similarly in the direction. Deviations from linearity will change the shape of the pulse as time passes. A similar construction can model the transverse expansion.

Various forms of the longitudinal shape factor .

Various forms of the longitudinal shape factor .

Expansion rates and longitudinal pulse profiles calculated by a variety of models. The following parameters, typical of current UED systems, were used: electrons, , initial pulse FWHM, initial , initial (top-hat transverse, Gaussian longitudinal), propagation time , zero initial temperature, and zero initial transverse expansion. The plasma frequency for the peak initial electron density comes to , so the simulation runs for a time equal to . Models are as specified in Table II, plus the original mean-field model . “Central” refers to the longitudinal profile for those electrons within of the axis; otherwise all electrons are included (“full” profiles). The notation only refers to restricted -electron simulations which only include longitudinal acceleration (all other models except the 2DCP radial coupled are similarly restricted). -electron results are accumulated from 25 runs.

Expansion rates and longitudinal pulse profiles calculated by a variety of models. The following parameters, typical of current UED systems, were used: electrons, , initial pulse FWHM, initial , initial (top-hat transverse, Gaussian longitudinal), propagation time , zero initial temperature, and zero initial transverse expansion. The plasma frequency for the peak initial electron density comes to , so the simulation runs for a time equal to . Models are as specified in Table II, plus the original mean-field model . “Central” refers to the longitudinal profile for those electrons within of the axis; otherwise all electrons are included (“full” profiles). The notation only refers to restricted -electron simulations which only include longitudinal acceleration (all other models except the 2DCP radial coupled are similarly restricted). -electron results are accumulated from 25 runs.

Longitudinal phase space plots calculated by a variety of models, at the end of the simulation time. See Fig. 5 and Table II for parameters and abbreviations. (b)–(e) add a linear offset to accentuate the differences among models. (b) and (c) contrast continuum and multiparticle approaches to the same problem. (e) is a close-up of the turnaround region in (d).

Longitudinal phase space plots calculated by a variety of models, at the end of the simulation time. See Fig. 5 and Table II for parameters and abbreviations. (b)–(e) add a linear offset to accentuate the differences among models. (b) and (c) contrast continuum and multiparticle approaches to the same problem. (e) is a close-up of the turnaround region in (d).

Projections of the final phase space density for electron bunches with the following initial conditions: electrons, , , initial FWHM pulse , initial FWHM , and Gaussian initial longitudinal distribution. Projections are (first column) in the plane, (second column) on the axis, and (third column) on the plane (with a linear offset to accentuate the differences). Initial conditions are (a) top-hat initial transverse profile, (b) Gaussian initial transverse profile, (c) Gaussian transverse profile with an added temperature (equilibrated before acceleration), (d) top-hat transverse profile with no temperature but with a initial divergence angle, and (e) as (d), but with a Gaussian transverse profile.

Projections of the final phase space density for electron bunches with the following initial conditions: electrons, , , initial FWHM pulse , initial FWHM , and Gaussian initial longitudinal distribution. Projections are (first column) in the plane, (second column) on the axis, and (third column) on the plane (with a linear offset to accentuate the differences). Initial conditions are (a) top-hat initial transverse profile, (b) Gaussian initial transverse profile, (c) Gaussian transverse profile with an added temperature (equilibrated before acceleration), (d) top-hat transverse profile with no temperature but with a initial divergence angle, and (e) as (d), but with a Gaussian transverse profile.

## Tables

Numerical calculations of the relationships between transverse and longitudinal time constants for a cylinder of constant charge density , length , and radius , expressed as a set of dimensionless numbers for each aspect ratio . The scaled electric fields at the surface are related to time constants according to the equation with for transverse motion and for longitudinal motion.

Numerical calculations of the relationships between transverse and longitudinal time constants for a cylinder of constant charge density , length , and radius , expressed as a set of dimensionless numbers for each aspect ratio . The scaled electric fields at the surface are related to time constants according to the equation with for transverse motion and for longitudinal motion.

Summary of the modeling approaches considered in this work. The original mean-field (MF) model fits in the 2DCP category, using a scaling parameter to distinguish different profiles. The last six rows indicate the success of the model in reproducing a given physical effect, with a blank entry indicating that the model makes no attempt. “Partial” modeling of transverse expansion means it can either be neglected (in favorable cases), externally specified (useful in experimental situations where lenses are adjusted to compensate transverse space charge expansion), or calculated using a coupled transverse expansion model, as discussed in the text. In these partial cases, if the transverse expansion is significant and left uncorrected then the “good” entries should be replaced with “medium.” Semiquantitatively, good implies correct within a few percent and medium within a few tens of percent for typical UED parameters, while “poor” could easily be off be a factor of 2 or more.

Summary of the modeling approaches considered in this work. The original mean-field (MF) model fits in the 2DCP category, using a scaling parameter to distinguish different profiles. The last six rows indicate the success of the model in reproducing a given physical effect, with a blank entry indicating that the model makes no attempt. “Partial” modeling of transverse expansion means it can either be neglected (in favorable cases), externally specified (useful in experimental situations where lenses are adjusted to compensate transverse space charge expansion), or calculated using a coupled transverse expansion model, as discussed in the text. In these partial cases, if the transverse expansion is significant and left uncorrected then the “good” entries should be replaced with “medium.” Semiquantitatively, good implies correct within a few percent and medium within a few tens of percent for typical UED parameters, while “poor” could easily be off be a factor of 2 or more.

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