^{1,a)}and J. E. Sipe

^{1}

### Abstract

We present a comparison of a three-dimensional analytic Gaussian (AG) model of electron bunch propagation with numerical simulations of quasi- and non-Gaussian distributions. Quasi- and non-Gaussian distributions are a good representation of electron bunches used in ultrafast electron diffraction (UED) experiments, and we show that the AG model is successful in predicting the evolution of such freely propagating bunches. The bunch parameters in our comparisons are the bunch size, the total momentum spread, and the local momentum spread. In the case of the local momentum spread, which is related to the bunch coherence length, we compare the predictions of the AG model with three methods for calculating the local momentum spread from numerical data. This comparison also highlights the difficulties of calculating the evolution of the local momentum parameter from -body simulations. The AG model shows good agreement with -body simulations of different distributions for all the bunch parameters and is therefore a convenient tool for refining the UED experimental design.

We would like to thank Eugene Ya. Sherman and R. J. Dwayne Miller and members of his research group for their insightful discussions on UED experiments. This work was supported by the Natural Sciences and Engineering Research Council of Canada.

I. INTRODUCTION

II. PHASE-SPACE VARIANCES

III. TEST DISTRIBUTIONS

IV. ANALYTIC GAUSSIAN MODEL

V. DATA COMPARISON RESULTS

A. Global bunch parameters

B. Local momentum parameter

VI. CONCLUSION

### Key Topics

- Numerical modeling
- 22.0
- Cathodes
- 7.0
- Coherence
- 7.0
- Cumulative distribution functions
- 5.0
- Phase space methods
- 5.0

## Figures

Snapshots of bunch evolution during free propagation in the bunch reference frame. Shown are the spatial coordinates for a electron bunch, with an initial energy spread of at FWHM. The initial spatial distribution at is Gaussian, and the initial momentum distribution is a cosine-squared distribution. The time corresponds to the approximate duration of free propagation before the bunch hits the target. At the target the spatial distribution is no longer Gaussian, and the bunch is strongly chirped.

Snapshots of bunch evolution during free propagation in the bunch reference frame. Shown are the spatial coordinates for a electron bunch, with an initial energy spread of at FWHM. The initial spatial distribution at is Gaussian, and the initial momentum distribution is a cosine-squared distribution. The time corresponds to the approximate duration of free propagation before the bunch hits the target. At the target the spatial distribution is no longer Gaussian, and the bunch is strongly chirped.

Spatial and momentum variances for a distribution in phase space, where or .

Spatial and momentum variances for a distribution in phase space, where or .

Initial spatial distributions: (a) Gaussian and (b) waterbag, shown in the plane. Both distributions have equivalent spatial widths , shown in the same scale of units. A typical experimental cross section is about .

Initial spatial distributions: (a) Gaussian and (b) waterbag, shown in the plane. Both distributions have equivalent spatial widths , shown in the same scale of units. A typical experimental cross section is about .

Initial position of momentum vectors with energy and momentum spread . The magnitude follows a Gaussian distribution, while the angles and are subject to isotropic or cosine-squared angular distributions.

Initial position of momentum vectors with energy and momentum spread . The magnitude follows a Gaussian distribution, while the angles and are subject to isotropic or cosine-squared angular distributions.

Initial momentum distributions above the surface: (a) isotropic and (b) cosine squared, both shown in the plane. Both distributions have the same energy spread , shown with the same scale of units, in the center of momentum frame.

Initial momentum distributions above the surface: (a) isotropic and (b) cosine squared, both shown in the plane. Both distributions have the same energy spread , shown with the same scale of units, in the center of momentum frame.

Global variances: (a) velocity and (b) spatial size for a electron bunch with an initial energy spread (FWHM) and initial average energy . The closed points correspond to an initial Gaussian spatial distribution, while the open points correspond to a waterbag initial spatial distribution. The relative error is given by Eq. (16).

Global variances: (a) velocity and (b) spatial size for a electron bunch with an initial energy spread (FWHM) and initial average energy . The closed points correspond to an initial Gaussian spatial distribution, while the open points correspond to a waterbag initial spatial distribution. The relative error is given by Eq. (16).

Global variances for (a) velocity and (b) spatial size, electron bunch with an initial energy spread of (FWHM) and initial average energy .

Global variances for (a) velocity and (b) spatial size, electron bunch with an initial energy spread of (FWHM) and initial average energy .

Comparison of the results of the AG model with 3D binning calculation from -body simulations according to Eq. (4). Results are for a electron bunch, with initial energy 0.2 eV and initial (FWHM).

Comparison of the results of the AG model with 3D binning calculation from -body simulations according to Eq. (4). Results are for a electron bunch, with initial energy 0.2 eV and initial (FWHM).

Local momentum value calculated from -body data as a function of the bin size. Bin size is taken as a percent of the FWHM dimensions of an electron bunch at . Also shown is the average number of electrons per bin .

Local momentum value calculated from -body data as a function of the bin size. Bin size is taken as a percent of the FWHM dimensions of an electron bunch at . Also shown is the average number of electrons per bin .

Comparison of 3D binning results from -body simulations with extracted from 1D binning and emittance. Shown also are the predictions of the AG model. Results are for a electron bunch, with initial energy 0.2 eV and initial (FWHM). The data points are calculated from 3D binning according to Eq. (4), from 1D binning, and from emittance as defined in Eq. (18).

Comparison of 3D binning results from -body simulations with extracted from 1D binning and emittance. Shown also are the predictions of the AG model. Results are for a electron bunch, with initial energy 0.2 eV and initial (FWHM). The data points are calculated from 3D binning according to Eq. (4), from 1D binning, and from emittance as defined in Eq. (18).

Representative phase-space plots of the (a) Gaussian- distribution and (b) the waterbag- distribution in the transverse direction. This is a sample plot of electrons per bunch at .

Representative phase-space plots of the (a) Gaussian- distribution and (b) the waterbag- distribution in the transverse direction. This is a sample plot of electrons per bunch at .

Local velocity variance calculated for a electron bunch with (a) and (FWHM), and (b) and (FWHM).

Local velocity variance calculated for a electron bunch with (a) and (FWHM), and (b) and (FWHM).

Local velocity variances calculated for electrons per bunch and initial (FWHM).

Local velocity variances calculated for electrons per bunch and initial (FWHM).

## Tables

A summary of the different electron distributions used in -body simulations for comparison with the AG model. The simulations were done for two different numbers of electrons, and . Two different energy initial conditions were also chosen, one with an initial average photoemitted electron energy of with an initial energy spread of (FWHM) and the other with with (FWHM). These energy values are in the laboratory frame.

A summary of the different electron distributions used in -body simulations for comparison with the AG model. The simulations were done for two different numbers of electrons, and . Two different energy initial conditions were also chosen, one with an initial average photoemitted electron energy of with an initial energy spread of (FWHM) and the other with with (FWHM). These energy values are in the laboratory frame.

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