Point-to-point Coulomb effects in high brightness photoelectron beam lines for ultrafast electron diffraction

In an effort to increase spatial and temporal resolution of ultrafast electron diffraction and microscopy, ultrahigh-brightness photocathodes are actively sought to improve electron beam quality. Beam dynamics codes often approximate the Coulomb interaction with mean-field space charge, which is a good approximation in traditional beams. However, point-to-point Coulomb effects, such as disorder-induced heating (DIH) and the Boersch effect, cannot be neglected in cold, dense beams produced by such photocathodes. In this paper, we introduce two new numerical methods to calculate the important effects of the photocathode image charge when using a point-to-point interaction model. Equipped with an accurate model of the image charge, we calculate the effects of point-to-point interactions on two high-brightness photoemission beam lines for ultrafast diffraction. The first beam line uses a 200 keV gun, whereas the second uses a 5 meV gun, each operating in the single-shot diffraction regime with ${10}^5\mathrm{electrons}/\mathrm{pulse}$. For the beam lines simulated in this paper, assuming a zero photoemission temperature, it is shown that including stochastic Coulomb effects increases the final emittance by over a factor of 2 and decreases the peak transverse phase space density by over a factor of 3 as compared to mean-field simulations. We then introduce a method to compute the energy released by DIH using the pair correlation function and approximate the contribution DIH has on the emittance, which may serve as a reasonable estimate for the effects of DIH beyond the cases studied in this work. This DIH energy was found to scale very near the theoretical result for stationary ultracold plasmas, and it accounts for over half of the emittance growth above mean-field simulations.

Figure 1

Depiction of PMP three-step space charge calculation. Filled in ellipses represent mean-field calculation of electric fields, and ellipses filled with dots represent a Barnes-Hut calculation of electric fields.

Figure 2

Layout of the cryocooled dc gun UED beam line used in the following simulations.

Figure 3

Layout of the NCRF gun beam line used in the following simulations.

Figure 4

(a) Spot size and (b) transverse normalized rms emittance comparison between the PMP method, Barnes-Hut method without a cathode, and mean-field space charge simulations of the dc UED beam line with 0 meV MTE.

Figure 5

(a) Spot size and (b) transverse normalized rms emittance comparison between the PMP method, Barnes-Hut method without a cathode, and mean-field space charge simulations of the NCRF UED beamline with 0 meV MTE.

Figure 6

Depiction of emittance vs particle fraction selection. Ellipses are drawn such that they represent the phase space area occupied by the beam using only a given fraction of the total number of particles. Ellipse dimensions are selected such that the emittance is minimized for each particle fraction.

Figure 7

Transverse normalized rms emittance vs particle fraction plots and phase space comparison between PMP and mean-field simulations of the two UED beam lines at the respective emittance minimum near the end of the beam lines. Subfigures (a), (c), and (e) correspond to the dc beam line and subfigures (b), (d), and (f) correspond to the NCRF beam line. Phase space portraits from the mean-field simulations are shown in subfigures (c) and (d) and for PMP simulations in (e) and (f) phase space portraits are shown with linear $x–p_x$ correlation removed.

Figure 8

Core emittance comparison between PMP and mean-field simulations of the dc UED beam line with 0 meV MTE.

Figure 9

Core emittance comparison between PMP and mean-field space charge for the NCRF UED beam line with 0 MTE.

Figure 10

Radial distribution function comparison between PMP and mean-field simulations of the NCRF UED beam line with 0 meV $\mathrm{MTE}\sim 3\mathrm{mm}$ away from the cathode. Only a small $r$ portion of the distribution is plotted to show the creation of the Coulomb hole when point-to-point space charge is used. For comparison, the distributions were normalized such that the mean of the radial distribution functions from 1.5 to $3.0\mu \mathrm{m}$ is equal to 1.

Figure 11

Energy from disorder-induced heating as calculated from $g(r)$ in the NCRF beam line with 0 MTE and a smaller initial density of ${10}^{17}{\mathrm{m}}^{-3}$. The density was reduced by increasing the initial radial size of the electron beam at the cathode.

Figure 12

Radial distribution function comparison between of the NCRF beam line with an MTE of 0 meV and an initial density of ${10}^{17}{\mathrm{m}}^{-3}$ before and after the beam waist. For comparison, the distances were normalized by the average interparticle distance, $a$, and the radial distribution functions, $g(r/a)$, were normalized such that $g(r/a=1.25)=1$.

Figure 13

Disorder-induced heating as calculated from Eq. (2) compared to the result calculated from simulation.

Figure 14

(a) Spot size, (b) transverse normalized rms emittance, and (c) core emittance comparison between the PMP method and mean-field space charge simulations of the dc UED beamline with 150 meV MTE.

Figure 15

(a) Spot size, (b) transverse normalized rms emittance, and (c) core emittance comparison between the PMP method and mean-field space charge simulations of the NCRF UED beamline with 150 meV MTE.

Figure 16

Transverse normalized rms emittance comparison for point-to-point, mean-field, and modified image charge simulations of the dc UED beam line with 0 meV MTE.

Figure 17

Transverse normalized rms emittance comparison for point-to-point, mean-field, and modified image charge simulations for the NCRF UED beam line with 0 meV MTE.