Lossless Monochromation for Electron Microscopy with Pulsed Photoemission Sources and Radio-Frequency Cavities

Resonant radio-frequency cavities enable exquisite time-energy control of electron beams when synchronized with laser-driven photoemission. We present a lossless monochromator design that exploits this fine control in the one-electron-per-pulse regime. The theoretically achievable maximum beam current on target is orders of magnitude greater than state-of-the-art monochromators for the same space-time-energy resolution. This improvement is the result of monochromating in the time domain, unconstrained by the transverse brightness of the electron source. We show analytically and confirm numerically that cavity parameters chosen to minimize energy spread perform the additional function of undoing the appreciable effect of chromatic aberration in the upstream optics. We argue that our design has significant applications in ultrafast microscopy, as well as potential for use in nontime resolved microscopy, provided future photoelectron sources of sufficiently small size and laser sources of sufficiently high repetition rate. Our design achieves in simulations more than two orders of magnitude reduction in beam energy spread, down to single digit milli-electron volts. Overcoming the minimum probe-size limit that chromatic aberration imposes, our design clears a path for high-current, high-resolution electron-beam applications at primary energies from single to hundreds of kilo-electron volts.

Figure 1

(a) Energy selecting monochromator as described in Ref. [34]: magnetic prisms disperse the energy spectrum of the electron beam in the transverse direction and a narrow acceptance slit selects the desired bandwidth at the cost of lost current. (b) Our lossless monochromator design: photoemission is triggered by a laser pulse and the beam energy spectrum disperses in the longitudinal direction. Time-correlated acceleration in the pair of cavities, indicated by force vectors F_z in the figure, equalizes the energies of the dispersed beam. The average current in our design is limited by the cavity frequency.

Figure 2

Normalized energy gain versus the single particle time of arrival, expressed as a fraction of the full rf period. The quadratic coefficient of the energy gain is subtracted, leaving only unwanted higher-order terms. Vertical lines show the initial pulse length $\mathrm{\Delta }{t}_{\ell }$ (solid) and final pulse length $\mathrm{\Delta }{t}_f$ (dashed). The duty cycle in Eq. (2) is $f\mathrm{\Delta }{t}_f$. Inset shows the full sinusoidal energy gain versus the scaled time of arrival, with the quadratic term restored.

Figure 3

(a) Simulated source and monochromator layout in the $x$-$z$ plane. The left axis is the transverse scale for the optical elements. All elements are axially symmetric. The transverse beam size is shown by the green curve, with the scale indicated by the axis on the right. The cathode and anode are at $z=0,10\mathrm{mm}$, respectively. At $z=50\mathrm{mm}$ is a focusing solenoid; at $z=160,240\mathrm{mm}$, dumbbell silhouettes approximate cavity cross sections. (b) Electrode profile and axial field $E_z(z)$ in the gun for a voltage of 50 kV.

Figure 4

Evolving correlation between the energy, radial position $r$, and time of arrival $\mathrm{\Delta }t$ in a dc electron gun. Results plotted are of single particle trajectory simulations. The accelerating field is uniform with a gradient of $5\mathrm{MV}/\mathrm{m}$. Each subplot shows the cross section of radial position and time of arrival at the exponentially increasing values of $z$ indicated on the top axis. Color indicates particle energy $\mathrm{\Delta }\mathcal{E}$ relative to the minimum energy in the statistical ensemble; $\mathrm{\Delta }t$ is defined relative to the last arriving particle. As the uncertainty in time and position grows, the correlation with energy tightens. Uncertainty in time of arrival asymptotically approaches the value predicted by Eq. (5) as the energy gained in acceleration comes to dominate the initial kinetic energy. The initial statistical ensemble in all our simulations, unless otherwise specified, is uniformly distributed in energy, time of emission, and solid angle over the forward hemisphere [42].

Figure 5

Cross section of the $3\mathrm{GHz}$ TM010 mode cavity used in the energy equalization device described in this paper.

Figure 6

Simulation cavity field map: (a) $z,r$ cross section of the radially symmetric cavity ${\mathbf{E}}^{\mathrm{rf}}$ field at nominal amplitude and phase; (b) the axial cavity field $E_{0,z}^{\mathrm{rf}}$ and its leading spatial derivatives, from which we construct the simulation field map.

Figure 7

Simulation of a beam with $\mathrm{\Delta }{t}_l=30$ fs and only longitudinal momentum spread. Transverse size and momentum spread are set to zero. In (a), an idealized, uniform field gun is used. In (b), the realistic gun field shown in Fig. 3 is used. Blue points denote the energy-time correlation just after the gun. Red points show the result after transiting a single cavity with amplitude set by the solution of Eq. (20). White lines are the predictions of Eqs. (11) and (24), where in (b) we use the field at the cathode for $E_z^{\mathrm{cat}}$.

Figure 8

Simulation of a beam with initial size ${\sigma }_x=12$ nm and nonzero transverse momentum spread. Here the beam has no longitudinal momentum spread and has zero duration. Blue points show the energy-space correlation just after the gun, and red points show the result after transiting a single cavity with amplitude set by Eq. (21). The white line is the prediction of Eq. (11). Because of thick lens effects, the cavity settings predicted by Eq. (21) slightly overcorrect the energy-space correlation. The green curve is the result of numerical optimization of the cavity amplitude.

Figure 9

Simulated evolution of energy correlations on transiting two rf cavities, with an initial energy spread of $1\mathrm{eV}$, vanishing initial transverse source size, and vanishing initial pulse length: (a) post gun but before the first cavity, the paraboloid distribution predicted by Eq. (11); (b) between the two cavities, the first cavity imparts a hyperboloid distribution as predicted by Eq. (18), with energy increasing in time and space; (c) after the second cavity, the residual energy spread involves cubic corrections to Eq. (11), shown in Eq. (12).

Figure 10

Results of simulating the beamline shown in Fig. 6. Particle ensembles initially have uniform energy spreads of $1\mathrm{eV}$, distributed uniformly in solid angle over the forward hemisphere. Probability density functions (PDF) for two accelerating voltages are shown: $10\mathrm{kV}$ (blue), $50\mathrm{kV}$ (red). Dashed horizontal and vertical lines indicate the FWHM.

Figure 11

Simulation results showing beam sizes and emittance as a function of beamline position for three different initial energy spreads; all particle ensembles have initial transverse size of $12\mathrm{nm}$ rms and momenta distributed uniformly over the forward hemisphere with initial uniform energy spreads of $1\mathrm{eV}$ (red), $500\mathrm{meV}$ (blue), and $250\mathrm{meV}$ (yellow). Cavity settings are chosen to achieve the smallest final energy spread. (a) Transverse normalized emittance; (b) rms energy spread; (c),(d) rms pulse duration.

Figure 12

Simulated energy distribution in the presence of white rf phase noise at the same 5 fs rms scale reported experimentally in Ref. [28]. Four sets of initial conditions are shown, with initial energy spreads of 1 eV in all. Colors indicate two sets of primary energies: 50 keV (blue), 100 keV (red). Line style indicates the initial pulse length: 30 fs (solid line), 200 fs (dotted line). (a) Probability density per milli-electron volt downstream of the rf monochromator. Inset shows the distribution of the timing noise in the simulation, with the noise identically and independently distributed in each cavity. (b) Cumulative probability function (CDF) with dashed horizontal lines indicating the interquartile range. (c) Probability of time of arrival per picosecond relative to the mean, downstream of the gun.

Figure 13

Simulations of cavity timing jitter. The initial particle ensemble has a uniform $1\mathrm{eV}$ energy spread distributed uniformly in solid angle over the forward hemisphere. Plots show the ensemble after transiting two $3\mathrm{GHz}$ cavities with timing offset from the optimal solution by ${\varphi }_1$ and ${\varphi }_2$, respectively. Offsets add in quadrature to $10\mathrm{fs}$. Left: Histograms of final energy show single milli-electron volt movement in peak location and $10\mathrm{meV}$ scale movement in tails. Right: Energy-time correlation are shown as a function of cavity phase, where different phases are offset on the time axis for clarity.

Figure 14

Correlations between particle divergence and time of arrival in particle tracking simulations of a uniform accelerating gradient of $5\mathrm{MV}/\mathrm{m}$. The initial conditions of the particle distribution are a 1 eV uniform energy spread distributed uniformly in solid angle over the forward hemisphere, and vanishing transverse size and pulse length. The time-dependent divergences predicted by Eq. (37) are shown by the white lines at constant time increments of $50\mathrm{fs}$.