Demonstration of eigen-to-projected emittance mapping for an ellipsoidal electron bunch

Phase-space partitioning offers an attractive path for the precise tailoring of complex dynamical systems. In beam physics, the proposed approach involves (i) producing beams with cross-plane correlations to control kinematic-moment invariants known as eigenemittances and (ii) mapping them to invariants of motion associated with given degrees of freedom via a decoupling transformation. Here we report on the direct experimental demonstration of the mapping of eigenemittances to transverse emittances for an electron beam. Measured phase space density confirms the generation of beams with asymmetric transverse emittance ratio $>200$ consistent with the initiated eigenemittance values. In addition, the bunches were produced via photoemission in the space-charge-driven expansion regime ultimately resulting in the generation of uniformly charged three-dimensional ellipsoidal bunches. The ellipsoidal character of the distributions is experimentally confirmed.

Figure 1

Overview of the AWA beam line showing only components relevant to the present experiment (a) with simulated emittance evolution in the skew-quadrupole channel (b), and associated beam (c) and phase-space (d) distributions simulated at X1 and X2. In (a) the labels “$\mathrm{SQ}i$” and “$\mathrm{X}j$” respectively correspond to skew-quadrupole magnets and insertable scintillating screens. The diagnostics station located at X2 also comprises scanning slits for emittance measurements. In (d) the phase-space simulated at X1 is cylindrical symmetric so that only $(x,x^{\prime })$ distribution (blue) is reported while both $(x,x^{\prime })$ (red) and $(y,y^{\prime })$ are reported at X2. In (b) we introduce the uncorrelated emittance as ${ϵ}_u\equiv \sqrt{{ϵ}_{4d}}$ for convenience.

Figure 2

Domain of CS parameters $(\alpha ,\beta )$ associated with the incoming magnetized beam that ensures the beam can be decoupled with the RFBT installed at AWA. The color map represents the settings of skew-quadrupole magnet gradients that can decouple the beam for a given value of the couple $(\alpha ,\beta )$. Field gradient values $B^{\prime }>10\mathrm{T}{\mathrm{m}}^{-1}$ along with regions where there is no solution for decoupling appear as white areas. Sequences (a)–(c) and (d)–(f) correspond to the two solutions producing a vertically flat beam. The green dots in (d)–(f) indicate the operating point selected during our experiment. Plots [(a),(d)], [(d),(e)], and [(c),(f)] give the magnetic gradient settings of quadrupole magnets SQ1, SQ2, and SQ3.

Figure 3

Photocathode-to-end simulation of the evolution of the averaged magnetization, 4D emittance (a) and eigenemittances (b) for two values of the magnetic field on the photocathode $B_c$. Plot (c) shows the evolution of the magnetization associated with longitudinal slices within the bunch upstream (blue trace, left axis) and downstream (green trace, right axis) of the RFBT for the case $B_c=140\mathrm{mT}$. The inset in (a) is a zoomed-in view of the RFBT section. The beam line layout is shown on the top of plot (a) with magenta, green, and blue rectangles respectively representing to accelerating cavities, solenoidal lenses, and skew-quadrupole magnets.

Figure 4

Transverse distribution of the magnetized (a) and round (b) beams $(\zeta ,x)$ PDF (c) downstream of the RFBT. These simulations were performed using the measured laser distribution on the photocathode. Plots (d), (e) and (f) respectively give the on-axis longitudinal electric $E_z(r=0,\zeta )$ and the transverse electric field $E_x(x,\zeta )$, $E_y(y,\zeta )$ inside the bunch computed at the bunch center ($\zeta =0$) and at the longitudinal positions at $\zeta =\pm {\sigma }_{\zeta }$.

Figure 5

Representative measurements of the laser distribution on the virtual cathode (a), and transverse electron-beam distribution recorded at X1 for the coupled beam (b) and at X2 for the decoupled beam (c). The lower row displays the horizontal (blue) and vertical (green) projection computed from the upper row. The solenoid lens “LF” was set to produce $B_c=140\mathrm{mT}$ and switching between coupled and uncoupled configuration is achieved by powering the skew quadrupole magnets (SQ1-3) to their nominal values.

Figure 6

Measured phase-space PDFs of the electron bunch in coupled [(a) and (b)] and uncoupled states [(c)–(f)] for $B_c=79\mathrm{mT}$ [(a), (c), and (e)] and $B_c=140\mathrm{mT}$ [(b), (d), and (f)]. Plots (a) and (b): $(y,y^{\prime })$ PDF of the round beam. Plots [(c) and (d)] and [(e) and (f)] are respectively the $(x,x^{\prime })$ and $(y,y^{\prime })$ PDF of the final uncoupled flat beams.

Figure 7

Measured eigenemittances associated with the incoming coupled beam $({ϵ}_{+},{ϵ}_{-})$ and corresponding final projected emittances $({ϵ}_x,{ϵ}_y)$ after decoupling for different magnetic-field values at the photocathode $B_c$ (a). Transfer of the uncorrelated emittances (b) and emittance ratio (c). The identity line is shown as a gray-dashed line in (b) and (c) and the orange line with shaded area respectively corresponds to a linear regression with associated $1\sigma$ confidence interval.

Figure 8

Transverse beam distribution (false-color image) with horizontal (blue traces) and vertical (green traces) projections. Images (a), (b), (c) and (d) respectively correspond to magnetic field on the photocathode of $B_c=79$, 100, 120, and 140 mT. The solid traces correspond to projections directly computed from the images while the dashed traces are parabolic fits using Eq. (16).

Figure 9

Measured spatiotemporal (a) and longitudinal phase space (b) PDF for the produced flat beam ($B_c=140\mathrm{mT}$). The solid traces (blue and green) are the projections of the PDF along the axis, and the dashed traces (blue and green) are the parabolic fit of the projections. The red dashed line in (b) is a linear fit to the PDF.