Chromatic effects in quadrupole scan emittance measurements

A reliable transverse emittance measurement for high-brightness electron beams is of utmost importance for the successful development of fourth generation light sources and for the beam transport in plasma-based accelerators. When the beam exhibits a significant energy spread, typical quadrupole scan emittance measurements may be affected depending on the beam properties and on the quadrupoles arrangement. The emittance degradation induced by chromatic effects in measurements involving magnetic lattices is evaluated analytically for different configurations. Analytical and numerical calculations compared with measurements have been used to evaluate the consequent error on the emittance value measured for single and double quadrupole schemes and for typical operating conditions at the SPARC facility.

Figure 1

Layout of SPARC high energy experimental area. Emittance measurements are performed by measuring the transverse beam size at the screen $F_1$, which holds a YAG:Ce screen.

Figure 2

Left picture: retrieved ellipse in the trace space enclosed by tangent lines corresponding to the set of measurements. Right picture: measured horizontal rms beam size (blue dots) compared to the ${\chi }^2$ fit (red curve) as a function of quadrupole gradient $g$ (measurements in the two-quadrupole configuration; 300 pC, 147.5 MeV, ${\epsilon }_n=1.83(0.11)\mathrm{mm}\mathrm{mrad}$).

Figure 3

Variation of the rms spot size on the measurement screen for a single quadrupole scan with (red solid line) and without (black dashed line) energy spread for typical SPARC operations (${\sigma }_x=2\mathrm{mm}$, ${\sigma }_{x^{\prime }}=100\mu \mathrm{rad}$, ${\sigma }_{x^{\prime }x}={\sigma }_{x{x}^{\prime }}=0$, $L=5\mathrm{m}$, ${\sigma }_{\gamma }=1\%$).

Figure 4

Variation of the rms geometric emittance in a doublet: the first quadrupole is converging (diverging) in the blue solid (red dashed) line (${\sigma }_x=1.7\mathrm{mm}$, ${\sigma }_{x^{\prime }}=76\mu \mathrm{rad}$, ${\sigma }_{x^{\prime }x}={\sigma }_{x{x}^{\prime }}=0.1284\mathrm{mm}\mathrm{mrad}$, $L_{12}=0.45\mathrm{m}$, ${\sigma }_{\gamma }=1\%$).

Figure 5

Variation of the rms spot size on the measurement screen for a doublet quadrupole scan with (red solid line) and without (black dashed line) energy spread, being the first quadrupole focusing (defocusing) in the left (right) plot. (${\sigma }_x=2\mathrm{mm}$, ${\sigma }_{x^{\prime }}=100\mu \mathrm{rad}$, ${\sigma }_{x^{\prime }x}={\sigma }_{x{x}^{\prime }}=0$, $L=4.5\mathrm{m}$, ${\sigma }_{\gamma }=1\%$).

Figure 6

Horizontal emittance as a function of gradient at the quadrupole exit for the three cases of Table . The blue circles are the simulation results, while the black dashed line is the analytical curve with no correlations included [i.e. Eq. (16)] and the red solid line accounts for correlation [i.e. Eq. (20)]; (a) Case 1, (b) Case 2, (c) Case 3.

Figure 7

Case 2: normalized horizontal and vertical emittance at the exit of the second quadrupole as a function of quadrupole gradient in a single (a) and doublet quadrupole (b) scan. In the doubler scan, the horizontal axis refers to the gradient in the first quadrupole where positive gradients are focusing in the horizontal plane.

Figure 8

Case 3: normalized horizontal and vertical emittance at the exit of the second quadrupole as a function of quadrupole gradient in a single (a) and doublet quadrupole (b) scan. In the doubler scan, the horizontal axis refers to the gradient in the first quadrupole where positive gradients are focusing in the horizontal plane.

Figure 9

Virtual emittance measurement (TSTEP simulations) for a two-quadrupole scan (‘‘$-$, $+$” configuration). Top (bottom) plots show the spot size scan and the corresponding trace space reconstruction for a quadrupole gradient sampling step of $0.16(0.08)\mathrm{T}/\mathrm{m}$.

Figure 10

Horizontal trace space for a 200 pC VB compressed SPARC beam with ${\sigma }_{\gamma }=0.860(0.030)\%$, ${\sigma }_{x,F_0}=1.400(0.020)\mathrm{mm}$ beam (analogous to Case 1 of Table ). The same trace space is reconstructed from a single quadrupole scan (left plot) and from doublet quadrupoles ‘‘$+$ $-$” (‘‘$-$ $+$”) scan in the center (right) plot.