Tune variation and transverse displacement as a diagnostic for profiling beam energy

In this paper we describe two methods that demonstrate an energy evolution at the ends of a long bunch. The first method is a nondestructive diagnostic that extends the response matrix technique, to provide insight into the time-sliced transverse behavior of a coasting bunch in a ring. This method uncovers intense beam physics from the resulting longitudinal energy variations at the ends of a rectangular bunch. Measurements of the tune profile, using this method, agree with analytical predictions resulting in a percent error of 0.32% and 2.45% (head and tail respectively). Measurements and calculations presented in this paper are presented for a beam with an intensity of 0.901 (beam current of 21 mA). The head and the tail tunes, averaged over four ring chambers, also illustrate the intensity variation axially along the bunch. This is exhibited through tune-shift measurements at the ends of the bunch. As confirmed through calculations, the measured tune shift at the head is smaller than the tail. This results from the sum of both the transverse coherent tune shift from image forces with the component from longitudinal space charge. The sum of the two components produces a smaller tune shift at the head of the beam and a larger tune shift at the tail. The second method presented in this paper optically illustrates the centroid “cork-screwing” motion within the beam ends, at a single location on the ring. The motion is resolvable using this diagnostic since it is capable of axially time-slicing regions of the beam with different energies. The difference between the two methods presented is that the first is a nondestructive diagnostic that can be extended to multiple turns whereas the second method is a destructive first turn imager.

Figure 1

Beam charge line density and energy profile snapshot illustrations as a function of time, starting from an initially rectangular beam distribution at $s=0$ (shown in blue). The curves in black are the same as in blue but at a distance $s=s_1$ from the source.

Figure 2

Lattice optics diagram with RC9, RC11, RC13, and RC15 circled.

Figure 3

Analytical current and velocity profiles calculated using the self-similar one-dimensional solution on the first turn, at RC9, RC11, RC13, and RC15 for the 21 mA beam.

Figure 4

Analytical zero-current tune profiles calculated on the first turn for the 21 mA beam, at RC9, RC11, RC13, and RC15.

Figure 5

Analytical tune profile (including coherent contribution) calculated on the first turn for the 21 mA beam, at RC9, RC11, RC13, and RC15.

Figure 6

Measured perturbed horizontal centroid motion (within the midregion of the beam using a 2 ns window) around the ring using a single beam position monitor (BPM at RC15, zero radians). The measured tune is $6.541\pm 0.062$ with a fit goodness of 0.99. The average error per point is $0.1\mathrm{mm}/\mathrm{A}$.

Figure 7

Top BPM plate at RC9, RC11, RC13, and RC15.

Figure 8

Horizontal tune as a function of beam length, measured using a single beam position monitor at RC9, RC11, RC13, and RC15.

Figure 9

3 ns gated camera images of the 21 mA beam head, measured at RC15 as a function of time along the beam pulse.

Figure 10

3 ns gated camera images of the 21 mA beam tail, measured at RC15 as a function of time along the beam pulse.

Figure 11

Centroid measurements from the 3 ns gated camera images of the 21 mA beam, measured as a function of beam length at RC15. We subtract the position where (0, 0) refers to the centroid of the beam center from all data points. (Calibration was $0.07\mathrm{mm}/\mathrm{pixel}$ in both $x$ and $y$.)