Wake Measurements of a Dechirper Jaw with Non-Zero Tilt Angle

The RadiaBeam/SLAC dechirper at the Linac Coherent Light Source (LCLS) is being used as a fast kicker, by inducing transverse wakefields, to e.g. facilitate Fresh-slice, two-color laser operation. The dechirper jaws are independently adjustable at both ends, and it is difficult to avoid leaving residual (longitudinal) tilt in them during set-up. In this report we develop a model independent method of removing unknown tilt in a jaw. In addition, for a short uniform bunch passing by a single dechirper plate, we derive an explicit analytical formula for the transverse wake kick as function of average plate offset and tilt angle. We perform wake kick measurements for the different dechirper jaws of the RadiaBeam/SLAC dechirper, and find that the agreement between measurement and theory is excellent.


INTRODUCTION
The RadiaBeam/SLAC dechirper at the LCLS is being used as a fast kicker, to facilitate the Fresh-slice, two-color scheme of generating X-rays [1]. In this mode of operation, after the final linac, the beam is made to pass close by to one jaw of a dechirper module, in order to send the tail of the bunch on a different trajectory than the head on the way to the undulator. During alignment of the jaws, each end is moved by an independent motor.
Thus, in general, a jaw will tend to have an offset as well as some residual tilt with respect to the beam trajectory.
Typically, for Fresh-slice, two-color operation the dechirper jaw is moved toward the beam and adjusted while observing the size of the effect on the induced, downstream betatron oscillation of the beam. The adjustment is not precise and is done somewhat by feel. There may come a time, however, when it is important to accurately know the location and orientation of the jaw with respect to the beam. In a recent report on wakefield measurements on the dechirper at the LCLS, the agreement between measurement and calculation was found to be excellent, after a slight adjustment to the gap parameter in the theory (in two-plate measurements) [2,3]. However, because of the possibility of an unknown tilt in the jaws, one could not simply conclude that the discrepancy implied an error in measurement or theory.
This report uses a model independent method of removing unknown tilt in a jaw of the RadiaBeam/SLAC dechirper. The idea of the method is simple. The average transverse kick (or center of mass kick) experienced by a beam on passing by a dechirper plate depends on a strong inverse power of the offset of beam from plate (minus the third power for short bunches). If we run a procedure that fixes the beam offset at the center of the plate (longitudinally, in z) while varying the tilt angle in both positive and negative directions, the average wake kick will trace out a curve that has a minimum at the condition of zero tilt angle. This is precisely the experiment that we have performed and report on here.
In this report we also develop an analytical formula for the wake kick experienced by a short bunch on passing a single dechirper plate, as function of average beam offset and plate tilt angle. This allows us to perform a more precise comparison with measurement than was done before [2,3].

THEORY
The geometry of three corrugations of the RadiaBeam/SLAC dechirper is shown in Fig. 1.
The parameters are (typical) half-gap a = 0.7 mm, h = 0.5 mm, p = 0.5 mm, and t = 0.25 mm. Let us consider the case of a beam passing by a single dechirper jaw, with the other jaw far away and not interacting with the beam. The ends of the dechirper jaws are independently adjustable. Thus, in general, the configuration of beam and jaw can be characterized by just two parameters, average offset b and extra offset at the jaw ends ±d (see Fig. 2; or, equivalently, jaw tilt angle tan θ = 2d/L ≈ θ, with L the dechirper jaw length).
In the measurements to be presented below, the wake strength is quantified by the average transverse kick induced in the beam during its passage near a jaw; this quantity is proportional to the average of the bunch wake, i.e. the kick factor, κ x . The bunch at the end of the LCLS linacs is short with an approximately uniform distribution. The kick factor for a short, uniform bunch of full length , passing by a single dechirper plate at offset b (with no tilt), is [3][4][5] with Z 0 = 377 Ω, c the speed of light; with The beam (blue ellipse) moves in the z direction below the dechirper jaw (red), at average offset b; the jaw tilt (with respect to z) is defined by the change in offsets at the jaw ends, ±d. Note that the corrugation size and tilt angle as sketched are much larger than in reality.
For a plate with a small angle tilt, with average offset b and offset at the ends b ± d, we approximate the total kick factor by averaging the analytical formula along the dechirper plate: Substituting in Eq. 1, and performing the integral we obtain with where and Ei(x) is the exponential integral function. Note that κ xt is symmetric with respect to the variable d, as it should be. In Appendix A we present the corresponding derivation for the longitudinal wake of a tilted plate.
The kick factor for the tilted configuration normalized to the non-tilted case, κ xt /κ x , as function of d/b is shown in Fig. 3 (in blue). Here we have used as corrugation parameters those of the RadiaBeam/SLAC dechirper, full bunch length = 17 µm, and average offset of plate from beam, b = 1 mm. For comparison, the longitudinal effect, i.e. the change in relative loss factor, κ t /κ, is given in red. We see that the longitudinal wake is a less sensitive function of the tilt than the transverse wake. This is because the longitudinal wake has a weaker dependence on offset of beam from plate, b; for a short bunch it varies as b −2 instead of the b −3 of the transverse case. The beam position at downstream BPM 590-assuming the beam is initially traveling parallel to the z-axis and that there is no intervening magnet-is simply  for each dechirper plate.
For the measurements described below, the beam was kept steady and the dechirper jaws were moved. For each data set one jaw was moved near the beam trajectory (while the three others were moved far away from it), following the sequence the horizontal jaws-North then  [3], these longitudinal results are extremely noisy, are not much help in the analysis of the measurements, and will not be discussed further.) During the measurements the charge Q = 160 pC and energy E = 13.24 GeV. In Fig. 5 we display the bunch distribution as obtained by the transverse cavity, XTCAV; the head of the bunch is to the left. We see that the distribution is approximately uniform; the uniform distribution with the same area and rms length has peak current I = 2.7 kA and full length = 18 µm. The transverse beam sizes at the dechirper are σ x = 14 µm, σ y = 40 µm; our theory assumes that the beam size is small compared to the distance between beam and plate, which is satisfied for our measurements.
where L (= 2 m) is the dechirper plate length and l, the distance between the LVDT sensors and the ends of the plates. We believe the measured variation in the δb i is fairly accurate, though with a possible, relatively large unknown shift, b 0 ; i.e. that for the i th measurement point (corresponding to d i ) the actual average offset is For each plate, the theory was fit to the data using a nonlinear model fit, with fitting parameters: shift in average offset b 0 , shift in tilt parameter d 0 , and shift in reading on BPM 590, x 0 (for North and South) or y 0 (for Top and Bottom); the function variables in the fit were d and δb. The δb i that were measured are shown in Fig. 6. We see that the functions were relatively flat during the North and Bottom measurements, but that they had significant variation during the South and Top ones.  Table I    In addition, we derive an explicit analytical formula for the transverse wake kick of a single dechirper plate, as function of plate offset and tilt angle with respect to the beam orbit. We present wake measurements with the different LCLS dechirper jaws and show that, for the kick factors, agreement with theory is excellent. Compared to previously reported single plate wake measurements that assumed the tilt angle was small and not important [3], the measurements reported here are a more sensitive test and stronger confirmation of the theory.
Having demonstrated the accuracy and sensitivity of this measurement procedure for orienting a dechirper jaw, we propose incorporating it routinely in the set-up and alignment of the RadiaBeam/SLAC dechirper at the LCLS. The procedure is relatively simple and quick to perform.
In Appendix A we derive an explicit analytical formula for the longitudinal wake kick of a single dechirper plate, as function of plate offset and tilt angle with respect to the beam orbit.

APPENDIX A: LONGITUDINAL EFFECT
For a uniform bunch distribution, the loss factor-the average of the longitudinal bunch wake-is given by [3] with s 0s = 2b 2 t πα 2 p 2 and For a plate with a small angle tilt, with average offset b and offset at the ends b ± d, we approximate the total loss factor as Substituting in Eq. 11, and performing the integral we obtain with where In Ref. [3] single plate wake measurements were performed and compared with theory.
There the plate was assumed to have negligible tilt and the transverse wake kick was measured as function of plate offset from the beam, b. Simultaneously, the longitudinal wake effect was also measured as function of b. As here, it was assumed that the measured offsets agreed to within 20 µm, for both jaw measurements, gave the authors confidence that the jaw tilt for these measurements was indeed small, and that the measurements confirmed the theory.
However, as a stand-alone measurement of the transverse wake of a single plate, the measurement described in the present report is much superior to that of Ref. [3]. To see why, consider Fig. 9, which simulates the earlier type of measurement. The corrugation parameters used were those of the RadiaBeam/SLAC dechirper; the bunch distribution assumed was uniform of full length = 18 µm. Fig. 9 shows the simulated kick factor κ xt as function of beam offset b assuming no tilt in the jaw (the blue solid curve). On the same plot we present results for tilt parameter d = 0.2 mm (red dashes), which differ significantly from the blue curve. However, when we shift the abscissa of these last results by b 0 = −70 µm, we obtain the curve of the gold dashes, which is now close to the blue curve. Thus, there is significant correlation between the parameters d and b 0 , and using them as fitting parameters for such a measurement will not reliably find their separate values. Consequently, this kind of measurement is not a good way to find the offset and tilt of a dechirper with respect to the beam.
In contrast, consider Fig. 10, which is a simulation of the type of measurement described in the present report. We plot κ kt vs. tilt parameter d for cases average offset b = 0.9 mm (red dashes), 1.0 mm (blue solid line), and 1.1 mm (brown dashes). The dashed curves have been shifted vertically so that all have the same minimum value. One can easily see that there is little correlation between average offset b and a vertical shift in κ xt , and that this measurement is very sensitive to changes in b. We can further conclude that the measurements for the North, Top, and Bottom dechirper plates (Fig. 7, top plot; Fig. 8) [6] indicate that the formula for the kick factor of a single corrugated plate, as functions of beam offset and plate tilt (Eq. 5), is correct to very good accuracy.