Matching small beta functions using centroid jitter and two beam position monitors

Matching to small beta functions is required to preserve emittance in plasma accelerators. The plasma wake provides strong focusing fields, which typically require beta functions on the mm-scale, comparable to those found in the final focusing of a collider. Such beams can be time-consuming to experimentally produce and diagnose. We present a simple, fast, and noninvasive method to measure Twiss parameters using two beam position monitors only, relying on the similarity of the beam phase space and the jitter phase space. By benchmarking against conventional quadrupole scans, the viability of this technique was experimentally demonstrated at the FLASHForward plasma-accelerator facility.

Since the focusing field from an exposed ion column in a plasma accelerator is typically very strong, beams must be tightly focused for the beam size not to oscillate, as this would lead to significant and unacceptable emittance growth [16]. In terms of Twiss or Courant-Snyder parameters [17], the beta function needs to be matched to where E is the beam energy, n is the plasma density, 0 is the vacuum permittivity, and e is the electron charge. Injecting a GeV-level beam into a typical plasma accelerator requires beta functions on the mm-scale. While plasma density ramps [18][19][20] can relax the matching condition by increasing β m at the entry and exit of the accelerator stage, it will nevertheless be challenging and time-consuming to experimentally produce and diagnose the required tightly-focused beams. Conventional beam-focus diagnostics include wire scanners and high-resolution screens around the focal point, or downstream quadrupoles that point-to-point image the beam onto a screen-all of which require nontrivial experimental setups and careful data analysis. This can be inconvenient when matching beams into a plasma accelerator-a slow multi-parameter optimization process where fast feedback will be crucial.
In this paper, we present an alternative method for simple, fast, and noninvasive measurement of small beta * carl.a.lindstroem@desy.de functions by using two beam position monitors (BPMs) to measure the centroid jitter. The technique is based on the observation that the phase space of the jitter often has similar Twiss parameters to that of the beam, and can therefore be used as a proxy. While the method is approximate in nature, it allows online monitoring and iterative adjustment of the waist location and beta function. This technique was successfully implemented and experimentally demonstrated at the FLASHForward [21,22] plasma-accelerator facility at DESY.

II. BEAM AND JITTER PHASE SPACES
The phase space of a beam consists of its particle distribution in x-x space. Similarly, the phase space of the beam centroid jitter (the jitter phase space) is the distribution of beam centroid offsets in x-x space when integrated over a large number of shots. Therefore, the jitter has its own Twiss parameters and emittance.
The central assumption underpinning this technique is that the Twiss parameters of the jitter are similar to those of the beam. The significance of this connection is that it is possible to simply and noninvasively measure the phase space of the jitter using BPMs, which then acts as an approximate measurement of Twiss parameters of the beam. It should be noted that this technique is generally not able to measure the beam emittance, but this is also not required for matching (see Eq. 1).
While the similarity of the beam and the jitter is not inevitable, it is motivated by both experimental observation and theoretical considerations. Linear accelerators usually have FODO-like lattices with beta functions on the 1-10 m scale. This means that magnets and accelerating cavities-sources of jitter-are typically distributed across a range of phase advances. As a consequence, the jitter-phase-space ellipse gradually expands while it rotates to acquire a similar shape to the beam-phase-space ellipse. Even if the beam and jitter phase spaces are moderately mismatched, both will evolve and be focused similarly in a linear-optics lattice-also in the case of strong focusing, as demonstrated by the example in Fig. 1. The most interesting quantities in the context of matching are the location and beta function of the focus waist. How inaccurate should we expect the jitter-based measurement to be? Consider a lattice that focuses the beam to a small waist, where the beam size is demagnified by a factor B. Starting from matched Twiss parameters β 0 and α 0 , the resulting waist beta function would be β 0 /B 2 . The transfer matrix of such a lattice [17] can be expressed as where the phase advance ψ is a free parameter. Consider then a mismatched jitter with an initial Twiss matrix where γ = (1 + α 2 )/β is the Twiss gamma function. The overall mismatch can be quantified by the mismatch parameter [23] whereβ e = β/β 0 ,α e = α − α 0 β/β 0 andγ e = (1 +α 2 e )/β e quantify the normalized error of each Twiss parameter. The mismatch parameter M is invariant in a linear-optics lattice, whereas the individual Twiss errors are not.
We can transport the mismatched jitter to the beam waist location (i.e., the end of the lattice) using The Σ 11 element corresponds to the jitter beta function at the beam waist location. However, the beam waist does not generally coincide with the jitter waist, and therefore Σ 11 does not correspond to the waist beta function of the jitter. Instead, assuming that the focus region consists only of a drift, the waist beta function equates to the inverse gamma function (1/Σ 22 ), which can be expressed as Similarly, the shift of the jitter waist location is given by the ratio of the alpha and the gamma function (−Σ 12 /Σ 22 ), which is derived to be δs w = β 0 B 2 α e cos 2ψ + 1 2 (β e −γ e ) sin 2ψ α e sin 2ψ +β e sin 2 ψ +γ e cos 2 ψ .
Although lengthy, it is easy to see that if the Twiss errorsα e ,β e , andγ e are all of order one (i.e., moderately mismatched), the brackets in both Eqs. 6 and 7 become numerical factors also of order one, regardless of the free parameter ψ. This means that the waist beta function of the mismatched jitter remains similar to the waist beta function β 0 /B 2 of the matched beam. Moreover, it implies that the offset of the waist location is also approximately β 0 /B 2 -on the order of the waist beta function itself.

III. TWO-BPM MEASUREMENT METHOD
Having connected the phase space of the beam to that of the jitter, the problem has been reduced to measuring the phase space of the jitter. This can be done quickly and noninvasively with a multi-shot measurement using two BPMs separated ballistically (i.e., with no magnets between them)-see Fig. 2 for a conceptual setup. Correlated offset data is required to measure the position and angle of each shot via where x 1 and x 2 are the upstream and downstream centroid offsets, respectively, and ∆s is the separation of the two BPMs. Both transverse planes can be measured simultaneously. As the number of shots increases, the jitter phase space will gradually build up, assuming that the optics remains unchanged. Whenever the optics does change, the measurement must be restarted. Given that no scan is performed, the data can be analyzed immediately from the start of the measurement, then re-analyzed with every additional shot, gradually increasing the precision. As the number of shots N increases, the relative measurement error of Twiss parameters and jitter emittance will be approximately 1/ √ N . Since the connection between the beam and the jitter phase space is only approximate, it will rarely be necessary to require better than about 10% precision (corresponding to 50-500 shots). In a typical accelerator with a 1-10 Hz repetition rate, this allows quasi-online monitoring on a few-second timescale.
In addition to being quick, the method is relatively insensitive to calibration errors. A correctly calibrated two-BPM measurement requires only an accurate measurement of the BPM separation and an accurate calibration of the BPM offset reading. This is simpler than for instance a quadrupole scan, which needs accurate measurements of quadrupole locations and field strengths, beam energy, as well as screen location and resolution.

IV. RESOLUTION LIMITS
The main limitation of this technique stems from the finite resolution of BPMs. In measuring the jitter-phasespace ellipse, the width of each angle-slice (i.e., the position jitter at the waist) must be well resolved, which limits how small a waist beta function can be measured.

A. Analytic model
To calculate this resolution limit, we consider the apparent covariance matrix of the jitter at the upstream BPM location where σ is the BPM resolution. The true covariances of the jitter can be expressed in terms of its waist parameters as x 2 = (β w + s 2 w /β w ), xx = − s w /β w , and x 2 = /β w , where is the geometric jitter emittance and s w is the distance from the upstream BPM to the jitter waist.
The measured jitter emittance for this finite BPM resolution is given by the determinant of Eq. 9 Employing the same logic as in Sec. II (for Eqs. 6 and 7), we can find the measured waist beta function from the inverse of the measured gamma function whereˆ can be substituted from Eq. 10, as well as the measured waist location from the ratio of the measured alpha and gamma functionŝ Equations 10-12 establish three resolution regimes: (1) well-resolved, (2) distorted, and (3) fully saturated. To avoid any distortion whatsoever, the BPM resolution must be better than found by requiring the quadratic σ 2 -term in Eq. 10 to be smaller than the constant 2 -term. To avoid saturation (i.e., noise dominating the signal), the resolution should be better than found by demanding the σ 2 -term in the denominator of Eq. 11 be smaller than the -term. Encouragingly, the measurement of the waist location is not affected by the distortion limit, and instead only by the significantly larger saturation limit. This is because the waist location is only related to the phase-space correlation and not its area. These regimes are demonstrated by the example in Fig. 3, which also shows exact agreement with Monte Carlo simulations of two finite-resolution BPMs. In a typical case where the waist beta function is small compared to the BPM separation (β w ∆s) and the waist is approximately half way between the BPMs (s w ≈ ∆s/2), Eq. 13 simplifies to σ √ 2 β w -therefore the BPM resolution should be smaller than the position jitter at the waist. This limit informs the choice of BPM technology required for the application in question.

B. Overcoming the resolution limit
Using state-of-the-art BPMs such as cavity BPMs, the resolution is around 1 µm. This is just on the edge of the distortion limit for plasma-accelerator parameters, where mm-scale beta functions and sub-µm jitter emittances are required. In such cases, measuring the waist beta function requires going beyond the resolution limit. This is in principle possible to do, if the emittance of the jitter is already known.
Just like the emittance of the beam, the jitter emittance is preserved in a linear-optics lattice, therefore one of two alternative measurement techniques can be utilized. First, we could relax the strength of the focusing until the jitter waist is well resolved-giving different Twiss parameters, but the same emittance. Second, we could simultaneously perform a similar measurement with two other BPMs just upstream or just downstream, where the focusing is relaxed compared to the focus region. The former method requires the jitter emittance to persist in time, whereas the latter requires it to persist in space.
When the jitter emittance is known, the analysis simplifies greatly. The waist beta function can be calculated using based on the variance of the angle jitter (Eq. 8), and the waist location is simply where ∂x2 ∂x1 is the slope of the correlation between the two BPM readings.

V. MEASUREMENTS AT FLASHFORWARD
Experimental demonstration of the two-BPM method was performed at the FLASHForward facility at DESY, which uses a 1 GeV electron bunch from the FLASH free-electron-laser facility [24]. FLASH provides high-charge (up to 1 nC), low-emittance (1 mm mrad) beams with relatively small centroid jitter. After an approximately 150 m long linac, the bunches are diverted into the FLASHForward beamline. Here, a dispersive section allows for advanced energetic collimation [25], then a final-focusing section [depicted in Fig. 1(b)] tightly focuses the beam into a plasma accelerator. Downstream  Fig. 4. (a) The beam was imaged onto a spectrometer screen from a range of different object planes within the focus region. (b) The variation of horizontally projected beam size for each object plane indicates that the beam was focused to a small waist beta function (34 mm) close to the center of the focus region-in good agreement with the two-BPM measurement. Error bars represent the standard error.
of the plasma is a suite of beam diagnostics, in particular a dipole spectrometer with quadrupoles for point-topoint beam imaging.

A. Comparison to quadrupole scans
To test the assumptions in Sec. II and the applicability of the method, a detailed comparison of the measured jitter and beam phase spaces was performed. A strongfocusing optic was set up to focus bunches with an energy of 678 MeV down to a cm-scale beta function at the location of the plasma accelerator module (which had been removed from the beam path). Surrounding this focus region were two cavity BPMs with a resolution of 0.5-1 µm [26,27], separated by 1.06 m, and approximately equidistant from the nominal focus point.
Two datasets were collected, using slightly different final-focusing optics with the beam focused at two locations 60 mm apart. Figure 4 shows the measured jitter phase space for each of these two settings using the two-BPM method. The presence of outliers (as seen in Fig. 4) can significantly skew the calculation of phase-space parameters, and thus an outlier-cleaning method was applied: (i) translate the jitter to the waist location from the BPM correlation (Eq. 16), (ii) perform Gaussian fits of both the x and x distributions, (iii) remove all shots beyond ±4σ, and then (iv) undo the translation from (i).
At the same time, an object-plane scan was performed with the downstream quadrupoles (after the second BPM), imaging the beam onto a LANEX screen with a sufficiently high resolution (50 µm). Figure 5 shows the corresponding measurement of the beam waist. Note that the spectrometer limits the measurement to the horizontal plane, as the dipole disperses vertically. No chromaticity was observed on the screen.
The two-BPM measurement agrees with the quadrupole scan measurement to an acceptable level. The waist beta function of the jitter (26-27 mm) differs from that of the beam (34 mm) by about 25%, and the jitter waist location is offset from the beam waist location by 14-15 mm-on the same scale as the waist beta function, as expected. Based on these numbers, the beam-jitter mismatch parameter was calculated to be M = 1.7-implying that the phase space of the jitter was indeed closely matched to that of the beam.
As an additional cross-check of the measurements, the centroid jitter was also measured directly on the spec- trometer (see Fig. 6) and shows that the two methods agree on the waist location of the jitter to within 3 mm, verifying the accuracy of the calibrations used.
FIG. 7. Chromaticity measurement using an energy-slice scan. (a) Jitter phase space for a range of energy slices. Significant dispersion (energy-dependent offsets) can also be observed. Each step consists of 500 shots. (b) The corresponding evolution of the beta function for each slice in the focus region, indicating a highly chromatic focus.

B. Slice-by-slice measurements
Chromaticity, where Twiss parameters change with energy [28,29], can be a concern when tightly focusing beams of finite energy spread [30]. This is especially important in energy-chirp-based two-bunch experiments where a trailing bunch needs to be exactly matched into the plasma wake behind a different-energy driver bunch.
Measuring chromaticity with the two-BPM technique requires it to be combined with an energy filter-each energy slice sufficiently narrow to have an achromatic focus. At FLASHForward this is accomplished using an energetic collimator [25]. Moving both the high-and low-energy collimators together, thin slices with 0.1% root-mean-square (rms) energy spread could be made. Figure 7 shows the result of such an energy-slice scan around a mean energy of 1120 MeV, indicating a highly chromatic focus in the vertical plane. The waist beta functions are relatively consistent (10-20 mm) across all slices, whereas the waist location shifted significantly between the highest and lowest energy slice (by 80 mm). In the horizontal plane (not shown in Fig. 7), the waist location spanned only 10 mm-considerably less chromatic. This asymmetric chromaticity is expected in a quadrupole-based final-focus system, where the beam is more strongly defocused in one plane (typically the vertical plane) before being focused to a waist.
Taking into account all the information gathered in an energy-slice scan, we can extract a partial 5D beam tomography. As seen in Fig. 7(a), the average position and angle of each individual energy slice is also measured, and therefore both the beam centroid and the (emittance-normalized) beam size of each slice is known in both planes. This in-situ tomography allows not only slice-specific matching, but also measurement and removal of any bunch dispersion. For a linearized longitudinal phase space, dispersion corresponds to a bunch tilt or curvature, which in a plasma wake leads to emittance growth [31,32] and potentially a hosing instability [33][34][35]. Finally, combining such a two-BPM tomography with longitudinal-phase-space data from a transverse-deflecting cavity allows a 6D phase space to be reconstructed-important for realistic simulations and detailed optimization of the external injection process.

VI. CONCLUSIONS
We have shown that centroid jitter measured by two BPMs can be used to quickly estimate Twiss parameters in a region of strong focusing. While being an approximate measurement, this facilitates a significant speedup of complex and delicate beam setup such as matching into a plasma accelerator, and allows noninvasive online monitoring of the beam focus. Experiments were successfully performed at FLASHForward to verify this technique, by comparing the two-BPM measurement to a conventional quadrupole scan. Already in routine use for plasma-wakefield experiments at FLASHForward, it is clear that the power of this method lies in its simplicity.