Record high-gradient SRF beam acceleration at Fermilab

The 1.3 GHz SRF accelerating cavities were originally developed in the context of the TESLA linear-collider project [20] and were included in the baseline design of the ILC [21]. Such a cavity, as depicted in figure 3, is a 9-cell standing-wave accelerating structure operating in the TM_010;π mode. Total cavity flange-to-flange length is 1.247 m, while the active length (weighted with electric field) is 1.038 m. The cavity operates at 2 K with a design average acceleration gradient E_acc = 31.5 MV m⁻¹ and quality factor Q₀ ≥ 0.8×10¹⁰. Each cavity has two end group sections—one with a port for coupling RF power from the power source into the structure, and the other with a port for a field sampling probe used to determine and control the accelerating gradient. Each of these ports accepts an electric field antenna required for qualification and operation. In the process of building a cryomodule, these cavity structures are cleaned/processed, tested and dressed. Dressing involves welding a helium jacket around the cavity for cooling, attaching the tuner assembly including both coarse and fine tuners for adjusting the frequency of the structure, magnetic shielding to minimize the cavity losses, a variable fundamental power coupler for powering the cavity, an electric field sampling antenna and two HOM electric field antennas. Eight of these dressed cavities are connected into a string and are a subcomponent of a 12.6 m long superconducting cryomodule.

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Each cavity is equipped with a so called 'blade tuner' system that transforms azimuthal rotation into longitudinal motion and changes the length of a cavity with it. The tuner allows both slow mechanical adjustment of the frequency of the cavity to bring it on resonance (static tuning) and a fast 'pulsed' adjustment using a piezo system to dynamically compensate Lorentz force detuning during the RF pulse. The tuning range of the cavity resonant frequency is of order 1 kHz.

The SRF cryomodule, CM2, installed in the FAST facility is the second of two ILC-style 8-cavity cryomodules built at Fermilab. The cryogenic infrastructure of the facility allows 2 K superfluid helium cooling of the cryomodule [22]. All cavities were fabricated in industry and extensively tested cold prior to assembly into the cryomodule. While the first cryomodule was assembled by Fermilab staff with oversight by collaborators, parts were provided as a 'kit' by DESY, CM2 was largely assembled by Fermilab with parts also obtained by Fermilab. A single 5 MW peak 1.3 GHz klystron provides the necessary power to drive the cryomodule. A waveguide distribution system provided by SLAC National Accelerator Laboratory [23] is attached to the cryomodule fundamental power couplers and splits the incoming power amongst the eight cavities. The low-level RF drive system was developed based on previous systems for SRF test beds and includes provision for both amplitude and phase feedback control within the ILC specifications. Figure 4 compares the gradient performance of all cavities. Vertical and horizontal tests are conducted on bare single cavities first and then after dressing, respectively. At FAST single cavity tests were first carried out on each cavity individually, being driven by the RF source, and finally as a unit. Single cavity testing allowed the entire available RF power to be delivered to a single cavity while 'unit' tests necessarily limit the peak RF power available to a single cavity to a maximum of 1/8 of the total output of the klystron.

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CM2 has now been operated for two extended operations periods—initial commissioning (without beam) from June 2013 through August 2015 and most recently for re-commissioning followed by beam operation beginning in July 2017 through November 2017. The main activities and accomplishments of the first run are previously described [15, 24].

Table 2 summarizes the peak gradients achieved and measured cavities' quality factors Q₀ at the design gradient, 5 Hz repetition rate, temperature of 2 K and 1.57 ms long RF pulse length with 0.97 ms flat-top. Most of the resonators are demonstrated to operate above 32 MV m⁻¹ and are limited at that point either by quench or available RF power. Cavity number 6 is the limiter and can operate only below 30 MV m⁻¹ before quenching. This limit was demonstrated during both the first commissioning run—see purple line in figure 4 taken during no-beam test with the cavity off the resonance and without the regulation of the pulse flat-top-and verified during the most recent beam run. Since all eight cavities are now driven by a single RF source, individual cavity performance is limited by the available power and the amount going to the 'weakest' cavity. As conditions permit, re-orienting the RF distribution system would conceivably allow for higher performance of cavities based on their most recent demonstrated performance. At FAST the peak gradient was administratively limited to 31.5 MV m⁻¹ during initial single cavity assessment in order to avoid any possible damage prior to all cavities reaching this level. This limit has since been raised to 35 MV m⁻¹ and the gradients in all the cavities but one, #6, were limited by the available RF power. Though it was not critical for the CM2 operation at FAST, it is noteworthy that the majority of the cavities had measured quality factors Q₀ above the ILC Technical Design Report [12] specification of 0.8×10¹⁰.

During the most recent test run in 2017 the commissioning protocol established previously was used albeit abbreviated: the klystron output remained oriented to drive all cavities simultaneously. This included a period of warm coupler conditioning followed by cool down, moving the cavities on resonance, on resonance conditioning, power calibrations, and finally peak gradient determination. The LLRF system was re-commissioned once cool down was completed and employed to move cavities on resonance. Feedback operation was re-established and gains were found to be virtually unchanged from previous operation. Similarly, adaptive feedforward compensation was rapidly re-deployed with minimal calibration time.

Perhaps not surprisingly, CM2 performance was found to be little changed even after a nearly 2 year hiatus at room temperature. Peak gradients, particularly that of cavity 6, were found to be essentially identical within normal error to that of before. The suite of radiation field emission detectors installed for initial commissioning was no longer available, but the remaining subset show indications of the same behavior as before. Q₀ measurements have not been repeated.

In order to achieve the desired accelerating voltage despite cavity number 6's limitation, a few techniques were undertaken to maximize the sum voltage. These steps were done to cavity 6 in order to maintain its operating gradient below its quench limit while keeping the total RF voltage constant and included: slightly lowering its loaded quality factor Q_L, from 3.5×10⁶ to 3.1×10⁶ and de-tuning its resonant frequency by about 50 Hz off the drive frequency f = 1.3 GHz. Collectively the duty factor was lowered from 5 to 1 Hz and the flattop length reduced from the nominal 590 to 100 μs in order to reduce the average power/heating. Combined these steps allowed sufficient additional voltage to achieve the total CM2 electron energy gain above 250 MeV.

All in all, performance of CM2 during FAST beam operation has been quite reliable. Some activity in the RF power couplers, field emission and attendant vacuum, has been observed, but has not greatly affected beam stability.

The ILC SRF cavities are susceptible to micro-scale mechanical deformations due to the relative thinness of their walls, 2.8 mm. Such deformations are induced by the force of the accelerating electromagnetic field on the walls as well as fluctuations in the pressure of the surrounding helium bath and microphonics induced by external noise sources. These mechanical deformations can change the resonant frequency of the cavity by Δf and lead to fluctuations of the phase and amplitude of the accelerating field beyond the ILC specifications of 0.35° and 0.07%, correspondingly [12]. For high-gradient, pulsed cavities operating in super-fluid helium, the dominant effect is the LFD due to the radiation pressure of the electromagnetic field during the RF pulse which deforms the cavity and pulls it off resonance proportionally to the square of accelerating field E_acc. For the ILC cavities the static detuning is approximately Δf ∼ −E²_acc × 1.0 Hz/(MV/m)² [25] but response to the pulsed force is complicated by excitation of mechanical vibrations, e.g. of the main resonant mode with about 200 Hz frequency. A reliable automated system to compensate for the LFD is needed, otherwise the ILC cavities would dynamically detune by several times the cavity's bandwidth of f/(2Q_L) ∼ 200 Hz and significant excess RF power would be required to maintain the accelerating gradient at the design level of 31.5 MV m⁻¹.

The use of piezo actuators to compensate for LFD was pioneered at DESY but has since been adopted widely [26]. The DESY team employed the technique of exciting the piezo actuator with a half cycle of sine wave prior to the arrival of the RF pulse. The duration, delay and amplitude of the half sine wave are optimized manually by trial and error. This method became standard and is used by many groups [27]. While this standard approach can provide acceptable compensation for LFD, the mechanical response of individual cavities to the Lorentz force and to the piezo actuator can differ. Changes in cavity operating conditions, e.g., the changes in the gradient or bath pressure, can require corresponding changes in the compensating waveform. Operating multiple cavities for extended periods will require control systems that can automatically determine the best parameters for each cavity and adapt to changing operating conditions. Because the cavity detuning does not respond linearly to the changes in some parameters of the standard unipolar pulse, the adaptive capability that can be incorporated in LFD systems based on this standard approach may be limited.

The Fermilab team developed an innovative method to automatically determine an optimal piezo actuator waveform for each individual cavity based on an adaptive feed-forward algorithm [28]. As the first step, the cavity reaction to the piezo impulse is measured at a gradient of several MV m⁻¹ below the maximum [29]. A series of low-amplitude unipolar 0.5 ms long drive pulses are sent to the actuator. The first pulse is timed to arrive some 10 ms in advance of the RF pulse. For each subsequent RF pulse, the delay between the piezo drive and the RF pulse is reduced by 0.5 ms until the piezo pulse follows the arrival of the RF pulse. The forward, probe and reflected RF waveforms are recorded at each delay and used to determine the detuning. The least squares algorithm is then used to find the combination of the delayed pulses that minimizes the frequency detuning from this data. Due to the nonlinear response of the system, the response needs to be measured over the range of the bias voltages on the piezo actuator. The voltage is thus stepped from pulse to pulse in small increments over the full range of the actuator and again the RF waveforms are recorded for each pulse. To measure the detuning due to the Lorentz force alone, the RF waveforms are recorded while no drive signal is sent to the piezo actuator. As a final step, the response matrix is used to calculate waveform (including bias) that will be applied to each cavity's piezo actuator. As operating conditions vary, the RF waveforms can be used to measure any residual detuning. The same response matrix can then be used to calculate the incremental waveform required to cancel that residual detuning.

Each dressed cavity, equipped with tuner, that was integrated into CM2, was tested first at horizontal test stand at maximum operating gradient in the range of 30–35 MV m⁻¹ [30, 31]. A single-cavity LFD compensation system based on the Fermilab adaptive algorithm was deployed and actively used during the HTS testing [32]. A similar system extended to operate with eight cavities was successfully deployed to compensate the LFD in the CM2 cryomodule during its commissioning and, later, beam operation [33]. Figure 5(a) shows the detuning with and without compensation of a 9-cell elliptical cavity (TB9AES008) at the HTS during qualification tests. Because of limitation in the power of the HTS klystron without LFD compensation, cavities cannot be operated above E_acc = 27 MV m⁻¹. The amplitude of the cavity's LFD during 0.5 ms RF pulse 'flat top' was almost 600 Hz. This implies that at 35 MV m⁻¹ LFD without compensation could reach ∼1000 Hz. After turning on the adaptive LFD compensation system it was possible to raise the gradient on the cavity up to 35 MV m⁻¹. The LFD compensation system suppressed the detuning to less than ±20 Hz during whole length of RF pulse. Figure 5(b) shows the piezo drive waveform used to compensate the LFD on this cavity.

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Cavities can also be detuned by variations in the pressure of the helium bath or due to the microphonics. In addition to compensating for the rapid detuning induced by the Lorentz force, Fermilab's adaptive LFD algorithm monitors the cavity resonance frequency and adaptively adjusts the DC bias on the piezo actuator to stabilize the resonance frequency. The resonant frequency of the CM2 cavities over a period of several hours drift by up to 20 Hz when adaptive compensation is not active. When adaptive compensation is on the drift was suppressed to less than one Hz.

In summary, an adaptive LFD compensation system was successfully used to compensate for large (up to 1000 Hz) LFD in all 8 cavities in CM2 during commissioning and operation.

The optimization of RF phases between the gun, CC1 and CC2 is fairly simple, given that each has its own LLRF and high level RF systems, and maximizing the total energy gain is relatively easy with only three cavities. This is not the case for the CM2, in which 8 cavities are powered by one RF source and controlled by a common LLRF system, resulting in a vector sum of individual cavity energy gains U_i

$$\begin{eqnarray}&&{\rm{\Delta }}W=\displaystyle \sum _{{i}=1}^{8}{U}_{{i}}\,\cos ({\varphi }_{{i}})\end{eqnarray} \tag{ 1 }$$

that can rapidly become difficult to deconvolve as the LLRF control loop is forcing the vector sum to be equal to the set point and each relative phase and amplitude change could impact the remaining cavities. To find the optimal position for the individual cavity phase φ_i tuners, we employed a beam-loading-based tuning method. Beam loading is the effect of interaction of the beam current with the cavity. If the beam is accelerated by the cavity, it results in the increase of the beam energy and corresponding reduction of the energy of electro-magnetic fields stored in the cavity. Should the beam be out of phase with cavity's RF field it gets decelerated and therefore its energy is transferred to the cavity. For a short bunch with charge q, and the amplitude of the beam-induced wakefield in the cavity is given by [34]:

$$\begin{eqnarray}&&{\rm{\Delta }}U=q\left(\displaystyle \frac{R}{Q}\right)\displaystyle \frac{\omega }{2},\end{eqnarray} \tag{ 2 }$$

where (R/Q) is the cavity geometry dependent parameter of about 1000 Ohms for the ILC-type cavities, and ω is the frequency of the exited mode, 2π × 1.3 GHz in our case. Cavity monitors of the RF field thus would report the variation of the total RF voltage amplitude depending on relative phase of the RF and the beam Δφ = φ_i − φ_b. Adjustment of either the LLRF phase or the phase delay tuning motor could be used to shift the RF phase change corresponding to the beam loading. A phase shift over the complete RF cycle will include both maximum deceleration and maximum acceleration through the cavity, the latter corresponding to the maximum beam loading, as shown in figure 7 for cavity 1. Choosing to maximize the acceleration for all cavities to a LLRF phase set-point of 0, the phase delay tuning motor was adjusted to maximize the peak beam loading in the cavity.

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This process was repeated for the remaining seven cavities and optimal phases were determined and set for all of them. Once all eight CM2 phase delay tuners were optimized, the beam was accelerated down the high energy beamline to the energy of 298.0 ± 3.4 MeV measured after the 15° dipole D600, which was used as the high energy spectrometer. There are two sources of errors in the energy measurement. First, it is the uncertainty in the beam angles before and after the dipole, measured by corresponding BPMs, which is estimated to be about 0.9% (2 mrad for the total bend angle of 15°). The second is the uncertainty of magnetic field calibration of the D600 magnet at the level of 0.7% rms. Stability of the energy past CM2 even with open-loop LLRF stabilization system was better than 0.1% rms, as derived from the beam position jitter in the BPM past D600.

The final beam energy is composed of the two parts gained in the low-energy section before CM2 and in CM2 itself. The first part was precisely measured with spectrometer magnet D122. That magnet was calibrated and had the NMR probe in it to measure actual magnetic field. Two pairs of BPMs are located immediately upstream and downstream of the D122 to measure the change of the beam trajectory angle. The resulting injector beam energy was found to be 42.9 ± 0.2 MeV for that particular run, therefore, the CM2 total acceleration was 255.1 ± 3.4 MeV.

Careful control of the beam size and trajectory is needed to assure lossless acceleration and transport of electron bunches from the gun to the high energy absorber, and, in the future, deliver suitable beam for the injection into the IOTA ring. Beam dynamics in a linear accelerator is defined by the initial conditions at a starting point and by the external perturbations in the downstream lattice. In a simplified form, the beam trajectory x(s) deviates from a perfect, e.g. straight, line, due to transverse angular kicks θ_k caused by imperfections, such as misalignment of magnets, RF cavities, etc in the preceding sections of the beamline [35]:

$$\begin{eqnarray}&&x(s)=\displaystyle \sum _{k}{\theta }_{k}\sqrt{\beta (s){\beta }_{k}}\times \,\sin (\varphi (s)-{\varphi }_{k})\end{eqnarray} \tag{ 3 }$$

while the rms beam size varies as:

$$\begin{eqnarray}&&\sigma (s)=\sqrt{\beta (s){\varepsilon }_{n}/\gamma }.\end{eqnarray} \tag{ 4 }$$

Here, β(s) is beta-function of the focusing lattice at the location of observation s—see figure 8—and β_k are its value at the locations of the kicks, ε_n is the normalized beam emittance determined mostly by the electron source, θ_k is the dipole kick due to the kth imperfection, and the betatron oscillation phase φ(s) = ∫ds/β(s). After the initial manual tuning of propagation of the beam to the absorber, alternating steps of beam-based focusing lattice and trajectory corrections were made. The trajectory correction was aimed at minimizing beam offsets from the magnetic axis of quadrupoles. The analysis and correction of the focusing optics was done using the linear optics from closed orbit technique [36, 37]. Operationally important aspects of the focusing lattice tuning were optimization of the beam shape distortions at low energies (4.5–20 MeV) to minimize space-charge induced degradation of emittances at high bunch charge [38], careful control of the orbits through slightly misaligned capture cavities CC1 and CC2 and proper correction of the dispersion achromat in the dogleg formed by the two 15° dipoles (at around 110–120 m in figure 8) [39]. Figure 9 presents the beam profile images on the transverse profile monitor downstream of the D600 dipole before and after the beamline tuning showing a great deal of improvement resulting in loss-free transportation of optimal emittance beam.

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