2.1 Adjust RF Timing

When an RF station is under conditioning, it is operated on a different time slot from the arrival time of electron bunches so that the RF conditioning will not disturb the beam operation. When applying the RF station on beam operation, the timing of the RF station should be adjusted to have a good overlap between the RF pulse and the beam arrival time.

SwissFEL employs an event based timing system [14]. The event number of the RF station should be firstly assigned to be the one used for beam operation, and then the delay setting in the Event Receiver (EVR) should be adjusted to roughly align the RF pulse with the beam. Due to the uncertain time of the RF pulse, it usually takes quite long time to overlap the RF and beam for the first time by manually adjusting the EVR delay. To facilitate the RF timing setting, an algorithm was developed to determine the EVR delay automatically as in Fig. 2.

Fig. 2.Full-screen View

Algorithm to determine EVR delay to align RF and beam.

In Fig. 2, it is assumed that a reference RF station has been properly applied on beam operation, and the total delay of the reference RF station counting from the fiducial time of the Event Generator (EVG) can be calculated as:

where ${\widehat{T}}_{\text{p}}$, ${\widehat{T}}_{\text{EVR}}$ and ${\widehat{T}}_{\text{fine}}$ are the delays of the reference RF station caused by the timing signal path, the EVR delay (with a resolution of about 0.5 µs) and the LLRF firmware fine delay (with a resolution of about 4 ns). In order to apply the new RF station on beam acceleration, the required total delay can be estimated as

where L is the distance between the RF station and the reference RF station and v is the speed of the electron bunch which equals approximately to the speed of light in vacuum. The term L/v corresponds to the beam flight time between the two RF stations. In equation (2), the timing signal path delay T_p is known from the timing system design and then the delay settings for the EVR and LLRF firmware can be determined.

The measurements show that the method above works well to guarantee the rough overlap between the RF and beam. In order to fine optimize the RF delay, the LLRF firmware fine delay can be scanned and correlated with the beam energy. Normally the delay is optimized by maximizing the beam energy. Fig. 3 shows the delay-energy correlation of a C-band RF station at Linac1.

Fig. 3.Full-screen View

Optimization of RF delay for maximum beam energy. The curve was fitted as a second-order polynomial to find the delay for maximum beam energy.

2.2 Calibrate Beam Phase Offset

The phase measurements of the RF signals picked up from the accelerating structures have arbitrary values due to the unknown signal paths. The phase offset needs to be calibrated to get the beam phase from the view point of beam dynamics (e.g. beam phase is defined to be zero degree with on-crest acceleration).

The calibration is done by scanning the RF phase for maximum beam energy [15],[16]. When the RF station is applied on beam operation for the first time, the beam phase is unknown and the downstream beam optics might not be able to accept large energy variations. To avoid beam losses, the amplitude of the RF station can be set to a smaller value (e.g. 10 % of the desired accelerating voltage) within the dynamic range of the downstream beam optics, and the RF phase can be scanned for a full cycle. Fig. 4 shows the full-cycle phase scanning of the second RF station in Linac1 correlated with the beam energy.

Fig. 4.Full-screen View

Beam phase offset calibration. The "Vsum Phase" is the vector sum phase measured from the RF signals picked up from the travelling wave structures. The cosine function fitting is also shown in the figure.

The relation between the beam energy variation and the phase measurement can be described as

where V_acc is the accelerating voltage, φ_mea is the phase measurement and φ_off is the phase offset calibration. The beam phase is the sum of φ_mea and φ_off. For the nth measurement point in the phase scanning, equation (3) can be rewritten as

where

The parameters a and b can be calculated by multi-variable linear fitting according to Eq. (4), and then we get

Equation (6) shows that the accelerating voltage of the RF station can also be calibrated if the beam energy measurement is accurate. The phase offset calibration is always valid regardless the accuracy of beam energy measurement.

The beam phase describes the relative timing between the RF and beam arrival time. If the beam arrival time is changed, the phase offset calibration should be updated correspondingly. For a deterministic change of beam flight time from the RF gun to the specific RF station, like due to the adjustment of a chicane, the phase offset calibration can be updated by adding a correction to compensate the change of beam flight time. This situation is depicted in Fig. 5.

Fig. 5.Full-screen View

(Color online) Change of beam flight time when passing through a chicane. A, B, C and D are dipole magnets.

The phase offset calibration is performed when the electron bunch passes through the chicane in a straight line (A to D), for which the phase offset is labeled as φ_off0. After the chicane is switched on, the beam passes along the path A, B, C to D, resulting in an increase of the beam flight time (Δt_f). To keep the beam phase reading still valid, the phase offset calibration should be updated taking into account the delay of beam arrival time:

where f₀ is the working frequency of the RF station. As an example, the beam flight time through the laser heater of SwissFEL is 3.77 ps, which requires a correction of 4.07 degree for the phase offset calibrations of the S-band RF stations in Booster2.

2.3 Identify RF-Beam Interaction Time

The measurements of RF amplitude and phase should match the RF fields that the beam feels when passing through the accelerating structures. With such accurate RF measurements, the beam can be stabilized by stabilizing the RF and the beam jitters can be estimated by measuring the RF jitters. For a travelling wave structure, the RF pulse width is usually larger than the filling time of the structure. The most important step to achieve the accurate RF measurements is to determine which part of the RF pulse is used for beam acceleration.

Fig. 6 shows the filling procedure of a travelling wave structure. The RF power at the early part of the pulse (T₁) enters the structure and travels to the loaded end with a decaying group velocity along the constant gradient travelling wave structure [3]. When the RF power injected at T₁ arrives at the load and the power at T₂ just enters the structure, the beam is injected. And because the beam travels at a speed (close to the speed of light in vacuum) much higher than the RF group velocity, the beam flight time through the structure can be neglected. Then the RF pulse region from T₁ to T₂ is the RF-beam interaction region used for beam acceleration and T₂–T₁ equals to the structure filling time. The RF field felt by the beam can be calculated as

Fig. 6.Full-screen View

(Color online) Filling procedure of a travelling wave structure.

where V_RF(t) is the measurement of the RF pulse as a time varying vector which can be represented as in-phase (I) and quadrature (Q) components: $V_{\text{RF}}\left(t\right)=I\left(t\right)+jQ\left(t\right)$. Equation (8) has assumed that different parts of the RF pulse have equal interaction with the beam, which is not true due to the variations of the group velocity (the RF power at time T₂ interacts stronger with the beam than the RF power at T₁), but it is still a good approximation of the RF field felt by the beam when we care more the fluctuations of V_acc.

In order to identify the RF-beam interaction region in the RF pulse (it is usually wider than the filling time of the structure), a single point in the LLRF DAC output table (DAC works at 250 MSPS) can be reduced to generate a negative bump in the RF power fed into the accelerating structures, see Fig. 7. The time of the RF power bump can be scanned by changing the time of the DAC bump. When the RF power bump appears in the time region between T₁ and T₂, the beam energy will be reduced. By comparing the beam energy profile when scanning the time of the RF power bump, the RF-beam interaction region in the measurements of the RF pickup signals can be identified. The pulse width of the RF power bump will be larger than the DAC clock cycle due to the band-pass filtering by the RF components in the RF system [11]. When the RF power bump is much earlier than T₁, the beam energy will not be affected; when it approaches to T₁, the tail of the bump will start to affect the beam energy and after it fully enters the RF-beam interaction region, the beam energy will be affected by the entire bump (see the point A in Fig. 7(c)). Within the region between T₁ and T₂, the power bump will reduce more beam energy when it approaches to T₂ because the bump will appear in the location of the accelerating structure with higher group velocity. After the power bump fully exits the RF-beam interaction region (see the point B in Fig. 7(c)), the beam energy will return to the original value. From the analysis above, the RF-beam interaction region can be identified by finding the point A and B in the beam energy profile.

Fig. 7.Full-screen View

Method to identify RF-beam interaction region in the RF pulse. (a). Time scan of a single-point variation (bump) on DAC output signal; (b). RF power bump generated by the DAC bump appeared in the pickup signals of accelerating structures (both the drive RF and the pickup from RF load); (c). Beam energy variations when scanning the time of DAC bump.

The algorithm was applied to the last S-band RF station in Booster2. The beam energy was measured by the spectrometer after the first bunch compressor. Fig. 8 shows the results of beam energy for different time of the RF power bump. Compared to Fig. 7(c), the RF-beam interaction region can be estimated as T₁ = 1.30 µs and T₂ = 2.05 µs.

Fig. 8.Full-screen View

Identification of RF-beam interaction region.

With Eq. (8) the amplitude and phase of the RF signal can be calculated by integrating the RF pulse within the RF-beam interaction region. The vector sum of the V_acc measured from the input (picked from the waveguide before the input coupler) and the output (picked from the RF load) of the accelerating structure were used to represent the RF field felt by the beam. To verify the RF-beam interaction region, the beam energy stabilities were measured with the amplitude and phase feedbacks of the last S-band RF station in Booster2 switched on. The effects of feedbacks were compared with different integration windows in the RF pulse, see Fig. 9. In the figure above, the beam energy is represented by the x-position of the electron bunch measured with the BPM after the injector spectrometer. The variation of the beam energy is calculated with the equation

Fig. 9.Full-screen View

Beam energy stabilities with different integration windows for the RF measurement. The amplitude and phase feedbacks were switched on. When the integration window equaled to the RF-beam integration region as in plot (a), the beam energy stability was ΔE/E₀ = 5.95e-5 RMS; while when shifting the integration window by 150 ns as in plot (b), the achieved energy stability was ΔE/E₀ = 6.57e-5 RMS.

where E₀ is the beam energy when the bunch passes through the BPM center (x = 0), ΔE is the energy variation, x is the beam position as shown in Fig. 9 and η is the dispersion length of the spectrometer at the BPM location, which has a value of η = -0.85 m for the injector spectrometer.

The results in Fig. 9 show that the RF based feedbacks can help more if the measurement of RF fields matches what the beam feels. As a follow up plan, the algorithm using Eq.(8) will be improved to take into account the variations of group velocity along the structure.

2.4 Optimize RF Feedbacks

The SwissFEL LLRF system employs pulse-to-pulse integral feedbacks to stabilize the RF amplitude and phase [10]. The RF station can be viewed as a static system for pulse-to-pulse control because the adjustments of amplitude and phase after one pulse will be fully applied to the next pulse. The block diagram of the feedback loop is shown in Fig. 10 with the system gain normalized to G = 1.

Fig. 10.Full-screen View

Block diagram of the pulse-to-pulse feedback loop. r is the amplitude or phase reference value, d is the disturbance to the RF station, y is the amplitude or phase of the RF output and n is the measurement noise. The amplitude and phase are measured for each RF pulse with Eq. (8). The sampling time of the discrete transfer functions is the period of RF repetition.

The pulse-to-pulse amplitude and phase feedbacks have been successfully adopted in SwissFEL to achieve accurate RF settings and maintain long-term RF stabilities. Fig. 11 shows the suppression of the amplitude and phase fluctuations in the RF Gun field with feedbacks. The Gun phase is more sensitive to the cooling water temperature changes than the amplitude.

Fig. 11.Full-screen View

Suppression of RF Gun amplitude and phase fluctuations with pulse-to-pulse feedbacks. The Gun works at a repetition rate of 10 Hz.

The parameters of the feedback loops (e.g. the feedback gains) should be optimized aiming to provide maximum RF stabilities. In order to do the optimization, a brief analysis of the noise transfer model of the feedback loop in Fig. 10 will be firstly performed.

The closed-loop bandwidth of the integral feedback loop will be determined by the gain (k) of the discrete integrator C. The selection of the feedback gain will be a trade-off between the characteristics of the disturbance and measurement noise, the required response speed to reference changes and the feedback stability criteria [17]. The transfers from the disturbance and measurement noise to the system output in frequency domain is shown in Fig. 12 (a) with the gain assigned to k = 0.5. With higher gains, the closed-loop bandwidth, which is the frequency at which the magnitude of the sensitivity function (transfer from d to y) is -3 dB, will be higher. The higher gain will result in more suppression of low-frequency disturbances, faster response to reference changes, but more amplification of high-frequency disturbances and more transfer of measurement noises. For a special case, when the disturbance and measurement noise are both white noises, which have constant power intensities within the Nyquist bandwidth of the RF repetition rate, the RMS jitters in the system output y generated by the disturbance d and measurement noise n for different feedback gains are plotted in Fig. 12 (b). Here the RMS values of d and n are normalized to 1. It can be seen that if there are no special low-frequency disturbances, the feedbacks will always increase the pulse-to-pulse jitters at the RF output. For a specific low-frequency disturbance (e.g. temperature fluctuation of the cooling water), the closed-loop bandwidth should be larger than the frequency of the disturbance. In practice, the feedback gain can be scanned and the optimal value can be selected for minimum RF or beam jitters as a trade-off between the disturbance rejection and jitter amplification as in Fig. 12 (b).

Fig. 12.Full-screen View

(Color online) Contributions of disturbance and measurement noise to output jitters. (a). Frequency responses of closed-loop transfer functions from n to y and from d to y with a feedback gain k = 0.5; (b). Contributions of disturbance and measurement noise (both are white noises with RMS values normalized to 1) to system output RMS jitters with different feedback gains. The feedback gain should be smaller than 2 to keep the loop stable.

The feedback gain optimization algorithm was tested at the X-band RF station of SwissFEL. The RF feedback gains were correlated with both the RF jitters and the bunch length jitters measured after BC1 which have strong correlations with the X-band RF [5]. The test results are shown in Fig. 13. The RMS jitters of the amplitude, phase and bunch length were calculated from 100 pulses with an RF repetition rate of 10 Hz, which covers the noise power from 0.05 Hz to 5 Hz (Nyquist bandwidth of the RF repetition rate).

Fig. 13.Full-screen View

Feedback gain optimization of the X-band RF station referring to both the RF jitters and bunch length jitters. The bunch length jitter is more sensitive to the X-band phase jitter than to the amplitude jitter. This is why the bunch length jitter shows less correlation with the amplitude feedback gains (right-top plot). The larger X-band phase jitter with higher phase feedback gains results in larger bunch length jitter (right-bottom plot) but the correlation is not so strong. This is because the bunch length jitter also depends on the phase jitters of the S-band RF stations upstream to the X-band RF station.

Fig. 13 shows that the feedbacks will amplify the RF jitters when the gains are larger than about 0.4. With the spectrums of the X-band amplitude and phase jitters in open-loop (see Fig. 14), it can be seen that the noises higher than 0.05 Hz are close to white noise, resulting in amplification of the RF jitters with a large feedback gain. Based on the tests and analysis above, the pulse-to-pulse feedback gains of the X-band RF station should be set with the following guidelines:

Fig. 14.Full-screen View

Spectrums of the open-loop amplitude and phase jitters of the X-band RF station.

1) Both the amplitude and phase feedback gains should be small (e.g. 0.1) to only suppress the low-frequency drifts (slower than 0.01 Hz in Fig. 14).

2) When scanning the RF amplitude and phase set points, the feedback gains can be temporarily increased to have faster responses. Feedforward is an alternative to accelerate the response to set point changes.

3) When the RF station is used as an actuator for beam based feedback, the LLRF system can work in open-loop if the beam based feedback can cover the same bandwidth as the RF based feedbacks. This helps to avoid the instabilities caused by the cascaded feedback loops if they have similar closed-loop bandwidths.