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Towards Sub-ppm Shot-to-shot Amplitude Stability
of SwissFEL Resonant Kicker
Martin Paraliev, Christopher Gough
Accelerator Concepts and Development
Paul Scherrer Institute
5232 Villigen PSI, Switzerland
[email protected]
Abstract—The development of a fast electron beam switching
system for Swiss Free Electron Laser[1] (SwissFEL) is in its final
phase. Two high stability resonant kicker magnets followed by a
septum should separate two closely spaced electron bunches
(28 ns apart) and send them to two separate undulator lines.
Extremely high shot-to-shot amplitude stability will ensure
minimal shot-to-shot variation of the generated X-ray pulses. As
previously reported, the prototype system met the project
requirements, reaching 3 ppm rms shot-to-shot amplitude
stability[2]. During final system optimization better than 1 ppm
rms shot-to-shot amplitude stability (1e-6) has been achieved.
Keywords— Electron beam switching; XFEL; Kicker magnet;
Low amplitude jitter
I. INTRODUCTION
The SwissFEL is a linac based X-ray Free Electron Laser
[3] user facility under construction at the Paul Scherrer
Institute (Switzerland). It will be capable of producing short
(2 to 20 fs) and high brightness X-ray pulses (up to 6·1035
photons·mm-2·mrad-2·s-1) covering the spectral range from
1 to 70 Å [4]. To increase the facility efficiency the main linear
accelerator will operate in two bunch mode. At beam energy of
3 GeV the two closely spaced electron bunches (28 ns apart)
will be separated by a precision beam switching system [5].
Two high stability resonant deflecting magnets (kickers) will
create the initial vertical beam deflection (~1.8 mrad) and then
a Lambertson septum magnet will give the final horizontal
deflection (2°). The two separated bunches travel to two
different beamlines which will feed simultaneously two
experimental stations. In order to ensure high X-ray amplitude
and pointing stability, the deflection system has to operate with
very low jitter.
During the development, the measurements of the
prototype system showed amplitude shot-to-shot stability of 3
ppm (3·10-6) rms, far below the value needed for the
accelerator project [2]; this opens possibilities to separate
multiple closely spaced electron bunches with low jitter.
II. MOTIVATION
Using multiple bunches in a single RF pulse is very
attractive way to increase efficiency of a linear accelerator but
it always needs certain engineering compromise. From one
hand, the Radio Frequency (RF) accelerating pulse is not with
constant amplitude and the closer the electron bunches are the
more similar accelerating conditions they see. Theoretically the
time separation between bunches could be reduced down to the
RF period of the accelerating structures. In our case, due to the
usage of several different RF acceleration frequencies, the
smallest separation is limited down to the period of the highest
common sub-harmonic frequency (7 ns). From the other hand,
the closer the electron bunches are the more difficult is to
separate them with the necessary pointing stability. In this
respect the kicker magnets are the most demanding element in
such beam switching system. They have to provide reliably fast
changing and highly repetitive magnetic field. The practical
dimensions of lumped parameters kicker magnets put a lower
limit to their inductance. This, combined with the requirement
for fast change of the magnetic field pushes the operating
voltages to tens of kilovolts making standard pulsed kicker
magnets a demanding engineering problem. A novel resonant
approach is taken in order to have more effective (lower
operating voltage) and better controlled (higher stability)
current through the kicker magnets.
III. AMPLITUDE STABILITY
A. Measurement system
A dedicated high resolution balancing measurement system
was developed to measure the pulse-to-pulse amplitude
stability of the resonant kickers. A sensing loop, magnetically
coupled to the resonator, is used to pick up fraction of the
kicker’s magnetic field, creating an electrical signal Um
proportional to the time derivative of the coupled magnetic flux
Φ m.
𝑈𝑚 = −
𝑑Φ𝑚
𝑑𝑡
(1)
In steady state regime the magnetic field in the resonator is
a sinusoidal function with respect to time t as well as the
coupled magnetic flux Φm(t)
Φ𝑚 (𝑡) = Φ0 sin⁡(𝜔0 𝑡 + 𝜑)
(2)
were ω0 is the angular resonance frequency of the resonator
and φ is initial phase angle. Substituting the magnetic flux in
eq. (1), for the voltage of the pick-up signal we have:
𝑈𝑚 = −𝜔0 Φ0 cos(𝜔0 𝑡 + 𝜑)
(3)
Since the resonant frequency ω0 is constant, the pick-up
signal is proportional to the resonator’s magnetic field and it is
used to directly monitor the field in the magnet.
In the measurement system a high stability DC reference is
subtracted from the picked up signal and the difference is
amplified, limited and digitized. The system gives high
resolution amplitude measurement around the peak of the sine
wave but has limited dynamic range (about ±100 ppm).
Programming the DC reference signal makes possible to adapt
the system to the actual working point of the magnet.
In addition, a full range measurement is also implemented
using fast 16 bit ADC (resolution 15 ppm). This measurement
is used for global amplitude monitoring and calibration of the
high resolution measurement system [6].
B. Calibration of high resolution measurement system
The sine wave amplitude of the magnetic field around
magnet’s working point is continuous monotonic function of
the magnets’ driver charging voltage Uch.
Using small changes of Uch around the working point
(±1%) and the full range measurement system, the response of
the kicker system could be linearized and reduced to a single
linear transfer coefficient k. A high stability programmable
voltage regulator gives Uch with ppm resolution. Based on this
linearization, a precise amplitude step (in this case ±50 ppm)
could be programmed to calibrate the high resolution
measurement system.
To characterize the measurement systems’ performance,
two identical and fully independent high resolution
measurement systems were operated simultaneously. Figure 1
shows the response of the two systems to ±50 ppm calibration
amplitude steps.
Fig. 2. Running standard deviation of the amplitude of 100 consequent
pulses measured simultaneously by high resolution measurement system A
and B.
D. Noise floor estimation
Analyzing statistically the results of the two independent
measurement systems, we can separate the noise contributions
of the signal and the high resolution measurement. From
variance properties it is known that the variance of normally
distributed random value does not depend on the expected
value. Using this we could leave the expected value of the
amplitude out of the equations and work only with the noise
contributions.
The noise of two independent measurement systems is
represented with A and B, and K represents the kicker noise,
with their respective variances⁡𝜎𝐴2 , 𝜎𝐵2 and⁡𝜎𝐾2 . From the two
independent high resolution measurement systems, we have the
measured noise MA and MB with their respective variances
2
2
𝜎𝑀𝐴
and⁡𝜎𝑀𝐵
. Using MA and MB we can create a mean value
MC:
𝑀𝐶 =
𝑀𝐴+𝑀𝐵
(4)
2
Returning to Figure 2, we can calculate its standard
deviation 𝜎𝑀𝐶 for the same period as used to find 𝜎𝑀𝐴 and
𝜎𝑀𝐵 ; this gives 𝜎𝑀𝐶 = 1.29⁡ ppm. The measured noise MA and
MB are sums of the kicker noise K and the noise of the
respective measurement system or:
Fig. 1. Response of high resolution measurement system A and B to
calibration steps of ±50 ppm.
C. Measurement results
Figure 2 shows the measured running standard deviation of
the amplitude of 100 consequent pulses. The averaged values
for the shown period are 𝜎𝑀𝐴 = 1.52 ppm and 𝜎𝑀𝐵 =
1.61 ppm measured by system A and B respectively. The
kicker was running at 50 Hz repetition rate synchronized with
the mains frequency. The time period shown in Figure 2 is
about 10 seconds.
𝑀𝐴 = 𝐴 + 𝐾⁡
(5)
𝑀𝐵 = 𝐵 + 𝐾⁡
(6)
We can express the variances of MA, MB and MC as
following:
2
𝜎𝑀𝐴
= 𝑉𝑎𝑟(𝑀𝐴) = 𝑉𝑎𝑟(𝐴) + 𝑉𝑎𝑟(𝐾)⁡
(7)
2
𝜎𝑀𝐵
= 𝑉𝑎𝑟(𝑀𝐵) = 𝑉𝑎𝑟(𝐵) + 𝑉𝑎𝑟(𝐾)⁡
(8)
2
𝜎𝑀𝐶
= 𝑉𝑎𝑟 (
𝑀𝐴+𝑀𝐵
2
)=
𝑉𝑎𝑟(𝐴)
4
+
𝑉𝑎𝑟(𝐵)
4
+ 𝑉𝑎𝑟(𝐾) (9)
From these 3 independent equations we can express the
variances of each system as following:
2
𝜎𝐾2 = 2𝜎𝑀𝐶
−
2
𝜎𝑀𝐴
2
−
2
𝜎𝑀𝐵
2
⁡
𝜀𝜑 = |
(10)
2
𝜎𝐴2 = 𝜎𝑀𝐴
− 𝜎𝐾2 ⁡
(11)
2
𝜎𝐵2 = 𝜎𝑀𝐵
− 𝜎𝐾2 ⁡
(12)
Using the measured standard deviations 𝜎𝑀𝐴 , 𝜎𝑀𝐵 and
𝜎𝑀𝐶 we get:
𝜎𝐾 = 0.94 ppm
(13)
⁡𝜎𝐴 = 1.19 ppm
(14)
𝜎𝐵 = 1.30 ppm
(15)
This puts the pulse-to-pulse amplitude stability of the
kicker just below unity ppm border.
cos⁡(0)
|
(16)
In our case the electron bunches arrive at the positive and
negative crest of the sinusoidal current of the resonant kicker
(with 180° separation). Using the measured phase noise value
we can calculate the phase noise driven amplitude error 𝜀180:
𝜀180 = |1 − cos(180° + 24 ∙ 10−3 °)| = 8.7 ∙ 10−8 (17)
Due to the fact that at the crest cosine derivative changes
sign the error function’s distribution is folded Gaussian. The
calculated value is much smaller than the rest of amplitude
instability and could be neglected.
For possible future use with more than two electron
bunches at the positive and negative peaks, we consider an
additional bunch either at 45° or at 90°, and calculate the
phase driven amplitude noise using the measured phase
stability.
IV. PHASE STABILITY
A. Measurement
The phase performance of the prototype kicker was
measured using a RF diode mixer. Figure 3 shows amplitude
envelope and the phase change during the RF pulse. At 57 µs
after the RF pulse start, the maximum amplitude is reached in
the resonator. The phase is measured at this time. After that
point the voltage droop in the driver’s storage capacitors
becomes larger than the resonance amplitude growth and the
resonator amplitude droops slightly.
cos⁡(φ)−⁡cos⁡(𝜑+∆𝜑)
𝜀45 = |
1
√2
− cos(45° + 24 ∙ 10−3 °)| = 295 ppm (18)
𝜀90 = |0 − cos(90° + 24 ∙ 10−3 °)| = 417 ppm (19)
The amplitude noise at these points is still too high to fulfill
the stability requirements for the present accelerator project.
Since the measured level of phase noise is more than
sufficient for the current project with two bunch operation, the
phase measurement was not fully optimized and the result
could be limited by the measurement noise floor of the setup.
There is some evidence (discussed in the next paragraph) that
suggests the resonator phase noise alone is lower than the
measured value. This should be further investigated because it
could open possibilities for stable multi-bunch separation.
V. LID VIBRATION
Fig. 3. Amplitude and phase waveforms of the RF mixer measurement of a
single pulse.
Calibration of the setup was done using 100 ps delay
element that corresponded to 1.8 mV shift in the phase plot.
The measured rms voltage noise was 64 µV corresponding to
3.6 ps or 24 millidegree (6.6 ∙ 10−5 relative to the resonator
sine wave period).
B. Phase noise driven amplitude instability
The relative single cycle phase noise driven amplitude error
εϕ can be expressed as ratio of the amplitude change due to the
phase misalignment Δϕ at the working angle ϕ and the
maximum deflection.
Experimenting with the prototype we found that driving the
resonator about 16 ppm above the resonance gives a minimum
value of amplitude jitter. Some amplitude instability was
caused by the fan air cooling. Even though the air velocity was
relatively low, the turbulent air passing through the resonator
tank caused the lid to vibrate. It was expected this would cause
detuning due to the lid displacement and respectively increase
of the amplitude jitter off resonant frequency. Nevertheless this
would not explain the frequency deviation from resonance for
the minimum of amplitude jitter. It was discovered that the lid
displacement modulates significantly the resonator’s Q-factor
as well, providing a second mechanism for amplitude change.
Using defined lid displacement steps the detuning and the losseffect were measured and modeled. On Figure 4 the dashed
line describes the expected amplitude jitter in function of
running frequency, combining the effects of frequency
detuning and Q-factor change due to the lid displacement. It
predicts a minimum at a frequency about 22 ppm above
resonance. At that frequency the two effects should completely
cancel out, zeroing the sensitivity to lid vibration. This model
is predicting qualitatively correctly the direction of frequency
shift but not quantitatively the frequency at which we measured
∆𝜑
minimum jitter value, indicated by the vertical dotted line (at
~16 ppm).
Another source of amplitude jitter is the low frequency
electrical phase noise of the driver electronics for exciting the
resonator. In Figure 4, the dash-dot line indicates the expected
phase noise driven amplitude jitter in function of running
frequency that clearly has its minimum aligned with the
resonance frequency (zero deviation). The two noise sources
(one mechanical and the other electrical) are uncorrelated and
could be summed in quadrature (the result is approximate in
the vicinity of the points where the derivatives of the sensitivity
functions change sign due to the folded Gaussian error
distribution)
2𝜋
=
∆𝑓
𝑓0
𝑡𝑓0
(22)
For one period 𝑡 = 𝑓0 −1 the relative phase change
equal to the relative frequency deviation
∆𝑓
𝑓0
∆𝜑
2𝜋
⁡ is
or finally for the
one cycle relative phase stability we have:
∆𝜑
2𝜋
∆𝑓
= ( ) = 6 ∙ 10−8
𝑓0 𝐸
(23)
This then suggests that the inherent resonator phase noise
alone is much lower than the directly measured value.
VI. CONCLUSION
Fig. 4. Amplitude jitter as a function of frequency deviation from resonance.
The continuous line in Figure 4 shows the combined effect
of the two noise sources. Fitting the relative amplitude of the
∆𝑓
two noise sources in Figure 4 (vibration ( ) = 8 ∙ 10−8 rms
∆𝑓
𝑓0 𝑉
and electrical ( ) = 6 ∙ 10
𝑓0 𝐸
−8
rms) we get very close
approximation of the measured amplitude jitter function of
frequency, indicated with triangles.
The lid displacement and the phase noise of the resonator
could be estimated using the fitted relative noise amplitudes
and the measured sensitivities. For the lid rms displacement we
have:
The development of a high stability resonant kicker system
for SwissFEL is in its final phase. The currently measured
amplitude shot-to-shot stability after removing the noise
contribution of the measurement system is just below 1 ppm
rms. This result is well within the requirement for the present
accelerator project. The measured value of the relative single
cycle phase stability is⁡6.6 ∙ 10−5 . This phase stability level is
enough for the currently planned two bunch operation mode
but it is still not good enough for 3 or more bunch separation.
Mastering of such high stability system would open
possibilities for future multi-bunch operation.
Mechanical vibration was found to contribute strongly to
the overall amplitude jitter. The kicker’s amplitude sensitivity
to lid translation (due to Q-factor change) was found to be
1.8 ppm/µm. A mathematical model was developed to explain
the frequency dependence of the amplitude stability that agrees
with the measured values. This model gave an indirect estimate
of the single cycle relative phase stability of the resonator. The
estimated value was much smaller than the directly measured
one. This discrepancy needs further investigation because if the
true phase noise value is proven to be this low a multi-bunch
operation (more than two bunches) is fully feasible.
REFERENCES
[1]
[2]
∆𝑓
𝜎𝑑 = ( ) ∙ 1.5 ∙ 10−2 = 1.2 ∙ 10−3 mm
𝑓0 𝑉
(20)
For noisy harmonic oscillator ⁡𝑠𝑖𝑛(2𝜋𝑓0 𝑡 + ∆𝜑) we can
write the following relation between nominal frequency f0,
frequency deviation Δf, phase change Δφ and time t.
∆𝜑 = 2𝜋∆𝑓𝑡
(21)
Rewriting eq. (21) to separate the relative frequency and
phase change we have:
[3]
[4]
[5]
[6]
M. Paraliev, C. Gough, “Development of high stability resonant kicker
for Swiss Free Electron Laser” Proc. 2013 IEEE Pulsed Power and
Plasma Science Conference, pp. 1264-1267, San Francisco, CA, USA,
doi:10.1109/PPC.2013.6627606, 2013
M. Paraliev, C. Gough, “Stability Measurements of SwissFEL Resonant
Kicker Prototype”, Proc. 2014 IEEE International Power Modulator and
High Voltage Conference, pp. 322-325, Santa Fe, NM, USA, doi:
10.1109/IPMHVC.2014.7287273, 2014
http://www.psi.ch/swissfel
“SwissFEL Conceptual Design Report”, PSI, April 2012
https://www.psi.ch/swissfel/CurrentSwissFELPublicationsEN/SwissFEL
_CDR_V20_23.04.12_small.pdf
M. Paraliev, C. Gough, “Development of High Performance Electron
Beam Switching System for Swiss Free Electron Laser at PSI”,
IPMHVC 2012
M. Paraliev, C. Gough, “Comparison of High Resolution “Balanced”
and “Direct conversion” Measurement of SwissFEL Resonant Kicker
Amplitude”, 2015 IEEE Pulsed Power Conference, Austin, TX, USA,
doi:10.1109/PPC.2015.7297016, 2015