Improved waveforms for Barrier-Bucket systems

Barrier-Bucket (BB) systems provide a method to apply a short gap to a coasting beam. This is utilized for different applications, like ion cleaning, or to compensate the medium energy loss caused by internal experiments. BB-cavities are broadband cavities, and the applied signal is commonly a short sine burst, followed by a flat section at zero voltage. Since the transfer function of the BB-system is usually neither flat nor linear, it is common to predistort the signal to obtain the desired shape at the rf gap. Nevertheless, the resulting waveform still has a ripple in the flat section. This is due to the lowpass characteristic of the amplifier and the sharp edges at the ends of the sine, which lead to an infinite number of harmonics. This paper provides better suited BB-waveforms, which are designed with a finite number of harmonics from the beginning. It is shown that a much better flatness can be achieved than for a conventional BB-waveform, without sacrificing any performance. These advanced waveforms are currently used at the hadron synchrotron COSY at Forschungszentrum Jülich, leading to improved BB-bunch shapes, in particular for electron-cooled beams.


Introduction
Barrier-Bucket (BB) cavities offer a way to gain more control over the bunch shape in a synchrotron.They periodically apply short pulses to the beam, each shaped like one single sinusoidal period.The length of the pulse is only a fraction (e.g.20 %) of the revolution time, so most phases do not experience any accelerating or decelerating fields.This results in a very long bucket.In addition, most of the bucket is of one potential, so beams of low momentum spread homogeneously along the bucket.
At COSY Jülich, a BB system is used to compensate the energy loss caused by the internal experiments [1].A constant event rate is desired for the experiments.A bunched beam counteracts this need.The BB is a good compromise because the beam intensity is almost constant for a large fraction of each revolution.
For HESR at FAIR, two BB systems are planned to provide efficient longitudinal stacking [2].They are shifted against each other to open new buckets in longitudinal phase space.Furthermore, the large gap allows for ion cleaning of residual gases in the beam pipe.
Other applications for barrier buckets include pre-compression of the beam [3], reduction of beam loss during extraction [4], or aid for transition energy crossing [5].
BB systems typically consist of a signal source such as an arbitrary waveform generator, amplifiers, a loaded cavity, and long cables.Unlike acceleration cavities, such systems must have good broadband performance.The barrier signal can be described by its Fourier decomposition.Since short pulses result in high harmonics, the system must be able to handle frequencies in the order of at least 12 times the rotational frequency.The exact shape of the barrier signal is not clearly defined.Rather, it must meet several characteristics.The barrier part of the signal should be symmetrical.This is to achieve symmetrical bunches, but most importantly to be free of a DC component.The barrier width must not exceed a certain maximum value.In addition, the bucket section should be as flat as possible.A ripple in the the bucket can lead to micro bunches, as was observed at COSY during beam cooling.
In general, the system response is neither flat nor linear in phase.This results in completely distorted voltage shapes at the gap of the cavity.Various strategies have been evaluated to predistort the source signal in such a way that a good shape is obtained at the gap.Furthermore, it is possible to compensate the nonlinear behavior of the amplifiers in the high power regime [6].All strategies have in common that the system response is measured, so that each harmonic in the signal can be corrected in amplitude and phase.
However, one source of distortion that cannot be addressed by system response compensation is the low-pass nature of the system.Frequency components that are attenuated to zero by the system cannot be restored by increasing their spectral power at the input.Thus, a sinusoidal pulse signal will always result in a broadened pulse and a well-defined ripple in the bucket after low-pass filtering, no matter how much the signal is predistorted.
In this paper, this problem is circumvented by optimizing the undistorted barrier signal itself.Different construction strategies for band-limited BB signals are presented and compared.
Furthermore, an attempt is made to find the best possible signal by optimizing the individual Fourier coefficients.

Signal shapes
To find an appropriate band-limited signal shape, one must first define some quality attributes.Figure 1 shows two signals.The first is a perfect one-period sinusoidal burst, The signals are periodic in T .The displayed interval from 0 to T was chosen for a bucketcentered visualization.
Since s 1 is not differentiable at the ends of the barrier, the signal has infinite harmonics.The Fourier sinusoidal coefficients, are shown in the right plot for α = 0.2.Due to its odd symmetry, the spectrum consists only of sinusoidal components.
Truncating the spectrum after the ninth harmonic results in the signal s 2 , also shown in the left diagram.Due to its band-limited nature, it has a smooth transition between barrier and bucket part, and the barrier has become wider.To define the width w of the barrier in a very general way, the point was chosen where the value drops to 5% of the peak value.The unfiltered signal s 1 has a width of 19.7%, s 2 of 22.0%.Furthermore, the bucket is no longer flat.The resulting ripple is quantified by the maximum of its absolute value, r.For s 2 , it is 5.4%.
Obviously, truncation of sinusoidal-bust signals is not a reasonable way to generate bandlimited BB signals.A sharp edge at the end of the barrier cannot be produced with typical bandwidth, nor is it really needed for pulse shaping purposes.A flat bucket, on the other hand, is critical for homogeneous particle distribution in beams with low momentum spread.Therefore signal shapes with a small ripple r must be found.
Here, three different soft-edge barrier strategies are compared: A raised cosine windowed sinusoidal burst, two Gaussian pulses, and the best solution found by an optimizer.Their shapes are illustrated in Figure 2.
For comparison, the unfiltered sinusoidal burst is used.The α parameter specifies the width of the window.This is not exactly the same as the 5%-width w.
The raised-cosine window is a common filter function in digital signal processing.It has a strong harmonic rejection.Applied to the sinusoidal burst, results in the desired barrier shape, but will reduce the width considerably.The third shape is constructed from two Gaussian pulses, one with a positive sign and one with a negative sign:  The standard deviation σ defines the width of the barrier.The choice of Gaussian pulses is motivated by the uncertainty principle of the Fourier transform.Short pulses in the time domain lead to wide pulses in the frequency domain and vice versa.The minimum of the product of the two widths is only achieved by normal distributions [7].Therefore, it seems reasonable that a train of periodic Gaussian pulses will have fast decaying harmonics, and consequently a truncation of the harmonics will have little effect on the signal shape.
As can be seen in the figure, a major difference between the two approaches is the behavior in the bucket.While s c is exactly zero, the Gaussian pulses have a soft transition.The fourier coefficients of s c oscillate around zero, but the coefficients of s g again have a Gaussian shape and decay monotonously.

Approach
The quality of the signal is limited by two parameters.The first is the number of harmonics h.It is determined by the frequency limit of the system divided by the revolution frequency of the beam.Second, the desired width w of the barrier.Narrower barriers require more harmonics, or lead to more ripple if h is kept constant.
Low-pass filtering of the ansatz functions leads to variable pulse widths.Thus, for each pair of (h, w 0 ), a root finding method is used to find the correct parameters α or σ that lead to the desired width w({α, σ}) − w 0 = 0.
To find the best solution, an optimization problem was defined.For a given desired barrier width w 0 and a number of harmonics h, the flattest bucket is sought.While the h coefficients s o perform significantly better.E.g., for w=20% and h=12, r s is about 6%, r c and r g are less than 2% and r o is less than 0.4%.For large values of h, the ripple decreases faster for the raisedcosine windowed signals than for the sine-burst signals.The ripple of the double-Gaussian pulse signals decays even faster.The optimizer was not able to find better solutions, but this is not of technical interest for such small values of r.
For realistic systems, a ripple of less than 1% is desirable.Much smaller values are generally difficult to achieve due to the limitations of pre-distortion.But such systems are also often severely limited in bandwidth.
For small harmonic numbers, neither the raised cosine nor the double Gaussian is always better for all h.But they are both much better than the sine burst.So if a strong minimization of the ripple is not very important, using one of the two smooth functions is already a huge improvement.
On the other hand, if minimizing the ripple is critical, optimizing the coefficients may be beneficial.For the given plots, the optimizer sometimes found no better solution at all, or failed completely.But in some regimes, such as w=20%, h=13, a signal with a ripple about three times smaller than the best analytical approach was found.
In conclusion, a soft-edged barrier shape should always be preferred.Further effort can be put into minimizing the ripple if it is worthwhile for the given application.For COSY, a new Gaussian or raised-cosine signal is manually generated for each experiment using the BB system, depending on the energy, barrier width and cooling requirements.For HESR, the bandwidth of the system is sufficiently large with respect to the rotation frequency, so a Gaussian-shaped signal will give satisfactory results.

Figure 2 .
Figure 2. Comparison of the unfiltered shapes of sinusoidal burst s s , raised-cosine windowed sine s c , and Gaussian pulses s g .The s s and s c signals have a parameter of α = 0.2.The Gaussian pulses in s g have a standard deviation of σ = π/(4 • 0.2).