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Effects of a 500 Mc s−1 additional cavity on spontaneous coherent synchrotron oscillations in the Super-ACO storage ring
Nuclear Instruments and Methods in Physics Research A 417 (1998) 413—427
Effects of a 500 Mc s\ additional cavity on spontaneous coherent synchrotron oscillations in the Super-ACO storage ring M. Bergher* LURE, BaL t. 209D, Centre Universitaire Paris-Sud, 91405 Orsay Cedex, France Received 6 May 1998
Abstract An additional 500 Mc s\ cavity, fifth harmonics of the main cavity, was installed on the Super-ACO storage ring in order to shorten the bunch length. This cavity was introduced to obtain shorter wavelengths in the UV for the FEL and shorter flashes of light for the experiments using the two-bunches-mode functioning for the time-resolved measurements. Spontaneous coherent synchrotron oscillations (SCSO) were profoundly modified by the presence of this cavity. These instabilities are probably the consequence of the formation of micro-bunches. The vanishing of these micro-bunches is associated to the emission of coherent synchrotron radiation, which gives the SCSO a cyclic character. The identification of the resonant elements responsible for these cyclic instabilities can help us to suppress or substantial ameliorate the SCSO by acting selectively on these resonant elements. This method can be applied to other storage or damping rings that show the same type of instabilities. 1998 Elsevier Science B.V. All rights reserved. PACS: 07.85.Q; 29.17; 29.20.D Keywords: Coherent radiation; Damping and storage rings; Micro-bunches; Parasite resonant element; Spontaneous coherent synchrotron oscillation; Wakefields
1. Introduction In a previous paper we showed that the bunches stored in a ring can substantially modify the “natural” synchrotron frequency by inducing wakefields on a strong parasite longitudinal impedance [1]. These results allowed us to hypothesize that each bunch was transformed into a group of microbunches. Measurements of spontaneous coherent
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synchrotron oscillations (SCSO) were performed again on the Super-ACO ring in the presence of a new 500 Mc s\ cavity under two different regimes of functioning with the same set-up described in Ref. [1]: (a) the cavity excited only by the beam, i.e. in a passive mode, and out of tune; (b) the cavity powered at 250 kV, a voltage that should lead to a shortening of the bunches by 2.7 times. Before the 500 Mc s\ cavity was installed, the two bunches traveling in the ring and in the same direction but at the opposite of each other (180°)
0168-9002/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 7 5 2 - 9
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showed always stable SCSO of low amplitude, which were dipolar in phase and in energy. The amplitude of the SCSO increased with the stored current. After the installation of the new cavity, in a passive mode, the SCSO became cyclic, with a irregular period, but close to the time constant of the synchrotron radiation damping time. During a cycle, the amplitude of the SCSO rose rapidly, decreased slowly and disappeared for a certain time. When the current changed, the two bunches oscillated either in phase or in opposite phase, although their frequency of oscillation remained approximately the same for a given current range. The measured oscillations had frequencies higher than the theoretical synchrotron frequency, similar to the situation before the new cavity was installed, but for a narrow current range, their values were close to that of the expected synchrotron frequency. Under these particular conditions the amplitude of the oscillations were larger and of opposite phase. This change in phase of the oscillation of the two bunches can be ascribed to a higher order mode (HOM) of the 500 Mc s\ cavity. The large oscillations at the synchrotron frequency can also be ascribed to the HOMs of the new cavity. When the stored current decreased, the adjustment of the plunger position to tune the 500 Mc s\ cavity, induced changes in the resonance frequency of some HOMs of this cavity, which will eventually become close to a harmonic of the rotation. A small improvement in the lifetime of the stored current can be noticed in the presence of the new cavity, probably because of the presence of cyclic instabilities. These instabilities are probably the consequence of the formation and vanishing of micro-bunches, because their formation is linked to the potential formed on the parasite resonant element. Their gradual vanishing is concomitant to the appearance of the coherent synchrotron radiation produced by the micro-bunches. At very low stored current, when the 500 Mc s\ cavity was powered and developed a potential of 250 kV, the shortening of the bunch was close to the predicted behavior. Above a threshold value of the current, the two bunches oscillated at a frequency smaller than the predicted synchrotron frequency. They oscillated with opposite phases and with an amplitude that strongly increases with current.
2. Measurement methods Measurements were performed at the exit window of a bending magnet, where the horizontal dispersion was 0.3 m. At this point one can simultaneously detect the variation in phase and energy of the beam, whose transverse dimensions are px" 370 lm and pz"270 lm. The image of the beam is formed on the photodetector, with a magnification of one, using an optical configuration consisting of one spherical and one flat mirror. All measurements were performed when the ring was running with two bunches circulating at the opposite of each other. Before the 500 Mc s\ cavity was installed, two measuring systems were used to study the synchrotron instabilities. In the first system we used a fast photodiode (8 Gc s\ band pass) with an active surface of 70;80 lm. With this photodiode and a spectrum analyzer, we could show that the SCSO were dipolar and in phase on both bunches and with a constant amplitude, for a variety of synchrotron frequencies which depended on the stored current. The second system was simpler, it consisted of an array of photodiodes, and an analog oscilloscope. The photodiode array was of a smaller band pass (20 Mc s\) but sufficient to see both bunches independently. The photodiodes were 0.9 mm wide, separated by 0.1 mm, and 2.4 mm high. By using an oscilloscope we visualized the modulation in energy of the particles in the bunch. The results obtained with the second system corroborated the results obtained with the first system. After the installation of the 500 Mc s\ cavity, we used only the second system for measurements because it could better detect the rapid cyclic variations of the synchrotron instabilities of the bunches.
3. Results 3.1. Measures performed before the installation of the 500 Mc s\ cavity When considering all the synchrotron bands around a higher harmonic of the frequency of rotation, one can see that the amplitude of each band
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Fig. 1. (a) Bessel functions for the first 3 orders. (b) Bessel function for the J order at the toric resonance frequency.
corresponds to a Bessel function: J ( ) corresponds to the harmonic of the frequency of rotation (zero order), J ( ) to the first pair of synchrotron bands, J ( ) to the nth pair of synchrotron bands, where L
is the argument of the Bessel functions, that corresponds to the peak amplitude of the phase oscillation at a given harmonic of rotation (Fig. 1). The higher the harmonic of observation the easier will be to detect the synchrotron bands around each harmonic of rotation because they will rise from the noise level. This is particularly true for the results showed in Fig. 2. The first two side bands that appeared for stored currents of 172 and 87 mA corresponded to the real synchrotron frequencies of 39 and 27 kc s\, instead of the predicted frequency of 14.5 kc s\ for the Super-ACO ring. All other higher order side bands were harmonics of these first frequencies, as previously reported [1]. The first two bands on each side of the rotation band do not have the same amplitude because they were recorded with a photodiode positioned at the hori-
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zontal end of the beam image corresponding to the exterior of the ring, as described in Ref. [1]. The modulation in the signal amplitude is reflecting the modulation in energy, the latter out-of-phase by 90° from the phase modulation. Because the oscillations were small, modulations of both phase and amplitude were sinusoidal and therefore all pairs of bands should have shown different amplitudes. Still, because the horizontal movement is larger than the transverse dimensions of the photodiode this distorts the sinusoidal signal amplitude, and therefore all band pairs of the spectrum, the first and the following, have their amplitude distorted. We also presented previously the spectra of synchrotron oscillations around the even and odd rotation harmonics [1]. We noticed that the synchrotron bands and the rotation band were not sufficiently decreased for odd harmonics. According to theory, when two bunches are revolving at the opposite of each other, and when the synchrotron oscillations are in phase and are of the same amplitude, for odd harmonics the amplitude of the rotation band and the synchrotron bands should depend only on the difference in the current between the two bunches. For all results presented in this paper the difference in the current between the two bunches was much smaller than 1%. The presence of bands at odd harmonics can only be explained by a phase difference between two groups of micro-bunches which have replaced the two initial bunches. This phase difference corresponds precisely to one period among micro-bunches, or a multiple of it. The higher the rank of the harmonics, the smaller becomes the difference between the odd and even harmonics of the rotation (Fig. 3). Furthermore, somewhere between the rank 722 and 1442, the amplitude of odd harmonics of rotation becomes even larger than the even harmonics (Figs. 2 and 3). Therefore the two bunches cannot be exactly opposite to each other (180°). This strengthens our hypothesis on the presence of a single resonant element which is inducing the described dephasing, the formation of microbunches, and the advent of SCSO. This resonant element, that has a high quality factor, is located either in the main cavity, as a HOM, or in the vacuum chamber. It permits to attain the longitudinal toric resonance. It has a resonant frequency
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Fig. 2. Spectral analysis of the spontaneous synchrotron side bands for different harmonics of rotation and stored current (I). (a) Harmonic 1442, I"99.6 mA. (b) Harmonic 1443, I"99.6 mA. (c) Harmonic 1442, I"119 mA. (d) Harmonic 1443, I"119 mA.
very far both from the harmonic of the main cavity, and from the harmonic that corresponds to the first zero value of the J function. Still, this single reson ant element must have a frequency that is a multiple of the main cavity frequency which is the 24th harmonic of the rotation. This is the element that induces the synchrotron instabilities in the ring functioning with 24 bunches [2].
For the values of current stored below 110 mA, where the amplitude of the oscillations decreases, the spectral analysis of the synchrotron bands shows that they are divided into fine bands, as described in Ref. [3] (Fig. 4). These fine bands are the consequence of the presence of micro-bunches. From the spectral analysis of the synchrotron band which has the largest number of fine bands, and
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Fig. 3. Spectral analysis of the even and odd harmonics and their first spontaneous synchrotron bands. (a) Harmonic 239, I"100 mA. (b) Harmonic 240, I"100 mA. (c) Harmonic 721, I"125 mA. (d) Harmonic 722, I"126 mA.
taking into account the length of the bunch measured independently for the given current, one can conclude that the distance among the microbunches is of the order of 11 ps. This corresponds to a frequency of 91 Gc s\, which is close to the first toric longitudinal resonance of the Super-ACO vacuum chamber. It is important to mention that the toric resonance depends on the simultaneity of the propagation rate of the phase of the longitudi-
nal field and the propagation rate of the microbunches that have created this field [4]. With the changes in stored current the multiple band appear, disappear and reappear again, under consecutive ranges of current. For each range where they are present their number remains constant, but among different current ranges the number of multiple bands changes in a stochastic way. Each range where the multiple bands are present
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Fig. 4. Spectral analysis of the fine bands of the synchrotron frequency F at the harmonic 1442 of rotation. (a) I"100.8 mA, F "27 kc s\. (b) I"38.9 mA, F "42.41 kc s\. (c) I"36.6 mA, F "41.58 kc s\. (d) I"34.8 mA, F "41.43 kc s\.
would correspond to a toric resonance frequency of the ring. The changes in the dispersion in energy of the particles in a bunch, which are dependent on the changes in current, lead to a change in orbit length of 0.19 mm and an apparent *E/E of 1.8;10\. The toric resonance frequency is very sensitive to this length variation [4] and can lead to change from one even harmonic of rotation to another [3].
At 91 Gc s\, the relative change in toric frequency will be 9;10\. It implies that the selectivity of the toric resonance frequency has to be much smaller than 10\. These conditions are too restrictive for this type of resonance. The orbit length variation of 0.19 mm corresponds to a variation of the toric resonance frequency of several hundreds Mc s\. The amplitude of the phase oscillation at the latter frequency leads to the oscillation of the function
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J (Fig. 1). The consequence of the above is the successive formation, disappearance and reappearance of the fine synchrotron bands. One can notice that, although the global dispersion of synchrotron frequencies remains approximately constant for any number of fine bands, the gap between them is self-adjusted to remain constant. The formation of micro-bunches, and the presence of fine synchrotron bands linked to two distinct resonant cavities, can be described in two different ways, as shown in Fig. 5. The classical explanation assumes that all micro-bunches have the same energy, and that they are aligned at a potential corresponding to the energy loss per turn. The slope that each micro-bunch sees is different. They have, therefore, different synchrotron frequencies which are not equally distributed. The current of each micro-bunch induces an increase in potential in the parasite resonant element, therefore an increase in the slope seen by the micro-bunch, which will then have a frequency of synchrotron oscillation higher than the preceding micro-bunch. All these conditions together should explain a constant gap among frequencies of the fine synchro-
Fig. 5. The interaction of the micro-bunch with the RF voltages. The sum of two RF voltages comprising the fundamental (» ) and its 100th harmonic (» ) is plotted against time. The horizontal line º corresponds to the voltage that compensates for the energy loss on each turn. One can see that the slope of voltage seen by each micro-bunch is different when the bunches have the same energy. By contrast, on the oblique line representing the voltage » , the slope seen by each bunch remains constant only when they have different energies.
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tron bands, but still we think it is not plausible for all cases when synchrotron fine bands appear. By contrast to the previous hypothesis, we can also postulate that micro-bunches have each a different energy, and that each of them is composed of particles with a small energy dispersion. (The hypothesis of bunches circulating with different energies applies only to almost isochronous rings where the momentum compaction is close to zero [5]. This is not the case for Super-ACO in which the momentum compaction is 1.48;10\.) The different micro-bunches are aligned on the potential delivered by the main cavity. The slope of the potential seen by all micro-bunches is identical. Each micro-bunch will have a different rotation frequency. The separation among these frequencies corresponds to the separation among fine synchrotron bands. One should then be able to see fine side bands around the line of the rotation harmonic. Their number and separation in frequencies is the same as for fine synchrotron bands. This is actually the case shown in Fig. 6. We can detect the frequency corresponding to this separation among the fine bands if we add a peak detector on the signal coming from the fast photodiode, which is composed of pulses modulated in amplitude. These pulses represent the passing over of groups of micro-bunches. The synchrotron oscillation is seen
Fig. 6. Spectral analysis of the fine bands of the synchrotron frequency (F ) and of the fine bands of the harmonic 1442 of the rotation frequency, for I"48.8 mA.
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Fig. 7. Double modulation of the signal amplitude coming from the fast photodiode for I"69.4 mA, F "27 kc s\, and the distance among the fine synchrotron bands equal to 622 c s\.
as the amplitude modulation of the pulses. The amplitude modulation of the synchrotron oscillation corresponds to the identical frequency gap among the different frequencies of the microbunches rotation, i.e. to the beating at this frequency (Fig. 7). We can conclude that the dispersion of frequencies for a number of fine synchrotron bands does not correspond to a real dispersion in particles energy of the “bunches”, as measured before for different currents. This is possible because, at the toric resonance frequency, each micro-bunch follows the field that it has created. This is not the case with a resonant cavity positioned in the ring. Above 110 mA, the dispersion and oscillation in phase and energy become too large, they cover a large number of harmonics of rotation and therefore the fine bands are not visible any more (Fig. 8). 3.2. Measures performed after the installation of the 500 Mc s\ cavity (a) ¹he cavity is used in a passive mode. The plunger is inserted to change the tune of the 500 Mc s\ cavity further away from the 120th harmonic of rotation. The two “bunches” show cyclic SCSO with a period close to the time constant of the longitudinal radiation damping time, which is 10 ms. The amplitude of the SCSO in-
Fig. 8. Spectral analysis of the spreading of the lower spontaneous synchrotron band at the harmonic 1442 of rotation. I"126 mA, F "38 kc s\.
creases rapidly, followed by a slower decrease. Then, the SCSO disappear all together for a long period of the cycle (Fig. 9). The origin of this instability will be discussed later. When the stored current is decreasing, the SCSO show frequencies and phases which depend on the range of the current present. From 200 to 100 mA, the two “bunches” oscillate in phase. Their frequency of oscillation, 39 kc s\, is identical to the one measured before the installation of the 500 Mc s\ cavity (Fig. 10). Still, in a small region, between 150 and 143 mA, the amplitude of the oscillations increases and their frequency, 13 kc s\, becomes smaller than the theoretical synchrotron frequency and the two bunches oscillate in opposite phases (Fig. 11). This is probably linked to the movement of the plunger which is designed to tune the frequency of the 500 Mc s\ cavity and acts on the frequency of a HOM of the cavity. From 100 to 80 mA, the frequency of oscillation is close to 26 kc s\, and the two bunches oscillate again in opposite phase (Fig. 12). This behavior is probably also due to a HOM of the 500 Mc s\ cavity; the latter can be different from the one mentioned above. (b) ¹he cavity is powered and has a potential of 250 kV at 500 Mc s\. The length of the bunch should be shorter under this condition by a factor of
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Fig. 9. Cyclic instabilities of the SCSO, F "38 kc s\. (a) I"117 mA. (b) I"112 mA.
Fig. 10. Measurements of synchrotron oscillations. Peak modulation of the two signals of two adjacent slow photodiodes centered in the beam image. The energy transverse movement of the bunches in the bending magnet is presented, with one signal inverted (lower trace) and the sum of both signals present in the middle. Stored current I"180 mA, F "38.5 kc s\.
2.7, and the synchrotron frequency should rise to 39 kc s\. This was actually the case only at very low current. Above the threshold of 1 mA, the SCSO of the two bunches appear. They have a constant amplitude, and are opposed in phase, probably because of a HOM of the added cavity. The amplitude of SCSO increases rapidly with the increase in stored current. The frequency of oscillation of the two bunches under these conditions, 33 kc s\,
Fig. 11. Same as Figs. 9 and 10 for I"145 mA, F "13.3 kc s\.
is lower than the expected synchrotron frequency. For 23 mA of stored current, the amplitude of oscillation in energy becomes 1% peak-to-peak (Fig. 13). The installation of the 500 Mc s\ cavity fulfilled only partially the expectations concerning the bunch shortening. Still, it could be improved by adding either of the following: an appropriate system of feed-back; a selective tuning of the HOMs, to separate them from the harmonics of rotation [6]; selective damping couplers, to reduce the effects of the HOMs [7]; a single trapped mode resonator, comprising large wave guides that would
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damp the HOMs. The latter was proposed and checked on a prototype [8].
4. Discussion
Fig. 12. Same as Figs. 9 and 10 for I"88 mA, F "26 kc s\.
The discussion thereafter is related to measurements performed on Super-ACO compared to other storage or damping rings, because we believe the detected cyclic instabilities have a similar origin. Interpretation of the cyclic regime of SCSO. The potential induced by the bunch on the high frequency parasite resonant element modifies the density inside the bunch. When the potential is sufficient the formation of micro-bunches occurs. This leads to a modification of the spectrum and induces
Fig. 13. Same as Fig. 10, but with the 500 Mc s\ cavity powered at 250 kV. (a) I"20 mA. (b) I"22 mA. The time axis is expanded to be able to observe the movement of the two bunches separately. (c) I"23 mA.
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an increasing potential on the parasite resonant element. The self-formation of micro-bunches will proceed round after round. The phenomenon will be accompanied by an apparent lengthening of the bunch, because of the filling of some of the “empty buckets” on both sides of the ones already present [9]. This is seen as an apparent lengthening of the bunch because the band-pass of the measuring device is not sufficient to see the micro-bunches. This self-consistent phenomenon will be limited as soon as micro-bunches are totally formed, which corresponds to the presence of maximal power at the resonance frequency of the parasite element. If the length of the micro-bunches is sufficiently short and corresponds to the wavelength lower than the cutoff wavelength of the vacuum chamber of the bending magnet, the micro-bunches will produce coherent synchrotron radiation at this wavelength. The radiation emitted will be much stronger (proportional to the square of the number of particles) than the incoherent synchrotron radiation that would be produced at the same wavelength by a bunch of the global length corresponding to the sum of all micro-bunches that composed it. The presence of coherent synchrotron radiation produced by each micro-bunch leads to a decrease of their length and a modification of their shape [10,11]. This self-consistent effect is cumulative and is adding to the effect which formed the microbunches. With the shortening of the micro-bunches, the incoherent synchrotron radiation is transformed into coherent radiation of progressively shorter wavelength. This brings an important loss of energy for all particles of each micro-bunch that have emitted coherent radiation. If the potential well produced by the parasite resonant element is sufficient, the effect of shortening will be accelerated, and still shorter wavelength (therefore of higher energy) will be added [10]. The following two phenomena will then occur: (1) If the parasite resonant element is not tuned to the harmonic of rotation, the micro-bunches will show synchrotron instabilities. The shorter they become the larger will be the amplitude of phase oscillation. This will lead to a limitation, or even a decrease in amplitude of the rotation harmonic line which corresponds to the resonance of the idle
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cavity. Under these conditions the voltage that has created the micro-bunches is limited and can even decrease. (2) The potential wells are not sufficient to keep all the particles in each micro-bunch. The particles that have emitted too much energy, in a coherent way, will leave the bucket and will start to follow complex trajectories around some other buckets, i.e. some other micro-bunches, which will lead to their gradual destruction. The Fig. 14 shows an example of trajectories of particles that have experienced a large change in energy and therefore are crossing the separatrixes of the different buckets. Each bunch has a strong dispersion in energy and keeps a length close to the apparent length of the bunch following micro-bunch formation. This cyclic phenomenon could be repeated only when the damping via synchrotron radiation restores back the initial characteristics of the bunches both in length and energy dispersion. During the time of synchrotron damping, which corresponds to the half of the cyclic instability, the micro-bunches disappear, the bunches become longer, and their transverse dimensions increase. All
Fig. 14. Example of a system of separatrixes obtained with two cavities having identical voltage. The frequency of the second cavity was chosen to be ten times higher than the first one in order to illustrate clearly the complexity of the particles moving outside the buckets.
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these modifications are the consequences of an increase in energy dispersion among the particles. The Touchek effect is decreased, and therefore, the lifetime of the beam is increased. In addition, the losses are decreased in the vacuum chamber and at higher modes of the cavities, which brings to a change of phase of the bunches compared to the phase of the main RF. This type of cyclic instabilities, named sawtooth instabilities, have been observed in different storage and damping rings. However, the conclusions drawn from the measurements were not always the same. Concerning Super-ACO, the installation of the 500 Mc s\ cavity in a passive mode leads, paradoxically, to a shortening of the bunch, although one would expect a slight lengthening under these conditions. A HOM of the cavity is tuned to a harmonic of the rotation by changing the position of the plunger. The induced potential on this HOM shortens the bunches [12]. The end result are shorter micro-bunches than the ones before the installation of the 500 Mc s\ cavity. Shorter micro-bunches produce coherent synchrotron radiation of multiple wavelengths which induces sawtooth instabilities. On the ELECTRA ring a cyclic instability with a sawtooth form was also present. It is a longitudinal oscillation whose amplitude increases rapidly, then decreases slowly. Concomitantly to the decrease in amplitude of oscillations, the dispersion in energy of the beam is increased and then damped by synchrotron radiation [13]. This behavior can be interpreted as the presence of micro-bunches which are destroyed by their coherent radiation, as described above. The coherent synchrotron oscillation phenomenon has also been observed on the BEPC ring when a single bunch was stored, with a frequency roughly equal to the triple of the synchrotron oscillation. A dipolar movement can be responsible for the same frequency of oscillation. The pulse peak of the bunch remains constant. Sometimes, the pulse lengthens and the peak amplitude decreases during the observation, which can also be an indication of the formation of micro-bunches. When this phenomenon occurs in the presence of a bunch of electrons and a bunch of positrons, an abrupt decrease in luminosity (20%) is seen [14]. If we postu-
late the presence of micro-bunches under these conditions, because of the b function, only the micro-bunches at the point of interaction will be seen by the luminosity detector. On the rings SOR and SURF II, instabilities with a sawtooth form have been observed for a long time [15,16]. In SOR the cyclic instabilities were eliminated efficiently and at the very beginning by the addition of a Landau cavity. This cavity will eliminate the cyclic instabilities linked to the potential presence of micro-bunches by spreading out the synchrotron frequency, but this leads simultaneously to a lengthening of the bunch and to a disappearance of the micro-bunches. Old measurements on SURF II have shown that during SCSO the two bunches were lengthened ( 30 cm). Paradoxically, recent measures [17] showed that coherent synchrotron radiation appears during the lengthening at wavelengths between 2 and 4 cm. Measures of the spectrum of coherent radiation showed that the contours of the different bands of the rotation harmonic measured during the fast increase phenomenon has a modulation in amplitude similar to the one presented in Appendix A. This seems to confirm the presence of a few microbunches as short as 2 cm, which would explain the seen lengthening. Unfortunately, the measured frequency bands in the coherent radiation spectrum did not reveal the frequency of the parasite resonant element which induced the formation of microbunches. The first frequency maximum, which would represent the inverse of the repetition time period of the micro-bunches was outside the spectrum shown. The SCSO decreased slower than they increased, and then they disappeared. Concomitantly to the decrease, the coherent radiations gradually disappeared and the dispersion of energy became large. The end of the cycle, comprising the synchrotron radiation damping time of the particles, showed a decrease in energy dispersion concomitant to a shortening of the bunches. All the measurements performed on the SLC damping rings, both with the old vacuum chamber and after its modification, showed the same type of sawtooth instabilities. Very different types of measurements are all in line with the hypothesis of the presence of micro-bunches. The lengthening of the bunches is accompanied, after a delay, by
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a smaller dissipation of energy [18]. The measured frequencies of the coherent synchrotron oscillations [18,19] look like the SCSO on Super-ACO. The position chosen for measurements led to different interpretations, although the presence of microbunches was again very probable. The first frequency of longitudinal toric resonance of the rings (at 193 Gc s\) [4], would imply maximally 5 ps between different micro-bunches. Therefore, each bunch would be composed of 15—20 micro-bunches. Their presence is suggested in the authors’ statement: “Turbulent bunch lengthening is often referred to as the microwave instability because the bunch length oscillation modes have a characteristic wavelength at some small fraction of the bunch length.” [18]. Cyclic instabilities were observed on all cited rings. The cycle begins by the SCSO whose frequency can be different from the synchrotron frequency. Their amplitude increases rapidly accompanied by an apparent lengthening of the bunches. Then their amplitude slowly decreases. The bunches maintain the same length but with a strong dispersion of energy. The remaining of the cycle comprises the reconstitution of the initial characteristics of the bunch length and energy dispersion by synchrotron radiation damping. The only hypothesis that can explain the described phenomenon is the formation of micro-bunches. This hypothesis is corroborated by the detection of coherent radiation on SURF II, the presence of SCSO on Super-ACO, and all the measurements performed on the damping rings of the SLC, in particular the decrease in energy dissipation when SCSO appeared. The remedies by which one could influence the different phases of the instability are the following: (a) To decrease the ring energy at a constant main RF potential and identical stored current. Under these conditions the bunches are shortened creating shorter micro-bunches. The synchrotron emission and the energy dispersion of the particles are decreased and the potential wells deeper. In the spectrum of synchrotron radiation numerous bands of coherent wavelength will be emitted. (b) To decrease the coherent synchrotron instabilities by modifying the frequency of rotation, in
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order to bring closer the higher harmonic to the parasite resonant circuit which contributes to the formation of the micro-bunches and to their instabilities. This represents a small change in frequency of rotation. (c) To install a special tunable cavity on the rings, passive [20] or powered, dedicated to create micro-bunches, with a new RF voltage sufficient to maintain them in the presence of coherent synchrotron radiation. The resonant frequency has to be close to the cutoff frequency of the vacuum chamber in order to eliminate the HOMs which induce the synchrotron instabilities. Under these conditions, the micro-bunches will be able to emit coherent radiation of a large number of wavelengths. The use of a super-conducting cavity will increase the efficiency and therefore will preclude several harmonics of rotation. This would be important especially for rings of larger sizes. The frequency of the cavity could be chosen to favor defined wavelengths of coherent synchrotron radiation.
5. Conclusion The installation of an additional 500 Mc s\ cavity on the Super-ACO storage ring led to a significant modification of SCSO. In the passive mode of the cavity the oscillations become cyclic and of the sawtooth type. They are similar to the ones detected on other storage or damping type rings. The comparison of the measurements performed on Super-ACO to the observations obtained on other rings allowed us to propose an explanation of the cyclic instabilities of the sawtooth type. We also suggested several means to eliminate them, with the additional bonus of producing coherent synchrotron radiation.
Appendix A The presence of micro-bunches replacing the bunches can be described as the product of two functions, G and G . First, in the equation G (t)"exp(!1/2(t/p ))
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the width of the bunch is represented by p , i.e. about 3 times p , the width of the bunch before the formation of micro-bunches which can be obtained from the equation G (t)"exp(!1/2(t/p )). The width of the micro-bunch is defined as p given by the equation G (t)"exp(!1/2(t/p )). In Fig. 15 the full line is the product of G and G , the dotted line the function G and the dashed line the function G . Fig. 16 is the representation of two groups of micro-bunches circulating into the ring and which have the same number of micro-
bunches. Fig. 17 is the square of the Fourier transform of the G ) G product. All the lines are not presented because their number depends on the choice of the number of bunches injected. The frequency difference is the inverse of the time separating two groups of micro-bunches. The Fourier transform of the function G is also presented in this figure by dots. On the spectrum presented in Fig. 17, the difference among maxima defines the inverse of the time separating the micro-bunches. The first maximum is the frequency resonance of the idle cavity. The envelope joining all the maxima represents the Fourier transform of G . The reson ant frequency of the idle cavity, or the chosen frequency of an installed cavity, will define the set of
Fig. 15. A representation of the micro-bunches. The solid line is the product of G and G , the dotted line the function G and the dashed line the function G .
Fig. 16. A representation of two groups of micro-bunches circulating into the ring.
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Fig. 17. Square of the Fourier transform of the G ) G product.
synchrotron lines, including those which are coherent. With the shortening of the micro-bunches p decreases, and the spectrum of coherent syn chrotron radiation widens towards higher frequencies i.e. towards shorter wavelengths. The Fourier transform of the distribution of the micro-bunches in time has a number of maxima of intermediate amplitude that corresponds to the number of micro-bunches minus two. The intermediate maxima are alternatively negative and positive. For example, if a parasite resonant circuit has a frequency equal to a positive maximum, then it can induce instabilities of the beam. If one adds one micro-bunch to a group of micro-bunches, the Fourier transform will also have an additional intermediate maximum. Under these new conditions, the frequency of the resonant circuit will be equal to a negative maximum or close to the zero level, therefore the instabilities of the beam will disappear.
References [1] M. Bergher, Nucl. Instr. and Meth. A 395 (1997) 259. [2] M. Raynaud, Lure, RT/96-03, Orsay. [3] M. Bergher, Micro Bunches Workshop, Upon, NY, AIP Conf. Proc. 367 (1995) 199. [4] R.L. Warnock, P. Morton, Part. Accel. 25 (1990) 113.
[5] L. Farvacque, T. Gu¨nzel, J.L. Laclare, C. Limborg, A. Ropert, U. Weinrich, Micro Bunches Workshop, Upon, NY, AIP Conf. Proc. 367 (1995) 245. [6] H. Ego, M. Hara, Y. Kawashima, Y. Ohashi, T. Ohshima, H. Suzuki, I. Takeshita, H. Yonehara, Nucl. Instr. and Meth. A 400 (1997) 195. [7] Y. Yamazaki, K. Takata, S. Tokumoto, IEEE Trans. Nucl. Sci. NS-28 (1981) 2915. [8] G. Conciauro, P. Arcioni, Proc. EPAC vol. 1 (1990) 149. [9] M. Bergher, Proc. PAC, IEEE 5 (1993) 3708. [10] M. Bergher, Proc. PAC and ICHEA Conf., Dallas, Texas, IEEE vol. 5, 1995, pp. 2928-2931. [11] K. Bane, S. Krinsky, J.B. Murphy, Micro Bunches Workshop, AIP Conf., Proc. 367 (1996) 191. [12] G. Flynn, Private communication. [13] C.J. Bocchetta et al., PAC and ICHEA, vol. 1, 1995, pp. 309—311. [14] Z.Y. Guo, X. Bai, G.X. Li, J. Qin, Q. Qin, G. Xu, C. Zhang, X.L. Zhang, Z.T. Zhao, Proc. PAC 5 (1993) 3237. [15] S. Asaoka, G. Isoyama, H. Mikuni, Y. Miyahara, H. Nishimura, Nucl. Instr. and Meth. 215 (1983) 493; Y. Miyahara, S. Asaoka, G. Isoyama, A. Mikuni, H. Nishimura, K. Soda, H. Kanzaki, Jpn. J. Appl. Phys. 22 (11) (1983) L733. [16] G. Rakowsky, IEEE Trans. Nucl. Sci. NS-32 (5) (1985) 2377. [17] G.T. Fraser, A.R. Hight Walker, U. Arp, T. Lucatorto, Characterization of the coherent microwave emission from the SURF II Electron Ring (1997), to be published. [18] P. Krejcik, K. Bane, P. Corredoura, F.J. Decker, J. Judkins, T. Limberg, M. Minty, R.H. Siemann, Proc. PAC IEEE 5 (1993) 3240. [19] K. Bane et al., Proc. EPAC (1996) 3109. [20] P. Marchand, L. Rivkin, Proc. PAC IEEE 2 (1991) 780.
Effects of a 500 Mc s\ additional cavity on spontaneous coherent synchrotron oscillations in the Super-ACO storage ring M. Bergher* LURE, BaL t. 209D, Centre Universitaire Paris-Sud, 91405 Orsay Cedex, France Received 6 May 1998
Abstract An additional 500 Mc s\ cavity, fifth harmonics of the main cavity, was installed on the Super-ACO storage ring in order to shorten the bunch length. This cavity was introduced to obtain shorter wavelengths in the UV for the FEL and shorter flashes of light for the experiments using the two-bunches-mode functioning for the time-resolved measurements. Spontaneous coherent synchrotron oscillations (SCSO) were profoundly modified by the presence of this cavity. These instabilities are probably the consequence of the formation of micro-bunches. The vanishing of these micro-bunches is associated to the emission of coherent synchrotron radiation, which gives the SCSO a cyclic character. The identification of the resonant elements responsible for these cyclic instabilities can help us to suppress or substantial ameliorate the SCSO by acting selectively on these resonant elements. This method can be applied to other storage or damping rings that show the same type of instabilities. 1998 Elsevier Science B.V. All rights reserved. PACS: 07.85.Q; 29.17; 29.20.D Keywords: Coherent radiation; Damping and storage rings; Micro-bunches; Parasite resonant element; Spontaneous coherent synchrotron oscillation; Wakefields
1. Introduction In a previous paper we showed that the bunches stored in a ring can substantially modify the “natural” synchrotron frequency by inducing wakefields on a strong parasite longitudinal impedance [1]. These results allowed us to hypothesize that each bunch was transformed into a group of microbunches. Measurements of spontaneous coherent
* Tel.: (33) 1 64 46 82 57; fax: (33) 1 64 46 41 48; e-mail: [email protected].
synchrotron oscillations (SCSO) were performed again on the Super-ACO ring in the presence of a new 500 Mc s\ cavity under two different regimes of functioning with the same set-up described in Ref. [1]: (a) the cavity excited only by the beam, i.e. in a passive mode, and out of tune; (b) the cavity powered at 250 kV, a voltage that should lead to a shortening of the bunches by 2.7 times. Before the 500 Mc s\ cavity was installed, the two bunches traveling in the ring and in the same direction but at the opposite of each other (180°)
0168-9002/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 7 5 2 - 9
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showed always stable SCSO of low amplitude, which were dipolar in phase and in energy. The amplitude of the SCSO increased with the stored current. After the installation of the new cavity, in a passive mode, the SCSO became cyclic, with a irregular period, but close to the time constant of the synchrotron radiation damping time. During a cycle, the amplitude of the SCSO rose rapidly, decreased slowly and disappeared for a certain time. When the current changed, the two bunches oscillated either in phase or in opposite phase, although their frequency of oscillation remained approximately the same for a given current range. The measured oscillations had frequencies higher than the theoretical synchrotron frequency, similar to the situation before the new cavity was installed, but for a narrow current range, their values were close to that of the expected synchrotron frequency. Under these particular conditions the amplitude of the oscillations were larger and of opposite phase. This change in phase of the oscillation of the two bunches can be ascribed to a higher order mode (HOM) of the 500 Mc s\ cavity. The large oscillations at the synchrotron frequency can also be ascribed to the HOMs of the new cavity. When the stored current decreased, the adjustment of the plunger position to tune the 500 Mc s\ cavity, induced changes in the resonance frequency of some HOMs of this cavity, which will eventually become close to a harmonic of the rotation. A small improvement in the lifetime of the stored current can be noticed in the presence of the new cavity, probably because of the presence of cyclic instabilities. These instabilities are probably the consequence of the formation and vanishing of micro-bunches, because their formation is linked to the potential formed on the parasite resonant element. Their gradual vanishing is concomitant to the appearance of the coherent synchrotron radiation produced by the micro-bunches. At very low stored current, when the 500 Mc s\ cavity was powered and developed a potential of 250 kV, the shortening of the bunch was close to the predicted behavior. Above a threshold value of the current, the two bunches oscillated at a frequency smaller than the predicted synchrotron frequency. They oscillated with opposite phases and with an amplitude that strongly increases with current.
2. Measurement methods Measurements were performed at the exit window of a bending magnet, where the horizontal dispersion was 0.3 m. At this point one can simultaneously detect the variation in phase and energy of the beam, whose transverse dimensions are px" 370 lm and pz"270 lm. The image of the beam is formed on the photodetector, with a magnification of one, using an optical configuration consisting of one spherical and one flat mirror. All measurements were performed when the ring was running with two bunches circulating at the opposite of each other. Before the 500 Mc s\ cavity was installed, two measuring systems were used to study the synchrotron instabilities. In the first system we used a fast photodiode (8 Gc s\ band pass) with an active surface of 70;80 lm. With this photodiode and a spectrum analyzer, we could show that the SCSO were dipolar and in phase on both bunches and with a constant amplitude, for a variety of synchrotron frequencies which depended on the stored current. The second system was simpler, it consisted of an array of photodiodes, and an analog oscilloscope. The photodiode array was of a smaller band pass (20 Mc s\) but sufficient to see both bunches independently. The photodiodes were 0.9 mm wide, separated by 0.1 mm, and 2.4 mm high. By using an oscilloscope we visualized the modulation in energy of the particles in the bunch. The results obtained with the second system corroborated the results obtained with the first system. After the installation of the 500 Mc s\ cavity, we used only the second system for measurements because it could better detect the rapid cyclic variations of the synchrotron instabilities of the bunches.
3. Results 3.1. Measures performed before the installation of the 500 Mc s\ cavity When considering all the synchrotron bands around a higher harmonic of the frequency of rotation, one can see that the amplitude of each band
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Fig. 1. (a) Bessel functions for the first 3 orders. (b) Bessel function for the J order at the toric resonance frequency.
corresponds to a Bessel function: J ( ) corresponds to the harmonic of the frequency of rotation (zero order), J ( ) to the first pair of synchrotron bands, J ( ) to the nth pair of synchrotron bands, where L
is the argument of the Bessel functions, that corresponds to the peak amplitude of the phase oscillation at a given harmonic of rotation (Fig. 1). The higher the harmonic of observation the easier will be to detect the synchrotron bands around each harmonic of rotation because they will rise from the noise level. This is particularly true for the results showed in Fig. 2. The first two side bands that appeared for stored currents of 172 and 87 mA corresponded to the real synchrotron frequencies of 39 and 27 kc s\, instead of the predicted frequency of 14.5 kc s\ for the Super-ACO ring. All other higher order side bands were harmonics of these first frequencies, as previously reported [1]. The first two bands on each side of the rotation band do not have the same amplitude because they were recorded with a photodiode positioned at the hori-
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zontal end of the beam image corresponding to the exterior of the ring, as described in Ref. [1]. The modulation in the signal amplitude is reflecting the modulation in energy, the latter out-of-phase by 90° from the phase modulation. Because the oscillations were small, modulations of both phase and amplitude were sinusoidal and therefore all pairs of bands should have shown different amplitudes. Still, because the horizontal movement is larger than the transverse dimensions of the photodiode this distorts the sinusoidal signal amplitude, and therefore all band pairs of the spectrum, the first and the following, have their amplitude distorted. We also presented previously the spectra of synchrotron oscillations around the even and odd rotation harmonics [1]. We noticed that the synchrotron bands and the rotation band were not sufficiently decreased for odd harmonics. According to theory, when two bunches are revolving at the opposite of each other, and when the synchrotron oscillations are in phase and are of the same amplitude, for odd harmonics the amplitude of the rotation band and the synchrotron bands should depend only on the difference in the current between the two bunches. For all results presented in this paper the difference in the current between the two bunches was much smaller than 1%. The presence of bands at odd harmonics can only be explained by a phase difference between two groups of micro-bunches which have replaced the two initial bunches. This phase difference corresponds precisely to one period among micro-bunches, or a multiple of it. The higher the rank of the harmonics, the smaller becomes the difference between the odd and even harmonics of the rotation (Fig. 3). Furthermore, somewhere between the rank 722 and 1442, the amplitude of odd harmonics of rotation becomes even larger than the even harmonics (Figs. 2 and 3). Therefore the two bunches cannot be exactly opposite to each other (180°). This strengthens our hypothesis on the presence of a single resonant element which is inducing the described dephasing, the formation of microbunches, and the advent of SCSO. This resonant element, that has a high quality factor, is located either in the main cavity, as a HOM, or in the vacuum chamber. It permits to attain the longitudinal toric resonance. It has a resonant frequency
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Fig. 2. Spectral analysis of the spontaneous synchrotron side bands for different harmonics of rotation and stored current (I). (a) Harmonic 1442, I"99.6 mA. (b) Harmonic 1443, I"99.6 mA. (c) Harmonic 1442, I"119 mA. (d) Harmonic 1443, I"119 mA.
very far both from the harmonic of the main cavity, and from the harmonic that corresponds to the first zero value of the J function. Still, this single reson ant element must have a frequency that is a multiple of the main cavity frequency which is the 24th harmonic of the rotation. This is the element that induces the synchrotron instabilities in the ring functioning with 24 bunches [2].
For the values of current stored below 110 mA, where the amplitude of the oscillations decreases, the spectral analysis of the synchrotron bands shows that they are divided into fine bands, as described in Ref. [3] (Fig. 4). These fine bands are the consequence of the presence of micro-bunches. From the spectral analysis of the synchrotron band which has the largest number of fine bands, and
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Fig. 3. Spectral analysis of the even and odd harmonics and their first spontaneous synchrotron bands. (a) Harmonic 239, I"100 mA. (b) Harmonic 240, I"100 mA. (c) Harmonic 721, I"125 mA. (d) Harmonic 722, I"126 mA.
taking into account the length of the bunch measured independently for the given current, one can conclude that the distance among the microbunches is of the order of 11 ps. This corresponds to a frequency of 91 Gc s\, which is close to the first toric longitudinal resonance of the Super-ACO vacuum chamber. It is important to mention that the toric resonance depends on the simultaneity of the propagation rate of the phase of the longitudi-
nal field and the propagation rate of the microbunches that have created this field [4]. With the changes in stored current the multiple band appear, disappear and reappear again, under consecutive ranges of current. For each range where they are present their number remains constant, but among different current ranges the number of multiple bands changes in a stochastic way. Each range where the multiple bands are present
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Fig. 4. Spectral analysis of the fine bands of the synchrotron frequency F at the harmonic 1442 of rotation. (a) I"100.8 mA, F "27 kc s\. (b) I"38.9 mA, F "42.41 kc s\. (c) I"36.6 mA, F "41.58 kc s\. (d) I"34.8 mA, F "41.43 kc s\.
would correspond to a toric resonance frequency of the ring. The changes in the dispersion in energy of the particles in a bunch, which are dependent on the changes in current, lead to a change in orbit length of 0.19 mm and an apparent *E/E of 1.8;10\. The toric resonance frequency is very sensitive to this length variation [4] and can lead to change from one even harmonic of rotation to another [3].
At 91 Gc s\, the relative change in toric frequency will be 9;10\. It implies that the selectivity of the toric resonance frequency has to be much smaller than 10\. These conditions are too restrictive for this type of resonance. The orbit length variation of 0.19 mm corresponds to a variation of the toric resonance frequency of several hundreds Mc s\. The amplitude of the phase oscillation at the latter frequency leads to the oscillation of the function
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J (Fig. 1). The consequence of the above is the successive formation, disappearance and reappearance of the fine synchrotron bands. One can notice that, although the global dispersion of synchrotron frequencies remains approximately constant for any number of fine bands, the gap between them is self-adjusted to remain constant. The formation of micro-bunches, and the presence of fine synchrotron bands linked to two distinct resonant cavities, can be described in two different ways, as shown in Fig. 5. The classical explanation assumes that all micro-bunches have the same energy, and that they are aligned at a potential corresponding to the energy loss per turn. The slope that each micro-bunch sees is different. They have, therefore, different synchrotron frequencies which are not equally distributed. The current of each micro-bunch induces an increase in potential in the parasite resonant element, therefore an increase in the slope seen by the micro-bunch, which will then have a frequency of synchrotron oscillation higher than the preceding micro-bunch. All these conditions together should explain a constant gap among frequencies of the fine synchro-
Fig. 5. The interaction of the micro-bunch with the RF voltages. The sum of two RF voltages comprising the fundamental (» ) and its 100th harmonic (» ) is plotted against time. The horizontal line º corresponds to the voltage that compensates for the energy loss on each turn. One can see that the slope of voltage seen by each micro-bunch is different when the bunches have the same energy. By contrast, on the oblique line representing the voltage » , the slope seen by each bunch remains constant only when they have different energies.
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tron bands, but still we think it is not plausible for all cases when synchrotron fine bands appear. By contrast to the previous hypothesis, we can also postulate that micro-bunches have each a different energy, and that each of them is composed of particles with a small energy dispersion. (The hypothesis of bunches circulating with different energies applies only to almost isochronous rings where the momentum compaction is close to zero [5]. This is not the case for Super-ACO in which the momentum compaction is 1.48;10\.) The different micro-bunches are aligned on the potential delivered by the main cavity. The slope of the potential seen by all micro-bunches is identical. Each micro-bunch will have a different rotation frequency. The separation among these frequencies corresponds to the separation among fine synchrotron bands. One should then be able to see fine side bands around the line of the rotation harmonic. Their number and separation in frequencies is the same as for fine synchrotron bands. This is actually the case shown in Fig. 6. We can detect the frequency corresponding to this separation among the fine bands if we add a peak detector on the signal coming from the fast photodiode, which is composed of pulses modulated in amplitude. These pulses represent the passing over of groups of micro-bunches. The synchrotron oscillation is seen
Fig. 6. Spectral analysis of the fine bands of the synchrotron frequency (F ) and of the fine bands of the harmonic 1442 of the rotation frequency, for I"48.8 mA.
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Fig. 7. Double modulation of the signal amplitude coming from the fast photodiode for I"69.4 mA, F "27 kc s\, and the distance among the fine synchrotron bands equal to 622 c s\.
as the amplitude modulation of the pulses. The amplitude modulation of the synchrotron oscillation corresponds to the identical frequency gap among the different frequencies of the microbunches rotation, i.e. to the beating at this frequency (Fig. 7). We can conclude that the dispersion of frequencies for a number of fine synchrotron bands does not correspond to a real dispersion in particles energy of the “bunches”, as measured before for different currents. This is possible because, at the toric resonance frequency, each micro-bunch follows the field that it has created. This is not the case with a resonant cavity positioned in the ring. Above 110 mA, the dispersion and oscillation in phase and energy become too large, they cover a large number of harmonics of rotation and therefore the fine bands are not visible any more (Fig. 8). 3.2. Measures performed after the installation of the 500 Mc s\ cavity (a) ¹he cavity is used in a passive mode. The plunger is inserted to change the tune of the 500 Mc s\ cavity further away from the 120th harmonic of rotation. The two “bunches” show cyclic SCSO with a period close to the time constant of the longitudinal radiation damping time, which is 10 ms. The amplitude of the SCSO in-
Fig. 8. Spectral analysis of the spreading of the lower spontaneous synchrotron band at the harmonic 1442 of rotation. I"126 mA, F "38 kc s\.
creases rapidly, followed by a slower decrease. Then, the SCSO disappear all together for a long period of the cycle (Fig. 9). The origin of this instability will be discussed later. When the stored current is decreasing, the SCSO show frequencies and phases which depend on the range of the current present. From 200 to 100 mA, the two “bunches” oscillate in phase. Their frequency of oscillation, 39 kc s\, is identical to the one measured before the installation of the 500 Mc s\ cavity (Fig. 10). Still, in a small region, between 150 and 143 mA, the amplitude of the oscillations increases and their frequency, 13 kc s\, becomes smaller than the theoretical synchrotron frequency and the two bunches oscillate in opposite phases (Fig. 11). This is probably linked to the movement of the plunger which is designed to tune the frequency of the 500 Mc s\ cavity and acts on the frequency of a HOM of the cavity. From 100 to 80 mA, the frequency of oscillation is close to 26 kc s\, and the two bunches oscillate again in opposite phase (Fig. 12). This behavior is probably also due to a HOM of the 500 Mc s\ cavity; the latter can be different from the one mentioned above. (b) ¹he cavity is powered and has a potential of 250 kV at 500 Mc s\. The length of the bunch should be shorter under this condition by a factor of
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Fig. 9. Cyclic instabilities of the SCSO, F "38 kc s\. (a) I"117 mA. (b) I"112 mA.
Fig. 10. Measurements of synchrotron oscillations. Peak modulation of the two signals of two adjacent slow photodiodes centered in the beam image. The energy transverse movement of the bunches in the bending magnet is presented, with one signal inverted (lower trace) and the sum of both signals present in the middle. Stored current I"180 mA, F "38.5 kc s\.
2.7, and the synchrotron frequency should rise to 39 kc s\. This was actually the case only at very low current. Above the threshold of 1 mA, the SCSO of the two bunches appear. They have a constant amplitude, and are opposed in phase, probably because of a HOM of the added cavity. The amplitude of SCSO increases rapidly with the increase in stored current. The frequency of oscillation of the two bunches under these conditions, 33 kc s\,
Fig. 11. Same as Figs. 9 and 10 for I"145 mA, F "13.3 kc s\.
is lower than the expected synchrotron frequency. For 23 mA of stored current, the amplitude of oscillation in energy becomes 1% peak-to-peak (Fig. 13). The installation of the 500 Mc s\ cavity fulfilled only partially the expectations concerning the bunch shortening. Still, it could be improved by adding either of the following: an appropriate system of feed-back; a selective tuning of the HOMs, to separate them from the harmonics of rotation [6]; selective damping couplers, to reduce the effects of the HOMs [7]; a single trapped mode resonator, comprising large wave guides that would
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damp the HOMs. The latter was proposed and checked on a prototype [8].
4. Discussion
Fig. 12. Same as Figs. 9 and 10 for I"88 mA, F "26 kc s\.
The discussion thereafter is related to measurements performed on Super-ACO compared to other storage or damping rings, because we believe the detected cyclic instabilities have a similar origin. Interpretation of the cyclic regime of SCSO. The potential induced by the bunch on the high frequency parasite resonant element modifies the density inside the bunch. When the potential is sufficient the formation of micro-bunches occurs. This leads to a modification of the spectrum and induces
Fig. 13. Same as Fig. 10, but with the 500 Mc s\ cavity powered at 250 kV. (a) I"20 mA. (b) I"22 mA. The time axis is expanded to be able to observe the movement of the two bunches separately. (c) I"23 mA.
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an increasing potential on the parasite resonant element. The self-formation of micro-bunches will proceed round after round. The phenomenon will be accompanied by an apparent lengthening of the bunch, because of the filling of some of the “empty buckets” on both sides of the ones already present [9]. This is seen as an apparent lengthening of the bunch because the band-pass of the measuring device is not sufficient to see the micro-bunches. This self-consistent phenomenon will be limited as soon as micro-bunches are totally formed, which corresponds to the presence of maximal power at the resonance frequency of the parasite element. If the length of the micro-bunches is sufficiently short and corresponds to the wavelength lower than the cutoff wavelength of the vacuum chamber of the bending magnet, the micro-bunches will produce coherent synchrotron radiation at this wavelength. The radiation emitted will be much stronger (proportional to the square of the number of particles) than the incoherent synchrotron radiation that would be produced at the same wavelength by a bunch of the global length corresponding to the sum of all micro-bunches that composed it. The presence of coherent synchrotron radiation produced by each micro-bunch leads to a decrease of their length and a modification of their shape [10,11]. This self-consistent effect is cumulative and is adding to the effect which formed the microbunches. With the shortening of the micro-bunches, the incoherent synchrotron radiation is transformed into coherent radiation of progressively shorter wavelength. This brings an important loss of energy for all particles of each micro-bunch that have emitted coherent radiation. If the potential well produced by the parasite resonant element is sufficient, the effect of shortening will be accelerated, and still shorter wavelength (therefore of higher energy) will be added [10]. The following two phenomena will then occur: (1) If the parasite resonant element is not tuned to the harmonic of rotation, the micro-bunches will show synchrotron instabilities. The shorter they become the larger will be the amplitude of phase oscillation. This will lead to a limitation, or even a decrease in amplitude of the rotation harmonic line which corresponds to the resonance of the idle
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cavity. Under these conditions the voltage that has created the micro-bunches is limited and can even decrease. (2) The potential wells are not sufficient to keep all the particles in each micro-bunch. The particles that have emitted too much energy, in a coherent way, will leave the bucket and will start to follow complex trajectories around some other buckets, i.e. some other micro-bunches, which will lead to their gradual destruction. The Fig. 14 shows an example of trajectories of particles that have experienced a large change in energy and therefore are crossing the separatrixes of the different buckets. Each bunch has a strong dispersion in energy and keeps a length close to the apparent length of the bunch following micro-bunch formation. This cyclic phenomenon could be repeated only when the damping via synchrotron radiation restores back the initial characteristics of the bunches both in length and energy dispersion. During the time of synchrotron damping, which corresponds to the half of the cyclic instability, the micro-bunches disappear, the bunches become longer, and their transverse dimensions increase. All
Fig. 14. Example of a system of separatrixes obtained with two cavities having identical voltage. The frequency of the second cavity was chosen to be ten times higher than the first one in order to illustrate clearly the complexity of the particles moving outside the buckets.
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these modifications are the consequences of an increase in energy dispersion among the particles. The Touchek effect is decreased, and therefore, the lifetime of the beam is increased. In addition, the losses are decreased in the vacuum chamber and at higher modes of the cavities, which brings to a change of phase of the bunches compared to the phase of the main RF. This type of cyclic instabilities, named sawtooth instabilities, have been observed in different storage and damping rings. However, the conclusions drawn from the measurements were not always the same. Concerning Super-ACO, the installation of the 500 Mc s\ cavity in a passive mode leads, paradoxically, to a shortening of the bunch, although one would expect a slight lengthening under these conditions. A HOM of the cavity is tuned to a harmonic of the rotation by changing the position of the plunger. The induced potential on this HOM shortens the bunches [12]. The end result are shorter micro-bunches than the ones before the installation of the 500 Mc s\ cavity. Shorter micro-bunches produce coherent synchrotron radiation of multiple wavelengths which induces sawtooth instabilities. On the ELECTRA ring a cyclic instability with a sawtooth form was also present. It is a longitudinal oscillation whose amplitude increases rapidly, then decreases slowly. Concomitantly to the decrease in amplitude of oscillations, the dispersion in energy of the beam is increased and then damped by synchrotron radiation [13]. This behavior can be interpreted as the presence of micro-bunches which are destroyed by their coherent radiation, as described above. The coherent synchrotron oscillation phenomenon has also been observed on the BEPC ring when a single bunch was stored, with a frequency roughly equal to the triple of the synchrotron oscillation. A dipolar movement can be responsible for the same frequency of oscillation. The pulse peak of the bunch remains constant. Sometimes, the pulse lengthens and the peak amplitude decreases during the observation, which can also be an indication of the formation of micro-bunches. When this phenomenon occurs in the presence of a bunch of electrons and a bunch of positrons, an abrupt decrease in luminosity (20%) is seen [14]. If we postu-
late the presence of micro-bunches under these conditions, because of the b function, only the micro-bunches at the point of interaction will be seen by the luminosity detector. On the rings SOR and SURF II, instabilities with a sawtooth form have been observed for a long time [15,16]. In SOR the cyclic instabilities were eliminated efficiently and at the very beginning by the addition of a Landau cavity. This cavity will eliminate the cyclic instabilities linked to the potential presence of micro-bunches by spreading out the synchrotron frequency, but this leads simultaneously to a lengthening of the bunch and to a disappearance of the micro-bunches. Old measurements on SURF II have shown that during SCSO the two bunches were lengthened ( 30 cm). Paradoxically, recent measures [17] showed that coherent synchrotron radiation appears during the lengthening at wavelengths between 2 and 4 cm. Measures of the spectrum of coherent radiation showed that the contours of the different bands of the rotation harmonic measured during the fast increase phenomenon has a modulation in amplitude similar to the one presented in Appendix A. This seems to confirm the presence of a few microbunches as short as 2 cm, which would explain the seen lengthening. Unfortunately, the measured frequency bands in the coherent radiation spectrum did not reveal the frequency of the parasite resonant element which induced the formation of microbunches. The first frequency maximum, which would represent the inverse of the repetition time period of the micro-bunches was outside the spectrum shown. The SCSO decreased slower than they increased, and then they disappeared. Concomitantly to the decrease, the coherent radiations gradually disappeared and the dispersion of energy became large. The end of the cycle, comprising the synchrotron radiation damping time of the particles, showed a decrease in energy dispersion concomitant to a shortening of the bunches. All the measurements performed on the SLC damping rings, both with the old vacuum chamber and after its modification, showed the same type of sawtooth instabilities. Very different types of measurements are all in line with the hypothesis of the presence of micro-bunches. The lengthening of the bunches is accompanied, after a delay, by
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a smaller dissipation of energy [18]. The measured frequencies of the coherent synchrotron oscillations [18,19] look like the SCSO on Super-ACO. The position chosen for measurements led to different interpretations, although the presence of microbunches was again very probable. The first frequency of longitudinal toric resonance of the rings (at 193 Gc s\) [4], would imply maximally 5 ps between different micro-bunches. Therefore, each bunch would be composed of 15—20 micro-bunches. Their presence is suggested in the authors’ statement: “Turbulent bunch lengthening is often referred to as the microwave instability because the bunch length oscillation modes have a characteristic wavelength at some small fraction of the bunch length.” [18]. Cyclic instabilities were observed on all cited rings. The cycle begins by the SCSO whose frequency can be different from the synchrotron frequency. Their amplitude increases rapidly accompanied by an apparent lengthening of the bunches. Then their amplitude slowly decreases. The bunches maintain the same length but with a strong dispersion of energy. The remaining of the cycle comprises the reconstitution of the initial characteristics of the bunch length and energy dispersion by synchrotron radiation damping. The only hypothesis that can explain the described phenomenon is the formation of micro-bunches. This hypothesis is corroborated by the detection of coherent radiation on SURF II, the presence of SCSO on Super-ACO, and all the measurements performed on the damping rings of the SLC, in particular the decrease in energy dissipation when SCSO appeared. The remedies by which one could influence the different phases of the instability are the following: (a) To decrease the ring energy at a constant main RF potential and identical stored current. Under these conditions the bunches are shortened creating shorter micro-bunches. The synchrotron emission and the energy dispersion of the particles are decreased and the potential wells deeper. In the spectrum of synchrotron radiation numerous bands of coherent wavelength will be emitted. (b) To decrease the coherent synchrotron instabilities by modifying the frequency of rotation, in
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order to bring closer the higher harmonic to the parasite resonant circuit which contributes to the formation of the micro-bunches and to their instabilities. This represents a small change in frequency of rotation. (c) To install a special tunable cavity on the rings, passive [20] or powered, dedicated to create micro-bunches, with a new RF voltage sufficient to maintain them in the presence of coherent synchrotron radiation. The resonant frequency has to be close to the cutoff frequency of the vacuum chamber in order to eliminate the HOMs which induce the synchrotron instabilities. Under these conditions, the micro-bunches will be able to emit coherent radiation of a large number of wavelengths. The use of a super-conducting cavity will increase the efficiency and therefore will preclude several harmonics of rotation. This would be important especially for rings of larger sizes. The frequency of the cavity could be chosen to favor defined wavelengths of coherent synchrotron radiation.
5. Conclusion The installation of an additional 500 Mc s\ cavity on the Super-ACO storage ring led to a significant modification of SCSO. In the passive mode of the cavity the oscillations become cyclic and of the sawtooth type. They are similar to the ones detected on other storage or damping type rings. The comparison of the measurements performed on Super-ACO to the observations obtained on other rings allowed us to propose an explanation of the cyclic instabilities of the sawtooth type. We also suggested several means to eliminate them, with the additional bonus of producing coherent synchrotron radiation.
Appendix A The presence of micro-bunches replacing the bunches can be described as the product of two functions, G and G . First, in the equation G (t)"exp(!1/2(t/p ))
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the width of the bunch is represented by p , i.e. about 3 times p , the width of the bunch before the formation of micro-bunches which can be obtained from the equation G (t)"exp(!1/2(t/p )). The width of the micro-bunch is defined as p given by the equation G (t)"exp(!1/2(t/p )). In Fig. 15 the full line is the product of G and G , the dotted line the function G and the dashed line the function G . Fig. 16 is the representation of two groups of micro-bunches circulating into the ring and which have the same number of micro-
bunches. Fig. 17 is the square of the Fourier transform of the G ) G product. All the lines are not presented because their number depends on the choice of the number of bunches injected. The frequency difference is the inverse of the time separating two groups of micro-bunches. The Fourier transform of the function G is also presented in this figure by dots. On the spectrum presented in Fig. 17, the difference among maxima defines the inverse of the time separating the micro-bunches. The first maximum is the frequency resonance of the idle cavity. The envelope joining all the maxima represents the Fourier transform of G . The reson ant frequency of the idle cavity, or the chosen frequency of an installed cavity, will define the set of
Fig. 15. A representation of the micro-bunches. The solid line is the product of G and G , the dotted line the function G and the dashed line the function G .
Fig. 16. A representation of two groups of micro-bunches circulating into the ring.
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Fig. 17. Square of the Fourier transform of the G ) G product.
synchrotron lines, including those which are coherent. With the shortening of the micro-bunches p decreases, and the spectrum of coherent syn chrotron radiation widens towards higher frequencies i.e. towards shorter wavelengths. The Fourier transform of the distribution of the micro-bunches in time has a number of maxima of intermediate amplitude that corresponds to the number of micro-bunches minus two. The intermediate maxima are alternatively negative and positive. For example, if a parasite resonant circuit has a frequency equal to a positive maximum, then it can induce instabilities of the beam. If one adds one micro-bunch to a group of micro-bunches, the Fourier transform will also have an additional intermediate maximum. Under these new conditions, the frequency of the resonant circuit will be equal to a negative maximum or close to the zero level, therefore the instabilities of the beam will disappear.
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