SYNCHROTRON SIDEBANDS IN BETATRON OSCILLATIONMEASURED BY MEANS OF RF KNOCKOUT IN THE PF STORAGE RING
T. Katsura, Y. Kamiya and S. Shibata
National Laboratory for High Energy Physics
Oho-machi, Tsukuba-gun, Ibaraki-ken, 305, Japan
Incoherent Longitudinal Oscillation

ABSTRACT

If only the incoherent phase motion presides, Eq.

Both incoherent and coherent synchrotron sidebands
were observed in the betatron oscillation by using theThe oscillation was detected byRF knockout system.
receiving the synchrotron light from the beam to aposition sensing photodiode. The oscillation signal isResults are comanalyzed with a spectrum analyzer.
pared with a theoretical model.

(4) is described with a = ai and the ensemble averageof xi is expressed with a term only for m = n, since ai
The sideband resonances can
is randomly distributed.
be seen at the driving frequency wd = wS + nw . andare spaced by w .. The ratio of the n-tn°amplittae tothe fundamental steak (n = 0) is given by.)><}(~A/vnINTRODUCTION

s~<} (~A/v . )>The betatron sidebands in the PF storage ring wereobserved by use of the RF knockout (RFKO) system, which
had earlier been introduced for the measurement of tune
By exciting the beam with transverse
parameters.
magnetic field, we observed transverse beam oscillations with a position-sensing photodiode array which
detected the synchrotron light from the beam. · The
photodiode signal was fed to a spectrum analyzer toobtain the frequency distribution. Both incoherent andcoherent longitudinal oscillations were distinguishedin the distribution and their amplitudes were comparedwith theoretical predictions. Also their chromaticity
dependence was obtained to test the theoretical model.

o+1((5)s~

Coherent Longitudinal Oscillation
If only the coherent phase motion exists, Eq. (4)in commonis interpreted with a = S and 6w/ w = $/ v
for every particle and all the sidJbands ca~cbe excitedby driving any one of the resonances. For the driving
wSo + nwsc, the spectrum includes two
freuqnecy wd
components:

Theoretical foundation on the subject is given ina paper by D.E.P. Mohl and P.L. Morton!. They indicatethat it should be possible experimentally to obtain
both incoherent and coherent longitudinal oscillation
frequencies by measuring the coherent transverse oscillation response.
Equation of Motion

andAn off-momentum particle with momentum p = p + 6p
will have a shift in the betatron frequency wS gi~en byl<x(n)>l n cr <J n2 (~B/v sc )> at wd

Usually ~Bhence dominate(1)with ~ = (p/w )(aw /3p) and revolution frequency w as
well as a loJgitu~inal phase oscillation of frequ~ncy
21rf .
w
Thes equation of transverse motion for the ~-thsparticle is described as<<v

scThefundamental

modewill(6) •EXPERIMENTAL SETUP
Figure 1 shows a block diagram of the experimental
setup. The system has already been used for measuring
Only atune parameters during operation periods 2 .
The observation
brief description is given below.
system consists of two parts, one to excite the beamand the other to detect its oscillations.
If the excitation frequency is, for example, set
at f *6Q where f is the revolution frequency and 6Q isthe fractional p~rt of the betatron wave number Q, thedetection system gives a peak at the chosen frequency.

(2)

The right hand side is the driving force from the RFKO
at frequency w •The longi'\:udinal motion of the 9,-th particle isgiven by

where A , w . and a,e, are the amplitude, frequency, andphase of thJl~incoherent longitudinal motion of the 9,-thand S are of the coherent
particle and where B, w
motion in common for allsfhe particles. Using (1) and(3), the solution of Eq. (2) is approximately given by

RF Knockout System
FourA RFKO system is used to excite the beam.
rods placed parallel tq the beam are driven with theirindividual current sources to make a magnetic field
The phase relation among thetransverse to the beam.
rods decides the direction of the field in either
horizontal or vertical. The signal generator produces
continuous sinusoidal signal at a fixed frequency with
an amplitude suitably adjusted for the excitation ofCare must be taken not to give excessive
the beam.
driving power and force the beam to oscillate at any
given frequency, which is named here "through".

FJn(6w/ws)

where expression 6wSi!n. ~ :€)2n2

and approximated for x << 1.

THEORETICAL MODEL

2wSo(wSo + nws- wd)

s~with v . = w • I w
rs~s1.
Assuming the 6p/p distribution to be Gaussian andsetting x = ~a /E v . with a /E = (6p/p) r.m.s., the€s~ratio is given €X

6w cos(wst +a) is used.

286

~~~

1 ~

DCL

. fS1GNALI~ ~~Y RFKO RODSSTORAGE RING

Typical spectrum taken with the driving
frequency at 626 kHz and the generator power
level at 100 dB. The scope settings are Hor.
200 kHz/div. and Ver. 10 dB.div.

Fig. 2

Fig. 1

Experiment al setup: RF knockout system with
its signal generator at the top and theanalyzing system with a position-se nsing
photodiode at the bottom.

obtain a peak above the background , the excitation
power level often had to be adjusted depending on where
the peak is; when it is in the hill region of the background, the driver power level must be raised comparedDue to this background , we often
to other region.
observed the beam being forced to oscillate at any
frequency. When this much power level was required, we
also took data called "through" by shifting the driving
frequency aside so as to later subtract the "through"signal from the measured signal at the aimed sideband.
The peaks obtained above with different driver
power levels are normalized to one specific value of
power level and corrected for the frequency response of
the RFKO and photodiode system.
The data are taken for different current values
with both uniform and partial fillings of RF buckets2.
Figure 3 shows a spectrum for sidebands obtained withbeam partially filled and its current of 120 mA. For
f , peaks are seen clearly at ±f and +2f but the peaka£ -2f seems to be hidden in t«e backgr5und hill. Forf , si'aebands up to ±2f are clearly seen. But theyb~rely stick out of the ~ackground at ±3f • Only theirs
upper limits are shown with arrows.
Theoretica l values of spectrum strengths are also
shown in dB with respect to the main peak with the mode
number attached.
In this high current data, the first sidebands areespecially broadened in both horizontal and verticaldirections , which is indicated with triangular peaks in

Oscillation Detecting System
To detect the oscillation of the beam, the synchrotron light from the beam was led to a photodiode on
which the beam image is focused with optical lenses.
The photodiode detector is an array made up of four
Four outputs from the diode are orderly
quadrants.
added and subtracted in an analog amplifier to separate
both horizontal or vertical signals of the oscillating
beam.
Either of the two signals is chosen correspondi ng
to the direction of excitation field and fed to thespectrum analyzer.
Calibration of System
As we have to compare the strengths of sidebands,
we calibrated both RFKO and detection systems over therange of frequencies necessary for the measuremen t.
The RFKO system has irregular response for such a wide
The frequency response of the RFKO
frequency range.
system is taken into account in the data processing .
The detector system, on the contrary, shows much
simpler frequency response which varies only monotonically within 10% over the frequency range concerned.
MEASUREMENTS AND RESULTS
Three kinds of measuremen ts were made. First, theandbeam was excited at betatron frequencies f , f
their sidebands stepped by ±nf , Second, tlteir ychromaticity dependence was observgd at f ±f by changing
the current of the defocusing sextup~le magnets.
Third, there are such cases that the main peak is enhanced more strongly than the sideband at a new driving
frequency differing only by a few kHz from that used inThis is the casethe first measuremen t made above.
when the longitudin al phase oscillation is coherent.

120 mA
BEAM CURRNTPARTIAL FILLING0(:x =- 9.6

ey = 8.5

Efyn•Ofx,----·-O dBr----CD"0n=O--OdB-fsw -50

0-fs

::::>

::::'_ja..

Betatron Sidebands Due to Incoherent Longitudin alOscillation

-3fs

::;;:
<[Figure 2 shows a typical picture of the spectrum
taken with the spectrum analyzer. The excitation frequency was set at the horizontal betatron frequency f .
Four sharp peaks are seen sticking above two oth~rsmooth hilly peaks wide as 200 kHz or more, which isbackground of the photodetec tor system including theamplifier. The peak at the lefthand side is the "zero"
The second sharp peak is the mainfor the scanning.
peak at f and the third one is also the main peak atf -f . Tlte right most peak gives merely f •r xThe excitation frequency was changed Ey step of fand sideband peaks are searched near at each step. Tos-100

0Fig. 3287

IT

200+fs.!l:..!__ 37+2fs +3fs

I+fs~-35+2fsn•2 -76n•2 -80
600400FREQUENCY (kHz)800

Synchrotron sidebands of f and f taken withbeam current of 120 mA. Tfleoretita l values areshown with horizontal broken lines for mode n.
Broad peaks are shown in triangular shape toindicate their widths.

the figure. No broadening was, however, observed in
low current data. This fact hints there may be iontrapping3. But, it would also be possible to attributeit to so called "758" instability pertinent to the RFsystem.
Further study will be needed to determine
which is causing the broadening. An independent observation of ion-trapping is discussed in another paper
submitted to this symposium 3 •Table 1
Coherent sideband amplitudes measured at f and f ±nf when the driving frequency was set at fdy= f ±tf
fo~ n = 1, 2 and the chromaticity E; = 8. ~Th~
definition of R(f) is given in the text. Theoretical
calculation is given for the amplitude a = 1.5 ~ 0.2.
Symbol * indicates that the data was taken with RFphase modulation on.

Chromaticity Dependence

MODE NO.

To test the theoretical model, the strength offirst sidebands were measured for different values ofchromaticity. Figure 4 shows the amplitudes of firstvertical sidebands f± f
when the chromaticity
E;
was changed. Theo¥eticalsvalues from Eq. (5) aresKown with solid lines.
The amplitude of f± f
are normalized to the amplitude of the main peakY.
sCalc.

R ( fy)dB

I---381! -36---76-37-18
Expt.*-23
1--'---+---· .
Expt.
-44
Cole.
-42...., -77
n=2 1-E-xp-t.-+-----33··
In =IR(fy ±nfs) dB--··---=c----+---~·------!

. --·-------51-84,...., -154·····-_-57--

1 FillUNIFORM
...

3.0
4.2

PARTIAL

17.5

UNIFORM

0ID
"0>.

-·

-20
Calc.

~
<(.......

---40XX1/)

+I

:>.

0
6.

-60

6.
0

<('----'c::r>

.3-80

0N

Fig. 5-100-10

Fig. 4-8-4

4810

within the parameter a = 1.5 - 0.2. Data with n = 2
exhibit some tendency similar to those with n = 1 but
do not agree with the prediction to the same extent.

Chromaticity dependence of sidebands f ± ftaken with uniform or partial beam filiing.s
Theoretical values are shown with solid curves.
Amplitudes are shown by assigning them toA(f ) = J<x(O)>J and A(f ± f ) = J<x(1)>J in
Eq.Y(s).

YSpectrum with the driving frequency at
293.8 kHz (f + f ) while RF phasemodulation i~ app~ied.

ACKNOWLEDGEMENTS.
We are indebted to greatly thank Prof. K. Huke,
head of the light source division for his useful
discussion. We also express our gratitude to the crew
for operating the storage ring and their invaluable
cooperation.

sBetatron Sidebands Due to Coherent Longitudinal
Oscillation

REFERENCES

According to the theoretical consideration, it ispredicted there exists sideband frequencies which enchance the main peak higher than the sidebands at thedriving frequency. Searching carefully near the sidebands, we found that the main peak pronounced higher
than the sidebands at frequencies differing by a few
kHz from that of the incoherent case. Table 1 summarizes these cases. Mode number n in Column 1 indicatesthe driving frequency is set at f.d = f 8 + nf • Twokinds of spectrum amplitudes are taKen at freque~cies f= f 8 and f& ± nf and given as R(f) in Column 2 and 3.
All the ampLitudls (in dB) are normalized to the amplitude of the main peak taken with the driving frequency
at fd = fe.
Data are also taken with RF phase modulation
turned on. The RF phase was modulated with a 56 kHz cw
source 5 • Figure 5 shows the spectrum with the l!lain
peak as well as sidebands at f± f • Results arelisted in the table with symbol ~ to c8mpare with data
without the phase modulation.
By defining a as the amplitude normalized to cr /E,
theoretical predictions from Eq. (6) are also added tothe table. They lie near to the experimental results

1.
2.
3.
4.
5.

288

D.E.P. Mohl and P.L. Morton, PEP-68, 1973 PEP
Summer Study (1974).
PHOTON FACTORY ACTIVITY REPORT, p.IV-27 andp.IV-42 (1982/83).
M.E. Biagini et al., Observation of ion-trapping atADONE, 11th International Conference on High Energy
Accelerators, CERN, p.687 (1980).
Y. Kamiya et al, Vertical instability caused byion-trapping in KEK-PF, contributed to thissymposium;
H. Kobayakawa, Y. Yamazaki and Y. Kamiya, KEK-83-8(1983).