Beam instrumentation - Home - Walter Scott, Jr. …projects-web.engr.colostate.edu/.../AY14/beam/LittauerAPC000869.pdf · Raphael Littauer Cornell University Ithaca, New York 14853

Beam instrumentationRaphael Littauer Citation: AIP Conf. Proc. 105, 869 (1983); doi: 10.1063/1.34244 View online: http://dx.doi.org/10.1063/1.34244 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=105&Issue=1 Published by the American Institute of Physics. Related ArticlesNew Products Rev. Sci. Instrum. 83, 029501 (2012) Code-division multiplexing for x-ray microcalorimeters Appl. Phys. Lett. 100, 072601 (2012) Development of an alpha/beta/gamma detector for radiation monitoring Rev. Sci. Instrum. 82, 113503 (2011) Note: Continuing improvements on the novel flat-response x-ray detector Rev. Sci. Instrum. 82, 106106 (2011) Detection efficiency vs. cathode and anode separation in cylindrical vacuum photodiodes used for measuring x-rays from plasma focus device Rev. Sci. Instrum. 82, 103507 (2011) Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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BEAM INSTRUMENTATION

Raphael Littauer

Cornell University

Ithaca, New York 14853

869

TABLE OF CONTENTS

I. BEAM SPECTRA . . . . . . . . . . . . . . . . . . . . . . . 872

II.

III.

IV.

I. Single Particle on Central Orbit . . . . . . . . . . . 872 2. Betatron Oscillations . . . . . . . . . . . . . . . . 875 3. Momentum Spread . . . . . . . . . . . . . . . . . . . 876

4. Bunched Beam . . . . . . . . . . . . . . . . . . . . . 878

5. Coherent Motion of Many Particles . . . . . . . . . . 882

6. Multi-Bunch Modes . . . . . . . . . . . . . . . . . . 892

7. Coherent Colliding-Beam Modes . . . . . . . . . . . . 893

SIGNAL PICKUPS . . . . . . . . . . . . . . . . . . . . . . 897

i. Some Specific Pickup Configurations ......... 898 2. Beam-Position Measurement . . . . . . . . . . . . . . 907

SIGNAL PROCESSING . . . . . . . . . . . . . . . . . . . .

i. 2.

3.

4.

5. 6. 7.

8.

910

General Comments . . . . . . . . . . . . . . . . . . . 910 Narrow-Band Processing . . . . . . . . . . . . . . . . 913 Broad-Band Processing . . . . . . . . . . . . . . . . 918

Beam-Position Measurement . . . . . . . . . . . . . . 922

Spectrum Observation . . . . . . . . . . . . . . . . . 925 Spectrum Analyzers . . . . . . . . . . . . . . . . . . 927 Lock-In Tune Measurement . . . . . . . . . . . . . . . 931 Beam Transfer Function . . . . . . . . . . . . . . . . 932

SYNCHROTRON RADIATION . . . . . . . . . . . . . . . . . . 942 i. Performance Data . . . . . . . . . . . . . . . . . . . 942

2. Utilization . . . . . . . . . . . . . . . . . . . . . 944

3. Limits of Resolution in the Image . . . . . . . . . . 946 4. Detection Equipment . . . . . . . . . . . . . . . . . 949

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 952


870

It is evident that instruments which detect, locate, and quantify

the beam of accelerated particles play a vital role in the running of

any accelerator. They monitor the results achieved and contribute

toward the diagnosis of any problems encountered. These instruments

can be classed together as beam monitors. Their development has ben-

efited from much effort and ingenuity, and the techniques they employ

span a wide range. This paper makes no pretense of providing complete

coverage, nor will it deal with specific circuit implementations or

styles of instrumentation, which in any case evolve rapidly as the

state of the art advances. Instead I will consider some of the basic

aspects regarding the central sector of beam monitoring, namely, the

detection and processing of the electromagnetic effects produced by

the beam's charge. This leaves to one side the information which can

be obtained through the beam's interaction with targets, e.g., with the

residual gas, gas jets, scanning wires, thin foils, scrapers or fluo-

rescent screens. It also soft-pedals what can be learned by destructive

or non-destructive examination of an extracted beam.

To set the stage, some of the beam parameters which we may wish to

monitor are listed in Table i.

The paper is divided into the following sections:

I. Beam spectra

II. Signal pickups

III. Signal processing

IV. Synchrotron radiation

0094-243X/83/I050869-85 $3.00 Copyright 1983 American Institute of Physics


INTENSITY:

POSITION:

EMITTANCE :

LATTICE FUNCTIONS:

ENVIRONMENT:

RF:

INSTABILITIES:

INJECTION/ EXTRACTION:

871

Table 1

SOME BEAM PARAMETERS TO BE MONITORED

Absolute measure

Relative between bunches

Time dependence (during acceleration or storage: lifetime)

Equilibrium orbit

Momentum dependence (dispersion)

Time dependence (instabilities, damping after stimulus)

Variation with lattice parameters (beam bumps, lens alignment)

Transverse (betatron)

Longitudinal (bunch shape)

Momentum spread

Magnetic aperture

Betatron tunes

B function

Dispersion function D

Chromaticity

Octupole moment

Coupling

Transverse impedance

Longitudinal impedance

Effect of feedback systems

Synchrotron tune

Parasitic power losses

Potential well distortion

Diagnosis of process

Stimulated instabilities

Damping times

Particle trajectories

Available aperture


872

I. BEAM SPECTRA

The signal induced in an electromagnetic pickup by the beam carries

information that we wish to extract. Some of that information is easily

accessible--for example, the amplitude of the signal, which may be a

direct measure of the total charge in the beam (the beam inter~ity).

Other types of information may be contained in more subtle aspects of

the signal, such as the detailed time structure or spectral content. It

seems appropriate therefore to begin with a summary of the signals pro-

duced by various modes of beam motion.

i. Sin$1e particle on central orbit

Suppose that a single particle, of charge e = • x 10 -19 C, cir-

culates on an orbit of circumference C, returning to a specified detector

location at time intervals T o = C/v o = C/8oC. Here v o is the speed of

the particle, and the subscripts o signify that the particle has the

nominal "central" momentum. The line charge density at the detector can

he represented in the time and frequency domains by

I = ~c ~(t - s o) = cosn~o t (I)

~ = - ~ n=_~

where ~o = 2~/To is the mean angular velocity of the particle and

R o = C/2~ = BoC/~o is the mean radius of the orbit. As shown in Fig. 1,

the time-domain signal consists of a chain of delta-function impulses;

in the frequency domain this produces equal-amplitude spectral lines

spaced by ~o" The line at ~ = 0 is the dc component of the signal,

X T _2rr DC A I o_ o Com oien' 1 ~ ~o

LL LIllII1 0 t 0 w Time Domoin Frequency Domoin

12 - 82 4429A 1

Figure I

representing the particle's charge

e spread around the complete

circumference of the orbit. The

remaining lines are the

orbital harmonics. Because

cos(-n~ot) = cos(+n~ot) the

negative-frequency lines are in-

distinguishablefrom those at


873

corresponding positive frequency; the combined amplitude of the orbital

harmonics is thus 2e/2~Ro, twice the dc component. It is however instruc-

tive to retain the negative-frequency notation, as will become apparent

below.

The signal picked up by the detector is not exactly proportional

to % because the detector views more than a single point along the par-

ticle's trajectory--it smears out the delta function of charge into an

impulse of finite duration. Similarly, if instead of a single particle

we consider a rigid bunch of finite azimuthal extent, the longitudinal

charge distribution of this bunch contributes to lengthening the indi-

vidual impulse signals at each passage. If these lengthening effects

are minor, the lower-order orbital harmonics are unaffected: the

spectrum of harmonics merely tapers off at very high frequencies, with

a characteristic "cutoff" frequency given roughly by ~c = i/o, where

o is a measure of the effective bunch duration as it passes the detector.

The detector signal from %, modified only by the longitudinal smear-

ing just described, is often called the longitudinal signal or intensity

signal.

2.

ecutes betatron oscillations about that orbit.

ment can be written

Y = Yo + Y cos ~Bt

where, for completeness, a fixed displacement Yo has been included to

represent the (usually undesired) offset of the equilibrium orbit, at

the detector's location, with respect to some ideal central position.

is the betatron amplitude and ~8 the betatron frequency; the phase

constant has been set to zero for simplicity. We usually specify ~8

by stating the betatron tune Q:

Betatron oscillations

A particle displaced transversely from its equilibrium orbit ex-

The transverse displace-

(2)

~8 = Q~o = (k + q)~o (3)


874

Q is the number of waves of betatron oscillation in one complete turn

around the orbit. Often it is convenient to talk about the fractional

tT~ne q, separately from the integer part k.

Having written y for the displacement, let's use this to represent

either horizontal motion (more specifically, x) or vertical motion (z).

I'Ii use the symbol s for the longitudinal coordinate.

To monitor the transverse beam motion we need a position-sensitive

detector. Most such detectors still retain their sensitivity to beam

intensity, so that their output signal is proportional to the beam's

llne dipole density

d-- %y

This signal is commonly called the A-signal.

For a single particle we have, in the frequency domain,

co co

e'~ E d --.2TrRo cos n mot + ~ (n + Q)~o t (4) 2wRo cos

n=-~ n=-~

The first term indicates that the A-slgnal contains longitudinal infor-

mation if the equilibrium orbit has nonvanishing offset (Yo # 0). We'll

ignore this information for the most part. The second term shows that

the betatron oscillations, by modulating the chain of delta-function

impulses into the detector, produce betatron 8ideband8 of the orbital

harmonics.

Figure 2 shows this sideband structure on a frequency scale

normalized to ~o" The negative frequencies are indistinguishable from

"mirror" positive frequencies (shown dotted), so that the spectrum as

seen by a spectrum analyzer, for example, contains betatron sidebands

A both below and above the orbital J - O --~i

i

J harmonic lines. i

Figure 2

Although observation at a

single point along the orbit

cannot distinguish between negative

and positive frequencies, the beam's


87S

complete behavior is different for the two cases. Betatron waves

propagate along the beam in space and time, and should be represented

more completely by

y = y cos(mt - Ks)

where K is the wavenumber (in rad/m). Once the spatial part of the wave

is taken into account, the patterns for +~ and -m are evidently different.

The betatron waves propagate with phase velocity m/K: thus they go in

opposite directions for positive and negative frequencies. Borrowing

the terminology from travelllng-wave tubes, waves with m < 0 are called

8lowwaves and those with m > 0 are fast waves.

To provide a more intuitive picture of how the betatron side bands

arise, Fig. 3 shows a symbolic sketch of a particle's trajectory over

several turns, marking the points which represent its successive passages

past the detector. The detector effectively samples the particle's

motion at time intervals To; by the well known Nyquist sampling theorem,

a high-frequency signal sampled too infrequently yields an alias fre-

quency--in fact, a whole set of such aliases. Reconstructions for the

�9 = "SAMPLED" PARTICLE POSITION-THE SAME ON ALL SKETCHES

TO

\A

u

,q,

n i

I,~/\x

A tVVY

A VI

I ~/ I

, f - Q ~0 t

Q =4,25 q = 0.25

q COot

(l+q)~ot Fast Waves

~ .~( I-q)COO!

__1(2-q)(~ot Slow Waves

1 z - e 2

4 4 ~ 9 A J

Figure 3


876

first few of them are sketched: it is clear that these fit the sampled

points perfectly well, so that a spectrum analyzer connected to the

detector would retrieve these frequencies quite innocently. Note, however,

that the slope of the reconstructed trajectories is wrong for the slow

waves; this point will play a role when we come to consider the coherent

motion of particles within an extended bunch.

The numerology of the sideband frequencies can be handled using the

fractional tune alone:

I~nl = l(n + Q) I~ ~ = (n' • q)~o (5)

showing that the rower sidebands of the orbital, harmonics are the slow

waves.

3. Momentum Spread

A particle with momentum different from the nominal Po generally

travels on an equilibrium orbit displaced by D 6p, where D is the

dispersion function (in m) at the detector's location. We will ignore

this orbit shift for the moment. In addition, the revolution frequency

is in general changed, both because the particle's velocity may be

different and because the orbit circumference L is changed. The lattice

parameter accounting for the way L depends on p is the momentum com-

paction factor, ~p:

d L = ~P dp (6) L p

(ap is usually a small number of order I/Q2.) Taking relativistic

account of the velocXty change that goes with a given momentum change,

we obtain for the orbital frequency [I]

(7)

where y is the particle energy in units of its rest energy. Defining

the transition energy YTR by i/y~ R ~ ~p identifies the energy for which

velocity change and orbit-length change just cancel, leaving the

orbital frequency independent of momentum. For an extreme-relativistic

beam we have


877 d~ =_ ~P dp (y§ ~o Po

Note that the dispersion D is often denoted by n~ this letter is

-2 _ y-2 in (7), which also commonly used to stand for the quantity YTR

merges with ~p as y § =. For convenience I'll adopt the latter conven-

tion (and continue to use D for dispersion); we can then write

dp 1 1 mod~-~ = _ n --po ; n E -~--- -~- ~ ap (as y § ~) (7a)

YTR Y

In addition to the change of orbital frequency with momentum, there

is in general also a shift of betatron frequency. This effect, known as

chromaticity, arises in part from the way the focusing properties of

magnetic elements change with particle momentum (or magnetic stiffness).

Chromaticity is usually described by the parameter ~:

d_QQ = ~ d_p_ (8) Qo Po

To m a i n t a i n t he s i g n i f i c a n c e of Q as the number of b e t a t r o n w a v e l e n g t h s

per o r b i t a l r e v o l u t i o n , we now w r i t e ~B = Q~ f o r t he off-momentum p a r t i -

c l e . Us ing t h i s d e f i n i t i o n i n c o n j u n c t i o n w i t h (8) and ( 7 a ) , we o b t a i n

f o r t he f r e q u e n c y of a s e l e c t e d b e t a t r o n s i d e b a n d , ~n = n ~ + ~fl, an o f f -

momentum shift , dun = [(~ - ~)Qo - nn] dP mo (9)

Po

A beam of particles normally consists of an ensemble containing a

distribution of different momenta characterized by a momentum spread 8p.

The frequencies of a particular betatron sideband then also show a spread

according to (9) which varies with the order, n, of the sideband under

consideration. Sidebands adjacent on either side to a given orbital

harmonic correspond to positive and negative values of n (fast and slow

waves); for these nearby sidebands we have the two terms in (9) either

adding or subtracting, numerically, so that the frequency spreads can be

quite different.

Many writers refer mB instead to the nominal orbital frequency ~o, a fixed quantity: Q= mB/mo. On this basis eq. (9) takes the slightly simpler form

dmn = (Qo~ - nn) d--P ~ o (9a) Po


878

Thls effect is clearly visible when a beam of particles with finite

momentum spread is observed as a whole. The situation is simplest for

a coasting beam, i.e., for a beam without phase focusing, where each

particle follows its own trajectory freely without being subject to any

attempt to restore its momentum to the nominal value. In such a coasting

beam the particle arrival times at thedetector are distributed randomly,

as also are the phases of any superimposed betatron oscillations. Thus,

on the average, both the longitudinal signal and the A-signal of a

coasting beam tend to be zero. However, statistical fluctuations give

rise to signals called Schottky noise by the same mechanism that pro-

duces Schottky noise on a nominally steady de current. To evaluate this

noise we add signal powers from the individual particles, resulting in

a signal whose rms voltage is proportional to ~ for N particles.[2]

When N is large the Schottky signal is small compared to any

coherent signals, which would go proportionally to N itself. However,

it is possible to observe the Schottky noise on a coasting beam. A

spectral scan of this noise yields the frequencies present in the

population of the beam and thus serves to monitor several important

parameters. Figure 4 shows examples of Schottky spectra obtained from

Slow Wove a coasting proton beam in the ISR;

~=(43-0)~o these remarkably clean curves are

I ~ the result of repeated spectrum

scans which have been combined in

ii: a signal averager.[2] The longi-

tudinal signal shows the spread in

LONGITUDIN revolution frequencies directly;

Fost Wove if n is known this can be translated 12-82 ~=(26+Q) WO-76kHz 4429A4

into a spread of beam momentum. Figure 4

The A-signal contains contributions

from both n and from the chromaticity ~; hence the latter can be evaluated.

The different frequency spreads for the fast and slow waves are clearly

visible in Fig. 4.

4. Bunched Beam

Existence of a stable beam bunch presupposes phase focusing: off-

momentum particles which would move longitudinally away from the nominal


879

particle because of their different revolution frequencies are brought

back to the vicinity of the nominal bunch by the action of the RF system.

In fact, through phase focusing all particles are constrained to travel

with the same orbital frequency, on average--namely, that frequency which

maintains synchronism with the RF (at the desired harmonic). In the

course of the phase focusing the particles execute phase oscillations

(syne~otron oscillations), which produce a periodic excursion of the

particle arrival times at the detector location. The resulting phase

modulation is evident on the longitudinal signal in much the same way

that frequency modulation is evident on a carrier sine wave. The phase

modulation also affects the A-signal, of course.

For simplicity we can treat the phase oscillations as linear, in

which case the arrival times deviate from the normal "orbital clock"

times (spaced regularly by To) by an amount

: ~ cos(~st + ~) (11) where

ms ~ Qsmo (12)

is the synchrotron frequency, and Qs is called the synchrotron tune.

The amplitude ~ and phase constant ~ for the oscillation are determined

by external factors, for our present purposes.

The line charge-density signal of a single particle now becomes

% = Bec ~ 6(t- T- ~To) (13)

or, in the frequency domain,[3]

j [ (n ~o +m ~s) t +m*] e j-m Jm(n mo 9)e (14)

= 2~Ro

(Writing this in complex notation makes it easier to keep track of the

phases, which we'll need in a moment.) This rather forbidding-looking

expression represents the spectrum shownin Fig. 5. The index n

describes the orbital harmonic, as before. Now, however, each harmonic

is accompanied by a cluster of sidebands produced by the phase modulation

of the beam. Index m is the order of the sideband which is spaced mm s


880

A j

1 2 - 8 2 ~ - - ~ ( '~0 4 4 2 9 A 5

Figure 5

by the peak time deviation ~. (This

from the harmonic. (This sideband

has a phase constant m~.) The side-

band amplitude is given by Jm'

the Bessel function of order m;

the argument of Jm is n~o@ , which

is just the peak phase deviation

produced at the harmonic frequency

quantity is analogous to the

modulation index specified for FM in communications.) Clearly the

higher-order sidebands (m large), with their higher-order Bessel

function Jm, require a larger argument n~o~ to achieve appreciable

amplitude: larger values of m are best seen at large values of n.

To distinguish the FM sidebands from those caused by betatron

oscillations, the former are often called syne~otron satellites.

It remains to introduce betatron oscillations into the picture,

and to examine the spectrum seen by a A-signal detector, sensitive to

the beam's line dipole density d = %y. For this we'll let the particle

be off-axis by

j(~Bt + $) y : y e (15)

where, for future use, I propose to keep track of the betatron phase

constant ~. In the frequency domain the signal then turns out to be

ey Z j-m eJ(~nm t + ~ +m~) d = 2~Ro Jm(~nT) (16)

n,m=-~

where the frequencies

= (n + Q)m + mm nm o s

(17)

could now be called synchrotron satellites of the betatron sidebands.

The argument for the Bessel-function envelope involves the frequency

n : (n + Q - ~)~o : (n + Q)~o - ~ (18)


881

which is the frequency of the "carrier" offset by

~ ~ ~Q ~o (19)

the chromatic frequency. (This change in the argument of the envelope

function isn't easy to understand intuitively; I'll give a hand-waving

illustration of it when we come to consider coherent motion of many

particles.) The effect of nonvanishing chromaticity is thus to change

the relative amplitudes of the various orders of satellites; in par-

ticular, the higher-order satellites can now have nonvanishing amplitudes

near zero frequency.

On the other hand, chromaticity does not enter into the observed

frequencies, ~nm" This comes about because the phase focusing forces

all particles to the same average orbital frequency. Chromaticity

plays an important part in the operation of accelerators with bunched

beams, especially insofar as it affects bunch stability in the head-tail

effect. However, it is usually easy to measure chromaticity by varying

the average momentum of the bunch (e.g., by varying the RF frequency),

and so the loss of its direct visibility in the beam spectrum isn't very

serious.

It is important to remember that there is a high degree of coherence

in the orbital motion of a bunched beam: the arrival times of the

particles at a detector are by no means random, but are instead tightly

grouped around the centroid of the bunch. Thus there are coherent

longitudinal signals which tend to overpower any Schottky noise we

might wish to examine.

For a A-signal the coherent longitudinal signal is in principle

irrelevant, but in practice the longitudinal signal leaks in via the

average beam offset Yo included in eq, (4). Careful beam centering may

reduce the longitudinal contribution greatly, but it usually leaves an

overpowering signal at the orbital-harmonic frequencies nevertheless.

This signal must be greatly suppressed by use of frequency-selective

filters before it becomes practicable to observe Schottky noise in the

A-signal. An example of a successful measurement is shown in Fig. 6,


882

~ UNBUNCHED

BUNCHED I I I I I

u - e 2 -2 -I 0 I 2 ~ m 4429A6

where the upper trace is the

Schottky spectrum from a coasting

proton beam, while the lower trace

shows the spectrum from the same

beam bunched by an RF system.[4]

Ideally the satellite lines in the Figure 6

lower trace should be infinitely

narrow; the upper trace is broadened in much the way that we saw in

Fig. 4.

An easier (and generally more fruitful) approach to the study of

these spectra is to produce some external coherent stimulation of the

beam and thus to enhance the signal artificially. This leads into the

measurement of the be~ transfer function, i.e., the input-output char-

acteristic of the beam regarded as a general network element. We will

return to this topic later. For the moment it is necessary to examine

what is meant by coherent beam motion.

5. Coherent Motion of Many Particles

We think of coherent motion as a steady-state pattern so contrived

that the signal contributions from all the individual particles add

constructively. This requirement cannot necessarily be satisfied for

several different frequency components of the signal at once: we may

therefore arrive at different coherent patterns for different frequencies.

These correspond to the normal modes of the ensemble of particles.

The motion of a single particle generates a complete manifold of

spectral lines, as we have seen. This motion therefore corresponds to

the superposition of all the normal modes, with appropriate amplitudes

and phases. When we now consider an ensemble of particles, we can excite

their motions in different relative patterns so as to produce coherence

for a selected mode (or frequency). It is helpful, as a matter of fact,

to visualize how such mode patterns can be excited coherently; remember,

though, that the coherent mode is defined quite independently of the

manner in which it is excited.


883

A device for driving transverse (betatron) oscillations is the shaker:

at its simplest, a pair of electrodes producing a time-dependent transverse

field. All particles in the beam suffer the same angular deflection in

passing through the shaker. If the shaker is driven sinusoidally at

frequency ~ex, the particles all experience a transverse perturbation

locked in phase to ~ex" Evidently this phase-locked situatlon permits

the recovery of a coherent A-slgnal from the beam. Shaking and A-slgnal

detection are in a sense complementary processes: what represents a

coherent mode to one does so also to the other. This assures us that

all the spectral lines we've been considering so far are in fact accessible

as coherent modes to excitation by a shaker.

Coherent mode patterns can be excited even without external help by

the reaction of the beam's field back on itself. For example, in some

environments the beam produces a transverse field proportional to the

displacement of the beam centroid from the central position. Coherent

motion in a particular mode can thus generate a "shaker" field at the

appropriate frequency in the section of beam environment under consider-

ation. If this field has the right phase and enough amplitude, the

coherent motion can grow exponentially, leading to a aoherent instability

of the beam. Such instabilities constitute one of the major limitations

in accelerator operation and must thus be studied in detail. That is

the main reason for dwelling on the spectra of coherent modes and on

the methods for their detection.

It is of course possible to excite the beam longitudinally as well

as transversely. A longitudinal shaker could consist, for example, of a

short section of drift tube which can accelerate or decelerate particles

during their times of transit. In bunched-beam machines, longitudinal

excitation is most readily produced by phase or amplitude modulation

of the accelerating RF voltage. Such a longitudinal excitation is the

complement of the picking-up of a longitudinal signal (i.e., an intensity

signal).

I will begin by discussing transverse coherent modes, however, because

they are somewhat easier to visualize.


884

The situation in a coasting beam is relatively straightforward.

Such a beam--in equilibrium--has its charge uniformly distributed in

azimuth. We need only picture the traveling betatron waves impressed

on such a ribbon of charge as it passes through a shaker to see what

the coherent mode patterns are.

For a bunched beam, motion of the individual particles within the

bunch is unavoidable (requirement of phase stability~). This internal

motion complicates the situation because the particles carry their

transverse oscillation with them as they shuffle to and fro in the

bunch. (I am reminded of Little Bo Peep's sheep, which carried their

tails behind theme) To produce a steady-state coherent pattern we must

evidently require that this internal shuffling continually cause the

pattern to evolve back into itself. ap

Particle motion due to phase

oscillations is most easily repre-

3 ~ ~ sented on the phase plane, as shown

in Fig. 7. A particle with a given

...... ~ " ' .... time amplitude ~ follows an Figure 7

elliptical "orbit" on this plane.

The bunch population (with given T) is evenly distributed around such

an orbit in a steady-state situation, so that the motion of the particles

leaves their density distribution (viewed as a projection on the T-axis)

unchanged. For example, particles i, 2, 3, and 4 take each other's place

cyclically after one quarter-cycle of phase oscillation (with frequency

ms). The complete bunch of course contains particles with various

amplitudes T, and separate phase orbits apply to each such group. The

relationship between populations with different values of T is, for the

moment, left open: they do not trade places with each other.

(a) Zero chromaticity. Let's start with this case, which yields the

simplest situation: all particles oscillate with the same betatron

frequency mB regardless of their instantaneous momentum (i.e., of their

position along the phase orbit). Then a coherent pattern evidently

results from having all particles on a given phase orbit oscillate with


885

the 8~e betatron amplitude y and phase ~. As they shuffle around, the

particles carry betatron oscillations which exactly match those of the

particles they replace. The phase motion is therefore "invisible" from

the point of view of betatron oscillation, just as it is invisible insofar

as the particle density is concerned.

To complete the description of the coherent mode, we need to state

how the betatron oscillations of particles on different phase orbits

(different ~) are related. They should evidently all have the same

phase ~; but their betatron amplitude y is left undetermined. In the

simplest case, where y is the same for all particles, regardless of T,

we have a bunch which executes betatron oscillations as a whole--a

rigid-bunch mode.

The fact that the phase oscillations do not interact with the

betatron oscillations implies that the coherent signal has frequency ~B'

with no reference to m : the mode is coherent for the value m = 0. s

Only the hetatron sidebands themselves are visible; none of the synchro-

tron satellites can be seen. [Note that all betatron sidebands (n+Q)~ o

are excited, even though the shaker is driven with the frequency mex

which corresponds to one arbitrarily selected value of n. This comes

from the frequency aliasing produced by the sampling action of the

detector.]

A more formal approach to this situation notes the phase constants

in the time-dependent exponential factor in eq. (16), which is

exp j(mnm t + ~ + m~). Now the particles in the bunch are distributed

around the phase orbits with all values of ~ from 0 to 2~: if all

these particles are to contribute to the signal at frequency mn

coherently they must have the same phase. In other words, we need

+ m~ = constant (20)

For simplicity we can set the constant equal to zero, so that we require

= -m~ (21)

The mode pattern we considered a moment ago had ~ = 0 throughout, and

thus was compatible only with m = 0 (since ~ covers all values).


886

Equation (21) makes it clear what the criterion for coherence is,

and that it applies only to a selected value of m. As an example, let's

consider the group of modes with

Ap I

" ~ - T I " - I ~ j A --;,J,

4 12 -- 82 /%

4 4 2 9 A 8 T

m = i. We must excite betatron

phases ~ = -~ in the particles

with various positions ~ around

the phase orbit; all particles

on a particular orbit must have ^

the same value of y. This situ-

ation is shown in Fig. 8. Note

that for a matched pair of

synchrotron phases, ~I and -~i' Figure 8

which describe particles with the

same posi%ion along the bunch (the same T = ~i ) , the betatron excursions

average to y = �89 -j~ + ye +j~) = 9 cos ~, a real factor indicating a

reduction of effective amplitude by cos ~ but no phase difference. This

is characteristic of a standing-wave pattern for which all segments

oscillate in phase, albeit with different (and, for some, reversed)

amplitudes. This standing wave results from the superpositlon of two

traveling waves, one along the top half of the phase orbit, the other

back along the bottom.

The mode we have just described is the dipole mode: the front and

back of the bunch make betatron oscillations in antiphase. This mode

generates A-signals with m = i, i.e., synchrotron satellites spaced by

u s from the betatron frequencies. (You can see how this "changed"

frequency arises: as the particles circulate around the phase orbit,

particle i is replaced by particle 4 as occupant of the front of the

bunch; this advances the phase of betatron oscillations there by ~/2.

For each complete cycle of phase oscillations, the betatron phase is

advanced by one cycle relative to m8' so that we observe the frequency

~B + Us')

The complete mode pattern would involve two further points for its

specification: (i) how is the bunch population distributed among phase

orbits with various amplitudes ~? And (2) how does y depend on ~?


887

Whatever the choice, the net betatron motion must be zero at the center

of the bunch--that's the location of a node on all phase orbits. Typical

distributions might look as in Fig. 9, where I have taken y to be con-

stant. Similar remarks apply to other modes; Fig. i0 illustrates this

with A-signal traces of patterns for m = O, m = i, and m = 2. Several

successive passages of the bunch past the detector are superimposed.

/ k

m = O

T

m= I

1 2 - 82 4429A9 6 - 8 3 m = ~ 4 4 2 9 A I 0

Figure 9 Figure i0

Coherent modes can thus be recognized both in the time domain (via

pictures such as those of Fig. I0) and in the frequency domain (via the

order of the synchrotron satellite).[5,6] In any event, it is important

to recall that coherence patterns apply to particular values of m, but

that spectral lines with all values of n will be present:

= (n + Q)~o + m~ . nm s

(b) Non-zero chromaticity. A nonvanishing chromaticity implies that

particles of different momenta oscillate with different frequencies m B.

Take for example a machine with ~ > O, n > O: then, in the phase orbit

Fig. ll, mB is larger in the top half of the diagram, smaller in the


888

Zip

T

r!

A -T

"X

A T

1 2 - 8 2 4429Al l

Figure ii a given pos~.t~on ~[ within the bunch,

whether this position is reached along

the top or the bottom of the orbit. In fact ~ varies linearly with AT,

the time of the particle relative to the front of the bunch (T = T).

lower half. Start a particle at point 1

with betatron phase ~ = 0 and watch it

advance around the orbit. Relative to

the phase it would have maintained at

point i, ~ is constantly increasing as

the particle swings, with positive Ap,

across the top of the orbit. At point 2

it has returned to nominal momentum;

thereafter ~ starts to slip back (be-

cause e8 is now lower than nominal).

When the particle returns to point 1

after one complete orbit, the betatron

phase ~ has dropped back to exactly

zero. We can plot ~ versus position

along the phase orbit, as sketched,

and note that a given value occurs for

If we now populate the phase orbit with particles whose betatron

phases ~ just match the curve we've just described, they can then travel

around the orbit and always arrive at any given position with the phase

appropriate to that point. To account for nonvanishing chromaticity,

therefore, we have to introduce a phase slewing down the length of the

bunch: this works out to be

~ = ~Qmo " AT = ~AT (22)

which involves the chromatic frequency ~ we encountered previously in

eq. (19). Another measure of the effect of chromaticity is to state

the phase difference between the head and the tail of the bunch, which

is

X = ~(2~) (23)


889

Beyond this chromatic phase slewing, the coherent mode patterns are

similar to the ones we've considered before.

In its effect on the mode spectra, chromaticity displaces the

envelope functions by a frequency offset ~: these envelopes are now

Jm[(n+Q)~o-~)T]' asymmetrical with respect to zero frequency. In

particular, the m = 0 modes have maximum amplitude at or near ~, instead

of around zero frequency, and the higher modes have finite amplitude

In the time domain, chromaticity

alters the shape of the A-signal

because of the phase slewing down

the length of the bunch. Some

typical patterns are shown in Fig. 12,

which again superposes many successive

traces for multiple passages of the Figure 12

bunch past the detector.[5] If you

follow any one trace you can see the complicated shape; however, the

overall envelope of these traces still has m nodes regularly spaced

down the length of the bunch.

The frequency offset by

~ in the envelope functions

(but not in the actual signal

frequencies themselves) can

be rather puzzling. In

Fig. 13 I have sketched an

exaggerated situation intended

to suggest how phase slewing

can affect the amplitudes of

different frequency components.

We're examining a selected

signal at frequency (n+Q)mo,

for example with a receiver

tuned to that frequency. A

Figure 13 single particle (or short

near zero frequency.


890

bunch) excites this frequency readily, as suggested at A. If instead

we have a long bunch with uniform dipole moment, as at B, the net receiver

excitation may be exactly zero if the bunch length is exactly one cycle

of the signal frequency. Now superpose a phase slewing of ~ on the bunch

and--prestol--the signal reappears, as is evident by examination of

trace C. On the other hand, trace C has zero effectiveness in exciting

a receiver tuned to a very low frequency.

We need to dwell on chromaticity--its measurement, and diagnosis of

its effects--because it plays an important role in several of the insta-

bilities that can trouble us. In particular, the phase shift that it

enforces between the head and the tail of a bunch permits interaction

between these two parts of the bunch with a quadrature component of

phase, which is just what's needed for energy transfer and thus for

instability.

The chromaticity of most accelerator lattices is naturally negative,

but it can be modified by including sextupole lenses and is usually under

the operator's control over some limited range. Chromaticity can be

measured quite peacefully, for example by determining how the betatron

tune Q varies with beam momentum. It's another matter to identify what

role chromaticity plays in any particular beam blowup that may be sent

to trouble us~ We can try to characterize the instability--if it is a

coherent one--by its patterns in the time domain, or we can try to

identify the spectral lines it involves. In either case it may be more

profitable to operate at beam intensities just below the threshold of

the instability, and to experiment with stimulating the instability in

a controlled manner.

(c) Longitudinal oscillations. The complicating factor here is that

the "oscillation" is in addition to the normal phase oscillation that

takes place within the bunch; it's a modification of that phase motion,

representing in effect nothing but a change in the population pattern

on the phase plane. Individual particles still follow their standard

"orbits" on the phase plane--all we can do is to modify the density of

particles in different regions.


891

The most immediately striking insight from this is that there can

be no m = 0 mode for longitudinal motion! If there were, this would

imply equal perturbations of population density all over the bunch:

but the total number of particles is fixed, so that's impossible. [An

alternative argument leading to the same result starts with the spectral

lines associated with various values of m. If we take m = 0, the

resulting frequency is that of a pure betatron sideband, with no

synchrotron satellite at all. Thus the longitudinal motion is irrel-

evant when m = 0.]

To picture the population patterns on the phase plane, it's con-

venient to scale the axes in such a way that the normal phase orbits

become circular. An equilibrium bunch distribution thus covers a

x( t )

m=l m=2 m=3 12 82 4429A14

Figure 14

circular patch, the population

rotating within that circle at

angular velocity ~ . If such a s

circular patch is displaced with

respect to the origin, as sketched

at top left of Fig. 14, it will

rotate as a whole on the diagram

(as a result of the rotation of

each of its constituent points).

Half-a-cycle of phase oscillation

later, the pattern has come to the orientation shown dotted; after a

full cycle it returns to its original place.

The net bunch density, as a function of T (or, equivalently,

position along the bunch) is the projection of the phase-plane patch

onto the T-axis. The detailed shape of this projection depends on just

how the phase plane population is arranged, but it might look like the

lower diagram I've sketched. The rotating motion on the phase plane is

now reflected as a fore-and-aft motion of the bunch as a whole, because

the projection of the circular patch remains unchanged in the course of

its rotation. We have rigid-bunch oscillation, corresponding to m = i

(the front of the bunch sees an increase of population when the back

sees a decrease). A longitudinal detector (sensitive to the line charge

density ~) receives a signal modulated by the oscillation frequency u s.


892

Higher-order modes result from more complicated population "patches"

on the phase plane. These patterns must have m-fold rotational symmetry

if they are to generate pure frequencies mms, of course. Figure 14

shows possible patterns for m = 2 and m = 3, illustrating also how the

longitudinal signal picked up by a detector would appear. (The dotted

contours in each case are drawn i/2m cycle of phase oscillation after

the solid contours.)

6. Multi'Bunch Modes

A beam consisting of B similar and equally-spaced bunches can

oscillate coherently in B different modes, depending on the phase rela-

tionship between the oscillations of each of the bunches. The extra

degree of freedom introduced by this can have far-reaching consequences

on accelerator operation. There are two ways of viewing this situation,

each providing some useful insight:

(i) In the time domain, we need to worry whether an oscillating bunch

will produce fields in the environment which can later act on another

bunch, and so on around the complete ring. Since successive bunches

can have different phases, the possibility of a regenerative interaction,

which leads to instability, is greatly increased. Note that the fields

"trailing" behind any bunch decay with time; if the decay is so great

that the next bunch experiences no significant effect, the multi-bunch

instability I've just alluded to becomes moot.

(ii) In the frequency domain, we find that coherent multi-bunch oscil-

lations select the "orbital harmonics" around which the betatron sidebands

and synchrotron satellites can group themselves. When these frequencies

excite appropriately susceptible structures in the beam's environment,

we can produce instabilities. Different multi-bunch modes thus open up

different regions of the frequency spectrum. In this domain, rapid

decay of fields trailing behind a bunch corresponds to a broad ("low-Q")

impedance characteristic of the beam environment; once you have broad-

band structures, the detailed distribution of spectral lines is no

longer important. Thus the multi-bunch possibilities fade away.


893

A beam consisting of B similar bunches, equally spaced, delivers

pulses to a detector at intervals To/B; thus it generates orbital

harmonics which are multiples of Bmo, i.e., only at every B th line in

the spectrum from a single bunch. If the bunches carry unequal charges

or are not equally spaced, this symmetry breaks down. We must then

analyze the actual bunch pattern into components each of which has a

given symmetry. For example, a string of B-I bunches with the B th

bunch absent can be considered to consist of the full B-bunch string

plus a single bunch, and has the appropriate spectral components--the

widely spaced harmonics with relative intensity (B-I) times larger than

the narrowly spaced group, from the single (absent) bunch.

The coherent oscillations among B bunches are characterized by the

phase shift between successive bunches, or equivalently by the total

phase advance going once round the complete ring. This total phase

advance must evidently be an integer multiple of 2~ to obtain a

uniquely defined phase; setting it equal to p(2~) we obtain a phase

difference between adjacent bunches of A~ = 2np/B. Coherent oscillation

sidebands then occur around the frequencies [6]

(riB + p)m ~ (24)

where letting p be any integer from 0 to (B-I) covers all situations.

Note that n still runs from -~ to ~. Any resulting negative frequen-

cies are "reflected" as positive frequencies (in a spectrum analyzer,

for example); thus the numerical frequencies take the form (InlB• P)~o "

It is sometimes possible to vary B in order to emphasize or

eliminate particular spectral lines, and thus gain information on what

frequency region is involved in a particular instability or other aspect

of beam behavior.

7. Coherent Colliding-Beam Modes

The beam-beam interaction between bunches circulating in opposite

directions can establish coherent patterns between these bunches. Two

single bunches cross at two diametrically opposite points of the


894

accelerator. At these points the beam-beam forces depend crucially on

whether the two beams are moving in the same phase or in phase opposition.

(Evidently more complicated combined modes arise when there are more than

just two bunches involved. I will not explore this here.)

The two colliding beams represent coupled oscillators. To focus

attention on a specific case, let me consider two bunches of opposite

polarity counter-rotating in the same guide field, and let's suppose

that in the absence of coupling forces they have the same frequencies.

The coupling then splits the spectrum into pairs of lines whose separation

is a measure of the coupling strength; in fact, coupling strength is

most usefully expressed in terms of the "Q shift" it produces in the

tune of the beams. The two frequencies of each pair represent the two

normal coherent modes. In the zero mode the two bunches oscillate in

phase at their interaction points; in the pi mode the oscillation is in

antiphase.[7]

These two normal modes exist also when the two bunch intensities

are unequal. We then have the bunch of smaller charge moving with

correspondingly larger amplitude: in the pi mode, for example, this

again yields a nonoscillating centroid of total charge.

To consider the overall coherent pattern we would evidently need

to think about the population within each bunch, which can become a

rather formidable analytical task.[8] I'ii discuss only the artificial

case where each bunch moves rigidly; this still sheds light on some

important points.

In the zero mode two rigid bunches exert no oscillatory coupling

forces on each other, because there is no relative displacement between

them where they meet. Thus the zero-mode frequencies in this case are

the same as the individual frequencies of the beams. In the pi mode,

on the other hand, the effective coupling forces are doubled by the

coherent but opposite displacements. We thus expect the pi-mode tune

shift to be twice the shift that occurs without the coherence, i.e.,

twice the incoherent tune shift (the shift experienced by a single

particle of one bunch due to the stationary field of the opposing beam).


895

Observation of the coherent modes can thus help to measure the

beam-beam coupling, although for an exact evaluation we need to work

with a more realistic model than that using rigid bunches.

The coherent signals from the two bunches can interfere constructively

or destructively. Consider, for example, the zero mode: with two bunches

of opposite polarity, the A-signals at the interaction point cancel

exactly. This assumes the position detector to be sensitive to the

E-field of the beams; if, by contrast, the detector picks up its signal

from the B-field, the two bunch contributions add instead of subtracting,

because opposite charges circulating in opposite directions constitute

similar currents. At a location where an E-field detector picks up zero

signal for a certain mode, it is likewise impossible to excite that

mode by means of an electrostatic shaker: such a shaker is constrained

to deflect oppositely charged particles in opposite directions, which

doesn't make a zero mode. (Of course a magnetic shaker works perfectly

well for the zero mode at the interaction point.)

Converse remarks to those just made evidently apply to the pi mode:

this cannot be detected or excited magnetically at the crossing point,

but only electrically.

What I've said for the relative phase of signals from the two beams

IP

12 - - 82 4429A I5

Figure 15

at the crossing point can be projected

to other azimuthal locations around the

ring, as shown in Fig. 15. Let ~ be

the betatron phase advance from the

interaction point IP to the detector D,

going one way; and let T be the transit

time of the bunch from IP to D. When we

observe a A-signal at a selected frequency

Cobs we obtain a phase shift ~, but an

amount mobs T of this is attributable just

to the transit delay. Referred to a fixed

time origin the phase change in the signal

is only ~ - ~obs T.


896

The corresponding quantities for the beam going the other way round

from IP to D are (2~Q - ~) and (T o - T). Thus the relative phase of the

two signals at D, referred to their relative phase at IP, for frequency

mobs, is

A~ = (~ - mobs T) - [(2~Q - ~) - mobs(T o - r)]

= (n + Q)m o and T o = 2~/mo, we obtain mobsT o - 2~Q = 2wn, which With mob s

simplifies the above expression to

A~ = 2(~ - mobsT), modulo 2~ (25)

This permits us to identify locations of D relative to IP which have

similar or opposite symmetries with respect to the normal beam modes.

At locations away from the crossing point--i.e., at places where

the two bunches pass at different times--it is possible to pick up

signals from one bunch only or to excite one bunch exclusively. If

such time gating is employed the relative phase of the two coherent

signal components becomes irrelevant, and each mode can then be detected

(or excited) by selection of the appropriate frequency. The bunch

that's been "gated out" still participates in the motion, of course,

because of the coupling that exists at the crossing point.


II. SIGNAL PICKUPS

897

To pick up information from the electromagnetic field of the beam

we use electrodes, loops, transformers, cavities, or the like. These

devices may be mounted inside the vacuum chamber, but they must of

course be kept clear of the working aperture required by the beam. If

the pickup is placed outside the vacuum envelope, it is usually

necessary to provide an insulating section of envelope to permit the

fields to escape; the only exception applies to extremely low-frequency

or dc fields, which penetrate a metal chamber. However, even here at

least a short insulating gap may be needed in order to interrupt stray

circulating currents in the metal chamber which might otherwise falsify

the signal.

The electrodes form part of the beam environment and thus participate

in the formation of image charges and currents which occurs everywhere.

Often the operation of the electrodes can most easily be understood by

considering these image charges and currents. Figure 16 shows a typical

bunch, of line charge density %B, propagating along its trajectory s

with speed v = Bc. The total charge in the bunch is

= • = ~f%BdS (26) QB

X8

~ - ~ : ~ : > v = B c

~<<l

-7-2. __~L ~

FWHM ,~ 1.4o

Figure 16

and the instantaneous current is

I B = hBV = %BBC (27)

The image charges from a pointlike

bunch are spread longitudinally

along the wall with a distribution

whose full width at half-maximum is

roughly 1.4a/x, where a is the radial

dimension of the vacuum chamber and

y is the relativistic dilation factor

(i _~2)-i/2. At very high energies

(y + =, ~ § i) the image charges have

essentially the same distribution as


898

the bunch itself: l W = -XB, where ~ of course stands for the total

line charge density on the wall at any location s. The distribution

of image charge azimuthally (i.e., around the beam axis) depends on

the shape of the chamber and the centering of the beam within it. Note

that I'll be using the word "azimuthal" here to refer to distributions

in a plane normal to the beam's trajectory, as contrasted with its use

to describe the azimuth of any given point of the trajectory on the

plane of the accelerator as a whole. For this latter location, cor-

responding to the coordinate s, I'll use "longitudinal."

i. Some Specific Pickup Configurations

(a) Transformer. This picks up the changing magnetic flux produced by

the beam, and thus cannot distinguish in polarity between two oppositely

charged beams traveling in opposite directions. The transformer often

has a high-permeability core (ferrite or strip-wound steel) and is

operated in the current-transformer mode, in which the magnetizing

amp-turns of the beam and the secondary load current cancel each other

closely (Fig. 17). The equivalent circuit is then as shown; the output

current I 1 varies inversely as the number n of turns in the winding, and

the inductance L of the winding increases in proportion to n 2. The

presence of L imposes a low-frequency cutoff with time constant T L = L/R 1

on the signal. The way this modifies the pulse shape for a square pulse

of beam current is indicated; if the accelerator works with many bunches

occupying a significant fraction of the circumference, the pulse under-

shoots from individual bunch signals combine to give an equilibrium

situation as shown in Fig. 18. Interestingly, the peak negative

n Turns; Inductance L Approx imate

Equivalent Ck.f: TB

N t nI Eq'uol

n 2 TL= L /R I ~ R~ '-'

( U)TL>>I ): Vl =18 Rl/n . . . . . . . .

Figure 17

H NNNNrlrirINNr - u u u u L I U L I L I I J L l

]~,]~.~,. -F- IDC

Figure 18


899

level of the resultant signal is a measure of the average (or dc) beam

current circulating.

If the transformer were placed around a conductive beam pipe, the

pipe would constitute a secondary winding. Not only would the unavoidable

ground connections produce a short-circuit turn effect, but also the stray

currents picked up by the pipe acting as loop antenna would be liable to

introduce unacceptable noise into the signal. Thus a short insulating

gap in the vacuum pipe is normally required.

In an effort to improve the low-frequency response of the transformer,

one might increase the number of turns, n, of the winding. With a given

required output voltage signal level V I (requiring R 1 = n) this gives

T L = L/R I ~ n. However, the higher impedance level, taking into account

stray capacitance and leakage inductance, degrades the high-frequency

performance. T L can also be increased by reducing RI, which can be done

without loss of effective signal level through the use of operational

feedback in the amplifier following the transformer.

High-frequency performance is important where the longitudinal

bunch profile %B(S) is to be observed: I return to this question in a

moment. If, instead, we are interested chiefly in monitoring the total

charge in a very short bunch, we can use the transformer's finite

risetime--controlled and tailored, if desired, with a supplementary

low-pass filter--to produce an output pulse whose amplitude is

12 - 82

U \ EquoP Areas

Figure 19 4 4 2 9 A 1 9

effectively independent of the bunch

profile. The peak signal V can then

be made a convenient total-charge

monitor, largely independent also of

beam centering within the magnetic

structure.[9] This is illustrated

in Fig. 19. Note that counter-

rotating bunches of opposite charge

yield similar polarities and must

therefore be distinguished on the

basis of timing. To avoid interaction

of the undershoot from one pulse with


900

the absolute level of another, sample-and-hold measurements of the

pulse level at points such as A and B on the waveform can be subtracted;

alternatively, with peak measurements (B) alone, a calculated correction

for the interaction can be introduced.

For accurate monitoring of the total circulating current on an

absolute scale, the most accessible technique uses a "dc transformer"--

a transformer whose core senses the dc component of magnetizing force

via the second- (or higher-order) harmonic components produced when

this magnetization is not zero.[10] The ampere-turns of the beam are

canceled by a reference winding through which a measurable current is

sent; this current is varied until the transformer's core detects that

a magnetic balance exists. The system constitutes a specialized

magnetic amplifier with external feedback.

Where high-frequency accuracy is important, it is common to use

single-turn transformers, with the single turn taking the form of a

continuous cavity-like enclosure around the iron, showing just a short

longitudinal gap. Such a structure looks very much like an accelerating

cavity~

(b) Wall-Current Pickup. It is a short step from this arrangement to

using the conductive vacuum chamber effectively as secondary winding of

a i:i transformer. This yields the gap monitor sketched in Fig. 20,

more readily viewed as a way of forcing the wall current ~ = -I B to

[ ] [ ] I I t

7-" 18 I I ~Iw

I 2 - 8 2 4 4 2 9 A 2 0

Figure 20

flow through a monitoring resistor R I.

I W ceases to be an image of I B at low

frequencies and dc, of course; the low-

frequency response can be extended by

placing high-permeability cores around

the beam pipe to increase the inductance

included within the ground connections.

The "stubs" of beam pipe included in the

pickup circuit act as resonant transmission

lines; use of lossy magnetic materials, and

careful tailoring of any exterior grounded


901

shield structures, can control the resultant pulse ringing. Carefully

designed gaps of this type are a favorite type of wide-band pickup.[ll]

The load R 1 usually consists of many resistors in parallel, physically

distributed around the azimuth of the gap. With a stray capacitance C

of about 30 pF, for example, and R 1 = 1 ~, a high-frequency time con-

stant of 30 ps can be achieved, corresponding to a cutoff at about 5 GHz.

(c) Short Electrode. Consider now introducing a short section of "wall"

independently of the vacuum enclosure. For the moment, let this separate

electrode completely enclose the beam. Let's think of the image charge

distribution ~ which is intercepted by this electrode; ~ may be

smeared out longitudinally with respect to the bunch charge %B when B < i.

We'll suppose that the electrode length ~ is short compared to the

longitudinal extension of ~. Then the charge on the electrode is ~%,

and the equivalent circuit in Fig. 21 shows that the signal across the

-Xw~ + XB --C-- Ri

I I "l- C ' -~ f F'- Xw~ -XB

Figure 21

load R I monitors (-s with a

low-frequency cutoff time constant

T L = RIC , where C is the ground

capacitance of the electrode.

Since C is roughly proportional to

Z, lengthening the electrode doesn't

change the signal level in the pass

band.

Typical values might be C = I0 pF, R 1 = 50 ~, T L = 0.5 ns. This

implies that bunch profiles longer than about 1 ns fall into the low-

frequency cutoff region and are effectively differentiated by such a

short electrode. (The situation

I Short Bunch Long Bunch

Figure 22

is radically different if R 1 is

made much larger; however, this

precludes direct connection to

an output cable and involves head

amplifiers placed close to the

accelerator proper--not always a

desirable arrangement.)

The short-bunch and long-bunch

situations are sketched in Fig. 22,


9O2

not drawn to the same time scales. Differentiation of the long pulse

produces a bipolar output; the short pulse is reproduced quite accurately,

except that a long trailing undershoot follows it--required to keep the

total de component of the output signal equal to zero (no net charge is

delivered to the electrode).

The "ring" electrodes I've considered so far need an output connec-

tion at one (or perhapsseveral) points on their azimuth. Wall-charge

signals must propagate azimuthally to these connection points, and this

sets up azimuthal waves and reflections in the electrode. Once the

circumference of the electrode cannot be considered "short" on the

desired time scale, these reflections spoil the output waveform--usually

irretrievably. The answer is to use an electrode with only a small

azimuthal width, such as is shown in cross-sectlon normal

to the beam in Fig. 23. I'll call the angle subtended by

the electrode at the beam e; if the rest of the chamber

has circular cross section and the beam is on axis, the

electrode intercepts a fraction e/2n = ~ of the wall

charge ~. In fact, I'll adopt the symbol ~ to denote

i~-82 4429a23 the azimuthal fraction of wall charge intercepted in

Figure 23 general: note that ~ depends on beam position and--

especially in a vacuum chamber of odd cross section--

loses its direct geometrical significance. The variation with beam

position is a nuisance when you're trying to monitor longitudinal

signals alone, but on the other hand it is highly useful if a A-signal

(position sensitive) is desired.

A short, narrow pickup electrode is usually called a button.

(d) Long Electrode. As soon as the electrode length becomes comparable

to the bunch length we need to consider signal propagation along th~

electrode in detail; that's best done by treating the electrode as a

transmission line.[12] The wave speed along this line is usually c (if

no dielectrics or magnetic materials are introduced). If the bunch speed

is lower, the situation becomes complicated: I won't tackle that problem

here, but only the case where 8 = 1--bunch and wave travel together.

Then also ~ and %B have the same longitudinal profile.


903

-~I8 .....

//////////////////////////////////// "'.

I z = Vz /Z o

12 - - 8 2

current I Z

Figure 24 shows the situation at

the upstream end of the line, where it

intercepts the wall current -~I B. This

current flows into the line through the

external load resistor R 1 and produces

a voltage wave V Z on the line. If the 4 4 2 9 A 2 4

characteristic impedance of the line is Figure 24

Zo, the propagation of V Z requires a

= Vz/Z o to enter the line, also through R I. The u~d

+ Vz/Z o. By Ohm's law this is also -Vz/RI, current in R I is thus -il B

so that we obtain

~I B V Z = ~IB(RII I Z ) (28)

O Rll + Zol

This expression leads to the circuit model of the situation shown

in Fig. 25. Ignoring the beam and its wall currents in detail, we model

their effect by injecting a current +~I B directly onto the upstream end

of the line; this current divides between the line impedance Z o and the

external load R I as sketched, yielding the pulse V Z on the line which

we've just calculated.

We can evidently model the downstream end of the transmission line

in analogous manner, extracting a current ~I B from the line at that end

(Fig. 26).

~ ooo

"///////////~/////////// RI

12 - 8 2 4 4 2 9 A 2 5 12 - 8 2

Figure 25 Figure 26

The overall performance can now be found from these models.

give two examples:

RZ

4 4 2 9 A 2 6

Let me


904

(i) R I = Z o. With the upstream end of the line terminated in a matching

load, the pulse which propagates down the line is �89 This pulse

reaches the downstream end just as the beam extracts a current ~I B there.

Let's use superposition to see what the combined effects are. It simplifies

the argument greatly if we suppose (temporarily) that the downstream end

is also correctly matched, i.e., that R 2 = Z o. In that case the pulse

coming down the line is absorbed perfectly, generating no reflections;

after producing a signal �89 in R2, this pulse has no further effect.

The contribution from the current ~I B extracted--or, equivalently, -~I B

injected--at the downstream end is easily obtained, since this current is

shared equally between R 2 and the line. Thus a signal -�89 is generated

in R2, which exactly cancels the signal ~(~IBZ o) produced by the incoming

pulse along the line: R 2 receives zero net signal. Also, a pulse -�89 o)

is generated on the line and propagates back toward the upstream end (for

all the world looking like the inverted reflection of the first pulse which

would have been formed by a short-circuit termination).

In fact, though, there wasn't a short-circuit termination at the

downstream end: we had put R 2 = Z o for ease of analysis, but actually

found zero signal generated in R 2. This implies that zero signal would

be observed in any value of terminating resistor R 2 whatever. The upshot

of the argument: it doesn't matter what termination is placed at the

downstream end; no signal appears there. An inverted reflection of the

first pulse [�89 appears at the upstream end, after a total transit

delay corresponding to twice the length of the transmission line. This

reflection is in effect created by the wall current which travels with

the pulse as it leaves the downstream end of the line.

There is thus some bipolar nature to the signal from this pickup

system, satisfying the requirement for zero net dc component imposed by

the absence of any actual charge transfer from the beam. However, by

adjusting the length of the transmission line, we are able to change

the spacing between the primary signal and its inverted reflection, thus

making a separation between them on the basis of timing more accessible.


905

(ii) R 1 = 0. In this situation, no pulse is generated on the line from

the upstream end, all the injected current ~I B going into the short

circuit at the input. At the downstream end, the current -~I B divides

between R 2 and the line, generating a negative signal across R 2 and also

sending a negative pulse back up the line. This pulse is inverted by

reflection at the input short circuit, thus producing a positive signal

in R 2 after a delay of twice the transit time. If further reflections

bouncing around are to be avoided, we had better make R 2 = Zo, in which

case all the pulse amplitudes are �89 the signals at the down-

stream end look very much like those we previously saw at the upstream

end, only with inverted polarity.

You can see that the transmission-line pickup has many of the

properties of a directional coupler. In fact, if both ends are

terminated in Zo, a bipolar pair of pulses appears only at the upstream

end: and if there are counter-rotating bunches, each direction of

bunch propagation has its own "upstream" signal port~ Before getting

too enthusiastic about this way of separating oppositely moving bunches,

let's note that the idealized performance just outlined suffers somewhat

in practice from the difficulty of bringing the transmission line ends

out cleanly through the vacuum envelope. There are mismatches and

radial pieces of conductor forming loops, so that the actual directivity achieved may be quite unimpressive.

We've assumed throughout that the bunch propagates with the speed

of light, in which case its E-field is compressed into a thin radial

"pancake"--it is almost purely radial. This resembles the field

within a coaxial cable, and in fact a short pulse sent down a thin

wire strung down the center of the vacuum chamber can serve very nicely

to simulate the signals generated by a corresponding beam pulse.[13]

This is very useful for absolute calibration and for playing with the

terminations of the line to obtain optimum pulse shapes.

Figure 27 summarizes our result for the case of matched upstream

termination. If we picture the pickup electrode as a strip line of

azimuthal angular width ~ in a pipe of circular cross section (cf. Fig. 23),


906

V2=O

and if its spacing from the wall is

A, an approximate expression for the

characteristic impedance is

60A Z o =-~- (2) (29)

which yields

Figure 27 showing that the voltage signal is

independent of the line's width ~; this result might have been expected

on the basis of Faraday's law--the induced voltage depends only on the

amount of magnetic flux from the beam intercepted in the space A.

The delayed, inverted reflection of the first pulse signal is some-

times a nuisance. It can be further removed from the desired signal by

lengthening the line, or it may be "tailored" by tapering the line or by

including lossy ferrite loading down its length.[ll] Whenever the reflec-

tion is thus modified, we expect some negative signal components, of course,

to keep the net dc component zero; the negative signal might however take

the form of a long, slow undershoot.

(e) Cavity or Wavegu~de. Structures with well defined field patters (at

some resonant frequency or over a limited span of frequencies) can interact

with the beam in a highly specific manner, as is evident from their use as

accelerating or deflecting devices. The reciprocal use suits them as signal

pickups. I won't go into detail here, except for one or two comments.

The usual "accelerating cavity" acts as intensity pickup, often

highly insensitive to beam centering within it. The signal represents

the spectrum component at the cavity frequency, which should therefore

be adjusted to an appropriate orbital harmonic. If the Q of the cavity

is not too narrow, longitudinal coherent signals from the beam will be

evident as phase modulation of this orbital harmonic frequency, and can

be processed by familiar communications techniques such as phase-

sensitive detectors or phase-locked loops. Amplitude standardization

(by limiters or AGC amplifiers) can permit processing of signals from

beams of widely different intensities.


907

A cavity with a "deflection mode" acts as position-sensitive pickup

device; its output is proportional to beam intensity and displacement

from the axis, i.e., to the beam's electric dipole moment d of eq. (4).

Signals from a waveguide pickup oriented transversely to the beam

can be taken from opposite ends of the waveguide, and will show opposite

phase shifts caused by the different propagation times when the beam is

displaced from the center line. Decoding this phase information can

therefore provide us with position information independent of the beam

intensity, over a suitable operating range.[14]

2. Beam-Position Measurement

Most pickup structures deliver a signal which varies with the

beam's position to some extent. Where an absolute intensity measurement

is desired this effect is a nuisance; it can be compensated to some

extent by taking the average of the signals from two electrodes placed

on opposite sides of the beam pipe. On the other hand, when the position

of the beam's charge centroid is to be monitored, the position dependence

of the signal can be turned to advantage. (Alternative approaches, such

as the phase encoding just mentioned, will not be considered further here.)

The signal from some pickups goes through zero when the beam is

correctly centered, reversing polarity for position deviations on either

side; an example might be a cavity pickup using a deflection mode. How-

ever, for most electrodes and loops the signal always retains the same

polarity, merely varying in amplitude: V 1 ~ %(1 + ky). To eliminate

the part proportional to intensity alone, we can subtract the signals

from two pickups located on opposite sides of the beam. Such a sub-

traction yields a pure A-signal proportional to the beam's dipole

moment %y, which is ambipolar in nature (i.e., it reverses polarity

depending on beam position).

The beam displacement y is itself the sum of a fixed equilibrium-

orbit error Yo and any superposed coherent oscillation, YB" The emphasis

in processing the position information shifts depending on whether an

absolute measurement of Yo is required or whether we are interested only


908

in the oscillatory part. In the latter case a pure subtracted A-signal

may not be required: we need merely separate out the oscillatory

component of amplitude modulation in the pickup signal V 1. For equilib-

rium-orbit survey, on the other hand, oscillatory parts should be

rejected; an important consideration is then the precision and stability

with which the exact balance point of the A-signal can be determined.

We will return to these points of emphasis when we consider the signal

processing chain in the following section.

The position dependence of the signal from a pickup can arise in

two ways:

(a) Proximity Effect. Buttons or strip-line electrodes involve an

azimuthal geometrical factor ~ which varies with beam position. A

vA vA

~ - a2 VB VB ,,429A28

pair of electrodes, in a pipe

of circular cross section, is

shown in Fig. 28; the electrode

nearer the beam subtends a larger

azimuthal angle:

nc~ = A_Z (31) G o a

In addition to this geometrical

Figure 28 effect, we also find that the

field lines tend to cluster more tightly toward the nearer of the two

electrodes. In the geometry shown, this effect turns out to produce

an exactly equal fractional effect, so that the signal changes accord-

AVA = 2Ay

VAo a

a VA - VB

Ay = 2 V A + V B

ing to

which yields

(32)

(33)

to

In other geometries, we can define a geometrical coefficient y* according


, VA - V B

Ay = y VA + VB

and either calculate or measure y* in order to obtain the position

sensitivity of the electrode pair.

Note that the Ay defined in (34) is obtained by taking the dif-

ference between two signals and normalizing this difference to the

beam intensity (by dividing by the sum signal). The A-signal we

previously discussed (in Part I, proportional to the beam's dipole

moment d) was not so normalized.

The proximity effect becomes nonlinear if the beam displacements

are large; i.e., the coefficient y* isn't really a constant. Moreover,

y* usually varies if the beam moves in the direction perpendicular to

the one being considered for y. These nonlinearities have to be

processed numerically where accurate beam-position information is

needed and relatively large displacements are involved.

(b) Shaped Electrodes. If the bunch is much longer than the electrode,

the signal picked up is proportional to s (cf. Fig. 21). We can thus

I shape the electrode to vary s with

beam position, gaining a certain

degree of freedom in tailoring the

desired response. Such shaped

electrodes (illustrated in Fig. 29)

usually have symmetrical partners

on either side of the beam, in the

direction normal to that for which Figure 29

beam position is to be monitored,

permitting first-order cancellation of any undesirable proximity effect

in that direction.

909

(34)


910

III. SIGNAL PROCESSING

A recurrent point in the discussion of processing techniques will

be the question of whether a difference signal is to be manufacturered

and handled, or whether the signals from the electrodes are to be treated

independently--with subtraction possibly to be performed later, at the

numerical stage. Difference signals are ambipolar, which may restrict

some of the methods by which they can be treated; on the other hand,

independent handling followed by later subtraction imposes more stringent

requirements on the gain stability of the processor, so that the null

point in the position signal may be fixed with adequate precision. I

will focus on processing of the position information in a later paragraph;

however, the considerations cannot be separated completely.

i. General Comments

The pulse signal from a bunched beam is usually short and sharp--it

contains a wide band of frequencies. To preserve this pulse waveform

requires processing at large bandwidth, which can be expensive and

inconvenient in view of the imperfections of lumped-parameter circuit

elements and possible difficulties of shielding. In most cases such

full-bandwidth processing is restricted to one or at most a few monitors,

with the remainder handled less lavishly. Because of high-frequency

losses in long cables, the broadband device may need to be placed near

the pickup electrode in direct proximity to the accelerator--often an

undesirable choice.

To reduce the signal bandwidth we can use some sort of storage con-

version device, which trades response time for bandwidth. For example,

a wide-band scope can be viewed remotely via TV or with the help of a

scan converter. Alternatively, the front end of a sampling 08ci~108eope

can be placed near the pickup, with the remainder of the display section

linked via long cables, which need handle only lower frequencies. Any

such scheme for compressing the information on the pulse shape into a

narrower transmission channel is likely to be expensive and somewhat

inflexible; I will not go into further details here.


911

For the monitoring tasks which do not require faithful reproduction

of the pulse profile, we might begin by representing the bunch by a

delta function in space or time. However, occasionally this obscures

some interesting points, in which case we can use a Gaussian bunch

profile as our model instead.

Table 2

Bunch charge QB = +Ne

X(s) = - -

ql(s) = ~ = - -

ql(t) =

dq I I I (t) = dt =

i I (t = -o T) = - -

QB -s2/2~ e

GL2~

QB ~ ~ -s2/2~ e

OL2~

QB ~ ~ -t2/2~ e

~COT 2~

QB ~ ~ e-t2/2~ 2

- 1 / 2 QB ~ ~ QB ~ ~ e ~ 0.24 BCOT2 ~ 8COT2

Table 2 summarizes the nota-

tion; o L and G T are the rms

widths of the bunch in space

and time, respectively.

When such a short bunch

passes a short pickup elec-

trode, the signal produced

depends on the capacitance of

the electrode and the load

resistance. The two extreme

cases are shown, together with

their appropriate equivalent

circuits, in Fig. 30:

--For RC >> o T we obtain an

output nearly proportional to

the instantaneous wall charge

ql on the electrode--a charge

signal.

Vl C

i v' t (R)

I

=RC = "'=--" �9 (RC l a r g e )

--For RC << o T the signal is

differentiated--it is a cur-

rent signal proportional to

I I = dql/dt.

Vl

12-82 4429A30

~i- dt

= "--: =

(RC small)

Figure 30

The spectrum of the

longitudinal charge signal

consists of equally spaced

lines at the orbital harmon-

ics. At the low-frequency

end the spectrum is cut off


912

because of the electrode's RC time constant, but we will ignore this

feature for the moment. The high-frequency end rolls off at ~c = I/~

which can again be ignored if

this frequency falls beyond

A(w) A(w) the range of interest for the

ql I, ~ processing chain. With these

simplifications the spectra of

IIII IIIIII IIII11 ,11~]1 the charge and current signals W

look as shown in Fig. 31. The 12 82 4429A31

line spacing is close because Figure 31

the orbital frequency is usually

low; even narrow-band systems often encompass several adjacent lines, so

that we can then regard the spectrum as effectively continuous.

An important point to note is that, for the current signal, the

spectral density falls in proportion to frequency--a consequence of the

RC differentiation at the electrode. We are often motivated to use the

low-frequency part of the spectrum, however--partly to stay clear of mc

(to avoid dependence on the precise beam profile), partly because low-

frequency processing is more precise and less expensive. The total

spectral energy--the product of bandwidth and spectral density--varies

with the square of the bandwidth employed; thus there is a surprisingly

sharp decrease in the signal level as the bandwidth is restricted.

To avoid such a sharp loss we are tempted to use the charge signal

instead. Making RC >> ~ usually requires the use of a head amplifier

with high input resistance, however. Note that, even in this case,

there can be no dc component in the signal because no charge transfer

actually occurs from the beam; thus the charge signal is followed by

a long undershoot characterized by the time constant RC (Fig. 30).

More commonly head amplifiers are avoided and the electrode is

connected directly to a 50-~ cable. This produces a small RC and

differentiates all but the shortest beam pulses. To put matters into


913

a practical context, here are some numerical values typical of the situ-

ation in an electron storage ring:

Electrode length ~ = 0.02m

Azimuth factor i = 0.05

C = 3 pF; RC = 150 ps

N = 2 x i0 II particles

QB = 32 nC

s T = 67 ps; o L = 0.02 m

The peak signal from the electrode in this situation is i I = 5 A, pro-

ducing a 250-V pulse in a 50-~ load. Though such a signal is actually

uncomfortably large from some points of view (voltage ratings of

attenuators or terminating resistors), restricting the bandwidth that

is utilized may yet result in a relatively small useful signal yield.

This is particularly true when the desired narrow-band signal has to

compete with parasitic feedthrough of the original fast pulse!

Information beyond the beam's intensity is contained in modulation

of the successive pulses, both in phase and in amplitude. In the spectrum

this modulation appears as sidebands and satellites, as described in

Section I. To avoid obliterating such information we must evidently

retain sufficient bandwidth in the processing chain; a bandwidth cor-

responding to one orbital-harmonic interval will intercept at least one

representative of these additional frequencies. However, some modes

generate sidebands under Bessel-function amplitude envelopes which occupy

only certain parts of the spectrum; and multi-bunch operation (B bunches)

can generate mode patterns spaced by Bm o instead of ~o--Cf. eq. (24).

2. Narrow-Band Processin$

This approach frankly gives up all information about the bunch pro-

file; the signal pulses are shaped instead into a series of damped pseudo-

sinusoids. We can picture this as resulting from the use of a bandpass

filter (Fig. 32) or a shock-excited tuned circuit. (In talking about the

quality factor of such a tuned circuit, I'ii place the usual symbol "Q"

in quotes to distinguish it from the symbol for charge.) Since, for a

circular accelerator, the beam pulses recur at time intervals To, suc-

cessive decaying sinusoids can overlap if the "Q" is sufficiently high;

then the relative phase of successive pulses begins to matter, which is


914

O.Sns

ELECTRODE SIGNAL Somewhat Broadened by Cable

IOV Peak

After 500 MHz Bandposs Filter (A f~30 MHz)

Figure 32

0.4V Peak

1 - 8 3 4429A32

converts signal amplitude into a

tantamount to saying that the system's

bandwidth is getting to be so narrow

that it needs to be tuned accurately to

a selected orbital harmonic frequency.

Why should we use such narrow-band

processors? Their main advantage lies

in the ease and accuracy with which

narrow-band signals can be handled.

Amplitude and phase information can

be extracted precisely with such devices

as phase-sensitive detectors, double-

balanced mixers, or phase-locked loops.

Frequencies can be transposed by

heterodyning--an extreme example being

transposition to zero frequency (dc),

which is how a phase-locked detector

steady signal, ready for the digitizer.

Note that phase-locked detectors have naturally ambipolar outputs, so

that they can readily handle a A-signal (which can take either polarity);

by contrast, wide-band digitizers, often preceded by diode pulse stretchers,

can become awkward in this context. Thus narrow-band processing has been

used particularly for accurate intensity determination and for obtaining

position information,

An intuitive insight into the way narrow-band signals behave is

obtained via the model of a shock-excited tuned circuit, resonating at

the band center frequency Jr" Taking the driving signal to be the current

signal, we can model it as two opposite delta functions, separated by

At = 2.5 ~T" We find the ringing response of the tuned circuit to the

first shock excitation; then we combine it with the slightly delayed

and opposite second excitation, using the phasor representation for the

resulting pseudo-sinusoids. (The delay between the two stimuli corre-

sponds to a phase difference A@ = ~rAt.) The overall response is smaller

than the response to a single stimulus by the factor ~rAt, assuming this

to be much less than unity, and it is in phase quadrature. Figure 33


915

V I =IIR

- - ~ V2

~r = I / .~ - ; Zo = ,V"~'C >> R

Resultant ~ S e c o n d Shock _~ 4 ~ m ~ . . ~ xcitoti~

A~ =oJ r A t

First Shock Excitation

1--83 4429A34

For one delta function:

[ fv,,]Zo fv,, For bipolar pulse:

= m At f v I dt= m At L

= 2.5 m r ~ RQB

Figure 33

illustrates this result with a simple circuit model: the amplitude of 2

the doubly shock-excited ringing is proportional to m r, and it is inde-

pendent of the "Q" of the circuit (higher "Q"--larger resonant gain--

smaller segment of signal spectrum captured). For a given total bunch

charge, the output amplitude is independent also of o T, as a consequence

of our assumption m << m of course. r c

Although the value of "Q" does not affect the peak amplitude of

ringing, it changes the total duration; subsequent circuits may yield

a larger output for a longer signal.

Where choice of a low ~ is desired, the dependence on m 2 may become r r

objectionable. We might then consider using the charge signal i~stead

of the current signal as primary stimulus, thus avoiding one power of ~r"

Such a scheme is shown in Fig. 34, where the ringing circuit makes use

of the pickup electrode's capacitance. This avoids the need for a

terminating resistor. A head amplifier can be eliminated by suitable

transformer coupling of the signal into the 50-~ output cable; this

avoids placing active elements near the accelerator proper. The system

lends itself to analog subtraction of signals from a pair of opposite

electrodes, simply by making the complete ringing circuit "push-pull."


916

' '

Cs ~ i L

I

C = CE+ C s

i I ~QB IL = ~fVldt = L 6 c C

C _- Vl % =

1-83 4429A33

Figure 34

Because the wide-band pulse signal has such enormous amplitude

compared to the narrow-band component to be extracted, any feedthrough

of the high-frequency components of the signal is liable to overload

subsequent processing stages and produce false results. (This is par-

ticularly true for systems incorporating "built-in" analog subtraction

in the narrow-band filter: when the beam is centered and the desired

signals are in balance, any asymmetry in parasitic feedthrough becomes

relatively more important.) It is therefore important to provide good

high-frequency rejection. Usually this is best done by use of ~lti-

stage filters. (The bandpass filter used for the waveforms of Fig. 32

is a multi-stage device.)

Once the signal itself has been "handicapped" by narrow-band

filtering, subsequent amplifiers should also be made frequency selective;

otherwise there is a risk of broad-band noise cumulatively gaining the

upper hand.

When beam intensity varies over a wide dynamic range, gain adjust-

ment of the processing chain is required. If this gain control reacts

slowly (manually switched attenuators or slow AGC), rapid amplitude

modulation of the signal will be retained. In the case of electronic

AGC, the AGC voltage required to maintain a constant output level is a

useful measure of beam intensity.


917

The pseudo-sinusoidal signal is usually rectified synchronously,

using a phase reference derived from the master oscillator which excites

the accelerator's RF system and thus fixes its orbital frequency.

C

o Vl~, ~ 8

"-7_- I v �9

As an example of such syn-

chronous rectification, Fig. 35

shows a diode bridge used as a

double-balanced mixer. With small

inputs, such a bridge produces an

output whose dc component is

V 3 = VlV2 COS ~ (35)

V

A r

V2

(smoll)

]2 - -82

Figure 35

V 3

On the other hand, if V 2 (the

reference signal) is large enough

to drive the diode bridge in a

switching mode, we can picture

it as alternately clamping points

C and D to point B, which is here

held at zero voltage. Then

V A = +V 1 and -V 1 alternately;

upon low-pass filtering this

yields the desired phase-

sensitive rectification of V I.

The effective bandwidth of the

demodulator is determined by the low-pass time constant: it can be made

extremely narrow, which would help to reduce noise. (In effect the

noise reduction comes from time averaging of the signal over the time

constant of the low-pass filter.)

In comparing the action of such a synchronous rectifier with that

of a simple diode peak rectifier, we note that the output V 3 passes

through zero linearly as the phase of input V 1 reverses--there is no

"diode pedestal." The synchronizing drive, in switching the diodes

externally, has eliminated the pedestal.

The double-balanced mixer, through its inherently symmetrical

structure, is relatively immune to feedthrough from either input port


918

alone. This feature can be exploited also with signals of pulse waveform

(rather than sinusoidal), making the mixer into a gate. However, the

longer duration of the rectified sinusoldal output gives it still greater

advantage over any residual feedthrough.

3. Broad-Band Processing

Here it is more revealing to contemplate the pulse waveforms instead

of the spectral content of the signal (we return to consideration of

spectrum analysis below). Because of the convenience of a direct con-

nection from the pickup to a 50-~ cable, we normally start with the

bipolar current signal (or a close facsimile of it). Such a bipolar

pulse can be characterized by the "dipole moment" of its two halves,

each of area A measured in volt-seconds. The dipole moment is thus

measured in units of V.s 2, confirming that the peak pulse amplitudes in

subsequent parts of the chain scale with the inverse square of the

risetime there.

R

o ~ 0 /

Vl C iT V2

0 l 0

k~- z~t

A

- A A I r 2

RC V2

1 2 - - 8 2 4429A36

Figure 36

If a simple RC low-pass filter

is driven by such a bipolar pulse,

its response is like that shown in

Fig. 36. There is a short, positive

pulse of amplitude A/T at the output,

followed by the desired long-lastlng

output, which is actually a pulse

undershoot and has amplitude AAt/T 2

and duration proportional to T.

Even if such an idealized response

could be realized with lumped-

constant circuit elements, further

filtering would be required to

remove the short positive pulse,

which would otherwise act like a

bull in a china shop further down-

line. In practice, because of the


919

imperfections of lumped elements (stray capacitance and inductance), the

pulse shapes don't often resemble the ideal. The lesson to draw from

this is once again that, for effective rejection of high frequencies,

multi-stage filters are needed. Long signal cables often contribute

usefully to this filtering function! (Table 3 and the associated

sketches summarize some properties of coaxial cables.[15])

While the signals are carried in the form of relatively narrow

pulses, they may be subjected to time gating, which permits selecting

out a desired bunch passage from others also present--e.g., in multibunch

accelerators or colliding-beam systems. The gate can take several forms

too numerous to mention here; I've already referred to the use of a

double-balanced mixer for such service.

The dynamic ran~je of a gate is often uncomfortably limited; this

requires gain control in the preceding section of the chain. (We can

use switched attenuators or p-i-n diode attenuators, for example.) A

gate also contributes parasitic outputs due to the opening command by

itself; this can be troublesome if it distorts the waveform or if, at

the repetition frequency in use, it produces harmonics that fall into a

range of interest. Sometimes opening the gate "blindly" at regular

intervals between the desired pulses can be used to produce parasitic

harmonics which cancel at the frequency of interest.

In most processing chains the signal progresses to slower risetimes

and wider pulse shapes, ultimately becoming tame enough for presentation

to a digitizer. This progressive pulse shaping involves dramatic

reduction of amplitude because of the feeble low-frequency content of

the original signal. Restoring this loss with amplifiers alone is

possible but not desirable, mostly for reasons of cost, stability, and

susceptibility to noise. A more attractive procedure uses pulse

stretchers or sample-and-hold circuits, which can increase the duration

of a signal by a large factor without loss of amplitude. (In effect,

these devices augment the low-frequency content of the signal by the

action of their nonlinearities, which can be thought of as producing

beat notes between all the high-frequency spectral lines.)


920

TABLE :5

Skin-Effect Losses in Cables:

Attenuation (dB/m) cc co I /2 : "risetime" c(( length) 2

1.0 0.9

0.5

....----INPUT

.•• OUTPUT * ( match ed)

to.5 to.9

to. 9 ~ 6 0 to. 5 This extremely slow approach

to asymptotic level isn't

usually noticed, because of other pulse-shaping elements or because input isn't a long

enough step function.

*Drawn for DC Loss = 0

Some Examples ( lO0- foot lengths):

TYPE to. 5

RG-58/U 1.8 ns RG-8/U 350 ps RG- 19/U 55 ps

Delta-Function response is obtained from step function

by subtracting two closely

spaced waveforms. If input is bipolar impulse, do one

more such subtraction.

Tracing of output from 200 ' RG I74/U(very thin cable};

input was beam pulse, where

cr t .~ 70ps

RC = 120ps

NOTE: OC 'oss: o/W s) if m a t c h e d .

RS= DC resistance of cable.

C ~ Negative Undershoot

Makes Net Area = 0

1.3ns


921

C

Figure 37

An elementary stretching

circuit is shown in Fig. 37.

~%en the input pulse is very

short, the diode's imperfections

(shunt capacitance, charge stor-

age, finite forward "cut-in"

voltage) limit the operation

severely; in particular, pulses much smaller than 1 V cannot be stretched

satisfactorily, and the input-output curve is quite nonlinear over much

of the range. More accurate stretching circuits, using feedback ampli-

fiers for example, tend to be unsuitable for fast signals and must

therefore be preceded by considerable pulse shaping.

Though pulse stretchers are very desirable in the progression from

a fast signal to a digitizer, we cannot perform analog subtraction (for

the A-signal) and then use a stretcher, since the ambipolar nature of

the A-signal gets in the way. Stretching first and then subtracting

leaves us with the relatively unpredictable gain characteristics of the

individual stretchers, which would have to be controlled by some process

of continuous calibration if absolute accuracy of the A-signal balance

point was to be maintained.

vi . ~ -VO C "J

IV l

0

Vo ( (dc)

1 2 - 8 2

<__ _3

Figure 38

An interesting variant of the

stretching circuit is the biased

pickoff circuit (also known as a

81ideback voltmeter) of Fig. 38.

The bias V o is adjusted until it

almost equals the peak voltage of

the input pulse, permitting only the

tiniest tip of the pulse to pass

through the diode. A sensitive amplifier determines whether this con-

dition is met, varying V ~ appropriately in response to an error signal.

The circuit is evidently limited to measuring repetitive inputs. As an

alternative, V o can be held fixed and the input V 1 varied with the help

of a p-i-n attenuator. In either case, if V ~ or the gain is constrained

to change only slowly, any rapid amplitude modulation of the V 1 pulses


922

emerges greatly magnified in the output of the error amplifier. This is

a popular method for monitoring coherent transverse beam oscillations,

and often serves as input for a transverse feedback system which drives

the beam in sueh a way as to damp the coherent oscillations.

4. Beam-Position Measurement

Determination of the equilibrium-orbit position is an important

requirement, both for circular accelerators and for high-intensity linacs.

Accuracies of a small fraction of a millimeter are often needed; moreover,

the measurements need to be referred to a precisely surveyed reference

line and must thus not be subject to significant zero drift. Other

aspects of beam monitoring which rely on beam-position information

include:

--measurement of dispersion (position change produced by momentum change);

--observation of the beam's coherent response to stimulation;

--diagnosis of coherent instabilities;

--supervision of deliberately introduced orbit distortions; and

--delivering an input signal to transverse feedback systems for beam

stabilization.

Most position-measurement systems rely on use of a A-signal. In

some cases this signal is normalized to beam intensity, e.g., by dividing

by the sum signal from opposite electrodes--eq. (34). To produce the

A-signal we can proceed in two different ways: (i) by computation,

(ii) by analog methods.

(i) The computational approach treats individual electrode signals

separately and encodes their amplitude digitally. Separate processing

chains can of course be used for each signal, but these must then have

highly stable relative gains. An alternative is to pass the two signals

from a pair of electrodes through the same chain sequentially, in which

case the gain of the processor is common to the pair and becomes less

critical. However, we still need an input switch which has equal

transmission for the signal from either electrode; and the beam signal

must of course be repetitive, preferably with an intensity that does


923

not fluctuate over the short time taken for the two successive measure-

ments. Because this requirement is readily met in a storage ring,

digital position calculation is a common choice here.

(ii) In analog processing we can perform the signal subtraction anywhere

along the processing chain, but the relative gains of the independent

parts preceding the subtraction must be well controlled. Thus we try to

subtract as early as possible, perhaps right at the electrodes themselves.

Note that the gains of any independent segments can be obtained through

calibration procedures--but it isn't always easy to simulate the beam

signals adequately with such a calibrator.

--Subtraction at the electrodes: The two signals can be fed into a

hybrid junction, which delivers their difference (and often also their

sum, at another port) directly. Precise balance may be a problem, and

the pulse shapes may be distorted. For narrow-band systems, a resonant

transformer (cf. Fig. 34) can be made in a balanced configuration.

Again, balance and asymmetrical feedthrough can be problems.

--Subtraction after a cable run: This is often preferred, since it

minimizes the amount of equipment near the accelerator proper. The

cables contribute some filtering of the high-frequency components,

determining the pulse shapes and reducing problems with feedthrough of

the fast pulses. However, it becomes important to match the transit

times of the cables, since unequal delays may offset the zero of the

system. In a narrowband system, for example, a relative delay ~t

corresponds to a phase difference mr~t at the band-center frequency mr;

this produces an unbalanced output (mr~t) times either signal alone.

(The phase of this unwanted output is 90 ~ different from the phase of

a true unbalance signal, and may thus be rejected if a phase-sensitive

detector is used.) From this point of view, the lower the frequency

of detection, the less stringent the requirements on time matching

become.--In a wide-band system, time differences produce unmatched

waveforms which may be detected as if they represented nonzero diff-

erence signals.


924

--Subtraction after some amplification and pulse shaping: The wider

and slower the signals, the easier subtraction becomes; for example,

operational techniques can be used once the signal bandwidth falls

within the useful response range of the amplifiers. However, by the

same token, the gains of the preceding independent sections must be

controlled more closely. (It is perhaps worth pointing out that

negative-feedback amplifiers, while they exhibit very stable gains

within their passbands, can have unstabilized gains at the band edges,

resulting from variable marginal stability. This would be important

if the amplifier's characteristics formed part of the frequency-defining

system.)

When the beam signals have a low repetition rate (as in linacs) or

are nonrepetitive, the processing techniques become more difficult.

Because of the possibility of pulse-to-pulse jitter in many cases,

sequential processing through the same chain may be unstable--parallel

processing or analog subtraction would then be needed.

For a real-time indication of beam position, such as in monitoring

coherent beam motion, analog methods are required. An actual subtraction

may not be necessary if only the oscillatory part of the beam motion

(not a slow drift of position) is of interest. As an example of a

1 2 - 8 2 4429A39

Figure 39

possible approach, consider the

sample-and-hold capture of succes-

sive beam pulses from a single

electrode, shown in Fig. 39. Such

an S/H circuit can be strobed

externally from a timing system

tied to the accelerator's master

oscillator. The output is pure dc

in the absence of any beam motion; if the signal is then passed through

an ac coupling, only beam oscillation is visible. One difficulty with

this approach arises from the need for the circuits preceding the S/H

stage to handle the full beam pulses, even if the amplitude fluctuations

represent only an extremely small fractional change. The relative depth


925 of this amplitude modulation can however be increased by the use of

pickoff circuits such as the one shown in Fig. 38.

A question often of interest concerns detector orthogonality: i.e.,

is a system nominally configured to respond to horizontal beam motion

goingto be sensitive to vertical motion as well? In many cases the

oscillation frequencies for the two directions of motion are different,

so that additional discrimination can be had by suitable filtering.

However, if we are investigating x-z coupling, frequency discrimination

evidently becomes meaningless. Then the detector must be orthogonal

(as must the shaker which drives the beam in one plane or the other).

Position-detector systems intended to locate the beam's equilibrium

orbit need to be calibrated by reference to a survey line or some other

criterion. Pickup electrodes can be excited by means of an antenna in

a suitable test fixture, determining their electrical center with respect

to certain fiducial points on the detector. These fiducial points are

later referred to the accelerator survey points when the detectors have

been installed. An alternative approach makes use of the fact that beam

centering is often required primarily with respect to the magnetic

centers of the focusing lenses (quadrupoles). An in 8itu determination

of the detector's electrical center can then be made by varying the

strength of the adjacent lens and noting whether any beam deflection

occurs as a consequence; if the beam is correctly centered in the lens,

it does not move as the lens excitation is varied. The motion corre-

Lated with lens current is readily detected with very high sensitivity,

so that even such small lens perturbations as are permissible in an

operating accelerator may be sufficient for the purpose of calibration.

5. Spectrum Observation

Control of the frequency response of the whole detector chain is

evidently of primary importance here. Note that pulse reflections

(caused by various mismatches) often occur with delays which place them

outside the time window under study, thus escaping detection; in the

frequency domain such reflections still cause peaks and dipsl Similarly,


926

the presence of "fellow travelers" (partly filled, unwanted beam bunches)

can modify the effective spectral response in dramatic ways--in fact, a

study of the broadband spectrum is often a highly sensitive way to check

for the presence of such fellow travelers. The lesson to be learned:

alway8 be aware of both the time-domain and the frequency-domain situa-

tion8 simultaneously, even if only one or other aspect is currently

under study.

Spectra reaching into the GHz range are best monitored with a wall-

current pickup [para. II(b)] or a carefully "choked" strip line [para.

II(d)]. For lower frequencies, progressively less care is needed in the

pickup itself.

Direct application of the pickup signal to a spectrum analyzer often

results in gross overloading or even damage, owing to the large amplitude

of the wideband signal. Suitable filters, known to be flat over the

frequency span of interest, must be inserted. Low-pass filters of course

reduce the signal level dramatically, as we have seen, and are therefore

required to provide extra isolation against the larger high-frequency

components. Often the low-frequency content of the signal can be

enhanced by the use of a nonlinear device such as a pulse stretcher;

however, such devices produce cross-modulation frequencies and are thus

liable to mislead us.

I 2 - 82 4429A40

Figure 40

Nonlinear processing opens up

some interesting possibilities. For

example, consider a bunch undergoing

longitudinal quadrupole oscillations

(m = 2, Fig. 14). Figure 40 shows

an exaggerated time-domain picture

of the bunch profile, which--even

though there is no centroid motion--changes its amplitude cyclically.

An analyzer sensitive only to the area under the beam pulses does not

detect such oscillations; if the pulses are peak rectified, the modula-

tion becomes detectable.

In general, observing the spectral lines from higher-mode oscilla-

tions requires use of the appropriate part of the spectrum, as dictated


927

by the envelope function, lhl, for the particular mode. [3] Finite

chromaticity displaces these envelopes along the frequency axis [cf.

eq. (!9)] ; sometimes detection of a certain mode can be enhanced by

deliberate change of the chromaticity. Note also that slow-wave and

lhl

Is ow l m= ;o

12-82 4 4 2 9 A 4 1

Figure 41

adjacent sideband amplitudes, perhaps while varying the chromaticity,

often helps in determining what particular mode number m is being

excited.

In the study of specific spectral ranges, all the techniques of RF

manipulation familiar from communications technology become available:

down- and up-conversion of the frequency band (using double-balanced

mixers) can be done with the help of frequency synthesizers or of

signals derived from the master-oscillator chain of the accelerator.

A variety of impressive filters, including very sharp crystal filters,

is at our command.

nearby fast-wave betatron side-

bands originate at different

points under the amplitude

envelope, and may thus show

greatly different response

(Fig. 41). Comparing such

6. Spectrum Analyzers

These fall into two general classes: (a) sweeping analyzers, and

(b) fast Fourier-transform computers (FFT). Each has some special

features on which I would like to comment.

(a) A sweeping analyzer Isbasically a tuned receiver whose center

frequency can be swept electronically; its bandwidth is adjustable and

its output response can be linear, square-law, or logarithmic. The

receiver detects the rms amplitude of the input spectrum within its

passband; if it is suitably configured and provided with a reference

signal, it may also display the relative phase of the signal (in which

case it's called a network analyzer). A sweeping analyzer is often

equipped with a tracking generator which delivers a sinusoidal output


928

of the same frequency as that to which the receiver is instantaneously

tuned; this can be used for stimulating the beam coherently, to measure

the beam transfer function.

The analyzer integrates and normalizes its rms response during the

time that it dwells on a given resolution interval; the slower the sweep,

the longer the corresponding dwell time and the lower the random back-

ground noise--a consequence of the effective signal averaging over the

dwell time. The sweep rate is limited also by the requirement for

resolution: the uncertainty relation

A~ At ~ i (36)

imposes a lower limit on the dwell time At if a given resolution A~ is to

be achieved.

Other considerations aside, what part of the frequency spectrum is

most advantageous for achieving a given resolution in minimum time? Most

spectrum features have the same absolute spread regardless of orbital-

harmonic order: hence their detection requires the same absolute

resolution and therefore the same dwell time. (An exception occurs in

the observation of the Schottky signal from a coasting beam [eq. (i0)],

where different harmonics show different frequency spreads. Here obser-

vations are most quickly made at high frequency.)

Most sweeping analyzers are not equipped to average the results of

several repeated sweeps, as might be desirable to enhance the signal-to-

noise ratio. They substitute a single, very slow sweep--and this is not

as desirable, since it is vulnerable to intervening drifts in the

accelerator itself. It is possible, of course, to equip a sweeping

analyzer with a signal averager to follow: this is how the clean

Schottky signals of Fig. 4 were obtained.

A useful feature of a sweeping analyzer is that it can be stopped

at a precise frequency setting previously determined from the spectrum

as a whole. The analyzer then becomes a fixed-tuned receiver, capable

of displaying the time variation of the signal within the selected

response band. As an example of this, Fig. 42 shows the coherent

response obtained from a vertical beam-position monitor after shock


929

12 --B2

Figure 42

excitation of the beam; the

analyzer is tuned to a vertical

betatron sideband. The coherent

damping of the signal can readily

be measured, particularly if the

analyzer is set to display a

logarithmic output. Note that

the shock excitation of the beam

drives modes with all values of m, each of which has a different coherent

frequency. Thus, by tuning the analyzer, we can check the damping of

each mode individually. Similarly, by measuring the damping as a func-

tion of beam intensity, we can determine the contribution of the beam's

environment to the damping (or instability).

(b) An FFT analyzer captures a time slice of the input waveform by

digitizing N equally-spaced samples of it and storing in digital memory.

The stored waveform is now subjected to a discrete Fourier transform and

the resulting spectrum is displayed. This procedure has the following

features :

--If the sampling frequency is fs' the duration of the time slice

("window") is N/f s. By the Nyquist sampling theorem, the highest

frequency which can be identified without al~a8 after such sampling is

fs/2--we need at least two samples per cycle.

--The analyzer assumes that the waveform repeats periodically outside

the sampling window; this introduces spectrum distortion ("leakage")

unless the window happens to intercept a whole number of cycles of the

frequency of interest. Leakage is reduced by softening the edges of

the sampling window, giving the data near the edges less relative

weight.

--The algorithm delivers both amplitude and phase information, referred

to the edges of the window. Where the window occurs at a random time,

the phase information has no value per 8e; but a two-channel analyzer

can display the relative phase of the Fourier components in the two

inputs, which can then be meaningful.


930

--From a single sample, the analyzer derives spectral information for

all of its frequency channels, typically 400 of them. Thus the effective

bandwidth at the input is 400 times wider than it would be if only one

resolution interval were processed at a time. The larger bandwidth

reduces the required dwell time by a corresponding factor. On the other

hand, time is "wasted" (as far as input receptivity is concerned) during

the computational interval, which typically does not overlap with

accumulation of a new sample of input data. Computation takes a frac-

tion of a second and becomes an important "tax" when the window time is

short, i.e., when high frequencies are being handled. Thus the FFT's

performance shines particularly at low frequencies.

--The high-frequency capability of the FFT analyzer is also limited by

the maximum digitizing rate its input sampler can achieve. Instruments

presently available tend to go only up to about i00 kHz; of course any

100-kHz frequency interval can be covered if it is appropriately shifted

by down-conversion.

--Most FFTs can operate in a bewildering variety of modes, since they

incorporate powerful computers and can readily be reprogrammed according

to need. For example, the analyzer can average successive samples in

the time domain before subjecting them to Fourier analysis, which is

useful if the sampling window can be synchronized effectively with the

signal under study. Alternatively, each input sample can be analyzed

separately and the resulting spectra can be averaged; this averaging

can cover such constructs as the relative phase or amplitude of the two

inputs in a dual-channel mode.

--The time-slice memory permits "post-triggering" of the analysis process:

the input is continuously rolled through the memory, and the desired

sample is frozen upon receipt of a trigger derived from some secondary

monitor (e.g., the detection of a sudden beam blowup).

--To match the broad-band input of the analyzer, we can stimulate the

beam with a broad-band signal also. For this purpose, the analyzer

delivers band-limited white noise--in a way the analog of the tracking

output available from a sweeping analyzer. If the stimulating noise


931

signal is presented to one channel of the analyzer and the beam's

response to the other, the correlation between the two tells us what

the beam's transfer function is over the whole broad-band spectrum

interval. Broad-band stimulation doses out the excitation to the beam

steadily, instead of in a single burst as the relevant frequency sweeps

by; thus, if there is damping continuously present in the beam, the total

(and sometimes deleterious) excitation may be reduced. On the other

hand, it may be technically more challenging to arrange for a broad-

band stimulating device.

7. Lock-ln Tune Measurement

An important requirement in many accelerators is the continuous

monitoring of the "tune"--either betatron or synchrotron frequencies.

Such continuous supervision may be needed during energy changes or

while the focusing lattice is being deliberately modified. It can also

help to diagnose many types of malfunction quickly. Ultimately, we may

wish to regulate the tune by negative feedback, using information

derived from the lock-in monitor and applying corrective action to

selected quadrupoles or RF parameters.

Shoker Pickup

earn

1 2 - - 8 2 4 4 2 9 A 4 3

Figure 43

The usual system is basically a

phase-locked loop (PLL) using the beam as

selective element. As indicated in Fig. 43,

we excite the beam with a sinusoid of

adjustable frequency, taken from a voltage-

controlled oscillator (VCO). We measure

the phase of the coherent beam response,

and vary the frequency until this phase

achieves a preset value (usually 90~ The

beam monitoring system shown is applicable

to transverse excitation, but the longi-

tudinal situation is very similar. In

either case a band-pass filter (BPF) is

used to isolate the desired frequency range,

so as to limit the dynamic range that the


932

phase-sensitive detector needs to accept. The whole loop's stability

is controlled with a low-pass filter (LPF) ahead of the VCO; the theory

describing the transient response is well developed in connection with

more conventional PLL systems.

Since the effective beam transfer function includes various phase

shifts due to beam transport time and cable delays, an adjustable phase

shift A~ is inserted to assure the loop locks onto the peak of the beam's

response.

When the loop is locked the VCO output is at the frequency of the

selected betatron sideband (or synchrotron satellite); it can be

processed in a standard frequency counter. The beam stimulation needed

to maintain this lock is often so small as to be indiscernible in the

operation of the accelerator. However, such lock-in loops are vulnerable

to losing lock as the result of some transient, and may then find and

lock onto another sideband; this spells disaster if the loop is part of

an overall tune feedback system!

8. Beam Transfer Function (BTF)

When the beam is stimulated by a sine-wave signal, its coherent

response has a component at the same frequency (as well as components

at the other frequencies tied to the same mode). The amplitude and

~ ~ : : ="~r---- =-------~:'_~ Longitudinol f (Cel~brotion ~ orA-Signel

/ \ co ,o,

I , , . : : : : :

Figure 44

phase of this response can be

determined, giving information

about the combined behavior of the

beam and of its electromagnetic

environment (the impedance of the complete vacuum chamber).[16]

Figure 44 shows the fundamental

setup for measuring such a BTF, in

this case by use of an FFT analyzer.

The BTF is defined by the amplitude

ratio A2/A 1 and the phase difference #2 - ~i' both of which are displayed

as a function of frequency by the analyzer. The system can be calibrated


933

if a suitable length of cable is substituted for the beam; the analyzer

will store the calibrating information and apply it as appropriate.

It is obvious that measurement of the BTF provides important

information about the beam's behavior. There are different levels of

sophistication at which this information can be extracted. At the

simplest level we regard the system merely as a means for stimulating

some coherent beam motion, permitting us to observe the characteristic

frequencies at our convenience. This stimulated spectrum study has an

important place in the day-to-day beam monitoring procedures.

More detailed information can be extracted from the amplitude and

phase of the BTF at the central betatron frequency. The amplitude of

transverse beam motion at any given point in the lattice is of course

proportional to the factor ~FB at that point; it is thus tempting to use

the observed amplitude to measure this factor. However, a mass of other

parameters enter into the amplitude of the observed signal--e.g., the

amount of beam stimulation and the gain of the processing chain and

analyzer. Thus a more realistic goal is to measure the relative values

of /8 at various points in the lattice by comparing the amplitudes of

the signals picked up there. Even this reduced objective still requires

great attention to detail if a reliable result is to be obtained. The

approach is not often exploited, in fact--especially because information

about /Sneeds twice the relative accuracy that a measurement of B

directly would require.

A somewhat cleaner measurement concentrates on the phase of the

pickup signal, which evidently contains buried within it the betatron

phase advance ~ to the location of the pickup station. Since ~ = /ds/8

we have here another approach to the lattice function 8. In particular,

we want to measure A~ between two nearby pickups and fit the result with

that calculated from our postulated ~ function. A suitable arrangement

for doing this is shown in Fig. 45. The beam is excited coherently,

either by a noise source or by a sinusoidal stimulus tuned to the desired

sideband frequency. A fixed reference pickup is connected to channel 1

of the FFT analyzer; channel 2 examines signals in turn from the two


934

stations A and B between which

the betatron phase advance is to

be measured. In displaying the

"transfer function" from channel i

to channel 2, the FFT focuses

attention on the relative phase

(and amplitude) between these two

inputs, thus eliminating the strong

overall variation imposed by the

[ I ~2 - 82 4 4 2 9 A 4 5

Figure 45

beam's o~ resonance be~vior. Thus, for ex~ple, if the frequency of

obse~at~n (or of stimulat~n, if a s~usoidal dr~er is employed)

should vary relative to the beam's betatron frequency, the signals to

both input channels would be changed dramatically, but their relationship

would be almost unaffected.

To disentangle the desired phase advance A~ from the other

par~eters which enter ~to the obse~ed p~se at the FFT, we need the

beam's transit time AT~ between the two pic~ps; this is generally ~o~

with great accuracy. We also need to ~ow the relative phase shifts

the separate processing segments which precede the A/B switch; with

suitable arrangements this may be reduced to a mere difference of cable

lengths, again easily cal~rated. Finally, we need to be sure that the

sig~is from the two pic~ps are sufficiently similar as to be treated

identically by the co~ part of the processing chain following the A/B

switch, which includes the FFT input itself. This processing equipment

is not sho~ explicitly in Fig. 45, but it might include such nonlinear

devices as pulse stretchers: in that case the effect~e phase shift

might well be a function of pulse ~plitude, and this would then require

equ~alent positioning of the beam at pic~p stat~ns A and B to produce

similar signals.

If a pre-exist~g system of ben monitors is to be used for such

betatron phase measur~ents, determination of cable lengths may be

somewhat cumbersome. In that case it ~y be poss~le to measure the

phase of a ~ n signal from the beam and use it to cal~rate the cables.

A bunched beam del~ers well ~o~ signals at the orbital revolution


935

frequency, with relative phases determined exclusively by the beam

transit delays AT. Phase comparisons made at mo are thus a good way to

calibrate relative cable lengths.

An interesting extension of this type of BTF measurements permits

determination of the coupling parameters which (in any nonideal machine)

connect the transverse oscillations in the vertical and horizontal planes.

(Such coupling can be produced, for example, by "skew" quadrupole lenses

or by longitudinal "solenoid" fields.) For determining the coupled BTF,

the beam is stimulated transversely in one plane and the response is

determined in the "crossed" plane as well as in the same plane. Relative

amplitude and phase measurements can then quantify the coupling.[17]

Evidently both the excitation system ("shaker") and beam-position pickup

must be orthogonal, i.e., capable of distinguishing cleanly between the

two transverse planes of oscillation.

Our attention so far has been focused on amplitude and phase of the

beam's response at a single frequency, namely the center of a particular

betatron sideband (or synchrotron satellite, for that matter). When the

beam consists of an ensemble of particles whose oscillation frequencies

spread over a certain range, the overall response function a8 a function

of frequency becomes of interest. A complete analysis of this situation

is beyond the scope of the present discussion; however, I include a

thumbnail sketch of the basics in order to show how measurement of the

BTF can help determine the stability limits of the beam in a manner

rather closely analogous to the way in which a network analyzer can

measure the stability margin of a feedback amplifier.[16]

(a) Response of particles to 8inusoidal excitation. Consider a group

of particles, each of which is an undc~ped oscillator of frequency Q~o"

If we excite them with an external drive at frequency w e they respond

with motion at the same frequency. Let's specify their position along

the orbit by means of the azimuthal angle 8; then we can form a coherent

pattern in the response if the displacement at @ is given by

Y = 9 eJ(~et - n@) (37)


936

with n any integer. This pattern streams past any stationary point of

observation with d@/dt = w o. The time variation of y is then found from

dy ~y + d@ ~y j -nm )y (38) dt = ~t d-~ 8@ = (We o

Place the external excitation at the point @ = 0, and let it impose

an acceleration Ge jmet on each particle at that point. The equation of

motion of the particle is then

d2y + Q2m2y = GeJWe t (39)

dt 2 O

which, with the help of (38), becomes

[-(w e - nmo)2 + Q2mo2] 9 = G

or -[(n+Q)mo - We][(n-Q)Wo - We ]9 = G (40)

This equation already shows the resonant response when the excita-

tion frequency w e coincides with either (n+Q)mo, a fast-wave frequency,

or with (n-Q)Wo, a slow-wave frequency. We'll suppose that w e is near

one or other of these frequencies, which we can call ~8• as appropriate;

one of the frequency factors in (40) then becomes (m8 - We) , while the

other (including the negative sign ahead of it all) is •

• ~ - We) ~ = G

= u i i (41) G 2Qm ~ me - m8

(b) Ensemble of particles of different natural frequencies. We now

suppose the natural frequencies to be distributed according to the

function f(ws)dms; the total number of particles, N, is evidently

N =If(m~)dw 8

Then the ~ruerage response of this ensemble is given by

f (m 8) dm 8 u 1 ~ A(We )

G = z-V~ool <=w s

(42)

(43)


937

where A(me) is the beam transfer function expressed in the units we

adopted here (i.e., excitation expressed as acceleration).

The integral in (43) is a dispersion integral which, on the face

of it, diverges because of the singularity at w e = m B. What saves us is

the continuous distribution function, which ensures that only a vanishing

number of oscillators has frequencies exactly coinciding with that of

the excitation. However, evaluating the integral is not straightforward

and exacts a certain penalty: we are forced to consider complex fre-

quencies, i.e., frequencies with imaginary parts which correspond to

growing or shrinking amplitudes. Now we have in fact neglected any

possible damping of the individual oscillators; so the appearance of

complex frequencies may seam strange. We must remember, however, that

the response A we're considering here is the average response of a whole

ensemble; the magnitude of this average response can grow or shrink as

the individual oscillators drift into or out of step with each other.

The dispersion integral is evaluated, as illustrated in Fig. 46, by

detouring around the singularity in a small semicircle. The semicircle

itself contributes an amount j~f(me )

Im w B to the integral, i.e., an imaginary

quantity determined solely by the

Re ~B value of the distribution function f

~e at the excitation frequency. (This

contribution is one-half of what a 2- B3 442~A46

Figure 46 complete circumnavigation of the pole

at w e would have yielded.) The rest of the integral comes from the

straight-line part along the real frequency axis--with a tiny gap left

where the circumnavigation occurs. Fortunately the integral so obtained

is independent of the size of this gap, provided it is kept very small.

Its computation is straightforward, once the distribution function f is

given: the (real) result is usually denoted by the abbreviation P.V.,

which stands for the Cauchy principal value; the complete dispersion

integral thus takes the form

f(mB)dmB

I ~e--~ = P.V. + j~f(me) (44)


958

You can see that, as the excitation frequency ~ is moved around in e

the vicinity of the region where the distribution function peaks, the

real and imaginary parts of the dispersion integral--and thus of the

transfer function A(me)--change. This is what a measurement of the BTF

displays directly.

The presence of the imaginary part gives rise to the possibility of

growing or decaying responses, i.e., to instability or to damping. This

phenomenon is known as Landau damping.

(c) Reaction of the beam to its own fields. The average motion of the

beam ensemble corresponds to a displacement of the effective beam charge.

It may produce electric or magnetic fields by interaction with the

environment, in such a way that the beam is deflected by these fields.

The resulting transverse acceleration must be combined with the external

G we've considered so far.

To characterize the beam environment, we consider a beam current I

to be displaced y from the center of the vacuum chamber and ask what

transverse deflecting fields result from this. Summing the effect over

the whole ring, we define the transverse impedance Z T according to the

conventional form [6]

(E + 8cXB)TdS

Z T E j~ ~I~ (45)

Z T is measured in units of ohms/meter (E.ds/l gives ohms, and the meters

come from the displacement y). To adapt our parameter to the actual

operating conditions, we further define

, elZ T

Z E 2~Rym (46)

with 2~R the circumference of the ring, m the rest mass of each particle,

and y the relativistic energy factor. Note that Z* is proportional to

the beam current I.

To the external excitation G we must now add an acceleration due to

Z*, given by G Z = Z*y/j. The transfer function then becomes


939

A* = ~ = A G i - A. jZ*

where A is the transfer function in the absence of Z*. This response is

just llke that of a negative-feedback amplifier of gain A, the feedback

path having gain jZ* (Fig. 47). The

Gc ~+/'~ ~ ~ measured beam transfer function

y represents A*, of course. From it,

information about A and Z* can be

deduced, especially if we note that

z-,, 4,,,,., Z* scales with the beam current I and Figure 47

can thus be controlled to some extent

(preferably without affecting the distribution function f at the same

timel).

A clearer display of these results is obtained by considering the

reciprocal of A*:

G= i Z* ~- j (47)

Lastly, to concentrate on energy transfer, we note (from: power =

force • velocity) that we're really interested in the velocity response,

v = j~ey , of the beam, not merely its displacement. This leads to

~G e i *

z (48) jA

showing how Z* combines directly with the term I/jA which is present

without the fields, i.e., at vanishing beam current. The real part of Z*

enters into the damping/antldamping balance; the imaginary part, on the

other hand, produces a frequency shift without directly changing the

damping. This frequency shift is due to the coherent motion of the beam

reacting back, via the environment, on the focusing forces controlling

the oscillations.

To end this quick summary, it is instructive to examine how Z*

collaborates with the Landau damping when we consider the possibility of

self-exclted instabilities. In this case the "stimulation" G arises


940

from the beam's coherent motion alone: we have to consider m to be e

determined by this motion, which implies that it moves around in response

to the imaginary part of Z* which produces the coherent frequency shift.

In Fig. 46 and eq. (44) we saw how the dispersion integral was affected

In particular, the imaginary part of A, eqs. (43) by the choice of m . e

and (44), is given by

- ~f (m e) + - -

2NQ~ o

(for fast and slow-wave frequencies, respectively) and therefore vanishes

if ~ falls outside the band in which the distribution function f is e

different from zero. In other words, Landau damping is lost if the

coherent frequency is shifted outside the band of natural ("incoherent")

frequencies present. In this indirect manner--by pulling the rug out

from under the Landau damping--the imaginary part of Z* can react back

on the overall stability of the beam, too.

Landau damping, which arises basically from the interplay of phases

among the members of an ensemble with different natural frequencies, is

of course most important in situations where such a spread of frequencies

is naturally present. This can arise inherently (e.g., in a coasting

beam with finite momentum spread) or it can result from nonlinear focus-

ing forces which make the frequency dependent on amplitude. This last

condition applies, for example, to longitudinal oscillations controlled

by a sinusoidal (not linear) RF voltage; or, in the case of transverse

oscillations, nonlinearity may be introduced through octupole lenses.

Most accelerators encounter coherent beam instability once the beam

current exceeds a particular threshold value. Stable operation can be

maintained at higher currents either by the deliberate introduction of

Landau damping via nonlinearities, or by applying an externally controlled

feedback force which responds suitably to the coherent motion of the beam.

Such feedback systems are in common use for both longitudinal and trans-

verse beam control. To supervise the situation, it is fruitful to

measure the beam's transfer function just below the instability thresh-

old, preferably as a function of beam current. One usually reduces the


941

data to the form used in eq. (48), i.e., to the reciprocal velocity

response; plotting them in this form yields what is known as the stability diagr~ for the beam. External feedback contributes to this plot in a

manner analogous to the way Z* enters. The modified stability diagram

then indicates directly how the feedback system is operating. J18]

Our whole development was, of course, based on consideration only of

frequencies in the vicinity of one of the beam's spectral lines--fast-wave

or slow-wave. We must remember, however, that coherent motion in any

particular mode involves a complete manifold of such spectral lines, as

set forth in section 1.5. Stability is determined by the algebraic sum

of damping terms for all of these lines. The relative weight of the

lines depends on their intensity in the spectrum, i.e., on their position

under the power-spectrum envelope. This envelope is in turn related to

the Bessel function of appropriate order m for the mode under considera-

tion; its detailed shape depends also on the particular form of the mode

being developed, which finally must be selected so as to produce a self-

consistent form for the fields in conjunction with the impedance function

Z*. The complete analysis is thus quite complicated. It is usually

approached by postulating some plausible frequency variation of Z* and

a convenient (though not necessarily exact) form for the charge profile

of the bunch. Measurements of the BTF can then be carried out at various

frequencies to arrive at parameters for the model and to correct its

basic form, if necessary. Additional information can often be obtained

by deliberate variation of the chromaticity, which displaces the spectrum

envelopes along the frequency axis (section 1.5b) and thus permits some

exploration of how Z* in fact varies with frequency.


942

IV. SYNCHROTRON RADIATION

The radial acceleration of charged particles in a circular segment

of orbit causes them to emit 8ynchmotron radiation. The effect becomes

important particularly at extreme-relativistic energies. Very high values

of y are readily attained by electrons: for protons, we must await the

achievement of energies in the TeV range before their synchrotron radia-

tion takes on an engineering significance.

The electromagnetic beam pickups we have considered in earlier

sections all utilize the near field of the charges. The radiated field

has some quite different properties. Synchrotron radiation provides an

entirely new view of the beam.

i. Performance Data

A particle of charge e, deflected by a transverse magnetic field

into an orbit of radius R, radiates a total power

= 2 reC E4 y4 PY 3 (me 2) 3 R 2 ~ 0.29 ~-~ eV/s (49)

where r = e2/4~e mc 2 = 2.8xi0 -15 m is the classical electron radius, and e o

R is measured in meters. This power is radiated into a forward-directed

cone of very narrow opening angle, because of relativistic effects. The

cone can be likened to the headlight beam from a locomotive traveling

along a curved track. At a fixed point of observation on the ground,

the observer receives a brief flash of light. The Fourier components

of this flash extend, in an effectively continuous spectrum, to very high

frequencies. The spectral distribution of power can be written in terms

of the critical frequency, ~ : c

p(m)d~ = -~Y S ~ dm (50) ~0C % 0~C i

where ~ = r: % = (51) c 2R c 3y3


943

~" 0 .4 ~ .

3 ~-" o.z

0 i I i ,. 0 0 . 5 1.0 1.5 2 . 0 2 .5

2 - 83 ~--'----" (~/(4)C 4429A++8

Figure 48

lished), for beam detection we are

quencies--the visible and near-ultraviolet. The spectral density in this

low-frequency range (m << mc ) is approximately

i/3

and S is a universal distribution

function of the normalized frequency

m/mc, sketched in Fig. 48.[19] Though

the high-frequency end of this spectrum

has proved of great value as a source

of far-ultraviolet and X-ray photons

(many laboratories utilizing such

synchrotron source8 have been estab-

usually interested in the lower fre-

8c

R ~

z-s3 ,.29~4, ~-- L ---~

Figure 49

The opening angle of the cone of

radiation is roughly e c = i/y. As shown

in Fig. 49, this implies that a fixed

point of observation views a short seg-

ment of the orbit, of length

2R L =-- (53)

Y

The duration of the pulse of light is far

shorter than might at first be suspected,

given the length L: the particle and the light travel at roughly the

same speed, and the radiation stays level with the particle except for

the fact that it goes along a straight cord while the particle follows a

circular arc. It's the resulting small path difference, proportional to

(I - cos e c) = e~/2, which spreads out the arrival time of the radiation

relative to the delta-function corresponding to a single-point radiating

particle. The pulse duration is roughly R/X3c, which can typically be

10 -18 s!

The vertical opening angle (i.e., for a horizontal orbit) in the

long-wavelength region is


944 0~. -~y (54)

Typically we might have % = i nm, which is about 10 -3 of the C

wavelength of visible light. Thus, despite the slow power-law dependence

in eq. (54), the opening angle for visible light can be much larger than

e = i/y. For example, in an electron ring at 5 GeV, e = 0.i mrad, while C C

visible light might spread by • mrad above and below the plane of the

orbit.

It's worth noting that synchrotron light is strongly plane-polarized:

the polarization (in the plane of the orbit) reaches 100% for emission at

zero angle.

2. Utilization [20]

Synchrotron light flies out tangentially, like mud from a spinning

wheel, in a thin region close to the plane of the particle's orbit. This

light can nevertheless be used to form an image of its source, as illus-

--83 4A29A30

Figure 50

trated in Fig. 50. The fan of light

is focused by a lens, forming an

image which corresponds to the apex

of the fan. However, the figure makes

it clear that the effective source has

considerable depth along the line of sight: this gives rise to depth-of-

field limitations in the sharpness of the image. There are also diffrac-

tion limits to the achievable resolution, as we'll see.

Within these limits, the lens forms an image of the beam's cross

section in the plane normal to the direction of viewing, permitting us

to see directly (and in "real" time) the transverse density distribution

of the particles. The instantaneous image yields a mapping of the beam's

incoherent emittance. When there is coherent motion of the beam, as in

the case of coherent instability or deliberate beam stimulation, that

motion is superimposed on the incoherent image and smears it out to an

extent depending on the averaging time of the imaging system. (TV

cameras, especially those using vidicons, have relatively long


945

averaging times.) For image detection with extremely short time

resolution, a streak camera may be used: this permits us to follow

changes of beam profile from one passage to the next of the bunch past

the point of observation. J21]

In electron machines it is common practice to display the beam's

image as picked up by a TV camera; this facilitates quick supervision

of performance and often permits a first diagnosis of beam instabilities.

(The polarization of the light allows us to use a rotatable polarizing

filter for intensity control.)

Since the pulse of light from a single particle is of negligible

duration, the time structure of the actual radiation is an excellent

replica of the longitudinal bunch profile imaged along the time axis.

With the advent of extremely fast photodetectors, the bunch length can

now be measured accurately with a resolution approaching i0 ps (3 mm of

length).[22]

Time and space discrimination can be combined if the beam's image

is observed on a photodetector through an adjustable slit. For example,

coherent beam motion produces corresponding intensity fluctuations if

the slit is offset from the center of the image. In view of the very

high intensity of synchrotron light, it might appear attractive to

examine the extreme fringes of the transverse beam profile by moving

the slit across the image; unfortunately most imaging systems are subject

to stray halos and parasitic reflections, caused by the constraints of

the vacuum-chamber geometry. Thus, unless very clean viewing conditions

can be specifically engineered, the outer parts of the image tend to be

unreliable for quantitative observations.

The advantages of such direct, nondestructive viewing of a beam are

so great that they arouse acute envy in operators of proton machines~

It is in fact possible to enhance the high-frequency spectrum radiated

by a particle at relatively low y, by artificially shortening the

duration of the radiation pulse. This can be achieved, for example,

by letting the particle radiate from a short segment of curved track.

Alternatively, the transition from a curved to a straight orbit causes


946

the radiation to go out in a sharp step-function in time, which also

contains many high frequencies. In this manner, usable amounts of

visible light have been gathered from protons in the 100-GeV range at

the edges of magnets in the guide field. However, the light is so feeble

that image intensifiers are needed for its detection.[23]

The gUide field can also be modified by including, in a nominally

straight piece of orbit, strong alternating wiggles crowded as close

together as possible. If the instantaneous transverse acceleration is

larger than that normally used in the circular arcs of the guide field,

greatly enhanced synchrotron radiation can be obtained. The wiggler can

consist of alternating fields in a single transverse plane, or it can be

a transverse field of constant magnitude, but twisting helically around

the beam axis.

If the angular deflection of a single wiggle is kept comparable to

the emission angle of the radiation, coherent effects from successive

wiggles can be utilized. Such structures are often called u~fulators.

Let's characterize the spatial alternations of the undulator field by

the wavelength % . Then the interference criterion linking the radia- u

tion's wavelength and the angle of emission is [24]

= ~o (I + y202) ; ~o E 2Y 2u (55)

The parameter I involves the matching of particle and radiation travel o

times. A typical situation might be: lu = 0.i m; y = 10b; lo = 10-9 m.

In this case, visible light (~ = 10 -6 m) would be distributed in a cone

of relatively large opening angle, making y0 >> i; then (55) becomes,

approximately,

i = ~ $2%u (56)

3. Limits of Resolution in the Image [24]

We have noted that the longitudinal distribution of the radiation

matches that of the bunch within extremely close limits, placing the

resolution limit in the detection equipment. In the transverse dimension


947

the situation is much less favorable. I wlll give only order-of-magnltude

arguments here; I'ii also concentrate on resolution in the vertical

direction, since most often we have to deal with horizontal ribbon-like

beams and therefore need the greatest precision in the vertical dimension.

The problems can be addressed under the two headings of (a) geometrical

and (b) physical optics.

(a)

focus of a lens of given aperture.

1111 .~

111 ~ .~" ..~" ~

J.-dc- ', ,' L_ L \ ^ ~ I I

I 2 , 2 ~ G I I

Geometrical Optics. The problem here lles in the limited depth of

The simplest demonstration of this

effect, as in Fig. 51, considers

what happens if you view the scene

from an angle e which corresponds

to the outermost edge of the imaging

lens, i.e., to the half-aperture of Figure 51

the system. An extended source of

length L in the viewing direction, but of negligible transverse dimen-

sions, then appears as a ribbon of height • where

i ~YG = ~ L~ (57)

(b) Physical Optics. As illustrated in Fig. 52, the limited emission

angle of the radiation implies that, at a distance D, the beam width is

given by H = 28%D. The angular resolution of such a beam is diffraction-

limited to %/H = %/28%D, which is equivalent (in the source plane a

distance D away) to a linear smearing by

% Ayp -" 28% (58)

i l Figure 52

Optimum resolution in the image

thus requires a compromise between

depth-of-focus problems (minimized

by small aperture) and diffraction

(minimized by large aperture). Let's

see what happens if we accept the

natural aperture limit set by the


948

emission angle 8%,

We then have a source length L = 20%R.

0%= %

which leads to

restricting our horizontal aperture to the same value.

(51) and (54) we obtain

(4__~.~ /3 2 /3 R1/3 2/3 113 Ay G = = 0.38 % R (59)

~ )1~/3 2/3 1/3 2/3 1/3 and Ayp = % R = 0.8 % R (60)

Thus, on this rough estimate, the geometrical and physical resolution

limits are similar, indicating that the choice of using the natural

aperture is reasonably close to an optimum compromise. (This situation

is not coincidental: image formation needs to be analyzed on a unified

basis, of course, not via separate geometrical and physical models.)

Given that the vertical aperture is naturally limited by the

emission angle, it is usually best to let the optical system have a

greater acceptance than strictly necessary: this minimizes the un-

pleasant side effects caused by minor misalignments. The horizontal

aperture must of course be limited by a slit (or by the edges of the

lens), since we have a fan-shaped beam illuminating the Optical system.

In typical situations the resolution achievable is of the order

of a fraction of a millimeter, which may be quite inadequate where the

beam is a ribbon only some tens of pm thick.

To achieve improved resolution, we are forced to consider the use

of shorter wavelengths in order to minimize the diffraction problem.

Unfortunately the opening angle also shrinks slightly as shorter

wavelengths are detected. Also, radiation in the ultraviolet is

relatively difficult to process. The depth-of-focus limitation of

course remains: restricting the horizontal aperture to reduce the

effective source length is ineffective once the smearing-out by the

angle 0% has become dominant.

From eqs.

-1/3 R


949

A significant improvement can be had by going all the way to X-ray

wavelengths, for which diffraction becomes a negligible limit. One can

then use the equivalent of a pinhole camera to form an image of the beam,

the resolution being degraded chiefly by edge effects at the "pinhole"

(or horizontal slit) if that geometrically limiting dimensions is made

too small. Filtering out the softer, scattered radiation can be helpful

here.

An alternative approach is to enlarge the emission angle artificially

by shortening the effective source length; this also relieves the geo-

metrical problem. For example, we could consider viewing light from a

very short bending magnet, or from the edge of a magnet. Unfortunately

it isn't easy to produce "short" magnetic elements on a scale compared

to the usual natural source length L, which is commonly around 0.i m:

the aperture needed for the beam prevents us from making very sharp

changes of magnetic field along the orbit.

4. Detection Equipment

(a) Total Intensity. Synchrotron light provides a very attractive way

of monitoring the average beam intensity in an accelerator. In a storage

ring with counterrotating beams, light is emitted in opposite directions

from the two beams, which naturally provides for independent monitoring

channels.

Once the critical frequency of the radiation is well above the

visible region to be monitored, combining eqs. (49)-(52) shows that

-2/3 1/3 p(~) ~ R co (for ~ << ~c ) (61)

independent of the particle energy xmc 2. Thus the visible light from a

constant beam current in an electron accelerator is independent of the

beam energy in this (easily attained) limit.

Limitations of this approach to intensity monitoring are of a

practical nature and hard to circumvent:

--Many photodetectors have inadequate long-term stability. Perhaps the

best candidates are photovoltaic detectors. Photomultipliers are


950

subject to fatigue and also to erratic gain changes, possibly caused by

charge accumulations on internal insulating surfaces.

--The effective aperture of the optical system may vary with beam posi-

tion or other conditions. (Though much of the radiation is concentrated

in a narrow cone, the large solid angle that lies far out can contribute

substantially to the overall detection yield.) Vacuum-chamber limitations

may prevent the use of a really generous collection aperture. In that

case, movement of the beam (for example due to manipulation of the lattice

in a storage ring) can produce false indications of intensity change,

sometimes sounding a beam lifetime alarm when no actual losses had

occurred.

--The power density on the primary mirror which deflects the radiation

out of the vacuum chamber (usually through a lateral viewing port) may

be uncomfortably high. If no mirror is used, an in-line port window may

be similarly exposed to an intolerable heating stress.

--Window materials may be darkened by the radiation, changing the light

transmission with age.

(b) Time Structure. Among photodetectors with good time resolution,

planar vacuum photodlodes have until recently occupied first place.

Such detectors can, with care, be matched quite well to transmission

lines. Reflections produced at the penetration through the vacuum

envelope can be made relatively harmless if the distance between the

feedthrough and the photocathode is large enough so that the reflection

arrives after the waveform of interest has already passed. Rise times

of order i00 ps are attainable.

Recent developments of silicon photodiodes have produced devices

with rise times of order i0 ps, although there appear to be problems

with charge storage so far. (These can be bypassed by suitable pulse

clipping with a shorted stub.)[22]

The observation of such fast waveforms normally requires the use

of a sampling oscilloscope, placed not too far from the detector itself.


9SI

Since the immediate vicinity of the accelerator is often inaccessible,

we have the choice of transporting the light to a remote detector or of

placing the sampling head of the scope near the accelerator, with the

rest of the control and display electronics further away. The trigger

for a sampling scope must be free from jitter on the time scale required

for the resolution: a trigger signal derived from the light signal itself

may be used, with an optical delay path ahead of the detector proper so

that the signal may be correctly displayed.

(c) Transverse Structure: Imaging. In the visible region, standard

optical techniques are applicable. Flexibility of instrumentation is

important: this requires the use of remotely controlled focus and field

adjustments, optical attenuators, etc. The image from a single high-

quality telescope may be split with pellicles to serve several different

detectors.[25]

Beyond the usual TVmonitoring cameras, we commonly use scanning

slits or fast image dissectors to quantify the transverse beam-density

information.[26] Streak cameras have been used to photograph successive

single passages of a bunch past the observation point, thus recording

transverse structural oscillations on a very short time scale.[21]


952

REFERENCES

The publications in this field are often relatively inaccessible,

taking the form of internal laboratory reports or brief conference

presentations. Unfortunately very few libraries are blessed with com-

plete sets of reports from the various major laboratories. I have

therefore chosen to cite, wherever possible, what is likely to be the

most widely distributed part of the literature: the Proceedings of

Accelerator Conferences. This selectivity often results in citation

of secondary sources instead of the original work. My apologies to the

originators of these ideasl Primary sources are usually quoted in these

papers and can be traced according to their local availability.

To abbreviate the references, International Accelerator Conferences

are cited by number, location, and year: e.g., Conf. IX, SLAC (1974).

Proceedings published by the IEEE as Transactions on Nuclear Science

are cited by volume and year: e.g., NS-26 (1979).


953

[i] J. J. Livingood, "Cyclic Particle Accelerators," D. Van Nostrand,

Princeton (1961), Sec. 6-4.

[2] J. Borer & al., Conf. IX. SLAC (1974), p. 53.

[3] J. L. Laclare, Conf. XI, Geneva (1980), p. 526.

[4] T. Linnecar and W. Scandale, NS-28 (1981), p. 2147.

[5] F. J. Sacherer, Conf. IX, SLAC (1974), p. 347.

[6] F. J. Sacherer, Proceedings International School Particle

Accelerators, Eriee: CERN 77-13 (1977), p. 198.

[7] A. Piwinski, NS-26 (1979), p. 4268.

[8] R. E. Meller and R. H. Siemann, NS-28 (1981), p. 2431.

[9] W. Radloff, NS-26 (1979), p. 3370.

[I0] K. Unser, NS-28 (1981), p. 2344.

[Ii] T.P.R. Linnecar, NS-26 (1979), p. 3409.

[12] Q. A. Kerns and D. B. Large, "Analysis of a Traveling-Wave Beam

Electrode," Lawrence Radiation Laboratory UCRL-II551 (1964).

[13] M. Sands and J. Rees, SLAC Report PEP-95 (1974).

[14] F. B. Kroes & al., NS-28 (1981), p. 2362.

[15] Q. Kerns & al., Counting Note CC2-I, Lawrence Radiation Laboratory

(1956, revised 1959).

[16] J. Borer & al., NS-26 (1979), p. 3405.

[17] J. P. Koutchouk, Conf. XI, Geneva (1980), p. 491.

[18] E. Peschard, Conf. XI, Geneva (1980), p. 506.

[19] M. Sands, "The Physics of Electron Storage Rings," SLAC-121 (1970),

Chap. V.

[20] A. Hofmann, NS-28 (1981), p. 2132.

[21] A. P. Sabersky and M.H.R. Donald, NS-28 (1981), p. 2449.

[22] E. B. Blum & al., Nucl. Instrum. Methods 207 (1983), p. 321.

[23] R. Bossart & al., Conf. XI, Geneva (1980), p. 470.

[24] A. Hofmann and F. Meot, CERN/ISR-TH/82-04 (1982).

[25] A. P. Sabersky, NS-28 (1981), p. 2162.

[26] W. Ebeling, NS-28 (1981), p. 2160.


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