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ENVELOPE DESCRIPTION OF QUASI-LAMINAR BEAMSUNDERGOING REVERSIBLE EMITTANCE TRANSFORMATIONS

Luca Serafini, INFN-Milano, Milan, ItalyJames B. Rosenzweig, UCLA Dept. of Physics and Astronomy, Los Angeles, USA

Abstract

A fully analytical description is presented for theenvelope behavior of intense, space charge dominatedbeams which are relativistic and quasi-laminar. It is basedon a particular solution of the envelope equation for anaccelerated beam, which is invariant under reversibleemittance transformations: this solution has been termedinvariant envelope . The treatment presented in this paperis applicable both to bunched beams in transport lines andin linacs, whenever the beam is space charge dominated.

The main interest is in maximizing the beambrightness achievable by Photo-Injectors[1] and preservingthe beam quality during the first stages of acceleration.Simple analytical formulae are provided to predict theperformances of these devices, when they are operated inthe space charge emittance correction regime [2].

1 EMITTANCE OSCILLATIONS IN AQUASI-LAMINAR BRILLOUIN FLOW

Quasi-laminar relativistic beams have become of greatinterest with the advent of laser-driven RF Photo-Injectors, which are able to produce electron beams

carrying current densities well in excess of 1 kA/cm2,

with a fast transition from the non-relativistic regime tothe relativistic one. Using accelerating gradient from 10MeV/m up to 100 MeV/m at RF frequencies in the 144MHz - 17 GHz range, these injectors accelerate the beamfrom rest at the photo-cathode emissive surface up torelativistic energy within a fraction of an RF wavelength,which is a distance comparable to one-half of plasmaoscillation period in the transverse plane. The trapping

condition is > 1/2, where eE0/2kmc2 is the

dimensionless amplitude of the vector potential associated

to the RF field, of frequency RF (k = 2RF /c) andamplitude E

0. From the transverse emittance point of

view, the usual operating conditions are such that therandom, thermal component to the transverse emittance ismuch smaller than the total rms emittance, which isdominated by the dilution of the transverse phase spacedensity caused by longitudinal-transverse correlations. Thebeam is fairly laminar both in the transverse plane, sothat trajectories do not cross each other, and in thelongitudinal one, where different slices do not mix eachother. For the purpose of analysis the beam may bebroken up into nearly independent longitudinal sliceswhich behave in a similar way as continuos beamlets.

Any beamlet is identified by its position = z vbt,relative to the bunch centroid position z, which moves

along the z-axis at a velocity vb = c .

In this section we clarify the concept of reversibleemittance transformations by using a simplified model: amono-energetic bunched beam drifting through a uniformfocusing channel, represented by an intense non-neutralcold plasma with slowly varying density longitudinally,so to satisfy the requirements of transverse andlongitudinal laminarity. The envelopes of the equivalentensemble of beamlets can be described by an extendedLawson's paraxial ray rms envelope equation [3]

+ Krks ()

3

3

=th

2

3 (1)

which governs the evolution of a cylindrically rmstransverse beam spot size (z,) under the effect of an

external linear focusing channel of strength

Kr = Fr /2mc2: = d2(z,) dz2 , while

mc2 is the mean beam energy. The defocusing space

charge term is proportional to the local beam perveance

k()= 2Ig()/I0 , where g() is a geometrical factorwhich contains the longitudinal dependence of thetransverse space charge field versus the internal bunch co-

ordinate ( I is the bunch peak current and I0 = 17kA isthe Alfven current for electrons). The RHS term of eq. 1

represents the outward pressure on the envelope due to thethermal emittance, which is expressed by

th =

1

2r2

r 2

r r

2(2)

where f

indicates a local average (i.e. on a slice at

= const.) of the quantity f(r,,z) over the transversephase space distribution in the (r, r ) plane (cylindricalsymmetry is assumed).

This is often called the rms slice emittance [4] and itrepresents the contribution arising from both random,thermalizing sources as well as the effects of nonlinear

macroscopic forces (e.g. spherical aberrations).The laminarity assumption is equivalent to neglect the

RHS term in eq.1, that is to consider a Brillouin flow,which represents the equilibrium condition predicted by

eq. 1 for th = 0 ,

eq()=2Ig()

I0Kr()3

(3)

Assuming that the beam is injected into the focusingchannel matched on the central slice, i.e.

0 (z= 0,)= eq( = 0), all other slices will bemis-matched by a quantity ()= 0 eq(). As far

as the initial mis-matching is small, i.e.

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may obtain the equation for small amplitude motion bylinearizing eq.1 about the equilibrium condition (3):

+ 2Kr = 0 (4)

which shows an oscillation frequency dependent on theexternal focusing but independent on the slice position

. It is this characteristic of the space charge dominated,quasi-laminar beam dynamics that allows emittancecompensation. As a matter of fact, all envelopes of thebeamlet ensemble oscillates with the same frequency butdifferent amplitudes, so that the total rms emittance of thebunch grows, but returns periodically to minimumvalues. Formally we have

(z,) = eq () + () cos 2Krz( ) (z,) = 2Kr()sin 2Krz( )

(5)

The total rms emittance (z) is defined as

(z)= 1

2r2 r 2 r r

2

(6)

where the averages are now performed over the whole

ensemble of beamlets, i.e. f(r,z,)

r(r,z,)f(r,z,)drd r(r,z,)drd . By

substituting eq.5 into eq.6 we find:

(z) = 2Kr eq2

2

eq 2

sin 2Krz( )

which becomes, taking into account only first order termsin ,

(z) = 2Kr0 eq sin 2Krz( ) (7)

It can be seen at a glance that the emittance minima

occur when the angles in phase space = tan1( /)are independent on the slice position , i.e. are vanishing

all over the beam. This happens at the maximum andminimum beam size. This type of behavior is in factsimilar to what observed in RF Photo-Injectors, as it canbe seen from multi-particle simulations, where the beamundergoes one envelope oscillation and two emittanceoscillations from the cathode to the injector end. Sinceacceleration is applied in this case, further emittanceoscillations are damped by diminishing space charge

forces through acceleration[7].

2 THE CONCEPT OF INVARIANTENVELOPE

The more general envelope equation for an axi-symmetric relativistic bunched beam in a linac reads:

+

+ / 8 + b2( )

2

ks ()

33=

th2

3(8)

where = 1+ z= 1+ kz and bis the ratio betweenthe magnetic field of an external solenoid lens and theaccelerating gradient. represents the ponderomotive RF

focusing term[5]: it is usually close to 1 for standingwave linacs while is almost negligible for travelling wave

ones. In order to find the equilibrium condition for a spacecharge dominated laminar accelerated beam, we transformeq.8 (under a laminarity assumption) from the real space

((z),z) into a Cauchy dimensionless space ((y),y),

where the equation becomes similar to eq.1 for a driftingbeam with an equivalent exponentially damped perveance:

d2

dy2

+ 2e

y

= 0 (9)

Here y ln(/0 ) , 0 /ks () and

/ 8 + b2 (0 is the initial beam energy). Due tothe exponential damping of the space charge term, theequivalent of Brillouin flow in the Cauchy space is nolonger a constant beam spot size transport mode, but it is

now given by an exact particular solution of eq.9,which is termed invariant envelope :

ey/ 2 / 3 / 8 + b2 (10)

which, in real space, becomes

=1

ks () / ( 3 / 8 + b

2) (11)

The reason for the name is the invariance of the phase

space angle / = ( /)(d/dy) = / 2 versusthe slice position , which, as explained in previous

section, is the basic condition to obtain a vanishing linearcorrelated emittance. It is interesting to notice that theinvariant envelope (IE ) exhibits a constant (versus z)

ratio p /Lg = 1 / 3 / 8 + b2 between two fundamental

scale lengths, the local plasma wavelength

p = 2c /p , p e n /0m3= (c /) ks () /

3

and the local incremental energy gain length Lg = / .In order to study the emittance behavior of the

ensemble of beamlets (slices distributed in ), we first

note that theIE is a stable region for small deviations inCauchy space: the small amplitude motion around is specified by

d2

dy2+ 22 +

1

4

=

0 (12)

which implies an oscillatory behavior for . This stablemotion has, like the small amplitude oscillationsdiscussed in previous section, frequency independent of thespace-charge strength, i.e. it is invariant for any slice.Going again through the same analysis for the rms

normalized emittance n(z) done in previous section

(here n(z) (z)), one finds a damped oscillatory

behavior as 1/ with oscillations at 2 times (for

b = 0) the frequency of the perturbations from theIE:

n ks

(3 + 8b2 )

00cos

02

sin

2

+

0

2cos

+

0 02

sin

2

(13)

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where ln /0( ) 1 / 2 + 2b2 and

0 = 0 () 0 () , 0 = 0 () 0 () are theinitial deviations of any beamlet from their own IE

conditions. The reason for the damping as 1 / is that

the equilibrium condition is now decreasing as 1 / ,while in Brillouin flow (see eq.7) the equilibrium spot

size eq is constant.Eq.13 shows that, as far as any beam slice is matched,

at injection, on its own proper IE , there are no further

emittance oscillations and n(z)= 0 . Unfortunately, thiscondition of full matching is in practice very difficult toachieve: if the beam is matched in a rms sense, i.e. theslice representing the rms beam behavior is matched onits ownIE, then all other slices which are not directly onthe IE will perform stable oscillations around the IE ,leading to damping of the rms normalized emittance of thefull beam ensemble.

It should be recalled that, in case of a standing wavelinac, the envelope is actually a secular one: actually, it isaveraged over the cell-to-cell oscillations caused by thealternating gradient focusing effect associated with thebackward component in the RF standing wave[5]. Whenthe beam leaves the accelerating structure one must add apositive (defocusing) kick, = /2 , to obtain thecorrect connection between the secular envelope in thelinac and the actual envelope outside. Since

= / 2 any beam transported on itsIE leaves thelinac at = 0, i.e. matched to a Brillouin flow in thefollowing drift. In this way, the linac+drift system isoperated under a global IE condition: this, again, points

out the analogy between theIE and the Brillouin flow.In case the beam is drifting at relativistic energies, as

usually at the exit of a RF gun, the envelope eq.1, withinlaminar conditions, becomes

1/= 0 (14)

where (z,) (z,) 3 /ks(). The generalsolution is given by

dx/ 0

2 + 2lnx1

/0

= (z z0)/0 (15)

while the small deviations around 0 are given by

+ /0

2 = 0 (16)

which implies stable oscillations with frequency 1/0around 0. The beam exiting a RF gun operated in the

emittance correction regime has typically a small negativedivergence, so that it goes through a waist and, if notfocused and/or accelerated, it diverges further away. If thebeam spot size at the waist is only slightly smaller than

0 , the drift space in the waist region becomes

comparable to a quasi-Brillouin flow with local stabilitycondition similar to the one described in sect.1. This isthe basic condition to achieve emittance correction in thedrift space. Since eq.14 is universally scaled, the conditionto achieve an invariant envelope like operation is the

invariance of the frequency 1/0 with respect to slice

position , which is equivalent[6] to

d0d

= 0 ;d 0d

= 0 (17)

3 EMITTANCE CORRECTION IN RFPHOTO-INJECTORS

We consider in this section a model for the applicationof the IE concept to the actual beam dynamics of RFPhoto-Injectors. A 1+1/2 cell RF gun is followed by adrift and a booster linac: here the RF field is taken as aTM010- resonant mode with pure first spatial harmonic

of amplitude (at the cathode, z= 0) E0 extending up to

z2 = 3RF /4, while the magnetic field of the externalsolenoid is constant with amplitude B0 from

zB1 = RF /8 up to zB2 =11RF /8 .We look for a solution of eq.17 in terms of four free

parameters, namely , b, , A , characterizing the RF

gun operation: Ip / 2r2 is the Cauchy current in

terms of the peak current Ip of the bi-gaussian electron

bunch and A r /z the laser pulse aspect ratio interms of its gaussian widths r and z (b cB0 /E0).

As shown elsewhere[6], it is possible to solve thesystem of two equations 17 in the four previousparameters, getting and b as functions of and A :an approximate solution is found to be

[kA] = 57.312.4+ 2.632 + 2.62A +

1.86A2 1.78A

b = 1.49 +1.67 / 2.071/ 4

REFERENCES

[1] J.S. Fraser et al., IEEE Trans. Nucl. Sci. NS-32(1985) p.1791

[2] B.E.Carlsten, IEEE Cat. 89CH2669-0 (1989) p.313[3] J.D. Lawson, The Physics of Charged Particle Beams

2nd ed. (Oxford Univ. Press, New York, 1988)[4] P.G.O'Shea, "Entropy and Emittance Growth in

Charged Particle Beams", subm. to Phys. Rev. Lett.X.Qiu et al, Phys. Rev. Lett. 76 (1996) p.3723

[5] J.B.Rosenzweig et al, Phys.Rev. E-49 (1994) p.1599[6] L.Serafini and J.B.Rosenzweig, to be published[7] J.B.Rosenzweig, E.Colby, AIP-CP 335 (1995) p.724