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ANALYTICAL MODEL FOR EMITTANCE COMPENSATION

IN RF PHOTO-INJECTORS

L.Serafini, INFN-Milan, Via Celoria 16, 20133, Milan, ItalyJ.B.Rosenzweig, Department of Physics, UCLA, Los Angeles, CA 90024

In this paper we present a new model to representanalytically the transverse beam dynamics in RF photo-injectors. It consists basically of an enhanced Kim's model [1],with incorporation of RF ponderomotive focusing effects,external magnetic focusing and a perturbative treatment ofspace charge along the beam envelope. Applying the resultingformulas it is possible to predict with high accuracy thetransverse beam envelope behaviour in a multi-cell RF gun, aswell as the operating conditions to achieve space chargeemittance compensation according to Carlsten's scheme [2].The agreement with sophisticated numerical simulations isreally quite satisfactory, as well as the match with

experimental measurements of the predicted operating range foremittance compensation.

I. BEAM ENVELOPE IN MULTI-CELL RF GUNS

The basic model adopted for calculating the beam envelopeconsists actually of an indefinitely long RF gun, supporting aT M 010- standing resonant mode (frequency ) withaccelerating field Ez = E0cos(kz)sin(t+0) (E0 is the peakfield at the cathode, located at z=0, k=/c, =2). The fieldis expanded linearly off-axis to find the transverse E r and Bcomponents [1]. An external solenoid is assumed to be foldedaround the first 2+1/2 cells of the RF gun cavity, producing a

constant magnetic field Bz = B0 from z=0 up to zC=(5/4) .Under the approximation > 1/2 , where = eE0/(2mc2k)is the dimensionless field intensity, it has been extensivelyshown elsewhere [3] that the beam envelope conditions 2 and2' at the second iris location z2=(3/4) can be written, for agaussian charge density distribution in the bunch, as

2 = cat 1 + SC B( )

orb =

2

cat pRF + SC 2B 1 + p

RF B

2 = orb

222

1)

where cat is the laser spot size at the cathode, pRF is theRF defocusing kick

pRF = 2 Log(1)

1 1Log(2)

82

Log(1 23 )

2(2 1)

,

B is the magnetic focusing term B = (1/2)[ksLog(2/B)]2

(ks=cB0/E0) and SC is the space charge defocusing kickSC = (1+SC)[1-Log(2)/(2-1)], calculated a la Kim . Thespace charge dependence on bunch and field parameters is

specified by SC = Z0 I / (8'E0 Ia 2cat) , where Z0=377ohm, I is the bunch peak current, Ia=Alfven current, ' = kis the dimensionless energy gain per unit length and specifies the dependence on the bunch aspect ratio A = r/z ,

= 1 / (2.45+1.82A1.25-0.55A1.5 ) . The beam energies 1and 2(in rest mass units) at the first and second iris are givenby 1 = 1 + /2 and 2 = 1 + 3 /2 (B = 1 + /4 ) .

After the second iris the beam envelope can be easilytracked, as far as the space charge can be assumed negligiblebeyond this point, by applying the RF focusing transportmatrix for relativistic beams [4] with initial conditions 2 and ' orb given by eqs.1 ('2 gives the secularorbit divergence andwill be used later as initial condition for the envelopeequation). The matrix gives at any position the beam spot ofthe average secular orbit and the beam divergence ' of theactual orbit.

The agreement between analytically predicted envelopes andnumerical simulation results are shown in Fig.1 for two10+1/2 cell guns at typical frequencies and peak fields.

0 100 200 300 400 500 600 7001

1.5

2

2.5

3

3.5

4

z (mm)

S-band 100 MV/m

1 nC

4 nC

0 250 500 750 1000 1250 1500

2

3

4

54 nC

0 nC

z (mm)

L-band 45 MV/m

(mm

)

Figure 1: Beam envelopes through two different 10+1/2 cellRF guns (=2.856 GHz upper diagram, =1.3 GHz lowerdiagram). Dashed lines give the secular orbits analyticallypredicted, while solid lines are numerical simulation results.

In the upper diagram the bunch aspect ratio is A=1.25 withcat=1.5 mm, corresponding to a peak current I=100 A at 1nC and 400 A at 4 nC (z2=79 mm , 2=8.7 , =1.64); in thelower diagram A=0.83 with caike problemsz2=174 mm ,

2=8.6 , =1.62). The simulations were performed with thecodes ATRAP [5] for the S-band gun and ITACA [6] for the L-

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band one. In this way the beam envelope can be predicted juston the basis of six main free parameters: three of themcharacterize the external fields, namely E0,, B0, while thebunch characteristics, determining the collective field, arespecified by cat, A, and I.

Eqs.1 are derived under the basic assumption of self-similarexpansion of the charge density distribution, which staysgaussian in the (r,z) space under the effect of all the external aswell as the collective forces acting on the bunch particle. Theaverage bunch phase is assumed to be = /2 ,corresponding to maximum acceleration in the gun.

In case the magnetic focusing is not applied (B0=0, as for

Fig.1), the beam envelope behaviour is such that the spacecharge effects can be considered negligible after the second iris.In presence of an external magnetic focusing we need adifferent treatment of the beam dynamics. The envelopeequation for a relativistic beam [7] seems the best approach:

+

+ Kr 2

(I/Ia)

3

n

2

3

2 = 0 2)

where '= d/dz , n is the rms normalized emittance andKr is the RF focusing gradient [4] Kr = (ks

2+1/8)('/)2 ,which incorporates the contribution (ks) from the magnetic

field. This equation actually holds for an un-bunched beam:since the bunch aspect ratio in its rest frame, given by A/,can be considered small enough beyond the second iris, wheretypically >5, we assume eq.2 can be taken as a goodapproximation. Moreover, since we are interested in studyingthe conditions which give rise to space charge emittancecompensation, the hypothesis of beam laminarity will be set

up, implying that the emittance term in eq.2 is negligible.This is equivalent to assume that the beam envelope will notgo through any crossover from the cathode up to the gun exit.

Under these assumptions, we can apply a Cauchytransformation to eq.2, setting y = Log(/2) (recalling that= 1 + kz = 1 + 'z ) and obtaining

d2

dy2+ 2 =

S

ey 3)

with = (y) and S = 2I / (Ia '2 2) . We solve eq.3following two different techniques, according to two differentdomains: the first one is defined by z2 < z < zc (0yc ,

+ /8 = Se-y

/ , in order toreduce all the parameters to dimensionless quantities.

By defining the dimensionless quantity / S , theenvelope eq. in the Cauchy dimensionless space (,y) reads

d2

dy2+

8=

ey

6)

which is a universal scaled equation, independent on anyexternal parameter. It is interesting to note that the function can be expressed as = 'p2/ : under this form it isclearly shown that scales like the ratio between the plasmawavelength p=2c/p and the energy gain length Lg = 1/'.Moreover, a particular exact solution of equation 6) can be

found to be * = 8/3 e-y/2 : this solution is characterized by

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a constant ratio between the two fundamental scale lengths, Lg

and p, i.e. p/Lg = 8/3 . The corresponding beam size *comes out to scale like 1/ , namely * = 8S2/(3) ,but, more relevant, * is the only solution displaying aconstant phase space angle , which is independent on initialconditions c and c in all of the three spaces (Cauchy

dimensionless, Cauchy, real) . In fact, * = * / = * / ='* / (') = -1/2 , so that in both Cauchy spaces the phasespace angle is a universal constant. The most relevantconsequence is that , on this particular envelope *, whichwill be called the invariant envelope, does not depend on thebeam current (whose dependence is embedded in the

expressions for c and

c): this is exactly the basic condition

to get a vanishing linear correlated emittance. In fact, it is wellknown that the emittance growth from linear space chargeeffects is due basically to the spread in phase space distributionof different bunch slices, which get different kicks from the

space charge field: these may be thought to be represented bydifferent current amplitudes in the envelope eq.2 .

cat

cat

0.8 1 1.2 1.4 1.6

0.09 0.10.11

B0 (T)

30

35

40

45

0.8 1 1.2 1.4

.09 ..

0.51

1.5

0.08 0.090.1 0.11

B0 (T)

30

35

40

0.51

1.5

.08 0.09. .

(mm)

(mm)

I = 150 A

I = 300 A

L-band

| |

Q (nC)

1 1.5

Q (nC)

|2

|||2 3 4

Figure 3: Solutions of eq.7 at =1.3 GHz, A=1, I=150 (upperdiagram) and I=300 A (lower diagram), represented by thesolid-dotted lines in 3-D sub-space (E0, B0, cat). Since thebunch aspect ratio is kept constant, the displayed bunch

charges are given by Q=2 Icat / A.

III. RF GUN OPERATING CONDITIONS FOREMITTANCE COMPENSATION

In order to find the operating points, as functions of the sixfree parameters, we must solve the equation:

[c 8S2/(3c) ]2 +[ c+ (1/2)8S2/(3c) ]2 = 0 7)whose solutions, in the 6-D space (, I, A, E0, B0, cat),

assure that c and

c match the initial conditions of the

invariant envelope. In this way the beam is transported fromthe cathode up to the gun exit with no space chargecorrelation, hence the emittance compensation is achieved.

In order to simplify the search for the solutions, we fixthree parameters, namely, I and the aspect ratio A. The rootsfound in the 3-D sub-space (E0, B0, cat) are plotted in Fig.3for a L-band gun at I=150 A and I=300 A. The projections onthe plane (B0, cat) are also plotted in the figure (solid lines),while the shaded surfaces set the limit of maximum charge

extractable from the cathode (only points on the right of thesurface are allowed).

Selecting one possible solution from Fig.3, namely =1.3GHz, A=1, I=150 A, E0=35 MV/m, B0=1.02 kG and

cat=0.84 mm (so that Q=1 nC), we plot the correspondingenvelope in Fig.4, for the nominal 150 A current and for lowerand higher currents, 110,130,170,190 A, respectively: it isworthwhile to note that envelopes corresponding to currentsdifferent from the nominal one spread out in the first cells (seeenclosed box), where the focusing action of the solenoid isdominant, but converge down into a common invariantenvelope along the second domain ( zc>330 mm) of eq.6.

2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5

100 200 300

1

2

3

z (mm)

L-band 35 MV/m 150 A

Figure 4: Envelopes corresponding to one of the solutionsshown in Fig.3, for various currents around the nominal one.

IV. REFERENCES

[1] K.J. Kim, NIM A275 (1989), p.201[2] B.E. Carlsten, IEEE Catalog no. 89CH2669-0[3] L. Serafini,Analytical Approach to Transverse Dynamicsin Multi-Cell RF Guns, Proc. of VI AIP Workshop onAdvanced Accelerator Concepts, Lake Geneva (WI), June 1994.[4] J.B.Rosenzweig, L.Serafini, Phys. Rev. E-49 (1994) 1599[5] J.M. Dolique and J.C. Coacolo, NIM A340 (1994), p.231[6] L.Serafini and C.Pagani, Proc. 1st EPAC, Rome, June1988, Ed. World Scientific, p.866

[7] J.D.Lawson, The Physics of Charged Particle Beams, 2nded. (Oxford Univ. Press, New York, 1988)