*Corresponding author. Fax.: 49-40-8998-4475.E-mail address: p#ueger@desy.de (J. P#uK ger)1On leave from FEL laboratory at Institute for High Energy

Physics (IHEP), P.O. Box 2732 Beijing 100080, Peoples Repub-lic of China.

Nuclear Instruments and Methods in Physics Research A 429 (1999) 386}391

Field "ne tuning by pole height adjustment for the undulator ofthe TTF}FEL

J. P#uK ger*, H. Lu1, T. Teichmann

Hamburger Synchrotronstrahlungslabor HASYLAB, at Deutsches Elektronen-Synchrotron, DESY, Notkestr 85, 22603 Hamburg, Germany

Abstract

The "eld of the undulator for the VUV}FEL at the TESLA Test Facility has to meet very tough tolerances in order toguarantee a close overlap between the electron beam and the laser "eld. Consequently the undulator was designed tohave height-adjustable poles in order to allow for "ne tuning of the vertical undulator "eld in such a way that thetrajectory is straightened. The signature of local pole height and gap changes on the "eld distribution was investigated. Itwas seen that changes are not restricted to the pole itself. Its e!ect can be seen up to the next eight neighboring poles. Inthis contribution we describe an algorithm in detail, which allows the prediction of required pole height changes in orderto correct for "eld errors. As input data "eld errors deduced from precise magnetic "eld measurements are used togetherwith the signatures of pole movements. A band diagonal system of linear equations has to be solved to obtain the poleheight corrections. For demonstration of the method the "eld of the 0.9 m long prototype structure was optimized tohave a straight trajectory. Since only a sparse band diagonal system of equations has to be solved, the method has thepotential to be used in very long undulators having 600 }1000 poles. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Undulator; Electron beam; Laser "eld

1. Introduction

At DESY in Hamburg a Free Electron Laser(FEL) for the VUV spectral range down to 6.4 nmusing the principle of self ampli"ed spontaneousemission (SASE) [1,2] is under construction. It willuse the electron beam generated by the TESLA testfacility (TTF) [3,4] and will be built in two stageswhich are described in detail in Refs. [3,5]. The

complete undulator system has a maximum lengthof about 30m. It is a "xed gap structure and isdescribed in detail in Refs. [6}9]. Table 1 repro-duces its magnetic parameters. The device is a com-bined function undulator which integrates twofunctions. First, it provides the sinusoidally-shapedundulator "eld so that the FEL process can takeplace. Second, an alternating gradient "eld of about$20 T/m for the FODO lattice which is superim-posed to the undulator "eld is generated.

The magnetic design was chosen to combine thefollowing properties:

1. It is a completely planar structure, which allowsfor very good access to the "eld region at the

0168-9002/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 1 1 2 - 6

Table 1Undulator parameters for the undulator for the VUV}FEL atthe TESLA Test Facility

Gap ("xed) (mm) 12Period length (mm) 27.3Undulator peak "eld (T) 0.5K-parameter 1.27Design gradient (T/m) 18.3Number of poles per undulator module 327Total length per module (mm) 4492.2Length of FODO quad section (mm) 136.5FODO period length (m) 0.9555Number of FODO periods per module 5Separation between undulator modules (m) 0.2853

beam position allowing for high accuracy "eldmeasurements as well as an easy installation ofthe vacuum chamber without breaking of anymagnetic circuits.

2. The gradient can be as large as +20 T/m.3. The exact value and the precise location of the

quadrupole axis is "ne tunable.4. Undulator and focusing "elds are decoupled.

This means that on the quadrupole axis the signand magnitude of the "eld gradient has no in#u-ence on the undulator "eld and vice versa.

The magnetic "elds of undulators for SASEFELs have to meet tough speci"cations [10]. Care-ful optimization and "ne tuning of an undulator isrequired after it has been assembled. Two steps areplanned. First the `nakeda undulator, i.e., the un-dulator without the focusing magnets will be mea-sured and optimized in such a way as to obtaina straight trajectory in the horizontal and verticalplane. In the horizontal plane the height adjustablepoles will be used to obtain an optimum B

y"eld

distribution. In the vertical plane "eld errors arenot expected to be serious. Good sorting of themagnets using simulated annealing will minimizeresidual error "elds [11] in any case. Small remain-ing "eld errors can be treated using very few suit-able shims.

In a second step the focusing magnets will beattached and their strength as well as the exactposition of quad centers will be "ne tuned as de-scribed in Ref. [12].

This contribution deals with the tuning of theBy"eld of the `nakeda undulator. A numerical

procedure which allows the prediction of pole ad-justments to obtain a straight trajectory from pre-cise magnetic measurements was developed and isdescribed in detail.

2. Magnetic measurements, experimental

The new 12 m long bench was used to character-ize the magnetic performance of the prototypestructure. It provides su$cient mechanical accu-racy for the magnetic measurements of the com-bined function undulator for the FEL at the TTF.Measurements could be performed with both highspatial and high-"eld resolution. For the "eldmeasurements the resolution is given by*B/B"5]10~4, the spatial resolution of the en-coder system is 1 lm.

The magnetic measurements presented in thiscontribution were made on a 0.9 m long prototypeof the undulator for the VUV}FEL with the sameparameters as in Table 1. The pole heights of thisstructure can be adjusted by tuning hex screws byan estimated $ 0.5 mm. The screws having a 0.7mm pitch act on the poles under an angle of 603 sothat there results a 3.9 lm pole height change fora rotation angle of 13 on the tuning screws. Thetuning screws were actuated manually using specialhex keys with degree scales. Due to backlash be-tween hex keys and the screws as well as to sticke!ects and elasticity in the 150 mm long key it wasestimated that the screws could be adjusted with anaccuracy of about $5}103 leading to a relativeinaccuracy of the pole heights of about 20}40 lm.

3. Description of the method

The idea for the error control by pole heightadjustment is simple: Error "elds are determined bymeans of precise magnetic measurements. Preciseknowledge of the in#uence of pole height changeson the "eld distribution, the so-called signature,can be obtained from magnetic measurements.Then an algorithm may be used to calculate therequired pole shifts, which minimize these errors.Similar approaches has been reported by Stonerand Beke" [13] who calculated shunt impedances

387J. Pyu( ger et al. / Nuclear Instruments and Methods in Physics Research A 429 (1999) 386}391

VIII. FEL TECHNOLOGY

Fig. 1. Example for generation of the error "elds on the poles forthe upper structure half. The "eld was measured 6 mm below thepoles of the upper structure half at the nominal beam position at12 mm gap. (a) Field data, pole d37 has been detuned onpurpose to demonstrate the e!ect. (b) After application of thej-"lter. The large end excursion at the ends is an artifact of thej-"lter. (c) Average error "eld at the poles. Due to the convolu-tion, the adjustment is now smeared out over next nearest-neighbors. This is clearly seen at pole d37. The end region hasbeen excluded.

to optimize the peak "eld homogeneity of a pulsedelectromagnetic short period undulator and Ram-ian et al. [14,15] who describe a robotic systemwhich is able to "ne tune the peak "eld of PMundulator systems. Due to the measurement tech-niques applied, only the peak "eld was optimized inRefs. [13}15]. In contrast, the goal of this work wasto optimize the complete trajectory. The treatmentneeds some assumptions and conventions, whichare described below. In a "rst step the error "eld oneach pole is determined by the `deviationa from theideal "eld. Each pole is characterized by one num-ber, the mean "eld deviation, which is the deviationaveraged over the half period length of that pole atits nominal position. We tried two di!erent ways todetermine the deviation from the ideal "eld:

1. Applying a j "lter, which means convoluting themagnetic "eld data with a square function ofwidth j, the period length. In this way all theinformation with periodicity j/n (n"1,2,2) isremoved from the data and only nonperiodicperturbations are left over. The convolutionhowever smoothes the data and smears out spa-tial information over one period length.

2. Alternatively, we tried various harmonic "ts tothe data up to order 11 to determine the periodiccontent and subtracted it from the "eld to obtainthe deviation. This method gives noisier data,but preserves spatial information.

Both methods were compared and gave compara-ble results. Fig. 1 demonstrates the steps in the caseof the application of a j "lter. Fig. 1(a) shows the"eld distribution of the upper structure half of a0.9 m long prototype structure. One pole, d37, hasbeen detuned on purpose to demonstrate how itshows up in the analysis. In Fig. 1(b) the j "lter hasbeen applied to the data of Fig. 1(a). The perturba-tion applied on pole d37 is now very clearly vis-ible. The large excursion at the ends is an artifact ofthe j "lter. This region therefore has been excludedfrom the analysis. Fig. 1(c) "nally shows the assign-ment of the "eld errors to the poles. The endpoleregion has been excluded. Spatial information issmeared out and the perturbation of pole d37 hasnow been split up on the next nearest-neighboringpoles as well. This is the input data to calculate thepole height adjustment.

In the second step we assume a linear relation-ship between the movement of a pole and the per-turbation induced at any other pole. This is anassumption which only holds for small pole shifts ofa few tenth of a millimeter. We assume an undula-tor consisting of N identical poles. Poles near theends again have to be excluded, since due to thetruncation they cannot be considered as identicalwith poles well inside the undulator. Excluding theoutermost 4}5 poles was found to be su$cient to

J. Pyu( ger et al. / Nuclear Instruments and Methods in Physics Research A 429 (1999) 386}391388

Fig. 2. Normalized response of a pole shift of the central poled 34 of the upper magnet structure. Dotted line: Measureddi!erence. Full line: Average "eld on the poles.

neglect both, artifacts from the j "lter and trunc-ation e!ects as well.

We are interested in the "eld change Sjon a pole

with index j if another pole with index i is moved byan amount p

i. In order to get the total "eld change

on pole j one has to sum over all contributions frompoles i. The S

jmay be interpreted as minus the "eld

change on pole j which is required to obtain a per-fect undulator "eld.

These considerations lead to the following linearsystem of equations:

S1"a

0) p

1#a

1) p

2#a

2) p

3#2#a

i~1) p

i

#2# aN~1

) pN

S2"a

~1) p

1#a

0) p

2#a

1) p

3#2#a

i~2) p

i

#2# aN~2

) pN

S3"a

~2) p

1#a

~1) p

2#a

0)p

3#2#a

i~3) p

i

#2# aN~3

) pN

)" ) )

)" ) )

)" ) )

Sj"a

1~j) p

1#a

2~j) p

2#a

3~j)p

3#2#a

i~j) p

i

#2# aN~j

) pN

)" ) )

)" ) )

)" ) )

SN"a

1~N) p

1#a

2~N) p

2#a

3~N) p

3

#2#ai~N

) pi#2# a

0) p

N.

The strength of the `interactiona is characterizedby the coe$cients a

mwhere m"i!j. These a

mco-

e$cients are the response of a pole shift pion the

"eld change Sj

on pole j normalized by the poleshift p

i. Their dimension is therefore [T/mm]. The

assumption of identical poles is re#ected in the factthat they depend only on the di!erence i!j andare assumed to be the same for all poles. Thecoe$cients a

mhave been determined experi-

mentally by measuring the response of the mag-netic "eld to a known pole shift (0.5 mm) of

a known pole (pole d34, the central pole at X"0.)of the upper structure half of the prototype struc-ture. In Fig. 2, the e!ect of this pole shift is shown(dotted line). The full line shows the average "eld onthe poles in the vicinity of pole d34. The index m isindicated. Under the pole itself there is the stron-gest e!ect, but up to the next four neighbors there isa measurable e!ect with opposite sign. Integratedover all neighbors this sign change eats up a con-siderable amount of the correction of the pole itself.The shape of the dotted curve is that of a dipolelayer of magnetic surface charges brought on thepole.

It is seen that the interaction on both sides arevery similar so that the assumption a

m"a

~mholds.

We assume the same interaction for all poles so thatthe coe$cient generated from pole d34 are alsovalid for all other poles. It is also seen that it issu$cient to treat only the four next nearest-neigh-bors. For larger distances the correction becomesnegligibly small. The linear system of equations istherefore greatly simpli"ed. Below and above themain diagonal there are only four side diagonalswhich have to be considered. Using a specializedequation solver for band diagonal systems of equa-tions reduces numerical e!ort and also minimizesmemory requirements dramatically. Large polenumbers can be treated at moderate e!ort on a PC.Up to 1000 poles have been tried in simulations.Even larger numbers may be possible.

389J. Pyu( ger et al. / Nuclear Instruments and Methods in Physics Research A 429 (1999) 386}391

VIII. FEL TECHNOLOGY

Fig. 3. Demonstration of the optimization of the upper structure half in three iteration steps. The second "eld integral is shown which isproportional to the beam excursion. At 300 MeV which will be used for Phase I 10 Tmm2 corresponds to 10 lm beam excursion.

There is no fundamental di!erence if a structurehalf, i.e. the top or bottom structure of an undula-tor or a full magnet structure with closed gap isoptimized. Only the a

mcoe$cients are di!erent and

have to be determined for both cases in the properway as described above. This means that for struc-ture halves the signature of a single pole movementhas to be evaluated and in the case of a closed gapthe poles in the upper and lower structure have tobe moved simultaneously resulting in a local gapchange.

For the prototype structure it was found veryhelpful to separately optimize the two structurehalves "rst and to do the "nal "ne tuning with thegap closed to the nominal position in a second step.This second step needs only minor additional ad-justments.

4. Results

The 0.9 m long prototype structure was used totest the optimization procedure. Again only theupper structure was used. The "eld was measured6 mm below the poles corresponding to the nom-

inal electron beam location with respect to theupper structure at a 12 mm gap. The resultingsecond "eld integral distribution from four iter-ation steps are shown in Fig. 3. The resulting elec-tron beam excursion can be calculated by

z (lm)"587/c ) I2

(Tmm2)

"0.3.I2

(Tmm2)/EK*/

(GeV)

where c is the kinetic electron energy divided by theelectron's rest mass energy. Emphasis was put ona straight trajectory in the undulator itself. It is seenin Fig. 3 that the endpoles are not adjusted proper-ly. This is of minor importance in this context sincethey are easily adjustable and they can be tuned toresult in a straight trajectory before and after theundulator. It is seen that within a few Tmm2 nodeviations can be seen corresponding to sub mi-crometer beam excursions. A full structure with 12mm gap results in twice the "eld. The "eld integraldeviations in this case therefore will be twice aslarge. For each iteration step the error "elds werecalculated and the resulting pole height adjust-ments were calculated. A computer printout wasgenerated containing the pole's number and theamount of adjustment needed on that pole. The

J. Pyu( ger et al. / Nuclear Instruments and Methods in Physics Research A 429 (1999) 386}391390

amount of adjustments was much larger in the "rstoptimization. For the last iteration step neededonly very little correction was needed.

5. Limitations, accuracy

The quality of the optimization is in#uenced bythree factors

1. the accuracy of the "eld measurements,2. the accuracy of the coe$cients a

m,

3. the accuracy of the pole height adjustment.

The measurements accuracy was conservatively es-timated to be: *B/B"5]10~4. In practice it mayeven be better.

Inaccurate coe$cients am

in#uence the conver-gence of the procedure more than the accuracy ofthe result, which means that more iterations areneeded to get the optimized result.

The main source of error is the modest accuracyof local pole height adjustment. $ 5}103 of angu-lar accuracy leading to an estimated inaccuracy ofthe pole height of $20}40 lm which in turn leadsto a "eld error of about 0.3}0.6%. This is by far thedominant contribution. To improve the accuracy ofpole height adjustment one has to measure thesechanges directly using a suitable dial gauge withmicrometer resolution instead of just counting thetuning screw angle. This improvement is underway.With this moderate e!ort it should be possible to

reduce the pole height adjustment error to 5}10 lmor better so that this contribution amounts to lessthan 0.075}0.15%.

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VIII. FEL TECHNOLOGY