Magnetic performance of a fast excitation wiggler ._ - -.

- -3ZFinverse free-ele c t r o n 1 asers

Juan C. Gallardo Department of Phy~ics, Bldg. 901-A

T. Romano, A. van Steenbergen

Brookhaven National Laboratory P. 0. Bos 5000

Wpton, New York 11973-5000

- NSLS, Bldg. 725- B

ABSTRACT

With the objective of performing an inverse free-electron laser accelerator experiment, an iron dominated (Vanadium Permendur) , fast excitation, high K planar wiggler has been built and measured. We present in this report an analysis of a coiistaiit period wiggler and several tapering configurations (gap=4 mm; 3.0 cm < A, < 5.0 cm) when we drive it to a peak field of B,,, ZN 1.4T.

I. INTRODUCTION

.A detailed description of the fast excitation wiggler shown in Figs. 1 and 2 has been given in several papers'-5 and additional magnetic measurements were presented in Ref. 6; Tb. I presents a summary of the different wiggler configurations that have been studied _up to now. The wiggler consists of magnetic material laminations (VaP) assembled in thickness substadis arranged in alternating up and down configuration and separated by non-magnetic material (Cu) which acts as eddy current induced *- field reflectors." Four straight current conductors, parallel to the axis of the magnet and interconnected only at the end of the assembly, form the current single excitation loop for tlie wiggler. The whole structure is held and compressed by simple tie rods.

111 this note we discuss the field errors and the imposed constraints to inaintaiii electron and laser overlap. The experimental setup allows the measurement of Lhe vertical component of the wiggler magnetic field in the horizontal plane at different vertical and horizontal positions in the gap, B,(z,y,r,) where zJ = A j with A = 0.15875cm corresponding to a complete turn of the driven shaft of the traveling measuremeiit system. The relative error in position of the probe is estimated to be around 0.02cm and the estimated error in the induced voltage signal after being integrated is about 0.1 9%.

To calculate the electron trajectory in this real wzggler it is necessary to make a complete reconstruction of the field In the gap from the measured values. From general principles' we express the magnetic scalar potential as

@(t. y,:) = S ( t ) Y ( y ) Z ( z )

- with I 3 ( z , y, :) = -e@(=, y. 2).

aid Z(-) = c o s ~ i k , ~ wlitrc tu = cslmswon for the scalar potential as

IT we assume symmetry in tlie horizontal plane and periodicity in the longitiidiiial direction, then X ( t ) = X ( - - E ) with A, tlie period and n thc harmonic number, we can write the most general

w t l i the constraiiit ki = k: Totice that there IS no median plane symmetry and its deviation can be represented with a single parameter r7

such that. By(z, -g 2 ) = qB,(z,g.z) where g is the gap distance This relation caii be understood inaking use of the Gauss theorem in the trans\erse plane (see Fig 1)

k: z kg( 1 + $1

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Simple algebraic manipulations lead to the expression of the field in the gap,

&(t, y, z ) = nkl sinh (nklz) [An exp (nk2y) + Bn exp (-nk2y)] cos(nk3z) By(=, y, z ) = nk2 cosh (nklz) [An exp (nk2y) - Bn a P (-nk2~)] cos(n+)

Br( t ,y ,z ) = -nk3cosh(nk1r)[A,exp(nk2y) + Bnexp(-nhy)] sin(nk32) (3)

the coefficients An and B, are determined by the boundary conditions at the poles and the peak vertical magnetic field on axis Ban; they read,

K a t we use the fields so determined to calculate the electron trajectory in the wiggler field, the spontaneous emission spectrum and by means of the Madey's theorem', the free-electron laser small-signal gain.

TABLE 1. Table of wiggler configurations studied. Data are normalized to L,=24 cm and I,,=4 kA (B,)b [kG] (BelC [kG] [kG/cm] $ [l/cm]

- 0.0168' % O 9 [kG/cm] A, [cml WP Icml (r).

WE 2.98 1.19 0.798 9.57 _. - 9.97 - var. A,; const r 4.29 1.73 0.806 - WIe 4.29 1.73 0.806 9.97 - 0.0168' S O

- - 10.37 - var. A,; const r 5.04 1.98 0.786 1 . O i 0.540 10.05 - 0.0168d 0.006 WII 3.96

~~ - - 10.45 - const A, 3.96 1.37 0.692 1.02 0.628 10.11 - 0.0072 -0.006 WIII 3.25 _ _ _ 1.02 0.478 - 10.28 - - const Wp 4.27

0.808 10.06 - *0.0068' % O WIVi 3.76 1.52 - 9.89 - - const A,, r 3.76 1.5? 0.808

--_ W, denotes the thickness of the VaP. w,

( A - P ) . * f =

b B , = B(r = 0) except for WIV where B, = B(z = 24) 'Be = B ( z = 34) except for WIV where 8, = B(r = 0)

e end effects without partial saturation, 0.0145

11. PERFORMANCE OF A CONSTANT PERIOD FAST EXCITATION WIGGLER

In Fig. 3 we show B, ( z ) vs. z measured on axis for a current of 6 kA. In Figs. 4a and 4b we display the scattering of positive and negative peak fields. Xotice that there is an apparent negative dipole offset. It has been verified that this is due to a small (0.35 %) systematic error related to the unequal response of the field measurement integrating circuit to positive and negative magnetic field.

The rms-fluctuation of the peak field og = d- about the average is under 0.15 % and the maximum scatter is 1.0%.

A complete measured field representation B,(r,y,z) for a single pole region (g = 4mm,X, = 37.6mm, I = 6 kA ) is given in Fig.5 for -1.2 5 y 5 1.2mm and -6.0 5 r 5 6.0mm.

A Fourier transform analysis of the entire field map on axis has been performed using the Fast Fourier pansform routine SSWD from the IMSL library with a Porzen window. From the power spectrum plot, the odd harmonics up to order 7th are clearly in evidence, a natural consequence of the periodicity of the field. The magnitude of the noise level is about and up to that level there is no indication of even harmonics. The 3rd. harmonic is about 3 % of the fundamental and the magnitude of the odd harmonics is fitted by a Laurent series -0.G: + - 9 + % - 5 in reasonable agreement with expectations.

The normalized horizontal velocity I' = is given essentially by the first integral of the vertical component of the

magnetic field B,(O,O. z ) = Bo(=) , 1.e..

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likewise, the position x is given by the second integral of the field,

where we have introduced li = 3.754 cm.

= 0.934Bmaz[T&(cm]. with Bm“ z 1.4T is the peak field and A, 2

Figs. 6a and 6b depict r’ vs. z i d z vs. z; we observe that the maximum z oscillation amplitude is zz 0.4mm The displacement and steering of--bhe-beam at the exit of the wiggler are approximately 0.001 mm and 3.4 mrad respectively.

Magnetic field statistical errors due to imperfections in the fabrication and assembly of the wiggler perturb the electron beam and leads to a deviation from the trajectory in an ideal (error free) wiggler. This deviation decreases the free-electron laser gain as a result of: a) loss of overlap between laser and electron beams, as a consequence of the electron random walk; b) loss of the resonance condition, as a consequence of the random phase variations. Several a u t l ~ o r s ~ ~ ’ ~ have shown that the centroid motion of the electron beam satisfy the relation (for uncompensofed wigglers,

< 622 >= -- 1 K2ki < (%)2 > x”(NA,)3 3 Y BY 2

and for a compensufed wiggler < 6z2 > is reduced by a factor of 4. The condition to minimize the random walk effect on the gain is to limit d x 5 O . h 0 where r, is the laser beam waist size. For our case Eq. 7 gives

I l 7 - < (-)? > zz 0.24 % which is larger than the measured dispersion of 0.15 %. Similarly, the random phase deviation is given by,

Degradation of the FEL gain becomes noticeable when < 6w >=z s. For our set of parameters, the limiting value oi 4- z 0.64 % is obtained, which is well met by the wiggler results cited above. z for a tapered period wiggler with

2.95 cm < A, < 5.04 cm and a current of 6 kA. For completeness we also present in Fig.7, a plot of the field B,(z) vs.

111. SPONTANEOUS EMISSION

The spontaneous emission spectrum“ of an undulator exhibit sharp peaks at wn = nG1(8), n = 1 . 2 , 3 , . . . where

(9)

The iindulator frequency is s, = and 0 ti the observation angle with respect to the oscillation plane (x-z). We stress that only in the forward direciion, B = 0. we expect to have odd harmonics, at any other angle the spectrum will also show evcn peaks.

Likewise, the spontaneous emission spectrum of a wiggler exhibit sharp peaks a t the harmonics wn(0) and the spectral flux (number of photons per uni t time and per unit solid angle) into a small bandwidth L i is given by,

1 d’W - d ’ 3 -- - - = a;V-y-( -)-F*(h-) ? , A W I hw dR’ - dR’ u e

where a = & is the fine-structure constant and the function F,, is defined as

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(10)

(11;

In the equation above we have used C = 4 ~ l + + K z + 7 2 ~ 2 ~ z . K' In practical units the spectral flux is

A plot of the function Fn(K) for odd harmonics shows that for high values of K the higher harmonics become more prominent and that the maximum." is achieved at the critical fiequcncy we = $- and the corresponding harmonic number is ne = = iK( 1 + $1'") a 50; as a consequence, we expect to see in the spontaneous emission spectrum sharp peaks at harmonics of w l ( 0 ) with a rather broad maximum at harmonics ne s 50.

In Fig. 8 we plot the spectral flux in arbitrary units near the fundamental w1 as computed by the code SIN-LUCE12. Notice the second harmonics with an intensity 50% below the fundamental, signaling a misalignment of the electron beam with the magnetic axis of the wiggler. This misaligument is caused by a combination of wiggler errors and improperly matched electron beam. Lastly, in Fig.9 we show the gain curve at the fundamental frequency w1(0 ) .

W 1

IV. CONCLUSION

It has been shown that the fast excitation driven, laminated Vanadium Permendur, wiggler with periodic interleaving of conductive copper field reflectors is capable of satisfying both the FEL and IFEL requirements. As indicated, a specific tapered period slope can readily be achieved. The wiggler structure, as presently executed is robust and simple to assemble and lends itself for easy modifications of period length, field magnitude and taper. These are very important features in actual FEL and IFEL experiments.

ACKNOWLEDGMENTS

The authors wish to acknowledge the contribution of J . Armendariz to part of the experimental work. We are also grateful to L. Giannessi for making the code SIN-LUCE available to us and for his continuous advice and assistance. Tllis research was supported by the U.S. Department of Energy under Contract No. DE-AC02-76CH00016.

V. REFERENCES

A. van Steenbergen, patent frTo.368618, (submitted June 1989) issued in August 1990. A. van Steenbergen, J. Gallardo, T. Romano, M. Woodle, 'Fast Excitation variable period wiggler," IEEE Particle Acceler- ator Conference in San Francisco. Val. 5 , pp. 2724-2716, May 1991. -4. van Steenbergen, J. Gsllardo, T. Romano, hl. Woodle,"Fast excitation wiggler development," Proc. Workshop Prospects €or a 1 AFEL. Sag Harbor, N.Y.. pp. 79-93, April 1990; BXL Report BNL-52273. ' E. Courant. A. Fisher. J . Gallardo. C. Pellegrini, J. Rogers. J. Sandweiss, J. Slieehan, A. van Steeabergen, S. Ulc, M. Woodle,

"Inverse Free-Electron Laser Accelerator Development," The Accelerator Test Facility Users' Meeting, October 15-16 1991, BKL Report BNL-47000, CAP #SI, ATF-SIP. ' A. Fisher. J . Gallardo, J. Saiidweiss, A. van Steenbergen. Inverse free-electron laser accelerator," Proc. Port Jeff. Workshop

June 1992: to be published. E J . Xrmendariz. J . Gatlardo. T. Romano, A. van Steenbergen. (i Fast excitation wiggler field measurement results," BlUL

Report BNL-47938. 1993. - G. Dattoli and A . Renier1.-Experimental and Theoretical aspects of the free-electron laser." Laser Handbook. Vol. 4 . Sorth-

Holland. Amsterdam. 1985. ' J . II J . Ll-Idey. "Relationship between mean radiated energy. niean squared radiated energy arid spontaneous power spectrum

i n L power series expansion of the equations of motion I n a free-electron laser .n Nuovo Cimento. Vol. SOB. pp. 64-88. 1979. ' B. iijncatd:-Karldorn errors i n undulators and thejr effects on the radiahon spectrum.- J . Opt. SOC. Am.. \'ol. B2, pp.

131-1306. 1985.

S P I f Vo/ 2013 ,' 785

lo J. Gallardo,"Randorn errors in iron-dominated microundulators," J. Appl. Phys., Vol. 70, pp. 1119-1120, 1991 and references therein. Kwang-Je Kh,'Characteristics of S w o t r o n Radiation," A.P.S. Conference Proceedings, Vol. 184, pp. 567433, 1989.

If R. Barbinii F-Ciocci, G. Dattoli and L Giannessi, Fortran 77 SIN-LUCE computer code, private communication (1992). . . . -

VI. FIGURES

FIG. 7. Details of a wiggler model

I - _ . 3 , zi=1 ; 0 20 20 43

Cowsmu Period. Fasr Exnraaoq Wi&r

1 = 3.76 cm. p = 0.40 an FIG. 3.

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i

I

1. ... . ...... . . . . . . . I

t '?,: -aA an 06 i l

,1-; I

FIG. 5. Wiggler field B 9 ( t , y , z ) for -6.0 5 t 5 6.mm,

-1.2 5 y 5 1.2mm, -0.25 5 e 5 0.25

0.000 p :',

FIG. 6 . a) First field integral ( v , vs. z of a representative electron):

b J Second field integral ( x vs. Z)

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E 0

t ' " ' i " " " " " ' " ' " ' ' ,

FIG. 8. Spectral flux as a function of fi neat the fundamental.

I " " i " " ~ I !

Gam

L I L

r

t

3.2 - t

0.0 1 F 1

.3.2 i-

FIG. 9. Free-electron laser gain function YS. Y ( Y = 2xN"*) computed using the Madey theorem

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