Nuclear Instruments and Methods in Physics Research A304 (1991) 753-758

753North-Holland

Focusing permanent magnet undulatorYoshiaki Tsunawaki a, Nobithisa Ohigashi b, Kitnioki Mima e, Tatsuro Akiba c,

Shinichiro Kuruma d, Kazuo Imasaki d, Sadao Nakai c and Luis R. Elias e

° Department of Electrical Engineering and Electronics, Osaka Sangyo University, Naka-gaito, Daito, 574 Osaka, Japanh Department of Physics, Kansai University, Yamate-cho, Suita, 564 Osaka, Japan` Institute of Laser Engineering, Osaka University, Yamada-oka, Sulta, 565 Osaka, JapanInstitute for Laser Technology, Yamada-oka, Suita, 565 Osaka, JapanCREOL, University of Central Florida, Research Parkway, Orlando, FL 32826, USA

A plane polarized undulator does not have the ability to focus an electron beam in the undulating plane ( X- Z plane) because themagnetic field decreases away from the X-axis . To focus in the bending plane, a quadrupole field must be added to the undulatingfield . This can be achieved using curved magnets or trapezoid-shaped magnets. In this article, we analyze a few magnet configurationswhich produce X-Z focusing . It has been found that semicircular magnets or trapezoid magnets having 1 .2 ° angle are suitable asfocusing permanent magnet undulator

1. Introduction

One of the most popular undulator configurationsused with free electron lasers (FELs) is the so-calledHalbach undulator. In such a plane undulator, an elec-tron beam propagating along the undulator axis (Z-axis)is defocused in the undulating plane (X-Z plane) be-cause the undulator's field strength diminishes withdistance from the X-Z axis .

In this work, the focusing effect has been studied bycomputing the magnetic field and electron trajectoriesin permanent magnet undulators having different geo-metrical structure. In particular, (1) magnets curved allover, (2) magnets curved only in their central part, (3)magnets with trapezoidal shape and (4) magnets withshapes of trapezoids and rhomboids have been analyzed .Fig. 1 illustrates the various configurations .

2. Procedure of calculation

Usually, permanent magnet undulators are made ofsamarium-cobalt or neodymium-iron-boron material .The permeability at the operating point on the demag-netization curve is 1 .02po to 1 .08,uo [1-3], where u, isthe vacuum permeability . This similarity of the permea-bility between the permanent magnet materials andvacuum simplifies the calculation of the undulator mag-netic field. The total field is the linear superposition ofthe field of each magnet . In this work, a current sheetmodel was used to calculate the magnetic field pro-duced by each magnet . Using the Biot-Savart law, the

0168-9002/91/$03.50 © 1991 - Elsevier Science Publishers B.V . (North-Holland)

X

Y

Fig . 1 . Permanent magnet undulator composed of (1) magnetcurved all over, (2) magnet curved in its central part, (3)magnet with trapezoidal shape, and (4) magnet with shapes of

trapezoids and rhomboids.

X. UNDULATORS

754

field of a rectangular magnet magnetized along the+ Y-axis is given by eq . (1) [4,5]; in eq . (1) a, b and care the x, y and z dimension of the magnet, respec-tively . J = B,/WO is the equivalent surface current den-sity on the magnet, B, is the remnant magnet field andA. is the period (or pitch) of the undulator. The numeri-cal computation was made using the formulae devel-oped by Hu [4,5] . As it is well known, the undulatormagnetic field reduces abruptly with increasing undula-tor gap g. Although the field increases with magnetheight (b), it saturates when b/X� is close to 0.5 . Themagnetic field of the whole undulator was obtained bysumming the contribution from each magnet under theconditions that a/X� , b/X �, c/X . and K (normalizedvector potential of the undulator) are 2, 0.5, 0.25 and 1,respectively . Also, the end magnet width is 0.5c .

The Lorentz force equation,

my dt,

-evX B,

(2)

was used to calculate electron trajectories without con-sidering space charge effects . m, e, v and y are respec-tively the electron rest mass, charge, velocity and thenormalized relativistic energy . B was calculated fromeq . (1) at each position of the electron's trajectory . Inorder to reduce calculation, it was assumed that theelectron beam was focused to a waist diameter of 0.1X �at the center of the undulator.

3. Results and discussion

Fig. 2 shows some trajectories of electrons with y of10 and with a transverse emittance of 1 .67mX � mrad inthe X-Z plane of a Halbach undulator. It is found thatthe gradual increase of the electron beam width along

Eq . (1)

JLOJ n/2a ,a/2X~B=-4m f

dy' ff

dx'n/2LT- /2x

Y Tsunawaki et aL / Focusing permanent magnet undulator

+ f-'/2x � dz'

-(Y/Xa-Y'/X � )x+(x/Xa-a/2X � )yc2a

3/2/

�

[(x/X. - a12Ä.) 2+(Y/X � - Y'/Xa)2+(ZIA . - Z'/X .)21

+ a/2X"dx'

-(z/X � +c/2Xa)Y+(Y/X � -Y'/X .)Z

4/2X �

3/2[(X/X . -x'/X � )2+(Y/X � -Y'/X �)2+(Z/Xu+c/2Xa)2]

+ `/2T' dz'

-(Y/X � -Y'/X � )x +(x/X � +a/2Xa)Y

Ifc/2Xu

3/2 1 .[(x/X � +a/2Xu)2+(Y/X � -Y'/X~)2+(Z/X � -Z'/Xa)2 ,II1~

the Z-axis is due primarily to emittance and secondarilyto defocusing .

When the magnets of an undulator are curved as infigs . 1 (type-1 and -2), the magnetic field on axis issmaller than its off-axis field, and focusing in the X-Zplane is expected as well as that denoted by Scharle-mann [6] . Fig. 3 shows the X-width variation of anelectron beam traveling along a type-1 undulator curvedover its full surface for various values of curvature. It isfound that the value of (curvature)/(undulator pitch)cannot be selected smaller than about 1 .5 under thecondition of K=1, due to mechanical contact betweenthe top and bottom magnets. As shown in the figure,furthermore, not much focusing is obtained with K= 1 .More focusing effects m the X-Z plane can be ob-tained with a steeper-gradient magnetic field close tothe undulator axis . The type-2 undulator curved over1/3 of its width provides this high-gradient field asshown in fig. 4. Fig. 5 shows the variation of electronbeam width in the X-Z plane m a type-2 undulator. Incontrast with the result for the type-1 undulator, thesmaller curvature of type-2 magnets results in strongerfocusing . For example, the strong focusing effect oftype-2 magnets with a curvature of 0.333X � is il-lustrated in fig . 6. The resulting X-Z betatron oscilla-tion can be seen . However, the reduction of the curva-ture causes a decrease of the magnetic field gradient onthe Y-axis as shown in fig. 7, thus reducing the focusingpower in the Y-Z plane. Fig. 8 summarizes the maxi-mum X-width, Y-width and betatron oscillation wave-length as a function of magnet curvature with y = 10and an emittance of 1 .67rtrX� mrad. For a curvature of0.37X � the X and Y beam width, as well as the betatronwavelength, are close to each other. It appears that for amagnet curvature of 0.37X � a symmetric electron beamcan be effectively transported through the undulator.

-(z/X � -c/2X � )y+(Y/Xa-Y'/Xu)Z

(x/X � -x'/X�)2+\�)2+(Y/X�-Y'/XJ+(Z/X� - c/2X � )2 13/2

rV

b 0.3

c

302

ro

a,

0.4

0

0.8

04

04

0.8

PeriodsFig. 2. Electron trajectories in a Halbach undulator. y and enuttance of the electron beam are 10 and l.671TA . mrad, respectively .

The ideal condition of beam transfer selected in this

article requires that there be approximately equal focus-

ing in the bending and the nonbending plane. That is,the dependence of By on X and Y should have the form

B_,=Bo [1 +a(X2+ Y2)] .(3)

Y. Tsunawaki et al. / Focusing permanent magnet undulator

curvature/p,tch00

10

0 2 4 6Periods

2 K=2

Fig. 3. X-width variation of an electron beam traveling along atype-1 undulator curved over its full surface for various valuesof curvature. y and emittance of the electron beam are 10 and

1 .67TrX � mrad, respectively .

i35

- Z

This condition was reached numerically by selecting anarbitrary radius of curvature for the magnets and thenplotting the resulting focusing parameters aX, ay [7],

(K )2

8x and

TTK)Z8

(4)y>XuY

~ uY

curvature/pitch

0 333

I

1

1

I1.0

05

0

05

10

(Distance)/(undulator pitch) on X axis

755

Fig. 4. Magnetic field distribution on X-axis at center of theundulator. Dashed and solid lines show the results for type-1undulator curved over its full surface and for type-2 undulator

curved over 1/3 of its width, respectively .

X. UNDULATORS

756

suaÔ

03

0.4

0 2 4

6Periods

Fig . 5

X-width variation of an electron beam traveling along atype-2 undulator curved over 1/3 of its width for variousvalues of curvature. y and emittance of the electron beam are

10 and 1.67Tra � mrad, respectively .

as a function of the radius of curvature. The optimumradius of curvature based on the assumed parameters is0.37X � .

Two other undulator geometries were studied in thisarticle . The geometry of the magnets has been il-lustrated in fig . 1 (type-3 and -4) . Undulator type-3consists of magnets cut into trapezoidal shape of angleB. The axis length of the magnet is c. Undulator type-4is made out of two types of magnets. As shown in the

0.10

0.05

0.05

0.10

0.15

Y. Tsunawaki et at /Focusing permanent magnet undulator

N

4

3vccriroE

vNro 2

ôz

04 02 0 02 04(distance)/(undulator pitch) on Y axis

Fig . 7. Magnetic field distribution on Y-axis at center of theundulator . Dashed and solid lines show the results for type-1undulator curved over its full surface and for type-2 undulator

curved over 1/3 of its width, respectively .

figure the configuration consists of alternate magnetshaving rhomboid and trapezoidal shapes . Both type-3and type-4 undulators generate a focusing quadrupolefield in the X-Z plane. Fourier analysis of the magneticfield showed a similar harmonic distribution with type-3and type-4 undulators for 0 smaller than 5 ° . Fig. 9shows the dependence of the undulator gap g on 0under the condition of K= 1 . It decreases with 0 due tothe reduction of effective volume as a result of magneticfield cancellation between adjacent magnets. Type-4

curvature/ pitch

PeriodsFig. 6. Electron trajectories m type-2 undulator curved over 1/3 of its width for curvature of 0.333X. y and emittance of the electron

beam are 10 and 1.67ma � mrad, respectively .

ä

03ÊA

ÔLV3EEEE

02

01

CU

aO

015

1 0

1.5

2.0(curvature )/(undu(ator pitch)

Fig. 8 . Maximum beam width and betatron oscillation wave-length of electrons in X- Z and Y- Z planes of type-2 undulator curved over 1/3 of its width. y and emittance of the

electron beam are 10 and 1 .67ir ;t . mrad, respectively.

undulators are more practical because a large magnetgap can be obtained for the same value of K= 1 . Fig .10 shows the beam width in the X-Z plane for type-3and type-4 undulators having various values of 0 . Theenergy and emittance of the electron beam are the sameas those in fig . 5 . It is seen that electron beam focusingincreases dramatically with 0 and its strength is almostthe same for type-3 and type-4 undulators . As in thecase of a curved undulator, the increase of focusingpower in the X-Z plane is accompanied by a reduction

Y. Tsunawaki et al. / Focusing permanent magnet undulator

80

51 2 3 40 (degrees)

Fig . 9. Undulator gap under the condition of K=1 for type-3and type-4 undulators.

n

ôN

05

r 0.4V

a

757

0

4

8

12Periods

Fig . 10 . X-width variation of an electron beam traveling alongtype-3 and type-4 undulators for various values of 0 . y andemittance of the electron beam are 10 and 1 .67ma � mrad,

respectively.

0 1 2 3 4 50

( degrees )

Fig . 11 . Maximum beam width and betatron oscillation wave-length of electron m X-Z and Y-Z planes of type-3 andtype-4 undulators . y and emmttance of the electron beam are 10

and 1.67mA � mrad, respectively .

X . UNDULATORS

758

of focusing in the Y-Z plane. This effect is betterillustrated by representing the above discussed undula-tor in terms of an equivalent plane undulator superim-posed by a continuous quadrupole magnet which focusesin the X-Z plane but defocuses in the Y-Z plane. Fig.11 shows focusing effects of either type-3 or type-4undulators as a function of trapezoid angle 0, in boththe X-Z and Y-Z planes . The betatron period is alsoplotted in the figure . The angle 0 = 1 .2' corresponds toa curvature 0.37N � for the type-2 undulator. Theseresults suggest that an electron beam can be transportedsymmetrically through undulators of type-3 and -4 of0 = 1.2 ° . This unique value of 0 = 1 .2 ° does not dependon energy or emittance for values smaller than 20 orsmaller than 1 .67mX� mrad, respectively . In practicaluse, we might have to select magnets having accuratesize very carefully and to construct them into an undu-lator with very uniform periodicity because the electrontrajectory is very easily affected by small deviations of0.

4. Summary

In this work, the focusing properties of a planeundulator composed of a permanent magnet with geo-

Y. Tsunawaki et al / Focusing permanent magnet undulator

metrically different shape has been investigated . It isconcluded that an electron beam can travel almostsymmetrically through an undulator which consists ofmagnets curved with an approximate semicircle on thecentral portion, or of magnets with the shape of atrapezoid having 1 .2 ° angle.

References

K Halbach, Nucl Instr . and Meth . 169 (1980) 1C A. Brau, Free-Electron Lasers (Academic Press, Boston,1989) chap . 6.N. Olugashi, in : Free-Electron Lasers and their Applica-tions, ed. Committee for FEL m Electron . Eng. Soc. Jpn.(Corona, 1990) chap . 5

[4] L R. Elias, R-C.J. Hu and G. Ramian, Quant. Inst . RepUCSB, QIFEL023 (1983) .R-C.J Hu, Ph.D . Thesis, University of California, SantaBarbara (1986) .E.T. Scharlemann, J. Appl . Phys . 58 (1985) 2154 .G. Dattoli and A. Remen, Laser Handbook, vol . 4, eds.M.L . Stitch and M. Bass (Elsevier, Amsterdam, 1985) chap .1 .

[5]

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