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Nuclear Instruments and Methods m Physics Research A304 (1991) 714-718North-Holland

Undulator fields and electron trajectories at the endof a helical undulator

Daniel Craun and W.B. ColsonPhiiits Department, Naval Postgraduate School, Monterey, CA 93943, USA

The magnetic field structure outside the entrance to a bifilar helically wound free electron laser (FEL) undulator is calculated andused to determine the incoming electron beam trajectories . The beam focusing or defocusing effects due to the stray magnetic fieldsare calculated and compared for different winding configurations to evaluate the optimum design .

1 . Undulator design characteristics

The key to free electron laser (FEL) performance isthe undulator design . One method of constructing aworking undulator is to use bifilar helical windings toproduce the periodic magnetic field . The bifilar helixdesign can be used to reach short wavelength light in aFEL with a compact undulator [1] . The magnetic fieldoutside the compact undulator cannot be expressedanalytically so that numerical integration of the actualwindings is pursued. Because of the small dimensions ofthe compact FEL design, the undulator winding designcannot be of a complicated nature. The small dimen-sions make the winding terminations at both ends of theundulator and the method of power lead connectionextremely important to the resultant magnetic fieldstructure at the entrance to the undulator. The leadwires at the end of the undulator carry sufficient currentto cause unwanted perturbations in the incoming elec-tron beam .

Three terminations are investigated in this article :the wire, loop and staggered termination schemes [2] . Inthe case of wire termination, the bifilar helical wires arecontinued radially outward when they reach the end ofthe undulator (see fig . la). In the case of loop termina-tion, a circle of wire is attached to the end of thecylinder formed by the helix with the windings attachedat opposite positions on the circle. Thus, the currentfrom one helical winding enters at one position on thecircle, flows equally around each side of the circle, andthen combines to flow in the opposite direction in theopposing helical winding (see fig . lb). Staggeredterminations are achieved by using wire or loopterminations as a current taper for the undulator byremoving a specified percentage of the current pertermination depending on its position in front of anextended undulator. This is easily visualized by extend-ing the bifilar helix windings past the entrance to the

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undulator and connecting them periodically by atermination which passes a percentage of the current inthe helical windings . Thus, the current flowing in thehelical windings is reduced stepwise by the number ofterminations added prior to the entrance of the undula-tor . This results in a tapering of the magnetic field .Tapering of the magnetic field at the entrance of theundulator can also be achieved by flaring the windingswhich is accomplished by using a progressively largerradius for the extended bifilar windings as distancebefore the original undulator entrance increases. Theloop termination scheme with current tapering (stagger-ing) vice flaring the windings results in the smallestmagnetic field spike at the entrance of the undulator [2] .This article investigates fields at the entrance of theundulator in the compact FEL. The effect of undulatorexit termination is explored as well, and the fields foundare used to determine the incoming electron trajectories .

Originally proposed by Warren (LANL), a compactundulator design uses a bifilar helically wound undula-tor with N= 10 periods, undulator period N o = 0.9 cm,gap 2g = 4 mm and an electron beam with a Lorentzfactor y = 30.35 . This article presents a simple andefficient computer program for rapid comparison ofdifferent winding configurations to find an optimumdesign . The electron beam trajectories will be used as

Fig. 1 Illustrations of termination geometry for (a) wiretermination and (b) loop termination .

the final determination for optimum design . Tapering inthis article will refer to current tapering at the entranceof the undulator described previously, and not thetapering along the undulator length to achieve higherefficiency .

2. Evaluation of fields

The Biot-Savart law is used to integrate the contri-butions from winding elements . By integrating over aspecific winding configuration, the magnetic field at anyparticular position is found. The integration techniquewas first verified for accuracy by computing the mag-netic field values inside the undulator and comparingthem to the accepted analytical solutions for the mag-netic field at the center of a bifilar helix [3] . Theintegration technique was then applied near the en-trance to the undulator. The structure of the fields isillustrated by contrast gradations corresponding to themagnitude of the magnetic field vectors. As one would

D. Craun, W.B Colson / Undulatorfields and electron trajectories

expect, the magnetic field values are large close to thewindings . Therefore, to see the fine structure of thefields, positions out to only half the radius of theundulator (g/2) are shown. Fig. 2 shows the structureof the magnetic fields for positions from five periodsoutside to the middle of the undulator. The z scale inthe middle shows the magnitude of the current in thewindings by a circle centered on successive periods witha radius that is proportional to the current. The figureshows an undulator with a two period taper that has awire entrance and exit configuration . The white linesare contour lines of constant magnitude. Due to thetaper, the electron experiences significant stray fieldsbefore the entrance to the undulator is reached. There-fore, the stray fields at the end of the undulator have alonger time to affect electron motion and cause deflec-tions . Although not shown, if the entrance is expandedto show more detail, the entrance fields can be seen tobe asymmetrical both axially and radially, Due to thesetwo effects, undesirable defocusing or severe bendingcould hamper the entry of any incoming electron beam .

** Undulator Magnetic Fields **N=10 g/1o=0 .2222 K(O,O,N/2)=1 .00

-N/2z

N/2

715

Fig. 2. Spatial plot of the magnetic field vector magnitudes for positions inside and outside the undulator with wire termination at theentrance and exit.

X. UNDULATORS

716

D Craun, WB Colson

3. Evaluation of trajectories

Using the magnetic field at any given position, wecan calculate the acceleration on an electron by theLorentz force equation (inks units),

eß=- (vxB) if y=0,ymc

where e and m are the electron charge magnitude andmass, B is the magnetic field felt by the electron, c isthe speed of light and v = is is the electron velocity .Since the incoming electrons are relativistic, we makethe approximation that when y >> 1, the transversevelocities are small (ßr, < << ft.) . The approximate equa-tions then become

=ymB,,

ßl --Ym Br "

ß_=0 .

(2)

The z component can be immediately integrated to get- zo + L-r, where T = et/L, and L= NX o is the undu-

lator length . Using the relations x -~ x/Xo, y -> y/Xo

Y

-g/2

-N/2

Undulatorfields and electron trajectories

z

and z ---> =/Xo to normalize, the nondimensional acceler-ations become

zx - Z y

K, (x),

y = - 2 y

K,(x),(3)

where ( ° ) = d( )/dT, and

K,,,, = eg oB,,,/2mmc . Thenew position of the accelerated electron is easily fountby integrating eq . (3) .

The optimum field design is determined by thetrajectories of an incoming electron beam . The beam isgiven an initial position and angle consistent with acharacteristic beam quality factor called "emittance" .The_definition for emittance is e, = 21TxB, and e, =2myO,, and the approximation used in this article is thate, = e, . The quantity x (y) is the rms initial positionspread of electrons along the x (y) direction and B,(0, ) is the rms initial angular spread of electrons fromthe axis of the undulator along the x (y) direction .Either the rms position spreads x, y or the rms angularspreads B,,B, can be changed by external focusing fieldsprior to entrance into the undulator, but their products

** Electron trajectories using K **N=10 y=30 .35

g/~,a0 .2222

#part .=5

N/2

Fig 3 Spatial plot of the magnetic field vector magnitudes including the electron trajectories with wire termination at the entranceand exit .

Ez,E, are fixed . Thus, arbitrary values for x,y and 0,0,are chosen such that x =y = 2 re where re is the electronbeam radius, and B = B, = 2r/N.

Fig. 3 shows the superposition of the electron beamtrajectories on the same design as in fig . 2 which haswire termination used at the entrance and exit of theundulator. The beam must enter in the correct positionand angle to order to get the beam to stay in theundulator. After some optimization, the input positionand angle parameters, starting back at zo = -7 awayfrom the undulator entrance, are found to be xo = 0.7g,y,,= -1 .0g, 0,=O' and 0,, = 3 ° where g=2 mm. Theideal injection angle is K/y = 2 ° . When additionaltapering is used, the input parameters become evenmore extreme. This result is surprising since additionaltapering has always been thought of as being the bestway to achieve the smallest input beam deflection [2,4] .If no offsets are used, the asymmetry in the entrancefields sends the beam into the side of the undulator atN=4 periods. The stray fields resulting from wiretermination at both ends are not suitable for practicalFEL application . Loop termination provides a moresymmetrical field composition [2] and should give asmaller deflection to the incoming electron beam .

-N/2

D Craun, W B. Colson / Undulator fields and electron trajectories

** Electron trajectories using K **N=I0 y=30 .35

g/ko0 .2222

#part .=5

z

71 7

If loop termination is used at the entrance of theundulator and wire termination is used at the exit, theresultant field structure shows better symmetry as seenin fig . 4. The contour lines show that basic symmetry isseen throughout the displayed length of the undulator.Although not shown, an expanded view of the entrancefields shows more symmetry also. The input angleparameters for this design, starting at z, = -7, become0, = -I ° and B,, = 3° . The loop termination reducesthe large magnitude field reversal in the x plane experi-enced by the electrons traversing the undulator entranceand thus, the resultant trajectories are much more likethose shown m the y plane (electron beam deflected bya single angle at the entrance to undulator) . Thus, onlythe angle offsets are given. The electron beam entrancerequirements are reduced, especially m the x plane aspredicted by Fajans [2] . Unfortunately, the y plane stillsuffers from about the same extreme position and angleoffsets . It is possible that, because of the compact FELdimensions, the exit leads (wire termination was used todirect the power leads out to the undulator powersource) are imposing an unexpectedly significant effecton the entrance stray fields . If loop termination couldbe applied to the exit leads, and the wires taken from

Fig . 4. Spatial plot of the magnetic field vector magnitudes including the electron trajectories with loopand wire termination at the exit .

N/2

termination at the entrance

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718

g/2

-g/2

g/2

Y

D Ciaun, WB. Colson / Undulator fields and electron trajectortes

** Electron trajectories using K **N=10 7=30 .35

g/Xa0 .2222

#part .=5

-g/2z

-N/2

N/2Fig 5. Spatial plot of the magnetic field vector magnitudes including the electron trajectories with loop termination at the entrance

and exit .

the undulator in a coaxial cable, the effects of the wireexit termnation would be reduced, and may amelioratethe entrance parameters .

Loop termination is applied to the undulator en-trance and exit leads shown in fig. 5. It is found that theminimum angle and position offsets are achieved whenno tapering is used . The input parameter angles, start-ing from --,, = -5 now due to the absence of tapering,becomes B, _ -0.6 ° and 0, =2° . The general magni-tudes of the fields, shown by the contours inside theundulator, appear more symmetrical than the previousexample. Unfortunately, there is a significant gradientto the field just outside the undulator due to the re-moval of the taper. If an expanded view is investigated,this gradient shows reduced but still significant asym-metry Also, there is significant asymmetry in the fieldat about one period inside the undulator that is not seento the previous design . The result is that the inputelectron beam parameters are slightly reduced from thevalues required from the previous design but are notreduced to the point of being insignificant .

All the designs considered show that stray fields atthe ends of bifilar helical undulators with realistic wireleads cause serious deflections of the entering electronbeam . Even with a termination that achieves improvedsymmetry m the magnetic field structure, loop termina-

tion, the discontinuity in the fields imposed by thetermination itself with any kind of current taperingresults m significant perturbations to the electron trajec-tories . The optimum design of those presented is theloop termination design at the undulator entrance andexit with no taper. This design achieves the smallestposition and angle offset requirements on the inputelectron beam, and would be the least complicated toconstruct .

Acknowledgements

The authors are grateful for support of this work bythe Naval Postgraduate School and the U .S . Office ofNaval Research .

References

[1] R.W Warren (LANL), private communication, September1989 .

[2] J. Fajans, J Appl . Phys 55 (1984) 43[3] J.P. Blewett and R. Chasman, J Appl . Phys . 48 (1977)

2692 .[4] L R. Elias and J.M Madey, Rev. Sci Instr 50 (1979) 1339