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Numerical Studies of Strong Focusing in Planar UndulatorsG. Travish and J. RosenzweigDepartment of Physics,University of California, Los Angeles, CA 90024

AbstractThe present trend towards shortwavelength operation with long undulators placestight requirements on the electron beam qualityand hence the need to maintain a well focusedbeam. This paper examines the performance ofalternating gradient (AG) sextupole focusing inplanar undulators 111. Numerical simulationresults of Free Electron Laser (FEL) performanceusing AG sextupole focusing are compared toresults using only natural focusing and to those

using quadrupole focusing.Introduction

Free Electron Laser performance is affectedby the overlap (in six dimensional phase space)between the electron beam and the optical beam.The extent of overlap (assuming perfect alignment)is dependent on electron beam size, emittance andenergy spread and optical beam size and Rayleighrange. In general, the brighter the electron beam,the better the FEL performance. The electron andoptical beam overlap is maintained by the wellknown optical guiding phenomenon. However,without focusing the electron beam diverges. Whenthe electron beam density is reduced, FELperformance is degraded. Hence, maintaining afocused electron beam over the distance of the FELundulator is paramount to high FEL performance.Obviously, for a given beam, the longer theundulator the more significant focusing becomes.Recent proposals for short wavelength devices havecalled for long undulators from 25 to over 60meters long.It is possible to make some simplequantitative statements about how electron beamfocusing affects FEL performance. For a high gain(exponential regime) amplifier, the power outputcan be written as

P( 2) = PoeZ/L, (11where PO is thealong the devicefolding) length.characterized by tp, [21

input power, z is the distanceand Lg is the power gain (e-The gain length can be.he fundamental FEL parameter,hL,=-i- 4&Tp (2)

e2k; +k2 --B, (4)x b -2E2,where x and y are the two transverse directions,B, is the undulator magnetic field, e is the electroncharge and Eb is the beam energy. For a planarundulator kpx=O or kpy=O; typically for a helicalundulator or curved pole faces, kpx=kpy Relation(4) sets a limit on the weak (constant gradient)

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where hu is the undulator period. The fundamentalFEL parameter scales with the third power of peakbeam current density, J1/3.A general feature of free electron lasers isthat if variations occur on a scales shorter than again length, performance is affected. For focusing,this rule of thumb implies a limit on the betafunction, p>k. A limit on the focusing strength asa function of the emittance is given byp> 2Pn

,(l+@)(31

where y is the beam Lorentz factor, en is thenormalized rms emittance and au is the usualnormalized (unitless) undulator parameter. Thislimit is derived in the one dimensional limitassuming no energy spread. It is useful for settingan approximate limit on how strong a given FELsfocusing channel can be. As is discussed in thefollowing sections, it is not merely the strength butalso the type of focusing that affects FELperformance.Sextupole Focusing

Free electron lasers provide weak naturalfocusing; in both planes for helical devices, butonly in one plane for planar undulators.Fortunately, this problem can be solved by usingScharlemanns curved pole faces 131. Constantgradient sextupole focusing (via curved pole facesor any other means), like natural focusing, has aconstant transverse velocity for each electron. Fora distribution of electrons there is a correspondingdistribution of velocities. Natural focusing does notperturb the transverse phase space: however, itcan not provide sufficient focusing for longerdevices.A relation between the focusing betatronwavenumbers is required by the Maxwellequations:

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focusing strength. A large kp2, and hence a strongfocus in one direction implies a negative kp2, andhence a defocus in the other direction.Consequently, strong focusing implies alternatinggradients.External quadrupoles have been proposedand used [4] to produce AG focusing in an FEL. Anexternal FODO lattice can provide strong focusingfor undulators. Unfortunately, only iron freeundulators can be used with conventionalschemes. Various novel schemes have beendevised in an attempt to overcome this limit:canted poles, Panofsky quads, edge fieldpermanent magnet arrays [5]. etc.. Regardless, ithas long been stated that quadrupole focusing candegrade FEL performance because it modulates thetransverse velocity of the electrons.Recently the idea of alternating gradientsextupole focusing in an FEL was studied. Theoriginal scheme called for alternating the curvatureof the undulator poles, thus producing a strongfocusing lattice. Other methods to achieve this typeof lattice have also been considered: externalsextupoles, side mounted magnet arrays, etc.. Thebeam dynamics and hence the FEL action are thesame (in the limit of averaging over an undulatorperiod) and so the exact fo rm of the sextupolefocusing is unimportant. Here the wavenumbersare allowed to be imaginary so that relation (4)does not limit the focusing strength. This isanalogous to quadrupole strong focusing and tothe feed down effect in circular machines.The transverse electron velocity is, ingeneral, different in a focusing section from that ina defocusing section. Of course a given e lectroncan be matched between the two sections, but adistribution of velocities precludes matching.This leads to the central question: Is thecontinues periodic oscillation of the transversebeam velocity caused by a quadrupole lattice betteror worse than the discrete changes caused bysextupoles? The answer is that quadrupoles arebetter. Upon reflection this seems to be intuitivelycorrect. Abrupt disruptions of the beam phasespace will tend to degrade the FEL action [6].Smooth changes which occur on scales greaterthan a gain length are less deleterious. It isperhaps easiest to show this by performing acomplete 3D simulation.

SimulationsThe code TDA3D [7] was modified to allowfor sextupole focusing. This code solves theaveraged FEL equations in 3D and takes intoaccount known phenomenon for the regime

studied here. The sextupole focusing is accountedfor in the simulation by modifying the vectorpotential of the undulator (au). Quadrupolefocusing is simulated by adding a term to theparticle equations of motion.The example parameter set discussed hereis the SLAC! based X ray FEL (8.91. The parametersare given in Table 1. It serves as a good test casedue to the long length of the undulator and lowbeam emittance. Notice that applying equation (3)yields a beta function of 5 meter for peak FELperformance.Table 1: SLAC X ray FEL parameters used in theimulations for thispaper:

Y Energy (mcL)en Emittance normalized(mm-mrad)Peak Current (A)Pulse Length (fS)au Undulator parameterLl Undulator period (cm)Al- Optical wavelength (m-n)0 FEL aarameter

140003 x 10-625001606

8.341.7 x 10-3In order to reliably compare sextupole andquadrupole focusing, identical lattices werecalculated (same period, beta function and phaseadvance per cell). Monoenergetic beams whereused (no energy spread). in a focus/ defocus (FD)lattice (no drifts). A study of the effect of the phaseadvance per cell was first done. Typically, a phaseadvance per cell of 90 degrees is used to minimizethe average beam envelope. However, this createslarge fluctuations in the beam size. As expected,simulations confirm that when the phase advanceis large and hence the beam is modulated a greatdeal, then the FEL action is degraded. To avoidthis added effect, subsequent comparisons whereperformed with a phase advance per cell of aboutten degrees.Figure 1 shows the results of a series ofsimulations. Three sets of data points are plotted:quadrupole lattice, sextupole lattice and 3D semi-analytic calculationslO. The quadrupole set iscIearly the best. As expected, there is an optimal

focusing strength. Peak quadrupole performanceoccurs close to, but not precisely at thetheoretically predicted 5 meter beta function. Afigure of merit for the effect of focusing on an FELis given by the variation of the phase over abetatron period. This is related to the extent ofdetrapping of electrons from the ponder-motivewell. For quadrupole cases, this effect is small. Forthe sextupoles used in this example, detrappingbecomes significant for a beta function 5 5 meters.

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ConclusionsStrong (alternating gradient) sextupoledoes not work as well as comparablefocusing. Strong quadrupole focusings very well in an FEL. Simulations indicateemittance limit on focusing can evenexceeded provided that a small phase advance

used. This implies a need for a high fieldfocusing systems. Such systems wouldfor the construction of shorter, higher gainelectron lasers.There may be occasions when sextupoleis advantageous, but this is unknown attime. An independent numerical confirmationthis work would be a useful endeavor.Acknowledgments

The authors thank Claudio Pellegrini for hisuseful comments. A great deal of this work isthe work of Ted Scharlemann. This worksupported by the US Department of Energy

References[l] For a detailed discussion of AG sextupolesee G. Travish and J. Rosenzweig, Strong

focusing fo r Planar Undulators, Proceedings of theThird Workshop on Advanced Accelerator Concepts(1992).[2] R. Bonifacio. C. Pellegrini, and L. M. Narducci,Opt. Commun. 50, ~373 (1984).[3] E. T. Scharlemann, Wiggle plane focusing inlinear wigglers, J. Appl. Phys. 58 6. p. 2154(1985).[4] T. J. Orzechowski, et al., High Ef/iciencyExtraction of Microwaue Radiation from a TaperedWiggler Free Electron Laser, UCRL-9484 1 ( 1986).151 R. Tachyn. Permanent Magnet Edge-FieldQuadrupoles As Compact Focusing Elements ForSingle-Pass Particle Accelerators, SLAC-PUB-6058. Also, see the Proceedings this conference.[6] C. Pellegrim, Nucl. Instr. and Meth. A272 ~364(1988).[7] T. M. Tran and J. S. Wurtele, Comput. Phys.Commun., 54 263 (1989).[8] H. Winick. et al., A 2-4 nm Linac CoherentLight Source (LCLS) Using the SLAC Linac.Proceedings this conference.[9] SLAC X-ray FEL simulations are presented in,Performance Characteristics, Optimization, andError Tolerances of a 4nm FEL Based on the SLACLinac. Proceedings of this conference.[ 101 Y. H. Chin, K.-J. Kim and M. Xie, Phys. Rev.A46, 6662 (1992).

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