Ashkenazy, J.

Bekefi, G.

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PFC/JA-87-18Permanent Magnet Helical Wiggler for Free ElectronLaser and Cyclotron Maser ApplicationsG. Bekefi and J. AshkenazyApril 1987Plasma Fusion CenterMassachusetts Institute of TechnologyCambridge, Massachusetts 02139This work was supported by the Air Force Office of Scientific Research,the National Science Foundation, and to one of us (J.A.) by the ChaimWeizmann Fellowship.Submitted for publication to Applied Physics Letters.- 1 -PERMANENT MAGNET HELICAL WIGGLER FOR FREE ELECTRON LASERAND CYCLOTRON MASER APPLICATIONSG. Bekefi and J. AshkenazyDepartment of Physics and Research Laboratory of ElectronicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139ABSTRACTA permanent magnet, "bifilar" helical wiggler for use in free electronlasers and cyclotron masers has been designed and tested experimentally. Itis composed of a cylindrical array of staggered samarium-cobalt bar magnets,transversely magnetized and held in place in an axially grooved hollow metalcylinder. High quality helically polarized fields of several kilogauss canbe readily achieved.- 2 -The magnetic wiggler, or undulator, is the principal ingredient in freeelectron lasers' (FELs) and high brightness synchrotron radiation sources. 2It is also useful in spinning up3 relativistic electron beams for use in gy-rotrons and cyclotron masers. At present, the favored permanent magnet sys-tem for FELs is composed of a linear array of Rare Earth Cobalt (REC) magnetsarranged in the Halbach' configuration. It produces a transverse, linearlypolarized wiggler magnetic field. A helically polarized wiggler composed ofglued segments of REC material has also been proposed,4 but, because of tech-nical difficulties, has not been widely used. To be sure, helical wigglersoffer advantages compared with linear wigglers. Because of their higher sym-metry, the electron motion and thus the electromagnetic radiation has a lowharmonic content. The FEL gain of the emitted circularly polarized radiationis larger than the corresponding gain associated with a linearly polarizedwiggler of the same strength. And finally, the fact that a helical wigglerprovides electron beam focussing in all transverse planes is a desirable fea-ture since it eliminates the need for placing focussing quadrupole magnetsor solenoids around the wiggler system.In this Note we describe the design and construction of a novel "bifilar",helically polarized wiggler system composed of a cylindrical array of stag-gered permanent magnets. A protype of the wiggler is illustrated in Fig. 1.Samarium-cobalt bar magnets with dimensions 0.4cmxO.4cmx4.8cm and magnetizedat right angles to one of their broad faces, are inserted in an aluminum cyl-inder grooved' on the outside with 12 straight channels running parallel tothe cylinder axis. An external nonmagnetic metal cylinder fits over thisstructure, thereby keeping the magnets in place.Figure 1(a) illustrates the direction of magnetization of our 12-periodsystem as would be observed at some arbitrary cut made perpendicular to thecylinder axis z. It is seen that the dipole moments of six of the magnets- 3 -point radially out, and the dipole moments of the remaining six magnets pointradially in. As oneproceeds along the z axis, this pattern remains invariantexcept for an azimuthal rotation governed by the pitch of the helix. Magnetsof length k yield a wiggler periodicity kw=2z.In order to achieve the desired pitch, the magnets are staggered in thez direction and their dipole moments alternated as is shown in Fig. 1(b). Thestagger is provided by nonmagnetic spacers placed at the beginning of eachchannel and differing in length by tw/N where N is the number of channels (12in our case). After filling the first 6 channels in this manner, a secondidentical set of spacers is used for the remaining six channels, except thathere the directions of the dipole moments are reversed. Stuffing the channelswith magnets is an easy task in view of the fact that neighboring magnets in agiven channel attract one another.Measurements of the magnetic field are shown in Fig. 2. These are car-ried out by means of a transverse Hall probe gaussmeter (Bell 610) mounted ona mechanical stage and motor driven along the wiggler axis. Figure 2(a) illus-trates the fields over the central 2-period length of a 5-period long wiggler,and represents a tracing made from an x-y recorder chart. Each of the seventraces corresponds to a different azimuthal orientation of the Hall probe madein angular steps 0 of 30*. The field profiles are seen to be very smooth, thefield amplitude is uniform and the phase relation between successive scans in6 are as expected. Figure 2(b) shows the wiggler behavior at one of its ends.As will be discussed below, our wiggler is a permanent magnet analogue ofa bifilar system of current carrying conductors. As such, the axial magneticfield along the wiggler axis z should be zero. Using an axial Hall probe, wefind that the amplitude of the axial field is indeed small, less than '50G.This is to be compared with the transverse magnetic field amplitude of 1.16kG(see Fig. 2(a)). The observed Bz is attributed in a large part to the finite- 4 -transverse dimension of the Hall probe (lmm).We note that the results of Fig. 2 are obtained with samarium-cobalt mag-nets5 whose remanence Br=900OG can vary by as much as ±1.5 percent and whosedirection of magetization (direction of the "easy axis") can vary by as muchas ±30. No attempt has been made to sort or arrange the magnets to minimizeerrors, as is generally done for planar wigglers.6 The good performance ofour wiggler is attributed in part to the large number of magnets per period,and in part to the overlapping of magnets along the axial direction, thus lead-ing to smoothing and averaging over field inhomogeneities. As a check, we re-place one full magnet in the wiggler center by a nonmagnetic spacer. The ef-fects of this rather large perturbation are shown in Fig. 2(c). We see thatthe field profile remains quite smooth, although the local field amplitudedrops by about 10 percent.Because of the approximate straight line relationship' beteen B and POHvalid for REC magnets, and thus applicability of the principle of linear super-position of vacuum fields, one can obtain an approximate expression for thewiggler amplitude Bw on axis. The magnetizaion of each barmagnetcan be rep-resented by an effective circulating surface current density J :B r/pO as isillustrated in Fig. 1(b). Superposing these currents in the continuum limitas N -. and the magnet widths and magnet spacings go to zero, one obtains he-lical current sheets flowing in opposite directions, and separated axially bya distance equal to tw/ 2. The current sheets are infinitely thin in the axialdirection, and have a thickness (r2-rl) equal to the bar magnet width in the rdirection. Using the well-known result' for a helical wiggler composed of in-finitely thin current carrying conductors, and integrating over the thicknessr2-rl, yields the following expression for the field amplitude on axis of ourREC magnet wiggler:Bw = (2Br/1) U(kwrl) - U(kwr2)F (1)- 5 -Here Br is the remanence for our material (9kG) and kw=2n/R, is the wigglerwave number; U(x)=xKi(x)+Ko(x) where Ko and K, are modified Bessel functionsof the second kind. Figure 3 shows a plot of U(x) as a function of x.The above equation is approximate. It fails to take account of the dis-crete properties of the magnets and thus can give no information concerningthe amplitude of higher spatial harmonics. However, to allow for gaps be-tween the magnets due to the finite width of the aluminum teeth separatingmagnet channels, we include in Eq. (1) a semiempirical filling factor F =IN(r2-ri)/w(r 2+ri)I equal to the cross sectional area of magnet material,divided by the total area subtended between radii r, and r2 (see Fig. 1(a)).With ri=1.00cm, r2=1.42cm, and N=12, F=0.64. Inserting this value of F inEq. (1) gives Bw=1.28kG for our wiggler periodicity tw=9.6cm. This is to becompared with the experimental value Bw=1.16kG. When we cut all of our mag-nets and thereby reduce the periodicity to 4.6cm, and again arrangethe magnets in accordance with Fig. 1, we obtain Bw= 0.93kG. Equation (1)predicts a value equal to 1.05kG. Thus, we infer from the above comparisonsand from several measurements in which we changed N and/or (rl-r 2), thatEq. (1) yields a reasonably good estimate of the wiggler field amplitude, andcan be used for purpose of scaling and system optimization.In summary then, this Note reports on a novel REC magnet bifilar wigglerconfiguration which is capable of giving a high quality helically polarizedmagnetic field. As yet, no attempt has been made in these preliminary studiesto optimize the system. For example, to increase the filling factor F onecan envision bar magnets with a trapesoidal rather than square cross section.And to further increase Bw, one could decrease r, and increase r2 (see Fig.1) so as to increase the quantity of magnetic material and thereby optimizethe function U(kwr )-U(kr 2) of Eq. (2). This could be accompanied by a re-duction in N which would allow the entire magnet system to come nearer the z- 6 -axis where the electron beam is located. On the other hand, too large a reduc-tion in N can result in an unacceptably large harmonic content. We believethat such helical REC magnet wigglers with wiggler strengths of several kilo-gauss are readily achievable. When one compares a permanent magnet helicalwiggler with a corresponding current carrying system, the former has two obvi-ous advantages. First, there is no need for a power supply. Secondly, wig-gler amplitude and/or period tapering for purposes of adiabatic beam injec-tion 8 and FEL efficiency enhancement 9 can be accomplished much more easilywith the present arrangement.We conclude by noting that REC magnets are not easily demagnetized, andtherefore our wiggler can be safely inserted' in an axial guide magnetic fieldas high as ".lOkG. Such a combination of wiggler and guide fields is usefulfor FEL operation in the Ubitron and Raman regimes 1 where the beam current isrelatively high and the beam voltage low. It is also useful in giving elec-trons rotational motion as is required in gyrotrons and cyclotron masers.'This work was supported by the Air Force Office of Scientific Research,the National Science Foundation, and to one of us (J.A.) by the Chaim WeizmannFellowship.- 7 -REFERENCES1. P. Sprangle, R.A. Smith, and V.L. Granatstein, Infrared and MillimeterWaves, edited by K.J. Button (Academic, New York, 1979), Vol. 1, p. 279,references therein.2. B. Kincaid, J. Opt. Soc. Am. B2, 1294 (1985) and references therein.3. P. Ferguson and R. Symons, IEDM Digest 198 (1981); also R.H. Jackson andC.A. Sedlak, Mission Research Corporation Report No. MRC/WDC-R-061 (1983).4. K. Halbach, Nucl. Inst. and Methods, 187, 109 (1981); also Proc. 1982Bendor FEL Conf., J. Physique (Paris) 44, C1-211 (1983).5. G. Bekefi, R.E. Shefer, and B.D. Nevins, Lasers '82, Society for Opticaland Quantum Electronics, SOQUE, 1982 (STS, 1982), p. 136; also G. Bekefi,R.E. Shefer, and W.W. Destler, Appl. Phys. Lett. 44, 280 (1984).6. D.H. Nelson, M.I. Green, K. Halbach, and E. Hoyer, J. Physique (Paris)45, C1-957 (1984); E. Hoyer, T. Chan, J.Y.G. Chin, K. Halbach, K.J. Kine,H. Winick, and J. Yang, IEEE Trans. Nucl. Sci. NS-30, 3118 (1983).7. B.M. Kincaid, J. Appl. Phys. 48, 2684 (1977).8. J. Fajans, J. Appl. Phys. 55, 43 (1984).9. N.M. Kroll, P.L. Morton, and M.R. Rosenbluth, IEEE J. Quantum Electron.17, 1436 (1981).10. J. Fajans, and G. Bekefi, Phys. Fluids 29, 3461 (1986); also J. Fajans,J.S. Wurtele, G. Bekefi, D.S. Knowles, and K. Xu, Phys. Rev. Lett. 57,579 (1986).- 8 -FIGURE CAPTIONSFig. 1. Schematic of a permanent bar-magnet helical wiggler: (a) cross sec-tional view (to scale) showing the direction of magnetization of mag-nets; (b) side view (not to scale) after unrolling the cylinder.Fig. 2. Magnetic field amplitude as a function of axial distance for differ-ent Hall probe rotations, 6=0, 300, 600, .... 1800; (a) at wigglercenter; (b) at wiggler end; (c) after removal of 1 bar magnet.Fig. 3. The function U(x) of Eq. (1).(a)JssAwS ,N N sN -' NS N S - N(b)Fig. 1Bekefi & Ashkenazyo (a)(kG)Z(cm)Fig. 2Bekef i & Ashkenazy3.53.0-2.5-2.0-U(X)1.51.0-0.5-0.0 0 5 1.0 1.5 2.0 2.5 3.0xFig. 3Bekefi & Ashkenazy

Permanent magnet helical wiggler for free electron laser and cyclotron maser applications

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Permanent magnet helical wiggler for free electron laser and cyclotron maser applications

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