Analysis and measurements of permanent magnet and "bifilar" helical wigglers

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Vfj(;/J A-87-415

Analysis and Measurements of Permanent Magnet 'Bifilar' Wigglers

J. Ashkenazy and G. Bekefi

November 1987

Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology

Cambridge, Ma. 02139

This work was supported by the Air Force Office of Scientific Research, the National Science Foundation, and one of us (J.A) was supported by the Chaim Weizmann Fellowship.

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Analysis and Measurements of

Permanent Magnet 'Bifilar' Helical Wigglers*

J. Ashkenazy and G. Bekefi

Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology

Cambridge, MA. 02139

Abstract

Analysis and measurements of the permanent magnet 'bifilar' helical wiggler [1}, composed of a cylindrical array of staggered bar magnets, are presented. The discreteness of the magnets is taken into account, thus enabeling a calculation of the field strengths of the fundamental and higher spatial harmonics. Calculations show

lhat

the performance of this wiggler is expected to be comparable or better than that of a planar wiggler [2] with similar parameters. In addition, we have studied experimentally and analytically samarium-cobalt-iron hybrids in which soft iron tubes were used to enhance and/or taper the wiggler field amplitude.

* This work was supported by the Air Force Office of Scientific Research, the National Sci-ence Foundation, and one of us (J.A) was supported by the Chaim Weizmann Fellowship.

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1. Introduction

Recently, we proposed a novel 'bifilar' helical wiggler for free-electron laser and cy-clotron maser applications, composed of a cylindrical array of staggered rare-earth cobalt (REC) bar magnets [1]. Such a wiggler is illustrated in Fig.1. An aluminum tube is grooved on the outside with channels running parallel to its axis. The bar magnets, magnetized at right angles to their broad faces, are inserted in each groove at alternating order so that the magnetic moment of one magnet points inward, the next one outward, etc.. The stagger is provided by precision machined nonmagnetic spacers placed at the beginning of each channel and differing in length by lu/N, where N is the number of parallel grooves and

4.

is the desired periodicity. An external nonmagnetic tube (not shown) fits over this structure, thereby keeping the magnets in place. Figure 2 shows schematically the magnet arrangment (a) in cross sectional view and (b) in side-view. It is seen to be composed of two helical strips made of discrete bar magnets. The two strips have opposite magnetiza-tion and a periodicity 1., = 21, where I is the length of each bar magnet. An approximate analogy, found between this wiggler and the current carrying bifilar wiggler, enables a simple though approximate calculation of the field strength [1].

In the present paper we expand the above analysis to include the effects of the discrete-ness of the bar magnet arrangement. A comparison with the performance of the planar wiggler [2] is also made. In addition, we have investigated hybrid configurations which use soft iron to enhance and/or taper the field strength.

The approximate calculation, based on an analogy to the bifilar wiggler, is summarized in section 2. A more elaborate analysis, which takes into account the fact that the wiggler is built of discrete bar magnets, is presented in section 3. It leads to an analytical solution for the magnetostatic field from which an expression for the wiggler field amplitude on axis is derived. In section 4, spatial field harmonics are discussed and their relative field amplitude on axis is calculated. This is useful in wiggler design, whose goals are typically, to maximize the field strength while keeping the field harmonics as low as possible. In section

5, the REC helical and planar wigglers are compared. It is shown that for comparable

design parameters both will have roughly the same field strength.

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was the case in previous sections), is analysed in section 6. The field strength achieved by this arrangement is somewhat smaller than for the radial configuration. However, it provides an additional degree of freedom for varying the field strength. In section 7 a hybrid configuration, in which the wiggler is inserted inside a soft iron tube, is discussed. For radially oriented magnets it is found to enhance the field strength. Since this effect depends on the internal radius of the soft iron tube, it also provides a simple way to taper the field amplitude for the purpose of efficiency enhancement of free electron lasers [31. In section 8 we present results of measurements of magnetic field amplitude and field quality of four different wigglers. In addition, adiabatic tapering can be achieved by insertion into the wiggler a soft iron tube with a tapered wall thickness, as is discussed in section 9. A summary of the results is presented in section 10.

2. Continuum approximation for the field amplitude

As a first approximation, the wiggler can be represented by two continuous helical strips of opposite magnetization. Then, the magnetization is replaced by an effective surface current density at the interfaces between the strips (see eq.(A8) of the appendix and Fig. 2(b)). What we get is equivalent to a bifilar wiggler, made of two helical current sheets having a radial thickness equal to that of the magnets. The current density is equal to 2B,/po where B, is the remanent magnetization. Thus, using the expression for the field of a bifilar wiggler with an infinitely thin wire [4], and integrating over the thickness of the magnetic material in the radial direction r2 - rl, we get:

B f [2E K (z)dzW VjV{Ih(nk~,r)ain[n(O - kz)1} (1)

where k,, = , n = 1,3,5..., x, = nk1,rl, X2 = nk~,r2, In(z) and Kn(z) are the modified

Bessel functions of the first and second kinds and Kn(z) = -K4-(z). From eq.(1) we derive the expression for the field strength on axis:

2Br

B,, z -{U(k ,r1 ) - U(kwr2)}F.,.m (2)

7r

where U(z) = zK1 (z) + Ko(z). Fmi is a semiempirical filling factor, which takes acount

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divided by the total area between r, and r2. For bar magnets with a square cross sectional area (see Fig.2a),

Fei N(r2 - r) (3)

7r(r 2 + r,)

3. Field amplitude calculation for a discrete magnet distribution

Eq.(1) is an appproximation which fails to take account of the discrete properties of the magnets and thus can give no information concerning the amplitudes of the higher spatial harmonics. This is important because typical design goals would be to maximize the field strength on axis while keeping the field harmonics as low as possible. While for the planar wiggler usually only four magnets per row per period are used [2], this number can vary for the helical wiggler and is in fact an additional design parameter. Thus, the dependence of the field profile on this parameter is important.

In the case of the discrete magnet problem the surface current model of section 2 becomes less useful (see Figure 2(b)). Therefore we use the approach of the effective charge distribution (see appendix). The bar magnets are assumed to have a trapezoidal cross section as shown in Fig. 3. When the magnets are thin compared to the radius i.e. r2 - r,

<

rl, this is a good approximation to bar magnets with square cross section.

In addition, when thick magnetic material is desired in order to achieve stronger fields, the use of trapezoidal bar magnets rather than square ones, also yields a higher filling factor. We take the magnetization of each bar magnet to be constant and radial. Then, the magnetization in the volume between r, and r2 is a function of 8 and z only: M = iM(O, z). This corresponds to an effective magnetic charge distribution p,, = - M('z), for which the magnetic scalar potential in the region r < ri can be solved (see appendix). M(6, z) can then be written as a sum over the contributions of each groove, shifted in z according to the pitch. The contribution to M(6, z) of each groove is a square-wave in z multiplied by a rect function in 6: N-1 _It 2r M(O, z) = - h(z - -)( - ) (4) if y O N( h(z ) = I if 0 < z - 1wk < 1w/2 (5 ) 1-1 if -l/2 < z - lk< 0

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where k = 0, ±1, ±2, ..., and

( =

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.N

N

<

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/N

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0 otherwise.

As can be seen from Fig. 3, c is the filling factor for this configuration.

Our purpose is to write our solution in a form which is as close as possible to that of the continuous case. In a sense, this is equivalent to finding the eigenfunctions of a discrete helix, or, to use the terminology of antenna theory, the 'array factor' of this structure. Substituting Fourier series for h(z) and g(6) and after some manipulations, one obtains:

4B, ain(en'r/N) 1

M(8, z) = -4r -in[n'+nkz] (7)

oln n (n 'r/N) n

with the selection rules:

n= 1,3,5,.. n +n' =iN i integer

Inserting eq.(7) in eq.(A4) of the appendix, we find for the field in the region r < ri:

4B X , sin(en'r/N)

B = -=

)

xK., (x)dz (r/N) 2V{fI (nker)in[n'0 + nk~z]} (8)

where z1 = nkeri, X2 = nk.r 2 and where the same selection rules as for eq.(7) apply here

also.

We notice that for n' = -n (i = 0) we get the expression for the continuous helix, which differs from eq.(1) only by the quantity 'in(enr/N) _(nir/N) This quantity represents the filling factor for trapezoidal magnets, corresponding to the semiempirical filling factor of eq.(3) applicable to square magnets. Thus, the field strength on axis is given by:

2B, sin(exr/N) (9

BW = {U(k.rl) -- U(k.r ₂)} (F/N)(9)

T (7/N)

4. Field harmonics on axis

The terms in eq.(8) with n + n'

#

0 represent the deviation from the field of a

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even, a non-vanishing contribution to the field on axis comes from the terms with n' = ±1 i.e. n = iN T 1. From eq.(8) we get for the field strength of the two lowest harmonics on axis, n = N F 1:

BN1 _ U[(N F l)k.ri] - U[(N F 1)k.r2]B, (10)

B NF 1 U(keri) - U(ker₂)

Because of the exponential behavior of U(x), BNFI depends strongly on the number of grooves. As an example, we take a wiggler with 1w = 9.6cm, r1 = 1.0cm and r2 =

1.4cm, which represent experimental parameters discussed in reference 1 and in section 8 below. For N = 6 we get Bw, z 0.05B. and B., - 0.01B,, while for N = 12 we have B L11 < 0.001Bw. In our prototype wiggler N = 12, and it is therefore not surprising that field distortion due to higher harmonics was not observed in these measurements.

5. Comparison with the planar wiggler

For the planar wiggler the field strength on axis of the fundamental can be written as [2]:

2B,. sin(er/N)

B, =B -- {U,(k.ri) - U,(k.r ₂)} (jr11)

(7r/N)

Here U,(x) = ire~, 2r, is the gap width between the lower and the upper rows of magnets,

r2-r, is the thickness of the magnets, e is the filling factor due to the gap between adjacent magnets and N is the number of magnets per period (see Fig. 4). In most practical planar wiggler designs constructed to date, N = 4.

In Fig. 5 U,(x) is compared to U(z) the characteristic function of the helical wiggler. As we can see, both behave almost identically in the region of interest. By comparing eq.(9) and eq.(11) we conclude that a helical wiggler with the same periodicity

4.,

the same magnet thickness r2 - r, as a planar wiggler, and with inner diameter equal to the

gap between the two magnet rows of the planar wiggler, will yield approximately the same field strength on axis.

The lowest harmonic term for the planar wiggler is N + 1 (5"' for N = 4). Using the

expression for the field of the planar wiggler [2] we find:

BN+1 _ sin[(N + 1)?/N] I U,[(N + 1)kwr1] - U,[(N + 1)kr2 Bwp (12)

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Since typically e ~~ 1, the value of the first factor on the right side of eq.(12) is also of the order of unity. Then, by comparing eq.(12) with eq.(10), it is seen that for similar design parameters, a helical wiggler with more than six grooves will have a lower harmonic content than a planar wiggler with N = 4.

6. A wiggler with azimuthally oriented magnets

When the magnets in the wiggler are rotated by 90 degrees, we obtain again a helical arrangement as shown in Fig. 6. In order to solve for this configuration, the magnets are again assumed to be of trapezoidal cross-section with constant azimuthal magnetization: M = iM(e, z) for r, < r < r2. M(O, z) is given by eq.(7). In this case the effective magnetic charge density is (see appendix): p,. = - , while surface term in eq.(A4)

vanishes. Following the same procedure, as applied in section 3 for the radially magnetized configuration, we find for the azimuthally magnetized configuration:

4B 2

sin(en'ir/N)n~'

B = 2 K,(x)d} ('I/N) n V{I(nkr)co8[n'O+nk,.z}} (13)

7rk,1, n '( 7 )W

where x1 = nkwri, X2 = nkr 2, r < r1 and where the same selection rules as for eq.(7)

apply here also.

For the field strength on axis we get:

B. = -- '-{Ko(k.r ) - Ko(kr2) 7r/N) (14)

K ) (if/N)

Comparing eq.(9) and eq.(14), we see that since xK 1(x) is a monotonically decreasing

function, the radial configuration will always yield a larger field strength than the azimuthal one for the same set of design parameters B,, 1W , ri, r2, N and c. As an example we take:

B, = 9kG, L,. = 9.6cm, r, = 1.0cm, r2 = 1.4cm, N = 12 and e = .64. We get for the

radial case B,. = 1.28kG but only 0.86kG for the azimuthal case.

The azimuthal configuration provides a means to vary the wiggler strength while using the same structure and magnets. In fact, one can envisage a scheme in which all the magnets are rotated in their grooves by the same arbitrary angle other than 90*, thereby changing continuosly the wiggler strength. However, it should be noted that this cannot

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be used in a given wiggler, as a means of tapering the field strength. This is because the phase shift associated with the rotation will distort the helical field shape.

7. A hybrid samarium-cobalt-iron wiggler configuration

The field strength of the radially magnetized wiggler can be enhanced by inserting the whole wiggler inside a soft iron tube (Fig. 7). If the permeability of the tube is very large (y -+ oo), the boundary condition at the tube surface is (r = a, 0, z) = 0, and the outer tube radius plays no role. 4m is the magnetic scalar potential and a is the internal radius of the soft iron tube. A magnet, oriented perpendicular to a surface on which the potential vanishes, has an image pointing in the same direction, thus increasing the field strength.

In order to calculate the field strength of this configuration, we have to replace the term _ , in eq.(A4) by the Green's function of a point charge at r' inside a hollow cylinder of radius a. Using standard methods [5], we find for this Green's function:

G (r, r') = 27r2 E (2 - &5mo)cos[m(6 - 0')] cos[k(z - z')jg]m(k, r, r')dk (15a)

where for r < r':

K,, (ka)

9M(k, r, r') = 47r{Km(ckr') - Im(ka)Im (kr')}Im(kr) (15b)

Proceeding as in section 3, the expression for the field is found to be similar to eq.(8) where the integral

f

rK',(x)dz is replaced by f[zK () - I(x)dx, = nk, 1j and X 2 = nk~,r2. This leads to an expression for the field strength on axis similar

to eq.(9), with U(x) replaced by:

K1(kca)

Ui(x) = xKl(x) + Ko( ) - { (Ii(x) - Io(x)}, z = k.r (16) Ii(ka)

From eq.(16) it is clear that he effect of the soft iron is maximal when a is as small as possible i.e. a = r2. As a becones larger, U; and correspondingly the field strength B., approach the values obtained wit no iron cover. The calculated dependence of B, on a

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is shown in Fig. 8 for two wigglers. As we can see, in one case the field strength can be enhanced by as much as 50%.

Fig. 8 suggests that amplitude taper for the purpose of efficiency enhancement [3], can be easily accomplished by varying a. This method seems convenient especially since it does not involve variation of the strengths, positions or orientations of the bar magnets themselves. Thus, it is very well suited, for example, for experimental optimization of the taper profile.

8. Measurements

A 12 channel system was machined to serve as a prototype for testing some of the configurations described in this paper. Samarium-cobalt bar magnets with B,. = 9kG, cross sectional area of 0.4cm x 0.4cm and length I = 1,/2 were used. The variation of the strengh of the magnets as specified by the manufacturer is ::1.5%, while the variation of the direction of the magnetization (the easy axis) is ±3%. The aluminum tube is grooved

so that r1 = 1.0cm and r2 = 1.4cm. For N = 12, the corresponding filling factor (eq.3) is F,.ei = 0.64.

Results of 4 tested wigglers are summarized in Table 1. The first two cases are wigglers with radially oriented bar magnets but differing in their periodicity. The third case is a wiggler with azimuthally oriented magnets. The forth case is for a radially oriented magnet wiggler inserted in a tube of "1010" steel (p/po ;: 5000). The magnetic field measurements were performed with a transverse Hall probe gaussmeter (Bell 610), mounted on a mechanical stage and motor driven along the wiggler axis. The discrepency between calculated and measured values of B, is seen to be about 10% for all four cases. This can be attributed in part to the semiempirical filling factor F,m,, for the square cross sectional bar magnets and in part to the fact that in our calculations we assumed the permeability of the REC magnets to be equal to that of vacuum, while in reality it is higher by a few percent [2].

Measured field traces of the first wiggler of Table 1, made from an x-y recorder chart, are shown in Figure 9. Figure 9(a) illustrates the fields over the central 2-period length of

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the 5-period long wiggler. Figure 9(b) shows the wiggler behavior at one of its ends. Each of the seven traces corresponds to a different azimuthal orientation of the Hall probe made in angular steps of 30*. Similar measurements, performed for the three other wigglers, also show very smooth field profiles with uniform amplitude and appropriate phase relations between successive scans.

The r.m.s variations of the field amplitude (see Table 1) were calculated by sampling the peaks of all the traces for each wiggler. It should be emphasized that no attempt was made to sort or arrange the magnets in order to minimize errors. We expect that using sorting procedures, the r.m.s error may well be reduced by about one order in magnitude. The relative amplitude of the lowest harmonics on-axis, B 1/Bw, was calculated from eq.(10) for the first and second cases in Table 1. For the last two cases, equivalent equations were used, with U(z) being replaced by Ko(x) in the third case and by U(x) in the forth case.

Figure 9(c) shows the field profile when one bar magnet at the center of the wiggler is replaced by a nonmagnetic spacer. Although the local field amplitude drops by as much as 14.5% as a result of this rather large perturbation, the field profile remains quiet smooth. This illustrates the smoothing effect due to the large number of ovelapping magnets per period.

When the wiggler with the missing magnet is inserted inside the soft iron tube, the perturbation (14.5%) remains the same as in the absence of the iron tube. In another test, one magnet in the center of the wiggler was rotated by 90*. In that case the field profiles remained smooth but the drops in the local field amplitude were 15.5% and 20% for the wigglers with and without soft iron cover, respectively. These later results indicate thit a soft iron cover may improve the field quality as far as variations due to errors in the direction of magnetization are concerned. A magnet whose magnetization is not properly directed, will have an azimuthal component of magnetization in addition to the radial com-ponent. As dicussed in section 7, a cylindrical cover of soft iron will effectively enhance a radially directed magnetization while reducing the strength of azimuthally directed mag-netization. Thus, the net result will be a magnetization vector aligned more radially. As a result, a soft iron cover is expected to reduce field inhomogeneities, caused by errors in the

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direction of the easy axis, of REC magnet helical wigglers, in addition to yielding higher wiggler field strengths. Thus, the action of the iron in this case is somewhat similar to the use of iron in planar REC wigglers [6}.

9. Adiabatic amplitude tapering

In most free electron laser application, the amplitude of the first few wiggler periods has to be tapered smoothly from zero to the maximum value, to enable adiabatic electron beam injection. When a tube made of permeable material is inserted inside the wiggler, it will tend to screen the magnetic field. The magnitude of this effect will depend on the yA of the material and the wall thickness of the tube. If the tube is thick enough, the field inside will be zero. Thus, a tube with a tapered wall thickness is a simple method to taper the wiggler field entrance. As in the case of the taper of the outside soft iron tube, proposed in section 7 for the purpose of efficiency enhancement, the present technique likewise does not involve varying the bar magnet distribution.

This method was tried with one of our tested wigglers (the first case in Table 1). The available material was "1010" steel which has a high u (: 5000) and a saturation field of a 16kG. The tube, 33.8cm long, was machined to an outside diameter of 1.59cm to fit inside the wiggler. The inside of the tube was divided into 7 straight sections, each 1/2 (4.8cm) long. Different drills, in steps of 1/64 inch, were used for each section. The inside diameter of the first section was 1.27cm, and that of the last one was 1.51cm. Thus, the tube wall thickness varied from 0.16cm at its thickest end to 0.04cm at its thinnest end.

The measured field on axis over a taper length of 3.5 periods is shown in Fig. 9. Each of the traces corresponds to a different azimuthal orientation of the Hall probe, made in steps of 30*. As can be seen, the field profile is smooth and undistored except for one point, which corresponds to the far end of the tapered tube. This may be the result of saturation at the thin edge of the tube. The tube wall thickness is 0.04cm at this point. The use of material with a higher saturation field and a lower ys may prevent this problem and it may also allow better control in machining the taper.

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10. Conclusions

In this paper we present a detailed study of the novel 'bifilar' REC helical wiggler first

described by us in reference 1. Additional schemes, such as the azimuthally oriented magnet wiggler and the soft iron hybrid wiggler, are introduced. These configurations as well as the original one [1] are solved analytically, including the effects of the discreteness of the magnets. The existance of analytical formulas for the field strengths and field harmonics facilitates design procedure. Our analysis shows that the field strength and quality of the REC helical wiggler is expected to be comparable or better than the equivalent planar wiggler (Halbach configuration [2]). Some of the ideas were tested experimentally on prototype wigglers and the results validate these concepts. A simple way to taper the wiggler entrance for adiabatic beam injection, was also tested successfully.

A promising configuration, in terms of field strength and quality and in terms of

versatility, seems to be the soft iron hybrid wiggler, with the possibility of tapering the external soft iron tube for efficiency enhancement and with an adiabatic electron beam injection provided by an internal tapered soft iron tube.

Heretofore all our experimental studies were made on a short (5 period) prototype system. We are in the process of designing similar permanent magnet helical wigglers to spin up relativistic electron beams on a cyclotrom maser (CARM) experiment to be performed at M.I.T at a frequency of 140 GHz [7]. Although all of our experimental studies were made on helical wigglers with periodicities of 4.6 and 9.6cm, using available magnets in our laboratory, we do not see any difficulties of constructing high quality permanent magnet helical wigglers in the more conventional range with periods of 2-4cm.

Appendix

Properties of REC permanent magnets

Throughout this work we use the simplifying assumptions [2]:

B11 = o(HII + M) _(A)

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to it. The magnetization M is equal to B,/po, where B,. is the remanent field. The above assumptions are a good approximation to the behavior of REC magnets over a large range of applied field strength. As a consequence of these assumptions, linear superposition of vacuum fields is applicable, and the magnetization plays the role of an external source distribution. Maxwell's equations can be solved either for H or for B. If one works with H, then the magnetization contributes an effective charge density and the equations are:

Vx H =0, VH=-M=pm (A2)

H can be derived form a magnetic scalar potential:

H =-V!m, V2 ' V.M . (A3)

The solution to eq.(A3) is given by [8]:

1 ifM -ds' V- M vl(4

de (A4)

47r r -rl r -

rl-The second integral is over the volume of the magnetic material while the first one is over its surface. The term _ , is the Green's function of free-space for a point charge at r'. In cylindrical coordinates (r, 0, z), it can be written as

[9J:

r -7 1 = ;E(2 - bm)cos[m(O -0)] cos[k(z - z')]Km(kr')Im(kr)dk (A5)

for the region r < r'.

When working with B, the magnetization contributes an effecive current density:

V- B = 0, V x B =pOV x M = IOJm, (A6)

Sometimes, V-M = 0 or V x M = 0 inside the magnetic material. It is then convenient to work with the effective surface charge density an (for H) or with the effective surface current density J. (for B) at the boundary between the magnetic material and free space. The boundary conditions for H are:

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The boundary conditions for B are:

. -(B2 - B1) = 0, h x (B2 - B) = -poh x M = poJ. . (A8)

The magnetic material is in region 1, while region 2 is free space and the unit vector i points from region 1 to region 2. If the surface under consideration is the interface between two REC magnets of identical magnetization and opposite orientation, the source terms in eq.(A7) or eq.(A8) add so that o, or J, have to be multiplied by a factor of two.

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References

1. G. Bekefi and J. Ashkenazy, "Permanent Magnet Helical Wiggler for Free-Electron Laser and Cyclotron Maser Applications", Appi. Phys. Lett. 51 (9), 31 August 1987, pp. 700-702.

2. K. Halbach, "Physical and Optical Properties of Rare Earth Cobalt Magnets", Nucl. Instrum. Methods 187 (1981), pp. 109-117.

3. N.M. Kroll, P.L. Morton and M.N. Rosenbluth, "Free-Electron Lasers with Variable Parameter Wigglers", IEEE J. Quantum Elec., Vol. QE-17, No. 8, August 1981, pp. 1436-1468.

4. J. Fajan, "End effects of a bifilar magnetic wiggler", J. Appl. Phys. 55 (1), 1 January 1984.

5. J.D. Jackson, "Clasical Electrodynamics", Wiley, New-York, 2nd edition, 1975, sect. 3.11.

6. K. Halbach, "Permanent Magnet Undulators", Proceedings of the 1982 Bendor FEL Conf., Journal de Physique 44, p. cl-211 (1983).

7. K.D. Pendergast, B.G. Danly, R.J. Temkin and T.M. Tran, "Design of a High Power 140 GHz CARM Amplifier", 12t Inter. Conf. on Infrared and MM Waves, Dec. 14-18, 1987, Orlando, Florida.

8. W.K.H. Panofsky and M. Philips, "Classical Electricity and Magnetism", Addison-Wesley, 2"d edition, 1969, p. 141.

9. W.R Smyth, "Static and Dynamics Electricity", McGraw-Hill, New-York, 3'd edition,

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Table 1. A comparison of four tested wigglers with twelve axial grooves (N = 12). orientation soft-iron of magnets radial no radial no period 41' 9.6 cm 4.6 cm 9.6 cm 9.6 cm no yes B₀ calc. 1.28kG 1.05kG 0.86kG 1.75kG meas. 1.16kG 0.93kG 0.78kG 1.60kG rms variations measured 0.9% 1.2% 1% 0.7% azimuthal radial calculated 0.001 5 -10-10-4 5.-10-4

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Figure Captions

Fig. 1. The 'bifilar' REC helical wiggler for radially oriented magnets, showing the placement of magnets in the grooved nonmagnetic (aluminum) cylinder.

Fig. 2. Schematic of a REC helical wiggler with radially oriented bar magnets: (a) cross-sectional view; (b) side-view after unrolling the cylinder (figure (a) only is to scale). Fig. 3. Bar magnets with trapezoidal cross-sectional area, assumed in the analytical

calcula-tion; e is the filling factor.

Fig. 4. 2D cross-section of planar REC wiggler [2].

Fig. 5. A comparison between the characteristic functions of the helical wiggler U(x) = xK,(x) + Ko(x) and the planar wiggler U,(z) = 7re-z (see text).

Fig. 6. Schematic of a REC helical wiggler with azimuthally oriented bar magnets: (a) cross-sectional view; (b) side-view after unrolling the cylinder.

Fig. 7. Schematic of a hybrid wiggler: tross-sectional view of a REC helical wiggler inside a soft iron tube.

Fig. 8. Calculated magnetic field amplitude for two hybrid REC helical wigglers as a function of the internal radius of the soft iron tube, showing substantial field enhancement for small a.

Fig. 9. Magnetic field amplitude measurements, on the first wiggler in Table 1, as a function of axial distance for different Hall probe rotations,

e

= 0', 30*,60*, ..., 180*: (a) at wiggler center; (b) at wiggler end; (c) after removal of 1 bar magnet; B,. = 9kG,

= 9.6cm, r, = 1.0cm, r2 = 1.4cm, N = 12 and F.,, = 0.64.

Fig. 10. Amplitude tapering showing magnetic field measurements as a function of axial

dis-tance for different Hall probe rotations,

e

= 0,300,60, ..., 180* when a soft iron tube, 3.5 wiggler periods long with tapered wall thickness, is inserted inside the wig-gler; B,. = 9kG, L. = 9.6cm, r₁ = 1.0cm, r2 = 1.4cm, N = 12 and Fem, = 0.64.

Analysis and measurements of permanent magnet and "bifilar" helical wigglers

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