Some limitations on the design of plane periodic electromagnets for undulators and free electron lasers

Nuclear Instruments and Methods 176 (1980) 487-495 © North-HoUand Publishing Company

SOME LIMITATIONS ON THE DESIGN OF PLANE PERIODIC ELECTROMAGNETS FOR UNDULATORS AND FREE ELECTRON LASERS M.W. POOLE and R.P. WALKER Science Research Council, Daresbury Laboratory, ICarrington I¢A4 4AD, UK Received 11 March 1980

Undulators and free electron lasers are potentially important new sources of intense quasi-monochromatic or coherent radiation produced by the action of a periodic magnetic field on a beam of relativistic particles. It is shown that field variations in the transverse plane arise as a direct result of the periodic nature of the magnet and that this restricts the minimum period that can be obtained. A parameter is introduced to describe the field variation and its effect on the linewidth of the output radiation. Magnetic field computations have been carried out for a range of magnet geometries and the results are compared with useful analytic approximations. The restrictions on reducing the magnet period are then examined in detail. Consideration is also given to the more pronounced saturation effects that occur in such periodic magnets.

1. Introduction

linear field polarization). An electromaglaet using steel poles to generate the magnetic field has a number o f advantages compared with the alternative helical winding geometry that is likely to require superconducting technology to produce acceptable field values [9]. These advantages include the relative simplicity o f conventional technology and the flexible design o f magnet geometry provided, for example, b y interchangeable pole pieces; there is also likely to be much better access to constituents o f the magnet, such as vacuum components. Although a plane magnet may have a highly uniform magnetic field in the plane of bending o f charged particles passing through it, inhomogeneities remain in the transverse direction normal to this due to the intrinsic periodicity of the magnet. The effect of such a field variation may have serious consequences on the performance of an undulator or free electron laser. During the course o f the F E L I X studies it became clear that certain criteria could be established that would be of assistance to the successful design o f periodic magnets for either type o f device. These conclusions are described here, together with numerical examples to illustrate their application. Both analytic and numerical calculations have been carried out, and the latter have been used to assess the validity of several convenient approximations.

Considerable interest exists in the use of undulators and free electron lasers as line sources o f radiation. Undulators have very high brightness and a quasi-monochromatic spectrum essentially produced by an interference phenomenon; when installed in a synchrotron radiation source they are able to make a significant contribution to the range of experiments that the facility offers. Although many undulators are either planned or under construction, few have so far been operated [ 1 - 6 ] and some of these were demonstration experiments that had designs that were not optimal for their full utilization. Free electron lasers are related to undulators, but the beam o f electrons is stimulated by the radiation field to emit a coherent monochromatic output; extremely high power and wide tunability "should be possible, and at output wavelengths at which no other such source exists. However few experiments have yet been undertaken [3,7]. Both types of device incorporate a periodic magnet which may have either circular or linear field polarization. However there is little experience in the design, construction and performance of such special magnet structures. A study o f a possible free electron laser experiment (FELIX) at the Daresbury Laboratory o f the Science Research Council [8] has included the design of a plane periodic magnet (i.e. 487

488

M.W. Poole, R.P. Walker/Plane periodic electromagnets

2. Some results from undulator theory The theory of undulator radiation, which is produced by electrons travelling in the z direction through a spatially periodic magnetic structure, has been treated in detail by a number of authors [5,10, 11 ]. The transverse field in a plane periodic magnet is of the form By = Bo sin(27rz/X0)

and the associated particle trajectory (fig. 1) is x = q;o(Xo/2rr) sin(2~rz/Xo) with )to the period of magnet (and particles); Bo the maximum transverse magnetic field; and fro the maximum deflection angle of particles from z axis. It is usual to express fro in terms of the field (deflection) parameter K = ~o3` = eBoXo/27rmoc

(1)

with 3` = (1 -/32) -1/2 and mo = particle rest mass. In general the undulator radiation consists of a series of harmonics with a fundamental wavelength Xol/1 K 2 ) X = 23`2 \ + ~-- + 3`202

(2)

with 0 the angle between radiation vector and z axis. Choice of K therefore determines the fundamental wavelength but it also prescribes the number and intensity of the higher harmonics contained in the spectrum. For K ' ~ 1 the source of radiation is an oscillating dipole in the frame moving with the average velocity of the particles along the z axis (the socalled "dipole approximation"); in this case only the fundamental appears. As K increases other harmonics become important and for K >~ i the greater part of the radiation intensity appears in these higher harmonies (although the transverse particle motion is not, as is sometimes stated, relativistic since still K "~ 3`). It should be noted that as K continues to increase

so does the fraction of the radiation appearing at the higher harmonics, and eventually for K >> 1 the envelope of these harmonics forms the familiar synchrotron radiation spectrum and the device becomes a "multipole wiggler" rather than an undulator. The fractional linewidth (fwhm) of the fundamental at any angle 0 is given by (3)

AX/X ~ 0.9/N

where N is the number of magnet periods. This is termed homogeneous broadening since it is present even for the radiation emitted by a single electron, but in practice the observed linewidth will always be increased by other effects. Eq. (2) shows that both angular and energy spreads in the beam of particles will contribute, as will also the finite acceptance angle of any radiation detectors. Furthermore the transverse field inhomogeneities of the undulator magnet in conjunction with the finite cross section of the particle beam will have an important effect given by Kz AB X -1+K212 B

AX

(4)

In general it may be stated that all such inhomogeneous broadening of the spectral lines of an undulator is undesirable, but as will be seen later there may be severe practical difficulties in minimising these effects. The performance of an undulator source may be characterized by its on-axis (0 = 0) brightness function, the energy radiated into the fundamental per unit solid angle and unit frequency interval, to a good approximation given by d2I

N2e 2 3`2

K2

dco d~2- 4rreoc (1 +K2/2) 2"

The brightness of any synchrotron radiation source is often expressed alternatively in terms of the photon flux per unit solid angle in a given fractional bandwidth but in either case the brightness is maximized for K = x/2 (reduced slightly by exact calculation). However a different optimum is obtained if the total energy per unit solid angle in the fundamental is to be maximized. Using the linewidth from eq. (3) dI Ne23`4 K2 d - ~ - eoXo (1+K2/2) 3"

Fig. I. Particle trajectory in a plane periodic magnet.

(5)

(6)

Clearly the undulator is required to have the smallest feasible value of magnet period, )to, and an optimum value of K = 1, for a given value of 3'. Alternative optimization procedures can be employed, especially if

M. W. Poole, R.P. Walker/ Plane periodic electromagnets the electron beam source is designed to suit the undulator, and the choice of K at which any undulator is operated must therefore be related to the specific application for which it is required. In practice any undulator will be designed to operate over a wide range of value of K up to some maximum value of magnetic field. The severe problems associated with achieving small values of ;% will be discussed in later sections, but at this point it may be noted that for a magnet period Xo = 10 mm the required field level is ~1 T for K = 1, indicating the difficulties to be expected with a conventional (non-superconducting) electromagnet. It should be noted that similar expressions can be obtained for the operating parameters of a helical undulator [9,12], replacing factors K2/2 by K 2 and therefore resulting in optimum values of K smaller by l/x/2 than those of the plane magnet considered in this paper. Similar radiation is produced except that it has circular not linear polarization and no harmonics appear at 0 = 0. 3. Some results from free electron laser theory The theory of free electron lasers is closely related to that for undulators but the dominant mechanism becomes the stimulated emission of coherent radiation in the presence of an existing radiation field. In particular it may be shown [3,13] that the gain of a free electron laser (FEL), determined by the difference between the stimulated emission and absorption rates, is proportional to the derivative of the spontaneous emission lineshape [~(AX/X)-2]; the importance of a narrow linewidth is evident, and the broadening mechanisms pointed out in section 2 must be minimized. FEL operation can be obtained in a "super-radiant" mode, where the laser amplifies either an externally injected radiation beam or its own initially spontaneous undulator radiation. However a more likely mode of operation would be as an oscillator using mirrors to create an optical cavity within which the radiation intensity builds up to the~desired value. A small optical mode area maximizes the coupling between the electron beam and the radiation field, and it is for this reason that the proposed FELIX project has adopted a confocal arrangement. In this case peak gain, G, for a single pass through the laser has the following form [8]

~::2

489

with L the magnet length (N'Ao) and I the peak electron beam current. Once again the advantage of a small magnet period, ;ko, is clearly demonstrated and the gain can be readily shown to be maximized for K = 2. In practice a FEL is likely to employ variation of K to extend the output wavelength tuning range beyond that achievable with electron energy alone [see eq. (2)].

4. Magnetic field computations Magnetic field computations have been carried out with the object of investigating the field imhomogeneities in the direction normal to the plane of bending of the electrons and determining the consequent restrictions on reduction of magnet periodicity. Although the chosen geometry was based on the preliminary design for the proposed FELIX magnet [8] this geometry is typical of plane magnet designs for undulators of FELs that are likely to be constructed. Fig. 2 shows the chosen geometry for the computations, including an end pole to ensure zero net deflection of an electron beam passing through the magnet. A return yoke between the top and bottom halves is not required for this type of magnet and so has not been included here. Results have been obtained with a wide range of magnet gaps (h = 3 0 - 1 0 0 mm) and periods (X0 = 100-200 mm), with particular emphasis on the parameters of the FELIX magnet (table 1). In this section are described those for which the pole width, p, and separation, s, were equal and the maxi-

[Coil

I hl

K2

G cc ;k~/~ (1 + K2/2) 3/2 IL2

Z

(7)

Fig. 2. Geometry used in magnetic field computations.

M.W. Poole, R.P. Walker/Planeperiodic electromagnets

490 Table 1 Proposed FELIX magnet parameters Nominal field Period No. of periods Length Pole width Pole separation Field parameter

Bo Xo N L p s K

J1Term /

10

/II

0.1 T 200 mm 25 5m 50 mm 50 mm 2

8 o

mum field on the median plane, Bo, was 0.1 T to avoid saturation effects; variations in these parameters will be considered briefly in the next section. The computations have been carried out with the program MAGNET [14] which has been employed extensively for magnet design at Daresbury Laboratory and elsewhere, since it is easy to use and requires relatively small amounts of storage and execution time. A square mesh is employed which is adequate for the simple geometry under investigation. The program proceeds by solving alternately for the air and iron regions, modifying the boundary condition on the interface at each change o f region in order to maintain field continuity. Figure 3 shows By/Bo on the median plane plotted against z over one magnetic period for the FELIX magnet case. It can be seen that with p/s = 1 the distribution is to a very good approximation sinusoidal. Fig. 4 shows the variation in By against y directly under the pole (z = )%/4). For small y the field varies approximately as y2. The variation over the whole gap is large, almost 10%, and by eq. (4) this would result in a significant inhomogeneous broadening o f the output radiation linewidth.

~

/

ii/

////

/ /

.-'-

0.6

08

,, / / ,, / / . . "

O0

02

04

Terms

////

III

6

2

t ~ ----12 Terms

10

2y/h

Fig. 4. Variation of transverse field component with y.

Previous work [15] has shown that the ratio between the maximum field on axis and that at the pole tip, Bo/Bp, varies with the ratio between the period and the gap, Xo/h. Fig. 5 shows the results for various gaps and periods plotted in this way. The points lie approximately on a single smooth curve although this is somewhat different to the one given in ref. [15]. The rapid drop in Bo/Bp as ~to/h decreases not only adversely effects the output linewidth but also imposes a restriction on the minimum period that can be achieved for a given maximum value of Bp and minimum h. This problem will be returned to in section 6. Maxwell's equations have a solution compatible with the magnet boundary conditions with the fol-

10. 10 • •

/s= X 3.0

"

06-

/

// /

/

CO 04" -/../

.

/ -0 2,

- -

////'"

-----

oo

oo,oo

ooo,,oo,

Equation 9

-10 50

tO0 z(mm)

150

200

Fig. 3. Variation of transverse field component along axis of FELIX magnet.

oo

i

~,

3

4.

5

6

ko/h Fig. 5. Ratio of maximum transverse field on axis to field at pole tip as a function of ratio of magnet period to gap height.

M. W. Poole, R.P. Walker/Plane periodic electromagnets

491

lowing field components:

The expression for Bo/Bp is therefore

By = B 0 sin(kz) cosh(ky)

Bo/Bp- c o s1h ( ~ ) ( ~ --[sinh(~)/3 -~~]/" sinh(3~)] ~

Bz = -Bo cos(kz) sinh(ky)

(8)

with k = 27r/Xo. This is valid strictly only for small y since no boundary conditions at y = h/2 have been included. Fig. 4 includes the field variation according to eq. (8) for comparision with the accurately computed values. As expected there is good agreement only at small y. It follows directly from the equation above that

I 1 B°/BP - cosh(kh/2)- cosh(~)

B m sin(mkz)cosh(mky),

Fig. 5 shows that this curve is in excellent agreement with the data. As an alternative to using Bo/Bp to measure the amount of field variation across the entire gap a more useful approach is to describe the variation of field at small y over realistic electron beam sizes. A parameter, a, can be defined by

By =Bo(1 +ay 2) (9)

with ~ = lr/(Xo/h). Although this is not expected to be valid the relation above does nevertheless approximate the shape of the true variation (fig. 5). A better analytic approximation to the true solution can be obtained by including higher harmonics which also satisfy the symmetry requirements [16]. The resulting expressions are

By= ~

(12)

(10)

(atz = ko/4)

(13)

and so the simplest approximation using eq. (8) predicts 112rr~2 a = 2 ~,~-o-o/

(14)

which is equivalent to ah 2= 2~ :. Using the more exact 2-term approximation it follows from eq. (1 1) that Bo

By = ~

[cosh(ky) - f cosh(3ky)]

(15)

m odd

Bz = ~

with f = sinh(~)/3 sinh(3~). It follows that for small y

Bm cos(mkz) sinh(mky) ,

rn odd

ah 2 = 2~2 1 - 9f 1 -f

3 2/2oNI sin(m~r/4) where B m - - 7rXo m sinh(mkh/2) with NI the ampere turns per coil. In this analysis the coefficients have been obtained by approximating the variation of B z with z at y = h/2 as a square wave of the appropriate amplitude. The summations are therefore over an infinite number of terms. The expression for By at y = 0 converges very rapidly, in most cases two terms being sufficient, but this is not the case for By in general where up to 12 terms can be required. The question arises therefore of how many terms need be included. As far as By is concerned it has been shown that the main error in approximating its behaviour by eq. (8) is the variation with y, which is clearly not as cosh(ky). Adding higher order terms only significantly affects the y dependence of By and in fact two terms gives the best fit to the data (see fig. 4). The argument for terminating the series after two terms is therefore not based on convergence, rather that this gives the best approximation to the computed field distribution. The resulting expression for By at z = )%/4 is then

3 212oNI (cosh(ky) B y - ~---~Xo \ sinh(~)

cosh(3ky) 3 sinh(3~)/"

(16)

20

15

c~ 10 ¸

• -....

~, %,

Computed values Equation 16 Equation 14

5,

.',,\

o(

i

2

3

a

S

6

?

Xolh

(1 1)

Fig. 6 Transverse field inhomogeneity as a function of ko/h.

M.W. Poole, R.P. Walker/Plane periodic electromagnets

492

Fig. 6 shows the results from the computations with MAGNET together with the analytic curves of eqs. (14) and (16). The agreement in the latter case is excellent. The rapidly increasing field inhomogeneity as )to/h decreases in once again clearly shown. The main variation with )to/h is seen to be described well by the simple approximation eq. (14), the agreement being very good for )to/h ~<2. This result implies that the field variation at small y depends essentially on )to and is influenced only to a small extent by the magnet gap. This is easily understood in terms of the following relation derived from Maxwell's equations

cost of introducing higher harmonics into the field variation along z with some consequent effect on the output radiation. However as the ratio p/s is increased the space available for coils is correspondingly reduced. As has already been pointed out there will generally be a requirement to minimize Xo in any magnet design and this makes it unlikely that values of p/s even as high as 2 will be acceptable. Nevertheless this technique could prove most valuable on occasions where the demand for good field homogeneity is particularly critical.

~2By _

~2By

5.2. High field

Oy2

Oz~

linking the field variation in the two planes, from which eq. (14) follows.

5. Further investigations

5.1. Variation o f pole width and separation For several values of Xo and h the computations have been repeated with different values of the pole width, p, and separation, s. Fig. 3 shows as an example the case p/s = 3 for comparison with p/s = 1 ; the increased field flatness exhibited also results in better uniformity in the y direction. Fig. 7 shows the variation of the transverse field coefficient, a, with p/s for the geometries that have been investigated. A reduction in the value of a by as much as a factor of 3 can be achieved by such increased pole widths, albeit at a

02

The fact that the magnetic field must be increased if smaller period magnets are to be designed (in order to keep an optimum K value) has prompted an investigation of the extent to which the previous results can be applied to high field magnets. Computations have been carried out with increased current densities for the FELIX magnet case, with corresponding values of Bo up to 0.7 T. Above this field level it proved impossible to achieve satisfactory convergence in the MAGNET computations, apparently due to large steel saturation effects even at field levels that would be considered modest for more conventional magnet geometries. Indeed significant saturation was observed in the upper regions of the pole at fields (Bo) above 0.5 T and 'at the highest computed field levels the overall mmf drop around the steel circuit was ~20%. Despite this the results suggest that tow field data on Bo/Bp and ah 2 (figs. 5 and 6) are still valid for all the computed field values.

to

(a) Xo-140, h-30 (b) ~o = 200, h=50 (c) h o - 2 0 0 , h - 3 0 C - core geometry IE E

geometry

o05 u3

oo

i

2

3

4

p/s

Fig. 7. Transverse field inhomogeneity as a function of ratio of pole width to separation.

OC

1'0

2'0 x 103

NI(At)

Fig. 8. Excitation curves for a periodic and equivalent C-core geometry.

M. W. Poole, R.P. Walker /Plane periodic electromagnets

In order to compare the extent of saturation in a periodic magnet with that in a magnet with a single pair of poles, computations have been carried out with a C-core geometry with the same gap, pole size and coil arrangement and the excitation curves for both magnets are shown in fig. 8. The earlier onset of saturation in the case of the periodic magnet is clearly seen. This problem should therefore be borne in mind when considering the design of high field periodic magnets.

5.3. Effect of coil geometry The results presented so far were obtained with the coil geometry shown in fig. 2. In order to assess whether the magnetic field distribution depends significantly on coil dimensions computations were also carried out with in one case large coils filling the entire space between the poles and in another small (5 mm X 5 ram) coils situated near the pole face, for the FELIX magnet (table 1). The results for the large coils were identical to those obtained previously, but the results for the small coils were slightly different. The effect of the small coils was to eliminate flux crossing directly between adjacent poles but to increase local flux circulation around each coil. In general this reduced field levels in the steel by as much as 40%, the exception being at the pole tips adjacent to the coils where the field was increased, but by a smaller amount. As in a conventional magnet geometry therefore this arrangement of coils is beneficial from the point of view of reducing saturation effects. However a further effect of the small coils was to decrease B0 and increase Bp, and therefore to increase the field inhomogeneity in the y direction; the coefficient a increased from 0.0311 to 0.0362% mm -2 and from fig. 7 we can deduce that this is equivalent to the po!e width being reduced from 50.0 mm to 47.4 mm. In practice no magnet is likely to be constructed with such small coils, which have very high current density and fail to utilize efficiently the available coil window between poles. The results presented with larger coils appear to be valid for all realistic coil geometries.

6. Choice of periodic magnet parameters The advantages of achieving as small a value of ko as possible in magnets for both undulators and free

493

xlO 2

6

5 ¸

~.

4-

2 ¸

1

0

i

~

3

a

6

6

Fig. 9 M a x i m u m obtainable K value for given m a g n e t gap and m a x i m u m field at pole tip as a f u n c t i o n o f ~.o/h.

electron lasers have already been stressed in sections 2 and 3. A further reason may be the desire to achieve output radiation with as short a wavelength as possible with the maximum electron energy available [see eq. (2)]. An example of this is the proposed European Synchrotron Radiation Facility (ESRF) [17] where the maximum energy of the proposed electron storage ring is determined by the requirement to produce 1 A radiation from installed undulators. For any chosen value of )to the curve in fig. 5 shows that Bo/Bp increases as h decreases, so that the minimum value of h and maximum value of Bp will determine the maximum value of K that can be achieved with any such plane periodic magnet. Conversely, if a specified value of K (e.g. the optimum) is to be possible then the minimum allowed value of Xo is defined. To see this more clearly, eqs. (1) and (9) can be used to express K/hBp as a function only of)to/h, and this relationship is shown in fig. 9. To illustrate the use of this curve one can assume a maximum Bp = 1.5 T and a minimum h = 20 mm, which appear to be close to the feasible limits of a conventional electromagnet installed in an electron storage ring. For K = 1 the minimum value for X0 is found to be 26 mm; even with a 5 GeV electron beam the shortest output wavelength of the fundamental is then about 2 A. Shorter output wavelengths would require such an undulator to be operated at a reduced value of K where the brightness is no longer optimized. Alternatively an increased value of K could be chosen and a higher

M.W. Poole, R.P. Walker/Plane periodic electromagnets

494

harmonic of the resultant spectrum selected; for K = 2 the minimum allowed Xo becomes 32 mm and the fifth harmonic is at ~1 A. It should be noted that in general a free electron laser will require to be operated at higher values of K than an undulator, necessitating somewhat longer values of X0 than the latter. It is useful to introduce a figure of merit for a periodic magnet, defined as the number of ampere-turns per coil required for a given K value. Some rearrangement of eq. (11) gives

NI K -

1.18 X 10 a sinh(~) (1

sinh(~) 3-~)1

"

(17)

This function is plotted in fig. 10 against values of Xo/h, and the dramatic increase in the magnet excitation current required as Xo/h decreases can be clearly seen. This demonstrates that for a specified magnet gap any attempt to reduce the magnet period while maintaining the same K value demands rapidly increasing current. Since the space between poles for coils is also decreasing a situation is soon reached where the current density becomes so large that superconducting coils are essential. For a gap of 20 mm and K = 1 the required current density rises from 10 A/mm 2 at Xo/h = 3.3 to 100 A/mm 2 at Xo/h = 1.7, assuming coils completely filling a square aperture between adjacent poles. Furthermore fig. 10 neglects

xlO 3 20-

18-

the saturation effects that have been shown to be of some importance in this type of magnet; in the superconducting magnet being installed in the ACO storage ring ~50% additional ampere-turns are required to overcome losses in the iron at an operating level of only 0.4 T [15]. The above treatment allows the fundamental technical limitation on the performance of a periodic magnet to be assessed. However any practical design must also generate radiation with a spectral linewidth that leads to acceptable brightness of an undulator or gain of a free electron laser. From eqs. (4) and (14) the inhomogeneous broadening due to magnetic field variation is A)t_ K2 1 (27r] 2 X (1 +K2/2) (AY)2

2\~o1

(18)

with Ay the electron beam half-height. This is the full width of the broadened line, and the FWHM is assumed to be ~50% of it, allowing the following ratio, R, between inhomogeneous and homogeneous contributions to the total linewidth to be determined [using eq. (3)] L K2 R = 7r2 X3(1 +t~,-2,~, /z) (AY)2

(19)

For a fixed magnet length the minimum Xo is seen to be dictated by the specified values of K and Ay if the permitted value of R has been determined. As an example, for FELIX a value R ~<0.5 has been assumed, so that the increase in total linewidth due to magnetic field variation is restricted to ~<12%. In this case, using parameters from table 1

16-

Xo ~> 50.9(Ay) 2/3 14"

and with a maximum electron beam radius of 7 mm the minimum allowed ~ is then 186 mm. Equation (18) has assumed the simplest approximation for the field coefficient, a, and the exact value depends on the choice of ~/h. The reasons for a choice ~ / h 2 3 have already been discussed and for such a case a more accurate calculation using eq. (16) permits a reduction of ~10% in the value of ~ predicted by (19).

12.

10"

7 v

(Ay, Xo in mm)

8"

7. Conclusion 1

2

3

4

5

6

7

Xo/h

Fig. 10. Required number of ampere-turns per unit K as a function of Xo/h.

The highest spontaneous power output from an undulator and maximum gain in a FEL are each obtained with the smallest magnet period at particular

M. I4I.Poole, R.P. Walker / Plane periodic electromagnets optimum K values. However it is in decreasing ~ that fundamental design limitations are reached. Because the amount of field variation across the gap increases rapidly as Xo/h decreases a lower limit on the value of )to is set by the smallest feasible aperture and by the maximum allowed field at the pole tip or the number of ampere-turns that can be provided. In certain applications the requirement for a small output linewidth may restrict Xo further. The inhomogeneous broadening due to the transverse field variation over realistic electron beam sizes, described by the parameter a, and the homogeneous broadening ( " q / N ) depend principally on Xo rather than Xo/h and so they determine a minimum value of Xo independently o f h . Any one of these criteria may be the most important limiting factor depending on the particular application. Special problems are likely to be encountered in the design of high field periodic magnets and in this category must be included magnets with modest Bo but with Xo/h ~< 2, since the onset of saturation effects is more rapid than is the case with more conventional magnet geometries. Careful optimization of coil geometry using a reliable computer program will then certainly be required, but the more general conclusions of this paper will still be valid. Some of the technical problems arising in the design of periodic electromagnets can be overcome by the use of permanent magnets constructed from materials such as rare earth cobalt. Since such magnets avoid the need for space for coils they might be used to achieve very small periods, although these will still be restricted by the permissible field variation across the gap to meet the linewidth specificatiori of an undulator or free electron laser. In a permanent magnet the maximum field on the axis will be determined by the remanent field of the material and also by the value of Xo/h (in a" similar way to an electromagnet); this limits the value of Bo to less than 0.5 T and imposes a requirement for a very small gap if correspondingly small magnet periods are to be obtained. A particular disadvantage is that Bo can only be varied by changing the magnet gap; although such a feature can be incorporated in the magnet design, there will inevitably be a corresponding variation in field homo-

495

geneity. However there will probably be occasions when a permanent magnet will be competitive with an electromagnet on b o t h technical and economic grounds, and in this case the general conclusions of this paper will still be of use. In conclusion, therefore, the results presented here should allow the main parameters of a periodic electromagnet to be determined with some confidence for any particular undulator or FEL application, although detailed computational work will be required to optimise the design, particularly in the case of "high field" magnets. The authors are indebted to the encouragement they have received from Dr. D.J. Thompson, Head of the FELIX Study Group at Daresbury Laboratory.

References [1] [2] [3] [4]

H. Motz et al., J. Appl. Phys. 24 (1953) 826. A.I. Alikhanyan et al., JETP Lett. 15 (1972) 98. L.R. Elias et al., Phys. Rev. Lett. 36 (1976) 717. M.M. Nikifin, Proc. Wiggler Meeting, Frascati, June 1978. [5] D.F. Alferov et al., Part. Accel. 9 (1979) 223. [6] A.S. Artamonov et al., Institute of Nucl. Phys., Novosibksk, to be published. [7] D.B. McDermott et al., Phys. Rev. Lett. 41 (1978) 1368. [8] M.W. Poole, Daresbury Laboratory Report, to be published. [9] G. Chu, Stanford Report, SSRP 77/05, Stanford (1977) IV-74. [10] A. Hoffman, Nucl. Instr. and Meth. 152 (1978) 17. [11] D.F. Alferov, Yu.A. Bashmakov and E.G. Bessonov, Synchrotron radiation, Lebedev Phys. Inst. Series 80, ed. N.G. Basov (New York Consultants Bureau, 1976) p. 97. [12] B.M. Kincaid, J. Appl. Phys. 48 (1977) 2684. [13] J.M.J. Madey, H.A. Schwettman and W.M. Fairbank, IEEE Trans. Nuel. Sci. NS-20 (1973) 980. [14] C. Iselin, CERN Program Library Long Write-up, T600 (1971). [15] J. Perot, Proc. Wiggler Meeting, Frascati, June 1978. [16] S. Krinsky, Proc. Wiggler Meeting, Fraseati, June 1978. [17] European Synchrotron Radiation Facility, Supplement 2, eds. D.J. Thompson and M.W. Poole (ESF, Strasbourg, May 1979).