The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October2001

PROPORSAL FOR A NEW TYPE WIGGLER WITH FOCUSING FOR SASE
M. Fujimotot, R. Kato, G. Isoyama, Institute of Scientific Industrial Research, Osaka University
K. Tsuchiya, S. Yamamoto, High Energy Accelerator Research Organization

Abstract
We have studied focusing properties of a planarwiggler for FEL and SASE in the infrared region. A
simple rectangular magnet model can be applied tocalculate the electron trajectory in the wiggler, if thefringing field at the edge is taken into account. A newtype of wiggler can be also used in SASE in the shortwavelength region using a high-energy electron beam.

1 INTRODUCTION
Self-Amplified Spontaneous Emission (SASE) has
attracted considerable attention as a source of high power
coherent X-rays since the power of SASE grows
exponentially in a wiggler up to the saturation level
without an optical resonator. A wiggler of length overmany tens of meters is required for such a shortwavelength SASE and an electron beam with high energy(~GeV) must be transported through it. It is, however, noteasy to transport the electron beam through such a longwiggler in keeping the brightness high, because a planar
wiggler has no focusing in the oscillation plane. To solve
this problem, focusing of the electron beam in a wiggler
has been studied for SASE in the short wavelength regionand some proposals for focusing schemes have been made,including focusing using the sextupole field [ 1].
We are studying focusing properties of a wiggler for
FEL and SASE in the far infrared region. It is widely
known that a planar wiggler has the focusing force in thedirection perpendicular to the oscillation plane, which is
proportional to the inverse square of the electron energy
and to the square of the wiggler field. The focusing forceof the wiggler is weak for the higher energy electron
beam, while it is strong enough to focus the lower energy
electron beam used for FEL and SASE in the infrared andthe far infrared regions.
In this paper, we report analysis of the vertical focusing
in a planar wiggler of the horizontal oscillation type. We
also propose a new type of wiggler with the strong
focusing force, which may be used for SASE in theshorter wavelength region.

2THEORYWe have numerically calculated the electron orbit in
the realistic magnetic field of the wiggler and evaluatedthe effective focusing force from the betatron wavelength
in the vertical plane. Note that there is no focusing in thehorizontal plane for the ideal wiggler. Fig. 1 shows thevertical betatron wavelength as a function of thet fri25@ sanken.osaka-u.ac.jp

oscillation amplitude y0 . The solid curve is the betatronwavelength calculated for an 11.5 MeV electron and thewiggler parameters listed in Table 1. The betatronwavelength is 0.94 m for y0 = 0, which indicates that thefocusing force is so strong that electrons oscillate 2 times
vertically in the 1.92 m long wiggler. The wavelength isnot constant and decreases as the oscillation amplitude
increases due to the non-linearity of the field.
A simple model is sometimes used to estimate effects
of a wiggler on the electron beam, in which the wiggler is
described as a series of rectangular bending magnets in
opposite directions separated by a drift space. The
trajectory of the electron beam can be calculated using thetransfer matrix method and the betatron wavelength thus
calculated is shown by the dotted line in Fig. 1 [2]. Since
the field is linear in this model, the betatron wavelength isindependent of y0 . The wavelength is approximately 10 %shorter than that calculated with the realistic magnetic
field for y 0 = 0 and therefore the focusing force isoverestimated in this model. In order to reproduce thefocusing force in the realistic field, we introduce thelinear fringing field to the edge focusing [2]. The linear
fringing field is determined such that it crosses zero at thesame point as the realistic magnetic field and thus their
1.2 r - - - - , - - - - . - - - - - . , - - - - - - ,Linear Fringing Field
0.9 ----E

Sharp Edge
Wiggler Field.......... 0.6«.c::J...
0.3

E = 11.5 MeV 8 0 = 0.37 T
0.0 ' - - - - - - - - ' - - - - - - ' - - - - - ' - - - - - - - '03

6912Yo [mm]Figure 1: Batatron wavelength A- 13 calculated as a functionof the oscillation amplitude y0 .
Table 1: Wiggler parameters for calculations in this paper
60 mm
Aw
Magnetic Period
32
Nw
No. of Period
30 mmgMagnet Gap
Block Dimensions
lOOmm
2a
Horizontal
20mm
2b
Vertical15mm
2c
Longitudinall.lT
Residual Field
Br
0.37 T
Peak Field
Bo
1.47
K-value
K-461-

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

field integrals are equal. The resulting trapezoid is very
close to that of the realistic field. The restoring force
calculated with this model is given by

where e is the electron's charge, m0 its mass, r its kinetic
energy in units of the rest mass and B0 is the peakmagnetic field of the wiggler. The betatron wavelength
given by A,~ = 2n I .Jk is calculated to be 0.93 m usingparameters listed in Table 1 and shown by the broken linein Fig. 1. It agrees quite well with the wavelength
calculated with the realistic model in the region aroundy0 = 0 and the difference between them is only ~2.4 % for
y0 less than 3 mm. We will, therefore, use the linear
fringing field model in the following analysis.

3 EDGE FOCUSING WIGGLER

force, the wiggler has focusing at the same time in theboth directions (double focusing). But if the horizontal
focusing kh(v) exceeds the natural focusing k, kv(h)becomes negative and the defocusing force appears in thevertical (horizontal) direction.
The magnetic field in the wiggler is numericallycalculated and shown in Fig. 3, where we assume ~ = 2°
and the other parameters are the same listed· in Table 1.
The upper panel shows By calculated on the z axis. Theletters a), b) and c) show positions where transverse
variations of the magnetic field are calculated and themagnetic field at each longitudinal position is shown in
the lower panel. It is seen in the lower panel that the field
gradient dByfdx is constant in the region around x = 0 mm
and it is 0.7 ~ 1.0 Tim. This field exerts a quadrupole likefocusing on the electron beam.
Fig. 4 shows trajectories of 11.5 MeV electrons withdifferent initial displacements in the y and the x directions,and they are calculated in the field presented in Fig. 3. It
is apparent in the upper panel of Fig. 4 that the electrons
execute betatron oscillations in the y direction around y =0. It can be also seen in the lower panel that the electronsFig. 2 shows the schematic drawing of one period of
the Edge Focusing (EF) wiggler. It is the Halbach typewiggler whose magnetic block is cut with an edge angle ~·In a simple model, this is described as a combination ofbending magnets with the edge angle and drift spaces. An
electron injected parallel to the z axis is curved to the xdirection. Since the angle 8 between the direction ofmotion and the normal of the edge is smaller than that ofrectangular magnet model, the edge focusing in thevertical direction decreases and horizontal focusing isgenerated. The angle is given by 8 = 8 - ~' where 8 is themaximum deflection angle of the electron given by fi K/y.
The horizontal and the vertical focusing forces calculatedwith the simple model are given byba)c).._ 0.0

IIco>-0.5
0.0

0.5

1.0

1.5

0.

5a)

b)

\__/

7-\
---50

-50

i=' 0. 0

(2)

A-w
(3)

kv =k-kh,

where k is the natural focusing of the wiggler and givenby Eq. ( 1). When the horizontal focusing force has apositive value and is smaller than the natural focusing

0x [mm]500x [mm]50-50v....·.··

/Y Aw

$...

>!:}

+

J-10~
a-1'E_soX,J"g 2b46~

108

1216AA ~ "AA ~ ~ Av vv y ~14'!

wvvvz /'A

XFigure 2. Schematic drawing of a single period of theEdge Focusing (EF) Wiggler.

2-==i===~50~ = 20

1=::==-...
••lr.""'

0x [mm]Figure 3: Magnetic field calculated with parameterslisted in Table 1 and edge angle ~ = 2°. a), b) and c)shows the positions where transverse variation of themagnetic field in the lower panel are calculated.
10v

2.0z [m]-0.5kh =~ Bo x!L_moe r

0.5

Figure 4: Trajectories of thew 11.5 MeV electron
calculated with the field shown in Fig. 3.

-462-

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

50

1.5.s

E = 1 GeV
cp = 30°
1.0c:Q.

-30E0.5~

X o30~---L----~--~----~----~--~z [m)E [MeV)

Figure 5: Betatron function as a function of electronenergy. See text in details.

Figure 6: Trajectories of the 1 GeV electron in the EF
wiggler with Q> = 30°.

injected at points near but different from Xo executebetatron oscillation in the x direction around the wiggling
orbit of an electron injected at x = Xo· This indicates thatthe wiggler has focusing in the both directions. Betatronwavelengths obtained from the trajectories in Fig. 4 areAJ'Ix = 1.33 m in the horizontal direction and AJ'Iy = 1.28 m
in the vertical direction, which agree quite well withbetatron wavelengths AJ'Ix = 1.34 m and AJ'Iy = 1.28 m
calculated with Eqs. (2) and (3), respectively.

function in the conventional wiggler, while the verticalbetatron functions are almost same in the energy region
lower than 10 MeV, as seen in Fig. 5. The cross sectional
area of the electron beam can be made much smaller inthe EF wiggler than in the conventional wiggler in thelower energy region and the gain length of SASE
becomes shorter due to higher electron density. The
double focusing scheme with the EF wiggler makes it
possible to reduce the saturation length of SASE
considerably in the longer wavelength region.
The natural focusing becomes weaker as the electron
enrgy increases, as can be seen in Eq. (1). The edge
focusing force, however, becomes weaker with theelectron energy and besides it can be made stronger byincreasing the edge angle, as can be seen in Eq. (2). In thehigher energy region, the alternating focusing scheme canbe used to obtain double focusing. In order to show strong
focusing force with the EF wiggler in the higher energy
region, we calculated trajectories of 1 GeV electrons in
the EF wiggler with ~ = 30°, where the same wiggler
parameters in Table 1 are used. The upper and the lower
panels show the calculated electron trajectories in thevertical and the horizontal directions, respectively. The
horizontal focusing is so strong that electrons executebetatron oscillations with the wavelength AJ'Ix = 3.2 m,
while they are defocused in the vertical direction. The
horizontal focusing and the vertical focusing can beinterchanged with the negative edge angle ~ = -30°.
Combining such two types of wigglers alternately, we canmake a long wiggler with double focusing for the higher
energy electron beam. The alternating EF wiggler can beused for SASE in the shorter wavelength region.

4 APPLICATIONS TO SASE
The beam size can be kept at a constant value throughthe wiggler if there is the uniform focusing force and theelectron beam is injected with the appropriate beam size[3]. This matched condition is given by a 0 = 0 and Po =k- 112, where a 0 and Po are the Twiss parameters at theentrance of the wiggler and k is the focusing force. The
beam size is given by .fjk , where f3 is the betatronfunction and & is the emittance of the beam. Fig.5 shows
betatron functions as a function of the electron energy
calculated with the parameters listed in Table 1. The
dotted curve shows the vertical matched betatron function
Pv for a conventional planar wiggler with focusing force
given by Eq. (1) and the dotted line shows the averagevalue of the horizontal betatron function in theconventional wiggler, where we assume that is focused toph = 0.96 mat the centre of the 1.92 m long wiggler [2].
The vertical focusing force in the conventional wiggler isstrong enough to make the beam size small in the lower
energy region, but the focusing becomes weaker as theelectron energy increases and there is practically no
vertical focusing in the energy region higher than 100
MeV. The solid curves in Fig. 5 show matched betatronfunctions phEF in the horizontal direction and PvEF in thevertical direction, respectively, for an EF wiggler with theedge angle ~ = 2° which has the horizontal focusing force
given by Eq. (2) and the vertical one given by Eq. (3).
The natural focusing force of the EF wiggler is the sameas the vertical focusing force of the conventional wiggler.
The horizontal betatron function in the EF wiggler ismuch smaller than the average of the horizontal betatron

5 REFERENCES[1] G. A. Travish, PhD thesis, Univ of Califormia Los
Angels (1995) p. 253 (ch. 8)[2] H. Wiedemann, Particle Accelerator Physics (1993)(Springer-Verlag) p. 118 (ch. 5)[3] C. A. Brau, Free-Electron Lasers (1990) (AcademicPress) p. 174 (ch. 4)-463-