The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

X-ray FEL in a Small Emittance Storage Ring
Y. Miyahara
SPring-8, Kamigori, Ako-gun, Hyogo 678-12, Japan

Abstract

Table 1 X-ray FEL and undulator parameters------------------Wavelength
Beam energy
Peak current
Electron beam size
X-ray wave size
FEL gain
Optical cavity loss
Undulator
PeriodGap height
Peak field
K value
Number of periods
Total length

It is shown that aq X-ray FEL using a single crystal
optical resonator with a· normal Bragg reflection in astorage ring endorsed with a small emittance and narrowenergy spread electron beam can have an FEL gain of 24% at a wavelength of 0.206 om, which is higher than thecavity loss. It is al$o shown that the gain is enhanced bya factor of 4 in the SASE regime.

1 Introduction
By the recent progress of a photo-cathode RF gun togenerate a very small emittance and highly intenseelectron beam, a single pass X-ray FEL(SASE) using ahigh energy linac, as high as 10-15 GeV, is becominga realistic subject. The emittance decreases as the increaseof the beam energy. Main cause of the trend is not only
because of the progress of the RF-gun, but also becauseof the lack of a high reflective mirrors in X-ray region.
Previously, we have noticed that a high reflectivity of
X-rays can be obtained by the Bragg reflection with asingle crystal, and a high peak current and a small
emittance, as small as 46 prad-m, of electron beam canbe obtained in a storage ring fairly modified from theSPring-S storage ring [1]. It was also shown that the FEL
gain at a wavelength of 0.217 om can be higher than themirror loss at a beam energy of 3 GeV.
In the present paper, a further consideration on theoptical cavity FEL in the X-ray region is discussed.

A.

E
lpCJx,ywo
G

B
A.og
Bo
K
Nu
Lu

0.20592.94
120011312420om
GeV
A

138
0.248
0.301
300
3900

mm
mm
TJ.UD

Jlm%%mmby the use of a Fabry-Pellow resonator [2, 3, 4]. It is
expected that the phase shift of the reflected wave is 1t andnearly constant around the center of the normal reflection.
To stand against a high heat load of the X-ray FEL,
we prefer the use of a diamond crystal (the unit celllength is a=0.356688 om) because of a high thermal
conductivity (1000-2000 W/m/K) and a small thermal
expansion coefficient (0.8x10-6/K). A reflection as highas 90 % can be obtained by the Bragg angle reflection in
the wavelength region shorter than 0.3-0.4 nrp, so weselect tentatively (111) plane reflection at a wavelength
1..=0.2059 om. Figure 1 shows schematically the FEL
system made of the single crystal optical resonator with a2 Optical resonator FEL with Braggreflection
Previously, we have assumed a Bragg reflection at anangle of 9=n/4 of a pure Si single crystal. Thereflectivity at the FEL wavelength of 0.217 nm is
expected as high as 90 %. The high reflectivity is,however, limited in a narrow angle region about 10 J.l.I'ad,
and the reflected wave suffers a phase change from -1t to 1t
depending on the reflection angle in the full width of thehigh reflective angle region. Thus, the ring resonator withthe Bragg reflection is not appropriate as the opticalcavity for the FEL.
In the course of this study, it was noticed that theBragg reflection at 9=n/2 is also very high even in amuch wider angle range about a few mrad. This property
has already been investigated theoretically for an
application to a high resolution X·ray monochrometerFig.1 Schematic FEL system and radiation wave size.

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Table 2(=1.7xl0-3), which are the necessary conditions for theFEL oscillation as well as the gain>loss relation.

~mall emittance storage ring parameters

Beam energyCircumference
Periodicity
Natural emittanceCompaction factor
Radiation energy
Energy spread
Tune

Natural chromaticity
Damping time

E3
GeV
1440mp2446
pm-radExO
5.68x10-5<Xp235keV
uo
6.23x10-4
EE/E
84.40
Vx
68.45
Vy
1.39x10-3
Vs-124.9~X-106.0
~y65
ms
tx
123
ms'tyIll
mste
208.2
kHzfoh2442
508.42
MHzfRF
351
kV
VRF
1.0%
~Em axlE
20ps
O"t
10 rnA/bunch
Io48
Nbc

Revolution frequency
Harmonic number
RF frequency
RF voltage
Bucket height
Bunch length
Average currentNumber of bunches
Bending magnets (per cell)
Bending field
B0.3272
Tn-valuen1041.96
Bending radius
p30.558
mMagnets length {number) lB
0.8(9), 0.4(2) m
Number of bends
264
NB
Quadrupole magnets
Field gradient$16By•
T/m
Magnet length (number) lQ 0.8(8), 0.6(4), 0.3(6) m

confocal radius R=b=30 m, where b is the mirror distance.
The radiation wave size of the Gaussian distribution ofintensity is wo=(A.b/27t) 112=31.4 Jlm at the center of thecavity and w=-Y2wo=44.3 J.!ffi at the surface of the mirror.
Thus, the divergence of the radiation at the position of themirror is only about 1.4 Jlrad, which is much smaller
than the width of the high reflection angle range. Table 1
represents the parameters of the FEL and undulator, which
was estimated from the parameters of the storage ring
described below.
We have previously made a design consideration of asmall emittance storage ring, which is modified from theSPring-8 storage ring in such a way that the beam energy
is reduced from 8 GeV to 3 GeV, and the number ofbending magnets is increased from 88 to 264. This
modification reduces the emittance from 6 nrad-m to 46
prad-m. The storage ring parameters are given in Table
2. The emittance ex,y(=exo/2) is close to the diffraction
limit AJ47t(=16 prad-m), and the energy spread is muchsmaller than the undulator radiation width l/(2Nu)

3Gain enhancement

In the previous section, the gain was estimated on
the assumption that the increase of the radiation power isvery small, so that the power is almost constant in one
pass through the undulator. This is the so called Compton
regime. However, the estimated gain is 24 %. Therefore,
it is necessary to take into account the increase of theradiation power. The SASE theory [5] can be applied
when the emittance of the electron beam is approximately
smaller than the radiation limit, which is satisfied in thepresent case as mentioned before. In this case the powerincreases as P=Poexp(47tNuP) , and the gain is given by
Gs=exp(47tNuP)- 1. Here, p is the Pierce parameter
given bywhere <yo> and Yr are the average and resonance energyof the electron beam, np is the plasma frequency androo(=27tc/A.o) is the undulator frequency. The plasma
frequency is given bywith no. the electron density, e, the electron charge, eo.
the dielectric constant in vacuum and me, the mass ofelectron. Using the same parameters of the undulator andthe electron beam given in Tables 1 and 2, we obtain theparameters for the FEL in the SASE regime show.n i.nTable 3, and the gain is Gs=0.95 %. Thus, the gam ts
enhanced about 4 times. We cannot, however, expect ahigher gain with a longer undulator since the energyspread is larger than the Pierce parameter.
Table 3

Parameters for SASE regime

Wavelength
Number of electrons per bunch
Electron density
Plasma frequency
Undulator frequency
Pierce parameter
Enhanced gain

4A.
Nenonproop
Gs

nm
0.20
3.0x1oll
1.3xl023
4.4x107 Hz
1.4x1012 Hz
1.8xl0-4%
95

ConciUlsion

In the present paper, we have investigated thepossibility of X-ray FEL at a wavelength of 0.206 .nm on
the assumption of a storage ring with a small emittanceand a small energy spread of the electron beam. We have-526-The 11th Symposium on Accelerator Science and Technology, Harima Science Garden City, 1997

assumed an optical resonator with a normal Braggreflection by a single crystal, with which a highreflectivity in a wide angle range can be expected.
Estimated FEL gain is 24 % in the Compton regime, andenhanced to 95 % in the SASE regime, which is muchhigher than the mirror loss. In spite of the crude
estimation, the present investigation is very encouraging
for promoting detailed investigations of the storage ringFELs.

Acknowledgment
The author is grateful to Professor K.Kohra and Dr.
T.Ishikawa for the discussion on the normal incidence
Bragg reflection and the information on the references
related to the subject.

References[1] Y. Miyahara, to be published in Proc. of Particle
Accelerator Conference 1997, Vancouver.
[2] K. Kohra and T. Matsushita, Z. Naturforsh. 27a(1972) 484-487.
[3] 0. Brummer, H.R.Haeche,and J.Nieber, Phys. Stat.
Sol. a53, (1979) 565-570.
[4] A. Caticha and S.Caticha-Ellis, Phys. Rev. B25,(1982) 971-983.
[5] J. B. Murphy and C. Pellegrini, Nucl. Instrum. andMethods A237, (1985) 159-167.

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