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Figure-8 undulator as an insertion device with linear polarization and low on-axis power density
Nuc!e;rr Immmentsmd
Metiodsin~Physics
Research A 364 (1995) 368-373
Figure-8 undulator as an:insertion device with linear polarization and low on-axis power density T. Tanaka”~“, H. @ tamurab Received 2 March 1995
Abstract A new type undolator for linearly polarized radiation. figure-8 undulator, is proposed. It has the advantage that the on-axis power density is much lower than that of an ordinary planar undulator. Electrons inside the proposed device move along the trajectory which looks like a figure-8 when projected on the plane perpendicular to the beam axis. The spectral performance and the power density are calculated and compared to those of an ordinary planar undulator.
1. Introduction Although planar or helical undulaton have been expected as most powerful sources in third generation synchrotron radiation facilities. heal load problems are very serious plrticularly in case of planar undulators [ I 1. It is well known that the lower photon energy of the fundamental can be obtained by applying higher K value ( = 0.9348 (T&a) A, (cm) ). which causes the increase of unwanted higher harmonics. As a result, the on-axis power density becomes higher and the optical elements may be damaged by unreasonabk heat load. In case of helical undulators. however. no bigher harmo+ its are observed on axis even when very high K values are applied [ 2 1. Therefore. the on-axis power density is much lower compared to that in the previous case, which means that we can expect ideal utilization of undulalor radiation without any unreasona bk heat load. However. it should be noted that only circularly polarized radiation is avaiiabk. The purpose of this paper is to find Out new approaches r0r obtaining linearly polarized radiation with low on-axis power density. We have found two approaches so far. One is a tandem helical utidulator of interfetential type composed 0r two helical undulatoss having differmt helicities. The tinearty polarized radiation can be obtained as a coherent sum of right-hand and left-hand poiahd radiations, the principle of which is similar to that of the crossed undulator 131. Although no higher harmonics ate obsened on axis like an ordinary helical utilator, the degree of the obtainable linear polariulion may be degmdeJ easily by the finite beam emittance. -
. carrcpoadirrgaahol.
in this paper, the other approach is proposed for obtaining high degree of linear polarization. This novel device is called figure-8 undulator since the electron orbit projected on the transverse plane has an eight-figured shape.
2. Principk! As an ideal case. an example of the electron trajectory in a figure-8 undulator is shown in Fig. I. If the trajectory ic projected on the n-v. y-z or z-x plane, we obtain the orbit
----+X
a
2 fig. I.T~of~ck*mraovilgra~~~~~~.bl$c $cri&orau~orrsbdwn.
7: Tunoku. II. A’itamurdNucl.
Insrr. and hfcth. in Phw. Rrs. A 364 f 199.0 36X-373
369
magnetic held is given by the equations B, = -Bat sin
as shown in Figs. 2a-2c. respectively. The electron moves on a right handed and a left handed circle alternately ( Fig. 2a). therefore, the polarization is linear since the componcnl of circular polarization is canceled out However. it is difhcult to realize such a trajectory. because the magnetic field should be discontinuous at the point where L = nA. (n is an integer). which does not saiisfy the Maxwell equations. Then, let us consider another type of ttajectory as shown in fig. 3. The orbit projected on the r-v. v-c or 2-x plane is shown in Figs. Sa-3c and the relation between the relative vchrcities. fl, and j3,. is shown in Fig. 3d. In this case. the
5~ . ( A” >
(I)
(2)
B,
where Bal.d, arc the peak magnetic fields. It is easy to realize the above fields by using an undulator as shown in Fig. 4. A pair of magnet arrays located above and below the axis is an ordinary planar undulator which generates the vertical field (horizontal undulator). while the four outer arrays yrcnerale the horizontal field ( vertical undulator). It should bc noted that the period length of tk vertical undulator is twice longer than that of the horizontal unduiator. Solving the equations 0)’ motion, we obtain the positior and relative velocity as
tb)
K,:
I)
.
-----;s:n2wcIr 8swny
p(rt=(uLm,f. (Y K’+
I-
13)
,
%5~, Y
K’
-”
K’
- 2 w
2 -~COS2f&~ 4r?
4y’ *
>
MS c&d
(41
(6)
(20)
(7)
As shown in the above equations, integer/half-odd-integer harmonics are found to be horizontally/verticaIly polarized. The power density is calculated as
The spectral intensity is calculated as
-=dzP
c
dR dw
dP
Oc - d?P‘
(8)
dl1 do ’
hll2
d2Pk -= dll dw
e2 y2N2 K(fi
whctek=
f.l.f.2
f,(y6.e$,
=4q2s,recosd
,:&y4wN
j
d&($-$).
(21)
-n
(9)
+ rt). . . . . . n/2 ,...
D=l+X2+Y2.
(22)
x=ye,
(23)
and
Y= - KI(S2 +s-?)lPN.
+S-i)lPh.
~~,(ye,~)=~[2StryBsin~-K,(.Si
- K,cos2&.
ye, -
K, cos&,
(24)
Et = Ki sin’ & + 4Kz sin’ 2iJ.
(25)
Ez =4( K,Ysint+
(26)
(IO) (II)
2K,Xsin2f)‘,
(12) where 8, = 6, cos C$and 6,. = @sin r$.
PM =
sin( ~-NW/W)
nN( k - w/w,)
(13)
’
3. Example
47rcy2/ A. WI = II-(K;+Kf)/2+y:d2’
(14)
(IS) ,,’ -
i
co
3.1. High-field case (K > I )
X = 4tK,yt?sin&
( 16)
- 2KvyBcosq5
.
(17)
7 SKi .=-.
( 18)
4
Half-odd-integer harmonics are found to appear in the spectmm. which is derived from the vertical unduiator having the period length, 2A.. On axis. Eqs. ( IO) and ( I I ) are rewritten
-~K,PN
2 [J-+r-~fYl I=-‘*: + J-z,-A.I(Y)IJ,(Z).
JcCO.4) = k = integer. 0.
k = half odd integer.
f 19) k = integer.
0.
-ZK& fdO.4,
=
I J-.z,w2~+,,,.2~Y~ I--+ J-‘~~-,~~-I,;~(Y)IJ,(Z).
(
Let us compare the performance between the planar and figure-8 undulators. For example, the electron energy is set to 8 GeV and the beam cutrent 100 mA.
k = halfodd
integer.
First, high-field case ( K > I ) is considrmd. The period length is assumed to be IO cm and the number of periods 44. The photon flux densities for various harmonics and the power density are shown in Figs. Sa and 56 as a timction of the ratio of KI to Ky. l’be photon flux density for each harmonic is normalized by the value at KS/KY = 0. In the figures. the value, Vm. is set to be constant as 4.72 in order to fix the energy of the fundamental at 500 eV. From these figures. it is fotmd that the power density andrlnnecessary higher harmonics are reduced rapidly as .Ta,‘KI. white the degr&tiin of the fundamental is not so temarkable. Although the optimum value of KI/K, is not decided uniquely, it is sufficient to keep K*/K, more than 0.5 in order to suppms the heat bad. Fig. 6a is the spectrum obtained from the planar undulamr (K, = 0, K, = 4.72). No even-number harmonics an found and the intensities of o&J-number harmonics are much higher than the fundamental. If only the tin&mental is needed, then other higher harmonies give rise to unreasonabkheatloodontheopics.F~.6bis~spectrumolnainad from the figure-8 undulrtor ( K1 = K, = 3.34). ‘he intensity of the cUndamental at 500 eV is about two thitds of that of tbe planar undniator as well as some of half-odd-integer lwmmiis are found in the spectrum. However, the inter+ lies of harmoniis bigber #tan the tenth are negligibfy low. In this’ way, the figure-g trajeetcny efl&iveiy UrpqrrJres ttie higher h4aamnii GfL axis.
(00 rt
60
gw 5 8
t40 20 0
(b)figure-8
x10" -'
<'?
(b)figureS 0; 2 1:
__I; 12x10”
1 A.
Spatial d&i&ions of the power density of the planar and figuure-8 undulaton are shown in Figs. 7a and 7b. respectively. the undulator parameters are the same as those in Figs. 6a and 6b. in case of the planar undulator. tiu ,pocver density has a maximum of 98 kW/mnd? on axis. @I the other hand. the distribution of the figun-8 unduiator is asymmetric with respect to 8, and iuoks like a V figure. since j?, is asymmetric with respect to & as shown in Rg. 3d. Since most of the r&i&d power of the figure-8 tttt&iator distributes on the V figure. much kss heat load falls on axis than in case of the planar undulat0f. TIE on-axis pawa density is as low as 1.4 kW/nxadZ. which is impo?tam benefit for the soft x-ray optics.
i<
3.2. lm-fefd
0 0
‘mo
lww
case {K “r I )
Nex~.tkbw-fieMcasc(K - II iscomidendTbt petiodlengthisassumedtobe3.2cmandthCnanbn@f periods 140.
+++f Y 5th(63
3keV)1
3rd’38
(a)planar
\\
-
0 1x1o’8
OkeV’
+ + -+---+--.+ .-+ - +-..-+._.._ (b) 4x10’8
o i L -1.. 0
RATlO
.-_-A
-4 50000 PHOTON
and the lifth harmonics t 31 K,/ K, = I .OS). Thus, a certain harmonic can be removed sclcctively hy setting K,/K, to the node. Fig. 9a is the spectrum obtained from the planar undulator ( Kx = 0, K, = 1.0). Although the contribution of higher harmonics to the spectrum is less than that in caseof the high ticldcasc. the intensity of the third harmonic is as high as half of that of the tirst harmonic. In genctai, this is removed by detuning the second crystal of a monochromator [ 4 1. In case of the figure-8 undulator. the intensity of the third harmonic can bc controlled by adjusting K, value as shown in Fig. 8a. The spectrum obtained from the figure-8 undulator (K, = K, = 0.71 ) is shown in 6ig. 9b. It is found that the third harmonic is as low as one tenth of that of the fundamental. so that dctuning may bc unnecessary. Although the second harmonic hccomcsrcmarkahlc.this may be easily climinatcd by using an ordinary crystal monochromator (e.g. Si( I I I ) ).
4. Ellects by the angular divergence In the previous section. we have ignored the emittance of the electron beam. In practice. the cmitntncc is finite and atkcts not only the brilliance but the polarzation. In this section. the photon flux density and lhc dcgnx of polarization are calculated for the clearon beam having the linitc angular divergence. If the pmbability distribution km&on of the heam is of Gaussian type. lhc photon flux density and the degree of
linear polarizati~rn.
9.
-ts,,).
(27
s,, - SW s,,+
(28 1
mr s.,.,,
1OOOOO
ENERGY(eV)
arc siven by
d’l’, = C2Y2Ni 4nroc(SII dI1 dw 4. =
(b)ftgure-8
=
I 27ry~U,M-,~
J J u du
I,
xexp
-
(ucosn
Wt.,(u.a)
--I
- yflcosf$)’
2Y’4,
[ (Usina
- yBsin#)’ Zy”o$,
(29) I
where cr >I,,) are the horizontal and vertical angular divergences. The spectra and Pt. obtained fmm the figure-R undulator are shown in Figs. IO (high-field case) and I I ( low-field case). The natural cmittance is assumed to be 7.0 x IO-’ m rad. coupling constant 2%. the horizontal bctatmn value 24 m and tbc vertical bctatmn value 9 m, which are tbc parameters designed at tire Super Photon tit@-8 GeV (Wring-l). The parameters of the figure-S undulator in Figs. 10 and 1 I are the same as those in Figs. 6b and 9b. tqwctive!y. Fmm these figures., it is found that both in ttte soft X-ray region and in tk hard X-ray region, the degradation of tbc linear polarization is negligibly small.
11000 PHOTON ENERGYfeV) Fig. IO. On-axis speara and linear polwiwlion, figure-8 undulator cakuiatcd includmg the anguk (high-held caw).
12ooa
13000
PHOTON FNERGY(eV)
5. obtained from Ihe dwzrgcncc of the ban
5. Summary The proposed device, a figure-g undulator, is found to have a good performance as described below. First. the obtainable photon flux density of the fundamental is almost the same as that of a planar undulator. while the on-axis power density is much lower (high-field case). Second. unwanted higher harmonics can be suppressed. Third, the degree of polarization is not so sensitive to the angular divergence of the electron beam. Therefore, we can say that this device has great benefit for the utilization of undulator radiation in third generation light sources.
The authors thank Dr. H. Yamaoka and Dr. T. lshikawa for discussions on crystal monochmmators.
tII 121 131 141
H. Kicemucr Rev. So. lnm 56 (1995) (IO be pubbsbcd). B.M. Kincad, 1. Appl. Pbyn 48 f !9?7) 2684 KJ. Kim, Nucl. IIKU. and M. 219 (1984) 425. B.W. Brvmus and D.H. Bikrback. X-ray bm and Radialion. cd%. G. s. nllnm chap 4 m: Handbodr cm synchmwl Brown aad DE Moncton (Nonh-Holland. Amucrdam. I991 I.
Metiodsin~Physics
Research A 364 (1995) 368-373
Figure-8 undulator as an:insertion device with linear polarization and low on-axis power density T. Tanaka”~“, H. @ tamurab Received 2 March 1995
Abstract A new type undolator for linearly polarized radiation. figure-8 undulator, is proposed. It has the advantage that the on-axis power density is much lower than that of an ordinary planar undulator. Electrons inside the proposed device move along the trajectory which looks like a figure-8 when projected on the plane perpendicular to the beam axis. The spectral performance and the power density are calculated and compared to those of an ordinary planar undulator.
1. Introduction Although planar or helical undulaton have been expected as most powerful sources in third generation synchrotron radiation facilities. heal load problems are very serious plrticularly in case of planar undulators [ I 1. It is well known that the lower photon energy of the fundamental can be obtained by applying higher K value ( = 0.9348 (T&a) A, (cm) ). which causes the increase of unwanted higher harmonics. As a result, the on-axis power density becomes higher and the optical elements may be damaged by unreasonabk heat load. In case of helical undulators. however. no bigher harmo+ its are observed on axis even when very high K values are applied [ 2 1. Therefore. the on-axis power density is much lower compared to that in the previous case, which means that we can expect ideal utilization of undulalor radiation without any unreasona bk heat load. However. it should be noted that only circularly polarized radiation is avaiiabk. The purpose of this paper is to find Out new approaches r0r obtaining linearly polarized radiation with low on-axis power density. We have found two approaches so far. One is a tandem helical utidulator of interfetential type composed 0r two helical undulatoss having differmt helicities. The tinearty polarized radiation can be obtained as a coherent sum of right-hand and left-hand poiahd radiations, the principle of which is similar to that of the crossed undulator 131. Although no higher harmonics ate obsened on axis like an ordinary helical utilator, the degree of the obtainable linear polariulion may be degmdeJ easily by the finite beam emittance. -
. carrcpoadirrgaahol.
in this paper, the other approach is proposed for obtaining high degree of linear polarization. This novel device is called figure-8 undulator since the electron orbit projected on the transverse plane has an eight-figured shape.
2. Principk! As an ideal case. an example of the electron trajectory in a figure-8 undulator is shown in Fig. I. If the trajectory ic projected on the n-v. y-z or z-x plane, we obtain the orbit
----+X
a
2 fig. I.T~of~ck*mraovilgra~~~~~~.bl$c $cri&orau~orrsbdwn.
7: Tunoku. II. A’itamurdNucl.
Insrr. and hfcth. in Phw. Rrs. A 364 f 199.0 36X-373
369
magnetic held is given by the equations B, = -Bat sin
as shown in Figs. 2a-2c. respectively. The electron moves on a right handed and a left handed circle alternately ( Fig. 2a). therefore, the polarization is linear since the componcnl of circular polarization is canceled out However. it is difhcult to realize such a trajectory. because the magnetic field should be discontinuous at the point where L = nA. (n is an integer). which does not saiisfy the Maxwell equations. Then, let us consider another type of ttajectory as shown in fig. 3. The orbit projected on the r-v. v-c or 2-x plane is shown in Figs. Sa-3c and the relation between the relative vchrcities. fl, and j3,. is shown in Fig. 3d. In this case. the
5~ . ( A” >
(I)
(2)
B,
where Bal.d, arc the peak magnetic fields. It is easy to realize the above fields by using an undulator as shown in Fig. 4. A pair of magnet arrays located above and below the axis is an ordinary planar undulator which generates the vertical field (horizontal undulator). while the four outer arrays yrcnerale the horizontal field ( vertical undulator). It should bc noted that the period length of tk vertical undulator is twice longer than that of the horizontal unduiator. Solving the equations 0)’ motion, we obtain the positior and relative velocity as
tb)
K,:
I)
.
-----;s:n2wcIr 8swny
p(rt=(uLm,f. (Y K’+
I-
13)
,
%5~, Y
K’
-”
K’
- 2 w
2 -~COS2f&~ 4r?
4y’ *
>
MS c&d
(41
(6)
(20)
(7)
As shown in the above equations, integer/half-odd-integer harmonics are found to be horizontally/verticaIly polarized. The power density is calculated as
The spectral intensity is calculated as
-=dzP
c
dR dw
dP
Oc - d?P‘
(8)
dl1 do ’
hll2
d2Pk -= dll dw
e2 y2N2 K(fi
whctek=
f.l.f.2
f,(y6.e$,
=4q2s,recosd
,:&y4wN
j
d&($-$).
(21)
-n
(9)
+ rt). . . . . . n/2 ,...
D=l+X2+Y2.
(22)
x=ye,
(23)
and
Y= - KI(S2 +s-?)lPN.
+S-i)lPh.
~~,(ye,~)=~[2StryBsin~-K,(.Si
- K,cos2&.
ye, -
K, cos&,
(24)
Et = Ki sin’ & + 4Kz sin’ 2iJ.
(25)
Ez =4( K,Ysint+
(26)
(IO) (II)
2K,Xsin2f)‘,
(12) where 8, = 6, cos C$and 6,. = @sin r$.
PM =
sin( ~-NW/W)
nN( k - w/w,)
(13)
’
3. Example
47rcy2/ A. WI = II-(K;+Kf)/2+y:d2’
(14)
(IS) ,,’ -
i
co
3.1. High-field case (K > I )
X = 4tK,yt?sin&
( 16)
- 2KvyBcosq5
.
(17)
7 SKi .=-.
( 18)
4
Half-odd-integer harmonics are found to appear in the spectmm. which is derived from the vertical unduiator having the period length, 2A.. On axis. Eqs. ( IO) and ( I I ) are rewritten
-~K,PN
2 [J-+r-~fYl I=-‘*: + J-z,-A.I(Y)IJ,(Z).
JcCO.4) = k = integer. 0.
k = half odd integer.
f 19) k = integer.
0.
-ZK& fdO.4,
=
I J-.z,w2~+,,,.2~Y~ I--+ J-‘~~-,~~-I,;~(Y)IJ,(Z).
(
Let us compare the performance between the planar and figure-8 undulators. For example, the electron energy is set to 8 GeV and the beam cutrent 100 mA.
k = halfodd
integer.
First, high-field case ( K > I ) is considrmd. The period length is assumed to be IO cm and the number of periods 44. The photon flux densities for various harmonics and the power density are shown in Figs. Sa and 56 as a timction of the ratio of KI to Ky. l’be photon flux density for each harmonic is normalized by the value at KS/KY = 0. In the figures. the value, Vm. is set to be constant as 4.72 in order to fix the energy of the fundamental at 500 eV. From these figures. it is fotmd that the power density andrlnnecessary higher harmonics are reduced rapidly as .Ta,‘KI. white the degr&tiin of the fundamental is not so temarkable. Although the optimum value of KI/K, is not decided uniquely, it is sufficient to keep K*/K, more than 0.5 in order to suppms the heat bad. Fig. 6a is the spectrum obtained from the planar undulamr (K, = 0, K, = 4.72). No even-number harmonics an found and the intensities of o&J-number harmonics are much higher than the fundamental. If only the tin&mental is needed, then other higher harmonies give rise to unreasonabkheatloodontheopics.F~.6bis~spectrumolnainad from the figure-8 undulrtor ( K1 = K, = 3.34). ‘he intensity of the cUndamental at 500 eV is about two thitds of that of tbe planar undniator as well as some of half-odd-integer lwmmiis are found in the spectrum. However, the inter+ lies of harmoniis bigber #tan the tenth are negligibfy low. In this’ way, the figure-g trajeetcny efl&iveiy UrpqrrJres ttie higher h4aamnii GfL axis.
(00 rt
60
gw 5 8
t40 20 0
(b)figure-8
x10" -'
<'?
(b)figureS 0; 2 1:
__I; 12x10”
1 A.
Spatial d&i&ions of the power density of the planar and figuure-8 undulaton are shown in Figs. 7a and 7b. respectively. the undulator parameters are the same as those in Figs. 6a and 6b. in case of the planar undulator. tiu ,pocver density has a maximum of 98 kW/mnd? on axis. @I the other hand. the distribution of the figun-8 unduiator is asymmetric with respect to 8, and iuoks like a V figure. since j?, is asymmetric with respect to & as shown in Rg. 3d. Since most of the r&i&d power of the figure-8 tttt&iator distributes on the V figure. much kss heat load falls on axis than in case of the planar undulat0f. TIE on-axis pawa density is as low as 1.4 kW/nxadZ. which is impo?tam benefit for the soft x-ray optics.
i<
3.2. lm-fefd
0 0
‘mo
lww
case {K “r I )
Nex~.tkbw-fieMcasc(K - II iscomidendTbt petiodlengthisassumedtobe3.2cmandthCnanbn@f periods 140.
+++f Y 5th(63
3keV)1
3rd’38
(a)planar
\\
-
0 1x1o’8
OkeV’
+ + -+---+--.+ .-+ - +-..-+._.._ (b) 4x10’8
o i L -1.. 0
RATlO
.-_-A
-4 50000 PHOTON
and the lifth harmonics t 31 K,/ K, = I .OS). Thus, a certain harmonic can be removed sclcctively hy setting K,/K, to the node. Fig. 9a is the spectrum obtained from the planar undulator ( Kx = 0, K, = 1.0). Although the contribution of higher harmonics to the spectrum is less than that in caseof the high ticldcasc. the intensity of the third harmonic is as high as half of that of the tirst harmonic. In genctai, this is removed by detuning the second crystal of a monochromator [ 4 1. In case of the figure-8 undulator. the intensity of the third harmonic can bc controlled by adjusting K, value as shown in Fig. 8a. The spectrum obtained from the figure-8 undulator (K, = K, = 0.71 ) is shown in 6ig. 9b. It is found that the third harmonic is as low as one tenth of that of the fundamental. so that dctuning may bc unnecessary. Although the second harmonic hccomcsrcmarkahlc.this may be easily climinatcd by using an ordinary crystal monochromator (e.g. Si( I I I ) ).
4. Ellects by the angular divergence In the previous section. we have ignored the emittance of the electron beam. In practice. the cmitntncc is finite and atkcts not only the brilliance but the polarzation. In this section. the photon flux density and lhc dcgnx of polarization are calculated for the clearon beam having the linitc angular divergence. If the pmbability distribution km&on of the heam is of Gaussian type. lhc photon flux density and the degree of
linear polarizati~rn.
9.
-ts,,).
(27
s,, - SW s,,+
(28 1
mr s.,.,,
1OOOOO
ENERGY(eV)
arc siven by
d’l’, = C2Y2Ni 4nroc(SII dI1 dw 4. =
(b)ftgure-8
=
I 27ry~U,M-,~
J J u du
I,
xexp
-
(ucosn
Wt.,(u.a)
--I
- yflcosf$)’
2Y’4,
[ (Usina
- yBsin#)’ Zy”o$,
(29) I
where cr >I,,) are the horizontal and vertical angular divergences. The spectra and Pt. obtained fmm the figure-R undulator are shown in Figs. IO (high-field case) and I I ( low-field case). The natural cmittance is assumed to be 7.0 x IO-’ m rad. coupling constant 2%. the horizontal bctatmn value 24 m and tbc vertical bctatmn value 9 m, which are tbc parameters designed at tire Super Photon tit@-8 GeV (Wring-l). The parameters of the figure-S undulator in Figs. 10 and 1 I are the same as those in Figs. 6b and 9b. tqwctive!y. Fmm these figures., it is found that both in ttte soft X-ray region and in tk hard X-ray region, the degradation of tbc linear polarization is negligibly small.
11000 PHOTON ENERGYfeV) Fig. IO. On-axis speara and linear polwiwlion, figure-8 undulator cakuiatcd includmg the anguk (high-held caw).
12ooa
13000
PHOTON FNERGY(eV)
5. obtained from Ihe dwzrgcncc of the ban
5. Summary The proposed device, a figure-g undulator, is found to have a good performance as described below. First. the obtainable photon flux density of the fundamental is almost the same as that of a planar undulator. while the on-axis power density is much lower (high-field case). Second. unwanted higher harmonics can be suppressed. Third, the degree of polarization is not so sensitive to the angular divergence of the electron beam. Therefore, we can say that this device has great benefit for the utilization of undulator radiation in third generation light sources.
The authors thank Dr. H. Yamaoka and Dr. T. lshikawa for discussions on crystal monochmmators.
tII 121 131 141
H. Kicemucr Rev. So. lnm 56 (1995) (IO be pubbsbcd). B.M. Kincad, 1. Appl. Pbyn 48 f !9?7) 2684 KJ. Kim, Nucl. IIKU. and M. 219 (1984) 425. B.W. Brvmus and D.H. Bikrback. X-ray bm and Radialion. cd%. G. s. nllnm chap 4 m: Handbodr cm synchmwl Brown aad DE Moncton (Nonh-Holland. Amucrdam. I991 I.