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An undulator with a composite magnetic field
388
Nuclear Instruments and Methods in Physics Research A234 (1985) 388-393 North-Holland, Amsterdam
AN UNDULATOR WITH A COMPOSITE
MAGNETIC FIELD
M i c h i o N I W A N O * a n d I-Iideo K I T A M U K A
Photon Factory, National Laboratory for High-Energy Physics, Oho-machi, Tsukuba-gun, lbaraki 305, Japan Received 27 April 1984
A new type of undulator greatly improved in tunability as compared with a usual undulator, is proposed. The undulator has a composite magnetic field which is a linear superposition of two sinusoidaUy varying fields with different spatial periods. The wavelength can be changed by varying the field strength of the longer-period component while keeping that of the shorter-period component at the highest possible value. Numerical calculations of the spectral brightness were carried out. The proper ratios between the two spatial periods were found to be 3 : 1, 5 : 1 and 7 : 1.
1. Introduction
An undulator is a periodic magnetic device in which relativistic electrons subjected to periodic acceleration emit quasi-monochromatic synchrotron radiation [1,2]. Theoretical [1-10] and experimental [5,6,9,11-15] investigations of the properties of the undulator radiation have been extensively performed. It has been revealed that the tmdulator radiation has a higher spectral brightness than the normal synchrotron radiation emitted from the bending magnets of storage-rings. Therefore, the undulator has recently attracted considerable attention as an extremely promising new radiation source for investigations in various branches of the natural sciences. An interesting characteristic of the undulator radiation is its tunability. The characteristic wavelength of the k t h harmonic of the quasi-monochromatic radiation is given by xk =
1+
202 + - 7 -
'
k -- 1, 2, 3 . . . . .
(1)
where X u is the spatial period of the undulator field, k the harmonic number, ~, the electron energy in rest mass units (~/= E J m c 2 ) , K the field parameter proportional to the product of X u and the peak value of the magnetic field, and 0 the angle of observation relative to the longitudinal direction, i.e., the average direction of electron motion. The equation shows that the observed spectrum of the radiation can be tuned by varying 3', K or 0. In a practical application, the spectrum is tuned by changing the field strength and * Permanent address: Miyagi University of Education, Sendal 980, Japan. 0168-9002/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
hence the parameter K, because radiation with high brightness is emitted in the forward direction (0 = 0) and the electron energy is usually fixed in a storage ring. For a permanent-magnet undulator, the change in the field strength is easily accomplished by varying the gap height of the magnet. The variable range of wavelength is then determined by the minimum gap height, and correspondingly the maximum obtainable value of K. The above equation also shows that radiation with shorter wavelengths is produced with smaller values of h u. Accordingly, undulators having a short period are necessary to extend the spectrum to shorter wavelengths in a given storage ring. However, these is a minimum gap height of the magnet determined by the beam aperture requirements, and moreover, for a given gap height the peak value of the magnetic field tends rapidly to zero as h u decreases. These facts make it very difficult for a short period undulator to have a large field and hence a large value of K. If the maximum obtainable value of K is small, the variable range of wavelength is considerably reduced. In the present paper, we propose a new type of undulator which makes it possible to shorten the characteristic wavelength, and furthermore to change the wavelength over a wide range. This undulator has a composite magnetic field which is a linear superposition of two sinusoidally varying fields with different spatial periods. The strength of the longer-period fieid component is variable, while that of the shorter-period field component is kept constant. The idea of the proposed undulator is based on the fact that a change in the characteristic wavelength is closely related to a variation in the average longitudinal velocity ~t)ll) Of the passing electron. The characteristic wavelength of the fundamental (k = 1) is represented in
M. Niwano, H. Kitamura / Undulator with composite magnetic field
389
terms of (vii) as [11 X i = X u ( 1 - ( v " )' c o sc0 )
(2)
which is the same as eq. (1) in the case of 0 << 1. It follows from eq. (2) that a change in ( v , ) gives rise to a change in the characteristic wavelength. Fig. 1 shows schematically an example of the composite field, where a sinusoidally varying field with a shorter spatial period XA is superposed on the field with a longer spatial period XB. In such a field an electron moves along a trajectory similar to the form of the composite field, i.e., the linear superposition of two sine curves with spatial periods of, respectively, XA and XB. Then, the electron path length is readily changed by varying only the strength of the longer-period field component. This change results in a change in ( v , ) , because the total velocity of the electron is constant in a magnetic field. On the other hand, the shorter-period field component is left unchanged, producing quasi-monochromatic radiation with shorter wavelength. Thus, the longer-period field component mainly plays the role of tuning the spectrum, while the shorter period one makes the characteristic wavelength short. Because a longer spatial period makes it easy to produce a larger field, we can change the characteristic wavelength over a wide range by varying the strength of the longer-period field component. Rare-earth cobalt (REC) magnetic materials would be especially suitable for the magnets of the undulator of the present type. Because its magnetic permeability is close to unity, the material behaves magnetically very nearly like a vacuum with an impressed current [16]. This property helps to produce a composite magnetic field as is needed for the present undulator. Fig. 2 shows an example of the present undulator using REC materials. Because of the above property, the magnetic field produced by an inside pair of linear arrays of magnet blocks is not disturbed by the field produced by an outside one. The gap height h 2 of the magnets which produce the longer-period field is varied in order to change the wavelength. On the other hand, the gap height hi of the magnets which produce the shorterperiod field is fixed at the smallest possible value. In sect. 2 we describe the motion of a relativistic electron in a composite magnetic field. We obtain a formula for the spectral brightness in sect. 3. In sect. 4
Fig. 2. Conceptual illustration of the present undulator using REC materials. The arrows show the direction of magnetization, we discuss the tunability of the present undulator based on the results of numerical calculations of the spectral brightness. Comparison between the tunability of the present undulator and that of a usual undulator is also presented.
2. Motion of an electron in a composite magnetic field
We consider a composite magnetic field of the form: 2,rz . 2~z Hy = HAsin--~-A + HBSm---~B,
H•.
A A f~"
/Longer-period
A A,.=z
Fig. 1. Field distribution in the present undulator.
(3)
where HA,B are the peak values of the constituent fields and XA,B are the spatial periods. Here, we assume that XA = X 0 / m A ,
XB = X0//mB,
(4)
where mA,s are natural numbers, and mA: mB is an irreducible ratio. Under these conditions, the composite magnetic field is also periodic and its spatial period is X0. The ratio mA :mB determines the type of composite field. The field shown in fig. 1 corresponds to mA :mB =5:1. We assume that a relativistic electron ('t >> 1) moves mainly in the z direction in the composite field and the electron velocity vector lies in the x - z plane. On condition that qJA.a ~ 1, the transverse and the longitudinal components of the electron velocity are in good approximation given by Vx-" --Cfl(~bA COS mAtOot+~b B COS mBt%t } ,
(5)
v z = c f l { 1 --½(~A COS mAto0t+t~B COS mB~00t)2},
(6)
where ~bA
CompositeField
H x = H z = 0,
eHAXA 2~rmc2 y ,
eHBA B • B - 2~.mc2 Y ,
(7)
and o 0 = 2*rc/X 0. It is easily seen from eqs. (5) and (6) that the electron trajectory is nearly the linear superposition of two sinusoidal curves having spatial periods of, respectively, XA and XB. Furthermore, the average longitudinal velocity ( v , ) is given from eq. (6) by
(v,,> = (v.> = ¢~(1 ¢ ~ 4+ ~
)
(8)
390
M. Niwano, H. Kitamura / Undulator with composite magnetic field
Looking at eqs. (7) and (8), we see that, as mentioned in the introduction, Coil) can be changed by changing one of the field strengths HA.B while keeping the other constant.
E e is the electron energy in GeV and I is the electron beam current in mA. As can be seen from eq. (12), the brightness for the k th harmonic attains its maximum for oJk = kto~, that is,
3. Spectral brightness
h k = 2k't 2 [
We first obtain the expression for the electric field. The starting point is, as in ref. [10], the LienardWiechert potential:
which is the expression for the characteristic wavelength for the present undulator. Note that for the present undulator, K 2 in eq. (1) is replaced by K2A + K 2, and accordingly the characteristic wavelength can be changed by changing one of the two field parameters KA.s while keeping the other constant. In the practical use of undulator radiation, the central brightness, that is, the brightness on the undulator axis (0 = 0) is important. Its maximum value is given from eq. (12) by
A = ~
1 -- n. o//c
x0 (1 +
retarded time'
where R is the distance between the source and the observer, and n is a unit vector connecting them. The electron velocity v is given by eqs. (5) and (6). Since the radiation is mainly concentrated within an angle of the order of 1/3,, we only consider the case where the angle between n and the z axis 0 ,~ 1. In case the length of the undulator is L = N ~ 0 , where N is an integer, we obtain from the vector potential given by eq. (9) the k th harmonic component of the electric field polarized parallel or perpendicular to the orbital plane: E k , ( ~ ) = G,~['?OSo cos ¢p + ½(KASA + K B S s ) ] ,
(10)
Ek. ( o~) = Gt;yOSo sin qo,
(11)
where
G=4'"le'tsin[Ncr(t°-kt°l)/~Ol]iexp(ik_~), , = k ( 1 +~,202 q K 2 + K 2 ) - ' 2 601 = 2"t2~t%/k,
KA.a = "tlkA,B,
and ep Is the angle between the plane containing n and the z axis and the x-z plane. Here, KA,a are the field parameters for the present undulator which can be simply expressed as KA,n = 0.0934 HA.n(kG)- hA.B(cm). The functions S0, SA and S a are, as explicitly defined in the appendix, a dimensionless function of KA.B, mA.B, ~'0 and q0. It is clear from eqs. (10) and (11) that both components have the same phase, and accordingly the radiation is linearly polarized. This result is the same as for the usual undulator [10]. The spectral brightness ( p h o t o n s / s . mrad 2 - 1% bandwidth) for the k t h harmonic component is given from eqs. (10) and (11) by .^12~2.- sin2[N*r(t°/t°t - k ) ]
Pk((,O) = 1.74 X Iu l~ellk
,~2 (,o/,~ 1 - / 0
(12)
2
where fk = 4~2{[V 0 cos cpSo + ½(KASA + KBS B)]2 + y20 2sin2~S02},
(13)
y202
rl + r 2) ,
-t" ~
1.74 × lO12E2IrkN 2 ,
(14)
(15)
where we introduced the brightness function Fk ~-fk (~,0 = 0).
(16)
The function Fk is a dimensionless function of K A, K s, m A and m a.
4. Tunability From the standpoint of tnnability, we now examine the dependence of the brightness function F k, defined in the preceding section, on the field parameters K A and K s . Accordingly, in the following discussion, we only consider the case where ,/0 = 0. Assuming that m A > ms, we consider the case where the strength of the longer-period field component H B is varied, and the strength of the shorter-period field component H A is kept constant, the corresponding field parameter K A being less than 1.0. In this case, we are concerned with the math harmonic. When the longerperiod field vanishes, that is, K B = 0, the math harmonic is identical with the fundamental ( k = 1) radiation produced by a usual undulator having a spatial period hA(= ~0//mA), as is clear from eqs. (1) and (14). In a usual undulator, in a range of K < 1 most of the radiated power is concentrated in the fundamental harmonic. Thus, at K a = 0, the math harmonic is a predominant harmonic component and has high brightness. On the other hand, according to eq. (14), an increase of K s causes a change in the characteristic wavelength of each harmonic. However, as K B increases, the brightness for each harmonic also changes in various ways, depending mainly on the types of composite field, i.e., the ratios mA: m s. If the math harmonic loses very much of its brightness with increasing K s , the wavelength region where the brightness is large enough for practical use is reduced. In order to get a practically high tunability, it
M. Niwano, H. Kitamura / Undulator with composite magnetic fieM
101[
is therefore required that for a given ratio of m A : m e, high brightness of the m a t h h a r m o n i c remains w h e n K e increases. T h e brightness for each h a r m o n i c is estimated f r o m the numerical values of the brightness function F k. N u m e r i c a l calculations of the brightness function Fj, were carried out for various ratios of mA: m e a n d for various values of K A a n d K n. W e also considered the composite fields where the plus sign in eq. (3) is replaced b y the m i n u s sign, which c o r r e s p o n d s to K A < 0. A s a result of the calculations, the case m A : m e = 3 : 1, K A < 0 , the case m A : m e = 5 : 1 , K A > 0 a n d the case m A : m e = 7 • 1, K A < 0 have been f o u n d to satisfy the r e q u i r e m e n t m e n t i o n e d above. As a n example, the results of the calculations for m A : m e = 5 : 1 a n d 4 : 1 are s h o w n in fig. 3. I n the figure, the values of F k for the m a t h h a r m o n i c are plotted as a function of K B for IKAI = 0.5. I n the case of m A : m B = 4 : 1, the values of Fk are i n d e p e n d e n t of the sign of K A. As c a n b e seen from fig. 3, in the case of m A: m B = 5 : 1, K A > 0, the values of Fk for the fifth h a r m o n i c do not vary significantly with KB, whereas the characteristic wavelength of this h a r m o n i c changes b y a factor of a b o u t 4 b y increasing K B, for example, from zero to 2.5. O n the other h a n d , in the case of m A : m e = 4 : l a n d the case of m A:m n=5:l, K A < 0 , the values of F , drastically c h a n g e as g B increases. In these two cases, we can n o t o b t a i n a practically high tunability, even if K B can b e c h a n g e d widely. In fig. 4 are s h o w n the d e p e n d e n c e of the function F k for the m a t h h a r m o n i c o n K B for different values of K A in the above three cases. As is s h o w n in fig. 4, with even smaller values of K A, high brightness is o b t a i n e d in a wide range of K B. A value of K A of a b o u t 0.1 is large e n o u g h for practical use of the present undulator. This result is favorable to us, because it becomes more difficult to o b t a i n a large field strength a n d hence a large field p a r a m e t e r as the spatial period decreases. F u r t h e r more, it should b e noticed that in the region K B < 1.5,
I
[
I
,°' t
I
mA:mB
L~ I 0 "I
i
5:1
0
.
:~
u~10"I °" ~A:mB=3:1
,02f 101
,
,
"'",0.5 100 _
b
".. ,
.... •. 0.1
0.3 "'~~:-
L~10"110 -2 ~
B=5:1
II
J
i
i
I
101 'r:''~'~'~ 0, 5 ',, ".
10 0
-
C
"~'.,,.~.,0.3
.... "".0.1
~.~,,
t~ 10 "210 -! .....0'........ .; ) ! } i ~ ' ? ; . " - ' : " •1 10"$ 0
, z, 1
,
Ks
j 2
,
Fig. 4. Dependence of the brightness function F k on K B for different values of K^: (a) m A : m B = 3 : 1, K^ < 0; (b) m A : m 8 = 5 : 1 , K ^ > 0 ; (c) m A : m B = 7 : l , K A < 0 . The figures attached to the curves represent the absolute values of K^.
.
.
.
,
mA:mB=4:1
10-2f
i I tl
10-3
.
.
i \~
i
,,,
t
1°1 ~
.
L=- .....
lOOf ~
/
t
10-2
I
.
391"
J~: i I
i 2 KB (a)
i
10-31 0
,
i 1
I I i 2 KO (b)
i
Fig. 3. Dependence of the brightness function F, on K B for mA : m B = 5 : 1 (a) and mA : m B = 4:1 (b). The solid curves correspond to K A > 0 and the dotted curve to K A < 0.
M. Niwano, H. Kitamura / Undulator with composite magneticfieM
392
the values of F k for K A = 0.0 which correspond to a usual undulator having a longer spatial period h B, tend rapidly to zero as K B decreases. As is seen from eq. (1), an alternative way to get radiation with shorter wavelengths in a given storage ring, is to use higher harmonic components of the radiation emitted from a usual undulator. However, in a usual undulator, we can not obtain high brightness for higher harmonics at values of K less than about 1.5, as shown in fig. 4. This, in practice, reduces the variable range of wavelength for higher harmonics, even if the field parameter K can be changed over a wide range. On the other hand, the present undulator facilitates a significant increase in the brightness at those values of K a corresponding to shorter wavelengths. Thus we conclude that the present undulator extends the variable range of wavelength for higher harmonics to the shorter wavelength region by superposing a weak field with a shorter period on an existing field with a longer period. Finally, we compare the tunability of the present undulator with that of a usual undulator in fig. 5. We considered the case where the ratio m A : m B is 5 : 1, and the maximum obtainable values of K A and K B are 0.2 and 2.0, respectively. In fig. 5, the figures attached to the curves represent the harmonic numbers concerned, and the abscissa represents the photon energy E corresponding to the characteristic wavelength, normalized by E 0 4*ry2hc/hA, which is related to K A and K a by the equation E / E o = [1 + ( K 2 + K 2 ) / 2 ] -1. The solid curve represents the brightness function Fk in which K B is changed from zero to 2.0 with K A = 0.2. The dotted and the dashed curves represent the brightness functions F k in which K B is changed from zero to 2.0 with K A = 0.0, and K A from zero to 0.2 with K B = 0.0, respectively. The case K a = 0.0 and the case K A = 0.0
correspond to usual undulators having a short period h A and a long period h a, respectively. It is clearly seen that the present undulator has an extremely high tunability compared with a usual undulator.
5. Conclusion We have proposed a new type of undulator which extends the variable range of wavelength as well as to shorten the characteristic wavelength. The undulator has a composite magnetic field which is a linear superposition of two periodic fields with different spatial periods. The field with the longer period plays the role of changing the wavelength while that with the shorter period shortens the characteristic wavelength. F r o m the analysis of the spectral brightness, the proper ratios between the spatial periods were found to he 3 : 1, 5 : 1 and 7 : 1 . The authors wish to thank Prof. T. Sasaki for encouraging this work. We would like to thank Dr. S. Sato and Dr. T. Miyahara for helpful discussions and critical reading of the manuscript. Thanks are also due to Mr. A. Mishina for putting the machine programme at our disposal.
=
Appendix
s^=
u(p ,
SB=
• a--
So = • h
.1¢ 101
i
i
i
U ( P A , P a , qA, qB, r + , r
),
(18)
-I-m B
U(PA, PB, qA, qB, r+, r_),
(19)
where a = k +pAmA + p B m s + 2qAm A + 2qBm B
lO 0
+ r + ( m A + roB) + r _ ( m A - mB),
5 f",
0
"",
i "..5 ".,.,
~10-2 ill:
(20)
U(PA, PB, qA, qB, r + , r_)
"',.
i 10-1
'O-o2
07)
a--0
t
m A : m B = 5:1
03
,+, r),
a - - -l-m A
11 i i
0:,
0t6
E/Eo
018
I
XJq^ 4 m A ]
qB~ma
4+
mA+m B
1.0 X
Fig. 5. Comparison between the tunabilities of the present undulator (solid curve) and usual undulators (dotted and dashed curves). The abscissa represents the normalized photon energy corresponding to the characteristic wavelength (see the text). The figures attached to the curves stand for the harmonic number.
( ) KAKB• - Jr_ mA _ mB
,
(21)
and J/ is the ith order Bessel function. In eqs. (17), (18) and (19), the summation R~_sU(p A, PB, qA, qB, r+, r_) means that all the terms U(PA, PB, qA qB, r+, r_) in which a is equal to s, are summed.
M. Niwano, H. Kitamura / Undulator with composite magnetic field
References [1] M. Motz, J. Appl. Phys. 22 (1951) 527. [2] V.L. Ginzburg, Izv. Akad. Nauk SSSR, Set. Fiz. 11 (1947) 165. [3] B.M. Kincaid, J. Appl. Phys. 48 (1977) 2684. [4] A. Hofmann, Nucl. Instr. and Meth. 152 (1978) 17. [5] D.G. Alferov, Yu.A. Bushmakov, K.A. Belovintsev, E.G. Bessonov and P.A. Cherenkov, Part. Accel. 9 (1979) 223. [6] A.N. Didenko, A.V. Kozhevnikov, A.F. Medvedev, M.M. Nikitin and V.Ya. Epp, Sov. Phys. JETP 49 (1979) 973. [7] R. Coisson, Phys. Rev. A20 (1979) 524. [8] S. Krinsky, Nucl. Instr. and Meth. 172 (1980) 73. [9] Y. Farge, Appl. Opt. 19 (1980) 4021. [10] H. Kitamura, Jpn. J. Appl. Phys. 19 (1980) L185; Nucl. Instr. and Meth. 177 (1980) 235.
393
[11] H. Motz, W. Thon and R.N. Whitehurst, J. Appl. Phys. 24 (1953) 826. [12] C. Bazin, M. Billardon, D. Deacon, Y. Farge, J.M. Ortega, J. Perot, Y. Petroff and M. Velghe, J. Phys. Lett. 41 (1980) L547. [13] K. Halbach, J. Chin, E. Hoyer, H. Winick, R. Cronin, J Yang and Y. Zambre, IEEE Trans. Nucl. Sei. NS-28 (1981) 3136. [14] H. Kitamura, S. Tamamushi, T. Yamakawa, S. Sato, Y. Miyahara, G. Isoyama, H. Nishimura, A. Mikuni, S. Asaoka, S. Mitani, H. Maezawa, Y. Suzuki, H. Kanamori and T. Sasald, Jpn. J. Appl. Phys. 21 (1982) 1728. [15] H. Maezawa, S. Mitani, Y. Suzuki, H. Kanamori, S. Tamamushi, A. Mikuni, H. Kitamura and T. Sasaki, Nucl. Instr. and Meth. 208 (1983) 151. [16] H. Winick and J.E. Spencer, Nucl. Instr. and Meth. 172 (1980) 45.
Nuclear Instruments and Methods in Physics Research A234 (1985) 388-393 North-Holland, Amsterdam
AN UNDULATOR WITH A COMPOSITE
MAGNETIC FIELD
M i c h i o N I W A N O * a n d I-Iideo K I T A M U K A
Photon Factory, National Laboratory for High-Energy Physics, Oho-machi, Tsukuba-gun, lbaraki 305, Japan Received 27 April 1984
A new type of undulator greatly improved in tunability as compared with a usual undulator, is proposed. The undulator has a composite magnetic field which is a linear superposition of two sinusoidaUy varying fields with different spatial periods. The wavelength can be changed by varying the field strength of the longer-period component while keeping that of the shorter-period component at the highest possible value. Numerical calculations of the spectral brightness were carried out. The proper ratios between the two spatial periods were found to be 3 : 1, 5 : 1 and 7 : 1.
1. Introduction
An undulator is a periodic magnetic device in which relativistic electrons subjected to periodic acceleration emit quasi-monochromatic synchrotron radiation [1,2]. Theoretical [1-10] and experimental [5,6,9,11-15] investigations of the properties of the undulator radiation have been extensively performed. It has been revealed that the tmdulator radiation has a higher spectral brightness than the normal synchrotron radiation emitted from the bending magnets of storage-rings. Therefore, the undulator has recently attracted considerable attention as an extremely promising new radiation source for investigations in various branches of the natural sciences. An interesting characteristic of the undulator radiation is its tunability. The characteristic wavelength of the k t h harmonic of the quasi-monochromatic radiation is given by xk =
1+
202 + - 7 -
'
k -- 1, 2, 3 . . . . .
(1)
where X u is the spatial period of the undulator field, k the harmonic number, ~, the electron energy in rest mass units (~/= E J m c 2 ) , K the field parameter proportional to the product of X u and the peak value of the magnetic field, and 0 the angle of observation relative to the longitudinal direction, i.e., the average direction of electron motion. The equation shows that the observed spectrum of the radiation can be tuned by varying 3', K or 0. In a practical application, the spectrum is tuned by changing the field strength and * Permanent address: Miyagi University of Education, Sendal 980, Japan. 0168-9002/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
hence the parameter K, because radiation with high brightness is emitted in the forward direction (0 = 0) and the electron energy is usually fixed in a storage ring. For a permanent-magnet undulator, the change in the field strength is easily accomplished by varying the gap height of the magnet. The variable range of wavelength is then determined by the minimum gap height, and correspondingly the maximum obtainable value of K. The above equation also shows that radiation with shorter wavelengths is produced with smaller values of h u. Accordingly, undulators having a short period are necessary to extend the spectrum to shorter wavelengths in a given storage ring. However, these is a minimum gap height of the magnet determined by the beam aperture requirements, and moreover, for a given gap height the peak value of the magnetic field tends rapidly to zero as h u decreases. These facts make it very difficult for a short period undulator to have a large field and hence a large value of K. If the maximum obtainable value of K is small, the variable range of wavelength is considerably reduced. In the present paper, we propose a new type of undulator which makes it possible to shorten the characteristic wavelength, and furthermore to change the wavelength over a wide range. This undulator has a composite magnetic field which is a linear superposition of two sinusoidally varying fields with different spatial periods. The strength of the longer-period fieid component is variable, while that of the shorter-period field component is kept constant. The idea of the proposed undulator is based on the fact that a change in the characteristic wavelength is closely related to a variation in the average longitudinal velocity ~t)ll) Of the passing electron. The characteristic wavelength of the fundamental (k = 1) is represented in
M. Niwano, H. Kitamura / Undulator with composite magnetic field
389
terms of (vii) as [11 X i = X u ( 1 - ( v " )' c o sc0 )
(2)
which is the same as eq. (1) in the case of 0 << 1. It follows from eq. (2) that a change in ( v , ) gives rise to a change in the characteristic wavelength. Fig. 1 shows schematically an example of the composite field, where a sinusoidally varying field with a shorter spatial period XA is superposed on the field with a longer spatial period XB. In such a field an electron moves along a trajectory similar to the form of the composite field, i.e., the linear superposition of two sine curves with spatial periods of, respectively, XA and XB. Then, the electron path length is readily changed by varying only the strength of the longer-period field component. This change results in a change in ( v , ) , because the total velocity of the electron is constant in a magnetic field. On the other hand, the shorter-period field component is left unchanged, producing quasi-monochromatic radiation with shorter wavelength. Thus, the longer-period field component mainly plays the role of tuning the spectrum, while the shorter period one makes the characteristic wavelength short. Because a longer spatial period makes it easy to produce a larger field, we can change the characteristic wavelength over a wide range by varying the strength of the longer-period field component. Rare-earth cobalt (REC) magnetic materials would be especially suitable for the magnets of the undulator of the present type. Because its magnetic permeability is close to unity, the material behaves magnetically very nearly like a vacuum with an impressed current [16]. This property helps to produce a composite magnetic field as is needed for the present undulator. Fig. 2 shows an example of the present undulator using REC materials. Because of the above property, the magnetic field produced by an inside pair of linear arrays of magnet blocks is not disturbed by the field produced by an outside one. The gap height h 2 of the magnets which produce the longer-period field is varied in order to change the wavelength. On the other hand, the gap height hi of the magnets which produce the shorterperiod field is fixed at the smallest possible value. In sect. 2 we describe the motion of a relativistic electron in a composite magnetic field. We obtain a formula for the spectral brightness in sect. 3. In sect. 4
Fig. 2. Conceptual illustration of the present undulator using REC materials. The arrows show the direction of magnetization, we discuss the tunability of the present undulator based on the results of numerical calculations of the spectral brightness. Comparison between the tunability of the present undulator and that of a usual undulator is also presented.
2. Motion of an electron in a composite magnetic field
We consider a composite magnetic field of the form: 2,rz . 2~z Hy = HAsin--~-A + HBSm---~B,
H•.
A A f~"
/Longer-period
A A,.=z
Fig. 1. Field distribution in the present undulator.
(3)
where HA,B are the peak values of the constituent fields and XA,B are the spatial periods. Here, we assume that XA = X 0 / m A ,
XB = X0//mB,
(4)
where mA,s are natural numbers, and mA: mB is an irreducible ratio. Under these conditions, the composite magnetic field is also periodic and its spatial period is X0. The ratio mA :mB determines the type of composite field. The field shown in fig. 1 corresponds to mA :mB =5:1. We assume that a relativistic electron ('t >> 1) moves mainly in the z direction in the composite field and the electron velocity vector lies in the x - z plane. On condition that qJA.a ~ 1, the transverse and the longitudinal components of the electron velocity are in good approximation given by Vx-" --Cfl(~bA COS mAtOot+~b B COS mBt%t } ,
(5)
v z = c f l { 1 --½(~A COS mAto0t+t~B COS mB~00t)2},
(6)
where ~bA
CompositeField
H x = H z = 0,
eHAXA 2~rmc2 y ,
eHBA B • B - 2~.mc2 Y ,
(7)
and o 0 = 2*rc/X 0. It is easily seen from eqs. (5) and (6) that the electron trajectory is nearly the linear superposition of two sinusoidal curves having spatial periods of, respectively, XA and XB. Furthermore, the average longitudinal velocity ( v , ) is given from eq. (6) by
(v,,> = (v.> = ¢~(1 ¢ ~ 4+ ~
)
(8)
390
M. Niwano, H. Kitamura / Undulator with composite magnetic field
Looking at eqs. (7) and (8), we see that, as mentioned in the introduction, Coil) can be changed by changing one of the field strengths HA.B while keeping the other constant.
E e is the electron energy in GeV and I is the electron beam current in mA. As can be seen from eq. (12), the brightness for the k th harmonic attains its maximum for oJk = kto~, that is,
3. Spectral brightness
h k = 2k't 2 [
We first obtain the expression for the electric field. The starting point is, as in ref. [10], the LienardWiechert potential:
which is the expression for the characteristic wavelength for the present undulator. Note that for the present undulator, K 2 in eq. (1) is replaced by K2A + K 2, and accordingly the characteristic wavelength can be changed by changing one of the two field parameters KA.s while keeping the other constant. In the practical use of undulator radiation, the central brightness, that is, the brightness on the undulator axis (0 = 0) is important. Its maximum value is given from eq. (12) by
A = ~
1 -- n. o//c
x0 (1 +
retarded time'
where R is the distance between the source and the observer, and n is a unit vector connecting them. The electron velocity v is given by eqs. (5) and (6). Since the radiation is mainly concentrated within an angle of the order of 1/3,, we only consider the case where the angle between n and the z axis 0 ,~ 1. In case the length of the undulator is L = N ~ 0 , where N is an integer, we obtain from the vector potential given by eq. (9) the k th harmonic component of the electric field polarized parallel or perpendicular to the orbital plane: E k , ( ~ ) = G,~['?OSo cos ¢p + ½(KASA + K B S s ) ] ,
(10)
Ek. ( o~) = Gt;yOSo sin qo,
(11)
where
G=4'"le'tsin[Ncr(t°-kt°l)/~Ol]iexp(ik_~), , = k ( 1 +~,202 q K 2 + K 2 ) - ' 2 601 = 2"t2~t%/k,
KA.a = "tlkA,B,
and ep Is the angle between the plane containing n and the z axis and the x-z plane. Here, KA,a are the field parameters for the present undulator which can be simply expressed as KA,n = 0.0934 HA.n(kG)- hA.B(cm). The functions S0, SA and S a are, as explicitly defined in the appendix, a dimensionless function of KA.B, mA.B, ~'0 and q0. It is clear from eqs. (10) and (11) that both components have the same phase, and accordingly the radiation is linearly polarized. This result is the same as for the usual undulator [10]. The spectral brightness ( p h o t o n s / s . mrad 2 - 1% bandwidth) for the k t h harmonic component is given from eqs. (10) and (11) by .^12~2.- sin2[N*r(t°/t°t - k ) ]
Pk((,O) = 1.74 X Iu l~ellk
,~2 (,o/,~ 1 - / 0
(12)
2
where fk = 4~2{[V 0 cos cpSo + ½(KASA + KBS B)]2 + y20 2sin2~S02},
(13)
y202
rl + r 2) ,
-t" ~
1.74 × lO12E2IrkN 2 ,
(14)
(15)
where we introduced the brightness function Fk ~-fk (~,0 = 0).
(16)
The function Fk is a dimensionless function of K A, K s, m A and m a.
4. Tunability From the standpoint of tnnability, we now examine the dependence of the brightness function F k, defined in the preceding section, on the field parameters K A and K s . Accordingly, in the following discussion, we only consider the case where ,/0 = 0. Assuming that m A > ms, we consider the case where the strength of the longer-period field component H B is varied, and the strength of the shorter-period field component H A is kept constant, the corresponding field parameter K A being less than 1.0. In this case, we are concerned with the math harmonic. When the longerperiod field vanishes, that is, K B = 0, the math harmonic is identical with the fundamental ( k = 1) radiation produced by a usual undulator having a spatial period hA(= ~0//mA), as is clear from eqs. (1) and (14). In a usual undulator, in a range of K < 1 most of the radiated power is concentrated in the fundamental harmonic. Thus, at K a = 0, the math harmonic is a predominant harmonic component and has high brightness. On the other hand, according to eq. (14), an increase of K s causes a change in the characteristic wavelength of each harmonic. However, as K B increases, the brightness for each harmonic also changes in various ways, depending mainly on the types of composite field, i.e., the ratios mA: m s. If the math harmonic loses very much of its brightness with increasing K s , the wavelength region where the brightness is large enough for practical use is reduced. In order to get a practically high tunability, it
M. Niwano, H. Kitamura / Undulator with composite magnetic fieM
101[
is therefore required that for a given ratio of m A : m e, high brightness of the m a t h h a r m o n i c remains w h e n K e increases. T h e brightness for each h a r m o n i c is estimated f r o m the numerical values of the brightness function F k. N u m e r i c a l calculations of the brightness function Fj, were carried out for various ratios of mA: m e a n d for various values of K A a n d K n. W e also considered the composite fields where the plus sign in eq. (3) is replaced b y the m i n u s sign, which c o r r e s p o n d s to K A < 0. A s a result of the calculations, the case m A : m e = 3 : 1, K A < 0 , the case m A : m e = 5 : 1 , K A > 0 a n d the case m A : m e = 7 • 1, K A < 0 have been f o u n d to satisfy the r e q u i r e m e n t m e n t i o n e d above. As a n example, the results of the calculations for m A : m e = 5 : 1 a n d 4 : 1 are s h o w n in fig. 3. I n the figure, the values of F k for the m a t h h a r m o n i c are plotted as a function of K B for IKAI = 0.5. I n the case of m A : m B = 4 : 1, the values of Fk are i n d e p e n d e n t of the sign of K A. As c a n b e seen from fig. 3, in the case of m A: m B = 5 : 1, K A > 0, the values of Fk for the fifth h a r m o n i c do not vary significantly with KB, whereas the characteristic wavelength of this h a r m o n i c changes b y a factor of a b o u t 4 b y increasing K B, for example, from zero to 2.5. O n the other h a n d , in the case of m A : m e = 4 : l a n d the case of m A:m n=5:l, K A < 0 , the values of F , drastically c h a n g e as g B increases. In these two cases, we can n o t o b t a i n a practically high tunability, even if K B can b e c h a n g e d widely. In fig. 4 are s h o w n the d e p e n d e n c e of the function F k for the m a t h h a r m o n i c o n K B for different values of K A in the above three cases. As is s h o w n in fig. 4, with even smaller values of K A, high brightness is o b t a i n e d in a wide range of K B. A value of K A of a b o u t 0.1 is large e n o u g h for practical use of the present undulator. This result is favorable to us, because it becomes more difficult to o b t a i n a large field strength a n d hence a large field p a r a m e t e r as the spatial period decreases. F u r t h e r more, it should b e noticed that in the region K B < 1.5,
I
[
I
,°' t
I
mA:mB
L~ I 0 "I
i
5:1
0
.
:~
u~10"I °" ~A:mB=3:1
,02f 101
,
,
"'",0.5 100 _
b
".. ,
.... •. 0.1
0.3 "'~~:-
L~10"110 -2 ~
B=5:1
II
J
i
i
I
101 'r:''~'~'~ 0, 5 ',, ".
10 0
-
C
"~'.,,.~.,0.3
.... "".0.1
~.~,,
t~ 10 "210 -! .....0'........ .; ) ! } i ~ ' ? ; . " - ' : " •1 10"$ 0
, z, 1
,
Ks
j 2
,
Fig. 4. Dependence of the brightness function F k on K B for different values of K^: (a) m A : m B = 3 : 1, K^ < 0; (b) m A : m 8 = 5 : 1 , K ^ > 0 ; (c) m A : m B = 7 : l , K A < 0 . The figures attached to the curves represent the absolute values of K^.
.
.
.
,
mA:mB=4:1
10-2f
i I tl
10-3
.
.
i \~
i
,,,
t
1°1 ~
.
L=- .....
lOOf ~
/
t
10-2
I
.
391"
J~: i I
i 2 KB (a)
i
10-31 0
,
i 1
I I i 2 KO (b)
i
Fig. 3. Dependence of the brightness function F, on K B for mA : m B = 5 : 1 (a) and mA : m B = 4:1 (b). The solid curves correspond to K A > 0 and the dotted curve to K A < 0.
M. Niwano, H. Kitamura / Undulator with composite magneticfieM
392
the values of F k for K A = 0.0 which correspond to a usual undulator having a longer spatial period h B, tend rapidly to zero as K B decreases. As is seen from eq. (1), an alternative way to get radiation with shorter wavelengths in a given storage ring, is to use higher harmonic components of the radiation emitted from a usual undulator. However, in a usual undulator, we can not obtain high brightness for higher harmonics at values of K less than about 1.5, as shown in fig. 4. This, in practice, reduces the variable range of wavelength for higher harmonics, even if the field parameter K can be changed over a wide range. On the other hand, the present undulator facilitates a significant increase in the brightness at those values of K a corresponding to shorter wavelengths. Thus we conclude that the present undulator extends the variable range of wavelength for higher harmonics to the shorter wavelength region by superposing a weak field with a shorter period on an existing field with a longer period. Finally, we compare the tunability of the present undulator with that of a usual undulator in fig. 5. We considered the case where the ratio m A : m B is 5 : 1, and the maximum obtainable values of K A and K B are 0.2 and 2.0, respectively. In fig. 5, the figures attached to the curves represent the harmonic numbers concerned, and the abscissa represents the photon energy E corresponding to the characteristic wavelength, normalized by E 0 4*ry2hc/hA, which is related to K A and K a by the equation E / E o = [1 + ( K 2 + K 2 ) / 2 ] -1. The solid curve represents the brightness function Fk in which K B is changed from zero to 2.0 with K A = 0.2. The dotted and the dashed curves represent the brightness functions F k in which K B is changed from zero to 2.0 with K A = 0.0, and K A from zero to 0.2 with K B = 0.0, respectively. The case K a = 0.0 and the case K A = 0.0
correspond to usual undulators having a short period h A and a long period h a, respectively. It is clearly seen that the present undulator has an extremely high tunability compared with a usual undulator.
5. Conclusion We have proposed a new type of undulator which extends the variable range of wavelength as well as to shorten the characteristic wavelength. The undulator has a composite magnetic field which is a linear superposition of two periodic fields with different spatial periods. The field with the longer period plays the role of changing the wavelength while that with the shorter period shortens the characteristic wavelength. F r o m the analysis of the spectral brightness, the proper ratios between the spatial periods were found to he 3 : 1, 5 : 1 and 7 : 1 . The authors wish to thank Prof. T. Sasaki for encouraging this work. We would like to thank Dr. S. Sato and Dr. T. Miyahara for helpful discussions and critical reading of the manuscript. Thanks are also due to Mr. A. Mishina for putting the machine programme at our disposal.
=
Appendix
s^=
u(p ,
SB=
• a--
So = • h
.1¢ 101
i
i
i
U ( P A , P a , qA, qB, r + , r
),
(18)
-I-m B
U(PA, PB, qA, qB, r+, r_),
(19)
where a = k +pAmA + p B m s + 2qAm A + 2qBm B
lO 0
+ r + ( m A + roB) + r _ ( m A - mB),
5 f",
0
"",
i "..5 ".,.,
~10-2 ill:
(20)
U(PA, PB, qA, qB, r + , r_)
"',.
i 10-1
'O-o2
07)
a--0
t
m A : m B = 5:1
03
,+, r),
a - - -l-m A
11 i i
0:,
0t6
E/Eo
018
I
XJq^ 4 m A ]
qB~ma
4+
mA+m B
1.0 X
Fig. 5. Comparison between the tunabilities of the present undulator (solid curve) and usual undulators (dotted and dashed curves). The abscissa represents the normalized photon energy corresponding to the characteristic wavelength (see the text). The figures attached to the curves stand for the harmonic number.
( ) KAKB• - Jr_ mA _ mB
,
(21)
and J/ is the ith order Bessel function. In eqs. (17), (18) and (19), the summation R~_sU(p A, PB, qA, qB, r+, r_) means that all the terms U(PA, PB, qA qB, r+, r_) in which a is equal to s, are summed.
M. Niwano, H. Kitamura / Undulator with composite magnetic field
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393
[11] H. Motz, W. Thon and R.N. Whitehurst, J. Appl. Phys. 24 (1953) 826. [12] C. Bazin, M. Billardon, D. Deacon, Y. Farge, J.M. Ortega, J. Perot, Y. Petroff and M. Velghe, J. Phys. Lett. 41 (1980) L547. [13] K. Halbach, J. Chin, E. Hoyer, H. Winick, R. Cronin, J Yang and Y. Zambre, IEEE Trans. Nucl. Sei. NS-28 (1981) 3136. [14] H. Kitamura, S. Tamamushi, T. Yamakawa, S. Sato, Y. Miyahara, G. Isoyama, H. Nishimura, A. Mikuni, S. Asaoka, S. Mitani, H. Maezawa, Y. Suzuki, H. Kanamori and T. Sasald, Jpn. J. Appl. Phys. 21 (1982) 1728. [15] H. Maezawa, S. Mitani, Y. Suzuki, H. Kanamori, S. Tamamushi, A. Mikuni, H. Kitamura and T. Sasaki, Nucl. Instr. and Meth. 208 (1983) 151. [16] H. Winick and J.E. Spencer, Nucl. Instr. and Meth. 172 (1980) 45.