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On the spectral and polarization characteristics of an undulator
Nuclear Instruments and Methods 177 (1980) 235-238 © North-Holland Publishing Company
ON THE SPECTRAL AND POLARIZATION CHARACTERISTICS OF AN UNDULATOR Hideo KITAMURA Synchrotron Radiation Laboratory, Institute for Solid State Physics, Universityof Tokyo, TanashL 188, Japan
The spectral and polarization characteristics of undulator radiation are investigated. The radiation was found to be linearly polarized for each harmonic and the plane of polarization dependent on the direction of observation. The causes of the broadening of the spectrum of the undulator radiation are also discussed.
high enough (7 >> 1) and that the K value, which is an important parameter with which to characterize undulator radiation, is in a range o f K ~ 1. The undulator radiation is linearly polarized. However, the plane of polarization is not necessarily in the orbital plane since it is dependent on the direction o f observation [6]. Fig. 2 shows the dependence of the direction of polarization on 70 and ~ for the first harmonic with K = 1. In the figure, each circle shows a contour of 3'0, and each bar indicates the plane of polarization. The polarization is found to be almost parallel to the orbital plane for values of 3'0 smaller than 0.25, while for larger values the plane of polar-
1. Introduction An undulator or a flat wiggler is an improved synchrotron radiation device producing highly collimated quasi-monochromatic light [ 1 - 4 ] . An electron in the periodic magnetic field of an undulator is accelerated in the transverse direction as well as in the longitudinal direction [5]. The transverse acceleration produces the component of radiation parallel to the orbital plane while the longitudinal acceleration, the frequency of which is twice the transverse component, is related to the component of radiation perpendicular to the orbital plane. Therefore, undulator radiation has interesting features with respect to spectral and polarization characteristics [6]. In the present paper, the results of the calculation of undulator radiation are reported briefly.
K= 1 n = 1
2. Polarization characteristics of an undulator
90 ° ~ - ~ , ~
6O °
\ 300
Fig. 1 illustrates coordinates relevant to the observation of undulator radiation [6]. The direction of observation is defined by the azimuth ~ and the angle of deviation 0 from the axis of an undulator. We assume that the relativistic energy of an electron is
Y
Fig. 2. Dependence of the direction of linear polarization on the direction of observation for the fkst harmonic with K = 1 [6]. Each circle represents a contour of ~'0 dNided by the azimuth ¢~. The direction of each bar inserted represents the plane of polarization.
Fig. 1. Coordinates relevant to observation of undulator radiation [6]. 235
VI. WIGGLERS, UNDULATORS
H. Kitamura / Characteristicsofan undulator
236
ization rotates along the contour. The power radiated from a single period of an undulator in a bandwidth dco and a solid angle d~2 is given by P(nl)dco df2 4C2'~2~
-
fn
c
. sin2(Nnco/co z) Jn~_~-)2
=(HSI']"
2S2
\270 cos¢
dco d~2,
(1)
tively. I f N is infinite, the power radiated in d~2 is /o(2) d~2
16~2e274 ~2 -
ku
7rn
fn d a .
(S)
The C-dependence of (~2/Trn)fn for the first harmonic with K = 1 is shown for each value of 3'0 up to 2.5 in fig. 3, in the lower part of which the dependence of the angle between the plane of polarization and the orbital plane on ~bis also shown.
70~S, cos ~ ) ~
+ ~7202S] sin2~b , dfZ = 0 dO d~b,
(2)
3. Spectral distribution
(3)
The power radiated in an angular aperture dO is obtained by integrating eq. (1) with ~,
where = n(1 + 7202 + K2/2) -x ,
(4)
cox = 47rc72(1 + 7202 + K2/2)-I/Xu,
(5)
P~dO dco-
81 = E
(6)
where
8e272~Tr F
c
sin2(Nrrco/col)
nN~2~/~l-~)zO
dO dco,
(9) p=_oo
Jn+2p(270K~ cos c~)Jp(K2~/4),
$2 = ~
p= _oo
2n
1
oo
pJn+zp(ZTOK~ cos q~)Jp(K2~/4),
with N and n being the number of periods of the undulator and the order of a higher harmonic, respec-
Ye=.l
(lO)
(7)
"¢.~.25
Ye = .63
o When the radiation is detected on-axis using an angular aperture A0, we obtain the spectral distribu-
Ye = 1
¢e=1.5
Ye=2.5
I 0-3
1 0 -3 _
_
I0 -t,
J
/
90 m
S°'o ' 45
~
o " t~s
"
'
o ois
9o
f
/ /,5
90 0
/,5
90
(DEGREES)
Fig. 3. Dependence of the intensity (f;2/Trn)fn of the first harmonic on 0 for each q,0 value [6]. In the lower part of the figure, the dependence of the direction of polarization is also shown.
H. Kitamura/ Characteristicsofan undulator
237
tion 10 ~
AO
n=l
o I f N is infinite,
[P, ]N=oo= 4e 27rFn/c "
(12)
In this case, the frequency co is related to 0 as co = 4nc,/2n(1 + 7202
+K2/2)-I/Xu.
10 ~'
(13)
The power given in eq. (12) is transformed into the number of photons obtained in 1% bandwidth at a beam current of i mA during a second using numerical values,
N n = F n X 5.72
10 t
X 1012
OI
02
.05
(photons/s mA 1% bandwidth/undulator period) (14) Fig. 4 shows the spectrum of Zn=l 4 Nn with K = 1 obtained within an angular aperture of A0 = 57 -1. The abscissa represents the wavelength normalized by X = X./(2",/").
The higher harmonics are more conspicuous for large K values [5]. Therefore, the K value should be lowered if we require undulator radiation containing small higher-order components. Fig. 5 shows the
,, [
K= 1
0f -~s[ 10'
1
~
~
i
]
2 10'°
.o
i
st I! [,"",i
\~\ " 2[[! ~,~,, '~\ o ' [ l ! i !",, "\
o:[2[ I l l 0. l o ~ ~
.2
5
1
2
5
I0
K
Fig. 5. K-dependence of the intensities of the first and third harmonics obtained on-axis.
K-dependence of the intensities of the first and third harmonics obtained on-axis. As shown in fig. 5, the intensity of each harmonic reaches a plateau for a large K fimit, while in a range of K "~ 1 the intensity of the first or third harmonic is proportional to K 2 or K 6 , respectively.
4. Bandwidth
~
tI:'}",,i'\i..\"\\\
.1
"',,,"', ),,
.5 1 2 s lo io go 1 -2 X/x. (X.=~Xwr) Fig. 4. Spectrum of undulator radiation obtained with an angular aperture A0 = 53,-1 (K = 1, Z4n=l Nn). The abscissa represents the wavelength normalized by Xo = ku/(23,2).
= The spectrum of undulator radiation is broadened by the effects, finite number of periods, angular aperture of detection," angular divergence in the beam, beam size and energy spread in the beam, but the last two effects are not s~ serious [3]. If the number of periods N is infinite ahd the angular divergence in the beam, Ox, and O'y, is very small, the bandwidth is determined only by an angular aperture of detection. When the observation is made on-axis with an angular aperture A0,the bandwidth becomes
72(0)
"--~-/&o - 1 + K2/2
•
(15)
The above relation shows that a very narrow bandwidth would be obtained by a small angular aperture. Now, we consider the case N: finite number. The bandwidth determined by N is Aco I G9 IN
...L.
(16)
nN VI. WIGGLERS, UNDULATORS
238
H. Kitamura / Characteristics o f an undulator L5
Kn=:11
N~=o ~/.-'~
N = 100
I
~~':~~1.o l_2.541111~i! "7,
94
96
98
1
W/We:
Fig. 6. Spectra of undulator radiation obtained within various angular apertures of detection (N = 100, n = 1 and K = 1). (A0)op is defined as in eq. (17). The abscissa represents the frequency normalized by ~o0=0 = 4rrc~,2n(1 + K2/2) -1/Xu.
We define (AO)o p from the equation (Aco/CO)a0 = (ACO/W)N as
( 0)op:
(1, +n K /q! ''2 " N
(17)
When we detect the radiation with an angular aperture A0 = (A0)op, we obtain a bandwidth nearly
equal to that determined by eq. (16). For a larger angular aperture, the bandwidth becomes larger according to the relation (15), while for a smaller angular aperture it is not so improved and the resulting intensity decreases. Fig. 6 shows the change of the spectral profde due to the finite angular aperture A0. Here, we choose the parameters to be K = 1, N = 100 and n = 1. Now, we consider the finite angular divergence in the beam. When the angular divergence is large compared with (A0)op, the spectrum has a tail in a lower frequency region, and the resulting bandwidth becomes larger. We conclude that the angular divergence should be smaller than ( A 0 ) o p if we require a bandwidth as narrow as ~l/nN. This criterion means that the beam emittance in an electron storage ring dedicated to undulators should be designed to be as small as possible and that they should be located at the high43 straight section of the ring [3].
References
[1] H. Motz, J. Appl. Phys. 22 (1951) 527. [2] H. Motz, W. Thon and R.N. Whitehurst, J. Appl. Phys. 24 (1953) 826. [3] A. Hofmann, Nucl. Instr. and Meth. 152 (1978) 17. [4] R. Coisson, Wiggler magnets, SSRP Report 77/05 (May, 1977). [5] D.F. Alferov, Yu. A. Bashmakov and E.G. Bessonov, Sov. Phys. Tech. Phys. 18 (1974) 1336. [6] H. Kitamura, Jpn. J. Appl. Phys. Lett. to be submitted.
ON THE SPECTRAL AND POLARIZATION CHARACTERISTICS OF AN UNDULATOR Hideo KITAMURA Synchrotron Radiation Laboratory, Institute for Solid State Physics, Universityof Tokyo, TanashL 188, Japan
The spectral and polarization characteristics of undulator radiation are investigated. The radiation was found to be linearly polarized for each harmonic and the plane of polarization dependent on the direction of observation. The causes of the broadening of the spectrum of the undulator radiation are also discussed.
high enough (7 >> 1) and that the K value, which is an important parameter with which to characterize undulator radiation, is in a range o f K ~ 1. The undulator radiation is linearly polarized. However, the plane of polarization is not necessarily in the orbital plane since it is dependent on the direction o f observation [6]. Fig. 2 shows the dependence of the direction of polarization on 70 and ~ for the first harmonic with K = 1. In the figure, each circle shows a contour of 3'0, and each bar indicates the plane of polarization. The polarization is found to be almost parallel to the orbital plane for values of 3'0 smaller than 0.25, while for larger values the plane of polar-
1. Introduction An undulator or a flat wiggler is an improved synchrotron radiation device producing highly collimated quasi-monochromatic light [ 1 - 4 ] . An electron in the periodic magnetic field of an undulator is accelerated in the transverse direction as well as in the longitudinal direction [5]. The transverse acceleration produces the component of radiation parallel to the orbital plane while the longitudinal acceleration, the frequency of which is twice the transverse component, is related to the component of radiation perpendicular to the orbital plane. Therefore, undulator radiation has interesting features with respect to spectral and polarization characteristics [6]. In the present paper, the results of the calculation of undulator radiation are reported briefly.
K= 1 n = 1
2. Polarization characteristics of an undulator
90 ° ~ - ~ , ~
6O °
\ 300
Fig. 1 illustrates coordinates relevant to the observation of undulator radiation [6]. The direction of observation is defined by the azimuth ~ and the angle of deviation 0 from the axis of an undulator. We assume that the relativistic energy of an electron is
Y
Fig. 2. Dependence of the direction of linear polarization on the direction of observation for the fkst harmonic with K = 1 [6]. Each circle represents a contour of ~'0 dNided by the azimuth ¢~. The direction of each bar inserted represents the plane of polarization.
Fig. 1. Coordinates relevant to observation of undulator radiation [6]. 235
VI. WIGGLERS, UNDULATORS
H. Kitamura / Characteristicsofan undulator
236
ization rotates along the contour. The power radiated from a single period of an undulator in a bandwidth dco and a solid angle d~2 is given by P(nl)dco df2 4C2'~2~
-
fn
c
. sin2(Nnco/co z) Jn~_~-)2
=(HSI']"
2S2
\270 cos¢
dco d~2,
(1)
tively. I f N is infinite, the power radiated in d~2 is /o(2) d~2
16~2e274 ~2 -
ku
7rn
fn d a .
(S)
The C-dependence of (~2/Trn)fn for the first harmonic with K = 1 is shown for each value of 3'0 up to 2.5 in fig. 3, in the lower part of which the dependence of the angle between the plane of polarization and the orbital plane on ~bis also shown.
70~S, cos ~ ) ~
+ ~7202S] sin2~b , dfZ = 0 dO d~b,
(2)
3. Spectral distribution
(3)
The power radiated in an angular aperture dO is obtained by integrating eq. (1) with ~,
where = n(1 + 7202 + K2/2) -x ,
(4)
cox = 47rc72(1 + 7202 + K2/2)-I/Xu,
(5)
P~dO dco-
81 = E
(6)
where
8e272~Tr F
c
sin2(Nrrco/col)
nN~2~/~l-~)zO
dO dco,
(9) p=_oo
Jn+2p(270K~ cos c~)Jp(K2~/4),
$2 = ~
p= _oo
2n
1
oo
pJn+zp(ZTOK~ cos q~)Jp(K2~/4),
with N and n being the number of periods of the undulator and the order of a higher harmonic, respec-
Ye=.l
(lO)
(7)
"¢.~.25
Ye = .63
o When the radiation is detected on-axis using an angular aperture A0, we obtain the spectral distribu-
Ye = 1
¢e=1.5
Ye=2.5
I 0-3
1 0 -3 _
_
I0 -t,
J
/
90 m
S°'o ' 45
~
o " t~s
"
'
o ois
9o
f
/ /,5
90 0
/,5
90
(DEGREES)
Fig. 3. Dependence of the intensity (f;2/Trn)fn of the first harmonic on 0 for each q,0 value [6]. In the lower part of the figure, the dependence of the direction of polarization is also shown.
H. Kitamura/ Characteristicsofan undulator
237
tion 10 ~
AO
n=l
o I f N is infinite,
[P, ]N=oo= 4e 27rFn/c "
(12)
In this case, the frequency co is related to 0 as co = 4nc,/2n(1 + 7202
+K2/2)-I/Xu.
10 ~'
(13)
The power given in eq. (12) is transformed into the number of photons obtained in 1% bandwidth at a beam current of i mA during a second using numerical values,
N n = F n X 5.72
10 t
X 1012
OI
02
.05
(photons/s mA 1% bandwidth/undulator period) (14) Fig. 4 shows the spectrum of Zn=l 4 Nn with K = 1 obtained within an angular aperture of A0 = 57 -1. The abscissa represents the wavelength normalized by X = X./(2",/").
The higher harmonics are more conspicuous for large K values [5]. Therefore, the K value should be lowered if we require undulator radiation containing small higher-order components. Fig. 5 shows the
,, [
K= 1
0f -~s[ 10'
1
~
~
i
]
2 10'°
.o
i
st I! [,"",i
\~\ " 2[[! ~,~,, '~\ o ' [ l ! i !",, "\
o:[2[ I l l 0. l o ~ ~
.2
5
1
2
5
I0
K
Fig. 5. K-dependence of the intensities of the first and third harmonics obtained on-axis.
K-dependence of the intensities of the first and third harmonics obtained on-axis. As shown in fig. 5, the intensity of each harmonic reaches a plateau for a large K fimit, while in a range of K "~ 1 the intensity of the first or third harmonic is proportional to K 2 or K 6 , respectively.
4. Bandwidth
~
tI:'}",,i'\i..\"\\\
.1
"',,,"', ),,
.5 1 2 s lo io go 1 -2 X/x. (X.=~Xwr) Fig. 4. Spectrum of undulator radiation obtained with an angular aperture A0 = 53,-1 (K = 1, Z4n=l Nn). The abscissa represents the wavelength normalized by Xo = ku/(23,2).
= The spectrum of undulator radiation is broadened by the effects, finite number of periods, angular aperture of detection," angular divergence in the beam, beam size and energy spread in the beam, but the last two effects are not s~ serious [3]. If the number of periods N is infinite ahd the angular divergence in the beam, Ox, and O'y, is very small, the bandwidth is determined only by an angular aperture of detection. When the observation is made on-axis with an angular aperture A0,the bandwidth becomes
72(0)
"--~-/&o - 1 + K2/2
•
(15)
The above relation shows that a very narrow bandwidth would be obtained by a small angular aperture. Now, we consider the case N: finite number. The bandwidth determined by N is Aco I G9 IN
...L.
(16)
nN VI. WIGGLERS, UNDULATORS
238
H. Kitamura / Characteristics o f an undulator L5
Kn=:11
N~=o ~/.-'~
N = 100
I
~~':~~1.o l_2.541111~i! "7,
94
96
98
1
W/We:
Fig. 6. Spectra of undulator radiation obtained within various angular apertures of detection (N = 100, n = 1 and K = 1). (A0)op is defined as in eq. (17). The abscissa represents the frequency normalized by ~o0=0 = 4rrc~,2n(1 + K2/2) -1/Xu.
We define (AO)o p from the equation (Aco/CO)a0 = (ACO/W)N as
( 0)op:
(1, +n K /q! ''2 " N
(17)
When we detect the radiation with an angular aperture A0 = (A0)op, we obtain a bandwidth nearly
equal to that determined by eq. (16). For a larger angular aperture, the bandwidth becomes larger according to the relation (15), while for a smaller angular aperture it is not so improved and the resulting intensity decreases. Fig. 6 shows the change of the spectral profde due to the finite angular aperture A0. Here, we choose the parameters to be K = 1, N = 100 and n = 1. Now, we consider the finite angular divergence in the beam. When the angular divergence is large compared with (A0)op, the spectrum has a tail in a lower frequency region, and the resulting bandwidth becomes larger. We conclude that the angular divergence should be smaller than ( A 0 ) o p if we require a bandwidth as narrow as ~l/nN. This criterion means that the beam emittance in an electron storage ring dedicated to undulators should be designed to be as small as possible and that they should be located at the high43 straight section of the ring [3].
References
[1] H. Motz, J. Appl. Phys. 22 (1951) 527. [2] H. Motz, W. Thon and R.N. Whitehurst, J. Appl. Phys. 24 (1953) 826. [3] A. Hofmann, Nucl. Instr. and Meth. 152 (1978) 17. [4] R. Coisson, Wiggler magnets, SSRP Report 77/05 (May, 1977). [5] D.F. Alferov, Yu. A. Bashmakov and E.G. Bessonov, Sov. Phys. Tech. Phys. 18 (1974) 1336. [6] H. Kitamura, Jpn. J. Appl. Phys. Lett. to be submitted.