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Production of linear polarization by segmentation of helical undulator
Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591
Production of linear polarization by segmentation of helical undulator T. Tanaka*, H. Kitamura The Institute of Physical and Chemical Research (RIKEN), Koto 1-1-1, Mikazuki, Sayo, Hyogo 679-5148, Japan Received 23 April 2002; accepted 16 May 2002
Abstract A simple scheme to obtain linearly polarized radiation (LPR) with a segmented undulator is proposed. The undulator is composed of several segments each of which forms a helical undulator and has helicity opposite to those of adjacent segments. Due to coherent sum of radiation, the circularly polarized component is canceled out resulting in production of LPR without any higher harmonics. The radiation from the proposed device is investigated analytically, which shows that a high degree of linear polarization is obtained in spite of a finite beam emittance and angular acceptance of optics, if a sufficiently large number of segments and an adequate photon energy are chosen. Results of calculation to investigate practical performances of the proposed device are presented. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.60.Ap Keywords: Helical undulators; Linear polarization
1. Introduction It is well known that radiation from a helical undulator consists only of the fundamental radiation and contains no higher harmonics when observed on-axis [1]. This is a great advantage in two points of view. One is that the heat load brought by the higher harmonics is negligible and the other is that the radiation after the monochromator is never contaminated by them. The former makes the cooling system of optical elements in the synchrotron radiation (SR) beam*Corresponding author. Tel.: +81-791-582809; fax: +81791-582810. E-mail address: [email protected] (T. Tanaka).
line quite simple, while the latter is important for the SR users to analyze their experimental data accurately. It should be noted, however, that only circularly polarized radiation (CPR) is available if the helical undulator is adopted. If the SR users need linearly polarized radiation (LPR) and not CPR, a conventional linear undulator should be adopted, which in turn causes the two problems brought by the higher harmonics. In order to solve the heat-load problem, the figure-8 undulator [2] can be adopted. The electron moves along a trajectory which looks like a figure of eight when projected onto the transverse plane, which produces LPR and reduces considerably the on-axis power without sacrificing significantly the flux of the fundamental radiation.
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On the other hand, the contamination by the higher harmonic can be solved by a detuning technique of a double-crystal monochromator [3] usually used in the X-ray beam-line. Because the Darwin width of the second crystal is narrower for higher harmonics, tilting it slightly from the Bragg angle (detuning) results in reduction of higher harmonic intensity. Another solution to this problem is to adopt a quasi-periodic undulator (QPU) [4]. The spectrum of QPU consists of harmonics with energies which are not integer multiples of the fundamental energy, enabling us to obtain monochromatized radiation with much less higher harmonics than that obtained with the conventional undulator. We have recently, proposed a simple scheme to suppress harmonic intensity of undulator radiation (UR) [5]. The undulator is divided into several segments and the relative phase in between is adjusted (phase detuning) to shift the fundamental peak to higher energy without affecting significantly the peak locations of other higher harmonics. Extracting the photon beam at the shifted fundamental energy, higher-harmonic intensity after the monochromator is reduced considerably. Because the CPR can be regarded as superposition of two polarization states, i.e., the horizontal and vertical polarizations, we can obtain LPR if the peak energies corresponding to these components are shifted independently by the phase detuning. This is the basic idea of the scheme to be proposed in this article.
helical undulator with the number of periods N; as shown in Fig. 1. Let M be an even number. The two consecutive segments have the opposite helicity and the length of drift sections between the nth and ðn þ 1Þth segments, Ln ; is expressed as ( L1 þ L2 ; n ¼ odd Ln ¼ L1 L2 ; n ¼ even: Let the magnetic field be B ¼ Bx ex þ Bx ex ; where ex;y are the unit vectors in the horizontal and vertical directions. Introducing a complex variable B* ¼ Bx þ iBy ; we have the magnetic field in the nth segment as follows: ( eiku ðzðn1ÞL1 Þ ; n ¼ even B* ¼ B0 ik ðzðn1ÞL L Þ u 1 2 e ; n ¼ odd where 2p=ku and B0 are the periodic length and peak field of the undulator. Let us introduce a new coordinate system represented by two unit vectors ea;b defined as ! ! ! ex cos y sin y ea ¼ : sin y cos y ey eb Clearly, the new ða–bÞ coordinate is obtained by revolving the original (x–y) one by an angle of y around the z-axis, or the undulator axis. The magnetic field expressed in the a–b coordinate, * iy B* 0 ¼ Ba þ iBb ; is found by the relation B* 0 ¼ Be to be ( eiku ðzðn1ÞL1 yÞ ; n ¼ even 0 B* ¼ B0 ik ðzðn1ÞL L þyÞ 1 2 e u ; n ¼ odd:
2. Principle
Assuming y ¼ ku L2 =2; we have n B* 0 ¼ B0 eð1Þ iku ðzðn1ÞL1 L2 =2Þ :
Let us consider an insertion device (ID) composed of M segments each of which forms a
This equation means that Ba and Bb have the relative phase ku L1 and ðku L1 þ pÞ between
Fig. 1. Schematic illustration of an ID composed of helical undulators with opposite helicity. CW or CCW denotes the helical undulator segment in which the electrons move along a clockwise or counterclockwise helical trajectory, respectively. The sinusoidal solid line shows the vertical magnetic field, while the dotted line the horizontal field.
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591
adjacent segments, respectively. As shown in Ref. [5], the peak intensity of UR obtained from the segmented undulator having mismatched relative phase is shifted to the energy determined by the phase. It is therefore possible to separate the two polarization components created by the two magnetic fields, Ba and Bb ; by adjusting L1 in order to obtain LPR at a specific energy. In addition, the angle of the polarization plane is determined by the value of L2 : In order to calculate the flux obtained from this device, the Fourier transform E o of the electric field EðtÞ should be calculated. Let T1 and T2 be the time interval corresponding to the length L1 and L2 ; respectively. Taking into account the similarity of the electric field in each segment, we have E ox;oy ¼ E 1ox;1oy þ E 2ox;2oy with E 1ox;1oy ¼
M=21 X
Z
m¼0
2mðNT þT1 ÞþNT
2mðNT þT1 Þ
E x;y ðt 2mT1 Þeiot E 2ox;2oy ¼ 7
M=21 X
Z
m¼0
ð2mþ1ÞðNTþT1 ÞþT2
where o1 is the fundamental energy of UR. After mathematical manipulation, we have E kx;ky
k¼1
1 eið2pNþf1 ÞMo=o1 eið2pNðko=o1 Þ1Þ 1 e2ið2pNþf1 Þo=o1 iðko1 oÞ
E 2ox;2oy ¼ eid E 1ox;1oy with d ¼ ðf1 þ f2 þ 2pNÞo=o1 f1;2 ¼ o1 T1;2 :
Therefore, we finally have E ox ¼ E 1 ð1 þ eid Þ
E oy ¼
1 eið2pNþf1 ÞMo=o1 eið2pNðko=o1 Þ1Þ 1 e2ið2pNþf1 Þo=o1 iðko1 oÞ
sin d E ox : 1 þ cos d
ð1Þ
The above equations show that E ox;oy have the same phase, i.e., LPR is obtained. The angle of the polarization plane W is found to be sin d 1 W ¼ tan : 1 þ cos d The energy spectrum is calculated as 2pMN 2 2 2 jE ox j þ jE oy j ¼ 4 SN SM o1 with
2p T¼ o1
N X
In the case of the helical undulator, E kx and E ky satisfy the relation ( E1; k ¼ 1 E kx ¼ iE ky ¼ 0; kX2:
ð2mþ1ÞðNTþT1 ÞþT2 þNT
E x;y ðt ð2m þ 1ÞT1 T2 Þeiot
E 1ox;1oy ¼
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SN ¼
sin2 ½pNð1 o=o1 Þ p2 N 2 ð1 o=o1 Þ2
SM ¼
sin2 ½ðM=2Þð2pN þ f1 Þo=o1 : M 2 sin2 ½ð2pN þ f1 Þo=o1
ð2Þ
As shown in the above equation, the energy spectrum is dominated by the two functions SN and SM : It is easily understood that SN is the normal spectral profile function of UR, while SM denotes the effect of interference between segments. Neither SN nor SM is dependent on f2 ; meaning that it determines only the polarization properties but not the flux. On the other hand, variation of f1 causes a considerable change in the spectral profile as shown in Fig. 2(a) and (b) where the product SM SN is plotted as a function of Nðo=o1 1Þ ¼ NDo=o1 for two different values of f1 : In each figure, the spectrum is composed of several peaks. To know the polarization state at each peak, the angle of the polarization plane W is plotted for two different values of f2 : The
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From now on, f2 is assumed to be zero because the polarization angle is not important for the following discussions. Substituting d ¼ ðf1 þ 2pNÞo=o1 into Eq. (1), we have 2pMN 2 2 Ix;y jE ox;oy j ¼ 4 o1 with Ix;y ¼ SN SMx;My SMx ¼
sin2 ðMd=2Þ M 2 sin2 ðd=2Þ
SMy ¼ SMx tan2 ðd=2Þ:
Fig. 2. Energy spectrum for two values of (a) f ¼ 2p and (b) f ¼ 1:5p: In each figure, the angle of the polarization plane is plotted as a function of the photon energy for two values of f2 :
discontinuity is due to limitation of the angle between 01 and 1801: In each case of f1 and f2 ; the principal peaks have different polarization states. This means that the polarization performance will be spoilt by practical performances of the electron beam, undulator itself, and beamline components, the effects of which will be investigated in the next section.
3. Effects due to angular divergence and acceptance The possible sources to spoil the polarization performance of the proposed device are the energy spread and angular divergence of the electron beam, undulator magnetic field error, and angular acceptance of beamline optics. All of them cause peak broadening in the spectrum. It should be noted, however, that the effects caused by the energy spread and magnetic field error are usually much smaller than those by others. Therefore, we focus only on the effects caused by the angular divergence and acceptance.
ð3Þ
The function Ix represents the angular flux density of the horizontally polarized radiation (HPR), while Iy that of the vertically polarized radiation (VPR). Let us consider the peak energy of HPR. It is easily found that the peak width of SMx is about M times narrower than that of SN : Therefore, Ix has the local maxima at the energies determined by the resonance condition of SMx ; i.e., sinðd=2Þ ¼ 0: This condition can be simplified to f1 f1 Do þ pN þ1 ¼ np o1 2 2pN where n is an integer. There are infinite numbers of Do satisfying this condition. Among them, we should choose Do so that jDoj becomes the minimum in order to obtain a value of SN as large as possible. Assuming f1 =2pN51; we have the peak energy of HPR as follows: c o1 Do ¼ 1 ¼ Dop ð4Þ 2pN with c1 ¼ f1 2pl f þp l¼ 1 2p where ½x denotes the integer part of x: As is easily shown, the phase c1 ranges from p to p: The angle of p is in general equivalent to p; however, we should separate these two angles because c1 is connected to the peak energy shift as in Eq. (4). Let us investigate the angular profile of HPR and VPR at the fixed energy of op ¼ o1 þ Dop :
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591
Substituting Do ¼ Dop into Eqs. (2) and (3), we have sin O1 ðyÞ sin MO2 ðyÞ 2 Ix ðyÞ ¼ O1 ðyÞ M sin O2 ðyÞ Iy ðyÞ ¼ Ix ðyÞtan2 O2 ðyÞ;
ð5Þ
with
f1 l f1 þ pNY2 1 1 N 2 2pN f f O2 ðyÞ ¼ 1 þ 1 O1 ðyÞ 1 2pN 2
O1 ðyÞ ¼
Y2 ¼
g 2 y2 1 þ K2
where y is the angle of observation with respect to the forward direction, K the deflection parameter of the undulator and g the Lorentz factor of the electron beam. In order to estimate roughly the dependence of the flux density on y and f1 ; we neglect the terms f1 =2pN and l=N to obtain 2 sinðf1 =2 pNY2 Þ sin pMNY2 Ix ðyÞ ¼ ðf1 =2 pNY2 Þ M sin pNY2 Iy ðyÞ ¼ Ix ðyÞtan2 pNY2 :
ð6Þ
Substituting y ¼ 0; we have the on-axis flux density sin f1 =2 2 : ð7Þ Ix ð0Þ ¼ f1 =2 This is a decreasing function of jf1 j as long as jf1 jop: Thus jf1 j should be smaller to obtain higher flux. As for the p angular ffiffiffiffiffiffiffiffiffi profile, it is easy to find that when Y ¼ 1= MN ; i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ K 2 1 y¼ g ¼ ycoh ; MN Ix ðyÞ vanishes. In fact ycoh is equal to the angle at which the flux density vanishes for UR emitted from a normal undulator with the number of periods of MN: Now let us consider the effects of the angular divergence and acceptance. We assume the angular distribution function of the electron beam to be a one-dimensional Gaussian with a standard
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deviation of s0 ; i.e., 2 1 02 f ðyÞ ¼ pffiffiffiffiffiffi ey =2s 2ps0 and the optical slit to have a pinhole with an angular acceptance of Dy: In this case, we obtain the flux Px;y passing through the pinhole as follows: " ! Z 1 N Dy y Px;y ¼ Ix;y ðyÞ erf pffiffiffi 2 0 2 s0 !# Dy þ y þ erf pffiffiffi y dy 2s0 " ! Z 1 þ K2 N DY Y Ix;y ðYÞ erf pffiffiffi ¼ 2 0 2S0 !# DY þ Y þ erf pffiffiffi Y dY ð8Þ 2S0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where DY ¼ Dy= 1 þ K 2 ; S0 ¼ s0 = 1 þ K 2 ; and erfðxÞ is an error function defined as Z x 2 2 erfðxÞ ¼ pffiffiffi ex dx: p 0 The degree of linear polarization PL0 is calculated as Px Py PL0 ¼ : Px þ Py To compute the integrand in Eq. (8), the Eq. (5) but not the approximated one (6) should be used in order to obtain more precise value of PL0 : Let us consider three cases of S0 ¼ 0; 0:05g1 ; and 0:1g1 : We take 2s0p for an adequate angular acceptance of the pinhole ðDyÞ to obtain sufficient flux, where s0p is the standard deviation of the photon beam angular divergence calculated approximately as [6] 2 02 s02 p ¼ ðycoh =2Þ þ s :
Fig. 3(a)–(c) show the dependences of PL0 on c1 for the three values of S0 and various combinations of M and N with the product MN being fixed at 240. For reference, the peak energy shift Dop =o1 calculated using Eq. (4) is plotted as a function of c1 : Applying a larger number of segments improves the degree of polarization significantly. When S0 > 0; the phase c1 close to
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placed in tandem and the phase in between is adjusted to obtain CPR. Our device for M ¼ 2 is obtained by replacing the vertical and horizontal undulators with two helical undulators having opposite helicity. It is well known that the degree of polarization of UR emitted from this type of undulator is degraded easily by the finite angular divergence of the electron beam. In other words, a high degree of polarization is expected if the angular divergence is small enough. However, it is found from Fig. 3(a) that PL0 of only 0.5 is obtained when M ¼ 2; c ¼ 0 and S0 ¼ 0: This is due to the angular acceptance Dy ¼ 2s0p : We can expect a higher degree of polarization by applying a smaller angular acceptance, which in turn causes degradation of flux. Instead, we should apply a larger number of segments to obtain a higher degree of polarization even if the angular divergence of the electron beam is extremely small. Fig. 4(a) and (b) show the dependences of PL0 on M for the three values of S0 with the product MN being fixed at 60 and 240, respectively. The number of periods per segment, N; calculated as Fig. 3. The degree of polarization PL0 as a function of the phase c1 for the three values of the angular divergence of the electron beam under three combinations of M and N: (a) M ¼ 2; N ¼ 120; (b) M ¼ 6; N ¼ 40; and (c) M ¼ 16; N ¼ 15: In each figure, the peak energy shift Dop =o1 is plotted as a function of c1 :
p should be applied as well as the large number of segments to obtain a high degree of polarization. This means that the peak of HPR should be shifted to higher energies ðDop > 0Þ to avoid contamination by tailing of the VPR peak to lower energies. When S0 ¼ 0; PL0 is almost equal to 1.0 regardless of c1 for the two cases of M ¼ 6 and 15. On the other hand, applying M of 2 degrades the degree of polarization significantly. It should be noted that M ¼ 2 and c1 ¼ 0 means that two helical undulators with opposite helicity are placed in tandem and photons at the exact resonance energy is extracted. There is an ID having a configuration similar to this, which is known as a crossed undulator [7] to control the polarization state of UR. Two undulators to provide HPR and VPR (horizontal and vertical undulators) are
Fig. 4. The degree of polarization PL0 as a function of the number of segment M for the three values of the angular divergence of the electron beam with the product MN being fixed at (a) 60 and (b) 240. In each figure, the number of periods per segment is plotted as a function of M:
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591 Table 1 Maximum number of periods per segment to achieve PL0 of 0.95 for two values of MN: The phase c1 is fixed at p S0 ðg1 Þ
0 0.05 0.1
MN 60
240
15 10 7
60 30 24
N ¼ ½240=M; is plotted for reference. As S0 increases, the minimum number of segments necessary to achieve a certain degree of polarization becomes large. This means that smaller number of periods per segment should be adopted to obtain a higher degree of polarization. For example, the maximum number of periods per segment to obtain PL0 of 0.95 (dotted line) under various conditions are summarized in Table 1. It is found that smaller number of periods per segment is required for MN ¼ 60 than for MN ¼ 240; meaning that more drift sections are necessary per unit length, which degrades the ratio of the practical length of the device to that of the whole straight section, resulting in lower flux. Therefore, this device is well suited for installation in a long straight section (LSS), such as the 30-m section in the Super Photon ring-8 GeV (SPring-8) [8].
4. Example In this section, examples of calculation to estimate practical performances of the proposed device are presented. Let us consider an ID with the proposed scheme to be installed in the 30-m straight section in the SPring-8 and to have the periodic length of 10 cm and the total number of periods MN of 240. The accelerator parameters in the center of the straight section are summarized in Table 2. In order to calculate the magnetic field of the undulator, we should consider the design of the magnetic structure. We have adopted the design of the helical undulator developed at the SPring-8 [9] and calculated the magnetic field to be used in
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Table 2 Accelerator parameters used in the calculation Electron energy Average current Natural emittance Energy spread Coupling constant Horizontal betatron value Vertical betatron value
8 GeV 100 mA 5:9 nm rad 0.001 0.3% 24 m 10 m
estimation of the performances of UR, however, we skip the detailed specification of the device such as the dimension, magnet material, and support structure because they are not important in this article. All calculations are performed by the SR calculation code, SPECTRA, developed at the SPring-8 [10]. Fig. 5(a) and (b) show the energy spectra of the flux and degree of linear polarization for different numbers of segments. The K value is assumed to be 2.25 to obtain the fundamental energy at 1000 eV: The optical slit is located 60 m far from the center of the device and has a rectangular shape with dimensions of 4 mm 1 mm in horizontal and vertical directions, which are determined so that the angular acceptance is equal to 4s0p in each direction. The phase f1 is set at p by moving half-length magnets placed at the extremity of each segment to an adequate position along the longitudinal direction. This is a simpler scheme to adjust the relative phase with less drift sections than creating a chicane orbit or moving the undulator segment, the details of which are not mentioned in this article. As shown in the figure, the peak is divided into three parts, the center of which is mainly composed of VPR, while the other two are composed of HPR. The higher energy peak corresponds to the phase c1 of p; while the lower one to the phase c1 of p; which has a lower degree of polarization due to contamination by the tailing of the central peak. Fig. 6(a) and (b) show the energy spectra of the flux and degree of linear polarization for different values of c1 with M and N being fixed at 16 and 15, respectively. The peak energy op corresponding to each c1 is indicated by an arrow with the value of c1 =p: For clarity, the flux and PL0 at op in
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Fig. 5. Examples of spectra and degree of polarization calculated with different numbers of segments indicated by arrows.
Fig. 6. Same as Fig. 5 but with different values of c1 indicated by arrows and M being fixed at 16. The photon energy range is expanded for clarity.
Figs. 5 and 6 are summarized in Table 3. The flux is independent of m; while PL0 is improved by increasing m; although it almost saturates at mX16: As c1 becomes larger, the flux increases, while PL0 is degraded. This means that we have to compromise when determining the value of c1 : Table 4 shows the flux and PL0 at op for different values of o1 with M and c1 being fixed at 16 and p: The dimension of the slit is varied so that it becomes equal to 2s0p in each direction. The flux reaches the maximum at o1 ¼ 600 eV; while PL0 increases as o1 is lowered. This is caused by reduction of S0 upon higher K value application to lower the photon energy.
5. Discussions So far, we have considered the scheme to produce LPR. It is worth noting that the proposed device can produce CPR and switch the helicity, too. The simplest way is to change the helicity of the odd- or even-number segments to make identical CPR, which is usually realized by a phasing technique, or moving several magnet arrays of the helical undulator along the longitudinal direction. Another scheme is to divide all the segments into the first and second halves having opposite helicity and shift the fundamental energy of either of the two halves to o0 ¼ o1 þ Do: Let the first
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591 Table 3 Flux and PL0 at the energy of op under various conditions of M and c1 M
c1
Flux (1014 photons/s/0.1%BW)
PL0
8 12 16 20 16 16
p p p p 0:9p 0:8p
7.69 7.74 7.88 7.94 9.08 10.0
0.939 0.957 0.961 0.961 0.949 0.938
Table 4 Flux and PL0 at the energy of op for different values of o1 o1 (eV)
Flux (1014 photons/s/0.1%BW)
PL0
1000 800 600 400 200
7.88 9.39 9.44 9.86 7.49
0.961 0.963 0.968 0.974 0.974
half produce right-handed CPR (RCPR) and the second left-handed (LCPR). If RCPR is desired, the fundamental energy of the first half is shifted to o0 ; while that of the second is fixed at o1 : In this case, RCPR with a high degree of polarization is obtained at o0 if Do is large enough to separate the RCPR and LCPR. In order to shift the photon energy, the K value, i.e., the gap of the undulator is changed. In the point of view of available flux, the phasing method is better because the practical number of periods is twice as that of the other, while the time necessary to switch the helicity for the energy-shift method is much shorter than that for the phasing method. The gap change necessary to separate the LCPR and RCPR is just several millimeters. There is another method to shift the peak energy much faster than the gap movement. Because each half is composed of plural segments, we can use the phase detuning [5]. By installing electromag-
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nets to make chicane orbit between segments and changing its path length, the phase can be varied. In this case, much faster repetition rate than the other two would be achieved. In this article, helical undulators are used to produce LPR. It is also possible to use horizontal and vertical undulators to generate CPR and switch the helicity by adjusting the phase between segments. This is the same scheme as the crossed undulator, however, the proposed device can achieve a much higher degree of polarization than the crossed undulator. It should be noted that adopting horizontal and vertical undulators instead of helical undulators brings the two problems already described in Introduction. Especially, the heat-load problem is quite serious under high K value application. As for the contamination by the higher harmonic, we can solve the problem by applying a negative value of c1 to shift the fundamental peak to higher energies. In this case, the same effect as the phase detuning is obtained, resulting in effective suppression of the higher-harmonic intensity.
References [1] B.M. Kincaid, J. Appl. Phys. 48 (1977) 268. [2] T. Tanaka, H. Kitamura, Nucl. Instr. and Meth. A 364 (1995) 368. [3] B.W. Batterman, D.H. Bilderback, X-ray monochromators and mirrors, in: G.S. Brown, D.E. Moncton (Eds.), Handbook on Synchrotron Radiation, North-Holland, Amsterdam, 1991 (Chapter 4). [4] H. Hashimoto, S. Sasaki, JAERI-M Report 94-055, 1994. [5] T. Tanaka, H. Kitamura, J. Synchrotron Rad., submitted for publication. [6] K.J. Kim, Proc. 1986 US Particle Accelerator Summer School, 1986. [7] K.J. Kim, Nucl. Instr. and Meth. 219 (1984) 425. [8] H. Kitamura, T. Bizen, T. Hara, X. Mar!echal, T. Seike, T. Tanaka, Nucl. Instr. and Meth. A 467–468 (2001) 110. [9] T. Hara, T. Tanaka, T. Tanabe, X.M. Mar!echal, K. Kumagai, H. Kitamura, J. Synchrotron Rad. 5 (1998) 426. [10] T. Tanaka, H. Kitamura, J. Synchrotron Rad. 8 (2001) 1221.
Production of linear polarization by segmentation of helical undulator T. Tanaka*, H. Kitamura The Institute of Physical and Chemical Research (RIKEN), Koto 1-1-1, Mikazuki, Sayo, Hyogo 679-5148, Japan Received 23 April 2002; accepted 16 May 2002
Abstract A simple scheme to obtain linearly polarized radiation (LPR) with a segmented undulator is proposed. The undulator is composed of several segments each of which forms a helical undulator and has helicity opposite to those of adjacent segments. Due to coherent sum of radiation, the circularly polarized component is canceled out resulting in production of LPR without any higher harmonics. The radiation from the proposed device is investigated analytically, which shows that a high degree of linear polarization is obtained in spite of a finite beam emittance and angular acceptance of optics, if a sufficiently large number of segments and an adequate photon energy are chosen. Results of calculation to investigate practical performances of the proposed device are presented. r 2002 Elsevier Science B.V. All rights reserved. PACS: 41.60.Ap Keywords: Helical undulators; Linear polarization
1. Introduction It is well known that radiation from a helical undulator consists only of the fundamental radiation and contains no higher harmonics when observed on-axis [1]. This is a great advantage in two points of view. One is that the heat load brought by the higher harmonics is negligible and the other is that the radiation after the monochromator is never contaminated by them. The former makes the cooling system of optical elements in the synchrotron radiation (SR) beam*Corresponding author. Tel.: +81-791-582809; fax: +81791-582810. E-mail address: [email protected] (T. Tanaka).
line quite simple, while the latter is important for the SR users to analyze their experimental data accurately. It should be noted, however, that only circularly polarized radiation (CPR) is available if the helical undulator is adopted. If the SR users need linearly polarized radiation (LPR) and not CPR, a conventional linear undulator should be adopted, which in turn causes the two problems brought by the higher harmonics. In order to solve the heat-load problem, the figure-8 undulator [2] can be adopted. The electron moves along a trajectory which looks like a figure of eight when projected onto the transverse plane, which produces LPR and reduces considerably the on-axis power without sacrificing significantly the flux of the fundamental radiation.
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On the other hand, the contamination by the higher harmonic can be solved by a detuning technique of a double-crystal monochromator [3] usually used in the X-ray beam-line. Because the Darwin width of the second crystal is narrower for higher harmonics, tilting it slightly from the Bragg angle (detuning) results in reduction of higher harmonic intensity. Another solution to this problem is to adopt a quasi-periodic undulator (QPU) [4]. The spectrum of QPU consists of harmonics with energies which are not integer multiples of the fundamental energy, enabling us to obtain monochromatized radiation with much less higher harmonics than that obtained with the conventional undulator. We have recently, proposed a simple scheme to suppress harmonic intensity of undulator radiation (UR) [5]. The undulator is divided into several segments and the relative phase in between is adjusted (phase detuning) to shift the fundamental peak to higher energy without affecting significantly the peak locations of other higher harmonics. Extracting the photon beam at the shifted fundamental energy, higher-harmonic intensity after the monochromator is reduced considerably. Because the CPR can be regarded as superposition of two polarization states, i.e., the horizontal and vertical polarizations, we can obtain LPR if the peak energies corresponding to these components are shifted independently by the phase detuning. This is the basic idea of the scheme to be proposed in this article.
helical undulator with the number of periods N; as shown in Fig. 1. Let M be an even number. The two consecutive segments have the opposite helicity and the length of drift sections between the nth and ðn þ 1Þth segments, Ln ; is expressed as ( L1 þ L2 ; n ¼ odd Ln ¼ L1 L2 ; n ¼ even: Let the magnetic field be B ¼ Bx ex þ Bx ex ; where ex;y are the unit vectors in the horizontal and vertical directions. Introducing a complex variable B* ¼ Bx þ iBy ; we have the magnetic field in the nth segment as follows: ( eiku ðzðn1ÞL1 Þ ; n ¼ even B* ¼ B0 ik ðzðn1ÞL L Þ u 1 2 e ; n ¼ odd where 2p=ku and B0 are the periodic length and peak field of the undulator. Let us introduce a new coordinate system represented by two unit vectors ea;b defined as ! ! ! ex cos y sin y ea ¼ : sin y cos y ey eb Clearly, the new ða–bÞ coordinate is obtained by revolving the original (x–y) one by an angle of y around the z-axis, or the undulator axis. The magnetic field expressed in the a–b coordinate, * iy B* 0 ¼ Ba þ iBb ; is found by the relation B* 0 ¼ Be to be ( eiku ðzðn1ÞL1 yÞ ; n ¼ even 0 B* ¼ B0 ik ðzðn1ÞL L þyÞ 1 2 e u ; n ¼ odd:
2. Principle
Assuming y ¼ ku L2 =2; we have n B* 0 ¼ B0 eð1Þ iku ðzðn1ÞL1 L2 =2Þ :
Let us consider an insertion device (ID) composed of M segments each of which forms a
This equation means that Ba and Bb have the relative phase ku L1 and ðku L1 þ pÞ between
Fig. 1. Schematic illustration of an ID composed of helical undulators with opposite helicity. CW or CCW denotes the helical undulator segment in which the electrons move along a clockwise or counterclockwise helical trajectory, respectively. The sinusoidal solid line shows the vertical magnetic field, while the dotted line the horizontal field.
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591
adjacent segments, respectively. As shown in Ref. [5], the peak intensity of UR obtained from the segmented undulator having mismatched relative phase is shifted to the energy determined by the phase. It is therefore possible to separate the two polarization components created by the two magnetic fields, Ba and Bb ; by adjusting L1 in order to obtain LPR at a specific energy. In addition, the angle of the polarization plane is determined by the value of L2 : In order to calculate the flux obtained from this device, the Fourier transform E o of the electric field EðtÞ should be calculated. Let T1 and T2 be the time interval corresponding to the length L1 and L2 ; respectively. Taking into account the similarity of the electric field in each segment, we have E ox;oy ¼ E 1ox;1oy þ E 2ox;2oy with E 1ox;1oy ¼
M=21 X
Z
m¼0
2mðNT þT1 ÞþNT
2mðNT þT1 Þ
E x;y ðt 2mT1 Þeiot E 2ox;2oy ¼ 7
M=21 X
Z
m¼0
ð2mþ1ÞðNTþT1 ÞþT2
where o1 is the fundamental energy of UR. After mathematical manipulation, we have E kx;ky
k¼1
1 eið2pNþf1 ÞMo=o1 eið2pNðko=o1 Þ1Þ 1 e2ið2pNþf1 Þo=o1 iðko1 oÞ
E 2ox;2oy ¼ eid E 1ox;1oy with d ¼ ðf1 þ f2 þ 2pNÞo=o1 f1;2 ¼ o1 T1;2 :
Therefore, we finally have E ox ¼ E 1 ð1 þ eid Þ
E oy ¼
1 eið2pNþf1 ÞMo=o1 eið2pNðko=o1 Þ1Þ 1 e2ið2pNþf1 Þo=o1 iðko1 oÞ
sin d E ox : 1 þ cos d
ð1Þ
The above equations show that E ox;oy have the same phase, i.e., LPR is obtained. The angle of the polarization plane W is found to be sin d 1 W ¼ tan : 1 þ cos d The energy spectrum is calculated as 2pMN 2 2 2 jE ox j þ jE oy j ¼ 4 SN SM o1 with
2p T¼ o1
N X
In the case of the helical undulator, E kx and E ky satisfy the relation ( E1; k ¼ 1 E kx ¼ iE ky ¼ 0; kX2:
ð2mþ1ÞðNTþT1 ÞþT2 þNT
E x;y ðt ð2m þ 1ÞT1 T2 Þeiot
E 1ox;1oy ¼
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SN ¼
sin2 ½pNð1 o=o1 Þ p2 N 2 ð1 o=o1 Þ2
SM ¼
sin2 ½ðM=2Þð2pN þ f1 Þo=o1 : M 2 sin2 ½ð2pN þ f1 Þo=o1
ð2Þ
As shown in the above equation, the energy spectrum is dominated by the two functions SN and SM : It is easily understood that SN is the normal spectral profile function of UR, while SM denotes the effect of interference between segments. Neither SN nor SM is dependent on f2 ; meaning that it determines only the polarization properties but not the flux. On the other hand, variation of f1 causes a considerable change in the spectral profile as shown in Fig. 2(a) and (b) where the product SM SN is plotted as a function of Nðo=o1 1Þ ¼ NDo=o1 for two different values of f1 : In each figure, the spectrum is composed of several peaks. To know the polarization state at each peak, the angle of the polarization plane W is plotted for two different values of f2 : The
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From now on, f2 is assumed to be zero because the polarization angle is not important for the following discussions. Substituting d ¼ ðf1 þ 2pNÞo=o1 into Eq. (1), we have 2pMN 2 2 Ix;y jE ox;oy j ¼ 4 o1 with Ix;y ¼ SN SMx;My SMx ¼
sin2 ðMd=2Þ M 2 sin2 ðd=2Þ
SMy ¼ SMx tan2 ðd=2Þ:
Fig. 2. Energy spectrum for two values of (a) f ¼ 2p and (b) f ¼ 1:5p: In each figure, the angle of the polarization plane is plotted as a function of the photon energy for two values of f2 :
discontinuity is due to limitation of the angle between 01 and 1801: In each case of f1 and f2 ; the principal peaks have different polarization states. This means that the polarization performance will be spoilt by practical performances of the electron beam, undulator itself, and beamline components, the effects of which will be investigated in the next section.
3. Effects due to angular divergence and acceptance The possible sources to spoil the polarization performance of the proposed device are the energy spread and angular divergence of the electron beam, undulator magnetic field error, and angular acceptance of beamline optics. All of them cause peak broadening in the spectrum. It should be noted, however, that the effects caused by the energy spread and magnetic field error are usually much smaller than those by others. Therefore, we focus only on the effects caused by the angular divergence and acceptance.
ð3Þ
The function Ix represents the angular flux density of the horizontally polarized radiation (HPR), while Iy that of the vertically polarized radiation (VPR). Let us consider the peak energy of HPR. It is easily found that the peak width of SMx is about M times narrower than that of SN : Therefore, Ix has the local maxima at the energies determined by the resonance condition of SMx ; i.e., sinðd=2Þ ¼ 0: This condition can be simplified to f1 f1 Do þ pN þ1 ¼ np o1 2 2pN where n is an integer. There are infinite numbers of Do satisfying this condition. Among them, we should choose Do so that jDoj becomes the minimum in order to obtain a value of SN as large as possible. Assuming f1 =2pN51; we have the peak energy of HPR as follows: c o1 Do ¼ 1 ¼ Dop ð4Þ 2pN with c1 ¼ f1 2pl f þp l¼ 1 2p where ½x denotes the integer part of x: As is easily shown, the phase c1 ranges from p to p: The angle of p is in general equivalent to p; however, we should separate these two angles because c1 is connected to the peak energy shift as in Eq. (4). Let us investigate the angular profile of HPR and VPR at the fixed energy of op ¼ o1 þ Dop :
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591
Substituting Do ¼ Dop into Eqs. (2) and (3), we have sin O1 ðyÞ sin MO2 ðyÞ 2 Ix ðyÞ ¼ O1 ðyÞ M sin O2 ðyÞ Iy ðyÞ ¼ Ix ðyÞtan2 O2 ðyÞ;
ð5Þ
with
f1 l f1 þ pNY2 1 1 N 2 2pN f f O2 ðyÞ ¼ 1 þ 1 O1 ðyÞ 1 2pN 2
O1 ðyÞ ¼
Y2 ¼
g 2 y2 1 þ K2
where y is the angle of observation with respect to the forward direction, K the deflection parameter of the undulator and g the Lorentz factor of the electron beam. In order to estimate roughly the dependence of the flux density on y and f1 ; we neglect the terms f1 =2pN and l=N to obtain 2 sinðf1 =2 pNY2 Þ sin pMNY2 Ix ðyÞ ¼ ðf1 =2 pNY2 Þ M sin pNY2 Iy ðyÞ ¼ Ix ðyÞtan2 pNY2 :
ð6Þ
Substituting y ¼ 0; we have the on-axis flux density sin f1 =2 2 : ð7Þ Ix ð0Þ ¼ f1 =2 This is a decreasing function of jf1 j as long as jf1 jop: Thus jf1 j should be smaller to obtain higher flux. As for the p angular ffiffiffiffiffiffiffiffiffi profile, it is easy to find that when Y ¼ 1= MN ; i.e., sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ K 2 1 y¼ g ¼ ycoh ; MN Ix ðyÞ vanishes. In fact ycoh is equal to the angle at which the flux density vanishes for UR emitted from a normal undulator with the number of periods of MN: Now let us consider the effects of the angular divergence and acceptance. We assume the angular distribution function of the electron beam to be a one-dimensional Gaussian with a standard
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deviation of s0 ; i.e., 2 1 02 f ðyÞ ¼ pffiffiffiffiffiffi ey =2s 2ps0 and the optical slit to have a pinhole with an angular acceptance of Dy: In this case, we obtain the flux Px;y passing through the pinhole as follows: " ! Z 1 N Dy y Px;y ¼ Ix;y ðyÞ erf pffiffiffi 2 0 2 s0 !# Dy þ y þ erf pffiffiffi y dy 2s0 " ! Z 1 þ K2 N DY Y Ix;y ðYÞ erf pffiffiffi ¼ 2 0 2S0 !# DY þ Y þ erf pffiffiffi Y dY ð8Þ 2S0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where DY ¼ Dy= 1 þ K 2 ; S0 ¼ s0 = 1 þ K 2 ; and erfðxÞ is an error function defined as Z x 2 2 erfðxÞ ¼ pffiffiffi ex dx: p 0 The degree of linear polarization PL0 is calculated as Px Py PL0 ¼ : Px þ Py To compute the integrand in Eq. (8), the Eq. (5) but not the approximated one (6) should be used in order to obtain more precise value of PL0 : Let us consider three cases of S0 ¼ 0; 0:05g1 ; and 0:1g1 : We take 2s0p for an adequate angular acceptance of the pinhole ðDyÞ to obtain sufficient flux, where s0p is the standard deviation of the photon beam angular divergence calculated approximately as [6] 2 02 s02 p ¼ ðycoh =2Þ þ s :
Fig. 3(a)–(c) show the dependences of PL0 on c1 for the three values of S0 and various combinations of M and N with the product MN being fixed at 240. For reference, the peak energy shift Dop =o1 calculated using Eq. (4) is plotted as a function of c1 : Applying a larger number of segments improves the degree of polarization significantly. When S0 > 0; the phase c1 close to
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placed in tandem and the phase in between is adjusted to obtain CPR. Our device for M ¼ 2 is obtained by replacing the vertical and horizontal undulators with two helical undulators having opposite helicity. It is well known that the degree of polarization of UR emitted from this type of undulator is degraded easily by the finite angular divergence of the electron beam. In other words, a high degree of polarization is expected if the angular divergence is small enough. However, it is found from Fig. 3(a) that PL0 of only 0.5 is obtained when M ¼ 2; c ¼ 0 and S0 ¼ 0: This is due to the angular acceptance Dy ¼ 2s0p : We can expect a higher degree of polarization by applying a smaller angular acceptance, which in turn causes degradation of flux. Instead, we should apply a larger number of segments to obtain a higher degree of polarization even if the angular divergence of the electron beam is extremely small. Fig. 4(a) and (b) show the dependences of PL0 on M for the three values of S0 with the product MN being fixed at 60 and 240, respectively. The number of periods per segment, N; calculated as Fig. 3. The degree of polarization PL0 as a function of the phase c1 for the three values of the angular divergence of the electron beam under three combinations of M and N: (a) M ¼ 2; N ¼ 120; (b) M ¼ 6; N ¼ 40; and (c) M ¼ 16; N ¼ 15: In each figure, the peak energy shift Dop =o1 is plotted as a function of c1 :
p should be applied as well as the large number of segments to obtain a high degree of polarization. This means that the peak of HPR should be shifted to higher energies ðDop > 0Þ to avoid contamination by tailing of the VPR peak to lower energies. When S0 ¼ 0; PL0 is almost equal to 1.0 regardless of c1 for the two cases of M ¼ 6 and 15. On the other hand, applying M of 2 degrades the degree of polarization significantly. It should be noted that M ¼ 2 and c1 ¼ 0 means that two helical undulators with opposite helicity are placed in tandem and photons at the exact resonance energy is extracted. There is an ID having a configuration similar to this, which is known as a crossed undulator [7] to control the polarization state of UR. Two undulators to provide HPR and VPR (horizontal and vertical undulators) are
Fig. 4. The degree of polarization PL0 as a function of the number of segment M for the three values of the angular divergence of the electron beam with the product MN being fixed at (a) 60 and (b) 240. In each figure, the number of periods per segment is plotted as a function of M:
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591 Table 1 Maximum number of periods per segment to achieve PL0 of 0.95 for two values of MN: The phase c1 is fixed at p S0 ðg1 Þ
0 0.05 0.1
MN 60
240
15 10 7
60 30 24
N ¼ ½240=M; is plotted for reference. As S0 increases, the minimum number of segments necessary to achieve a certain degree of polarization becomes large. This means that smaller number of periods per segment should be adopted to obtain a higher degree of polarization. For example, the maximum number of periods per segment to obtain PL0 of 0.95 (dotted line) under various conditions are summarized in Table 1. It is found that smaller number of periods per segment is required for MN ¼ 60 than for MN ¼ 240; meaning that more drift sections are necessary per unit length, which degrades the ratio of the practical length of the device to that of the whole straight section, resulting in lower flux. Therefore, this device is well suited for installation in a long straight section (LSS), such as the 30-m section in the Super Photon ring-8 GeV (SPring-8) [8].
4. Example In this section, examples of calculation to estimate practical performances of the proposed device are presented. Let us consider an ID with the proposed scheme to be installed in the 30-m straight section in the SPring-8 and to have the periodic length of 10 cm and the total number of periods MN of 240. The accelerator parameters in the center of the straight section are summarized in Table 2. In order to calculate the magnetic field of the undulator, we should consider the design of the magnetic structure. We have adopted the design of the helical undulator developed at the SPring-8 [9] and calculated the magnetic field to be used in
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Table 2 Accelerator parameters used in the calculation Electron energy Average current Natural emittance Energy spread Coupling constant Horizontal betatron value Vertical betatron value
8 GeV 100 mA 5:9 nm rad 0.001 0.3% 24 m 10 m
estimation of the performances of UR, however, we skip the detailed specification of the device such as the dimension, magnet material, and support structure because they are not important in this article. All calculations are performed by the SR calculation code, SPECTRA, developed at the SPring-8 [10]. Fig. 5(a) and (b) show the energy spectra of the flux and degree of linear polarization for different numbers of segments. The K value is assumed to be 2.25 to obtain the fundamental energy at 1000 eV: The optical slit is located 60 m far from the center of the device and has a rectangular shape with dimensions of 4 mm 1 mm in horizontal and vertical directions, which are determined so that the angular acceptance is equal to 4s0p in each direction. The phase f1 is set at p by moving half-length magnets placed at the extremity of each segment to an adequate position along the longitudinal direction. This is a simpler scheme to adjust the relative phase with less drift sections than creating a chicane orbit or moving the undulator segment, the details of which are not mentioned in this article. As shown in the figure, the peak is divided into three parts, the center of which is mainly composed of VPR, while the other two are composed of HPR. The higher energy peak corresponds to the phase c1 of p; while the lower one to the phase c1 of p; which has a lower degree of polarization due to contamination by the tailing of the central peak. Fig. 6(a) and (b) show the energy spectra of the flux and degree of linear polarization for different values of c1 with M and N being fixed at 16 and 15, respectively. The peak energy op corresponding to each c1 is indicated by an arrow with the value of c1 =p: For clarity, the flux and PL0 at op in
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Fig. 5. Examples of spectra and degree of polarization calculated with different numbers of segments indicated by arrows.
Fig. 6. Same as Fig. 5 but with different values of c1 indicated by arrows and M being fixed at 16. The photon energy range is expanded for clarity.
Figs. 5 and 6 are summarized in Table 3. The flux is independent of m; while PL0 is improved by increasing m; although it almost saturates at mX16: As c1 becomes larger, the flux increases, while PL0 is degraded. This means that we have to compromise when determining the value of c1 : Table 4 shows the flux and PL0 at op for different values of o1 with M and c1 being fixed at 16 and p: The dimension of the slit is varied so that it becomes equal to 2s0p in each direction. The flux reaches the maximum at o1 ¼ 600 eV; while PL0 increases as o1 is lowered. This is caused by reduction of S0 upon higher K value application to lower the photon energy.
5. Discussions So far, we have considered the scheme to produce LPR. It is worth noting that the proposed device can produce CPR and switch the helicity, too. The simplest way is to change the helicity of the odd- or even-number segments to make identical CPR, which is usually realized by a phasing technique, or moving several magnet arrays of the helical undulator along the longitudinal direction. Another scheme is to divide all the segments into the first and second halves having opposite helicity and shift the fundamental energy of either of the two halves to o0 ¼ o1 þ Do: Let the first
T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 490 (2002) 583–591 Table 3 Flux and PL0 at the energy of op under various conditions of M and c1 M
c1
Flux (1014 photons/s/0.1%BW)
PL0
8 12 16 20 16 16
p p p p 0:9p 0:8p
7.69 7.74 7.88 7.94 9.08 10.0
0.939 0.957 0.961 0.961 0.949 0.938
Table 4 Flux and PL0 at the energy of op for different values of o1 o1 (eV)
Flux (1014 photons/s/0.1%BW)
PL0
1000 800 600 400 200
7.88 9.39 9.44 9.86 7.49
0.961 0.963 0.968 0.974 0.974
half produce right-handed CPR (RCPR) and the second left-handed (LCPR). If RCPR is desired, the fundamental energy of the first half is shifted to o0 ; while that of the second is fixed at o1 : In this case, RCPR with a high degree of polarization is obtained at o0 if Do is large enough to separate the RCPR and LCPR. In order to shift the photon energy, the K value, i.e., the gap of the undulator is changed. In the point of view of available flux, the phasing method is better because the practical number of periods is twice as that of the other, while the time necessary to switch the helicity for the energy-shift method is much shorter than that for the phasing method. The gap change necessary to separate the LCPR and RCPR is just several millimeters. There is another method to shift the peak energy much faster than the gap movement. Because each half is composed of plural segments, we can use the phase detuning [5]. By installing electromag-
591
nets to make chicane orbit between segments and changing its path length, the phase can be varied. In this case, much faster repetition rate than the other two would be achieved. In this article, helical undulators are used to produce LPR. It is also possible to use horizontal and vertical undulators to generate CPR and switch the helicity by adjusting the phase between segments. This is the same scheme as the crossed undulator, however, the proposed device can achieve a much higher degree of polarization than the crossed undulator. It should be noted that adopting horizontal and vertical undulators instead of helical undulators brings the two problems already described in Introduction. Especially, the heat-load problem is quite serious under high K value application. As for the contamination by the higher harmonic, we can solve the problem by applying a negative value of c1 to shift the fundamental peak to higher energies. In this case, the same effect as the phase detuning is obtained, resulting in effective suppression of the higher-harmonic intensity.
References [1] B.M. Kincaid, J. Appl. Phys. 48 (1977) 268. [2] T. Tanaka, H. Kitamura, Nucl. Instr. and Meth. A 364 (1995) 368. [3] B.W. Batterman, D.H. Bilderback, X-ray monochromators and mirrors, in: G.S. Brown, D.E. Moncton (Eds.), Handbook on Synchrotron Radiation, North-Holland, Amsterdam, 1991 (Chapter 4). [4] H. Hashimoto, S. Sasaki, JAERI-M Report 94-055, 1994. [5] T. Tanaka, H. Kitamura, J. Synchrotron Rad., submitted for publication. [6] K.J. Kim, Proc. 1986 US Particle Accelerator Summer School, 1986. [7] K.J. Kim, Nucl. Instr. and Meth. 219 (1984) 425. [8] H. Kitamura, T. Bizen, T. Hara, X. Mar!echal, T. Seike, T. Tanaka, Nucl. Instr. and Meth. A 467–468 (2001) 110. [9] T. Hara, T. Tanaka, T. Tanabe, X.M. Mar!echal, K. Kumagai, H. Kitamura, J. Synchrotron Rad. 5 (1998) 426. [10] T. Tanaka, H. Kitamura, J. Synchrotron Rad. 8 (2001) 1221.