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Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 Application Limit of SR Interferometer for Emittance Measurement Y. Takayama ,T. Okugi, T. Miyahara, Tokyo Metropolitan University, Tokyo, Japan S. Kamada, J. Urakawa , T. Naito, KEK, Tsukuba, Japan Abstract 2 AVAILABILITY OF THE VAN CITTERT-ZERNIKE THEOREM We investigate the application limit of the SR interferom- eter for emittance measurement at the KEK-ATF. We need The Twiss parameters and dispersion at the position where to consider two important problems, which are the limita- the electron beam size is evaluated, and emittance in the tion of the availability of the van Cittert-Zernike theorem horizontal and vertical directions are shown in the Table 1 and the diffraction effect due to a narrow vertical aperture [3]. Here, we assumed that there is no vertical dispersion of the SR extraction line. The former problem is analyzed and the coupling is 1 %. We investigate whether the van with the theory introduced in another paper [1]. The lat- Cittert-Zernike theorem is available in the horizontal and ter one is studied with the numerical calculation, where the vertical directions individually. The details of derivation of narrow aperture is assumed to be an optical slit with an ade- the conditions are shown in another paper [1]. quate vertical width. We show that these problems must be solved for the accurate measurement of the electron emit- tance, especially in the vertical direction. Table 1: Design value of the ATF damping ring at the point where the SR interferometer is installed. horizontal (x) vertical (y ) æ 0.3590 -1.1690 1 INTRODUCTION æ (m) 0.3796 2.8435 æ (1/m) 2.9739 0.8323 A measurement of the electron beam emittance is one of the 0.0491 0 0 most important theme for the accelerator physics. At the -0.1458 0 KEK-ATF damping ring, several attempts are performed to " (nm rad) 1.08 0.0108 estimate the emittance. Especially, the SR interferometer which measures the spatial coherence (visibility) has some advantages compared with other methods [2]. The elec- tron beam size can be obtained by performing the Fourier transformation of the spatial coherence, which is called as 2.1 Horizontal direction the van Cittert-Zernike theorem. However, it is not trivial In the horizontal direction, three conditions must be satis- whether this theorem is available for the bending magnet ﬁed for the electron beam size çx in order to use the van radiation. Recently, some conditions to judge whether this Cittert-Zernike theorem, which are theorem is available or not for the bending magnet radiation ç çæx2"x ; were derived by the authors [1]. We investigate whether çx (1) these conditions are satisﬁed for the SR interferometer at L the ATF damping ring. çæy "y çx ç æy "y ; (2) We need to consider another important problem, the ef- 2 L 1+ çp 2 fect of the narrow vacuum chamber at the SR extraction s line. If the light is cut by this vacuum chamber, the spatial æx ç"x coherence at the downstream changes. For an extreme ex- çx ç æx L ; (3) ample, if the width of the vacuum chamber is much smaller than the wavelength of light, the spatial coherence at the downstream is perfect for any electron beam parameters. where L = 7:04 m is the distance between the light source Therefore, in order to measure the electron beam size with and double slit, ç = 5:73 m is the bending radius. æ, the SR interferometer, we must investigate how the vacuum æ , æ and " are the Twiss parameters and emittance after integrating energy spread, respectively. æ is deﬁned as æ = æ , 2æL + L2æ . çp and çp are the beam size and chamber affects the spatial coherence. 0 In this paper, we investigate above two points in detail beam divergence of the light at the waist in the vertical di- and judge whether the SR monitor is available to measure rection which is emitted by a single electron. çp is deﬁned q the electron beam size at the ATF damping ring. as çp + L2 çp 2 . For the wavelength = 500 nm, çp and 2 0 çp are calculated as 19.4 0 m and 2.06 mrad, respectively. Email: takayama@phys.metro-u.ac.jp Using the parameters in Table 1, we have çx = 34:0 m. 0-7803-5573-3/99/$10.00@1999 IEEE. 2155 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 The conditions in (1), (2) and (3) are written as the vertical direction. For this purpose, we suppose that the narrow space is equivalent with the vertical entrance slit çx ç 5:20 10,2 m; with the width d, as shown in Figure 1. We put the dis- çx ç 7:56 10,5 m; tance between the light source and the entrance slit to be çx ç 3:42 10,2 m; L1 = 0:56 m and the distance between the entrance slit and the double slit to be L2 = 6:48 m, respectively. respectively. These condition are well satisﬁed for the de- φ φ φ φ B0 Bi Bf BD sign value and the van Cittert-Zernike theorem is safely used to estimate the beam size in the horizontal direction. y y y y 2.2 Vertical direction d D As same with the case of the horizontal direction, three double slit screen conditions are necessary in order to use the van Cittert- entrance slit Zernike theorem, which are L1 L2 v ç u "x "y æx æy Light source u çy ç L 2 t æ " ; (4) d 1+ D y y çp 2 B0 Bi B f BD L"y çy ç çp ; (5) s L " y æy çy ç 2k ç ; (6) Figure 1: Transformation of the phase space of light. p p + "y æy ç2 If we denote the electric ﬁeld on the entrance slit and where çy = 5.54 m is the electron beam size in the vertical on the double slit as Ei y and ED y , respectively, the direction. There is an extra condition on the divergence of correlation of the ﬁelds are related with the equation the light beam under which the light beam reaches at the Z observer points strongly enough, which is ,D y1 ; y2 = dya dyb T ya T yb çp çy é r ; ,ik r1a ,r2b ,i ya ; yb e r r (7) çy 2 ; 1+ 0 (8) çp 0 1a 2b 0 where the integration is performed on the entrance slit. k is where we put the electron beam divergence as çy p"y æy . = the wave number of light and As for the conditions in (4), (5) and (6), we have ,i y1 ; y2 = hEi y1 Ei y2 i; çy ç 1:98 10,3 m; ,D y1 ; y2 = hED y1 ED y2 i; q çy ç 5:25 10,3 m; r1a = L2 + y1 , ya 2 ; 2 çy ç 2:41 10,1 m; q r2b = L2 + y2 , yb 2 : 2 respectively. Therefore, these conditions are well satisﬁed. However, the condition (7) is written as T y is the transmittance function of the entrance slit. (8) is valid if the wavelength of the light is much smaller than the çy é 19:9 m; entrance slit. Intuitively, (8) means that the electric ﬁeld on the double slit can be obtained by summing up the spherical which is not satisﬁed in this case. This means that the wave emitted by the point sources on the entrance slit. The whole curve of the visibility can not be obtained due to spatial coherence on the double slit is written as the weak intensity of the light for large slit separation and the accurate measurement of the electron beam size is very æD =p ,D D=2; ,D=2 ; difﬁcult in the vertical direction. ,D D=2; D=2 ,D ,D=2; ,D=2 where D is the separation of the double slit. 3 EFFECT OF VERTICAL APERTURE In order to calculate the spatial coherence, the correla- Between the light source and the double slit, there is a nar- tion of the ﬁelds on the entrance slit is needed. This can row space of the vacuum chamber in the vertical direction be obtained by approximating the radiation ﬁeld with the whose width is only 4 5 mm. It is important to inves- Gaussian beam. With this approximation, the brightness tigate whether this narrow space affects the coherence in function is represented analytically. Since the brightness 2156 Proceedings of the 1999 Particle Accelerator Conference, New York, 1999 function B y; ç and the correlation of the ﬁelds , ya ; yb 1.6 1.6 1.4 1.4 Spatial coherence are related with the Fourier transformation [4]. , ya ; yb 2mm slit Spatial coherence 1.2 1.2 1 1 4mm slit is also obtained analytically. Using this property, the visi- 0.8 0.6 without slit 0.8 0.6 without slit bility is numerically calculated if we set the electron beam 0.4 0.2 relative intensity 0.4 0.2 relative intensity parameter at the emitting point [5]. 0 0 10 20 30 40 50 60 70 80 0 0 10 20 30 40 50 60 70 80 Slit separation (mm) Slit separation (mm) 1.6 1.6 3.1 Numerical calculation 1.4 1.4 Spatial coherence Spatial coherence 1.2 1.2 5mm slit 1 1 6mm slit We calculate the visibility with the formula discussed in 0.8 0.8 the previous section. We put "y = 0.01 nm rad. Figures 0.6 0.4 relative intensity without slit 0.6 0.4 without slit 2, 3 and 4 are plots of the numerical calculations for çp = 0 relative intensity 0.2 0.2 0 0 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 1.0 mrad, 2.0 mrad and 3.0 mrad, respectively. For each Slit separation (mm) Slit separation (mm) light divergence, four types of the entrance slits are used. In each ﬁgure, three curves are drawn. The curve with ”rel- Figure 3: Calculation for çp = 2.0 mrad. 0 ative intensity” is the plot of the intensity on the double slit normalized by that for D = 0. The curve with ”without 1.8 1.8 1.6 relative intensity 1.6 1.4 1.4 Spatial coherence Spatial coherence 2mm slit relative intensity slit” is the curve of the spatial coherence without the en- 1.2 1.2 4mm slit trance slit. The curve with ”d mm slit”, where d = 2, 4, 5 1 1 0.8 0.8 or 6, is the curve of the spatial coherence with the d mm 0.6 0.6 without slit without slit 0.4 0.4 0.2 0.2 entrance slit. The horizontal axis represents the separation 0 0 10 20 30 40 50 60 70 80 0 0 10 20 30 40 50 60 70 80 of the double slit D, and the vertical axis represents the Slit separation (mm) Slit separation (mm) 1.8 1.8 relative intensity or the spatial coherence. 1.6 1.6 Spatial coherence 1.4 1.4 Spatial coherence Although the intensity distribution on the double slit is 1.2 5mm slit 1.2 6mm slit 1 1 Gaussian, the behaviors of the relative intensity on the dou- 0.8 0.8 0.6 without slit 0.6 without slit ble slit are very complicated, especially for small entrance 0.4 relative intensity 0.4 relative intensity 0.2 0.2 slit and large light divergence. It is due to the diffraction 0 0 10 20 30 40 50 60 70 80 0 0 10 20 30 40 50 60 70 80 Slit separation (mm) effect. The vibrations of the spatial coherence with the en- Slit separation (mm) trance slit are caused by the Fraunhofer diffraction. For the region where this vibration remarkably appears, the spa- Figure 4: Calculation for çp = 3.0 mrad. 0 tial coherence with the entrance slit has great discrepancy with that without the entrance slit. Fortunately, this dis- mated by using the van Cittert-Zernike theorem at the ATF crepancy is hard to observe since the light intensity is ex- damping ring. There exists an experimental difﬁculty in the tremely weak for such region. vertical direction because of the weakness of light inten- As we pointed out in the previous section, it is difﬁcult sity. This difﬁculty will be overcome by using the vertical to obtain the exact curve of the coherence in the vertical bend as mentioned in another paper [1]. However, the ver- direction, since the intensity decreases rapidly before the tical intensity distribution on the double slit might have a coherence decreases. very complex form due to the vacuum chamber, which has 1.4 1.4 a chance to improve the spatial coherence. Therefore, the 1.2 2mm slit 1.2 4mm slit measurement of the intensity distribution on the double slit Spatial coherence Spatial coherence 1 1 0.8 0.8 is signiﬁcant. 0.6 without slit 0.6 without slit 0.4 relative intensity 0.4 relative intensity 0.2 0 0 10 20 30 40 50 60 70 80 0.2 0 0 10 20 30 40 50 60 70 80 5 ACKNOWLEDGEMENT Slit separation (mm) Slit separation (mm) 1.4 1.4 The authors would thank to all members of the ATF col- 1.2 5mm slit 1.2 laboration to make available discussions and support this Spatial coherence 6mm slit Spatial coherence 1 1 0.8 0.8 work. 0.6 0.6 without slit without slit 0.4 relative intensity 0.4 relative intensity 0.2 0.2 6 REFERENCES 0 0 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 Slit separation (mm) Slit separation (mm) [1] Y. Takayama and S. Kamada, Phys. Rev. E 59 Number 6, to be published (1999). Figure 2: Calculation for çp = 1.0 mrad. 0 [2] T. Naito et. al.,”Emittance Measurement at ATF Damping Ring”, in this conference. [3] K. Kubo, ”Beam Development In ATF Damping Ring”, Proc. 4 CONCLUSION of EPAC’98, (1998). [4] K-J. Kim, Nuc. Instr. Methods A246, 71, (1986). The calculation in this paper shows that the electron beam [5] Y. Takayama et. al., J. Synchrotron Rad. 5, 1187, (1998). size in the vertical and horizontal directions can be esti- 2157