Technical Note Amman, October 29, 2006 (draft) Note: M ... - SESAME

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Trajectory [ micrometer ] 30 Hor. Trajectory 25 Ave. Hor. Traj. Vert. Trajectory 20 Ave. Vert. Traj. 15 10 5 0 -600 -400 -200 0 200 400 600 -5 -10 -15 y [ mm ] Figure 18. The 2.5 GeV electron trajectory in the model magnet for the vertical undulator phase. 10 5 z [ micrometer ] -30 -25 -20 -15 -10 -5 0 0 -5 -10 x [ micrometer ] -15 Figure 19. The 2.5 GeV electron trajectory in the xz-plane for the vertical undulator phase. 12
4. INTERACTION BETWEEN EPU AND ELECTRON BEAM 4.1. FOCUSING POTENTAIL The transverse roll off of the magnetic flux results in an angular kicks and, hence a tune shift. This is expressed by a focusing potential given by [7, 9]; ∞ 2 1 e ' ' 2 ' ' 2 Φ( x , z) = − ( ) ∫(( ∫Bx ( x, z, y ) dy ) + ( ∫Bz ( x, z, y ) dy ) ) dy 2 γmc −∞ z −∞ where e is the electron charge, mc the electron momentum and B x and B z the horizontal and vertical components of the magnetic flux density. The integrals can be solved analytically if we keep only the dominant n = 1 Fourier component of the flux density. Φ L ⎛ ecλ 2 ⎜ ⎝ 2πE 2 z −∞ 2 Ψ ( ) ( x, z) = u 2 ( x, z) = − ⎜ ⎟ B ( x, z) + B ( x, z) e ⎞ ⎟ ⎠ 1, x where L is the length of the undulator, u the undulator period length, E e the electron beam energy and B 1,x and B 1,z the n = 1 component in the Fourier expansion. This is a very good approximation for PPM undulators, where the n = 1 component dominates. The potential is largest at minimum gap, where the flux density is largest, and decreases rapidly with increasing gap. The energy-independent part of the potential (x,z) at minimum gap is shown in Figures 20 – 22 for the Horizontal, Circular and Vertical modes of operation. 1, y E 2 e Focusing Potential 120000 110000 100000 Pot.HT 2 mm 3L 90000 - 20 - 10 0 0 XHmmL 10 2 - 4- 80000 4 2 ZHmmL 20 Figure 20. The focusing potential for the horizontal undulator phase at minimum gap 13
Page 1 and 2: Technical Note ____________________
Page 3 and 4: The inclined mode of operation has
Page 5 and 6: Table 2. Flux densities, k-values a
Page 7 and 8: Flux Density [ T ] 0.7 0.6 0.5 0.4
Page 9 and 10: First vert. integral [ G.cm ] 100 8
Page 11: Trajectory [ micrometer ] 55 45 35
Page 15 and 16: 4.2. ANGULAR KICK AND TUNE SHIFT Th
Page 17 and 18: Depending on the position x,z of th
Page 19 and 20: 0.025 Vertical Tune Shift 0.02 0.01
Page 21 and 22: OPTICAL FUNCTIONS 30 Beta Z Beta X
Page 23 and 24: Vertical Beta Beat 2 1.5 1 0.5 0 -0
Page 25 and 26: 10000 Vertical Force [Newton] 5000
Page 27 and 28: Brilliance [Photon/s/mm 2 /mrad 2 /
Page 29 and 30: REFERENCES [1] G. Vignola, M. Attal
Page 31 and 32: 0.004 0.003 Hor. Ang. Kick [T 2 m 2
Page 33 and 34: APPENDIX B FEASIBILITY OF HOUSING S
Page 35 and 36: Horizontal Beta Beat 3.5 3 2.5 2 1.

Technical Note Amman, October 29, 2006 (draft) Note: M ... - SESAME

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