X-Y COUPLING GENERATION WITH AC/PULSED SKEW QUADRUPOLE AND ITS APPLICATION

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1 X-Y COUPLING GENERATION WITH AC/PULSED SEW QUADRUPOLE AND ITS APPLICATION # Takeshi Nakamura # Japan Synchrotron Radiation Research Institute / SPring-8 Abstract The new method of x-y coupling generation with AC or pulsed skew quadrupole is proposed. With this method, no difference resonance is requested; therefore the horizontal and vertical tunes should not be the same value. The AC skew quadrupole is driven by the difference frequency of horizontal and vertical betatron frequency to convert the horizontal position frequency to the vertical kick frequency and vice versa, therefore the coupling with this method is on resonance and can be driven to full coupling strength. In this report, the principle, simulation result, and possible applications are descried.. (x-y ) AC Skew (AC-SQ) [] AC-SQ Touschek [2] x-y AC-SQ AC-SQ x-y ( x-y AC-SQ x-y x-y # nakamura@spring8.or.jp [3] AC-SQ AC- SQ 2. AC Skew x-y 2. AC x-y d 2 x dt 2 + ω x 2 x = y () d 2 y dt 2 + ω y 2 y = x (2) ω x ω y x-y t ()= cos( ω x ω y )t (3) x, y xt ()= x cos( ω x t +ϕ x ) (4)

2 yt ()= y cos( ω y t +ϕ y ) (5) () (2) t ()xt ()= x cos( ω x ω y )t cos( ω x t +ϕ x ) [ ( )] = 2 x cos ( ω y t +ϕ x)+ cos ( 2ω x ω y )t +ϕ x t ()yt ()= y cos( ω x ω y )t cos( ω y t +ϕ y ) [ ( )] = 2 y cos ( ω x t +ϕ y)+ cos ( ω x 2ω y )t ϕ y (t () (2 2ω x ω y =±ω y ω x = ω y ω x = difference resonance integer resonance 2 (), (2) d 2 x dt 2 + ω x 2 x = 2 y cos ω x t +ϕ y d 2 y dt 2 + ω y 2 y = 2 x cos ω y t +ϕ x (6) (7) ( ) (8) ( ) (9) AC-SQ x-y onresonance 2.2 normalized coordinate η x ( φ x )= x / β x, () η y ( φ y )= y / β y, () φ x ()= s s d s, (2) ν x β x d 2 η x dφ x 2 + ν x 2 η x = ν x d 2 η y dφ y 2 + ν y 2 η y = ν y β x θ x φ x β y θ y φ y (4) k= ( ) δφ ( x 2πk) (5) k= ( ) δφ ( y 2πk) ϕ x ϕ y ϕ x = ϕ y = 2πk, k=...,-,,,2,.. ϕ x ϕ y ϕ x = ϕ y = ϕ (4), (5) η x = η x e i ( ν x +Δν )φ (6) η y = η y e i ( ν y +Δν )φ (7) AC-SQ ( φ)= cos( Δν x Δν y )φ = ( 2 ei Δν x Δν y )φ i Δν + e ( x +Δν y )φ (8) ϕ = (t) AC-SQ θ x ( φ)= ( φ)y( φ) = 2 β y η y e i ( Δν x Δν y +ν y +Δν )φ i Δν + e x +Δν y +ν y +Δν (9) θ y ( φ)= ( φ)x( φ) = 2 ( )φ β x η x e i ( Δν x +Δν y +ν x +Δν )φ i Δν + e x Δν y +ν x +Δν (2) ( )φ δφ ( 2πk) 2π k= e ipφ (2) p= φ y ()= s s d s (3) ν y β y (4), (5) θ x θ y AC-SQ Δν η y = Δν η x = β x β y η x (22) β x β y η y (23)

3 Δν =± β x β y (24) T c = 2 T /Δν = 4π β x β y T (25) * Gaussian * * RF * * RF * T AC-SQ AC-SQ AC-SQ AC-SQ x-y Landau ( Landau Gaussian τ = 2 π ω σ ν (26) ω σ ν r.m.s. τ T c < τ (27) SPring-8 6GeV 5-4 Landau 2ms 8 GeV * AC-SQ * Table : Sample Ring and Magnet Parameters Parameter Symbol Value Energy E 6 GeV Revolution period T 4.79 µs Revolution frequency f 28.8kHz Betatron damping time τ β 8.3 ms Betatron tunes ν x / ν y.5 /.35 Betatron frequency f x / f y 3kHz / 73kHz Beta functions β x / β y 5m / 5m Betatron tune spread σ νx / σ νy 5-4 / 5-4 Landau damping time 2ms / 2ms Chromaticity ξ x / ξ y 2 / 2 AC-SQ Magnet Parameters ( coil winding : half-turn ) Drive frequency f y - f x 42 khz Strength. Energy exchange period T c.2 ms Length L m Bore radius b 3 mm Field gradient (peak) B'.2 T/m Drive current (peak) I 43 A Inductance L ~3.5µH Drive voltage (peak) V 32 V Fig. Fig.2 Fig. t = (25) Fig. 2 x-y Fig AC-SQ t = T c = ms =. Fig. 4 AC-SQ 2ms

4 Figure : Motion of a particle, kicked horizontally at t =. AC-SQ is continuously excited with =.. The energy is transferred between horizontal and vertical with the period T c. Tune spread and chromaticity are set to zero and radiation excitation is turned-off, to see the motion clearly Figure 2 : Emittance evolution after AC-SQ is turned on with =. at t =. Horizontal and vertical emittances are oscillating with the period Tc. Tune spread and chromaticity are set to zero. Figure 3 : Full coupling by AC-SQ, which is turned on at t = with =.. Tune spread is σ ν =5-4 and chromaticity is 2 for horizontal and vertical. c x or y posision x -9 2 x -9.5 x -9 x -9 5 x - 2 x -9.5 x -9 x -9 5 x - 2 x -9.5 x -9 x -9 5 x - Tc =., σ ν =ξ = Tc =., σ ν =ξ = Figure 4 : Full coupling by AC-SQ with is gradually turned on with time constant 2ms at t =. Oscillation is suppressed with gradual turning on. Fig. 5 AC-SQ x-y x y =., σ ν =., σ ν x -9.5 x -9 x -9 5 x - Figure 5: Emittance after AC SQ is turned off at t= with the same parameters as in Fig =., σ ν x-y Touschek [2] AC-SQ AC-SQ x-y Fig. 6 Pulsed AC-SQ CW AC-SQ Pulsed AC-SQ CW AC-SQ Fig. 6 m =. 2.5 kv m 5 AC-SQ ~ 2

Figure 6 : Example of stripline kicker k for pulsed operation of AC-SQ for cancelation of CW AC-SQ just at the neighborhood of injected bucket. Beam passes through the center of the figure. 4.

5 Figure 6 : Example of stripline kicker k for pulsed operation of AC-SQ for cancelation of CW AC-SQ just at the neighborhood of injected bucket. Beam passes through the center of the figure AC-SQ AC-SQ Fig. 6 AC-SQ t = T c / 2 AC-SQ T c ms AC 2 x -9.5 x -9 x -9 5 x - 4mm 2mm y =.2 =.2 =. =. σ ν Figure 6 : Horizontal emittance ( ) and vertical emittance ( ) exchange by pulsed operation of AC-SQ. AC-SQ is turned on with values ( =.2,.,.5) at t = and turned-off at t = T c / 2 corresponding to each values. Tune spread is σ ν =5-4 and chromaticity is 2 for horizontal and vertical. x =.5 = FEL [4] [4] 5. AC Skew x-y AC Skew 4 x-y JASRI [2] " X-Y USR " (23) [2] M. Borland, "Exploration of a Tevatron-Sized Ultimate Light Source", FLS22 (22). Borland.pdf [3] The emittance exchange schemes are summarized in A. Chao, "Gymnastics in Phase Space", SLAC-PUB-4832 [4] Y. Cai, Y. Ding, R. Hettel, Z. Huang, L. Wang, L. Xiao, "An X-ray Free Electron Laser Driven by an Ultimate Storage Ring", Synchrotron Radiation News, Vol. 26. No.3 (23), also slac-pub-538.pdf.

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