• Slides: 26

Simulation of the beam loading for the CLIC accelerating structures Oleksiy Kononenko

Contents • • • Introduction Bandwidth analysis of the tapered structures Pulse propagation in the CLIC structures Beam loading model and simulation Conclusions

Introduction: E-field in T 24 structure Considering T 18, T 24 CLIC structures

Bandwidth calculations • Rise time is proportional to 1/Δf • Easier to analyze 1 cell with the periodical conditions than the tapered structure • “First cell based” and “last cell based” structures are considered • Optimization of the matching cells is required

T 18 structure bandwidth calculations

T 24 structure bandwidth calculations t rise ~ 1 / BW

S-parameters in T 24 structure

Filling time in computation in T 24 t filling ( f ) = dφ(f) / dω, φ(f) = angle ( s 12 ( f ) )

Rectangular pulse

Ramped pulse

Rectangular pulse propagation in T 24 s 12(f) → [ ift ] → s 12 (t) → [conv prect(t)] → pout(t)

Ramped pulse propagation in T 24 s 12(f) → [ ift ] → s 12 (t) → [conv pramped(t)] → pout(t)

Unloaded gradient calculations Ez(z, f) → [ exp ( ± i *z *ω/c ) ] → G 0 (z, f) [ ift ] → G 0(z, t) → [conv p(t)] → G(z, t) → [∫ dz] ↑ G 0 (z, f) ↓ ↓ Vacc (t) ↑ [ ∫ dz ] → V 0 (f) → [ ift ] → V 0 (t) → [ conv p(t) ]

Unloaded Vacc in T 24 structure x 10 7 9 Accelarating Voltage, arbitrary units V V+ 8 V 7 6 5 4 3 2 1 0 -1 11. 7 11. 8 11. 9 12 f, GHz 12. 1 12. 2

Beam loading: steady-state d. P / dz = -ω *W(z) / Q(z) - G(z)*I z G(z)=G 0(z)[1 - ∫ I ω ρ(z) / ( G 0(z) vg (z) ) dz] 0 G 0(z) = g 0 * F (vg (z), Q(z) , ρ(z) ) *Beam loading for arbitrary traveling wave accelerating structure. A. Lunin, V. Yakovlev

Beam loading model Longitudinal discretization by cells: • Energy density: w(C) = ε 0 /2 ∫ |E(x, y, z, f 0)|2 d. VC / Clength • Loss factor: k’(C)=G (C, f 0)^2/( 4*w(C) ) • Averaged gradient: G (C, f 0)= ∫ G (z, f 0) d. Lc / Clength • Group velocity: vg(C)=Pin*Clength / wpec(C) ≈ 0. 8 -1. 6 % c

Beam loading model Time discretization: • • Tbunch per cell = Clength / c ≈ 0. 0278 ns TRF cycle = 1 / f 0 =3* Tbunch per cell ≈ 0. 0834 ns Tbunch separation= 6 * TRF cycle =18 * Tbunch per cell Tenergy per cell = Clength / vg (C) = wpec (C) / Pin ≈ 1. 5 -3 ns f max = 1 / T bunch per cell

Beam loading model • Energy lost per bunch per cell: Wbunch(t, C)=k’(C)* q 2(t, C) * Clength Wfield(t, C) =G(C)* q(t, C) * Clength • Energy moves with vg(C) • Wall losses Pwalls(C) = ω*W(C) / Q(C) • Total energy W(t, C) = ∑ Wb(t, C) • Gradient: G(t, C)=2*sqrt(k’(C) * W(t, C) / Clength)