# Simulation of the beam loading for the CLIC

- 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: steady-state

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)

Beam loading simulation

Accelerating voltage in T 24

Accelerating voltage in T 24

Accelerating voltage in T 24

Conclusions 1. Performed comparison of the BW for the tapered vs non-tapered structures 2. Pulse propagation in CLIC structures is investigated 3. Beam loading model is developed and simulations are carried out.

Further steps 1. Optimization of the pulse shape 2. More accurate beam loading calculations

Thank you for the attention!