CSR Microbunching: Gain Calculation

Transcription

1 CSR Microbunching: Gain Calculation Zhirong Huang, Kwang-Je Kim Argonne National Laboratory

2 Integral Equation for CSR Microbunching For bunching parameter b at mod. wavelength λ = 2π/k b( k; where kernel = b ( k; + b ( k; is the bunching without CSR K( s', = s ik( R ds' K( s', b( k'; s') 56 ( s' I( s') Z( k') exp(... ε, σ...) δ γi 4 A Landau damping Given any initial condition (density and/or energy modulation), this determines the final microbunching Calculate gain=b final /b initial, comment on initial conditions

3 Staged Amplification Typical chicane dipole separation L >> dipole length L b 3 Lb within the same dipole 2 ρ R56( s' ~ 2 L Lb from one dipole to another 2 ρ Ignore the induced bunching from energy modulation in the same dipole (Schneidmiller et al.) Consider staged amplification from dipole to dipole by setting K(s,=O(L b / L)= if s-s < L

4 Iterative Solution Integral equation can be solved by two iterations b( k; = b ( k; + ds' K( s', b ( k'; s') one - stage amplification s s no emittance Schneidmiller et al. s' + ds' K( s', ds'' K( s'', s') b ( k''; s'') two - stage amplification (no compression and = b ( k; + I f (1 3) + I f (2 3) dominant in low-gain 2 I f (1 2 3) high-gain

5 Berlin CSR Benchmark: No Compression σ δ = No emittance σ δ =3 1-5

6 Berlin CSR Benchmark: Compression ε n = µm, σ δ =2 1-6 ε n =1 µm, σ δ =2 1-6 ε n = µm, σ δ =2 1-5 ε n =1 µm, σ δ =2 1-6 before compression

7 LCLS Bunch Compressors (BC) Injector DL1 SC wiggler Linac Linac Linac DL2 Energy Compressor current Norm. emittance Energy spread BC1 25 MeV 48 A 1 mm BC GeV 4 A 1 mm (3 1-5 )

8 BC1 Gain of Density Modulation σ δ = , ε n =1 µm 1) 1 stage 2) 1+2 stage At low gain, one-stage amplification dominates

9 BC2 Gain of Density Modulation G f G f l 1) σ δ =3 1-5, ε n =1 µm 2) σ δ =3 1-5, ε n = µm 3) σ δ =3 1-6, ε n =1 µm l σ δ =3 1-6, ε n = µm Two-stage amplification dominates at high-gain Analytical solutions agree with the numerical solutions (from Heifets/Stupakov/Krinsky, dashed curve

10 Energy Modulation CSR Induced energy modulation in bends p( s I( s') ds Z( s') b( s')exp(... ε, γi A CSR microbunching can be generated by initial energy modulation of the bunch due to upstream wakefields (such as CSR) b p = ikr p) (exp one - stage amplification) 56 ( + Total gain of BC1+BC2 gain in BC1 X gain in BC2 + energy modulation from BC1 converted to density gain in BC2 σ δ...)

11 BC1+BC2 Wiggler off Wiggler on Reasonable agreement with P. Emma s Simulation

12 Dog Legs Total gain of two chicanes can be large (>1) even though the gain in each chicane is small (<3) LCLS has more bends than BCs, Dog Legs (DL for beam transport (DL1+DL2) DLs typically have very small R56 (compared to BC ignore density gain but keep energy modulation (same approximation made within bends of a single chicane) DL1 energy modulation can be turned into BC1 density modulation through R56 of BC1, cascading through the whole system and leading to more gain in CSR microbunching

13 DL1+BC1 Assume only initial density modulation before DL1 induced energy modulation at the exit of DL1 (in units of initial bunching) Turns into BC1 density gain through R56 at end of BC1

14 Estimated Gain of All LCLS Bends Wiggler off Wiggler on SC wiggler suppresses a short-wavelength, high-gain peak

15 Comments on Initial Condition From shot noise b eff ~ N 1 coherence length ~ ( ) with a gain less than 1, this should be a small effect From sharp current spike Nspike b How big? N total Other sources of energy modulation (wakefields ) Watch out for numerical noise in simulations!

16 Conclusion CSR microbunching in a bunch compressor is studied using the iterative solution of the integral equation Initiated by density and/or energy modulation Cascading effects of multiple chicanes Gain curves agree with numerical solution and simulation Significant gain is found for the LCLS system (DL1+BC1+BC2+DL2), can be suppressed by increasing the uncorrelated energy spread through a SC wiggler