Sven Reiche UCLA ICFA-Workshop - Sardinia 07/02

1. Comparison of the Coherent Radiation-induced Microbunch Instability in an FEL and a Magnetic Chicane Sven Reiche UCLA ICFA-Workshop - Sardinia 07/02 Sven Reiche - ICFA Sardinia

2. CSR SASE FEL I I Instability The Analogy A Typical FEL Beamline Linac Chicane Undulator Gun Linac Trajectory Sven Reiche - ICFA Sardinia

3. The Resonance Approximation The FEL model is based on the resonance approximation • The consequences of this assumption are: • Energy change per period is small • Electron motion can be averaged over the undulator period • Selection of a small bandwidth around central, resonant frequency • Radiation field is interacting with electron beam over entire undulator length, although the changes per period are small as well Sven Reiche - ICFA Sardinia

4. Universal scaling parameter The FEL Model (1D) FEL equations Deviation of mean energy g0 from resonant energy gR Pondemotive phase q=(k+ku)z-wt Deviation of particle energy from mean energy Linear in energy deviation Radiation field ampitude Linear in field amplitude Linear in bunching Space charge parameter Normalized position in undulator z=2kurs Sven Reiche - ICFA Sardinia

5. Solutions of the FEL Equations The ansatz A~exp[iLz] yields a dispersion function for L, with the initial energy distribution f(d) as argument. • In the simplest case (L3=-1) there are three roots, corresponding to • an exponentially growing mode, • an exponentially decaying mode, • an oscillating mode. • The model is only valid as long the resonance approximation is fulfilled. Sven Reiche - ICFA Sardinia

6. The Limit of the FEL Model What happened for r ~ 1 ? Technically the FEL model is based on perturbation theory in first order with r as the order parameter. Approaching unity requires higher order and gives poor convergence! Qualitatively the limit corresponds to a significant growth within one period. The explicit motion of the electrons has to be taken into account. Currently no such device exist! A chicane is different because the transverse offset is larger than the beam size. Radiation interacts for short time before leaving the bunch. This allows to model the radiation by an instantaneously acting wake potential. Sven Reiche - ICFA Sardinia

7. trajectory field The Motion in a Chicane (1D) The CSR potential: The equations of motion: Long. position Sum over all R56, reduces to well-known expression if d(s’) is constant. Energy deviation Amplitude and phase of current modulation Phase offset between modulation and wake Sven Reiche - ICFA Sardinia

8. The Low-Gain Model Because any change in energy has a delayed effect on the particle position, the energy modulation is accumulated with an almost constant rate. Approximation: b(s) ~ b(0) in energy equation. d(s) ~ s z(s) ~ F(s).sin[kz(0)+f(0)+p/3] Particle falls back due to growing bend radius. Polarity change shortens path length. Steadily growing radius is dominant effect. Path length reduction from bend 1 & 2 are combined. 1 2 3 4 Klystron-like Motion Sven Reiche - ICFA Sardinia

9. The Gain in the Low-Gain Model The final gain, including energy spread is with and Example: Generic LCLS chicane (g = 500, I0 = 100 A, R = 12 m, L = 1.5 m) a=0 a=0.003 a=0.015 a=0.05 Sven Reiche - ICFA Sardinia

10. Limitations The model is limited by • Negligible growth of the modulation in the first half of chicane. • Negligible change in the bunching phase. High-gain regime of microbunch instability Comparison of low gain model with self- consistent model by Heifets, Krinsky and Stupakov Low-gain model Heifets et al. model Sven Reiche - ICFA Sardinia

11. High-Gain Model Check for high-gain growth in a single dipole. Collective variables: Current modulation Energy modulation Differential equations: Linear in bunching 3rd order in energy modulation with Sven Reiche - ICFA Sardinia

12. Solution Dispersion equation for the ansatz B~exp[iLs] : • Dispersion equation has 4 roots corresponding to • 2 exponentially growing modes, • 2 exponentially decaying modes. The maximum growth rate is |Im(L1)|=(rcsr/R)sin(7p/24) and the characteristic length (gain length) of the instability is R/rcsr(for the FEL the characteristic length is 4plU/r). Sven Reiche - ICFA Sardinia

13. When Does ‘High-Gain’ Apply? The exponentional growth is limited by two effects: • Finite length of the dipole • Start-up lethargy Needs at least 4 gain lengths to show significant growth in modulation. Sven Reiche - ICFA Sardinia

14. Energy Spread With given expression for equations of motion, energy spread is difficult to incorporate (e.g.Vaslov equation). Qualitative Analysis: Energy spread is converted into phase spread as Phase spread is independent on modulation wavelength or bend radius in measures of the gain length. Estimate for high-gain threshold: Sven Reiche - ICFA Sardinia

15. Final Comparison Sven Reiche - ICFA Sardinia

16. Chicane Unknown FEL ~ 1 << 1 >> 1 Characteristic Parameter Conclusion Instabilities have same principle of interaction between electron beam and synchrotron radiation, but the signature is different for the different characteristic sizes of the devices. Presented model valid for special chicane layout (no drifts), but many results can qualitatively be applied to other cases. Sven Reiche - ICFA Sardinia