1. Sven Reiche UCLA ICFA-Workshop - Sardinia 07/02
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. Sven Reiche - ICFA Sardinia
The Analogy
A Typical FEL Beamline
Gun
Linac
Chicane
Linac
Undulator
Trajectory
Instability I
CSR
SASE FEL
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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
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4. The FEL Model (1D) FEL equations Universal scaling parameter
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
Universal scaling parameter
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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.
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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.
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7. The Motion in a Chicane (1D)
The CSR potential:
field
trajectory
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
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8. Sven Reiche - ICFA Sardinia
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
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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
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10. Sven Reiche - ICFA Sardinia
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
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11. Sven Reiche - ICFA Sardinia
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
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12. Sven Reiche - ICFA Sardinia
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).
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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.
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14. Sven Reiche - ICFA Sardinia
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:
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15. Final Comparison FEL Magnet Chicane Modes (1D) 3 4
Scaling Parameter ( r )
<< 1
>> 1
(low gain > 1)
Frequency Band
Narrow
Wide
Approximation
Resonance Approximation
Wake Potential
Electron Motion
Averaged
Explicit
Radiation Field
Continuous Overlap
Short Overlap
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16. Sven Reiche - ICFA Sardinia
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.
Characteristic Parameter
Chicane
FEL
Unknown
>> 1
~ 1
<< 1
Presented model valid for special chicane layout (no drifts), but many results can qualitatively be applied to other cases.
Sven Reiche - ICFA Sardinia