Free Electron Lasers

Synchrotron radiation is emitted when electromagnetic fields exert a force on a charged particle. This opens the possibility to apply external fields with specific properties for the stimulation of electrons to emit more radiation.

Unperturbed Electron motion in helical wiggler (in the absence of radiation field)

Energy change of electrons due to radiation field
Consider a circularly polarized electromagnetic wave (plane wave is an assumption for 1D analysis, which is usually valid for near axis analysis) propogating along z direction Energy change of an electron is given by To the leading order, electrons move with constant velocity and hence z = v z t − t 0 Detuning parameter: E 0 is the average energy of the beam.
Energy deviation:

Low Gain Regime: Pendulum Equation
We assume that the change of the amplitude of the radiation field, E, is negligible and treat it as a constant over the whole interaction.
Low Gain Regime: Similarity to Synchrotron Oscillation is the angle between the transverse velocity vector and the radiation field vector and hence there is no energy kick for ψ ψ = π / 2 Synchrotron Oscillation FEL 0 −π π Low Gain Regime: Qualitative Observation *Plots are taken from talk slides by Peter Schmuser.

Energy deviation
Energy deviation The average energy of the electrons is right at resonant energy: The average energy of the electrons is slightly above the resonant energy: With positive detuning, there is net energy loss by electrons.

Low Gain Regime: Derivation of FEL Gain
Change in radiation power density (energy gain per seconds per unit area):

Energy deviation at entrance
Pondermotive phase at entrance Low Gain Regime: Derivation of FEL Gain Assuming that all electrons have the same energy and uniformly distributed in the Pondermotive phase at the entrance of FEL: and .
Inserting the zeroth order solution back into eq. (1) yields the 1 st order solution: (1) The zeroth order solution for phase evolution is given by ignoring the effects from FEL interaction:   The gain is defined as the relative growth in radiation power: Growth in the amplitude of radiation field:

Cubic in FEL length
As observed earlier, there is no gain if the electrons has resonant energy.
1. Energy kick from radiation field + dispersion/drift -> electron density bunching; 2. Electron density bunching makes more electrons radiates coherently -> higher radiation field; 3. Higher radiation fields leads to more density bunching through 1 and hence closes the positive feedback loop -> FEL instability.
The positive feedback loop between radiation field and electron density bunching is the underlying mechanism of high gain FEL regime.
We seek the solution for vector potential of the form: Inserting eq. (2) and (3) into eq. (1) yields 2. Ignoring second derivative by assuming that the variation of is negligible over the optical wave length.
Multiplying both sides by e ik w z and neglecting terms proportional to e ik w z−ik z−ct ( ) since they will change fast over the FEL (same as the helicity argument).

Wave Equation
In order to relate the vector potential to the electric field, we use the Maxwell equation: After neglecting the fast oscillation terms, we get the following relation between the current perturbation and the vector potential of the radiation field: Finally, the relation between the radiatio field and the current modulation is obtained: is normalized longitudinal location along wiggler, is the 1-D Gain rate parameter is called Alfven current

1-D High Gain FEL Equation for Cold Beam and
Zero Detuning Can we reproduce the previously obtained low gain solution by taking the proper limit of the high gain solution?
The high gain solution indeed give identical solution when the undulator is shorter than the gain length. But it also tell us what happens if the undulator is long and hence it is more general than the low gain solution.

Bandwidth at High Gain Limit I
Zeroth order equation: It is sometimes hard to extract insights from the exact solution of the 3 rd order polynomial equation for the eigenvalue. Therefore, it is useful to get the approximate solution which is simpler but gives accurate results for the region that we are interested in.ˆ1