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Free Electron Lasers Lecture II

Free Electron Lasers Lecture II. Brian M c Neil, University of Strathclyde, Glasgow, Scotland. Electron bunching in a fixed radiation field. The electron-radiation interaction. The Lorentz force (electron dynamics). Maxwell wave equation* (radiation evolution).

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Free Electron Lasers Lecture II

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  1. Free Electron Lasers Lecture II Brian McNeil, University of Strathclyde, Glasgow, Scotland.

  2. Electron bunching in a fixed radiation field

  3. The electron-radiation interaction The Lorentz force (electron dynamics) Maxwell wave equation* (radiation evolution) Both equations must be solved together simultaneously (self-consistently) to fully describe the FEL interaction *Neglect static fields (space charge effects) – Compton limit

  4. Can calculate How the electron is effected by the resonant radiation The Lorentz Force Equation: Hendrick Antoon Lorentz The rate of change of electron energy

  5. Slow energy exchange The rate of change of electron energy: Consider plane-wave field: Assuming: Interacting with an electron on trajectory:

  6. Slow energy exchange The first sin term on RHS is a wave with speed in z direction of: Recall previous result for resonance: So, a resonant electron with average speed will have The second sin term on RHS is a wave with speed in z direction of:

  7. Resonant emission – electron energy change is +ve Energy of electron changes ‘slowly’ when interacting with a resonant radiation field. e- e- u

  8. is +ve is -ve Resonant emission – electron energy change Rate of electron energy change is ‘slow’ but changes periodically with respect to the radiation phase e- e- u

  9. is +ve Lose energy Gain energy Resonant emission – electron bunching Electrons bunch at resonant radiation wavelength – coherent process Axial electron velocity r

  10. Bunched electrons can exchange energy coherently with radiation If then the 2nd term >> 1st term as there are N 2 of them and results in coherent emission.

  11. Electron bunching in a self-consistent radiation field

  12. Basic FEL mechanism Radiation field bunches electrons Bunched electrons drive radiation

  13. Basic FEL mechanism Radiation field bunches electrons Bunched electrons drive radiation These equations are assumed ‘slowly varying’ i.e. any evolution is assumed slow with respect to the radiation/undulator period. They can be subsequently averaged over a radiation/undulator period.

  14. e- Conventional laser Vs FEL pulses Active medium Conventional laser pulse interacts with all of the active medium Partial form of wave equation describes slippage of radiation envelope through the electron pulse Undulator FEL radiation pulse interacts with only a section of the active medium

  15. Steady-state approx.: “No pulses” Linear analysis Assume that: Using: The steady-state approximation can be thought of as the continuous e-beam limit where the electron ‘pulse’ has no beginning or end. In this case one can see that the radiation field can only be a function of the distance through the undulator and no pulse effects can be present. Where:

  16. Linear analysis First assume resonance: Away from resonance: Differentiating linear equations: the dispersion relation is: 1 Real 2 Complex conjugate 3 Real

  17. Linear analysis Real parts give oscillatory solutions. Imaginary parts give exponential growth: and exponential decay:

  18. Gain as a function of detuning from resonance Oscillatory terms dominate Oscillatory & exponential +ve exponential term dominates

  19. Linear Numerical Constants of motion Two constants of motion can be obtained from these equations in the steady-state limit: Where the constant is the variables’ initial values. The first constant above corresponds to conservation of energy. The second, incorporating phase dependent terms is related to the Hamiltonian of the system. Opposite is plotted the linear and non-linear (numerical) solutions of the equations for a resonant interaction (δ= 0). From the definition of : and the saturated scaled field|Asat|~1, it is seen that ρis a measure of the efficiency if the interaction.

  20. The electrons can be thought of as a collection of pendula initially distributed over a range of angles with respect to the vertical. The radiation field is analogous to the gravitational field. The separartrix defines the boundary between pendula that librate and rotate. Of course in the FEL equations above, unlike a gravitational field, the radiation field can evolve in both amplitude a, and phase . The pendulum equation and phase-space separatrix

  21. Letting: 0 Can assume periodic BC over one potential well: F 1) e- begin to bunch about θ=3π/2 2) Radiation phase driven and shifts 3) Radiation amplitude is driven

  22. Letting: 0 Can assume periodic BC over one potential well: F 1) e- begin to bunch about θ=3π/2 2) Radiation phase driven and shifts 3) Radiation amplitude is driven

  23. FEL pulse effects

  24. z0 z0 FEL pulses starting from noise in a High-Gain amplifier (SASE) vz=c vz=c e- e- vz<c vz<c z1= z0+(c-vz)t Many regions of radiation pulse evolve independently from other regions

  25. Self Amplified Spontaneous Emission (SASE) SASE Power output: SASE spectrum:

  26. Seeded FEL e- e- Longitudinal coherence of radiation pulse is inhereted from that of seed if Pseed>>Pnoise

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