By S. Varro

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It governs the linear (quadratic) path-length dependence from the particle energy. In the following, we choose a longitudinal coordinate system such that the head of the bunch is at z<0. A chicane has R56 < 0 and T566 > 0 with this convention. A linac with energy gain eV sin φ (V and φ are the RF accelerating peak voltage and phase, respectively) imparts to the beam the following linear and quadratic energy chirp, ≈ δ0,u + hz0 + h z20 , with: δ = EΔE BC h= h = 1 dE EBC dz 1 dh 2 dz = =− 2π eV cos φ λ RF E0 +eV sin φ 2 2π λ RF (13) eV sin φ E0 +eV sin φ E0 and EBC are the beam mean energy at the entrance and at the exit of the linac, respectively.

1  31 c An important aspect of the FEL dynamics is the non-linear harmonic generation, resulting from the bunching mechanism, which itself is a consequence of the energy modulation mechanism (see Fig. 25). When the electrons are micro-bunched at a wavelength which corresponds to a submultiple of the fundamental wavelength, coherent emission is produced at higher order harmonics, throughout the mechanism of nonlinear harmonic generation. Fig. 25. Bunching mechanism induced by the FEL interaction.

The fundamental wavelength emitted is proportional to λu /γ2 , where λu is the undulator period, typically a few centimeters long, and γ is the relativistic Lorentz factor of the electrons, which typically reaches several thousand for X-ray emission. The main ﬁgures of merit of an FEL are extremely high brilliance, close to full transverse and longitudinal coherence, a bandwidth approaching the Fourier limit and a stable and well characterized temporal structure in the femtosecond time domain. We can identify two general ways to generate X-rays with an FEL.