Relativistic x-ray free-electron lasers in the quantum regime

We present a nonlinear theory for relativistic x-ray free-electron lasers in the quantum regime, using a collective Klein-Gordon (KG) equation (for relativistic electrons), which is coupled with the Maxwell-Poisson equations for the electromagnetic and electrostatic fields. In our model, an intense electromagnetic wave is used as a wiggler which interacts with a relativistic electron beam to produce coherent tunable radiation. The KG-Maxwell-Poisson model is used to derive a general nonlinear dispersion relation for parametric instabilities in three space dimensions, including an arbitrarily large amplitude electromagnetic wiggler field. The nonlinear dispersion relation reveals the importance of quantum recoil effects and oblique scattering of the radiation that can be tuned by varying the beam energy.

Figure 1

Schematic picture of different regimes for the FEL instability, showing the quantum regime $k_0>k_{0,\text{crit}}$, the Raman regime $k_0<k_{0,\text{crit}}$, and the Compton regime $a_0>a_{\text{crit}}$.

Figure 2

The maximum growth rate as a function of $a_0$ for $K_{\perp }=0$ and $\gamma =5$, using the full dispersion relation (8) (solid curve), and the approximations (14, 15, 16) (dashed, dashed-dotted, and dotted curves, respectively). The vertical dotted line indicates $a_0=a_{\text{crit}}=0.038$.

Figure 3

Resonant parallel and perpendicular radiation wave numbers $k_{z-}$ and $k_{\perp -}$ for $\gamma =5$ and $\gamma =36$ (top panels) with the corresponding growth rate $\Gamma \left({\mathrm{s}}^{-1}\right)$ shown in color (grayscale). The bottom panels show the growth rate as a function of $k_{z-}$. (The scalings of the vertical axes are enhanced in the top panels.) The vertical bars show the location of the largest resonant wave numbers $\approx -4k_0{\gamma }^2/{\gamma }_A^2$.