Figure 1

Schematic of the seeded FEL initial parameters. $\overline{z}$ is the longitudinal coordinate along the undulator. $L_e$ is the electron bunch length and $\delta$ the slippage.Reuse & Permissions

Figure 2

Evolution of the FEL longitudinal normalized optical field intensity $|A{|}^2(\tau )$ along the undulator according to Eq. (1). Seed parameters: (a) $S_{\mathrm{seed}}=0.25$ (with hyper-Gaussian shape, ${\sigma }_{\mathrm{seed}}=40$), (b) $S_{\mathrm{seed}}=10$ (with Gaussian shape, ${\sigma }_{\mathrm{seed}}=1$), and (c) $S_{\mathrm{seed}}=2$ (with Gaussian shape, ${\sigma }_{\mathrm{seed}}=5$), (continuous line) calculation of ${\overline{z}}_{\mathrm{sat}}$ according to Eq. (5). (d)–(h) Phase space of the electron beam at locations indicated on (a)–(c). Maximum seed field amplitude at $\overline{z}=0$: $A_0=0.15$. Electron bunch discretized along the $\tau$ axis. Phase space consisting of $200\times 200$ particles associated to each $\tau$ coordinate. Initially: ${\theta }_j$ uniformly distributed within $[-\pi ,+\pi ]$ and $p_j$ following a normal distribution centered around zero with standard deviation 0.2%-RMS. Electron beam profile: flat top using super Gaussian of parameter $m=4$ and $L_e=40$, corresponding to $S_e=0.25$.Reuse & Permissions

Figure 3

Evolution of the FEL pulse intensity along the undulator calculated with Eq. (1). Intensity integrated over the whole pulse and normalized with the value at $z=0$. (a),(b),(c) plots correspond to the (a),(b),(c) cases of Fig. 2.Reuse & Permissions

Figure 4

Local gain and optical field intensity along the undulator. $|A{|}^2(\overline{z},\tau =10)$ line and $|A{|}^2(\overline{z},\tau =20)$ bold line calculated with Eq. (1). $G(\overline{z},\tau =10)$ dashed and $G(\overline{z},\tau =20)$ bold dashed line. Electron beam shape: gate with $L_e=40$. $-5<\tau <50$. Seed shape: Gaussian with ${\sigma }_{\mathrm{seed}}=10$, $|A|(\overline{z}=0,\tau =20)=0.15$, $|A|(\overline{z}=0,\tau =0)=0.0025$.Reuse & Permissions