Characterization of soft x-ray echo-enabled harmonic generation free-electron laser pulses in the presence of incoherent electron beam energy modulations

Echo-enabled harmonic generation free-electron lasers (EEHG FELs) are promising candidates to produce fully coherent soft x-ray pulses by virtue of efficient high-harmonic frequency up-conversion from ultraviolet lasers. The ultimate spectral limit of EEHG, however, remains unclear, because of the broadening and distortions induced in the output spectrum by residual broadband energy modulations in the electron beam. We present a mathematical description of the impact of incoherent (broadband) energy modulations on the bunching spectrum produced by the microbunching instability through both the accelerator and the EEHG line. The model is in agreement with a systematic experimental characterization of the FERMI EEHG FEL in the photon energy range 130–210 eV. We find that amplification of electron beam energy distortions primarily in the EEHG dispersive sections explains an observed reduction of the FEL spectral brightness proportional to the EEHG harmonic number. Local maxima of the FEL spectral brightness and of the spectral stability are found for a suitable balance of the dispersive sections’ strength and the first seed laser pulse energy. Such characterization provides a benchmark for user experiments and future EEHG implementations designed to reach shorter wavelengths.

Figure 1

Main components of the EEHG scheme: first modulator (M1), strong first dispersive section (DS1), second modulator (M2), and weaker second dispersive section (DS2). Equations refer to quantities of the electron beam longitudinal phase space introduced in Eq. (1). In particular, $\mathrm{\Delta }{p}_{1,2}$ is the incoherent energy modulation [see Eq. (2)] and $b_{\mu 1,\mu 2}$ is the incoherent bunching factor [see Eq. (13)]. After DS2, the nano-bunched electron beam travels into the radiator (R) and emits coherent and powerful light pulse.

Figure 2

Left: Relative FEL rms spectral bandwidth vs LH-induced energy spread. Blue and red lines are experimental data (solid) and theoretical prediction [dashed-dotted, Eq. (8)] for $n=-1$ and $n=-2$, respectively. EEHG is tuned at ${\lambda }_{\mathrm{FEL}}=8.8\mathrm{nm}$. The dashed green line is from Eq. (6). In case of $n=-1$, the first seed energy is $8.3\mu \mathrm{J}$ ($A_1{\sigma }_E=0.88\mathrm{MeV}$) and for $n=-2$, it is $20\mu \mathrm{J}$ ($A_1{\sigma }_E=1.37\mathrm{MeV}$). Right: calculated energy modulation amplitude $p_1({\lambda }_{\mu })$ and $p_2({\lambda }_{\mu })$ from MBI modeling for ${\sigma }_{\mathrm{LH}}=20\mathrm{keV}$ (top) and 40 keV (bottom).

Figure 3

Comparison of standard deviation of frequency fluctuations of 50 shots $n=-1$ (blue) and 100 shots $n=-2$ (red) configurations in EEHG experiment respect to the different induced LH energy spread. EEHG harmonic is 30 (${\lambda }_{\mathrm{FEL}}=8.8\mathrm{nm}$). The FEL parameters are same as in Fig. 2.

Figure 4

Top: FEL intensity vs LH-induced energy spread. The FEL parameters are same as in Fig. 2. Bottom: square bunching factor. The calculated value of $p_1({\lambda }_{\mu })$ and $p_2({\lambda }_{\mu })$ in Fig. 2 are used to evaluate bunching factor.

Figure 5

Left: FEL intensity vs first seed laser pulse energy. Blue and red lines are experimental data for LH-induced energy spread ${\sigma }_{\mathrm{LH}}=24\mathrm{keV}$ and ${\sigma }_{\mathrm{LH}}=16\mathrm{keV}$. Right-top row: beam energy modulation amplitude as function of the (compressed) MBI modulation wavelength and first seed coherent energy modulation amplitude (seed energy), at the exit of the second modulator. Right-bottom row: bunching factor and product function vs first seed coherent energy modulation amplitude. Dotted lines with circles show ${\stackrel{`}{b}}_{-2,20}$, dashed lines with stars show $b_{-2,20}$ in Eq. (11). The two subplots refer to the LH inducing 24 keV and 16 keV rms energy spread, respectively.