Fig. 1. Scaling function of the gain length for the fundamental (dashed line) and third harmonic (solid line) for (red) and (blue), corresponding to scaled energy spread with optimal detuning.

Fig. 2. Schematic layout of the multi-stage harmonic cascade based on HGHG. The yellow and blue lines correspond to the fundamental FEL pulses of the first and second stages of the radiator, respectively. The purple line represents the FEL pulse of the desired wavelength, denoted as , which is amplified throughout the entire radiator. Each stage of the undulator is tuned to the subharmonic of the next stage.

Fig. 3. The distributions of electron beam bunching factor at wavelengths of 27 nm (red), 9 nm (yellow) and 3 nm (blue) after the first (a) and second (b) stages, respectively.

Fig. 4. The temporal evolution of energy spread along the radiator (right) and its distribution at the entrance of the first, second and third stages, respectively (left).

Fig. 5. The evolution of the weighted bunching factor (a) and pulse energy (b) at wavelengths of 27 nm (red), 9 nm (yellow) and 3 nm (blue) along the radiator. The shaded regions denote the RMS undulator parameters.

Fig. 6. The power profiles and spectra of FEL pulses emitting at wavelengths of 27 nm (a), 9 nm (b) and 3 nm (c).

Fig. 7. The power profiles (left) and spectra (right) of 100 FEL shots under the condition of . The pulse energies and spectrum bandwidths (), as well as their statistical information, are also depicted.

Fig. 8. The longitudinal phase space and current distribution of the electron beam.

Fig. 9. The start-to-end simulation results of the multi-stage harmonic cascade are depicted. Panel (a) illustrates the evolution of pulse energy along the radiator. Panel (b) presents the power profile and spectrum of the FEL pulse at 3 nm.