Fig. 1. Bunching factor versus energy modulation amplitude $A$ at various harmonic numbers. Each line corresponds to the maximum bunching factor at the optimal dispersion strength.

Fig. 2. Schematic layout of the self-modulation HGHG setup. A self-modulation HGHG includes extra dispersive chicane and self-modulator, further amplifying laser-induced energy modulation to obtain a higher harmonic bunching factor.

Fig. 3. The seed laser power ratio of the standard HGHG and self-modulation HGHG in different cases. The blue dot, red cross, and yellow circle correspond to the nominal case of beam size of $100\mu \mathrm{m}$ and peak current of 700 A, the second case of beam size of $50\mu \mathrm{m}$ and peak current of 700 A, and the third case of $100\mu \mathrm{m}$ and 1400 A, respectively. The dotted line corresponds to the scaling curve.

Fig. 4. The energy spread ratio of the standard HGHG and self-modulation HGHG in different cases. The blue dot, red cross, and yellow circle correspond to the nominal case of beam size of $100\mu \mathrm{m}$ and peak current of 700 A, the second case of beam size of $50\mu \mathrm{m}$ and peak current of 700 A, and the third case of $100\mu \mathrm{m}$ and 1400 A, respectively.

Fig. 5. Optimization for the $R_{56}$ of two chicanes by GENESIS simulations to obtain the 13th harmonic bunching factor of 8% and corresponding energy modulation amplitude $A_2$ of the entrance of the radiator in different beam sizes. (a), (b) Beam size of $100\mu \mathrm{m}$; (c), (d) beam size of $50\mu \mathrm{m}$.

Fig. 6. Comparison of the FEL performance between self-modulation HGHG (blue) and standard HGHG (red) at the 13th harmonic of the 266-nm seed laser in the cases of different beam sizes. (a), (b) Beam size of $100\mu \mathrm{m}$; (c), (d) beam size of $50\mu \mathrm{m}$.

Fig. 7. Optimization of the $R_{56}$ of two chicanes toward the 30th harmonic of the seed laser. The self-modulator resonates at (a), (b) the fundamental wavelength; (c), (d) the second; (e), (f) the third harmonics of the seed laser, respectively.

Fig. 8. The longitudinal phase space of the electron beam in one seed laser wavelength ${\lambda }_s$ at the entrance of the (a) self-modulator and (b) radiator, where the self-modulator is tuned at the third harmonic of the seed laser.

Fig. 9. The output FEL performance at the 30th harmonic of the seed laser in the third-harmonic self-modulation. (a) 8.87-nm radiation gain curve in the radiator. (b), (c) The power profile and spectrum after six radiator modules, respectively.

Fig. 10. Bunching factor after the second chicane as a function of the harmonic number in various cases, including without self-modulator and the resonance of the self-modulator tuned at the fundamental wavelength, second, and third harmonic of the seed laser, respectively.

Fig. 11. The typical setup of the SXFEL-TF adopted a cascaded EEHG–HGHG scheme. In the self-modulation experiment, modulator 1, with a period of 80 mm in the first stage EEHG, was used as the first modulator. Chicane 3 is the fresh bunch chicane used as the first chicane. A modulator of the second stage HGHG with a period of 55 mm was the self-modulator. Chicane 4 was regarded as the second chicane. X-band transverse deflection structure (XTDS) section was used to measure the longitudinal phase space of the electron beam.

Fig. 12. The measured intensity of the coherent radiation at various harmonic numbers in the first undulator segment of the radiator, under different $R_{56}$ values of the second chicane of (a) 0.038 mm and (b) 0.048 mm, respectively. The points represent the measurement results, and the curve represents the envelope obtained by smoothing the measurement data.