The realization of high-intensity coherent X-ray free-electron lasers (FELs) has been a longstanding aspiration among researchers and users in the field of accelerator-based light sources^[1^–7^]. In particular, the spectral range known as the water window, spanning from K-absorption edges of oxygen (2.3 nm) to carbon (4.4 nm), holds substantial promise for achieving high-contrast real-time imaging of biological organisms. The advent of an intense X-ray pulse within this water window region would enable in situ measurements of live cells, offering an opportunity for real-time analysis of cellular processes without significant damage to biological structures.

The potential to generate stable and coherent radiation at short wavelength is a notable advantage of external seeded FELs, which employ optical-scale manipulation of electron beam phase space. A typical operation mode of seeded FELs is high-gain harmonic generation (HGHG)^[8^–10^], which consists of two undulator sections separated by a dispersion section. The electron beam interacts with an external seed laser in the first undulator (modulator) to generate sinusoidal energy modulation, which converts into longitudinal density modulation (micro-bunching) after the dispersion chicane. Such an electron beam, containing frequency components at high harmonics of the seed laser, will generate coherent radiation at short wavelengths in the subsequent undulator (radiator). Nevertheless, the harmonic up-conversion efficiency of HGHG is constrained by the conflict requirements of the electron energy spread between harmonic multiplication and FEL amplification^[4^,11^,12^]. In 2020, FEL radiation at the 25th harmonic (10 nm) of the seed laser was observed at FERMI (Free Electron laser Radiation for Multidisciplinary Investigations)^[13^]. This could be the highest frequency up-conversion number achieved in single-stage HGHG.

To enhance the frequency multiplication efficiency of a seeded FEL while maintaining acceptable laser-induced energy spread, more complicated schemes such as echo-enabled harmonic generation (EEHG)^[14^,15^], phase-merging enhanced harmonic generation (PEHG)^[16^,17^], cascaded HGHG^[18^] and the echo-enabled harmonic cascade (EEHC)^[19^] have been developed. These advanced harmonic up-shifting methods extend the FEL wavelength to several nanometers. However, the aforementioned methods require specific machine configurations, which limit their implementation in existing facilities.

An alternative approach to reduce the FEL wavelength is harmonic lasing^[20^], which was initially proposed for FEL oscillators^[21^] and further explored experimentally in high-gain FELs^[22^]. Subsequently, this method was implemented and tested at the PAL-XFEL and European XFEL facilities, achieving FEL amplification at 1 nm^[23^] and 2.8 Å^[24^], respectively. These methods exhibit poor temporal coherence and intrinsic pulse energy fluctuations stemming from their reliance on self-amplified spontaneous emission (SASE)^[25^,26^]. The integration of harmonic lasing into seeded FELs shows potential for generating coherent ultra-high harmonic radiation, with two chief examples being the harmonic cascade EEHG^[27^] and the harmonic optical klystron (HOK)-based EEHG^[28^]. Apart from the increased complexity of these approaches, the preservation of the fine energy banding structure during harmonic lasing needs further theoretical and experimental validation.

This paper proposed a novel method for producing coherent X-ray FEL radiation within the water window region at single-stage HGHG. The approach originates from the concept of harmonic lasing self-seeded (HLSS) FELs^[22^,29^] and the superradiant cascade^[30^]. In this proposed method, the radiator is divided into three stages, wherein the second and third stages are tuned to resonate at the harmonics of the preceding one. Simulation results illustrate that the proposed technique has the capability to generate coherent 3 nm (90th harmonic) radiation pulses with peak power exceeding 1 GW using a 270 nm seed laser. Such a wavelength is unattainable with single-stage HGHG techniques and falls well within the water window region, enabling the real-time imaging and analysis of biological organisms in vitro. As an initial trial of the proposed method, a preliminary experiment was performed to assess the generation of both fundamental and harmonic lasing based on HGHG.

Harmonic lasing refers to the phenomenon where higher harmonics of the planar undulator are generated and amplified independently from the fundamental radiation. The theory of high-gain harmonic lasing has been intensively studied in Refs. [20,31–33] for odd harmonics and Refs. [34,35] for even harmonics. For the higher odd harmonics, the scaling function of the gain length can be written as follows^[31^,36^]:

$$\begin{array}{r}\frac{\mathrm{Im}\left({\mu }_n\right)}{D_n}=\frac{1}{2k_{\mathrm{w}}{L}_{\mathrm{gn}}{D}_n}:=G(k_{\mathrm{s}}\epsilon ,\frac{{\sigma }_{\gamma }}{D_n},\frac{k_{\beta }}{k_{\mathrm{w}}{D}_n},\frac{\omega -{\omega }_{\mathrm{r}}}{{\omega }_{\mathrm{r}}{D}_n}).\end{array}$$  (1)

Here, ${\mu}_n$ can be derived from the following:

$$\begin{array}{rl}& {\mu }_n{a}^2(1-e^{-{\chi }_n})-\frac{{\chi }_n}{n}[1-(1-{\chi }_n)e^{-{\chi }_n}]\\ & ={\int }_{-\mathrm{\infty }}^0\exp [-i(\frac{{\mu }_n}{D_{\mathrm{u}}}+\frac{\omega -{\omega }_{\mathrm{r}}}{{\omega }_{\mathrm{r}}{D}_n})s-2{\left(\frac{{\sigma }_{\mathrm{r}}}{D_n}\right)}^2{s}^2]\\ & \times (\frac{1-e^{-{\eta }_{+}}}{{\eta }_{+}}-\frac{1-e^{-{\eta }_{-}}}{{\eta }_{-}})\frac{s\mathrm{d}s}{\cos \left(k_{\mathrm{s}}/D_n\right)},\end{array}$$  (2)

$$\begin{array}{rl}{\eta }_{\pm }& =3is\left(\frac{k_{\beta }/k_{\mathrm{w}}}{D_n}\right)\left(k_{\mathrm{s}}\epsilon \right)+\frac{{\chi }_n}{2}[1\mp \cos \left(\frac{k_{\beta }/k_{\mathrm{w}}}{D_n}s\right)],\\ D_n& ={\left(\frac{2Z_{0}e}{\pi {mc}^2}\frac{I_0}{{\gamma }_0}\frac{K^2}{1+K^2/2}\right)}^{1/2}{\left[\mathrm{JJ}\right]}_n,\\ {\chi }_n& =-a\sqrt{{\mu }_n}\frac{H_0^{{}^{\prime }(1)}\left(a\sqrt{{\mu }_n}\right)}{H_0^{(1)}\left(a\sqrt{{\mu }_n}\right)},\end{array}$$  (3)

$$\begin{array}{r}\frac{\omega -{\omega }_{\mathrm{r}}}{\omega D_n}=-3\left(\frac{k_{\beta }}{k_{\mathrm{w}}{D}_n}\right)k_{\mathrm{s}}\epsilon .\end{array}$$  (4)

The scaling function $G$ only depends on four dimensionless scaled variables, which characterize the influence of emittance, energy spread, focusing of the electron beam and the diffraction and guiding of the radiation, respectively. Although obtaining an analytic solution is quite challenging, the scaling function of the gain length can be numerically determined from Equations (2)–(4). The results are illustrated in Figure 1 for both fundamental radiation and third harmonic lasing under the conditions of ${\sigma}_{\gamma }/{D}_n=0.1$ . For ease of notation, let $\Lambda ={k}_{\beta }/\left({k}_{\mathrm{w}}{D}_n\right)$ .

Under these conditions, it is evident that harmonic radiation shows a shorter gain length compared to fundamental radiation at small scaled emittance ( $4\pi \varepsilon /\lambda \ll 2$ ). This difference increases as scaled emittance decreases. Therefore, the fundamental radiation in the undulator can be maintained well below saturation to prevent nonlinear harmonic generation^[37^]. This provides the opportunity to perform multi-stage harmonic cascade.

The schematic layout of the proposed scheme is sketched in Figure 2, comprising a short modulator resonant at the seed laser wavelength ${\lambda}_{\mathrm{seed}}$ , a dispersion chicane and three stages of undulators. Each stage is tuned to the harmonic of the preceding one.

A 270 nm seed laser imprints a sinusoidal energy modulation onto the electron beam in the modulator. This energy modulation is converted to a density modulation with relatively low-order harmonic components (i.e., 10th harmonic). In the first stage of the undulator, the micro-bunched electron beam generates coherent radiation at wavelength of ${\lambda}_1$ . Here, harmonic lasing occurs within the exponential gain regime, while the fundamental radiation remains well below saturation. This harmonic lasing then serves as the seed in the second stage, which is tuned to a wavelength of ${\lambda}_1/{h}_2$ . The amplification process of harmonic lasing, similar to that in the first stage, repeats in the second stage. In the last stage, the fundamental is resonant at ${\lambda}_3$ , which is amplified as the $\left({h}_2\times {h}_3\right)\mathrm{th}$ harmonic in the first stage and the ${h}_3\mathrm{th}$ harmonic in the second stage.

To explore the feasibility of the proposed multi-stage harmonic cascade, a simulation was carried out utilizing the parameters listed in Table 1. The electron beam energy is 2.5 GeV with a relative energy spread of $8\times {10}^{-5}$ . The normalized emittance is 0.4 mm $\cdot$ mrad and the peak current is 800 A. A 270 nm seed laser with peak power of about 100 MW interacts with this electron beam in a 2-m-long modulator. The induced dimensionless energy modulation amplitude is approximately 5, which is sufficient to initiate coherent radiation at the 10th harmonic for HGHG. The dispersion strength ${R}_{56}$ is set to about 0.11 mm, which is slightly smaller than the optimal calculation^[38^]. The radiator comprises six 4-m-long variable-gap undulators with a period length of 50 mm. These undulators are grouped into three stages, resonating at wavelengths of 27 nm ( ${h}_1=10$ ), 9 nm ( ${h}_2=3$ ) and 3 nm ( ${h}_3=3$ ), respectively.

The simulations were performed with GENESIS4^[39^]. The bunching factor at the 10th harmonic of the seed is around $2.5\%$ at the entrance of the radiator. The determination of lengths for the first and second stages entails a trade-off between the energy spread of the electron beam and the bunching factor at the third harmonic. To ensure a sufficient third harmonic bunching factor while mitigating an excessive increase in energy spread, both the first and second stages are designed with only one undulator. Figure 3 depicts the bunching factor distribution at wavelengths of 27 nm (red), 9 nm (yellow) and 3 nm (blue) after the first and second stages. The third harmonic bunching factors after these stages are approximately $7.5\%$ and $2.6\%$ , respectively. These bunching factors are sufficiently large to suppress shot noise and promote the generation of temporally coherent radiation in subsequent stages.

Figure 4 illustrates the temporal evolution of energy spread along the radiator, as well as the corresponding distributions at the entrances of the first (z = 0 m), second (z = 5 m) and third (z = 10 m) stages. The relative weighted energy spreads at the entrances of each stage are $2.24\times {10}^{-4}$ , $2.96\times {10}^{-4}$ and $4.11\times {10}^{-4}$ , respectively, all of which are smaller than the Pierce parameter $\rho$ ( $\sim 1.69\times {10}^{-3}$ ). The small contribution to the final energy spread ( $\sim 7.21\times {10}^{-4}$ ) in the first two stages suggests that the fundamental modes remain notably below saturation during these stages.

The weighted bunching factors and pulse energies at 27 nm (red), 9 nm (yellow) and 3 nm (blue) along the radiator are depicted in Figure 5. The root mean square (RMS) undulator parameters, indicated in gray shading, have been carefully chosen to enhance the FEL lasing at 3 nm. Specifically, the optimized undulator gaps are about 9.35, 16.24 and 25.47 mm, respectively.

With the optimal undulator parameters, the detailed power profiles and spectra of FEL pulses at 27, 9 and 3 nm are displayed in Figure 6. The pulse energy at 3 nm reaches approximately 88.52 μJ, significantly surpassing the pulse energies obtained at 27 nm (10.18 μJ) and 9 nm (7.96 μJ). The peak power at 3 nm approaches nearly 1.5 GW, providing ample capability for performing experiments within the ‘water window’ region^[1^]. The relative bandwidths ( $\Delta \lambda /\lambda$ ) at different wavelengths are $8.31\times {10}^{-4}$ , $3.75\times {10}^{-4}$ and $4.56\times {10}^{-4}$ , respectively. This value indicates a considerably narrower bandwidth compared to that typically achieved with HLSS FELs, implying a superior temporal coherence.

One critical aspect of this approach is the formation of third harmonic bunching in the first two stages, which could be deteriorated by nonlinear effects such as longitudinal space charge (LSC) and intra-beam scattering (IBS). After the dispersion chicane, the energy modulation is converted into a density modulation, resulting in a peak current less than 2 kA in the central spike. This value is significantly lower than those used in current-enhanced schemes, where peak currents typically exceed 10 kA^[40^–43^]. The maximum LSC-induced energy loss is approximately ±70 keV/m. The phase space evolution resulting from FEL lasing would predominate over the alterations induced by this level of LSC (see more details in Appendix A). The IBS describes multiple Coulomb scattering in the electron beam, which leads to an increase in beam size and energy spread^[44^–46^]. In our case, the total energy spread induced by IBS is around 28.57 keV, which is significantly smaller than the energy spread depicted in Figure 4, rendering it negligible. In addition, separate simulations have been conducted with a 1.5-fold increase in normalized emittance and an increase in energy spread to $1\times {10}^{-4}$ . These modifications result in a reduction of the 3 nm FEL pulse energies to approximately 51.94 and 48.65 μJ, respectively. These findings also demonstrate the proposed scheme’s tolerance to variations in electron beam parameters.

Another important consideration is the robustness of the proposed scheme. Fluctuations in the electron beam energy cause deviations in the resonance condition, leading to variations in the intensities of the third harmonic during these two stages. Consequently, these fluctuations exert a substantial influence on the overall FEL performance in the proposed method. To validate the stability of this approach, we conducted multi-shot simulations based on the proposed scheme, with the relative deviation in electron beam energy ( ${\sigma}_{\mathrm{E}}/E$ ) set at 0.01%. Figure 7 displays the simulated power profiles and spectra, together with the corresponding pulse energies and bandwidths. The average pulse energies reach approximately 80.44 μJ with an RMS jitter of 7.17%. The analysis of these spectra shows an average bandwidth (full width at half maximum, FWHM) of $4.54\times {10}^{-4}$ and central wavelength jitter (RMS) of $9.40\times {10}^{-5}$ . Based on the simulation results, it is evident that the reduction in pulse energy remains well within acceptable limits, exhibiting an RMS jitter of less than 10%. In addition, the bandwidth of the radiation pulse will broaden, while maintaining a stable central wavelength of 3 nm. Such fluctuations in energy and spectral characteristics are generally considered acceptable during user experiments.

Besides simulations conducted with an ideal electron beam, a start-to-end simulation was performed using the specific parameters at the newly proposed Shenzhen Superconducting Soft X-Ray Free-Electron Laser (S³FEL)^[47^], which is about to begin with civil construction.

The longitudinal phase space and current distribution of the electron beam before modulation are shown in Figure 8. The seed laser power is around 80 MW with a pulse duration (FWHM) of 100 fs. After an optimized dispersion chicane ( ${R}_{56}\approx 0.12\;\mathrm{mm}$ ), the bunching factor at the 10th harmonic of the seed laser achieves 2.3%. The radiator is designed similarly to that employed in ideal beam simulations, with minor adjustments made to the RMS undulator parameters.

Figure 9 depicts the evolution of pulse energy along the radiator, accompanied by the final power profile and spectrum of the 3 nm FEL pulse. Such a simulated pulse, characterized by an energy of around 74.35 μJ and a narrow relative bandwidth (FWHM) of $2.71\times {10}^{-4}$ , holds considerable promise for experimental applications within the water window region due to its sufficient pulse energy and high spectral purity. Furthermore, the increase of radiation power at 3 nm can be attained by implementing reverse taper configurations in the first two stages^[27^].

Through the proposed multi-stage harmonic cascade approach, coherent radiation with pulse energy exceeding 70 μJ and a central wavelength falling within the water window region can be directly generated based on single-stage HGHG. A preliminary experimental investigation of the harmonic cascade was conducted at the Dalian Coherent Light Source (DCLS)^[48^] to evaluate the generation of both fundamental and harmonic lasing through single-stage HGHG (see Appendix B). It is important to emphasize that the proposed method, although not universally applicable across all FEL facilities due to its strong reliance on the characteristics of the electron beam, presents a promising pathway towards achieving ultra-high harmonic conversion. This enriches the practical alternatives available for conducting relevant experiments.

By adjusting the bunching factor after the dispersion chicane and controlling the undulator taper, it might become feasible to tailor the pulse energy at different wavelengths. This capability facilitates the simultaneous generation of multi-color pulses (e.g., 27, 9 and 3 nm) with comparable intensities^[49^,50^]. The relative intensities of these pulses could be potentially manipulated with the help of phase shifters^[51^]. Furthermore, through the implementation of an EEHG-based multi-stage harmonic cascade, it is possible to significantly extend the wavelength of coherent FEL radiation into the angstrom regime, which exhibits considerable potential for investigating ultrafine atomic and molecular structures with unprecedented resolution and precision.