1 Introduction
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[ Reference Ackermann, Asova, Ayvazyan, Baboi, Bähr, Balandin, Beutner, Brandt, Bolzmann, Brinkmann, Brovko, Castellano, Castro, Catani, Chiadroni, Choroba, Cianchi, Costello, Cubaynes, Dardis, Decking, Delsim-Hashemi, Delserieys, Di Pirro, Dohlus, Düsterer, Eckhardt, Edwards, Faatz, Feldhaus, Flöttmann, Frisch, Fröhlich, Garvey, Gensch, Gerth, Görler, Golubeva, Grabosch, Grecki, Grimm, Hacker, Hahn, Han, Honkavaara, Hott, Hüning, Ivanisenko, Jaeschke, Jalmuzna, Jezynski, Kammering, Katalev, Kavanagh, Kennedy, Khodyachykh, Klose, Kocharyan, Körfer, Kollewe, Koprek, Korepanov, Kostin, Krassilnikov, Kube, Kuhlmann, Lewis, Lilje, Limberg, Lipka, Löhl, Luna, Luong, Martins, Meyer, Michelato, Miltchev, Möller, Monaco, Müller, Napieralski, Napoly, Nicolosi, Nölle, Nuñez, Oppelt, Pagani, Paparella, Pchalek, Pedregosa-Gutierrez, Petersen, Petrosyan, Petrosyan, Petrosyan, Pflüger, Plönjes, Poletto, Pozniak, Prat, Proch, Pucyk, Radcliffe, Redlin, Rehlich, Richter, Roehrs, Roensch, Romaniuk, Ross, Rossbach, Rybnikov, Sachwitz, Saldin, Sandner, Schlarb, Schmidt, Schmitz, Schmüser, Schneider, Schneidmiller, Schnepp, Schreiber, Seidel, Sertore, Shabunov, Simon, Simrock, Sombrowski, Sorokin, Spanknebel, Spesyvtsev, Staykov, Steffen, Stephan, Stulle, Thom, Tiedtke, Tischer, Toleikis, Treusch, Trines, Tsakov, Vogel, Weiland, Weise, Wellhöfer, Wendt, Will, Winter, Wittenburg, Wurth, Yeates, Yurkov, Zagorodnov and Zapfe1– Reference Zhang, Zeng, Tian and Li7]. 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)[ Reference Yu8– Reference Yu, DiMauro, Doyuran, Graves, Johnson, Heese, Krinsky, Loos, Murphy, Rakowsky, Rose, Shaftan, Sheehy, Skaritka, Wang and Wu10], 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[ Reference Allaria, Appio, Badano, Barletta, Bassanese, Biedron, Borga, Busetto, Castronovo, Cinquegrana, Cleva, Cocco, Cornacchia, Craievich, Cudin, D’Auria, Forno, Danailov, De Monte, De Ninno, Delgiusto, Demidovich, Di Mitri, Diviacco, Fabris, Fabris, Fawley, Ferianis, Ferrari, Ferry, Froehlich, Furlan, Gaio, Gelmetti, Giannessi, Giannini, Gobessi, Ivanov, Karantzoulis, Lonza, Lutman, Mahieu, Milloch, Milton, Musardo, Nikolov, Noe, Parmigiani, Penco, Petronio, Pivetta, Predonzani, Rossi, Rumiz, Salom, Scafuri, Serpico, Sigalotti, Spampinati, Spezzani, Svandrlik, Svetina, Tazzari, Trovo, Umer, Vascotto, Veronese, Visintini, Zaccaria, Zangrando and Zangrando4, Reference McNeil, Robb, Poole and Thompson11, Reference Zeng, Feng, Gu, Wang, Zhang, Liu and Zhao12]. 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)[ Reference Penco, Perosa, Allaria, Di Mitri, Ferrari, Giannessi, Spampinati, Spezzani and Veronese13]. 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)[ Reference Stupakov14, Reference Xiang and Stupakov15], phase-merging enhanced harmonic generation (PEHG)[ Reference Deng and Feng16, Reference Feng, Deng, Wang and Zhao17], cascaded HGHG[ Reference Yu and Ben-Zvi18] and the echo-enabled harmonic cascade (EEHC)[ Reference Feng, Liu, Chen, Zhou, Zhang, Qi, Gu, Wang, Jiang, Li, Wang, Wang, Zhang, Feng, Li, Lan, Li, Zhang, Deng, Xiang, Liu and Zhao19] 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[ Reference Schneidmiller and Yurkov20], which was initially proposed for FEL oscillators[ Reference Colson21] and further explored experimentally in high-gain FELs[ Reference Schneidmiller, Faatz, Kuhlmann, Rönsch-Schulenburg, Schreiber, Tischer and Yurkov22]. Subsequently, this method was implemented and tested at the PAL-XFEL and European XFEL facilities, achieving FEL amplification at 1 nm[ Reference Nam, Min, Kim, Yang, Kim, Heo, Kwon, Park and Kang23] and 2.8 Å[ Reference Schneidmiller, Brinker, Decking, Froehlich, Guetg, Noelle, Scholz, Yurkov, Zagorodnov, Geloni, Gerasimova, Gruenert, Laksman, Liu, Karabekyan, Kujala, Maltezopoulos, Petrov, Samoylova, Serkez, Sinn and Fabris24], respectively. These methods exhibit poor temporal coherence and intrinsic pulse energy fluctuations stemming from their reliance on self-amplified spontaneous emission (SASE)[ Reference Kondratenko and Saldin25, Reference Bonifacio, Pellegrini and Narducci26]. 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[ Reference Zhang, Liu, Qi, Fu, Feng, Deng and Liu27] and the harmonic optical klystron (HOK)-based EEHG[ Reference Wang, Zeng, Zhang and Yang28]. 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[ Reference Schneidmiller, Faatz, Kuhlmann, Rönsch-Schulenburg, Schreiber, Tischer and Yurkov22, Reference Schneidmiller and Yurkov29] and the superradiant cascade[ Reference Mirian, Di Fraia, Spampinati, Sottocorona, Allaria, Badano, Danailov, Demidovich, De Ninno, Di Mitri, Penco, Ribič, Spezzani, Gaio, Trovó, Mahne, Manfredda, Raimondi, Zangrando, Plekan, Prince, Mazza, Squibb, Callegari, Yang and Giannessi30]. 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.
2 The principle of harmonic generation
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. [Reference Schneidmiller and Yurkov20,Reference Deng and Dai31–Reference Deng, Bei and Dai33] for odd harmonics and Refs. [Reference Zhukovsky34,Reference Kalitenko35] for even harmonics. For the higher odd harmonics, the scaling function of the gain length can be written as follows[ Reference Deng and Dai31, Reference Yu, Krinsky and Gluckstern36]:
$$\begin{align}\frac{\operatorname{Im}\left({\mu}_n\right)}{D_n}=\frac{1}{2{k}_{\mathrm{w}}{L}_{\mathrm{gn}}{D}_n}:= G\left({k}_{\mathrm{s}}\varepsilon, \frac{\sigma_{\gamma }}{D_n},\frac{k_{\beta }}{k_{\mathrm{w}}{D}_n},\frac{\omega -{\omega}_{\mathrm{r}}}{\omega_{\mathrm{r}}{D}_n}\right).\end{align}$$
Here,
${\mu}_n$
can be derived from the following:
$$\begin{align}&{\mu}_n{a}^2\left(1-{e}^{-{\chi}_n}\right)-\frac{\chi_n}{n}\left[1-\left(1-{\chi}_n\right){e}^{-{\chi}_n}\right] \nonumber\\& \quad ={\int}_{-\infty}^0\exp \left[-i\left(\frac{\mu_n}{D_{\mathrm{u}}}+\frac{\omega -{\omega}_{\mathrm{r}}}{\omega_{\mathrm{r}}{D}_n}\right)s-2{\left(\frac{\sigma_{\mathrm{r}}}{D_n}\right)}^2{s}^2\right]\nonumber \\&\qquad\times \left(\frac{1-{e}^{-{\eta}_{+}}}{\eta_{+}}-\frac{1-{e}^{-{\eta}_{-}}}{\eta_{-}}\right)\frac{s\mathrm{d}s}{\mathit{\cos}\left({k}_{\mathrm{s}}/{D}_n\right)},\end{align}$$
with
$$\begin{align}{\eta}_{\pm}&=3 is\left(\frac{k_{\beta }/{k}_{\mathrm{w}}}{D_n}\right)\left({k}_{\mathrm{s}}\varepsilon \right)+\frac{\chi_n}{2}\left[1\mp \cos \left(\frac{k_{\beta }/{k}_{\mathrm{w}}}{D_n}s\right)\right],\nonumber\\ {D}_n&={\left(\frac{2{Z}_0e}{\pi {mc}^2}\frac{I_0}{\gamma_0}\frac{K^2}{1+{K}^2/2}\right)}^{1/2}{\left[ \mathrm{JJ}\right]}_n,\nonumber\\{\chi}_n&=-a\sqrt{\mu_n}\frac{H_0^{'(1)}\left(a\sqrt{\mu_n}\right)}{H_0^{(1)}\left(a\sqrt{\mu_n}\right)},\end{align}$$
where
${Z}_0=377\;\Omega$
is the impedance of free space,
${\left[ \mathrm{JJ}\right]}_n$
is the coupling factor for the nth harmonic,
${k}_{\mathrm{s}}=2\pi /{\lambda}_{\mathrm{s}}$
and
${k}_{\mathrm{w}}=2\pi /{\lambda}_{\mathrm{w}}$
denote the wavenumber of fundamental radiation and the undulator field, respectively,
${k}_{\beta }$
is the betatron wavenumber without external focusing,
$a=\sqrt{2{k}_{\mathrm{s}}{k}_{\mathrm{w}}}{R}_0$
denotes the scaled beam size,
${R}_0$
is the electron beam size and
${H}_0^{(1)}$
is the Hankel function of the first kind. To yield near-maximum gain, the scaled factor
$\left(\omega -{\omega}_{\mathrm{r}}\right)/{\omega}_{\mathrm{r}}$
is calculated as follows:
$$\begin{align}\frac{\omega -{\omega}_{\mathrm{r}}}{\omega {D}_n}=-3\left(\frac{k_{\beta }}{k_{\mathrm{w}}{D}_n}\right){k}_{\mathrm{s}}\varepsilon .\end{align}$$
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
$\Lambda ={k}_{\beta }/\left({k}_{\mathrm{w}}{D}_n\right)$
.
Scaling function of the gain length for the fundamental (dashed line) and third harmonic (solid line) for
$\Lambda =1$
(red) and
$\Lambda =0.1$
(blue), corresponding to scaled energy spread
${\sigma}_{\gamma }/{D}_n=0.1$
with optimal detuning.
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[
Reference Tremaine, Wang, Babzien, Ben-Zvi, Cornacchia, Nuhn, Malone, Murokh, Pellegrini, Reiche, Rosenzweig and Yakimenko37]. This provides the opportunity to perform multi-stage harmonic cascade.
3 Simulations and results
3.1 The proposed scheme
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.
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
${\lambda}_3$
, which is amplified throughout the entire radiator. Each stage of the undulator is tuned to the subharmonic of the next stage.
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.
3.2 Simulation results
3.2.1 Ideal beam simulation
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[
Reference Yu and Wu38]. 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[
Reference Reiche39]. 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.
Simulation parameters.
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.
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 temporal evolution of energy spread along the radiator (right) and its distribution at the entrance of the first, second and third stages, respectively (left).
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.
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.
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[
Reference Ackermann, Asova, Ayvazyan, Baboi, Bähr, Balandin, Beutner, Brandt, Bolzmann, Brinkmann, Brovko, Castellano, Castro, Catani, Chiadroni, Choroba, Cianchi, Costello, Cubaynes, Dardis, Decking, Delsim-Hashemi, Delserieys, Di Pirro, Dohlus, Düsterer, Eckhardt, Edwards, Faatz, Feldhaus, Flöttmann, Frisch, Fröhlich, Garvey, Gensch, Gerth, Görler, Golubeva, Grabosch, Grecki, Grimm, Hacker, Hahn, Han, Honkavaara, Hott, Hüning, Ivanisenko, Jaeschke, Jalmuzna, Jezynski, Kammering, Katalev, Kavanagh, Kennedy, Khodyachykh, Klose, Kocharyan, Körfer, Kollewe, Koprek, Korepanov, Kostin, Krassilnikov, Kube, Kuhlmann, Lewis, Lilje, Limberg, Lipka, Löhl, Luna, Luong, Martins, Meyer, Michelato, Miltchev, Möller, Monaco, Müller, Napieralski, Napoly, Nicolosi, Nölle, Nuñez, Oppelt, Pagani, Paparella, Pchalek, Pedregosa-Gutierrez, Petersen, Petrosyan, Petrosyan, Petrosyan, Pflüger, Plönjes, Poletto, Pozniak, Prat, Proch, Pucyk, Radcliffe, Redlin, Rehlich, Richter, Roehrs, Roensch, Romaniuk, Ross, Rossbach, Rybnikov, Sachwitz, Saldin, Sandner, Schlarb, Schmidt, Schmitz, Schmüser, Schneider, Schneidmiller, Schnepp, Schreiber, Seidel, Sertore, Shabunov, Simon, Simrock, Sombrowski, Sorokin, Spanknebel, Spesyvtsev, Staykov, Steffen, Stephan, Stulle, Thom, Tiedtke, Tischer, Toleikis, Treusch, Trines, Tsakov, Vogel, Weiland, Weise, Wellhöfer, Wendt, Will, Winter, Wittenburg, Wurth, Yeates, Yurkov, Zagorodnov and Zapfe1]. 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.
The power profiles and spectra of FEL pulses emitting at wavelengths of 27 nm (a), 9 nm (b) and 3 nm (c).
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[
Reference Zholents40–
Reference Wang and Feng43]. 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[
Reference Dattoli and Sabia44–
Reference Tsai and Qin46]. 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.
The power profiles (left) and spectra (right) of 100 FEL shots under the condition of
${\sigma}_{\mathrm{E}}/{E}_0=0.01\%$
. The pulse energies and spectrum bandwidths (
$\Delta \lambda /{\lambda}_0$
), as well as their statistical information, are also depicted.
The longitudinal phase space and current distribution of the electron beam.
3.2.2 Start-to-end simulation
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)[ Reference Wang, Zeng, Shao, Liang, Yi, Yu, Sun, Li, Feng, Wang, Zhao, Jia, Huang, Zhang and Yang47], 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[
Reference Zhang, Liu, Qi, Fu, Feng, Deng and Liu27].
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.
4 Conclusion and discussion
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)[ Reference Yu, Liu, Yang, Wang, Shi, Ding, Tao, Tang, He, Chen, Tian, Dai, Wu, Zhang, Yang, Feng, Chen, Wang, Gu, Chen, Liu, Lan, Feng, Zhang, Zhong, Zhang, Li, Xiao, Li, Zhao, Chen, Zhu, Ding, Huang, Zhang, Lai, Yang, Wang, Xiang, Huang, Sun, Gao, Jiang, Zhou, Liu, Wu, Chen, Wang, Huang, Zhang, Deng, Li, Lin, Yu, Yan, Yan, Xia, Zhou, Liu, Gu, Gu, Fang, Leng, Yin, Wang and Zhao48] 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[ Reference Prince, Allaria, Callegari, Cucini, De Ninno, Di Mitri, Diviacco, Ferrari, Finetti, Gauthier, Giannessi, Mahne, Penco, Plekan, Raimondi, Rebernik, Roussel, Svetina, Trovó, Zangrando, Negro, Carpeggiani, Reduzzi, Sansone, Grum-Grzhimailo, Gryzlova, Strakhova, Bartschat, Douguet, Venzke, Iablonskyi, Kumagai, Takanashi, Ueda, Fischer, Coreno, Stienkemeier, Ovcharenko, Mazza and Meyer49, Reference Giannessi, Allaria, Prince, Callegari, Sansone, Ueda, Morishita, Liu, Grum-Grzhimailo, Gryzlova, Douguet and Bartschat50]. The relative intensities of these pulses could be potentially manipulated with the help of phase shifters[ Reference Schneidmiller and Yurkov51]. 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.
Appendix A LSC-induced energy loss
The modulated electron beam after the modulator-chicane system would experience relatively rapid debunching at high harmonics due to nonlinear LSC forces. Figure A1 presents the central energy (blue) and the energy loss induced by LSC (red) for the central slices, revealing that the maximum energy loss is approximately ±70 keV/m. To enhance clarity, Figure A2 illustrates the influence of energy loss on the longitudinal phase space during the first stage. Although there is a slight modulation in the energy distribution, this change is minimal, suggesting that the evolution of phase space due to FEL lasing takes precedence over the modifications induced by LSC in this case.
The electron beam central energy after the dispersion chicane (blue) and the LSC-induced energy loss (red).
Electron beam phase space before the first stage (left), after the first stage without LSC-induced energy loss (middle) and after the first stage with LSC-induced energy loss (right).
The layout of the DCLS operated in HGHG mode.
Measured spectra of the 150 nm (h = 2) (a) and the 75 nm (h = 2
$\times$
2) (b) FEL radiation before (blue) and after (green) the second stage.
Appendix B An experimental trial at the Dalian Coherent Light Source
As an experimental trial, we made a preliminary experimental exploration of the proposed method to generate coherent FEL pulses through HGHG-based harmonic cascade at the DCLS, which has the capability of generating FEL pulses at arbitrary wavelengths from 50 to 150 nm. The layout of the DCLS is shown in Figure B1. The electron beam is generated by a photocathode radio-frequency (RF) gun operating at a frequency of 2856 MHz and a repetition rate of 10 Hz. It can be boosted up to 300 MeV by a linear accelerator (linac), which consists of three 3-m-long S-band structures (L1) and four additional structures (L2). A compression chicane can compress the beam to approximately 0.6 ps, resulting in a peak current of about 300 A. The relative project energy spread is about
$1.25\times {10}^{-3}$
and normalized transverse emittance is around 1.5 mm
$\cdot$
mrad. The seed laser is tunable from 240 to 360 nm with a peak power of approximately 100 MW. This seed laser interacts with the electron beam in a 1-m-long modulator that has a period of 50 mm. The radiator section comprises five variable-gap undulators, each 3 m in length with a period of 30 mm. The spectra were acquired using a dedicated online spectrometer with the spectral coverage from 50 to 150 nm and a resolution of 0.02 nm at 116 nm. The FEL pulse energies were measured by photodiodes located downstream of the radiator. Optimizing for 50 nm FEL pulses with such an electron beam and radiator configuration yields an undulator parameter
$K$
of approximately 0.546, consequently resulting in a relatively low intensity of FEL radiation. Therefore, this experiment primarily focused on the generation of second harmonic lasing.
Due to constraints imposed by the electron beam quality and the design of undulator gaps, the radiator was partitioned into two stages, incorporating four and one undulators, respectively. The seed laser wavelength is chosen to be 300 nm, with the first and second stages resonating at 150 and 75 nm, corresponding to
${h}_1={h}_2=2$
. Through precise tuning of the phase shifter situated between the first and second stages, the pulse energy of 75 nm FEL radiation can be adjusted. Moreover, this phase shifter enables meticulous control over the time delay between fundamental and harmonic pulses, achieving adjustment precision at the attosecond scale. This capability is particularly valuable for investigating ultrafast phenomena such as Auger decay processes.
The measured spectra of the 150 nm (h = 2) and 75 nm (h = 2
$\times$
2) pulses are presented in Figure B2. The blue line in Figure B2(b) indicates the generation of second harmonic radiation in the first stage. After five undulators, the relative bandwidth (FWHM) for the first pulse is
$2.73\times {10}^{-4}$
with a central wavelength of around 150.08 nm, while these values for the second pulse are approximately
$3.59\times {10}^{-4}$
and 75.13 nm, respectively. The pulse energies are measured to be 77.78 μJ for 150 nm radiation and 2.78 μJ for 75 nm radiation. In the experiment, the pulse energy of 75 nm radiation is severely limited by the quantity and quality of the undulator.
Despite being constrained to two stages for generating extreme ultraviolet radiation in the experiment, the findings confirm the feasibility of producing fully coherent FEL pulses through an HGHG-based harmonic cascade, and we have directly observed and measured the second harmonic radiation. This experiment provides valuable insights for future investigations into harmonic cascades, as well as essential experience for future multi-stage harmonic cascade experiment at the S3FEL.
Acknowledgements
The authors would like to thank Xiaozhe Shen, Huaiqian Yi, Yifan Liang, Lingjun Tu and Hao Sun for useful suggestions on the simulations and experiments. The authors also extend their sincere appreciation for the support provided by the Dalian Coherent Light Source. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12305359 and 22288201) and the Scientific Instrument Developing Project of Chinese Academy of Sciences (Grant No. GJJSTD20220001).
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