Over the past few decades, structured light, driven by the ability to tailor light through spatial control of its amplitude, phase, and polarization, has attracted significant interest from researchers [1–3]. Foremost among the family of structured light fields is light-carrying orbital angular momentum (OAM), which is characterized by a helical wavefront $\exp (il\phi)$, where $\phi$ is the azimuth and $l$ is helicity. In this regard, OAM beams in the visible- and IR-wavelength regimes have already been used in diverse application areas such as micromanipulation, quantum information, and optical data transmission [2]. In the X-ray regime, the use of OAM beams can enable the direct alteration of atomic states through OAM exchange [4] and facilitate the development of new methods to study the quadrupolar transitions of materials [5]. However, practical applications of X-ray beams with OAM are currently limited, owing to the lack of suitable optics and the difficulties encountered to realize bright coherent light sources. Much attention has been focused on their efficient creation.

In general, OAM beams are easily created by inserting optical elements [2] such as programmable spatial light modulators, stepped phase plates, and spiral Fresnel zone plates into the propagation path of light. However, these direct optical manipulation approaches may not be practical or available for application to modern X-ray free-electron lasers (XFELs) [6–11], which are currently the brightest source of X-rays for scientific applications. In this context, researchers have studied and proposed several methods and techniques to directly generate OAM beams in a single-pass FEL with helical undulators [12–15]. One such practical technique involves higher-harmonic emission from a helical undulator [12]. However, external spectral filters or other techniques are required to separate OAM beams from the mixed radiation because the fundamental radiation also is predominantly emitted in the helical undulators. Moreover, in a single-pass seeded FEL, an alternative approach to produce OAM light at the fundamental harmonic can be achieved using a helically microbunched electron beam [13]. However, this scheme may fail at hard X-ray regimes because it requires harmonic modulation, which requires a very high power seed laser.

Unlike the single-pass XFEL, X-ray free-electron laser oscillators (XFELOs) are low-gain multipass devices that can produce intense, fully coherent hard X-rays at a high repetition rate [16–18]. In an XFELO, the X-ray pulses circulate in a low-loss optical cavity formed by multiple Bragg-reflecting crystals. Similar to a conventional optical laser, XFELOs can generate light beams with the OAM by shaping the gain and cavity loss of each transverse mode. For example, instead of direct phase-conversion elements, a special spatial amplitude mask can be used to form the desired OAM mode in the cavity [19–21].

Here, we present a straightforward, highly flexible method that essentially preserves the amplification of OAM beams at the fundamental harmonic and avoids the need for external optical elements. The schematic of our method represents a typical XFELO configuration, as shown in Fig. 1. This method is based on the coupling of longitudinal and transverse modes. The coupling stems from the important fact that the small-signal FEL gain spectrum for an individual transverse mode exhibits a shift owing to Gouy phase detuning [22]. Meanwhile, the Bragg mirrors of the XFELO cavity only reflect X-rays within a spectrum width (FWHM) of meV level, which is much narrower than the gain spectrum width. The combined effect allows XFELOs to operate in a spectral regime fixed by the Bragg reflection, wherein the radiation in a specific high-order transverse mode would obtain the maximum gain. As a consequence, by carefully controlling the coupling, an OAM beam is formed and amplified in multiple passes inside an XFELO to reach saturation.

 

Fig. 1. Scheme to generate X-ray OAM beams by using a typical configuration of the XFELO. Compound refractive lenses (CRLs) are used for X-ray focusing.

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To define OAM cavity modes, we start with the Laguerre–Gaussian (${\rm{LG}}_p^l$) modes that can possess arbitrary OAM [23], where $l$ denotes any real integer for the azimuthal mode and $p$ is zero or any positive integer for the radial mode. With the ${\rm{LG}}_p^l$ basis modes, the transversely dominated FEL radiation fields can be described as $E(r,\phi ,z) = \sum\nolimits_{p,l} {a_{p,l}}{u_{p,l}}$, and ${u_{p,l}}$ can be expressed as

In general, the radiation evolution during an XFELO pass consists of the FEL gain in the undulator followed by the reflection from the narrowband Bragg mirrors. This process then repeats on the next pass with a fresh electron bunch. Following the procedure in [16], the evolution can be described by the following partial differential equation:

Making use of the well-known small-signal gain, gain ${\tilde G_l}$ is found to be [25–27]

Another important factor to be considered in the gain is the extra phase term $(2p + |l| + 1)\mathop {\tan}\nolimits^{- 1} z/{z_R}$ in Eq. (1), which is the Gouy phase. The presence of the Gouy phase leads to a difference in the phase of the ponderomotive potential in the FEL equations, which means that the FEL resonant wavelength is slightly different for each ${\rm{LG}}_p^l$ mode [22]. For an XFELO, the averaged deviation over the undulator is given by $\langle {\Delta \lambda /{\lambda _r}} \rangle \approx 2(2p + |l| + 1)\mathop {\tan}\nolimits^{- 1} ({{L_u}/2{z_R}})/2\pi {N_u}$ [26,27], where ${L_u}$ denotes the undulator length. As a result, the gain spectrum for each mode is given by ${\tilde G_l}[{\nu - 2(2p + |l| + 1)\mathop {\tan}\nolimits^{- 1} ({{L_u}/2{z_R}})}]$. The theoretical prediction of the gain spectrums is shown in Fig. 2, and will be verified later by simulations without simplifications.

 

Fig. 2. (a) Normalized filling factor ${{\rm{F}}_l}/{{\rm{F}}_0}$. (b) Normalized gain for different $l$ modes, while the yellow line indicates the reflectivity of the sapphire (0 0 0 30) at 14.3 keV. Obviously, the reflectivity can select the gain. The parameters based on a realistic scenario: ${f_ \bot}(r,\phi) = \exp (- {r^2}/2r_e^2)$, ${f_E}(\eta) = \exp (- {\eta ^2}/2\sigma _\eta ^2)/\sqrt {2\pi} {\sigma _\eta}$, ${\sigma _\eta} = 0.01\%$, $2r_e^2/w_0^2 = 3$, ${z_R} = 15$ m, ${L_u} = 20$ m, and $p = 0$.

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Figure 2(a) illustrates the normalized filling factor as a function of the reduced electron beam size $2r_e^2/w_0^2$ and the mode order $l$. For convenience, ${{\rm{F}}_l}$ is normalized by ${{\rm{F}}_0}$. The filling factor between the electron beams and radiation decreases with $|l|$. This phenomenon occurs because the intensities of those $|l| \gt 0$ modes vanish on-axis, and the trend increases with increases in $|l|$. In the case of larger $2r_e^2/w_0^2$, ${{\rm{F}}_l}$ increases and merges to a value, which means that individual modes can be amplified more equally [26]. Hence, a wide electron beam, whose transverse size exceeds the radiation size, may be preferable to amplify OAM beams.

Figure 2(b) displays gain spectrums for different modes and the reflectivity of the sapphire (0 0 0 30) at 14.3 keV. The maximum gain is normalized to one. Notably, the width of the reflectivity is much narrower than that of the gain spectrum, seen in the small plot in Fig. 2(b). Therefore, in the high-reflectivity regime, the radiation only in the ${{\rm{LG}}^{\pm 1}}$ modes could obtain the maximum gain. This effect enables XFELO to amplify ${{\rm{LG}}^{\pm 1}}$ carrying OAM to reach saturation. it is important to note that Bragg reflection does not directly filter out the ${{\rm{LG}}^{\pm 1}}$ radiation from the mixed one, but fixes the spectral regime. The radiation in ${{\rm{LG}}^{\pm 1}}$ modes can reach saturation because it sustains the maximum gain over the other modes in each pass. As the gain process repeats, the mode competition effect provides the selection of the ${{\rm{LG}}^{\pm 1}}$ mode.

Obviously, both ${{\rm{LG}}^l}$ and ${{\rm{LG}}^{- l}}$ have the same transverse intensity distribution and gain spectrum, but exhibit different helical wavefront handedness. Therefore, conventional solid lasers are prone to yielding an output beam with a coherent superposition of ${{\rm{LG}}^l}$ and ${{\rm{LG}}^{- l}}$. However, the XFELO output would have a particular handedness. The handedness is randomly selected by the FEL amplification process because this process forms a helical microbunching, which is imprinted by the wavefront. Owing to the orthogonality, the microbunchings formed by different $l$ modes do not couple to each other. When the radiation is a coherent superposition of multiple modes, radiation in the mode with a higher intensity will receive a larger microbunching. Meanwhile, the microbunching in this mode suppresses the gain of the other modes due to the degradation of the electron beam quality. Consequently, as the amplification process repeats in the XFELO, the mode with higher intensity increasingly predominates over the other modes. And, the gain seen by other modes would be insufficient to overcome the round-trip loss. In short, the superposition of ${{\rm{LG}}^l}$ and ${{\rm{LG}}^{- l}}$ is not stable in the XFELO, unless they have exactly the same intensity or the microbunching effect is very weak. The detailed analysis and simulations are presented in Supplement 1.

It should also be noted that higher radial mode $p$ affects the optimal detuning of the gain, since the gain spectrum is a function of $2p + |l|$. In addition, the coupling efficiency between each $p$ modes is nonzero during FEL amplification, as presented in [24]. Consequently, this effect has an impact on the amplification of modes with $|l| \gt = 2$. For example, if the detuning is set by $2p + |l| = 2$, three modes can be found to share the optimal detuning: $p = 1,l = 0$ and $p = 0,l = \pm 2$. Therefore, the amplification of $l = \pm 2$ may be difficult without the suppression of modes of $p \gt 0$. Fortunately, for $|l| = 1$ modes with a detuning of $2p + |l| = 1$, $p$ must be zero, and its effect is naturally suppressed. Therefore, XFELOs can generate robust OAM modes with $|l| = 1$, as verified by the following simulations.

We performed numerical simulations of OAM beam generation in an XFELO with the proposed method, based on the Shanghai High repetition rate XFEL and Extreme light facility (SHINE), which is the first hard X-ray FEL facility in China (currently under construction and scheduled for completion within the next four years) [28,29]. The linear accelerator (LINAC) of SHINE, operating at 1 MHz repetition rates, will deliver 8 GeV electron bunches with a $0.4\;{\rm{mm}} \cdot {\rm{mrad}}$ normalized emittance and 0.01% relative energy spread. The typical electron bunch has a charge of 100 pC, and is compressed to a peak current of 500 A in this study. The total length of the planar undulator is 20 m, which is divided into four segments of 250 periods each with a period length of 16 mm. Each segment is spaced 1 m apart, and focusing quadrupoles are inserted in between. A symmetry cavity of size 150 m is formed by two sapphire crystal mirrors with total reflectivity of 80% at 14.3 keV. Meanwhile, two CRLs with a focal length of 57.7 m are placed 100 m apart. The 3D simulations were carried out using a combination of GENESIS [30], OPC [31], and BRIGHT [32]. In addition, a shot-to-shot angular deviation of 5 nrad (rms) was applied to the mirrors. In the simulation, the resonant wavelength deviation $\Delta \lambda /{\lambda _r}$ was 0.11%, which reasonably agrees with the theoretical prediction of 0.094% for the OAM modes with $|l| = 1$.

Figures 3 and 4 demonstrate the power and robustness of our approach. Figure 3(a) shows the pulse energy as a function of the round trips and the evolution of the transverse mode. The saturation cavity output reaches 120 µJ, approaching the level of conventional operation. In the early stages of XFELO operation, a larger number of transverse modes are excited, seen in Fig. 3(b). These modes originate from the spontaneous undulator radiation occurring when the coherent XFELO pulse is not yet built up. In the short period ranging from 100 to 400 round trips, there are still several transverse modes, and the mode competition occurs. A snapshot is illustrated in the subplot (200) of Fig. 3(b). Generally, only the mode with the highest growth rate (${{\rm{LG}}^{|1|}}$ mode) will remain at saturation. Therefore, the ${{\rm{LG}}^1}$ mode predominates over other modes and reaches saturation.

 

Fig. 3. (a) Cavity output energy growth. (b) Snapshots of transverse profiles. The evolution of the intensity profile into a doughnut-like one can be clearly observed. The total output energy is about 120 µJ.

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Fig. 4. (a) Transverse profile. (b) Phase distribution. (c) Longitudinal power profile. (d) Spectrum. The transverse profile and phase of the light reveal a OAM mode of $l = 1$ at saturation. A peak power of 400 MW with a spectral width (FWHM) of 11 meV can be obtained.

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At saturation, the characteristic hollow profile and helical phase can be found in Fig. 4. It is obvious from the phase distribution that the dominant mode is the ${{\rm{LG}}^1}$ mode. The intensity of modes $l \ne 1$ is negligible. The longitudinal power profile and the spectrum are shown in Fig. 4. While the output power increases up to 120 µJ, the peak power exceeds 400 MW with the FWHM spectrum width of 11 meV.

These results regarding the generation of the $l = 1$ OAM mode at the fundamental frequency of the XFELO suggest that an XFELO can natively control the transverse mode without the need for any additional elements that have been considered elsewhere. In addition, we believe this method could possibly be applied to tunable XFELO cavities [16], since the key points are the overlap of radiation and the detuning of the resonance. Moreover, this method may be able to achieve fast OAM switching by controlling the detuning with the electron beam energy.

To investigate the random selection of the mode handedness, we performed 451 XFELO simulations with different random numbers. Figure 5 shows a portion of the results. Overall, the ${{\rm{LG}}^1}$ mode reaches saturation 228 times, while the ${{\rm{LG}}^{- 1}}$ mode reaches saturation 223 times. The results demonstrate that the occurrence of ${{\rm{LG}}^1}$ and ${{\rm{LG}}^{- 1}}$ is essentially random. The control of the handedness may be achieved by inserting a short helical undulator that operates at the second harmonic. The short helical undulator would induce an extra amplification only in the mode of interest. As a result, that mode would predominate over the other modes after multiple FEL amplifications. Detailed analysis will be carried out in future work.

 

Fig. 5. Fifty XFELO simulations with different random numbers. The occurrence of ${{\rm{LG}}^1}$ and ${{\rm{LG}}^{- 1}}$ is random.

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In conclusion, for what we believe is the first time, to the best of our knowledge, we proposed OAM operation of an XFELO without the need for external optical elements. The proposed method is based on the coupling of longitudinal and transverse modes, which enables the Bragg crystal to serve as a “selector” in the transverse modes by maximizing the gain of the mode of interest. This allows the amplification and saturation of OAM beams in a typical XFELO configuration. The feasibility of the proposed design was verified through 3D numerical simulations based on the SHINE. The typical results for the two-mirror-symmetry cavity indicated that the OAM operation of an XFELO can generate 120 µJ pulses in the $l = 1$ Laguerre–Gaussian mode at 1 MHz. Without the need for any additional elements, the proposed method demonstrates the simplest approach to generate the $|l| = 1$ OAM modes in an XFELO. We believe that our approach can promote future applications of X-ray free-electron laser oscillators.