High-gain free-electron laser with orbital angular momentum seeded by an x-ray regenerative amplifier

High-gain high-mode generation (HGHMG) free-electron laser has been experimentally confirmed to produce vortex light from a relativistic electron beam and is proposed to generate fully-coherent, high-brightness hard x-ray carrying orbital angular momentum at modern free electron laser facilities. However, this requires a coherent x-ray seed laser with sufficient power to perform the helical modulation on the electron beam. In this contribution, we propose a promising scheme to generate a fully coherent x-ray seed laser for the HGHMG system. In this scheme, an x-ray regenerative amplifier is used to offer a fully coherent x-ray seed laser to modulate the electron beam in a helical undulator. With the proposed technique, high-power and high-repetition-rate x-ray with orbital angular momentum can be produced, which will open routes to scientific research in x-ray science.


Introduction
The orbital angular momentum (OAM) [1] of light is an important component of the angular momentum of a light beam.The light carrying OAM has a helical wavefront.At conventional laser wavelengths, OAM beams have many scientific applications in super-resolution imaging, microscopy, optical communication, and optical manipulation.At x-ray wavelengths, OAM beams have promising applications in expanding magnetic circular dichroism, probing chirality in biological materials, and research in scattering and spectroscopy.At conventional laser wavelengths, OAM beams can be generated by using optical shaping elements such as spiral phase plates, spatial light modulators, and q-plates.Some of these optical elements can be used to produce x-ray OAM light at synchrotron light sources.However, these techniques may not be available at x-ray free-electron lasers (FELs) [2] due to the high peak intensity or high repetition rates.X-ray FELs have become powerful tools for exploring the deep structure and behavior of matter at the atomic scale and femtosecond time scales.However, the basic transverse mode of FELs is the Gaussian mode due to the principle of FELs.Recently, some methods have been proposed to produce x-rays carrying orbital angular momentum in FELs.Some techniques aim to create helical microbunching through laser-assisted manipulation.[3,4,5,6].Another route is to create helical microbunching based on harmonic energy modulation in a helical undulator, which is also called high-gain high-mode generation (HGHMG) [7].This method has been experimentally confirmed to produce OAM light from a relativistic electron beam [8] and is proposed to generate fully coherent high-brightness hard x-ray carrying OAM at modern FEL facilities.However, this requires a coherent x-ray seed laser with sufficient power to perform the helical modulation on the electron beam.To avoid this limitation, self-seeding schemes including single-beam self-seeding and two-beam self-seeding may be used to offer the coherent x-ray seed laser.However, the monochromatization process of the self-seeding schemes significantly reduces the seed laser power, which affects the helical modulation in the HGHMG.
Cavity-based FELs, including x-ray free-electron laser oscillators (XFELO) [9] and x-ray regenerative amplifiers (RAFEL) [10], have been proposed to generate fully coherent x-rays at high repetition rates.Among them, the oscillator-amplifier scheme can be used to generate high-brightness x-ray beams [11].Motivated by this technique, we propose a promising scheme to generate a fully coherent x-ray seed laser for the HGHMG system.In this scheme, an x-ray regenerative amplifier is used to offer a fully coherent x-ray seed laser to modulate the electron beam in a helical undulator.With an energy-chirped electron beam, one part electron beam will not lase in the regenerative amplifier since the FELs resonance relationship is not satisfied.This part of the electron beam will be helically modulated in a helical undulator by the coherent x-ray from the previous regenerative amplifier.With the proposed technique, high power and high repetition rate x-ray with OAM can be produced, which will open routes to scientific research in x-ray science.

The layout
The schematic layout of the proposed scheme is shown in Fig. 1.The energy-chirped electron beam is sent into the x-ray regenerative amplifier to generate a fully coherent seed laser for the HGHMG.One part of the electron beam will not lase in the regenerative amplifier since the FEL resonance relationship is not satisfied.This part of the electron beam will achieve helical energy modulation in a helical undulator.After a dispersion section, the helical energy modulation will be transformed into a helical density modulation.Finally, the helical electron beam will generate x-ray beam with a helical phase in the amplifier.

simulations
Start-to-end simulation is carried out for this proposed scheme based on the parameters of the Shanghai High Repetition Rate X-ray FEL and Extreme Light Facility (SHINE).ASTRA is used to track the particles from the gun to the end of the injector, which considers longitudinal space charge and microbunching instability effects.ELEGANT is utilized to track the particles in the linac and bunch compressors, which conside the coherent synchrotron radiation (CSR) and incoherent synchrotron radiation (ISR) effects.In general, the electrons are usually accelerated off-crest in the Linac and achieves a large energy chirp, then the bunch is compressed longitudinally in a magnent compressor to obtain high peak current.The residual energy chirp compensation is achieved by using a corrugated pipe as a beam dechirper.In this paper, we need an energy chirped beam so we remove this dechirper [12].The parameters of the electron beam are presented in Table 1 and the longitudinal phase space, current, emittance distribution are shown in Fig. 2.
The FEL performances are performed with Genesis 1.3 [13] and the performances of x-ray in a cavity are simulated by the combination of Ocelot [14] and Bright [15].In this simulation, the round trip length of the cavity is 300 meters to match the repetition rate of the electron beam.The optical cavity is composed of four crystal mirrors and compound refractive lenses (CRLs) which count for focusing.Some detail parameters of the cavity are presented in Table 1.The simulations are carried out at 5 keV photons.The undulator in a cavity is 15.6 meters and the oscillator operates at high gain.The simulation results are shown in Fig. 3.The peak power will stabilize after 15 passes as shown in Fig. 3(a).As we can see from Fig. 3(b), the peak power exceeds 10 GW when the RAFEL enters the steady state.Fig. 3(c) shows the excellent time coherence of the FEL pulse with the calculated FWHM bandwidth of 8.5 × 10 −5 .
In the simulation, the out-coupling of the optical cavity is 95%.Therefore, fully coherent pulses with peak power close to 10 GW can be used for helical modulation in the helical undulator.The linearly polarized laser field excites a helical modulation at the second harmonic of the helical undulator.The relative electron energy at the helical modulator exit is where η 0 is the initial relative energy, ϕ is the electron's azimuth coordinate, r is the radial  position of the electron, s is the initial position of the electron, k l is the laser wave number, and the energy modulation amplitude is where L m is the helical modulator length, K is the rms undulator parameter, k w is the wave number of the helical modulator magnetic field, P is the peak power of the laser, and w 0 is the laser spot size.It can be seen that the energy modulation has a spiral shape via the phase dependence on ϕ.We simulate the modulation process based on the above equations.The laser parameters used for modulation come from the light field coupled by the upper stage RAFEL.The parameters of the helical undulator are listed in Table 1.The next dispersion section with a small R 56 = 0.24 µm can convert the energy modulation into a helical density modulation.The phase space and the three-dimensional distribution of the electron beam at the entrance of the amplifier are shown in Fig. 4. It can be seen that the three-dimensional distribution has a spiral shape.Finally, the helical electron beam is injected into the amplifier.Figure 5 shows the gain curve and the evolution of helical bunching factors along the amplifier.The helical bunching of azimuthal mode l = 1 has always been dominant along the amplifier.The transverse intensity

conclusion
In this contribution, we propose a promising scheme to generate x-rays with OAM at a high repetition rate.We presented the start-to-end simulation results of the HGHMG seeded by RAFEL.With the proposed technique, high-power and high-repetition-rate x-ray with orbital angular momentum can be produced, which will open routes to scientific research in x-ray science.

Figure 1 .
Figure 1.A possible design for generating X-ray with OAM.

Figure 2 .
Figure 2. (a) Longitudinal phase space of the electron beam (b) The current (magenta line) and emittance (orange and blue line) distribution of the electron beam.

Figure 3 .
Figure 3. (a) The relationship between the peak power with the pass number in the RAFEL.(b) The power distribution and the corresponding spectrum (c) of the FEL pulse when the RAFEL enters the steady state.

14thFigure 4 .
Figure 4. (a) The phase space distribution of the electron beam at the entrance of the amplifier.(b) The three-dimensional distribution of the electron beam at the entrance of the amplifier.

Figure 5 .
Figure 5.The FEL gain curve and the evolution of helical bunching factors along the amplifier.

Figure 6 .
Figure 6.The transverse intensity distribution and phase distribution of the FEL light field at the exit of the amplifier.

Figure 7 .
Figure 7. (a) The relationship between the peak power with the pass number in the amplifier.(b) The power distribution and the corresponding spectrum (c) of the FEL pulse when the amplifier enters the steady state.