Scheme for generating 1 nm X-ray beams carrying orbital angular momentum at the SXFEL

He-Ping Geng· Jian-Hui Chen · Zhen-Tang Zhao

Abstract Optical vortices have the main features of helical wavefronts and spiral phase structures, and carry orbital angular momentum. This special structure of visible light has been produced and studied for various applications.These notable characteristics of photons were also tested in the extreme-ultraviolet and X-ray regimes. In this article,we simulate the use of a simple afterburner configuration by directly adding helical undulators after the SASE undulators with the Shanghai Soft X-ray FEL to generate high intensity X-ray vortices with wavelengths ～1 nm.Compared to other methods, this approach is easier to implement, cost-effective, and more efficient.

Keywords X-ray · Orbital angular momentum (OAM) ·Synchrotron light source · Free-electron laser (FEL)

1 Introduction

Structured light, whose phase structures rotate when they propagate through space, has been a topic of intense research. With the development of tools and technologies for creating and detecting structured light, applications have begun to appear steadily[1].Among these studies,the optics and laser communities are interested in phase singularity [2] and visible photons carrying orbital angular momentum (OAM) [3], to provide new explanations of optical effects and develop distinct applications ranging from manipulations of small particles, driving micro-machines, and quantum computation, to optical communications [4–14].

The noteworthy attributes of photons were also demonstrated in the extreme-ultraviolet and X-ray or ever γ ray [15] regions, pioneered by Peele et al. [16, 17]. Triggered by this pioneering effort,various generation methods based on particle accelerators [18–25], high-power lasers[26–29], and plasma [30, 31], as well as possible novel applications [32–38], are growing steadily.

Undulators are the workhorses in third-generation synchrotron radiation light sources and free-electron lasers(FELs) for generating X-ray beams. Sasaki and McNulty[39] were the first to indicate that when electrons traveled along a spiral path in helical undulators, the harmonics produced by them carried a spiral phase. Therefore, it is proposed that the helical undulator could be used as a source for generating intense X-rays carrying OAM light(XOAM). Bahrdt and coworkers at BESSY-II [40] first measured the special helical structure in spontaneous undulator emission. This was later measured in the XUV(X-ray and extreme ultraviolet) region at UVSOR-III [41]by Tatsuo and his coworkers, and in coherent radiation emitted from energy and density modulated (bunched)electron beams [42] by the US group at SLAC. In 2014,Hemsing and coworkers described a simpler afterburner configuration based on the HGHG setup to generate OAM light[43].Based on Hemsing’s HGHG scheme,Ribic et al.[44]proposed the use of an optical phase mask to produce a transverse phase modulation in the incident seed laser,which could also produce an XUV or X-ray optical vortex.Three years later, in 2017, they demonstrated two techniques to obtain intense femtosecond coherent optical vortices at the FERMI FEL in the extreme ultraviolet[45].The first method used the helical undulator to produce spiral beams at the second harmonic, while the second method relied on a spiral zone plate by putting it directly into the propagation path of the transverse Gaussian optical beam.

In this study,we simulate the use of a simple afterburner configuration by directly adding helical undulators after the SASE undulators at the Shanghai Soft X-ray FEL(SXFEL)[46] to generate high intensity X-ray vortices with wavelengths of ～1 nm. This configuration has two distinct advantages. First, the setup is very simple, since it only requires the addition of a few more helical undulators to the existing polarization control configuration. Second, it does not need special devices to produce and preserve the spiral bunched e beam structure during e beam transport, compared to previous methods.

2 Theory and modeling of higher harmonics from helical undulators

As mentioned earlier,X-ray photons carrying OAM can be generated in particle accelerators by converting an X-ray beam [18–23] or using helical undulators [39]. This triggers a careful re-examination of the angular momentum properties of emission from charged particles in circular and spiral motion [47–51].

We know that there are different kinds of undulator magnetic fields and possibly various structures of external lasers in diverse undulator devices. When the relativistic electron beam (e beam) is injected into the undulator, its energy and the eventual density distribution are affected by the magnetic and laser fields. In other words, the energy and final density distribution of the relativistic e beam are determined by the three-dimensional structure of the resonant interaction between the e beam, undulator field, and external injected laser. After the resonant interaction, the microbunched e beam is produced. These microbunched e beams can then be used to radiate electromagnetic waves in different downstream radiation facilities. Special light beams can be generated by resonant coupling with higher order harmonics.Therefore,it may be used to radiate fields with special 3D structures by properly tuning the injected microbunching and harmonic resonances in the downstream radiator. The helical light beams carrying OAM,generated in Ref. [42], arise from the transmission of the structural information of the helical microbunched e beams to the optical field.

We use the standard linearized FEL equation to describe the interaction between the injected e beams and input laser in the planar undulator. After some discussion, the density perturbation n1（r） can be expressed as the sum of the expanded eigenmodes. The density modulation expansion is [52]:

where c is the speed of light,K is the undulator parameter,ω is the frequency, ^g⊥denotes the electron motion, F（h）q，q′indicates the overlay effect of the e beam and incident laser lights, and f（r⊥） represents the transverse density distribution of electron beams. After interacting with a typical harmonic field incident mode, the e beam modulation in Eq. (1) can be expressed as follows [52]:

In the above formula, the mode index q includes two coefficients, q=（p，l）, where p and l correspond to the radial and azimuthal modes,respectively.When the e beam distribution is axisymmetric f（r⊥）=f（r）, the overlay effect of the e beam and incident fields can be straightforward [52]

We can see the relationship between l, l′, h and the polarization direction of the undulator, ±, where l and l′represent the radial modes about the azimuth of the electron beams and electromagnetic fields. There are some simple cases. For example, when an incident electromagnetic wave in a random l′mode is injected into the undulator in the fundamental mode (h=1), the homologous azimuthal modes can be obtained in the electron beams l=l′.When an axisymmetric electromagnetic wave l′=0 is injected into the undulator working at the second harmonic mode (h=2), a spiral bunched electron beam is formed, where the helical undulator can be either righthanded (l=-1) or left-handed (l=1). Therefore, corkscrewed micro-bunched electron beams at the second harmonic can be produced by the interaction between an initially unmodulated electron beam and incident Gaussian electromagnetic field (laser field).

We now discuss the properties of emission from helical undulators.The time-dependent complex field amplitude A of radiation from a pure circular undulator is [39]:

Here, X =2nξγθK, ξ=1/（1+γ2θ2+K2）, K is the undulator deflecting parameter, θ is the viewing angle of the propagation axis, φ is the azimuth of the observer’s location in the transverse plane perpendicular to the direction of propagation, γ is the Lorentz factor, Jnis the nth order Bessel function of the first kind, and J′n（X）is the first derivative of Jnwith respect to X. It is clear that the above equation is typical of Laguerre–Gaussian (LG)modes with a topological charge of l=n-1. LG beams are a group of solutions of the paraxial wave equation in the cylindrical coordinate system, and they are characterized by the OAM of a photon withThe LG modes are expressed by [3]:

Similar to the above example, the last term on the right in Eq. (7) shows that there is an azimuthal component of the phase,where p is the radial index which decides the radial intensity distributions and l is the azimuthal index. The wavefront (or the phase structure) is a spiral surface when l/=0.

Now go, he ended, and look after the pigs! So the next day she took out the pigs, and found them such good feeding grounds that they grew fatter every day

As an example, we perform a simulation using the current Shanghai Synchrotron Radiation Facility (SSRF)[53–55]beam parameters(3.5 GeV beam energy,3 nm-rad horizontal emittance, and 1% coupling) and an actual magnetic field (K ～0.78) of the 32-mm-period helical undulator, whose total length is 4.5 m. The radiation properties of such helical undulators are simulated using the SPECTRA code [56].

The trajectory of an electron moving inside the undulator is a spiral along the axis in the direction of propagation. The radiation produced by the electron is emitted along the direction of propagation within a very small divergence angle. The spectral flux density of the light from the helical undulator is shown in Fig. 1, which indicates that the fundamental and second harmonic modes are at 2256 eV and about 4413 eV within two orders of magnitude, respectively. We then calculate the complex amplitude spatial profile of the radiation field on the cross section perpendicular to the direction of propagation, and estimate the phases of the first and second harmonic radiation using the real and imaginary parts of Exand Ey,which are shown in Fig. 2. We observe that the transverse flux density profile of the fundamental radiation is Gaussian, while the transverse flux density distribution of the second harmonic is nearly annular. The typical transverse phase distribution of the second harmonic light is a spiral surface along the direction of propagation,which indicates that the photons carry an OAM (in this case, a topological charge l=-1).

It turns out that a relativistic e beam will move spirally along the axis of propagation after entering the helical undulator. Owing to this special motion mode, the light produced by the moving electrons forms a spiral structure at higher harmonics indicating that the photons carry OAM.

3 Theory of self-amplified spontaneous emission in free-electron lasers

A revolutionary development in X-ray production was the progress of XFEL[57].When an electron beam enters a periodic magnetic structure, the coherent radiation is amplified and this is the basis of the FEL. The radiation generated in an XFEL interacts with the beam of electrons in return to get more excellent X-ray than light sources based on storage rings in terms of intensity and coherence.At present, the high-energy e beams produced by a linac can be used to drive high-gain XFEL amplifiers using a long undulator. The gain of the FEL amplifier can be high enough to allow the initial incoherent undulator radiation to evolve into a high intensity quasi-coherent electromagnetic field—SASE [58].

In principle, an FEL can work at any wavelength, limited only by the energy and quality of the e beams obtained by the accelerator, and thus, can be used to fill gaps in the spectrum that cannot be covered by other coherent sources.At present, FEL oscillators based on electrostatic accelerators,cyclotrons,and rf-linear accelerators have been built.FEL oscillators in the storage rings are suitable for producing coherent radiation from visible to ultraviolet wavelengths [58]. The main technique for producing coherent radiation at shorter wavelengths involves a high gain FEL that does not require a reflector.In the late 1990s and early 2000s, devices were first designed and built to produce high-intensity ultraviolet and vacuum ultraviolet radiation. In 2005, the FLASH installed in Hamburg,Germany,began offering soft XFEL to scientific users[59].In 2009,the Linac Coherent Light Source(LCLS)at SLAC ushered in the era of hard XFEL [60]. Subsequently, the XFEL devices in Japan (SACLA) [61] and Italy (FERMI)[62] were successfully debugged and provided services to users.At present,the soft XFEL device in Shanghai,China has been built and successfully tested [46]. We also mention the European XFEL in Germany [63], PAL XFEL at Posco in South Korea [64], and SwissFEL at PSI in Switzerland [65].

An FEL can operate in several modes like any laser,such as the FEL oscillator, amplifier, and SASE. The oscillator uses reflectors to constrain the radiation so that the radiation field passes through the undulator many times.An FEL can also function as a linear amplifier to amplify radiation with a central frequency close to the resonant condition. Owing to the need for seed light of the corresponding wavelength,this mode of operation is most easily implemented in the spectral region already covered by other types of light sources. FEL amplifiers extending to shorter wavelengths often require advanced techniques such as harmonic generation, so it is challenging to apply them to X-ray wavelengths.It is for this reason that the first scale and built XFEL devices were based on the SASE mode [58].

SASE originates from the growth and saturation of spontaneous emission in a long undulator, without the reflector and oscillator.Although the SASE radiation is not completely coherent, the gain process increases the brightness of the source to about 10 orders of magnitude higher than that of traditional synchrotron sources.In order to learn more about SASE, we consider the radiation field of the FEL process given by [58]

Equation (8) is approximately obtained considering a long propagation distance,where ^z ≫1.The first term in square brackets of the formula (8) represents the coherent amplification of the external radiation signal, while the second and third terms in square brackets represent the FEL radiation caused by density and energy modulation,respectively, of the electron beams. When these modulations occur due to the shot noises of e beams, the exponential growth is known as SASE. In Eq. (8), where the normalized ordinate is ^z ≡2kuρz, and ρ is the nondimensional Pierce parameter [66],

Equation (22)indicates that the output of the saturated FEL is much stronger than the undulator emissions. Their ratio is the number (Nlcoh＞105) of particles in the coherence time. The energy of the FEL is the product of the single electron radiation energy of the undulator, square of the number of electrons in the coherence length and number of coherent regions T/tcoh.

In conclusion, the SASE FEL emits radiation from shot noise and the electron beams in the undulator can be distributed in the scale of the radiation wavelength to form periodic density modulations and achieve coherent enhancement of radiation intensity.The density modulation originates from the resonant interaction between the electron beams and X-rays in the periodic undulator. Hence,when the undulator is long enough, current of the electron beams is sufficiently large,and electron beams are of good quality (low emissivity and low energy divergence), the electrons will generate periodic density modulation at the scale of the resonant X-ray wavelength after interacting with the radiation. This results in intensities higher than incoherent undulator radiation, even though the electron beam is much longer than the radiation wavelength.In this work, we use the SASE FEL to obtain periodic microbunching e beams for generating high-quality XOAM.

4 Scheme for generating coherent X-rays carrying OAM at the SXFEL

As indicated in the recent study of the interaction between atoms and XUV spiral light[68],it is necessary to observe the non-dipole transition optionally when the photon beam has a single clear phase singularity or the position of the sample atom is close to the phase singularity. It is suggested [68] that the diffraction-limited storage ring[69]generates such XUV or X-ray spiral light under special conditions and an FEL could produce coherent vortex beams taking advantage of diverse schemes.

The phased project SXFEL [46] mainly includes the SXFEL test facility (SXFEL-TF) and SXFEL user facility(SXFEL-UF). The SXFEL-TF is mainly used to promote the research of FEL in China, such as investigating the feasibility of seed XFEL with two-stage cascade HGHG–HGHG, proposing a brand-new plan based on an EEHGHGHG cascade, and promoting the research and development of critical XFEL technology. Finally, the Chinese Academy of Sciences decided to build the SXFEL-TF at the SSRF in Shanghai. The initial commissioning was successfully performed from 2014 until the beginning of 2020. The Shanghai Institute of Applied Physics (SINAP)cooperated with the Shanghai-Tech University to upgrade the SXFEL-TF to a water window SXFEL-UF, including the development of the scientific experimental line station,upgradation of the energy of the linear accelerator to 1.5 GeV,construction of the second SASE FEL facility,as well as the construction of other necessary public facilities.It is expected to open to the public user by the end of this year.

In the newly added second SASE FEL branch, a typical polarization control setup (using helical undulators as afterburners)is being planned.We use a configuration with minor modifications to the baseline design to generate Xrays carrying OAM, as shown in Fig. 3. This special method is analyzed in greater detail in Ref. [70]. The first SASE undulators are used for generating microbunching from the electron beam, and the microbunched electron beams are then sent to the downstream helical undulators to generate coherent X-rays carrying OAM with wavelengths of ～1 nm at the second harmonic of the helical undulators.The main parameters are summarized in Table 1.

We use the SIMPLEX code[71]to simulate this process with time dependence. First, we simulated the SASE FEL part with six planar undulators, and the gain curve of the first harmonic of FEL radiation at λ1=2 nm is shown in Fig. 4(left). In order to achieve better FEL performance downstream, we selected four SASE undulator segments owing to the best total bunching factor at z=18.9 m(shown in Fig. 4 right) and relatively lower energy spread,compared to the saturation point.

Table 1 Main parameters of the X-rays carrying OAM at the SXFEL

In our simulation,the focusing magnets are periodically arranged to be compatible with the layout of the undulator.We select the FUDU (QF-u-QD-U) lattice type to satisfy our requirements.Since the gain of the FEL is related to the e beam size and angular spread, there is an optimum betatron function that maximizes the gain.Thus,the lattice function should be selected to realize the optimum betatron function over the whole undulator line. The field gradient of the focusing quadrupole magnets in the upstream planar undulator segments is 6.50744 T/m and the field gradient of the defocusing quadrupole magnets is -3.50995 T/m.The focusing and defocusing quadrupole magnets are both 0.2 m long.In addition to the lattice type,the initial Twiss parameters at the entrance of the undulator are listed as follows. The horizontal (x) and vertical (y) betatron functions are 11.8631 m and 4.61501 m,respectively.while the horizontal (x) and vertical (y) alpha functions (corresponding to the derivative of the betatron function) are 1.59554 and -0.68688, respectively. The horizontal (x)and vertical (y)betatron functions along all the four SASE undulator segments are shown in Fig. 5(left). Moreover,the lattice type in the afterburner helical undulator segments is FUDU (QF-U-QD-U), and the field gradients of the focusing and defocusing quadrupole magnets are 4.26216 T/m and -4.27089 T/m, respectively. The focusing and defocusing quadrupole magnets are both 0.2 m long. The horizontal (x) and vertical (y) betatron functions are 11.462 m and 5.04241 m,respectively,while the horizontal (x) and vertical (y) alpha functions are 0.966439 and -0.520785. The horizontal (x) and vertical(y) betatron functions along the afterburner are shown in Fig. 5(right).

In the SASE FEL, there is no special distribution of e beams when they first enter the undulator. However, the unstable SASE FEL makes the electrons produce shot noises which act on the electrons after amplification and redistribute the e beams. The profile of the emission from the device is determined by the microbunching structure and geometric configuration of the FEL radiator.We know that the emission at the fundamental modes of both planar and helical undulators has a strong peak on the axis and the screw-type phase of the XOAM originates from the spiral microbunched e beams. Based on the above discussions,these fine-tuned microbunched e beams require precise control over the electron profile upstream of the circularly polarized undulator with a particular input laser field. The fine-tuned microbunched e beam is then injected into the downstream radiator to emit and amplify coherent XOAM light. Thus, the above method requires the careful preservation of the spiral structure of the microbunching e beams when propagating into the downstream undulator.

While it is relatively difficult to maintain the spiral structures of the microbunched e beams, our simulated method does not require the preservation of the spiral structures. High-quality periodic micro-bunched electron beams are obtained with the FUDU lattice design. The micro-bunched e beams are then injected into the downstream helical undulator. The helical undulator has the same periodic structural parameters as the upstream planar undulator to obtain X-ray vortices at the second harmonic at the wavelength λ2=1 nm. The gain curve along z,transverse intensity, and horizontal phase at the exit z=8.9m are shown in Fig. 6.The typical donut shaped spatial distribution of the intensity distribution and spiral phase indicate that the second harmonic carries OAM with an azimuthal mode index of l=-1. The X-ray OAM intensity is at the sub-μJ level. We use the SASE FEL to produce microbunched e beams with a specific periodic structure when the preservation of the spiral microbunched e beams is unnecessary.Hence,this is an easier method for generating high energy XOAM.

We also discuss the stability of the pulse energy of the second harmonic radiation using the same parameters for the SASE and helical undulator but different initial macroparticle distributions, i.e., random numbers auto seeding the SASE FEL, under the time-dependent state simulation mode. In the first section of the undulator,spontaneous radiation and periodic micro-bunched e beams are generated, which then act as source e beams for the X-ray carrying OAM generation process. After ten simulations with various initial shot noise seeding in the SASE section,we find that the evolution of the bunching factor at the first harmonic along the direction of propagation in the planar undulator and the pulse energy of the second harmonic radiation in the helical undulator are similar, and stable growth within the error range. These are shown in Fig. 7. Therefore, using the SASE FEL mode as the upstream setup, it is possible to generate similar wellbunched microbunching e beams, despite fluctuations of the initial macroparticle distribution producing shot noise seeding in the SASE. The pulse energy of the second harmonic of the helical undulator is relatively stable,guaranteeing the stability of this method for generating XOAM.

It is known that when microbunched relativistic electron beams pass through the helical undulator and produce resonant interactions, the axisymmetric incident laser field transmits the structural information to the electron beams.Thus, an electron beam propagating through a long helical undulator can generate many microbunches of electrons.Each microbunch of electron beams is a helix, which can produce OAM radiation in harmonics, whose radiation powers are relatively weak.To increase the intensity of the vortex light, the special helical microbunched electron beams could be regarded as sources of OAM emission in a radiator, while carefully maintaining the spiral density distribution during propagation in the downstream undulator. In our simulation, the well-defined microbunches of electron beams are generated in the upstream planar undulator, which are then injected directly into the downstream short helical undulator. This is tuned to the harmonic and thus,the bunched e beam radiates coherently in the afterburner. The resultant vortex radiation carrying OAM is generated on the harmonics. In the downstream afterburner,only the parameter K of the helical undulator is changed. This enables it to emit coherent radiation at the second harmonic of the helical undulator,but owing to the weaker interaction between the microbunched e beam and helical undulator radiation at the second harmonic, the intensity of the OAM radiation emitted at the second harmonic is smaller than that at the first harmonic.

In order to compare with the FERMI-like but different all-helical-undulator configuration [62], which is not externally seeded possesses the same set of parameters for the accelerator beam and undulators, and only differing in the use of a helical undulator rather than linear one, we simulate its FEL output at the second harmonic.It is shown that in the afterburner configuration (Fig. 3), the FEL intensity quickly saturates within 10 m, while in the FERMI-like all-helical-undulator configuration, the FEL intensity grows slowly to saturation in 25 m, with similar FEL intensity output.However,if we use the exact FERMI configuration with external seeding, the afterburner type setup can be easily mimicked to obtain fully coherent OAM. To summarize, there are two advantages of this simple afterburner setup using the existing devices in SXFEL at Shanghai. First, the short helical undulators can be directly added after the existing SASE planar undulators, without the need to generate and preserve the complicated helical electron distribution. Second, this afterburner configuration is more cost-effective and efficient since it is based on existing devices and there is no need to build a new all-helical undulator.

5 Conclusion

We simulate a straightforward and easy-to-implement helical undulator afterburner configuration at the SXFEL to generate high power X-rays carrying OAM with wavelengths of ～1 nm.We use four SASE segments from the SASE FEL line at the SXFEL to produce periodic microbunched e beams. These are a consequence of the interaction between the initial randomly distributed electron beams and shot noise in the SASE process. The microbunched e beams are then injected into the downstream helical undulator (two segments). The scheme does not need additional dispersion devices along the transport channel of the electron beams, which can then be used to obtain optical spiral radiation with topological charge l=-1.The minus sign of the topological charge is due to the right-handed helicity of the circularly polarized undulator.We select the FUDU(QF-u-QD-U)lattice type in the SASE planar undulator and downstream helical undulator to satisfy our requirements. Using different shot noises in the initial SASE undulator, the relatively stable results are obtained.After simulation,the short helical undulators can be added directly after the existing SASE planar undulator.This configuration is more cost-effective and efficient as it is based on existing devices.

As far as we know, this is the first demonstration of the generation of intense X-ray vortices whose wavelength is～1 nm at the SXFEL in China. We expect that exciting experiments with OAM carrying X-rays at the SXFEL will be realized as soon as the SASE facility is completed and the structure of the helical undulator is well processed.These X-rays with OAM property in the ‘‘magnetic window’’ can be used in various fields of research, especially for exploring magnetic dichroism [34] and detecting microscopic magnetic vortices [35].

AcknowledgementsWe appreciate the stimulating discussions with Nanshun Huang,Zhichu Chen,Hailong Wu,Bo Liu,and Dong Wang of Shanghai Advanced Research Institute, Chinese Academy of Sciences.