Twisted light or vortex light is light with a well-defined orbital angular momentum (OAM) equal to ℓħ per photon, where ℓ is the topological charge. Twisted light has a helical wavefront and a dark center associated with phase singularity. In 1992, Allen et al.¹ proposed that twisted light can be generated in the laboratory using astigmatic optical elements. Since then, growing attention has been paid to this fundamental property of light. The OAM of optical vortices can be transferred to atoms, molecules, and nanostructures,^2–10 thereby allowing diverse applications in the visible and infrared regimes, such as in quantum information,¹¹ microscopy,¹² and micromanipulation.^13,14 To bring these applications down to the nanometer scale,¹⁵ different methods have been proposed to generate extreme-ultraviolet and x-ray vortex beams via high-order harmonic generation in gas-phase atomic targets,^16–18 plasmas,^19–21 and helical undulators.^22–24 Recently, extensive studies have been devoted to the generation of γ photons,^25–30 which could have promising applications in nuclear physics and astrophysics. Twisted photons of MeV energies can be used to study nuclear resonance fluorescence and photonuclear reactions that depend on angular momentum selection rules³⁰ and to manipulate the rotation of nuclei by absorbing high-order twisted photons at energies below those of giant dipole resonances.²⁶ Twisted photons in the MeV to GeV range may play a significant role in investigating the spin puzzle³¹ and phenomena around rotating massive astrophysical objects.^32–36

To generate a γ-ray vortex, Compton/Thomson scattering of light with angular momentum off ultrarelativistic electrons is a promising method. It has been analytically demonstrated that the radiation field of an electron moving along the ideal spiral trajectory has a periodic azimuthal dependence, i.e., accommodates OAM, and the ratio of angular momentum to energy is ℓ∕ω₀ for all harmonics.³⁷ Physically, this is realized for electron motion in a circularly polarized plane laser field³⁰ or in a helical undulator.²² To upshift the energy of the radiation to the γ-photon energy range, nonlinear Thomson scattering of a circularly polarized laser beam on an ultrarelativistic electron is considered.³⁰ Photons of circularly polarized light carry ħ spin angular momentum (SAM), and during multiphoton Thomson scattering the SAM of ñ absorbed laser photons (ñħ) is transferred to the emitted photon. Since the total absorbed angular momentum is transferred to both the spin and orbital angular momentum of the radiation, each photon of the nth harmonic can only carry (n − 1)ħ OAM. Unlike light with SAM, twisted light with OAM can provide a more efficient way to generate high-order twisted photons. Each twisted laser photon has ℓħ (ℓ = ±1, ±2, …) OAM, which can be directly transferred to radiation. Linear Thomson/Compton scattering of twisted light/photons has been demonstrated with a single plane wave electron in a quantum description²⁶ and with an electron beam in a classical treatment.²⁹ The information of the OAM of the driving laser field is encoded in the transverse structure of the field, which can be successfully sensed by a single plane wave electron in the quantum description²⁶ or by the electron beam in the classical description.²⁹ Therefore, in both cases, the OAM of the driving field is transferred to the radiation field. A question arises as to how the OAM transfer from the driving field to the radiation in Thomson scattering will change in the case of a non-plane-wave single electron, described quantum mechanically as a limited transverse wavepacket, or equivalently in the case of a single electron in the classical description.

In this paper, we study nonlinear Thomson scattering of twisted light in the classical regime in the cases of a single electron and of an electron beam. The laser intensity parameter ξ = eE₀/mcω₀ = 1 is assumed, with laser field amplitude E₀ and frequency ω₀, which allows nonlinear interaction, while, at the same time, the formation length of the radiation can be comparable to the laser wavelength. Whereas in the single-electron case, only SAM of the driving field is transferred to the radiation, in the case of an electron beam with a large transfer profile, both SAM and OAM are transmitted to the emitted radiation. In the latter case, the twisted radiation consists of harmonic components that carry ℓnħ OAM per photon of the nth harmonic.

The twisted light considered here is a linearly polarized Laguerre–Gaussian (LG) propagating along the z direction. The laser beam under the paraxial approximation takes the form³⁸where k₀ = 2π/λ₀ = ω₀/c is the wavevector, τ_p is the pulse duration, T = 2π/ω₀ is the laser period, σ_z ≡ i(αβ∗ − α∗β) (with σ_z = 0 for linear and ∓1 for right/left-hand circular polarization), andwhere $w(z)=\sqrt{1+z^2/z_R^2}$, the Rayleigh length $z_R=\pi w_0^2/{\lambda }_0^2$ (with w₀ the laser waist size), andThe laser parameters are set as ξ = 1, λ₀ = 1 μm, ℓ = 1, w₀ = 2λ₀, and ω₀τ_p/2π = 6.

We consider that the laser beam is scattered by a counterpropagating ultrarelativistic single electron initially located at x = λ₀, y = λ₀ with energy γ = 10. The ratio of angular momentum to energy of the radiation is calculated as^39,40where $\langle \overrightarrow{S}\rangle =(c/4\pi )\mathrm{R}\mathrm{e}(\overrightarrow{E}\times {\overrightarrow{H}}^{*})$ is the time-averaged Poynting vector, $d\overrightarrow{\sigma }=r^2\sin \theta d\theta d\varphi \overrightarrow{n}$ is the surface element of the spherical surface at distance r from the origin, $\overrightarrow{n}=\overrightarrow{r}/r$ is the direction toward the observer, and Ω is the solid angle. For the nth harmonic,³⁷${\overrightarrow{E}}_n$ and ${\overrightarrow{H}}_n$ are the Fourier components of the electric and magnetic fields, which can be expressed aswhere k = nω₀/c is the wavevector of the nth harmonic, andwith electron coordinate $\overrightarrow{r}(t)$ and velocity $\overrightarrow{\upsilon }(t)$. In calculating the angular momentum of radiation, one cannot drop the terms of high order in 1/r, even in the far field. Otherwise, the radiation will be approximated as a transverse field, leading to a linear momentum along the propagation direction, with a vanishing angular momentum of radiation. High-order terms give a small component along $\overrightarrow{n}$, resulting in an azimuthal component of linear momentum. Therefore, a nontrivial angular momentum independent of distance r is obtained when 1/r² terms are included.

The ratio between the angular momentum and the energy of the fundamental frequency for a linear and circular Gaussian laser beam and Laguerre–Gaussian laser field in the LG₀₁ mode are calculated respectively with Eq. (4), and the results are shown in Fig. 1. The angular momentum of radiation for a circular Gaussian beam and a circular LG₀₁ beam is qualitatively consistent with the prediction of the analytical solution given in Ref. 37, indicating that SAM of the laser photons is transferred to the angular momentum of the radiation field. However, the ratio ${\omega }_0^{\prime }\langle d{L}_{nz}/dt\rangle /\langle d{W}_n/dt\rangle \ll 1$ for both the linearly polarized Gaussian and LG₀₁ beams. The latter indicates that in the case of a single electron, the OAM of the driving laser photons is not transferred to the radiation field, although the formation length of the photon emission is comparable to the laser wavelength.

For understanding of the last property, we analyze the electron trajectories. In Fig. 2, the trajectories of the electron initially located at x₀ = 0.5w₀, y₀ = 0.5w₀ (the peak of the LG₀₁ beam) are shown in different laser fields. It appears that the trajectories of an electron in twisted and nontwisted light are quite similar as long as the lasers have the same polarization. Specifically, an electron in a linear Gaussian or linear LG₀₁ beam oscillates along the x direction and moves in the z direction, as shown in Fig. 2(a). An electron in a circular Gaussian or circular LG₀₁ beam rotates in the polarization plane and moves in the z direction, as shown in Fig. 2(b). Therefore, in the case of a single electron, its classical trajectories are mostly determined by the polarization property of the driving laser field (which defines the SAM of the field), but not by the large-scale transverse property of the field (which defines the OAM of the field). SAM of the laser field photons can be transferred to radiation by a single classical electron, but OAM cannot. The spiral motion of an electron in a circularly polarized Gaussian beam can produce a twist in radiation. Meanwhile, the OAM is a property encoded in the phase structure of the laser beam. This property of the laser beam can be transferred to the radiation if the electron is exposed to the full phase structure of the laser beam with OAM. In the case of a single electron, this is possible only in a quantum description with a large size of the electron wavepacket encompassing the field structure.

For linear Compton scattering, Thomson/Compton scattering of twisted light by an ultrarelativistic plane wave electron (an infinite transverse wavepacket) has been investigated quantum mechanically in Refs. 25–27, and the results suggest that the projection of the angular momentum of the emitted twisted photon on the propagation direction is the same as that of the incoming photon. Nonlinear Thomson/Compton scattering of an intense twisted laser field is difficult to treat fully quantum mechanically, but classical simulation with an electron beam, which mimics the plane wave case, may shed some light on this problem.

In this section, we consider nonlinear Thomson scattering of a twisted strong laser beam off an electron beam. The laser considered here is a linearly polarized LG₀₁ beam, with ξ = 1, λ₀ = 1 µm, ℓ = 1, w₀ = 2λ₀, and ω₀τ_p/2π = 6. The 2 × 10⁴ electrons are initially distributed in a sheet located at z = 0. Electrons are randomly distributed with a transverse spatial Gaussian distribution $f(x)={(2\pi {\sigma }_{\perp }^2)}^{-1/2}\exp (-x^2/2{\sigma }_{\perp }^2)$, σ_⊥ = 1.2λ₀. The radius of the beam is 2λ₀, which means that only electrons with coordinates (x, y) that fulfill $\sqrt{x^2+y^2}<r$ are accepted. The initial energy of the electron is γ = 10. The radiation fields at z = −13.6λ₀ are shown in Fig. 3. For the chosen parameters γ ≫ ξ, the emission is well collimated and the frequency of the radiation is dominated by the Doppler shift corresponding to the forward emission. Harmonics in the not very strong intensity regime (ξ ∼ 1) are weaker than those at the fundamental frequency; see Fig. 3(b). Therefore, the radiation field is governed by the fundamental frequency of forward emission. In fact, the radiation spectrum in Fig. 3(b) is dominated by the corresponding peak at the upshifted fundamental frequency ω′ = 363ω₀ ∼ 4γ²ω₀. The transverse distribution exhibits a rotating pattern characterized by topological charge ℓ = 1, which is repeated in a period T′ ∼ T/363. More interestingly, the third harmonic with frequency ω′₃ ∼ 1 keV has an LG₀₃-like transverse pattern, as shown in Fig. 3. Therefore, the radiation field of the nonlinear Thomson scattering of twisted light on an electron beam is twisted, and the harmonic order is proportional to the topological charge (Fig. 4). Thus, twisted γ-ray radiation can be generated with ultrarelativistic electrons with energies in the several hundreds of MeV to GeV range, with a high OAM of radiation. One can conjecture from the classical calculation that nonlinear Compton/Thomson scattering of twisted photons off an ultrarelativistic single plane wave electron would provide high-order twisted photons with topological charge proportional to the harmonic order. The electron in the considered setup for OAM transfer can be quantum mechanically described as a plane wave, if the transverse size of the electron wavepacket Δr_⊥e is comparable to the characteristic transverse size r_⊥f of the driving field, which contains information on OAM. In this case, the uncertainty relation Δp_⊥Δr_⊥e ∼ ħ limits the momentum spread of the electron wavepacket, Δp_⊥ ∼ ħ/r_⊥f. Assuming r_⊥f ∼ λ₀ ∼ 10⁻⁴ cm ∼10⁴ a.u., one finds Δp_⊥ ∼ 10⁻⁴ a.u.

Nonlinear Thomson scattering of light with OAM is significantly different from that of light with SAM. We provide a classical description of radiation at Thomson scattering. In the case of a single classical electron, only the SAM of the driving laser field can be transferred to radiation. Meanwhile, in the classical description, the OAM of the laser field can be transferred to radiation only when an electron beam is used. In the latter case, the radiation field has well-defined OAM with $\langle d{L}_{nz}/dt\rangle /\langle d{W}_n/dt\rangle =l/{\omega }_0^{\prime }$. Each photon in the radiation field will be twisted, if, in addition, the formation length of the radiation is large enough that all phase structure of the driving field contributes coherently to the formation of the emitted photon. Otherwise, the produced photons are not twisted, and the OAM will constitute a collective property of the radiation beam. Moreover, a part of the OAM absorbed from the laser beam may also be shared with electrons.⁴¹

Based on these classical results, we make some conjectures regarding the quantum description. Specifically, in the quantum description, an electron with limited momentum spread has a large wavepacket and experiences the complete spatial structure of the laser beam. This corresponds physically to the classical case of an electron beam. In fact, quantum study of nonlinear Compton/Thomson scattering of twisted photons off a single ultrarelativistic plane wave electron^25,26 has shown the generation of twisted photons. In contrast, a single electron quantum wavepacket with a large momentum spread and a small wavepacket size cannot experience the complete structure of the laser beam, corresponding to the case of a single classical electron. In this case, the emitted photon will not be twisted.

Thus, nonlinear Thomson scattering of twisted light provides the possibility of generating highly twisted γ photons with much higher efficiency than schemes using circular polarized Gaussian beams. In fact, twisted light with topological charge l (=1, 2, 3, …) carries lħ OAM per photon, while each photon of a circularly polarized Gaussian beam has only ħ SAM. Thus, if the electrons absorb the same number of photons, twisted light can transfer more angular momentum to radiation. Moreover, in a circularly polarized Gaussian beam, the OAM of the radiated photon due to the n-photon Thomson process is only (n − 1)ħ, because the emitted photon itself has ħ SAM. Since in the considered classical regime ξ ∼ 1 the fundamental frequency component of the radiation is generally more intense than the harmonics, the emission of twisted γ photons with a circular Gaussian beam would be inefficient. The generated twisted γ photons may have potential applications in nuclear physics and laboratory astrophysics.