Introduction

Nowadays, laser intensity can increase up to 1022 W/cm2 1,2. Energetic protons have been obtained through different mechanisms, such as target normal sheath acceleration3,4,5,6,7,8,9,10,11,12,13,14,15,16,17, radiation pressure acceleration18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37, collisionless shock acceleration38,39,40,41,42, breakout afterburner43,44 and a combination of different mechanisms43,44,45,46. However, these ions are mainly accelerated in the laser propagation direction. Ion acceleration in azimuthal orientation is scarcely mentioned. A circularly polarized (CP) light may carry the angular momentum47. The main reason for this phenomenon is that a CP light carries an orbital angular momentum per photon. More than 70 years ago, the mechanical torque created by the transfer of angular momentum of a CP light was first observed in Beth's47 experiments. However, the small quantities of the optical angular momentum are difficult to detect in the CP light experiments.

A laser with a Gaussian mode, such as Laguerre–Gaussian (LG) mode, also possesses an orbital angular momentum48. A linearly polarized (LP) LG laser with a helical wave-front structure has a central phase singularity49. The angular momentum produced by such structure is sometimes referred to an orbital angular momentum, which is different from the spin angular momentum produced by the CP laser pulse47. LG laser pulse is circularly symmetric in the cross-section with respect to the optical axis [the direction of light propagation, Figs. 1(b) to 1(e)]. The mode of the LG laser pulse () is described by integer indices l and p, where l denotes the number of 2π phase cycles around the circumference and (p + 1) denotes the number of radial nodes in the mode profile. This study discusses the mode, where l ≠ 0 indicates the presence of an azimuthal phase term exp(-il ϕ) in the laser mode. laser carries an orbital angular momentum per photon.

Figure 1
figure 1

(a) Schematic arrangement of the 3D PIC simulation. The distributions of electric fields in the (y, z) plane at (b) x = 18 μm, (c) x = 18.2 μm, (d) x = 18.4 μm and (e) x = 18.6 μm.

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The nature of the orbital angular momentum of different LG modes has been investigated in optical trapping experiments. Allen et al.48 showed that an LG mode has a well-defined orbital angular momentum. They also observed the torque on suspended cylindrical lenses arising from the reversal helicity of an LG mode. He et al.50 demonstrated that the absorptive particles trapped in the dark central minimum of a doughnut laser pulse are set into rotation. Furthermore, the rotation particles are controlled using both the spin and the orbital angular momentum of light. In such case, the LG light is beneficial because it reduces the ac stark shift and the broadening of transitions at the trap center. Kuga et al.51 proposed to trap atoms along the beam center using an LG light. They exploited the spatial profile of LG modes with p = 0, which has the form of a ring of light. This feature is important in laser cooling and trapping experiments because the repulsive optical dipole force for blue detuned laser light restricts the atoms to the inner dark region of the laser beam, where photon scattering and the associated heating are minimized. Such hollow-structured LG laser can be used to investigate some difficult problems, such as generation of x-rays with orbital angular momentum52,53,54, plasma accelerators55,56, fast ignition for inertial confinement fusion57,58,59 and pulsars in the astrophysical environment60.

Results

In this letter, the doughnut LG laser is used for the first time in relativistic intensity laser plasma interaction. The LG laser rotates electrons and protons in the azimuthal orientation. Unlike conventional laser pulses, such as LP and CP, enhancement of the proton angular momentum along the longitudinal direction is obtained when an intense LP LG laser pulse irradiates on a thin foil. Three-dimensional (3D) PIC simulations are performed to investigate the LG laser interaction on a foil. In the simulation, the field amplitude E () of an LG laser with mode (l, p) is given by

where E0 is the peak amplitude of the electric field, r is the radius, W⊥ is the radius at which the Gaussian term falls to 1/e of its on-axis value, Wx is the pulse length in the x direction, 0 < t < 2 Wx/c, is the generalized Laguerre polynomial, k is the wave number, ω is the laser frequency, x is the distance from the beam waist, xr is the Rayleigh range, x/xr is the Guoy phase of the mode and ϕ is the azimuthal angle61. This study mainly discusses the mode of and thus, p = 0 and l = 1 are used in Eq. (1).

Fig. 2 shows the total angular momentum of the particles (electrons and protons) in 3D PIC simulations. The detailed simulation parameters are shown in the Methods section. To describe the rotation effects of mode on the plasma, the angular momentum of the particles in the x direction (the longitudinal direction) me (ypz − zpy) + mp (ypz − zpy) is calculated, where py,z = γvy,z is the velocity and is the relativistic factor. Fig. 2 further shows the simulation results at different times. The LG laser irradiates on the foil at t = 30 T and is totally reflected by the foil at t ~ 54 T. The angular momentum of the electrons and protons increases up to −1.55 × 10−17 kg·m2/s until the laser pulse is totally reflected. Thus, the trapped particles rotate by the helicity of the laser.

Figure 2
figure 2

Total angular momentums me (ypz − zpy) + mp (ypz − zpy) in the x direction.

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Fig. 3 shows the angular momentums of particles at 60 T for different laser amplitudes a0. The radius W⊥ and duration of the laser pulse remain constant for different a0. The angular momentums increase to −1.32 × 10−16 kg·m2/s for a0 = 14. From Fig. 3, a critical condition for the rotation of the particles is observed at a0 ≈ 5. Detailed simulations have proved this condition. The LG laser is almost reflected for a0 = 1 at t = 50 T (Fig. 4). The laser transmits through the foil at a0 = 5 and more portions of the laser pulse transmit at a0 = 10. Thus, the critical condition for the proton rotation may be related to the transmission of the foil18,35,36,62,63,64,65 and may be expressed by a0 ≈ πnd35,36,37, where the foil density n is normalized by nc and the foil thickness d is normalized by laser wavelength λ. The theory estimates that the laser starts to transmit through the foil at a0 ~ 6.28 for n = 2 and d = 1 according to a0 ~ πnd, which is larger than the simulation results in Figs. 4(d) to 4(f), where the laser begins to transmit through the foil at a0 = 5. This phenomenon is attributed to the enhancement of the transmission of the laser pulse caused by the self-focusing of the laser pulse and multi-dimensional instabilities, as demonstrated in our previous simulation study35,36. Fig. 3 shows that the angular momentums of particles increase when the LG laser transmits through the foil. The main reason is that more laser energy is absorbed by the foil when the laser transmits through the foil. In realistic interaction, the laser transmission may be realized by LG laser pulses with larger intensity and a larger angular momentum of the foil can be generated, similar to the case in Fig. 3.

Figure 3
figure 3

Angular momentums of particles at 60 T for different laser amplitudes a.

The initial foil density is n0 = 2 nc and the foil thickness is d = 1 μm.

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Figure 4
figure 4

LG laser interactions on the foils for (a–c) a = 1, (d–f) a = 5 and (g–i) a = 10.

The distributions of the electric fields (first column), electron density (middle column) and ion density (third column) are shown at t = 50 T.

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Fig. 5 shows the distribution of electric fields, electron densities and ion densities in the (y, z) plane inside the foil. The corresponding distributions in the (x, y) plane are shown in Figs. 4(g) to 4(i). The particles (electrons and protons) are rotated in the direction of the helicity of the beam at different position in the foil, just as shown in Fig. 5. The electrons in a ring are drilled out while a compressed point remains at the center at t = 50 T [Figs. 5(d) to 5(f)]. Such ring structure is related to the helical structure of the mode laser [Figs. 5(a) to 5(c)]. The trapping and compression of the foil at x = 30.5 μm are presented at different time window in Fig. 6, which clearly show the process of trapping and compression with time. Fig. 6(a) shows that the electrons are first dragged along the tangential direction at t = 35 T. The protons remains almost at rest due to their large mass [Fig. 6(e)]. The protons begin to be accelerated by the charge separation electric field between the electrons and protons as LG laser continues to rotate in the foil [Figs. 6(f)–6(h)]. At t = 50 T, both the electrons and protons are trapped and compressed into one point by the hollow-structured LG laser. In addition, some ripples are generated at the edge of the ring structure [Figs. 6(d) and 6(h)], which confirms that the mode laser drills in the plasma like a screw. Such hollow screw-like drill can uniformly trap and compress the plasma at the center [Figs. 5(e) and 5(h)], which may realize the screw-like drilling in the inertial confinement fusion and laser-driven particle accelerations.

Figure 5
figure 5

Distributions of (a–c) electric fields, (d–f) electron densities and (g–i) ion densities in the (y, z) plane at x = 30 μm (first column), x = 30.4 μm (second column) and x = 30.8 μm (third column) at t = 50 T.

The corresponding distributions in the (x , y) plane are shown in Figs. 4(g) to 4(i).

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Figure 6
figure 6

Distributions of (a–d) electron densities and (e–h) ion densities in the (y, z) plane at x = 30.5 μm for different time t = 35 T, 40 T, 45 T and 50 T.

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Discussion

The particles are rotated by the LG laser (Fig. 2). The transfer of angular momentum from the laser to the particles is then theoretically estimated in realistic cases. In terms of quantum mechanics, the rotation can be caused by the angular momentum of photons. The LG mode can be seen as the eigenmode of the angular momentum operator of Lz48 and carries an orbital angular momentum of per photon. The angular momentum carried by a photon of a polarized LG mode laser is , where σz is ±1 for the CP laser and 0 for the LP laser. The total angular momentum absorbed from the laser can then be approximately expressed as

where η is the absorbing ratio from the laser pulse during the interaction, and W⊥ are the radius at which the Gaussian term falls to 1/e of its on-axis value, h = 6.63 × 10−34 J·s is Plank constant, ν = 1/T is the frequency of light, hν is the energy of single photon and is the pulse duration. Pangular = 8.22 × 10−16 kg·m2/s is then obtained for an LP laser (l = 1 and σz = 0) with a0 = 10, W⊥ = 4 μm and . A ring of the target (the radius of the inner and outer ring is 2 and 4 μm, respectively) is assumed to be rotated by the hollow structure of the mode laser. Assuming that the angular momentum of the laser is totally transferred to the foil (η = 1), is obtained, where the foil density is 2 × 103 kg/m3, foil thickness is d = 1 μm, γ is the relativistic factor, ω is the angular velocity of the foil and r is the radius of the foil. Afterward, ω ≈ 1.09 × 1012 rad/s is obtained and the velocity of the particle at r = 4 μm is approximately 4.36 × 106 m/s. The angular momentum is proportional to the laser amplitude and laser duration , indicating that the angular momentum absorbed from the LG laser can be increased to a certain extent using a high-intensity long pulse based on Eq. (2). Notably, the increase of the angular momentums of particles mainly depends on the laser energy and the absorbing ratio from the laser pulse during the interaction η, as shown in Eq. (2). Clearly, a more accurate measurement is taken into account, which states that the total angular momentum is divided by the laser energy. Pangular/Plaser∝η is obtained when the laser energy is considered as . Previous studies have shown that hole boring is deeper for a higher a0 and the time of hole boring is longer for a larger 36,37. The particles can absorb more energy from the laser pulse (corresponding to a larger η) with the enhancement of the hole boring. Then, a larger Pangular is obtained according to Eq. (2). It should be noted that the total angular momentum transferred from laser to particles can be calculated from Eq. (2) when the laser pulse does not transmit the foil. In this case, the angular momentum Pangular is proportional to the laser energy absorbed by the foil (η S ). Less angular momentum may be generated when the foil is totally destroyed. In this case, a smaller η is obtained and less energy of the LG laser is contributed to total angular momentum of protons.

To show the difference of LG laser on the rotation of the particles, LP and CP laser pulses are also considered. The total angular momentum density per unit power has been defined by Allen et al., where the cases of LP (σz = 0) and CP (σz = ±1) are considered48. Fig. 7 shows that the angular momentums of the particles in the longitudinal direction me (ypz − zpy) + mp (ypz − zpy) for LG, LP and CP laser pulses. The amplitude of the LP pulse is (a0 = 7.4) and the amplitude of the CP pulse is (a0 = 5.2), where 0 < t < 2 Wx/c. The amplitude of the LG pulse is expressed by Eq. (1), where a0 = 10, W⊥ = 4 μm and Wx = 12 μm; d = 1 μm and n0 = 4 nc are used in three cases. The values of laser energy in three cases are similar. Compared with LP and CP lasers, the LG laser can generate larger angular momentum of particles (Fig. 7), indicating that the LG laser is beneficial for the rotation of the particles. Compared with case of CP, the doughnut-structured LG pulse (see Fig. 1) has an S-shape potential well just as shown in Fig. 1. More electrons and ions can be trapped in the potential well as the LG laser screw-like drill into the plasma. The LG laser also works as a fan to blow the plasmas forward, which may enhance the absorption of the laser pulse η (Fig. 5). Thus the total angular momentums of protons can be raised, just as shown in Fig. 7.

Figure 7
figure 7

Total angular momentums of protons in the longitudinal direction for LG (squares, a = 10), LP (circles, a = 7.4) and CP (triangles, a = 5.2) laser pulses at t = 60 T; d = 1 μm and n0 = 4 nc.

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In conclusion, the particles rotation alignment in the tangential direction is realized with the use of an intense LG laser pulse. Compared with LP and CP lasers, the enhancement of the proton angular momentum in the longitudinal direction is obtained when an intense LP LG laser pulse irradiating on a thin foil. The PIC simulations show that the LG pulse can drill into the plasma like a screw. The angular momentum of the particles produced by the LG laser is enhanced compared with that produced by the LP and CP pulses with similar pulse energies. It is also found that electrons and protons are trapped and uniformly compressed in the dark central minimum of the doughnut LG pulse. LG laser has been generated by several techniques. For example, a high-order Hermite–Gaussian (HG) mode can be generated by inserting an intra-cavity cross-wire into a laser cavity. An LG laser can then be obtained using a mode converter on this HG laser66. A spiral phase plate67 and a computer-generated hologram49 may be used to generate the LG modes from a fundamental Gaussian mode (TEM00). The intense LG laser exhibits potential applications in field of relativistic intensity, such as laser-driven plasma accelerators. Such LG laser can be applied in the TNSA or RPA experiments to generate a particle beam with angular momentum in the future.

Methods

The 3D simulations are performed with VORPAL, which is a relativistic, arbitrary and dimensional hybrid plasma and beam simulation code. It includes the utilities for data analysis and scripts for data visualization. Particle-in-cell (PIC) algorithm is used in VORPAL to describe the kinetic plasma model. A charge-conserving current deposition algorithm is applied in electromagnetic limit to enable the integration of Maxwell's equations without any additional divergence correction. The instantaneous charge distribution is used to calculate Poisson's equation at every time step in the electrostatic limit.

In our simulation, a 40 fs p-polarized laser pulse is incident from the left side on the foil. The dimensionless peak amplitude of the incident laser pulse is a0 = eE0/meωLc = 5 (the intensity is I = 3.4 × 1019 W/cm2), where ωL is the laser frequency, λ = 1 μm is the laser wavelength, c is the light speed in a vacuum and me and e are the electron mass and charge at rest. The radius of the laser is W⊥ = 4 μm and Wx = 12 μm. The laser front reaches the front surface of the foil at t = 30 T (T = λ/c). The foil thickness is d = 1 μm and the front surface of the foil is at x = 30 μm. The transverse range of the foil is −14 μm < y < 14 μm and −14 μm < z <14 μm. The foil is assumed to be fully ionized into protons and electrons before the arrival of the main pulse. The foil density is n0 = 2 nc, where is the critical density. A low-density step-like density profile is used to simplify the model and reduce the 3D PIC simulation time. The size of the simulation box is (60 × 60 × 60) μm and the cell number is 600 × 600 × 600. Each cell is filled with 10 protons and 10 electrons.