High-energy quasi-monoenergetic proton beam from micro-tube targets driven by Laguerre–Gaussian lasers

High-energy proton beams with low energy spread and angular divergence are highly valuable in various applications, such as proton therapy, ultrafast science, and nuclear physics. Laser-driven ion sources are being explored as a more compact and cost-effective alternative to conventional accelerators. However, producing high-quality proton beams using lasers has been a significant challenge. In this study, we propose a novel approach to achieve high-energy quasi-monoenergetic proton beams by utilizing a Laguerre–Gaussian (LG) laser to irradiate a solid hollow tube target. Our three-dimensional particle-in-cell simulations demonstrate that the LG laser efficiently pulls out and accelerates electrons. As these electrons exit the rear side of the tube, a charge-separation electric field for proton acceleration is self-established. Meanwhile, a considerable part of the electrons are confined to the laser axis by the ponderomotive force of the LG laser, generating a longitudinal bunching field and transverse focusing field. As a result, protons are accelerated and focused in these fields, producing a high- quality proton beam with high energy (247 MeV) and low energy spread (∼2%) using an LG laser with an intensity of 1.9×1021 W cm−2 and a tube target with an electron density of 40nc . In comparison to a standard Gaussian laser, the LG laser exhibits superior performance in terms of proton cut-off energy, energy spread, and angular divergence.

Although laser-based proton sources hold great promise, their potential applications are limited by the quality of the sources.One such limitation is seen in the TNSA scheme, in which protons are generated by charge-separation sheath fields induced by laser-heated hot electrons [25,26].Unfortunately, the hot electrons have a wide spread, causing the protons to also diverge, leading to a typical TNSA source with an average divergence angle of about 15 • [27].Moreover, the energy spectrum of the protons is broadband and exponential due to the absence of a bunching field.Consequently, additional focusing and energy-selection devices are needed to make the TNSA source suitable for tumor therapy [28,29], which increases costs and reduces proton numbers.As for the RPA scheme, a bunching field can be formed through a large number of electrons piled up by laser ponderomotive force, but the transverse instabilities [30][31][32][33][34] and finite laser spot size [35] can break the structure of the bunching field and can result in large divergence.Therefore, obtaining high-energy proton beams with low energy spread as well as narrow angular divergence through laser remains a challenge in the field.
One area of exploration that holds great potential for enhancing the proton spectrum is the use of different topological structures for the target.Studies have observed that by inducing a tube target instead of the conventional planar geometry, effective electron heating can be achieved when a Gaussian laser propagates inside the tube.The tube target focuses the laser to achieve higher intensity [36] and guides the electrons to be collimated [37][38][39], resulting in an enhancement in proton acceleration [40].However, due to the different topological structure of the intensity profile of the Gaussian laser and the tube target, most of the laser energy is not effectively coupled with the target.
In this context, the use of Laguerre-Gaussian (LG) or vortex lasers presents a promising solution.Unlike standard Gaussian lasers, LG lasers are capable of carrying orbital angular momentum [41,42], which makes them useful for a variety of applications in the relativistic regime [43][44][45][46][47]. Studies have shown that LG lasers have unique advantages in ion acceleration.For example, experiments have demonstrated that a smaller solid angle is emitted by the TNSA source when driven by an LG laser pulse compared to a Gaussian laser pulse [48].Another study using the MVA scheme found that LG lasers can confine electrons on the axis more effectively and then retard the fading of accelerating fields, resulting in higher-energy protons [49].In addition, LG lasers with intensities exceeding 10 22 W cm −2 can accelerate protons to GeV level with low divergence [50].While LG lasers have demonstrated the ability to increase proton energy and collimation in proposed schemes, the resulting proton beams still have a significant energy spread, which currently limits their use in proton therapy and other applications requiring strict proton source quality.
In this paper, we propose a new scheme that uses LG lasers to produce high-energy proton beams with low energy spread and narrow angular divergence.As illustrated in figure 1, our scheme consists of two stages, where the first stage involves irradiating an LG laser pulse onto the tube target from the left.Electrons are extracted from the tube surface and accelerated by a longitudinal electric field to energies of hundreds of MeV.In the second stage, these dense and energetic electrons travel to the rear of the tube, creating a strong electrostatic field that accelerates the protons originally in the CH layer at the rear edge of the tube.As this happens, dephasing electrons get decelerated and confined to the laser axis by the inward ponderomotive force exerted by the LG laser.As a result, protons are accelerated within an electron cloud, experiencing a bunching field that narrows the energy spread.The confined electrons also produce a strong focusing field, which transforms the proton beam from a ring-shaped distribution into a dense high-quality proton beam at the axis of the laser.The resulting proton beam is simultaneously accelerated, bunched, and focused, leading to the production of a dense, quasi-monoenergetic proton beam with low angular divergence.

Model and simulation parameters
To validate our proposed scheme, we conduct three-dimensional (3D) particle-in-cell (PIC) simulations using the EPOCH code [51].The simulation box has dimensions of 80λ 0 (x) × 30λ 0 (y) × 30λ 0 (z), with a resolution of 1600 × 1500 × 1500 cells.A circularly polarized (CP) LG laser pulse is incident from the left side of the box (x = −15λ 0 ) at t = 0T 0 , and is focused at the front edge of the target.The laser has a normalized amplitude of a 0 = 30 (equivalent to an intensity of I 0 = 1.93 × 10 21 W cm −2 ) and a wavelength of λ 0 = 0.8 µm, with a cycle time of T 0 ≈ 2.67 fs.Its focus spot radius is r 0 = 6λ 0 , and the full width half maximum (FWHM) duration is 18.9 fs.We use a hollow tube target (made of gold) located between 0 µm < x < 16 µm, with a radius of R 0 = 6λ 0 and a transverse thickness of d 0 = 0.4λ 0 .A hydrocarbon (CH) layer is placed at the rear edge of the tube to to serve as a source of protons.The CH foil has a longitudinal length of 0.4λ 0 and a transverse thickness of d 0 = 0.4λ 0 (same as the tube).We note that the choice of a longer CH layer compared to the contaminants typically encountered in real experiments is to ensure that protons are not individually accelerated as test particles but rather experience collective acceleration.Both targets are chosen to be relativistically overdense, with an electron density of n e = 40n c , where n c = π m e c 2 /e 2 /λ 2 0 is the critical density.In this expression, m e and e are the mass and charge of an electron, respectively, and c is the speed of light.The ion species in the main target is initialized as Au 51+ , while a combination of H + and C 6+ is used in the CH foil.The density ratio of proton to carbon ions is n p : n C 6+ = 1 : 1.We set the number of macroparticles per cell for electrons, gold ions, carbon ions, and protons to be 8, 1, 4, and 32, respectively.
The initial intensity profile I(x, r, t) ≡ | ⃗ E| 2 of the incident LG laser pulse pulse is given by where ⃗ E is the electric field of the LG laser pulse, Equation (1) indicates that the LG laser has a doughnut-shaped intensity distribution with a maximum at r max = √ |ℓ|r 0 = 6λ 0 .To ensure that most electrons in the tube experience the highest intensity of the incident laser, we have chosen the tube target radius to be equal to r max .This way, multiple electrons can be pulled out of the tube and accelerated by the longitudinal electric field.Further information on this process is provided in the next section.

Electron extraction and acceleration
The transverse electric field of the LG laser in cylindrical coordinates (r, θ) is expressed by where θ is the azimuthal angle, E L is the maximum electric field amplitude and ϕ = kx represents the laser phase.We have neglected focusing terms and time variation for simplicity.We can observe that ϕ = 0 defines a spiral wavefront where E θ vanishes and E r points radially outward.Electrons along this spiral wavefront are dragged out to the outer side of the tube plasma.Similarly, ϕ = π defines another spiral wavefront where E r points radially inward, and electrons are dragged out to the inner side of the tube by this field.Figure 2(a) presents the electron density profile at t = 32T 0 , displaying the helical structure of the electron beam.These dragged-out electrons are immediately accelerated to relativistic energy and are further accelerated forward by longitudinal electric field.A random sample of energetic electrons (with energy > 10 MeV) at t = 24T 0 is plotted in the phase space of (ϕ, p x ) where ϕ and p x are normalized by 2π and m e c, respectively.Figures 2(c) and (d) present sampled electrons as blue dots from inside (r < R 0 ) and outside (r > R 0 ) of the tube respectively.The periodic accumulation of electrons in the phase space of (ϕ, p x ) indicates a spiral distribution of electrons.The electron spiral outside the tube is longitudinally separated by λ 0 /2 from the spiral inside the tube, as seen in figures 2(c) and (d).The red curve in figure 2(c) presents the variation of longitudinal electric field inside the tube at y = 5.5λ 0 and z = 0, normalized by E 0 = 2π m e c 2 /eλ 0 .The energetic electrons are mostly located in the acceleration zone of E x < 0 (marked as grey areas).The same situation occurs outside the tube, as shown in figure 2(d), where the red curve presents the variation of E x /E 0 at y = 6.5 and z = 0.
When a laser pulse propagates through a vacuum, the longitudinal electric field of the same LG laser has a peak of approximately E x /E 0 ≈ 1.2, however the maximum value of E x /E 0 is significantly higher than 1.2 when the laser interacts with the target, as illustrated in figures 2(c) and (d).This increase in E x can be attributed to the excitation of surface plasma waves (SPWs) at the interface between the laser pulse and the target material.SPWs can be excited when a laser pulse is incident parallelly into a flat surface of an overdense plasma [52][53][54][55].They produce much stronger longitudinal electric field than the laser itself.When electrons are peeled off from the tube surfaces, they get accelerated by a SPW with amplitude supposed to be E SP .Since the laser and SP wavefronts travel with different phase velocities, they will stay in phase until a phase difference ∆ϕ = k 0 L d (v ph − v SP ph )/c = π is accumulated after they propagate over a distance L d .Here, v ph and v SP ph are phase velocities of the laser field and the SPW, respectively, and k 0 is the wave number of the laser field.By using , where ω the laser frequency (the waveguide effect on the phase velocity can be neglected under our parameters), we can calculate the dephasing length In our simulations, L d is approximately 37λ 0 , which is longer than the target length.Thus the energy gain of the electrons during the acceleration can be estimated by ϵ e ≈ eE SP L 0 where L 0 = 20λ 0 is the length of the tube.
The longitudinal electric field E SP generated by SPWs is responsible for the majority of the E x field observed in our simulation.In fact, the maximum E SP recorded in our simulation is approximately 6.1E 0 or 2.45 × 10 13 V m −1 , which corresponds to an estimated maximum electron energy of about 391 MeV.By t = 40T 0 , the front tip of the electron beam has reached the rear edge of the tube target, with an effective electron temperature of around 151 MeV and a maximum energy of approximately 316 MeV (see figure 2(g)), which is in good agreement with our above estimation.
The electron beam ejected into vacuum from the rear edge of the target maintains its structure and collimation for a long time.This can be seen in figure 2(b), which shows the electron density distribution at t = 52T 0 , where the helical shape is still clearly visible.Additionally, due to the inward ponderomotive force exerted by the LG laser, a considerable number of electrons are confined around the laser axis.Typical trajectories of these trapped electrons are presented in figure 2(e), where the changing colors represent the evolution of time.We can observe that most of these electrons come from inside the tube, since the electrons outside the tube will experience divergent ponderomotive force when being ejected into vacuum.We compare the work done by the longitudinal field w x = −e ´⃗ v x • ⃗ E x dt to the sum of work done by transverse fields w y + w z = −e ´⃗ v ⊥ • ⃗ E ⊥ dt where ⊥ indicates the directions perpendicular to the laser propagation as shown in figure 2(f).At first, w x is dominant in electrons' acceleration; after these electrons leave the target, they get immediately out of phase (we call them dephasing electrons) since E SP no longer exits in vacuum.Significant reduction in longitudinal velocity can cause the divergence of electrons, however, this consequence is prevented by the ponderomotive force of the LG laser.This inward force does a lot of negative work (see red curves in figure 2(f)) and transversely decelerates electrons to confine them around the laser axis.
To evaluate the performance of the LG laser against a standard Gaussian laser with equivalent parameters, we conducted a simulation using the Gaussian laser while keeping all other parameters constant (i.e. with the same laser intensity, spot size and pulse duration).The simulation results indicate that the LG laser is able to pull out a total electron charge of 518 nC from the tube surface at t = 32T 0 , while the Gaussian laser pulls out only 349 nC.These pulled-out electrons get accelerated more efficiently by the LG laser, as shown in figure 2(g).The LG laser also ejects 90 nC of electron charge into vacuum from the rear edge of the target, compared to only 61 nC for the Gaussian case.Furthermore, we present electron density profiles at t = 52T 0 in figures 2(h) and (i).In the case of the Gaussian laser, there are no electrons in the center due to the repelling force of the ponderomotive force.These results validate the high efficiency of using an LG laser in our scheme.

Proton acceleration
To generate a high-energy monoenergetic proton beam, a long-lasting negative-gradient accelerating field is required.In our scheme, the efficient electron acceleration leads to the self-establishment of such a bunching field when the dense and energetic electrons travel to the rear edge of the tube target.Figure 3(c) presents the profile of the longitudinal averaged field at y = 0 and t = 48T 0 with a maximum of 5.6 × 10 13 V m −1 .The proton energy spectrum initially has a wide spread.However, apart from the high-energy electrons that travel ahead, there are also relative low-energy electrons well confined by the LG laser near the rear edge of the target, as already demonstrated in figure 2(b).These confined electrons have a charge of 36 nC, which is well above the total charge of protons (4.9 nC).As a result, protons are accelerated in an electron cloud, where ∂E x /∂x = 4π e(n p − n e ) < 0. This generates a longitudinal bunching field that reduces the energy spread of the proton beam.As illustrated in figure 3(e), the peak energy of the proton beam increases from 64 MeV to 198 MeV, while the energy spread decreases from 6.0% to 1.2% during the time interval between 44T 0 and 80T 0 .The bunching process lasts for about 40T 0 , thanks to the effective confinement of electrons by the LG laser, as discussed in the previous subsection.
The accumulation of electrons around the laser axis also plays a key role in the development of a strong transverse focusing field, which is evident in the averaged field profile E z shown in figure 3(d) at y = 0 and t = 40T 0 .(The displayed field represents an average over four laser cycles.Fluctuations at around x = 30λ 0 is due to the distribution of electrons).Protons experience a force that pulls them towards the laser axis in this field, as a result of which their density profile evolves, as seen in figures 3(a) and (b).Initially, the protons maintain their ring-shaped structure, as shown in figure 3(a).However, the strong transverse focusing field quickly comes into effect as the protons leave the tube, and after 32 laser cycles, a dense proton beam is formed at the laser axis.The maximum density of this beam is 3.16n c , but the protons are still highly divergent.Nevertheless, the long-lasting transverse field keeps focusing new protons towards the laser axis, improving the collimation of the proton beam over time.At t = 120T 0 , a well-focused proton beam is generated at the laser axis, with an angular divergence of 4.6 • (0.080 rad) and an emittance of 0.0084π mm mrad.The maximum density of this beam is 0.52n c , with a total of 5.5 × 10 8 protons.
Again, we compare our acceleration scheme to the Gaussian laser case with all other parameters fixed.The Gaussian laser is unable to accelerate as many electrons as the LG laser and is also unable to confine dephasing electrons, resulting in a much shorter duration for the static acceleration field.This leads to a significant reduction in proton energy, as shown in the (x, p x ) phase space in figure 4(a) and the energy spectrum (the blue curve) in figure 4(b).
Using the LG laser, the maximum proton energy reaches up to 247 MeV with an energy spread of about 2.6% at t = 120T 0 , whereas for the Gaussian laser case, the maximum proton energy is reduced to 135 MeV with an energy spread of 3.9%.Additionally, the absence of dense electrons in the laser axis leads to a reduction in the transverse focus field, resulting in a more divergent proton beam, as shown by the angular resolved energy spectra in figures 4(c) and (d).
We observe that despite having the same laser intensity, spot size, and pulse duration, the Gaussian laser possesses lower energy compared to the LG laser.In order to effectively showcase the advantages of the LG laser, we conduct a simulation where we utilize a Gaussian laser with an equal laser energy, but with an increased spot radius of r 0 = 8λ 0 and FWHM duration of 28.3 fs.The resulting spectrum of the proton beam is depicted by the yellow curve in figure 4(b).However, even with these adjustments, the cut-off energy remains approximately 50 MeV lower than that achieved with the LG laser.
The advantage of our scheme lies in the intensity distribution of the LG laser.In the case of the planar scheme used in previous studies [52,55], the Gaussian laser only focuses its highest intensity at the center point, while our LG laser covers the entire front surface of the tube target with the most intense part of the laser.This unique feature of the LG laser increases the efficiency of electron and proton acceleration.Furthermore, the planar target scheme lacks a focusing field in the z-direction, resulting in a larger divergence in that direction.In contrast, our scheme utilizing the LG laser generates a focusing field that enables symmetric focusing of protons.To illustrate the differences, we conduct a comparison between our scheme and the planar target scheme.Simulations are performed using a planar target with dimensions x × y × z = 40λ 0 × 0.4λ 0 × 45λ 0 irradiated by a y-polarized Gaussian laser with parameters a 0 = 30, r 0 =  8λ 0 and FWHM = 57 fs which has the same energy of the CP LG laser in our scheme.In figure 4(b), the proton spectrum obtained from the planar target is represented by a green curve.Comparing with the LG case featuring the tube target, we observe that the tube target outperforms the planar target in terms of proton cut-off energy and particle number.Additionally, the planar target yields a large divergence of 8.6 • in the z-direction.On the other hand, protons from the LG case are distributed symmetrically with a divergence of 4.6 • , as mentioned earlier.
We further examine the impact of the tube radius on beam quality.We vary the tube radius R 0 from 3λ 0 to 6λ 0 and adjust the laser spot radius r 0 accordingly so that r 0 = R 0 for each case.The results at t = 92T 0 are illustrated in figure 5. Firstly, we control the laser intensity to be constant across all cases.The energy spectrum exhibits a noticeable peak even as the tube radius decreases, as shown in figure 5(a).The peak energy reduces from 200 MeV to 156 MeV as R 0 shrinks from 6λ 0 to 3λ 0 due to the reduction of laser energy, while the energy spread remains around 2%, as shown in figure 5(b).Next, we investigate the scenario where the laser energy is fixed.In this case, we find that the resulting proton spectrum for R 0 = 4λ 0 exhibits minimal changes compared to the case with R 0 = 6λ 0 , despite the difference in tube radius.This observation indicates that, with a fixed laser energy, altering the tube radius does not significantly impact the proton spectrum.We also investigate the effect of laser misalignment and incidence angle on the proton spectrum.Specifically, we test two scenarios: one with a laser misalignment of y d = 1λ 0 and the other with a laser incidence angle of θ = 2 • .As shown in figure 5(c), the resulting spectrum differs only slightly from the original.

Discussion and conclusion
In this study, we have compared our new laser-driven ion source with known sources and found several advantages.Firstly, the emittance of our proton beam is less than 0.01π mm mrad, which is one order of magnitude lower than that of a typical RPA source [18].Secondly, our scheme features a long-lasting bunching process that is attributed to dense confined electrons, rather than radiation pressure.This eliminates the transverse instability issues that are often observed in the RPA scheme.Thirdly, the proton spectrum produced by our scheme has a monoenergetic component with an energy spread of about 2%, whereas the proton spectrum produced by TNSA scheme exhibits an exponential decay.This characteristic of our scheme is highly desirable for many applications, as it allows for more precise control over the ion beam energy.It is crucial to note that the monoenergetic spectrum observed in our scheme primarily arises from the bunching field generated by the dense and collimated electron beam accelerated along the tube surface.It is not a result of the reduction in proton numbers.Even if the proton number is increased by extending the length of the CH layer, the monoenergetic peak remains unchanged.Finally, our scheme has an additional strong focusing field that improves the collimation of the proton beam.This feature makes our scheme particularly suitable for applications that require a well-focused ion beam with low angular divergence.
To generate LG laser pulse utilized in our scheme, a viable method involves employing a Gaussian laser to irradiate a spiral-shaped phase plate, as suggested in previous studies [56,57].Note that the primary objective of utilizing the LG laser in our scheme is to concentrate the majority of the laser energy onto the tube while inducing an inward ponderomotive force.Alternatively, one potential solution is to divide the Gaussian pulse into multiple beamlets, creating a roughly ring-like intensity distribution without directly generating an LG laser.However, effectively focusing all the beamlets onto the tube may pose a considerable challenge.
In conclusion, we have developed an innovative and efficient proton acceleration scheme, specifically designed for use with the LG laser.By utilizing a tube target that matches the topological structure of the LG laser intensity profile, we are able to accelerate electrons efficiently.As these dense electrons travel to the rear edge of the target, a longitudinal bunching field and a transverse focusing field are spontaneously established, allowing for the acceleration and focusing of protons.Through 3D PIC simulations, we have demonstrated that our scheme can produce a high-energy quasi-monoenergetic proton beam with low angular divergence using a CP LG laser pulse.This high-quality proton beam is desirable for a wide range of applications, including cancer treatment and particle physics research.

Figure 1 .
Figure 1.Schematic diagram of a dense quasi-monoenergetic proton beam generation by an LG laser irradiating a micro-tube target.

Figure 2 .
Figure 2. (a) and (b) Electron density distributions at times t = 32T0 and t = 52T0, respectively.The blue contours represent the electrons with energy above 50 MeV, and the isosurface values are ne = 0.8nc and ne = 0.4nc, respectively.The grey contour represents the isosurface of electron density of value ne = 10nc.The (x, y) projections are slices at z = 0, the (x, z) projections are slices at y = 0, and the (y, z) projections are slices at x = 1λ0 and x = 16λ0, respectively.(c) Randomly sampled electrons with energy above 10 MeV from the area of r < R0 in the phase space of (x/λ0 − θ/π, px/mec) at t = 24T0.The red solid line indicates the variation of Ex field at z = 0 and y = 5.5λ0.(d) Same as (c), but for electrons from r > R0 and Ex field at z = 0 and y = 6.5λ0.(e)Trajectories of selected electrons, where the color represents the evolution of time.(f) Electron trajectories in the phase space of (wx, wy + wz), normalized by mec 2 .(g) Energy spectra of electrons at t = 40T0 for the LG case and the Gaussian case.(h) Electron density profile at t = 52T0, averaged over a spatial range of x = 32λ0 to x = 34λ0 for the LG case.(i) same as (h) but for the Gaussian case.

Figure 3 .
Figure 3. (a) and (b) Proton density distributions at two different times during the acceleration process, specifically at t = 48T0 and t = 80T0, respectively.The blue contours represent the isosurface value of np = 0.15nc.The (x, y) projections are slices at z = 0, the (x, z) projections are slices at y = 0, and the (y, z) projections are slices at x = 21λ0 and x = 38λ0, respectively.(c) and (d) The electrostatic longitudinal Ex field and transverse Ez field at t = 48T0, respectively.(e) The evolution of the proton peak energy (red curve) and energy spread (blue curve) during the acceleration process.

Figure 4 .
Figure 4. (a) Proton phase space of (x, px) at t = 120T0 for both the LG laser case and the Gaussian laser case.(b) Energy spectra of proton beams at t = 120T0.(c) and (d) Angular resolved energy spectra of proton beams at t = 120T0 for the Gaussian laser case and the LG laser case, respectively.

Figure 5 .
Figure 5. Energy spectra and their characteristics of resulting proton beams in different cases at t = 92T0.(a) Proton spectra for target radii of R0 = 6λ0 and R0 = 4λ0.(b) Effects of target radius on the proton peak energy and spread.(c) Proton spectra considering a laser incidence angle of θ = 2 • and a laser misalignment of y d = 1λ0 compared to the original.