1. Introduction

Accelerating charged particles is of paramount importance in a wide variety of fields, ranging from medicine [1, 2] and material science [3] to being used to resolve the smallest structures of our universe [4, 5]. During the past decade, the rapid development of high-intensity laser systems which are likely to exceed in the near future 10²⁵ W cm⁻² [6, 7] rendered particle beam creation by laser-matter interaction feasible. Laser-driven accelerators offer the unique feature of ultra-high electric field gradients of several TV m⁻¹, outperforming those in conventional accelerators by more than six orders of magnitude and thus offering the possibility of compact and low-cost devices [8]. Electron [9] and proton beams have recently been generated by focusing high-intensity laser light onto solid targets. This mechanism of target normal sheath acceleration (TNSA) [10, 11, 12, 13] is realized by the strong quasi-static electric field induced by the ionization and acceleration of electrons by the intense laser field. A further laser-plasma-interaction process, the skin-layer ponderomotive acceleration (S-LPA) resulting from the huge electric potential gradient of the plasma leads to ion beams of high density [14]. To date the energy resolution and the ion kinetic energy of the generated beams merely reach the parameters required for skin-deep neoplasm, i.e. excluding deep-seated tumors. Furthermore there is still controversy [12] which beam quality, i.e. which total particle number, kinetic energy and energy spread will be accessible with the forthcoming multi petawatt-class laser systems [6, 7].

In this paper we investigate the vacuum post-acceleration of plasma-generated ions by means of lasers in a single or crossed-beams configuration. The latter scheme of laser particle acceleration was first proposed for electrons [15, 16, 17]. The basic idea is to send the protons originating from a TNSA or S-LPA experiment through the crossing point of two laser beams at a half-angle θ with respect to the z-axis (see Fig. 1b for a scheme and a coordinate system). Within the simple plane-wave picture and assuming the laser fields to have the same amplitude, frequency, phase and to be polarized as shown in Fig. 1b, the resultant electric field component along the symmetry axis of the set-up vanishes for all points on the x-axis. At the same time, the x-component, increased by constructive interference, violently accelerates the particles. Such a coherent combination of intense beams is experimentally feasible (see p. 52 of Ref. [6]). Subsequent motion of the charged particle ejected from the focal region and no longer interacting with the laser pulse may be taken as linear. The results of our theoretical simulations demonstrate that the protons gain kinetic energies larger than 200 MeV (employing two crossed beams each with a peak intensity of 1.9×10²⁴ W cm⁻²) with an energy spread of roughly 1%. Furthermore (assuming a realistic repetition rate of 10 Hz [18, 6]), the total number of generated protons reaches 10¹⁰ min⁻¹. For the first time, all requirements are fulfilled for broader radio-oncological use [1, 2] based on an optical accelerator.

 

Fig. 1 (Color online) (a) The protons, produced by laser-plasma interaction, are injected with the angle θ_i with respect to the propagation direction of a pulsed beam through its focus. The laser field polarization is denoted by E and the propagation direction is given by k. The protons are ejected out of the focus in the polarization direction E. (b) Here, the protons are injected through the intersection point of two pulsed beams with crossing half-angle θ. The laser field polarizations are denoted by E₁ and E₂ and their propagation directions are given by k₁ and k₂. The protons are ejected in direction of the resulting electric field E.

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2. Model and results

In order to generate ultra-strong accelerating fields [6, 7] of 10²⁴ W cm⁻², one needs to focus the laser field to a beam waist radius on the order of the laser wavelength [19], which also ensures that the Lawson-Woodward theorem is violated. This necessitates an accurate description of the fields beyond the widely-used paraxial approximation. The parameters of a linearly polarized Gaussian beam which propagates in the z-direction and is polarized in the x-direction and subsequently rotated by θ_i (Fig. 1a) will be used to model the fields, i.e. the beam waist radius w₀, the Rayleigh length $z_r=\pi w_0^2/\lambda$, where λ is the laser’s wavelength, and the diffraction angle is ε = w₀/z_r = λ/(πw₀). The expressions giving the Cartesian field components E_x, E_y, E_z, B_x, B_y, B_z, as well as the expression for the power of the fields to order ε¹¹ in the diffraction angle can be found in Ref. [20, 21]. For the intensity profile of the employed Gaussian beams see Fig. 2.

 

Fig. 2 (Color online) (a) Intensity profile of two crossed Gaussian beams in the propagation plane y = 0. For visualization purpose the crossing-half angle is set to θ = 35°. The brighter areas correspond to higher field intensities. (b) Vector field plot of the polarization plane z = 0 at t = 0.2 fs for the single beam scheme (blue arrows) with power P and crossed beam scheme (red arrows) with power P/2 for each beam. The constructive interference of the crossed beams results in a higher electric field strength in the intersection volume. The background of the graph shows a density map of the electric field strength |E| of two crossed beams in the polarization plane.

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High-intensity laser systems provide their energy in short pulses which are already sufficient to accelerate particles to high velocities [22]. Employing pulsed fields also ensures that the particles injected into the focus get captured rather than reflected. To the lowest order in time, this can be described by multiplying the electromagnetic field components with a Gaussian temporal envelope factor,

The motion of an ensemble of N identical particles of mass m and charge q in the electric and magnetic fields E and B, respectively, of a laser beam is considered classically, with randomized initial distributions. The use of laser systems of high intensity (exceeding 10²⁴ W cm⁻² for protons) requires a relativistic treatment of particle motion. Thus, the dynamics is governed by the coupled Newton-Lorentz (or energy-momentum transfer) equations (given in SI units):

To obtain the kinetic energy gained by interaction with a laser beam, numerical solutions of the equations of motions will be sought. A numerical integration of Eq. (2) yields ${\mathit{\beta }}_j^{\text{fin}}$ and, hence, ${\gamma }_j^{\text{fin}}$ at a later final time t^fin taken equal to many laser field cycles. Finally, one calculates the final kinetic energy of the particle from $K_j^{\text{fin}}={\gamma }_j^{\text{fin}}{mc}^2$.

Foremost, we carry out simulations based on the coupled equations of motion Eq. (2) for an ensemble of 50 particles at later used particle densities in order to determine the dominant nearest-neighbor contribution of proton-proton interaction effects on the resulting particle beam. Due to the dominating ponderomotive laser forces which lead to a fast drifting apart of the ensemble’s particles, for relativistic laser intensities, i.e. > 10²⁴ W/cm² for protons, it turns out that the dynamics, the energy gain and its spread are influenced negligibly by inter-ionic interaction. This is in contrast to the non-relativistic laser regime. Here, the repulsive Coulomb interaction is the prevailing part of the interaction, which is in this case non-negligible compared to the electromagnetic fields of the laser. As a consequence, the accelerated ions occupy a larger phase space volume. From radio-oncological point of view we are interested in proton beams of fully relativistic energies, hence for further calculations at these energies and densities it is sufficient to study the uncoupled equations of motions only, i.e. we set ${\mathit{B}}_j^{\text{int}.}={\mathit{E}}_j^{\text{inc}.}=0$ in Eq. (2).

The definition of a coordinate system for the crossed beams set-up depicted in Fig. 1b is given by the coordinate transformations x₁ = x cosθ – zsinθ, y₁ = y, z₁ = x sinθ + zcosθ for the first beam and x₂ = x cosθ + zsinθ, y₂ = y, z₂ = −xsinθ + zcosθ for the second beam, respectively. The resultant field components which enter Eq. (2) are

In order to simulate a realistic particle injection into the focal point of the laser pulses, we consider an ensemble of 5000 particles initially randomly distributed in a micron-scale volume V_focus oriented along the z-axis. Initially, we have to ensure that the ensemble is not already exposed to the laser fields. This is realized by starting the simulations at time t ≤ −5Δt. At these initial times the electromagnetic fields are damped by the Gaussian envelope factor such that the particles’ motion is only negligibly influenced by the external laser fields. Fig. 3 compares the motion of a proton starting at the same spatial point but having two different initial times. One can see that in the unphysical case when the particle is directly exposed to the laser fields (dashed line), it immediately gains energy and is ejected out of the focus after one laser cycle. In a realistic setting, we choose an initial time of t = −10Δt: the particle slowly starts to oscillate and then gets captured and accelerated by the approaching pulse (full line). Choosing the initial time to be t = 0 would lead to an energy gain overestimation by a factor of approximately three.

 

Fig. 3 Kinetic energy K for one proton initially being at rest and located at x = z = λ/30 and y = 0. The initial time is t_i = 0 (dashed line) and t_i = −10Δt (full line). The single proton dynamics is compared at same laser parameters.

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The volume initially containing the particle ensemble has a length of the order of the laser wavelength and a radius of tens of nanometers, dependent on the focus diameter of the applied laser system, ensuring that all protons are exposed to a homogeneous field. The particles will be assumed to possess initial kinetic energies distributed normally around a mean value K̄ and having a spread ΔK. As a source we take protons originating from laser-plasma interactions, such as the S-LPA mechanism, with K̄ = 17 keV [14] and assuming a large energy spread of ΔK = 100% or from the TNSA mechanism, with K̄ = 1.2 MeV and ΔK = 25% [10]. The total power and the beam pulse duration are varied. Note that the peak intensity of one linearly polarized 10 PW laser beam focused to w₀ = 1λ is already I ∼ 9.6 × 10²³ W cm⁻² [6, 7].

Our main interest is in the energy gain, or exit kinetic energy, of the nuclei, their trajectories and, hence, the aspects that determine the quality of an accelerated beam of such nuclei. In Tab. 1, simulation results for the laser acceleration are summarized. The single and crossed beams scheme are compared at same total laser power P, pulse duration Δt and focus radius w₀. The energy gain is ranging from 59 MeV to 233 MeV in case of the crossed beams setup and from 28 MeV to 113 MeV for injection of the ensemble into the focus of only one beam, respectively. Looking at the energy spread one can see that for the crossed beams it is always 1% and a little higher for the case of the single beam setup.

Moreover, one can see from Tab. 1 that for the systems studied here, the average exit kinetic energy K̄ obeys for constant injection angle and injection energy the rough scaling behavior $\overline{K}\propto I\propto P/w_0^2$. This behavior results from the fact that the optimal acceleration regime depends strongly on the pulse duration Δt rather than only on the electric field strength. It is achieved for the laser-particle interaction length being of the same order of the Rayleigh length. In Ref. [26] the same scaling behavior was derived for electrons. Therefore, in order to maximize the energy gain of the accelerated protons we first choose the laser power P and the focus radius w₀, and then adjust to the optimal pulse duration Δt.

Table 1. Average particle kinetic energy K̄ and its percentual spread for different laser system parameters. N_i = n_i · V_focus is the number of ions one can accelerate as one bunch with $n_i^{\text{S}-\text{LPA}}\approx {10}^{21}{\text{cm}}^{-3}$ and $n_i^{\text{TNSA}}\approx {10}^{19}{\text{cm}}^{-3}$ is the ion density of the source used and V_focus denotes the volume initially containing all ions. The crossing half-angle is θ = 3°. The optimal particle injection angle for the single beam set-up is θ_i = 3° for the S-LPA source and θ_i = 50° in case of the TNSA source, respectively. For two crossed beams the particles are injected with an angle θ_c with respect to the symmetry axis (z-axis) of the laser beam configuration. In case of the S-LPA source we have θ_c = 0° and for the TNSA source θ_c = 50°.

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A further scaling law can be derived for the particle number, which is proportional to the focus volume. Using $V_{\text{focus}}\propto w_0^2\cdot z_r$ and the definition of the Rayleigh length $z_r\propto w_0^2$, one obtains $V_{\text{focus}}\propto w_0^4$. The typical particle number needed for ion cancer treatment is 10⁶–10¹⁰ per shot with a repetition rate on the order of 5 Hz [27], depending on the ionic species. For a typical interaction volume, one needs at least an ion density of n_i = 10⁶/V_focus ≈ 10²⁰ cm⁻³ in order to render our scheme feasible for medical applications. Using plasma-generated protons one obtains with the TNSA mechanism an ion density up to the order of $n_i^{\text{TNSA}}\approx {10}^{19}{\text{cm}}^{-3}$ and the S-LPA ion source generates a density of $n_i^{\text{S}-\text{LPA}}\approx {10}^{21}{\text{cm}}^{-3}$ [14]. The latter yields particle numbers of 10⁷ per laser shot. Combined with lasers operated at 10 Hz repetition rate [6, 18] this ion number is sufficient for cancer therapy while for the TNSA mechanism a modest improvement would still be necessary.

Such improvement can e.g. be achieved by substituting the assumed titanium-sapphire laser by a super-intense CO₂ laser with a typical wave length of λ = 10.6 μm. Using the fact that the waist radius w₀ ∝ λ and hence the Rayleigh length z_r ∝ λ, the focal region thus increases by three orders of magnitude. Consequently, the ion density needed decreases by three orders of magnitude and the needed laser intensity by two orders of magnitude. In the near future the use of high repetition rate laser systems [6, 18] will further decrease the number of protons needed per shot.

The combination of the laser-plasma mechanism as a proton source and the post acceleration process by means of single or crossed laser beams places laser acceleration of particles on the cusp of medical feasibility utilizing present-day or near-future laser technology [6, 7, 18], while anticipated costs are presently on the scale of those for current synchrotron facilities (cf. [28] and [29]). The rapid advancement of laser technology renders significant reduction likely for the near future. All requirements needed may be achieved: sufficient proton density and exit kinetic energies, and sharp energy spread of approximately 1 %. The scheme that we introduce in the present work calls for tight focusing mechanisms [19] and relies on results of laser-plasma-interaction research [10, 11, 12, 13].