Stable laser produced quasi monoenergetic proton beams from interactive laser and target shaping

In radiation pressure dominated laser ion acceleration schemes, transverse target deformation and Rayleigh-Taylor (RT) like instability always develop quickly, which break the acceleration structure, limit the final accelerated ion energy, and lower the beam quality. To overcome these issues, we propose a target design named dual parabola targets (DPT) consisting of a lateral thick part and a middle thin part, each with a parabolic front surface of different focus positions. By using such a target, through interactive laser and target shaping processes, the central part of the thin target will detach from the whole target and a micro target is formed. This enables the stable acceleration of the central part of the target to high energy with high quality since usual target deformation and RT-like instabilities with planar targets are suppressed. Furthermore, this target design reduces the laser intensity required to optimize radiation pressure acceleration by more than one order of magnitude compared to normal flat targets with similar thickness and density. Two-dimensional particle-in-cell simulations indicate that a quasi-monoenergetic proton beam with peak energy over 200 MeV and energy spread around 2% can be generated when such a solid target (with density 400nc and target thickness 0.5λ0) is irradiated by a100fs long circularly polarized laser pulse at focused intensity 21 2 ~ 9.2 10 W/cm L I × . PACS numbers: 52.38.Kd, 41.75.Jv, 52.50.Jm, 52.65.-y Email: minchen@sjtu.edu.cn ‡Email: zmsheng@sjtu.edu.cn


I. Introduction
There is increasing demand worldwide for more effective cancer therapy techniques. Among all the known methods, ion beams show incomparable advantages due to their high cure rate and painless treatment, mainly due to its unique sharp Bragg absorption peak. Usually well controlled energy spectrum ( /~1% E E Δ ) proton beams with energy around 200MeV or carbon ions with energy around 400MeV/amu and flux ≥ 10 10 s -1 are essential for practical applications of this technique. Even though the traditional accelerator technology is able to get such ion beams currently, the huge cost in the construction and maintenance of a large ion accelerator and subsequently the large cost imposed to patient may limit its wide applications. On the other hand, with the rapid development of ultra-intense laser technology [1], there have been lots of studies on ion beam generation by using laser plasma interaction (see Ref. [2] and references therein) aiming at cancer therapy and related applications [3][4][5].The method of laser driven ion beams may allow potential simplification in beam control, avoiding gantry systems with the conventional accelerator technology. Laser plasma interaction has been proved to be a promising way to obtain high energy particle beams. For example, GeV level electron beams [6] and tens of MeV level ion beams have already been demonstrated in experiments [2].Currently one of the big challenges is to produce >200MeV proton beams under feasible experimental conditions. It is much more difficult to accelerate ions than electrons because of the large mass to charge ratio of ions.
When the target thickness is much larger than the skin depth of the laser pulse, it is in the so-called "hole-boring" regime, where electrons in the front area of the target are pushed into the target by the laser ponderomotive force. The resulting charge separation field induces an intense electrostatic shock wave, which reflects and accelerates the downstream ions. At the end of this stage, almost all the ions contained in the target are accelerated to the same speed: c a Zm n Am n υ ≈ , where b υ is the so called "hole-boring" speed [36], a is the normalized laser amplitude, Z is the ion charge, m e and Am p are the electron mass and ion mass, respectively, n c and n e are the critical density and electron density, respectively. This process can repeat even after the whole target bulk is pushed off from the original target position until the end of the laser irradiation, which gives multi-staged accelerations [37]. Another one is the relativistic "light sail" regime. In this regime, the target thickness is close to the skin depth and the laser intensity is strong enough that all the target electrons are pushed into a sheath layer with thickness of the skin depth ~/ p c ω at the target rear side within the first few laser cycles of interaction.
Thus the "hole-boring" stage is almost skipped and plasma sheath layer is immediately pushed away from its original position. Afterwards, it is steadily accelerated as a whole for a long time [31]. Qiao et al. proposed the "relativistic hole-boring" regime [38], which requires that the laser intensity to be as high as to satisfy 3 0.25 so that the hole-boring velocity approaches the speed of light, realizing smooth connection between the short "hole-boring" stage and "light sail" stage. Theoretically, the optimal target thickness is = /( π) opt D a n [31,34], where D opt is normalized by the laser wavelength in vacuum λ 0 , a is the laser amplitude normalized by m e ω 0 c/e and n is the target density normalized by the critical density n c . If one takes D opt =0.5, then one can estimate that for a thin foil with density 400 c n n = the hole-boring velocity is which is in fact in the "relativistic hole-boring" regime.
Earlier theoretical and experimental studies have demonstrated that the ion energy spectrum obtained from the TNSA mechanism is broad and its high energy part accounts only a small portion of the total accelerated ions. The beam quality is still far from the requirement of most practical applications. RPA can give a much narrower energy spectrum and it may be a feasible way to obtain high energy and quasi-monoenergetic ion beams. However, there are several problems that prevent the realization of the RPA scheme in experiments. Firstly, the RPA process is usually not stable in high dimensional geometries and the acceleration structure can be destroyed by the development of transverse instabilities such as Rayleigh-Taylor (RT) like instability and transverse target deformation [45]. The RT-like instability can rapidly destroy the interacting surface and prematurely terminates the acceleration process. In order to mitigate this problem, several schemes have been proposed recently such as by use of special laser modes or different target components [38,46,47,50,51]. Secondly, the part of the target that obtains effective acceleration is usually automatically selected by the interaction process, thus the total acceleration charge is relatively low. For example, recently Yan et al. [48] found a self-organizing, quasi-stable region which can produce 1GeV nano-Coulomb proton bunches from laser foil interaction with the laser intensity of as in Ref. [48], the required laser intensity should be as high as 23 2 3.4 10 W/cḿ (corresponding to the normalized laser amplitude
To overcome these issues in the RPA process, the use of micro high-density targets (often called as mass-limited targets with transverse size of tens of micrometers and thickness of a few nanometers) has been suggested theoretically and experimentally [49,50,51]. Firstly, with such targets, target deformation effects will be largely reduced due to the relative uniform distribution of the local laser intensity. Secondly, the RT-like instability will be suppressed if the target size is less than the typical wavelength of the instability. However, how to make such a target for experiments is also challenging. Even though cluster targets may have appropriate density and enough small size, laser pulses cannot be easily focused to a single cluster. Recently, a new levitating technique is demonstrated by Sokollik et al., in which a micro target with size of 8μm is made and isolated [49] for laser acceleration experiments. This gives a promising way for laser-micro target interaction. However, for the moment, the target thickness is still too large for RPA.
In this paper, we propose that with a proper target design, the interaction between the intense laser pulse and the target can result in the fast formation of micro density targets in a controlled way before the growth of the RT like instability. As a result, a stable acceleration structure is formed, leading to the production of high quality proton beams. Furthermore, with such target design the incident laser intensity required for RPA is reduced significantly (over an order of magnitude) from what has been predicted before for normal plane targets. Although the idea of using specific target curvatures or geometry to optimize laser ion acceleration is not new, e.g., in Refs. [52][53][54][55] based upon the TNSA mechanism, our target design is different essentially in principle as shown below.

II. The dual parabola target and its performance
Our target design named dual parabola target (DPT) is shown in Fig. 1. It consists of two parts. The lateral part (side target) has a parabolic inner wall and it is focused at F 1 . The middle thin part (marked in yellow color) also has a parabola-shaped front surface with focus at F 2 . We show the geometrical light path in Fig. 1(b). As the laser pulse interacting with the DPT target, the outer part of the laser pulse (marked as L 1 ) is focused by the parabolic surface of the side thick target to F 1 and then defocuses. The defocused pulse irradiates the surrounding area of the front surface of the middle target which heats the electrons there. The part of the laser pulse next to the outer part (marked as L 2 ) is reflected by the front surface of the middle target and focused to F 2 .
The local obliquely incident laser pulse heats the electrons there intensely as well. At the same time, the central part of the laser pulse (marked as L 3 ) steadily pushes the central area of the middle target. The reason that puts F 2 away from F 1 is to protect the main area of the middle target as shown in Fig. 1(b). It turns out that the "light sail" process is well maintained with this kind of target design according to two-dimensional (2D) particle-in-cell (PIC) simulations. The stable acceleration of the middle part of the target is just due to the suppression of the target deformation and RT-like instability there. As one can see, the transverse size of this central accelerated part is about 1λ 0 , which is less than or close to the typical RT-like instability modulation period (λ 0 ) [57]. The shape of this acceleration part is evolving during the acceleration process, from a convex shape to a concave shape but remaining at small size. This enables the middle part of the DPT target to be accelerated steadily for a long time. This scheme we propose here can be considered as an expansion to the previous work by Yan et al. [48], where a self-organized dense clump at the center of a plane target is developed via combined Weibel and RT-like instabilities. However, here we control the whole process actively through target design.
Besides, the parabolic design makes the pulse focusing automatically, which reduces the requirement on initial laser intensity as shown later.
We have studied the whole acceleration process in details with our target design by comparing it with the case of a normal plane target in Fig. 3. It is found that the acceleration with the DPT target can be considered as a three-stage process. The first stage starts from the initial interaction moment at t=11T 0 and ends at around t=22T 0 when the overall central region of the middle target is going to be pushed away from the target bulk. As shown in Fig. 3(a), the marginal regions of the middle target are first heated to higher temperature, which is completely different from the case of a plane target with the central region heated to higher temperature at first as shown in Fig. 3(e). between the DPT target case and the plane target case. There is only one concentrated proton group in the DPT target case. Correspondingly, there is only one peak in its energy spectrum as shown in Fig. 4(a). However, there are two concentrated proton groups for the plane target case with the reflected protons in the high-energy group. Correspondingly, there appear two peaks in the proton energy spectrum with a wide high energy part behind the low energy peak as shown in Fig. 4(d). This high energy part that we marked as "divergent protons" is caused by the electrostatic shock which has propagated into the accelerated plasma bulk, while the shock always reflects the whole plasma bulk in the DPT target case. The third stage ends around t=45T 0 . Figures 3(d) and 3(h) plot the phase space distribution at t=45T 0 , which show similar difference between the DPT target case and the plane target case as those at t=34T 0 . Correspondingly, the energy spectrum of the DPT target case still maintains monoenergetic feature as shown in Fig. 4(a) while that of the plane target shows a much broader spectrum at this time shown in Fig. 4(d). Figures 4(b) and 4(e) show the energy spectrum of the protons in the central region (29λ 0 <y<31λ 0 ) at later times. It is found that the monoenergetic peak for the DPT target case gradually moves to the high energy region and maintains the energy spread around FWHM peak / E E D~2%. The peak energy at t=72T 0 is around E peak =203MeV. The reduction of the proton numbers in the monoenergetic peak at later time is attributed to the fact that some energetic protons propagate out of the region with transverse coordinates 29λ 0 <y<31λ 0 , within which they are counted for the distribution. Figure 4(c) shows that a well collimated monoenergetic proton beam appears marked by a black dotted circle.

III. Robustness of the DPT target performance
To check our scheme in other plasma density conditions, we performed simulations with the target density chosen as n=100n c , n=200n c and n=600n c. For each of them, we run PIC simulation to get the minimum laser intensity by which the light sail acceleration still works effectively. The energy spectrum at t=64T 0 for all the simulations are shown in Fig. 5(a). It shows that monoenergetic peaks are observed in all of these simulation cases, which shows the robustness of our DPT target design. In the RPA regime, one knows if target density increases, the required laser intensity should almost linearly increase as well for a plane target. However, with our DPT target, the required intensity can be reduced considerably. To give a measure how much the laser intensity is reduced with our target design, we define a variable F=a/(nD), where a is the dimensionless laser field amplitude, n is the target plasma density normalized by the critical density n c , and D is the target thickness normalized by the laser wavelength. This factor actually describes the ratio between the laser ponderomotive force and the maximum electrostatic field that the target foil can establish. Normally in order to push ion acceleration in the RPA regime, it is required that F >1, implying high laser intensity required. Actually, it has been pointed out that there is an optimized value F =π for a plane target in 1D model according to [31,34]. With our target design, RPA can occur with F <1. Furthermore, with the increase of plasma density, the factor F can be reduced to even less than 0.3, as shown in Fig. 5(b). This implies that the required laser intensity for RPA can be reduced some 100 times from that estimated with a plane target in 1D model under the same target thickness and density. The main reasons for this can be explained as follows: on one hand the transverse dispersion effect significantly decreases the light sail density at the later stage, and on the other hand the focusing effect of the laser increases the on-target laser intensity, which then reduces the required laser intensity for large density target.
One may be concerned by the sensitivity of particle acceleration on the target parameters. In order to demonstrate the robustness of this DPT target with regard to its geometric parameters, such as the curvature of the top of the middle target and the distance between F 1 and F 2 , we have performed more 2D-PIC simulationruns. The results are shown in Fig. 5(c). We define 1 2 F F F Δ = − , where F 1 and F 2 are both on the center axis of the DPT target and F 1 is fixed at In Fig. 5(c), "Plane" means that the middle part of the target is a plane one, i.e., its curvature is infinity. " F 0 Δ = " is the case that both focuses are at the same position.
" is just the case that discussed in Figs. 2, 3, and 4. One can see that there is no monoenergetic peak in the cases of "Plane" and " F 0 Δ = ". This can be understood that if the central is a plane target, the heating effect of the laser field L 2 schematically shown in Fig. 1(b) is greatly weakened, which is very important for the formation of the light sail. While if F 2 and F 1 are coincided with each other, the curvature of the middle part of the target is so large that the oblique incidence of the laser field deforms the middle part of it, which is harmful to producing high radiation pressure for ion acceleration. The two cases of " showing monoenergetic peaks as the case of " 0 F 4.5λ Δ = " suggests that the moderate offset of F 2 relative to F 1 does not prevent the formation of the light sail. These demonstrate the robustness of the DPT target design.

IV. Summary and discussion
In summary, we have proposed a dual parabola target (DPT) design for generating high quality proton beams in the radiation pressure acceleration (RPA) regime with 2D particle-in-cell simulation. In the case of using a plane target, both transverse target deformation and the development of the Rayleigh-Taylor like instability are inevitable, which can prevent the formation of a light sail process for effective proton acceleration. While for the case of our proposed DPT target, the laser field is redistributed at the target front surface, which makes the main acceleration part of the target detach from the whole target through the laser and target interactive processes. As a result, a micro target is automatically formed in a controlled way. This process enables the main accelerated part of the target not be affected by the usual target deformation and RT-like instabilities. Ions there can be finally accelerated to high energy with narrow energy spread.
Furthermore, this new target design allows the RPA to occur under the significantly reduced peak laser intensity (such as 1-2 orders of magnitude) as compared to the plane target case for the same target thickness and density. 2D-PIC simulation results indicate that a quasi-monoenergetic proton bunch with peak energy larger than 200 MeV and energy spread around 2% can be generated when such a target with a reasonable plasma density n=400n c and target thickness D=0.5λ 0 is irradiated by a 100fs long circularly polarized laser pulse at the focused intensity of So far our results are limited to 2D simulation. In real 3D case, the laser focusing may change the laser intensity in a way different from 2D geometry, which can change the accelerated proton energy. In this sense, our 2D results are the qualitative, which illustrate the key features with the new target design such as the central target detachment from the main target and instabilities suppression due to the resulting small target, which are expected to occur also in 3D geometry. 3D simulation requires much more computational resource, which may be tested in the future whenever it is available.
In our simulation, the radiation loss effect is not considered, which are supposed to appear for intensities above10 22 W/cm 2 [58][59][60][61]. Such effect is usually found significant when there is laser interaction with large amount of colliding energetic electrons beams. In our scheme, since there are not so much return current (electrons moving opposite to the laser propagation direction), the radiation loss effect is not observable. To