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Fast proton bunch generation in the interaction of ultraintense laser pulses with high-density plasmas

T.Okada1, Y.Mikado and A.Abudurexiti

Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan

Energetic proton acceleration from concave targets, the fronts of which were irradiated with 40 fs laser pulses with an intensity of 1020 W/cm2, has been studied as a function of the depth of the concave shape. Three kinds of targets, a triangular concave target, a circular concave target and a parabolic concave target are considered. When the depth of the concave shape was varied, the peak proton energy showed a maximum. The underlying mechanism for the existence of a maximum peak proton energy is presented by tracing the proton trajectory. It is concluded that a parabolic concave target is the best, among the targets considered, for accelerating a proton beam, since a proton beam from a parabolic concave target goes through the strongest electric field. 1. Introduction

Energetic particles generated by laser-plasma interactions can be used in many applications, ranging from manufacturing to medicine, and for the initiation of tabletop particle accelerators for high-energy physics and fusion by fast ignition. The generation of energetic protons by the interaction of an ultrashort high intensity laser pulse with a plasma has been demonstrated in recent theoretical [1-3] and experimental [4-6] studies. It has been shown that the energy of a laser pulse can be efficiently converted into fast proton energy using foil targets. Simulations [1-3] have shown that the mechanisms for generating proton acceleration are the ambipolar field and Coulomb explosion. Energetic electrons ejected from the foil by the laser field create a strong electrostatic field, which is the main source of acceleration of protons ejected from the rear foil surface. Thus, a collimated proton beam can be produced by focusing an intense laser beam onto the surface of a solid film. It is clear that these protons can be focused if the foil surface has a curvature. This effect has been shown numerically for plasmas with sharp density gradients [2] and has also been experimentally demonstrated; however, the process has not been optimized. Most experimental high power lasers produce a pre-pulse, which generates a plasma layer with a smooth density gradient on the surface of the foil. This density gradient plays an important role in laser energy absorption [7,8]. We have reported high energy generation of an ion bunch from a triangular concave target and a circular concave target including the density gradient [10]. The effect of a plane target thickness on peak proton energy has also been described [11]. The proton focusing effect by the parabolic concave target has been considered in reference [2].

This paper reports the investigation of the influence of the depth of the concave shape on energetic proton acceleration by ultra-intense laser interaction.

2. High energy electron generation and energetic proton acceleration

We consider various types of targets to analyze the mechanism of proton acceleration in plasma layers with smooth density gradients. Based on the developed model and 2D particle-in-cell (PIC) simulations, the optimal foil target shape needed to produce maximal accelerated proton beam is investigated. Calculations of proton mobility were carried out for a plasma with the initial density profile shown in Fig.1. Simulations were performed for a laser wavelength λ of l.06 µm, beam spot radius R of 1.5λ , laser intensity I of 1020 W/cm2 and a

1Address; 2-24-16 Naka-machi, Koganei-shi, Tokyo 184-8588 Japan Fax; +81-42-388-7116, e-mail; Okada@cc.tuat.ac.jp

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pulse duration of τ of 40 fs. The laser pulse is linearly polarized, normally incident, and the intensity distribution has a Gaussian shape in space. The maximum electron density is ne = 4 nc, where nc is the critical density. The simulation region and number of particles were 118λ×17λ and 2.5×106 respectively.

Proton acceleration depends critically on the efficiency of the transfer of laser energy into high-energy electrons, since these electrons are the source of the required strong electrostatic field. Numerical simulations show that in the case of a short (40 fs) laser pulse with an intensity I of around 1019 W/cm2, the absorption coefficient is independent of the angle of incidence, with the absorption being about 14% in the absence of a pre-pulse. Similar absorption coefficient values and a plasma gradient L dependence of the absorption coefficient are reported in Ref. [7] for a 130 fs pulse with an intensity of 4×1019 W µm2/cm2. According to our simulations [9], as the laser intensity is raised to 1020 W/cm2, the absorption coefficient increases up to 30％, which demonstrates the influence of proton movement on absorption at high intensities even for a very short pulse. The analytical scaling [12] of the average electron absorption coefficient η (L,I) is close to the PIC simulation data:

0.818 18( , ) 0.05 (0.1 0.01 ) /(15 )L I L I Iη = + + + . (1)

Here, I18 is the laser intensity in units of 1018 W/cm2, L=Lω/c where L is the density gradient length, ω is the laser frequency and c is the speed of light.

In Fig.2, simulation results of the electron absorption coefficient with the plasma gradient are plotted with open circles for a slab plasma target. The solid line is the analytical prediction (Eq. (1)) and the dotted line is the numerical fit.

The absorbed laser energy is transferred to high energy electrons, which are ejected from the rear foil surface. These electrons produce an ambipolar field that causes a proton beam to be emitted from the rear foil surface. We consider the direction of a proton beam from the rear foil surface. The average energy of the high energy electrons ehε is then calculated from the ponderomotive potential for linear polarized laser radiation [13,14] as:

2 218( 1 / 2.74 1)eh em c I µε λ≈ + − (2)

where em is the electron mass and µλ is in microns (µ m). The maximal proton energy can be estimated from the following formula [15,16]

2 22 [ ( / 2 1 ( / 2 ) )]i eh pi ef E pi ef Eln t e t eε ε ω ω≈ + + (3)

where 24 /pi e iq n mω π= , im is the proton mass, q is the proton charge, 4e cn n= , eft τ≈

and 2.71828....Ee = The vector potential dependence of the laser beam intensity on the transverse coordinate y is assumed, as is an exponential drop in skin layer sl inside the plasma. 2 2

0( , ) exp( / / 2 )L sA x y A x l y R= − − (4) The maximum distance h from the foil that the electrons can travel is easily estimated by equating the electrostatic energy of the electrons to the total energy of the electrons yielding [16] 2 1/ 2( / 4 )eh ehh n qε π≈ (5) where /eh Dn Iη τ λ≈ and Dλ is the Debye radius. At distances larger than h, the electron bunch will expand radially, forming an electron cloud that continues to expand isotropically. Hence, h can be regarded as the critical distance from the surface beyond which the electron cloud can no longer be referred to as an electron beam.

The angle of ion divergence at the foil rear, which is determined by the electron cloud, can be estimated from the following formula:

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2 2

2iy eh e eh eh

i i i c e

v n e hR m c nRv n m c

εξε ε λ

= ≈ ≈ (6)

Here, the effect of the magnetic field is neglected compared to the electric field. Eq. (6) gives the angle 1sin 4α ξ−= ≈ o (at ehε ≈ 2.8 MeV, iε ≈ 7.5 MeV from Eq. ( 3 )

and 1.5R λ= ).

Fig.1. Target density profile.

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0.0 0.2 0.4 0.6 0.8 1.0

η (%

)

L/λ Fig.2. The electron absorption coefficient vs. plasma gradient length. Open circles represent the simulation results for 2010I = W/cm2. Analytical prediction obtained using Eq. (1) is shown by the solid line and the numerical fit by the dotted line.

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3. Energetic proton bunch simulation with concave targets

Proton focusing effects were verified using a model in which protons are ejected from concave targets, as shown in Fig.3. In this simulation, we assume that the length w of the concave target is constant at 3λ , the density gradient length L is constant at λ , and the depth D of the concave target is changed from 0 to λ . In Figs. 4(a) and (b), 5(a) and (b), and 6(a) and (b), spatial distributions of electron and proton densities at tω =350 ( t = 197 fs) are shown for the triangular concave target, the circular concave target and the parabolic concave target with D =λ , respectively. From these figures, bunches of protons can be focused at a given point in space by the curvature of the rear surface of the foil.

In Figs. 7(a) and (b), 8(a) and (b), and 9(a) and (b), proton beam trajectories and contours at tω =800 ( t = 449 fs) are shown for the triangular concave target, the circular concave target and the parabolic concave target, respectively. It can be obviously seen from these figures that the proton beam, accelerated by the strong electric field induced by the region of heightened electron density, is collected near the x-axis direction when y/λ is around 6～10. For the triangular concave target, as in Fig. 7(a), the peak proton energy is increased by changing D from 0 to 0.4λ because the proton beams go through the stronger electric field. However, after D=0.4λ, the peak proton energy decreased because the proton beams cross the stronger electric field, as seen in Fig. 7(b). For the circular and parabolic concave targets, the acceleration mechanism of the proton beam is the same as for the triangular concave target.

Fig.3. The Geometry of the target containing the concave cavity.

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Fig.4. Spatial distribution of (a) electron density and (b) proton density from a triangular concave target at tω = 350.

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Fig.5. Spatial distribution of (a) electron density and (b) proton density from a circular concave target at tω = 350.

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Fig.6. Spatial distribution of (a) electron density and (b) proton density from a parabolic concave target at tω = 350.

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Fig.7. Proton beam trajectories and contour of (a) D=0.4λ and (b) D=λ from a triangular concave target at tω = 800.

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Fig.8. Proton beam trajectories and contour of (a) D=0.6λ and (b) D=λ from a circular concave target at tω = 800.

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Fig.9. Proton beam trajectories and contour of (a) D=0.6λ and (b) D=λ from a parabolic concave target at tω = 800.

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4. Conclusions

There is a maximum for the peak proton energy as shown in Fig.10. For the triangular concave target, the peak proton energy is maximum at D=0.4λ which corresponds to θ ≈ 15°. For the circular concave target, the peak proton energy is maximum at D=0.6λ which corresponds to the radius of the circle r ≈ 2.2λ. For the parabolic concave target, the peak proton energy is maximum at D=0.6λ which corresponds to a ≈0.27 for the formula of the parabola of 2( 8 ) / 18x a y λ λ λ= − + . These results show that the parabolic concave target is the best for accelerating a proton beam among the three targets examined in this study [17].

Fig.10. Peak proton energy vs depth of the concave shape (D/λ) from the triangular, circular and parabolic concave targets at tω = 800.

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