2. TARGET CONFIGURATION AND SIMULATION PARAMETERS

The simulations are performed with a 2D3V PIC code KLAP2D (Chen et al., Reference Chen, Sheng, Zheng, Ma and Zhang2008b). In the simulations, 1500 cells longitudinally along the z-axis and 2000 cells transversely along the y-axis constitute a 15 × 20 μm² simulation box, with absorbing boundary conditions for particles. A laser pulse linearly polarized along the y-axis with a ₀ = 6 (intensity I ₀ ≈ 4.9 × 10¹⁹ W cm⁻²) and wavelength λ₀ = 1 μm is incident on a solid plain target comprised of electrons and protons. The front side of the plain target is located at z = 5 μm, of which thickness is 1 μm and width is 20 μm. To include the prepulse effect, we employ a linear density gradient in 0.5 μm at the laser illumination surface. Figure 1 shows a conceptual diagram of the single plain target and the foil–ramparts target studied in our simulation. In the case of the single plain target [see Figure 1(a)], the initial electrons and protons peak density is 50 n _c, and the foil–ramparts target is designed to have the same configuration and particle settings with the single plain one, and together with these, two horizontal 5 μm thick, 100 n _c dense ramparts with length of h and interval of 4 μm made up of Al³⁺ and electrons attached behind the foil [see Figure 1(b)]. The initial temperature of electrons is set to be 1000 eV. About (1.4–2.0 × 10⁶) superparticles are employed in our simulations. The laser pulse coming from the left boundary has a transverse Gaussian profile with beam waist r ₀ = 2.5 μm and a trapezoidal temporal profile of duration τ = 22 T, consisting of a plateau of 20 T and rising and falling periods of 1 T each, where T is the laser period. As has been studied by Yu et al. (Reference Yu, Ma, Chen, Shao, Yu, Gu and Yin2009) and Nakamura et al. (Reference Nakamura, Kawata, Sonobe, Kong, Miyazaki, Onuma and Kikuchi2007), the optimal width between the ramparts is of the order of the laser spot size and proton beam divergence and energy characteristics have little dependence on pulse duration; so we fix the laser parameters and rampart width all through this paper.

Fig. 1. The conceptual diagram of the single plain target and the foil–ramparts target studied in our simulation. (a) The single plain target with the initial electrons and protons peak density of 50 n _c, and (b) the foil–ramparts target, which is designed to have the same configuration and particle settings with the single plain one, and together with these, a hole with length of h = 5 μm and diameter of 4 μm surrounding by two 5 μm thick, 100 n _c dense horizontal ramparts made up of Al³⁺ and electrons. The front side of the plain target is located at z = 5 μm, and its thickness and width are 1 and 20 μm, respectively. To include the prepulse effect, we employ a linear density gradient in 0.5 μm at the laser illumination surface. The initial temperature of electrons is set to be 1000 eV. The target materials have been marked.

3. EFFECT OF FOIL–RAMPARTS TARGET IN RESHAPING SHEATH FIELD AND INDUCING TRANSVERSE ELECTRIC FIELD

The main acceleration mechanism we consider here is TNSA, which means the shape of sheath electric field determined by the accelerated hot electron cloud spread is vital to the quality of the subsequently generated proton beam. Firstly, the shape of sheath electric fields for both plain and foil–ramparts target are investigated. Figures 2(a)–(f) show distributions of cycle-averaged sheath field E _sheath (E _sheath is actually the longitudinal electric field E _z) at t = 10, 20, and 30 T for plain (top) and foil–ramparts target (bottom), respectively. One can notice that in Figures 2(a)–(c) for the plain target, at early time, say t = 10 T, the maximum sheath field $E_{{\rm sheath}}^{{\rm max}} {\rm \sim}\! 1.2 \times {10^{13}}\;{\rm V}\,{{\rm m}^{{\rm -} {\rm 1}}}$ is centralized around the laser incident axis with a diameter of about 10.3 λ and the further away from the central axis, the weaker E _sheath. As the accelerated hot electrons are further exploding into the vacuum, say at t = 20 T, E _sheath is expanding along both the transverse and longitudinal directions as well, resulting in a wider and longer bell shape. What's more, E _sheath is still y-axial symmetric, but is stronger ($E_{{\rm sheath}}^{{\rm edges}} {\rm \sim} 6.6 \times {10^{12}}\;{\rm V}\,{{\rm m}^{ - 1}}$) at the target top and bottom edges than that ($E_{{\rm sheath}}^{{\rm axis}} {\rm \sim} 4.5 \times {10^{12}}\;{\rm V}\,{{\rm m}^{{\rm -} {\rm 1}}}$) in the region nearer around laser axis. At later time points t ≥ 30 T, E _sheath has expanded more decentralized, and the maximum sheath field is obviously located at the top and bottom edges of the plain target, leaving the centraxonial E _sheath much more weaker. Add it all up, the transverse edge effect of E _sheath is one of the reasons which lead to proton beam divergence and large spot size. As shown in Figures 2(d) and (f), in the case of the foil–ramparts target, the transverse edge effect of E _sheath is suppressed by the horizontal ramparts made of Al, which forces E _sheath to distribute uniformly and locally inside the ramparts, and from our simulation results, we obtain E _sheath ~9.0 × 10¹², 4.8 × 10¹², 2.1 × 10¹² V m⁻¹ at t = 10, 20, 30 T, respectively. The accelerated hot electrons in the initial proton plain target can transport in the closely behind Al ramparts and propagate forward into the vacuum. These propagating hot electrons can set up the very strong sheath field E _sheath at the tip of ramparts and the vacuum, instead of the regions like that of the plain target. This is why the foil–ramparts target can help to eliminate the edge effect.

Fig. 2. Distributions of cycle-averaged sheath electric fields E _sheath at t = 10, 20, and 30 T for plain (a)–(c) and hole-target (d)–(f), respectively. The dashed black lines show the initial inner boundaries of the Al ramparts. The electric fields are normalized by m _eω₀c/e, where m _e, ω₀, c, and e are the electron rest mass, laser angular frequency, light speed in vacuum, and electron charge, respectively.

As the shape of E _sheath evolves over time, we pick up three snap shots of sectional view (the cross-section is at z = 6.5 μm at the same time points mentioned above), and the results are presented in Figure 3. Compared with E _sheath of the plain target (red line), which is much broader and has two sharp corners almost symmetrical about the central axis (y = 10 μm), E _sheath of the foil–ramparts target (black line) is limited just tightly around central axis, being local and uniform. Moreover, for the single plain target, as we discussed above, the two corners of E _sheath are moving outwards to the top and bottom edges of the plain target and meantime the central part of E _sheath is getting weaker. The maximum E _sheath shown in Figure 3 is smaller for the foil–ramparts target due to less accelerated hot electrons produced between the ramparts than that in vacuum of the plain target rear side.

Fig. 3. Sectional view of E _sheath for the cross-section located at z = 6.5 μm at time points, (a) t = 10 T, (b) t = 20 T, and (c) t = 30 T, where the red line is for the plain target and the black line is for the foil–ramparts target. The electric fields E _sheath are cycle-averaged and normalized by m _eω₀c/e.

The foil–ramparts target not only has an advantage over the plain one in controlling the shape of E _sheath, but also has a transverse electric field E _y induced between the ramparts which does not exist in the case of the plain target. As one can see in Figure 4, E _y has opposite directions, that is, in the upper side E _y is negatively along −y, while in the lower side E _y is positively along +y; thus this transverse field tends to focus and confine the protons in the transverse direction as a tight bunch. The transverse electric field can reach as large as 6.43 × 10¹² V m⁻¹ according to our simulation results, which is approximately consistent with

$${E_y}{\rm \sim} \displaystyle{{{\Phi _y}} \over {e{{\rm \lambda} _{\rm D}}}}{\rm \sim} \displaystyle{{\left( {\sqrt {1 + a_0^2} - 1} \right){m_{\rm e}}{c^2}} \over {e{{\rm \lambda} _{\rm D}}}} = 6.71 \times {10^{12}}\;{\rm V}\,{{\rm m}^{{\rm -} {\rm 1}}}.$$

where we estimate ${{\rm \lambda} _{\rm D}} = \sqrt {{{\rm \epsilon} _0}{T_{\rm e}}/{n_{\rm e}}{e^2}} $ as the Debye length of accelerated hot electrons with temperature ${T_{\rm e}} \approx {m_{\rm e}}{c^2}$$\sqrt {1 + 2{U_{\rm p}}/{m_{\rm e}}{c^2}} \ {\rm \sim} 2.23\;{\rm MeV}$, a ₀ = eE ₀/m _eωc is the dimensionless parameter and ε₀ is the vacuum permittivity.

Fig. 4. (a) Distribution of transverse electric field E _y in the hole at t = 20 T. (b) The sectional view of transverse electric field E _y for the cross-section located at z = 11 μm at t = 20 T, where the black dashed line indicates the cross-section. The electric fields E _y are cycle-averaged and normalized by m _eω₀c/e.

To have a clear understanding about the energy distribution characteristics of generated protons in the rear side of the targets, we use Figure 5 to show the distribution of proton energy density in the region beyond z = 5 μm at time t = 30 T for both plain and foil–ramparts target. One can ensure that thanks to both the reshaped E _sheath and the induced transverse electric field E _y, protons produced by the foil–ramparts target have been confined between the ramparts with energy more greatly concentrated compared with that of the plain target.

Fig. 5. Distributions of proton energy density in the region z ≥ 5 μm at t = 30 T for (a) the plain target and (b) the foil–ramparts target. Protons’ energy density is in unit of n _cm _ec ².

4. THE DEPENDENCE OF PROTON BEAM DIVERGENCE ON LENGTH OF THE RAMPART

In Section II, when declaring the simulation parameters, we fix the width between the ramparts (D = 4 μm) and leave length h as the only geometric variable. Here we investigate the dependence of proton beam divergence on h. Figure 6 demonstrates the rear sheath field E _sheath at t = 30 T at the cross-section z = 6.5 μm for the single plain target and foil–ramparts target with different lengths h = 0.6, 1, 3, 5, and 7 μm, which are given respectively by lines in different colors as stated in legend. As we have proved above, there is severe edge effect for the plain target, leading to proton divergence. All the foil–ramparts targets have their sheath electric fields limited between the ramparts, except for the difference that the field distributions in the hole formed by the ramparts and on the margin are quite distinct. In the case of h = 0.6 μm, marginal sheath field is the strongest among the five foil–ramparts targets, which may result in more accelerated electrons in the marginal region and thus large beam divergence; and furthermore, “marginal effect” makes E _sheath fluctuate more wildly between the ramparts. For h = 1, 3, 5, and 7 μm cases, the marginal fields are suppressed significantly to equal the level of the electric field between the ramparts, and therefore, the sheath field E _sheath is reshaped to be almost uniform and centralized.

Fig. 6. Distribution of rear sheath electric fields E _sheath at t = 30 T at the cross-section z = 6.5 μm: (a) for the single plain target and tailored hole-targets with different lengths h = 0.6, 1, and 5 μm, where the black line is for the plain target, and the red, green, and blue lines are for h = 0.6,1, and 5 μm, respectively; (b) for the tailored hole-targets with different lengths h = 5, 3, and 7 μm, where the blue, red, and green lines are for h = 5, 3, and 7 μm, respectively. The electric fields E _sheath are cycle-averaged and normalized by m _eω₀c/e.

Now we focus on the dependence of proton beam divergence on length of the ramparts. Figure 7 shows the divergence spectrum of accelerated forward-going protons in the rear side (z > 6 μm) at t = 30 T for the single plain target and foil–ramparts target with different lengths h = 0.6, 1, and 5 μm. Here divergence angle is defined by the following formula:

$${{\rm \theta} _{{\rm div}}} = \arctan \displaystyle{{{\,p_y}} \over {{\,p_z}}},$$

where p _y and p _z are protons transverse and longitudinal relativistic momentum, respectively. One can see clearly that the proton beam from the plain target has two divergence angle peaks in θ_div ≈ −3.95° and 1.91°, while proton beam from the foil–ramparts target with h = 0.6 μm has an obvious angular deviation from the central axis and one angle peak in θ_div ≈ 1.32°, and proton beams from the foil–ramparts targets with h = 1 and 5 μm each has angle peaks in θ_div ≈ 0.04° and 0.19°, respectively. Although the foil–ramparts target with h = 1 μm has a smaller peak divergence angle, on the whole its divergence spectrum shape is fatter than foil–ramparts target with h = 5 μm. The accelerated proton number in the case h = 5 μm from our simulation results is about 9.65 × 10⁹.

Fig. 7. Divergence spectrum of accelerated forward-going protons in the rear side at t = 30 T for the single plain target and foil–ramparts target with different lengths, where the black line is for the plain target, and the red, green, and blue lines are for h = 0.6, 1, and 5 μm, respectively. The divergence angle is in unit of degree, and the proton number is indicated in the vertical axis.

So far, we have learned the primary function of the ramparts is to eliminate the electric sheath field “edge effect”, confine the accelerated hot electrons and focus the subsequent protons, and we have demonstrated the dependence of proton beam divergence on length of the ramparts, and next, the length range should be figured out. As we know, the accelerated hot electrons temperature is ${T_{\rm e}} \approx {m_{\rm e}}{c^2}{(1 + 1/2a_0^2 )^{1/2}}$, and according to the following formula

$${T_{\rm e}} = e{E_{\rm l}}{L_{\rm n}},$$

we can estimate the minimum length of the ramparts as the local plasma scale length ${h_{{\rm min}}}{\rm \sim} {L_{\rm n}} = {C_{\rm s}}t \approx 0.47{(1 + 1/2a_0^2 )^{1/4}}$, where ${C_{\rm s}} = \sqrt {{T_{\rm e}}/{m_{\rm i}}} $ is the ion sound speed, E _l is the longitudinal accelerating field, and t is taken to be the incident laser duration. Now we make some assumptions about the maximum value of h. Wilks et al. (Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001) found that protons gain energy proportional to the electron temperature,

$${E_{\rm p}} = {\rm \alpha} {T_{\rm e}} = q{E_{\rm l}}({\rm \alpha} {L_{\rm n}}),$$

where q is the proton charge, and α is somewhere between 2 and 12, depending on the model, and from our simulation results, we can roughly estimate α ≈ 5. As a result, by a simple algebraic transformation, we can obtain $h_{\rm max} \sim {\rm \alpha} C_{\rm s}t \approx 2.35 {(1 + 1/2a_0^2 )^{1/4}}$. Figure 8 shows the estimated range of ramparts length, and we can see using our simulation parameters, h _min ≈ 0.98 μm and h _max ≈ 4.91 μm. It is reasonable that for incident laser with larger a ₀, one should use foil–ramparts targets with longer ramparts, and the minimum length shows little change.

Fig. 8. A rough estimation of the hole length ranges depending on a ₀ of the incident laser.