3.1 Beam–target fusion

To validate the algorithm, 2D beam–target simulations were performed to analyze $\alpha$ particle generation from ${\mathrm{p}}^{11}\mathrm{B}$ fusion reactions. The simulation domain was $40\;\unicode{x3bc} \mathrm{m}\times 40\;\unicode{x3bc} \mathrm{m}$ , divided into $800\times 400$ cells, with 50 macro-particles per cell. The time step was set to ${T}_0$ = 20 fs. A cold ${}^{11}\mathrm{B}$ target with a number density of ${n}_{{}^{11}\mathrm{B}}$ = $1\times {10}^{28}$ m ${}^{-3}$ was bombarded by a proton beam with a number density of ${n}_{\mathrm{p}}$ = $1\times {10}^{23}$ m ${}^{-3}$ . The proton beam duration was 1 ps. The beam entered the domain from the left-hand boundary along the x-axis, within the range y = –2.5 to 2.5 $\unicode{x3bc}$ m. A 5  $\unicode{x3bc}$ m thick boron target was located in the region 2 < x < 7 $\unicode{x3bc}$ m, extending along y = –5 to 5 $\unicode{x3bc}$ m. Periodic boundary conditions were applied to all particles to ensure accurate tracking of $\alpha$ particle generation.

3.1.1 The total yield of $\alpha$ particles

Simulations of total $\alpha$ particle yields were performed for three proton energies of 0.675, 1.37 and 2.64 MeV. For a thin target, the $\alpha$ particle yield, ${N}_{\alpha }$ , from the fusion reaction is expressed as follows:

(9) $$\begin{align}{N}_{\alpha }={N}_{\mathrm{p}}{n}_{\mathrm{B}}\sigma \left({E}_{\mathrm{p}}\right)d,\end{align}$$

where ${N}_{\mathrm{p}}$ is the number of protons, ${n}_{\mathrm{B}}$ is the density of ${}^{11}\mathrm{B}$ nuclei, $d$ is the target thickness and $\sigma$ (E ${}_{\mathrm{p}}$ ) is the fusion cross-section. The simulated results and theoretical predictions obtained with Equation (9) are summarized in Table 4. The deviation between them is within 4%. This discrepancy mainly arises from the non-ideal monoenergetic nature of the proton source and the minor variations introduced by the energy-dependent interpolation of the reaction cross-section during the iterative computation process.

Table 4

Comparison between simulated and theoretical $\alpha$ particle yields for ${\mathrm{p}}^{11}\mathrm{B}$ fusion reactions.

Table 4 Long description

The table is organized by Proton energy in M e V, with two primary reaction headers: 11 B open parenthesis p, alpha sub 0 close parenthesis 8 Be and 11 B open parenthesis p, alpha sub 1 close parenthesis 8 Be super asterisk. Each reaction contains four columns: Cross-section in m-squared, Simulated yield, Theoretical yield, and Deviation in percent.

* At 2.64 M e V:

- For alpha sub 0: Cross-section 5.80e minus 30, Simulated 3.34e6, Theoretical 3.26e6, Deviation 2.45 percent.

- For alpha sub 1: Cross-section 2.06e minus 29, Simulated 1.13e7, Theoretical 1.16e7, Deviation 2.59 percent.

* At 1.37 M e V:

- For alpha sub 0: Cross-section 6.56e minus 31, Simulated 2.69e5, Theoretical 2.66e5, Deviation 1.12 percent.

- For alpha sub 1: Cross-section 1.74e minus 29, Simulated 7.05e6, Theoretical 7.04e6, Deviation 0.14 percent.

* At 0.675 M e V:

- For alpha sub 0: Cross-section 4.81e minus 31, Simulated 1.42e5, Theoretical 1.37e5, Deviation 3.64 percent.

- For alpha sub 1: Cross-section 1.16e minus 28, Simulated 3.31e7, Theoretical 3.29e7, Deviation 0.06 percent.

3.1.2 The energy-angle distribution of $\alpha$ particles

For a proton energy of ${E}_{\mathrm{p}}=0.675\;\mathrm{MeV}$ at ${\theta}_{\alpha}^{\mathrm{lab}}=90{}^{\circ}$ (Figure 1(a)), the energy spectrum of secondary $\alpha$ particles shows a saddle-shaped distribution, consistent with experimental data. The $\alpha$ particles are mainly emitted in two energy groups: one centered around 4 MeV and the other near 1 MeV. The strong peak below 1 MeV observed in the experimental data (black curve), which results from elastic proton scattering, does not appear in the simulation. At ${E}_{\mathrm{p}}=1.37\;\mathrm{MeV}$ (Figure 1(b)), the simulated $\alpha$ particle spectrum agrees well with the measurements, exhibiting an ${\alpha}_1$ peak at 5.2 MeV. Overall, the spectral distribution simulated shows good agreement with the experimental data in terms of the major energy peaks, spectral shape and an overall trend. Although full three-dimensional (3D) simulations would be required for a fully quantitative comparison of absolute particle yields and detailed angular distributions, the above agreement in these key spectral features supports the use of 2D simulations for benchmarking the underlying physical mechanisms. Figure 2 shows the angular distributions of $\alpha$ particles for ${E}_{\mathrm{p}}$ = 2.64 MeV. The intensities of the ${\alpha}_1$ and ${\alpha}_0$ peaks decrease with the increasing detection angle. The simulated energy range agrees well with theoretical predictions. The energies of both ${\alpha}_1$ and ${\alpha}_0$ particles remain nearly constant within 0°–10° and 170°–180°, respectively. Due to the difference in reaction $Q$ values, ${\alpha}_0$ particles have higher energies than ${\alpha}_1$ particles. The simulated angular-energy distribution agrees closely with reference data, confirming the validity of the current EPOCH implementation.

Figure 1

Spectra of the simulated $\alpha$ particles (red curves) compared with the experimental data (black curves) for (a) ${E}_{\mathrm{p}}$ = 0.675 MeV at ${\theta}_{\alpha}^{\mathrm{lab}}=90{}^{\circ}$ and (b) ${E}_{\mathrm{p}}$ = 1.37 MeV at ${\theta}_{\alpha}^{\mathrm{lab}}=30{}^{\circ}$ . The curves in green and blue represent the primary and the secondary $\alpha$ particle energy spectra from the ${}^{11}\mathrm{B}{\left(\mathrm{p},{\alpha}_1\right)}^8{\mathrm{Be}}^{\ast }$ channel. The experimental data are from Refs. [Reference Stave, Ahmed, France, Karwowski, Mueller, Prior, Spraker and Weller63,Reference Li, Yuan, Lin, Zhang and Meng64].

Figure 1 Long description

Two panels labeled a and b show energy spectra with the x-axis as E sub alpha in M e V from 0 to 8 and the y-axis as Counts.

Panel a represents E sub p equals 0.675 M e V at theta sub alpha lab equals 90 degrees.

* A black solid line for experimental total data shows a sharp narrow peak near 0.5 M e V reaching 150 counts and a broad peak centered near 4.2 M e V reaching approximately 110 counts.

* A red stepped line for simulated total data follows the broad peak but lacks the sharp low-energy peak.

* A green dotted line for primary alpha particles forms a narrow peak within the broad peak at 4.2 M e V.

* A blue dashed line for secondary alpha particles forms a wider base under the broad peak from 3.5 to 5.0 M e V and a low plateau from 0 to 3.5 M e V.

Panel b represents E sub p equals 1.37 M e V at theta sub alpha lab equals 30 degrees.

* The black experimental line shows a broad peak from 0 to 2 M e V and a higher peak near 5.2 M e V reaching 150 counts.

* The red simulated line shows a high-intensity stepped peak at 0.5 M e V and another at 5.2 M e V.

* The green dotted primary alpha line contributes to the sharp peak at 5.2 M e V.

* The blue dashed secondary alpha line matches the simulated data in the 0 to 4 M e V range and forms a lower shoulder on the 5.2 M e V peak.

Figure 2

The angular distributions of the $\alpha$ particle beams with their energies for 2.64 MeV. The theoretical distributions for ${\alpha}_1$ and ${\alpha}_0$ emissions were calculated according to Equation (2). As mentioned above, ${\alpha}_1$ and ${\alpha}_0$ are produced by ${}^{11}\mathrm{B}{\left(\mathrm{p},{\alpha}_1\right)}^8{\mathrm{Be}}^{\ast }$ and ${}^{11}\mathrm{B}{\left(\mathrm{p},{\alpha}_0\right)}^8\mathrm{Be}$ reactions, respectively.

Figure 2 Long description

A two-dimensional heat map plot.

* The horizontal x-axis represents energy E sub alpha in M e V, ranging from 0 to 9.

* The vertical y-axis represents the angle theta in degrees, ranging from 0 to 180.

* A color bar on the right indicates intensity d N sub alpha over d E d theta, ranging from 0 in white to 0.2 in dark red.

* In the top-left corner, text indicates t equals 70 T sub 0.

* The plot contains scattered blue data points forming vertical bands.

* A black dashed line labeled theory alpha sub 1 starts at approximately 4.2 M e V at 180 degrees, curves rightward to a peak near 6.8 M e V at 0 degrees, following a dense blue data ridge.

* A pink dashed line labeled theory alpha sub 0 starts at approximately 6 M e V at 180 degrees and curves rightward to approximately 9 M e V at 0 degrees, following a lighter data ridge.

* Lower energy regions between 0 and 4 M e V show diffuse vertical bands of low-intensity blue data.

3.2 Thermonuclear fusion

To compare with theoretical predictions, we performed numerical simulations in a 2D domain. The simulation box had dimensions of ${L}_x=5\kern0.22em \unicode{x3bc} \mathrm{m}$ and ${L}_y=2\kern0.22em \unicode{x3bc} \mathrm{m}$ . It was divided into $500\times 200$ cells, with each cell containing an equal number of protons and ${}^{11}\mathrm{B}$ ions. The ion number density was ${10}^{25}\kern0.1em {\mathrm{m}}^{-3}$ . The simulation time was set to $t=100\kern0.22em \mathrm{fs}$ under different temperature conditions. The initial ion velocities followed the Maxwell–Boltzmann distribution. The theoretical fusion reaction rate is expressed as $\left\langle \sigma v\right\rangle ={\left(\frac{8}{\pi \mu}\right)}^{1/2}\frac{1}{(kT)^{3/2}}{\int}_0^{\infty } E\sigma (E)\exp \left(-\frac{E}{kT}\right)\mathrm{d}E$ ^[ Reference Tentori and Belloni ^{65 ]}. The reaction rate obtained from simulations is calculated as $\left\langle \sigma v\right\rangle ={N}_{\alpha }/{n}_{\mathrm{p}}{n}_{\mathrm{B}} tV$ , where ${N}_{\alpha }$ is the number of $\alpha$ particles and $V$ is the target volume. Periodic boundary conditions were applied. As a result, the ion velocity distribution remained nearly unchanged during the simulation, and the fusion reaction rate $\left\langle \sigma v\right\rangle$ stayed approximately constant over time. This allows a direct comparison between theoretical and simulated results to evaluate the reliability of the calculated reaction rates. Figure 3 shows the fusion reaction rates at different temperatures, which exhibit close agreement between the simulation and theoretical predictions.

Figure 3

Validation of the MC ${\mathrm{p}}^{11}\mathrm{B}$ fusion module implemented within the EPOCH framework. The ${\mathrm{p}}^{11}\mathrm{B}$ fusion reaction rates are calculated within a box with the sensitivity to temperature variations.

4.1 Laser-driven NWA configuration

With the ${\mathrm{p}}^{11}\mathrm{B}$ nuclear reaction module implemented, we performed self-consistent 2D PIC-MC simulations for femtosecond (fs) laser-driven NWA targets. A schematic of the simulation setup is shown in Figure 4. The simulation box size was 35 $\unicode{x3bc}$ m $\times$ 10 $\unicode{x3bc}$ m, divided into $3500\times 1000$ cells in the $x$ and $y$ directions, respectively. The laser wavelength was ${\lambda}_0=0.8\;\unicode{x3bc} \mathrm{m}$ . A linearly $y$ polarized laser pulse with a Gaussian profile propagated along the $x$ -axis from the left-hand boundary and was normally incident on the NWA targets. The pulse duration was ${\tau}_0$ = 30 fs, with a period of ${T}_0={\lambda}_0/c$ = 2.67 fs and a spot radius of ${\sigma}_0=4.5\;\unicode{x3bc} \mathrm{m}$ . The NWA target was composed of B ${}_{18}$ H ${}_{22}$ ^[ Reference Kriš, Ehn, Kozlová, Boult, Pokorný, Gajdoš, Dudzák, Renner, Guldan, Myska, Škoda and Londesborough ^{66 ]} nanowires positioned at ${x}_0$ = 15.0 $\unicode{x3bc}$ m. A solid substrate with a thickness of $d$ = 2.0 $\unicode{x3bc}$ m was attached to the NWAs to provide mechanical support. The target density was $\rho$ = 1.1 $\mathrm{g}/{\mathrm{cm}}^3$ , and the atomic ratio of ${}^{11}\mathrm{B}$ to ${}^{10}$ B was maintained at 4:1. The initial electron and ion densities were 288 ${n}_{\mathrm{c}}$ and 48 ${n}_{\mathrm{c}}$ , respectively, where ${n}_{\mathrm{c}}={m}_{\mathrm{e}}{\omega}_0^2/(4\pi {e}^2)$ = $1.7\times {10}^{21}$   ${\mathrm{cm}}^{-3}$ is the critical plasma density. Each cell contained 50 macro-particles for each ion species and 100 macro-particles for electrons.

Figure 4

Schematic diagram of fs laser interaction with NWA targets, where $L$ , $D$ and $S$ denote the nanowire length, diameter and spacing, respectively.

4.2 Optimization of interaction parameters

Laser-driven ${\mathrm{p}}^{11}\mathrm{B}$ fusion with NWA targets involves complex nonlinear physics. The fusion yield depends strongly on both laser and target parameters. To identify the optimal parameter set for maximum yield, large-scale parameter scans are usually required, which result in high computational costs. BO is suitable for this type of complex and computationally expensive optimization problem^[ Reference Dolier, King, Wilson, Gray and McKenna ^{67 ,} Reference Kim, Botton and Zigler ^{68 ]}. As a result, we employ BO to efficiently explore the high-dimensional parameter space and then to achieve intelligent optimization of the ${\mathrm{p}}^{11}\mathrm{B}$ fusion yield. The $\alpha$ particle yield per joule is defined as the objective function. The optimized interaction parameters include the laser intensity ( $I$ ) and the nanowire length ( $L$ ), diameter ( $D$ ) and spacing ( $S$ ). The laser and target configuration is the same as the one shown in Figure 4. In the BO process, random forests are used as the surrogate model, and the upper confidence bound acquisition function is employed to balance exploration and exploitation. This strategy ensures systematic and efficient optimization. The laser intensity is selected within the range achievable by hundred-terawatt laser systems. The parameters $L$ , $D$ and $S$ are chosen within typical ranges for NWA targets. The detailed parameter ranges are summarized in Table 5.

Table 5

Parameter space used in BO.

Table 5 Long description

The table consists of three columns: Parameter, Value range, and Units.

* Row 1: Laser intensity, I. Value range is open bracket 5, 50 close bracket dot 10 super 19. Units are W / cm squared.

* Row 2: Nanowire diameter, D. Value range is open bracket 100, 1000 close bracket. Units are nm.

* Row 3: Nanowire space length, S. Value range is open bracket 100, 1000 close bracket. Units are nm.

* Row 4: Nanowire length, L. Value range is open bracket 1, 10 close bracket. Units are mu m.

Figure 5 shows the optimization of the $\alpha$ particle yield as a function of the laser and target parameters $I$ , $D$ , $S$ and $L$ . After 60 simulation iterations, all four interaction parameters exhibit good convergence. The convergence ranges are approximated to be $I=(1.0\hbox{--} 1.5)\times {10}^{20}\;\mathrm{W}/{\mathrm{cm}}^2$ , ${D=100\hbox{--} 150\;\mathrm{nm}}$ , $S=250\hbox{--} 300\;\mathrm{nm}$ and $L=9\hbox{--} 10\;\unicode{x3bc} \mathrm{m}$ . The optimal interaction parameters for maximal fusion yields are obtained to be $I=1.4\times {10}^{20}$   $\mathrm{W}/{\mathrm{cm}}^2$ , $D$ = 140 nm, $S$ = 260 nm and $L$ = 9.4 $\unicode{x3bc}$ m. Since previous studies have shown that the smaller the diameter, the higher the laser absorption efficiency^[ Reference Calestani, Villani, Cristoforetti, Brandi, Koester, Labate and Gizzi ^{69 ,} Reference Cristoforetti, Londrillo, Singh, Baffigi, D’Arrigo, Amit, Lad, Milazzo, Adak, Shaikh, Sarkar, Chatterjee, Jha, Krishnamurthy, Kumar and Gizzi ^{70 ]}, it is worth revisiting the optimal interaction parameters by focusing the diameter down to tens of nanometers. As the parameter spaces ${I=(0.8\hbox{--}2.5)\times {10}^{20}\;\mathrm{W}/{\mathrm{cm}}^2}$ , $D=50\hbox{--} 250\;\mathrm{nm}$ , $S=150\hbox{--} 400\;\mathrm{nm}$ and $L=7.5\hbox{--} 10\;\unicode{x3bc} \mathrm{m}$ are employed, the optimal interaction parameters are found to be highly similar to those from previous optimizations.

Figure 5

Optimization of the $\alpha$ particle yield determined by the nanowire length ( $L$ ), diameter ( $D$ ), spacing ( $S$ ) and laser intensity ( $I$ ). The top panel shows the measured values of the $\alpha$ particle yield as a function of the iteration number (orange points), together with the model predicted optimum after each nuclear reaction (blue curve) as well as the final optimal value from the model (red vertical dashed line). The variation of each control parameter is shown in the lower plots (points and shaded region) along with the final optimized values (black horizontal dashed line), also as functions of the iteration number. The best individual parameter is indicated by the vertical red dashed line in each plot and it can be seen that all parameters approach convergence with an increasing number of Bayesian optimization iterations (i.e., they are close to the black horizontal dashed line).

Figure 5 Long description

The figure consists of five vertically stacked panels sharing a common x-axis labeled Iteration Number ranging from zero to sixty. A vertical red dashed line intersects all panels at iteration thirty-five.

* The top panel plots Yield in units of per Joule. Orange points represent measured values that fluctuate but generally increase. A blue solid line shows the model-predicted optimum, which rises in steps and plateaus near iteration twenty-five. A black horizontal dashed line marks the final optimal yield at approximately 1.5 times 10 super 6.

* The second panel plots D in nanometers. Data points and a pink shaded region show high initial variance between 100 and 900 nanometers, converging toward a black horizontal dashed line at approximately 150 nanometers.

* The third panel plots L in micrometers. Data points and a light green shaded region show fluctuations between 1 and 10 micrometers, converging toward a black horizontal dashed line near 9.5 micrometers.

* The fourth panel plots S in nanometers. Data points and a light blue shaded region show initial variance between 200 and 1000 nanometers, converging toward a black horizontal dashed line at approximately 250 nanometers.

* The bottom panel plots I in Watts per square centimeter. Data points and a dark green shaded region show significant oscillation between 0.5 and 5 times 10 super 20, eventually stabilizing around a black horizontal dashed line at 1.4 times 10 super 20.

Figure 6 presents the correlation matrix of the four parameters determining the $\alpha$ particle yield. The correlations are quantified using Pearson’s correlation coefficient, expressed as follows:

(10) $$\begin{align}r=\frac{\sum_{i=1}^n\left({x}_i-\overline{x}\right)\left({y}_i-\overline{y}\right)}{\sqrt{\sum_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}\sqrt{\sum_{i=1}^n{\left({y}_i-\overline{y}\right)}^2}}.\end{align}$$

Figure 6

The correlation matrix of four physical quantities determining the $\alpha$ particle yield.

As shown in Figure 6, the $\alpha$ particle yield is strongly correlated with the nanowire diameter, spacing and length. Smaller diameters and spacings are associated with higher yields, while longer nanowires also enhance the yield. In contrast, the laser intensity exhibits only a weak correlation with the yield.

4.3 Enhanced ${p}^{\mathit{11}}B$ fusion yields in NWA targets

After obtaining the optimal interaction parameters, we further investigate, through 2D PIC-MC simulations, an enhanced ${\mathrm{p}}^{11}\mathrm{B}$ fusion and $\alpha$ particle production in laser-driven NWA targets.

In the following, we present the simulation results together with their underlying physics. For comparison, additional simulations are conducted for the case of a planar target.

Laser absorption is a critical factor governing laser–matter interactions. Figure 7(a) compares the absorbed laser energy and the fractions converted into ions and electrons for both targets. The NWA target shows significantly higher absorption, with a total absorption ratio of 59.9%, which is about 10 times that of the planar target. The absorbed energy is nearly equally distributed between ions (29.4%) and electrons (30.5%) at $t$ = 100 ${T}_0$ , whereas in the planar target, energy transfer is dominated by electrons. This enhanced absorption is mainly attributed to the larger effective surface area of the NWA targets.

Figure 7

(a) Total laser absorption ratio and the fraction of the laser energy converted into ions and electrons over the simulation time for the planar and NWA targets. (b) Temporal evolution of the $\alpha$ particle yield (red) and production rate (blue) for the NWA targets, together with the $\alpha$ particle yield (black) for the planar target.

Figure 7 Long description

Panel a is a line graph with an X axis labeled t in units of T sub 0 ranging from 40 to 100 and a Y axis labeled Ratio in percent ranging from 0 to 80. Red lines represent N W A targets and black lines represent Flat targets. N W A underscore tot shows a rapid increase to 60 percent by t equals 50 and remains stable. N W A underscore e peaks at 50 percent then gradually declines to 30 percent. N W A underscore i shows a steady linear increase from 0 to 30 percent. All Flat target lines (Flat underscore tot, Flat underscore e, and Flat underscore i) remain significantly lower, plateauing below 10 percent.

Panel b is a dual Y axis line graph with the same X axis. The left Y axis is N sub alpha on a logarithmic scale from 10 super 8 to 10 super 12. The right Y axis is the partial derivative of N sub alpha with respect to t in units of 1 over s, ranging from 10 super 22 to 10 super 26. A red line for N W A underscore N sub alpha rises sharply from 10 super 8 at t equals 40 to nearly 10 super 12 by t equals 100. A black line for Flat underscore N sub alpha starts later at t equals 50 and reaches approximately 5 times 10 super 9. A blue line representing the production rate N W A underscore partial derivative of N sub alpha with respect to t peaks at 3 times 10 super 24 around t equals 50 and maintains a steady level through t equals 100.

Figure 7(b) shows the temporal evolution of the $\alpha$ particle yield and production rate for the NWA targets, along with the yield for the planar target. The NWA target exhibits a two orders of magnitude increase in $\alpha$ particle yield compared with the planar target. This improvement originates from its efficient laser absorption and subsequent conversion of absorbed energy into ion kinetic energy, as shown in Figure 7(a). The production rate curve of the NWA targets indicates two distinct growth stages. The first stage, from approximately 40 ${T}_0$ to 50 ${T}_0$ , corresponds to a nonlinear growth phase, during which both the yield and the production rate increase by two orders of magnitude within about 10 ${T}_0$ . The second stage, after 50 ${T}_0$ , represents a linear growth phase. Although the production rate decreases slightly (by about 5%) between 50 ${T}_0$ and 56 ${T}_0$ , this has a negligible effect on the total yield. In this phase, the production rate remains nearly constant at $2\times {10}^{24}$   ${\mathrm{s}}^{-1}$ until the end of the simulation at $t$ = 100 ${T}_0$ .

At the simulation time of $t$ = 42 ${T}_0$ , the value of electric field ${E}_y$ in the gaps between the NWAs oscillates along the $x$ direction. The ${E}_y$ distributions in Figure 8(a) show that a fs laser pulse can penetrate deeply into the NWAs. When the laser irradiates the NWA targets, nanowire electrons are stripped from the nanowire surfaces by the intense laser field and accelerated into the inter-nanowire gaps (Figure 8(b)). This acceleration is partly attributed to the excitation of surface plasma waves, as discussed in Refs. [Reference Gizzi, Cristoforetti, Baffigi, Brandi, D’Arrigo, Fazzi, Fulgentini, Giove, Koester, Labate, Maero, Palla, Romé, Russo, Terzani and Tomassini71,Reference Cristoforetti, Baffigi, Brandi, D’Arrigo, Fazzi, Fulgentini, Giove, Koester, Labate, Maero, Palla, Romé, Russo, Terzani, Tomassini and Gizzi72], which efficiently transfer laser energy to the stripped electrons. The resulting charge separation generates a strong sheath field, ${E}_{sy}$ , normal to the nanowire surfaces^[ Reference Curtis, Calvi, Tinsley, Hollinger, Kaymak, Pukhov, Wang, Rockwood, Wang, Shlyaptsev and Rocca ^{48 ]}, with an average strength reaching approximately 1.6 ${E}_0$ ( ${E}_0={m}_{\mathrm{e}}c{\omega}_0/{q}_{\mathrm{e}}=4\times {10}^{12}\ \mathrm{V}/\mathrm{m}$ ). Protons and boron ions subsequently gain energy from ${E}_{sy}$ and are laterally accelerated into the gaps, as illustrated in Figure 8(c).

Figure 8

Spatial distribution of the electric field ${E}_y$ (a), $x$ - ${p}_y$ phase diagrams of the electrons (b) and ions (c) at $t$ = 42 ${T}_0$ .

Figure 8 Long description

Panel a shows the spatial distribution of the electric field E sub y all over E sub 0. The x-axis ranges from 15 to 25 micrometers and the y-axis from negative 1 to 1 micrometers. A color scale on the right indicates values from negative 6 in blue to 6 in red. The plot shows a series of horizontal red and blue alternating bands extending from left to right.

Panel b is an x-P sub y phase diagram for electrons. The x-axis is x in micrometers and the y-axis is P sub y all over m sub e c ranging from negative 0.3 to 0.3. The color scale represents log sub 10 of N sub e in arbitrary units from 13 in blue to 15 in red. The data shows a cloud of particles narrowing into a dense horizontal line at P sub y equals 0 as x increases toward 25.

Panel c is an x-P sub y phase diagram for ions. The x-axis is x in micrometers and the y-axis is P sub y all over 10 super 3 m sub e c ranging from negative 0.2 to 0.2. The color scale represents log sub 10 of N sub i in arbitrary units from 11 in blue to 15 in red. Similar to panel b, the ion distribution converges from a wide vertical spread at x equals 15 to a highly concentrated horizontal filament at P sub y equals 0 near x equals 25.

In addition to sheath field acceleration, the Z-pinch effect plays a crucial role^[ Reference Kaymak, Pukhov, Hollinger, Bargsten, Shlyaptsev and Rocca ^{73 ]}. In the laser field, electrons experience a collective drift along the laser propagation direction due to the $v\times B$ force. The forward electron flow near the nanowires is balanced by a return current ${J}_x$ inside the nanowires, which preserves local quasi-neutrality (upper panel of Figure 9(a)). This current generates a quasi-static azimuthal magnetic field, ${B}_z$ , reaching an average strength of 1.27 ${B}_0$ ( ${B}_0$ = ${m}_{\mathrm{e}}{\omega}_0/{q}_{\mathrm{e}}$ = $1.33\times {10}^4$  T), as shown in the lower panel of Figure 9(a). The resulting inward $J\times B$ force compresses the nanowires. This compression forms a high-density layer at the nanowire tips. The central region of each nanowire contracts radially to approximately 50 nm, and the maximum proton density exceeds the initial value by more than a factor of six (Figure 9(b)). Boron ions are less strongly compressed due to their larger mass (Figure 9(c)). As the interaction progresses, more ions are accelerated into the gaps, which leads to charge neutralization and the gradual disappearance of the sheath field. At the same time, the values of ${J}_x$ and the associated ${B}_z$ decrease (Figure 9(d)), leading to the loss of magnetic pinching. The nanowires then undergo spontaneous radial expansion dominated by inertial effects. This expansion is driven by the radial kinetic energy accumulated during the preceding Z-pinch compression, promoting density homogenization across the NWAs (Figures 9(e) and 9(f)).

Figure 9

The return current ${J}_x$ on the top half of (a) and the magnetic field ${B}_z$ on the bottom half of (a), spatial distributions of the number density for protons (b) and boron ions (c) at $t$ = 50 ${T}_0$ . (d)–(f) Spatial distributions of the same physical quantities as in (a)–(c), but at $t$ = 80 ${T}_0$ .

Figure 9 Long description

A six-panel grid labeled a through f. All panels share an x-axis from 15 to 25 micrometers and a y-axis from negative 1 to 1 micrometer.

Top row at time t equals 50 T sub 0:

Panel a top shows return current J sub x in M A per micrometer-squared with red horizontal filaments. Panel a bottom shows magnetic field B sub z all over B sub 0 with alternating red and blue horizontal bands.

Panel b shows log base 10 of proton density n sub p in inverse centimeters-cubed, featuring five distinct yellow-orange tapered jets extending from x equals 25 toward the left against a blue background.

Panel c shows log base 10 of boron ion density n sub B with a similar jet structure but slightly lower intensity.

Bottom row at time t equals 80 T sub 0:

Panel d shows the same current and magnetic field quantities as panel a, but the filaments and bands are more diffused and less intense.

Panel e shows proton density where the jets have expanded and merged significantly, filling more of the spatial area with green and yellow hues.

Panel f shows boron ion density with a similar expansion and diffusion pattern compared to the earlier time step.

Figure 10 shows that the majority of $\alpha$ particle production occurs within the NWA targets at $t$ = 50 ${T}_0$ . The fusion dynamics inside the nanowires are governed by the interplay between sheath field acceleration and Z-pinch effects. As discussed above, sheath fields induced by charge separation efficiently accelerate protons and boron ions laterally into the inter-wire gaps, while the Z-pinch effect produces a transient compression of the nanowire structure. To clarify the relative contributions of these mechanisms to the fusion reactions, we analyzed the average kinetic energies of protons and boron ions at $t$ = 50 ${T}_0$ for an exemplary single nanowire around $y$ = 0 (Figures 10(b) and 10(c)). The results show that ions inside the nanowire interior do not acquire sufficient energy to efficiently trigger fusion reactions, indicating that direct fusion driven solely by in-wire Z-pinch compression can be negligible.

Figure 10

The number density of $\alpha$ particles generated within the entire NWA targets (a), the average kinetic energy for protons (b) and boron ions (c) at $t$ = 50 ${T}_0$ in an exemplary case of a single nanowire around $y$ = 0. The black dashed line indicates the initial position of all NWA targets.

Figure 10 Long description

A three-panel set of heat maps labeled a, b, and c. All panels share a horizontal x-axis from 15 to 25 micrometers and a vertical y-axis from negative 1 to 1 micrometer. Horizontal dashed lines represent the N W A targets, and a vertical dashed line at x equals 24.5 micrometers marks the initial position.

* Panel a: Displays n sub alpha in units of centimeters to the negative 3 power. High-density clusters (dark red) are concentrated at x equals 15 to 17 micrometers, aligned with the nanowire positions. The color scale ranges from 0 to 5 times 10 super 15.

* Panel b: Displays E sub p (proton kinetic energy) in M e V. The data shows two diverging plumes originating from y equals 0 at x equals 15, extending toward the right. The energy is highest (red) at the outer edges of the plumes, reaching approximately 0.9 M e V.

* Panel c: Displays E sub B (boron ion kinetic energy) in M e V. Similar to panel b, it shows two diverging plumes. The energy scale is higher, ranging from 0 to 5.5 M e V, with the highest energy concentrated at the leading edges of the plumes.

Figure 11 presents the energy spectra of protons and boron ions at $t$ = 50 ${T}_0$ for the same nanowire, together with the ${\mathrm{p}}^{11}\mathrm{B}$ fusion cross-section $\sigma (E)$ as a function of proton and boron ion kinetic energy in the laboratory frame. The shaded regions in Figures 11(a) and 11(b) correspond to the energy intervals of high-energy protons and boron ions entering the neighboring nanowires, as marked in Figures 10(b) and 10(c). Since ions inside the nanowires remain at relatively low energies and can be approximated as a quasi-stationary target, the fusion rate can be reasonably estimated using a proton beam–target picture. As shown in Figure 11, the fusion yield is mainly associated with energetic protons accelerated by sheath fields, with energies close to the peak of the fusion cross-section. These results demonstrate that the fusion reactions occurring inside the nanowires are produced mainly by energetic protons accelerated by sheath fields from neighboring nanowires.

Figure 11

Energy spectra for protons (a) and boron ions (b) at $t$ = 50 ${T}_0$ in an exemplary case of a single nanowire around y = 0. The black line represents the cross-section $\sigma (E)$ of ${\mathrm{p}}^{11}\mathrm{B}$ fusion, as a function of proton (a) and boron ion (b) kinetic energy in the laboratory frame. The cross-section focuses mainly on the ${}^{11}\mathrm{B}{\left(\mathrm{p},{\alpha}_1\right)}^8\mathrm{Be}^{\ast}$ channel, which dominates the ${\mathrm{p}}^{11}\mathrm{B}$ reaction. The shaded regions in (a) and (b) correspond to the energy intervals of high-energy protons and boron ions entering the adjacent nanowires, as shown in Figures 10(b) and 10(c), respectively.

Figure 11 Long description

Two side-by-side line graphs labeled a and b.

Panel a shows the proton energy spectrum. The horizontal axis is E sub p in M e V ranging from 10 super negative 1 to 10 super 1 on a logarithmic scale. The left vertical axis is d N sub p over d E sub p in M e V super negative 1 ranging from 10 super 13 to 10 super 17. The right vertical axis is sigma in barn ranging from 0 to 1.2. A red line representing the spectrum at t equals 50 T sub 0 shows a general downward trend from 10 super 16 to 10 super 14. A black line representing the cross-section shows a sharp peak reaching 1.2 barn at approximately 0.6 M e V. A vertical gray shaded region covers the energy range from approximately 0.4 to 0.8 M e V.

Panel b shows the boron ion energy spectrum. The horizontal axis is E sub B in M e V ranging from 10 super negative 1 to 10 super 1. The left vertical axis is d N sub B over d E sub B in M e V super negative 1. The right vertical axis is sigma in barn. The red line shows a steady decline from 10 super 16 at 0.1 M e V down to 10 super 13 at 5 M e V. The black cross-section line shows a major peak reaching 1.2 barn at approximately 8 M e V. A vertical gray shaded region is positioned between 3 and 5 M e V.

It should be noted that the 2D simulation adopted in this study corresponds to a parallel nanosheet structure in 3D space within the context of NWA target research. This simplified model successfully reveals physical mechanisms such as enhanced laser energy absorption, sheath field acceleration and Z-pinch compression, and provides clear trends for parameter optimization. However, the 2D geometry cannot describe the more complex phenomena present in real cylindrical NWAs, and may overestimate the ion acceleration field strength and maximum ion energy^[ Reference Gizzi, Cristoforetti, Baffigi, Brandi, D’Arrigo, Fazzi, Fulgentini, Giove, Koester, Labate, Maero, Palla, Romé, Russo, Terzani and Tomassini ^{71 ,} Reference Héron, Adam and Mora ^{74 ,} Reference Raynaud, Héron and Adams ^{75 ]}. Therefore, the yield enhancement factors and optimal interaction parameters presented in this paper should be regarded as quantitative estimates guided by qualitative trends. Future work should involve 3D simulations to quantitatively assess the precise effects of geometric dimensionality on laser coupling efficiency, ion energy spectra and fusion yield.