Relativistic proton emission from ultrahigh-energy-density nanosphere generated by microbubble implosion

In the past quarter century, the chirped-pulse-amplification (CPA) technique has increased the laser intensity more than ten million times.¹ Consequently, diverse research via laser-matter interactions has been pursued. Examples include fast ignition^2–4 and high energy particle acceleration for electrons and ions with respect to different applications.^5–8 These studies have been conducted under the relativistic electron regime,⁹ corresponding to the laser intensity I_L with $10 18 ≲ I L$ (W cm⁻²) $≲ 10 22$⁠.

Utilizing laser fields is, however, far from straightforward because of their rapid oscillations. This presents a fundamental difficulty, particularly when attempting to accelerate ions that can then be used for a range of promising applications. A common approach to circumvent this difficulty is to first heat electrons in a laser-irradiated target.^10–14 As the heated electrons expand, they generate a strong electric field at the target surface that causes ions to accelerate.^15–18 For example, Coulomb explosion is a well-known scheme for ion acceleration,^19–22 in which nm- to μm-sized clusters are irradiated by an intense laser pulse to blow off most of the electrons in a moment. The remaining ions then begin to spherically expand due to the strong Coulomb force. All these ion acceleration schemes strongly depend on the laser intensity applied on the targets.

As well as the particle acceleration, high field science such as vacuum physics^23–29 is also important and fundamental topics in high energy density physics via laser. In quantum electrodynamics (QED), the Schwinger limit, $E S = m e 2 c 3 / e ℏ ≃ 1.3 × 10 18$ V m⁻¹, is reported as the scale above which the electromagnetic field is expected to create electron-positron pairs spontaneously, where m_e is the electron mass, c is the speed of light in a vacuum, e is the elementary charge, and $ℏ$ is the reduced Planck constant. The electric field at the Schwinger limit corresponds to the laser intensity $I L ≃ 2.3 × 10 29$ W cm⁻². To date, many studies have focused on increasing laser performance^30–33 with regard to power and intensity. While the threshold for electron-positron pair production may be substantially below E_S for multiple focusing pulses,^34,35 the current “distance” in the laser intensity to the Schwinger limit is still roughly five to six orders of magnitude away even by taking such measures into account.

We have recently proposed a conceptually new approach to generate ultrahigh fields and resultant high energy protons, which is referred to as “microbubble implosion” (MBI),³⁶ and the envisioned picture is illustrated in Fig. 1(a). Suppose that a spherical micron-sized bubble, prepared artificially in a solid target, is forced to implode when placed into a heat bath composed of hot electrons [Fig. 1(b)], which are actively circulating in and outside of the bubble. For simplicity, the target is assumed to be pure hydrogen. The ion implosion continues until the ions become compressed to a nanometer scale such that their radial inward motion is halted by the resulting outward electric field. This means further compression of the original laser energy in space and time in the shape of an extremely dense ion core, which leads to generation of an ultrahigh electric field at the center. This field is much stronger than the field that initiated the implosion, and it causes a violent explosion of the compressed ions, with resulting energies many times higher than the energy gained during the implosion.

At this early stage of investigation, there are still many open questions about this novel physical concept. For example, is such a highly uniform implosion required for MBI really possible? Here, it should be noted that MBI is volumetrically driven by the long-range Coulomb force. That is, local nonuniform distributions of the electrons in and outside of the bubble exert force on the wall protons in a volume-integrated manner. In fact, particle-in-cell (PIC) and molecular dynamic (MD) simulations have demonstrated that the violent electromagnetic fields generated under irradiance of an intense laser pulse are quickly smeared, leaving only a strong and uniform electrostatic field only in the vicinity of the bubble wall to drive a symmetric bubble implosion.³⁶ 

The aim of this paper is not to precisely address this uniformity problem or to enumerate detailed numerical simulation results, but rather to construct a simple semi-analytical model, which encapsulates the important features obtained from multidimensional simulations. Not only can this model easily visualize the underlying physics of this novel phenomenon but also define the limiting performance. For example, it can approximate the extent of approaching the Schwinger limit within the framework of the idealized one-dimensional (1D) scenario. Our goal is to show that MBI holds promise as a unique plasma-optical device to generate ultrahigh fields, which remarkably exceed an applied laser field by orders of magnitude, and resultant energetic protons in the relativistic regime.

This paper is organized as follows: in Sec. II, one-dimensional (1D) hybrid simulation and 3D PIC simulation are conducted to demonstrate salient features of bubble implosion. In Sec. III, the maximum compression of the bubble-wall protons and generation of ultrahigh electric field upon the bubble collapse are described in terms of a simple model and 3D MD simulation. In Sec. IV, thermodynamic characteristics of MBI are discussed. Finally, a summary is presented in Sec. V.

When irradiating the target by an ultraintense femtosecond laser pulse with an intensity of I_L ≈ 10²¹–10²³ W cm⁻², hot electrons with temperatures of T_e ≈ 10–100 MeV are generated according to the ponderomotive scaling³ 

where I_L22 and the laser wavelength λ_Lμ are in units of 10²² W cm⁻² and μm, respectively. Hot electrons with long mean free paths move freely in the target to uniformly ionize part of the atoms to a density n_i0 at an average ionization state Z (= 1 in this paper) under an initial solid density $n s 0 ≃ 5 × 10 22$ cm⁻³. These hot electrons fill the bubbles in a very short period, $R 0 / c ≲$ a few femtoseconds. In many applications, the high mobility of hot electrons often results in unwelcome energy dissipation and an entropy increase. However, these features play crucial roles in the present scheme because hot electrons provide uniform electrostatic fields in and outside of bubbles to drive spherically uniform implosions.

Practically, the specific value of the ionization degree in space, $α = n i 0 / n s 0$⁠, results from the interplay between the laser and the target material. Hence, it depends on the external parameters such as the absorbed laser energy E_La and the target volume V_T. The rate α in space is then given as

where the ionization energy of atoms and ion kinetic energies are neglected in approximating the last term.

Hot electron distribution in the bubble is the key physical issue in MBI. We first conducted 1D hybrid simulations, in which electrons and ions are treated as an electrostatic field and particles, respectively. The electron distribution in the bubble is obtained by solving the Poisson-Boltzmann (P-B) equation, $∇ 2 ϕ = 4 π e [ n ec exp ( e ϕ / T e ) − n i ]$⁠, where ϕ is the electric potential and n_ec is the temporal electron density at the center. After normalizing the P-B equation, the present system is found to depend on a single dimensionless parameter defined by

where

is the ion Debye length. As a function of Λ, the P-B equation is numerically solved to give ϕ(r) and consequently n_e (r) under the appropriate boundary conditions. The potential profile ϕ(r) evolves over time in accordance with the temporal ion density profile n_i (r). The two dimensionless variables, α and Λ, constitute the analysis as control parameters.

The inset of Fig. 2 shows the normalized ion density $n ̃ i ( r ) = n i ( r ) / n i 0$ and the electron density $n ̃ e ( r ) = n e ( r ) / n i 0$⁠. These are obtained at t = 0 for different values of Λ, assuming that the bubble is surrounded by an infinitely thick target material. The electron profiles for $Λ ≲ 1$ are rather flat over the entire domain, but they are conspicuously reduced in the bubble with increasing $Λ ( ≳ 2 )$⁠. The smaller λ_Di, the more the electrons shield the electrostatic field of the ions to maintain quasi-neutrality.

The working window for Λ is on the order of unity to achieve a higher implosion performance such as accelerated proton energies and the high density compression. The scalings for the key physical quantities are given in very simple forms as functions of the total amount of electrons contained in the bubble

where $n ¯ eb$ is the average electron density in the initial bubble. For example, a maximum implosion energy, which corresponds to the energy expended by the electric field through the implosion $E 0 = ∫ 0 R 0 e E f d r$ with the electric field in the bubble $E f ≃ ( e N e 0 / R 0 3 ) r$⁠, is reduced to

The numerical results and fitting formula for $n ¯ eb$ are shown in Fig. 2, and the latter is given in the form (valid for $Λ ≲ 20$⁠)

where n_ec0 is the initial electron density at the bubble center (compare Fig. 2). At first glance, the approximation of Eq. (7), $n ¯ eb ≃ n ec 0$⁠, seems to be a somewhat pessimistic because slightly more electrons are in the vicinity of the bubble wall than at the center. However, the density of the wall protons (and consequently the surrounding electron density) slightly decreases after the implosion begins because of their expansion into the bubble. The simulation confirms that Eq. (7) is an acceptable approximation.

Figure 3(a) shows ion trajectories obtained by the 1D simulation with Λ = 1. The bubble is assumed to be filled with hot electrons already at t = 0. It is one of the characteristic features of MBI to alternate implosion and explosion. This behavior is possible due to the strong charge separation and can never be seen in neutral plasmas. Although the bulk ions repeat the bouncing motion, the innermost protons are strongly accelerated on the collapse of the bubble (proton flash). Figure 3(b) shows a magnified view of the small rectangle part in red of panel (a). The red curves denote the flashed protons, which have substantially higher energies than the maximum imploding energy. A bubble with strong charge separation, thus, behaves as what one can assume “nanopulsar.”

Figure 4 shows the temporal evolution of proton kinetic energies, which are normalized by the maximum implosion energy $E 0 = 1 2 m p v i . max 2$⁠, where m_p and $v i . max$ denote the proton mass and the maximum implosion velocity, respectively. The red curves correspond to those in Fig. 3(b). The green and blue curves are colored so that they can be easily differentiated from each other. The inset shows the time evolution of the velocities for the corresponding color groups normalized by $v i . max$⁠. The maximum energy after being reflected at the center reaches about six times $E 0$⁠. This can be explained as follows. At the end of the implosion phase, the innermost protons stagnate when their kinetic energy has all been converted into the potential energy at the center. The innermost ions then “find” that their following ions immediately behind them have built up an extraordinarily steep slope of Coulomb potential. Then, they slide down the potential slope with resulting energies many times higher than the energy gained during the implosion.

We performed 3D PIC simulations with an open-source code EPOCH.³⁷ We employed such a periodic boundary condition that a single bubble was located at the center of a cubic plasma volume with one side of L = 1200 nm and the minimum cell size of 6 nm. For simplicity, laser-matter interaction was not taken into account. Initially, the bubble of R₀ = 300 nm was set to be empty, and a uniform charge-neutral plasma surrounding the bubble was composed of hot electrons with T_e = 10 MeV and cold ions with density of n_i0 = n_e0 = 10²¹ cm^–3.

Figure 5(a) shows the snapshots of bubble implosion with the proton density distributions color-coded. The pulsating behavior of MBI is robust despite that the shrinking bubble is substantially deformed into a squared shape due to accumulated numerical errors under the Cartesian mesh employed in the PIC code [e.g., see panel 9 in Fig. 5(a)].

Figure 5(b) shows the temporal evolution of the proton energy spectrum. The pulsation period is estimated to be $T cyc = 2 π / ω pi 0 ≈ 150$ fs, which indeed agrees with the interval between the 1st (t = 70 fs) and the 2nd (t = 220 fs) proton flash. Furthermore, throughout the explosion phase, the maximum implosion energy of 230 keV is amplified up to 1.4 MeV with an amplification factor of ≈6, which agrees well with the results observed in Fig. 4.

The key physical quantities at the maximum compression can be roughly estimated as follows. First, the minimum radius r_min, where the innermost protons stagnate, is obtained from the energy balance between the imploding kinetic energy and the converted Coulomb potential energy as³⁶ 

Second, from the mass and momentum conservations by assuming that the imploding protons are comprised of a continuous fluid, the density profile of the compressed core at the stagnation is approximately given by³⁶ 

Third, from Gauss's law, the electric field, E_f = Q(r)/r², with the spatially integrated charge, $Q ( r ) = ∫ r min r 4 π r 2 e n i ( r ) dr$⁠, is reduced to

Equation (10) has a maximum value at r = 2r_min, which is given by

Figure 6(a) compares the density profiles at the maximum compression obtained by 3D MD simulations and the 1D model, Eq. (9). The fixed parameters are R₀ = 1 μm and $n i 0 = n ¯ eb = 5 × 10 22$ cm⁻³, assuming $Λ ≪ 1$⁠. The inset shows the initial configuration of the pseudoparticles used for the simulations, where we took the innermost four atomic layers into account. One thousand pseudoprotons are uniformly arranged on each of the four layered spherical surfaces³⁸ with an interatomic distance $d 0 = n s 0 − 1 / 3 ≃ 0.27$ nm. Each pseudoproton carries a mass and charge corresponding to about 10⁵ real protons. The electrons were treated as a uniform background. Here, it should be noted that in the 3D MD simulations, all binary collisions between the pseudoprotons are precisely computed. This feature is indispensable when evaluating proton dynamics on such an infinitesimally small scale as nanometer.

It is remarkable that the four atomic layers eventually stagnate after imploding down to the small radii $≲ 6$ nm. Upon the bubble collapse, the particles scatter around the 1D minimum radius, $r = r min ≃ 0.8$ nm, by random collisions, and a substantial part of them is further compressed into an even smaller central volume (1D-forbidden space). The characteristic time interval of proton stagnation at the center is about 10 as. The observed average proton densities inside the sphere for $r ≲ 1$ nm are roughly 10²⁸ cm^–3 as demonstrated in Fig. 6(a). Meanwhile, the maximum density predicted by the 1D model is 1.3 × 10²⁸ cm^–3, which together with the overall spatial profile agrees well with the simulation result.

Figure 6(b) compares the electric field obtained by the 1D model, Eqs. (10) and (11), with the 3D MD simulation. The 1D curve excellently reproduces the simulation curve in the core volume for $r ≲ 6$ nm. By contrast, for $r ≳ 6$ nm, the simulation curve decays more swiftly than the 1D curve according to the power law, $E f ∝ r − 2$⁠. This is because the almost all the protons are accumulated in the core volume at the maximum compression. This accumulated core size r_core ≈ 6 nm is explained by the 1D analysis as $r core ≈ r min + 6 × 3 d 0 ≃ 5.7$ nm (⁠ $d r 2 / d r 0 ≃ 6$³⁶). The inset shows the hemispherical perspective view of the protons distributed around the target center at the maximum compression, where distances of the proton from the center are color-coded. To the best of our knowledge, we do not know of any other principles published in the literature to achieve such unprecedented physical quantities on earth as the compressed density on the order of 10⁵–10⁶ times the solid density and the electrostatic field on the order of 10¹⁶–10¹⁷ V m⁻¹.

Now the system, Eqs. (1)–(11), is closed to draw curves for $E f . max$ as a function of the reduced laser intensity $I L λ L 2$⁠. Figure 7 shows the maximum electric field, where the parameters are α = 0.2, 0.4, and 0.8 as well as $R 0 = 0.3 μ$m, 0.9μm, and 2.7μm. The curves for $E f . max$ monotonically increase with $I L λ L 2$ but tend to level off at higher intensities. This behavior is attributed to the electron mean free path, which becomes longer as the electron temperature T_e increases, corresponding to $Λ ≪ 1$⁠. The electron density profile in the bubble then flattens, and consequently N_e0 should not increase any further (compare the inset in Fig. 2).

An interesting feature is the existence of limiting curves, which determine the maximum electric field at a given level of the laser intensity $I L λ L 2$⁠. This limiting curve is obtained as the envelope of the individual field curves with different R₀. The long-dashed line labeled $E f . max ∗$ in Fig. 7 depicts one such limiting curve for the fully ionized case, α = 1. When the limiting field $E f . max ∗$ is attained, corresponding values of $R 0 ∗$ and Λ* are also given as the following power laws together with $E f . max ∗$⁠:

The constancy for Λ^* given by Eq. (14) should be a useful guide when designing future MBI experiments. In addition, it is important from a target design viewpoint that the target size should be kept sufficiently small so that α (and the energy confinement efficiency) remains maximized. However, it is also important to maintain a suitable solid volume surrounding the initial bubble to guarantee a sufficient implosion uniformity. This competitive situation between the efficiency and the uniformity is quite similar to that of the hohlraum-target design in inertial confinement fusion (ICF),⁹ but the volume of an ICF hohlraum target is typically a trillion times larger than that of the microbubble.

As a good comparison to $E f . max$⁠, the laser electric field, $E Lf$ (V m⁻¹) $≃ 2.7 × 10 14 ( I L 22 λ L μ 2 ) 1 / 2$⁠, is also plotted in Fig. 7. For example, $E f . max ≃ 2.5 × 10 15$ V m⁻¹ at $I L λ L 2 ≃ 10 22$ W cm⁻² μm² with R₀ = 0.9μm and α = 1.0. This value is 10 times higher than the laser field at the same intensity. Recalling the scaling, $E Lf ∝ ( I L λ L 2 ) 1 / 2$⁠, MBI may realize an effective plasma-optical device, which optimally amplifies the laser intensity by a factor of two orders of magnitude. Here, it should be noted that the limiting curve given by the dashed line in Fig. 7, which corresponds to α = 1, is not the upper limit. The value of α can be further increased by, for example, coating the target surface with a high-Z material or using such a hydride as plastic (CH) as the target material. Consequently, the absolute numbers of hot electrons can be increased in the system.

The maximum radius, up to which the scaling (9) is expected to apply, is roughly estimated by solving, $n ¯ i ≈ n i 0 = n i ( r )$⁠, to give

where $g Λ = n ¯ eb / n i 0$ [compare Eq. (7) and Fig. 2]. The maximum kinetic energy of flashed protons corresponds to the potential gap built around the bubble center at the maximum compression, i.e., $E max ≈ ∫ 0 r max e E f ( r ) dr$⁠. Note that the integration domain is approximately set to be $0 ≤ r ≤ r max$ instead of $r min ≤ r ≤ r max$ because of the existence of high density protons in the 1D-forbidden space, $0 ≤ r ≤ r min$⁠, as was observed in Fig. 6. Then, with the help of Eqs. (8), (10), and (15), one can estimate the energy amplification factor, $E max / E 0$⁠, through the explosion phase as

For example, $E max / E 0 ≃ 4.5$–6.8 for R₀ = 0.3 μm–3 μm under $g Λ ≈ 1$ (or $Λ ≲ 1$⁠) and $n i 0 = 10 22$ cm⁻³.

Adding Eq. (16) to the system, Eqs. (1)–(11), curves for $E max$ are computed as a function of $I L λ L 2$ as shown in Fig. 8 with R₀ and α being the parameters. As a general feature, the curves for $E f . max$ monotonically increase with $I L λ L 2$ although they tend to level off at higher intensities. This is due to the same physical reason as that seen in Fig. 7. For comparison, the electron temperature T_e given by Eq. (1) is also plotted as the dashed straight line. The proton energies under R₀ = 3 μm are read off to go beyond 100 MeV and 1 GeV at $I L 22 λ L μ 2 ≈ 0.3$ and 30, respectively. One should keep in mind, however, that the larger the bubble radius R₀, the more the laser energy is required to be invested according to the scaling, $E L ∝ R 0 3$⁠.

In this section, MBI as a plasma device is evaluated to extract its unique and salient features, by comparing with other systems in terms of such basic plasma parameters as number density, temperature, and energy density. First, the extent that the 1D scenario is preserved from a kinetic viewpoint is estimated. One factor may be the lateral motion of the protons during the implosion process, which is induced by collisions with the hot electrons in the bubble. This lateral motion is directly related to the resultant ion temperature T_i. It is roughly estimated as $T i / T e ∼ ( 2 π ν ei / ω pi ) · ( m e / m i )$⁠, where $ν ei$ and $ω pi$ are the electron-ion collision frequency and the plasma ion frequency, respectively.³⁹ The characteristic time scale of the bubble implosion (∼a few 10s of femtoseconds) is on the same order as $2 π ω pi − 1$⁠. The factor, $2 π ν ei / ω pi$⁠, explains the acceleration ratio between the lateral and longitudinal (radial) directions, while $m e / m i$ determines the degree of energy transfer from an electron to an ion per single collision. Employing the solid density $n s 0 ≃ 5 × 10 22$ cm⁻³ and the electron temperature T_e ∼ 10–100 MeV, which are typical values for MBI, one finds $T i / T e ∼ 10 − 8$–10^–7.

Figure 9(a), which plots the density vs temperature diagram, compares the working domain of MBI with other plasma mechanisms. The two dashed lines labeled “ $k B T = E F$” and “ $n λ D 3 = 1$” represent the critical boundaries relevant to the Fermi degeneracy and the strongly coupled plasma,⁴⁰ respectively. The ultrahigh density, which can be achieved at the target center in MBI, is comparable to the interior of a white dwarf. MBI is a somewhat paradoxical scheme when viewing as a thermodynamic device. It generates an ultradense cold-proton sphere, which is driven by low-density extremely hot electrons.

According to Eq. (11), the maximum value of the electrostatic energy density, $ϵ f ( r ) = E f 2 / 8 π$⁠, is also attained at $r = 2 r min$ to give

This maximum field energy is understood more intuitively when compared with that observed on the bubble surface at the initial stage, $E f 0 = e N e 0 / R 0 2$⁠, as

Figure 9(b) plots number density vs energy density, comparing the working domain of MBI with other plasma mechanisms. The ultrahigh energy density $ϵ f . max$ of MBI labeled “Bubble imploded field,” which is typically 3–4 orders of magnitude higher than that of ICF, is achieved under the action of electrostatic attraction to the bubble center being driven by the PW-class laser-produced hot electrons, as indicated by the green dashed arrow. Note that the area labeled “White dwarfs” corresponds to the degenerate Fermi energy of electrons, and the others stand for thermal energies. The energy density in the 1D-forbidden space labeled “Bubble-imploded hot protons” (compare Fig. 6) is even higher than the field energy $ϵ f . max$⁠. For example, the protons that are flying in the 1D-forbidden space still have residual kinetic energies of 30–120 MeV in the case of Fig. 6. Then, together with a density of 10²⁸–10²⁹ cm^–3, the corresponding energy density of the hot protons is estimated to be ϵ ∼ 10²³–10²⁵ erg cm⁻³.

We propose the novel scheme, microbubble implosion, to achieve unprecedentedly high electrostatic field and energetic proton acceleration. A simple model, encapsulating the multidimensional simulation results, has been developed to evaluate the attainable electric field and the proton kinetic energy. As a result, the limiting curve and the relevant scalings have been obtained, in terms of the applied laser intensity $I L λ L 2$⁠, the spatial degree of ionization α, and the bubble radius R₀.

MBI holds promise in principle to achieve such an ultrahigh-energy-density state of matter that is higher by orders of magnitude than those expected in a fusion-igniting core of inertially confined plasma. Introducing high-Z materials to MBI as the target composition or a surface coating are expected to achieve even higher electric fields and resultant proton energies. MBI has potential as a plasma-optical device, which optimally amplifies the applied laser intensity by a factor of two orders of magnitude.

M.M. was supported by the Japan Society for the Promotion of Science (JSPS). A.A. was supported by the Air Force Office of Scientific Research under Award No. FA9550-17-1-0382. Simulations were performed using the EPOCH code (developed under UK EPSRC Grant Nos. EP/G054940/1, EP/G055165/1, and EP/G056803/1). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1548562 through allocation TG-PHY180033. Y.N. was supported by the Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase II, a project co-financed by the Romanian Government and the European Union through the European Regional Development Fund - the Competitiveness Operational Program (1/07.07.2016, COP, ID 1334).