2. Radiation pressure acceleration regimes and simulation set-up

We consider an intense laser interacting with a planar target with density $n_0>n_c$ (i.e. an overdense target), where $n_0$ and $n_c=m_e\omega _0^2/4{\rm \pi} \, {\rm e}^2$ are the initial plasma density and the critical density associated with laser propagation in the plasma, with $e$ being the elementary charge, $m_e$ the electron mass, and $\omega _0$ the laser frequency. The intense laser exerts a coherent ponderomotive force on the electrons at the surface of the solid target, creating a charge separation between the pushed electrons and the heavier ions, which in turn accelerates the ions. In practice, the laser radiation pressure acts as a piston pushing the plasma at the front surface inwards as the laser light is reflected from the surface. Using a one-dimensional (1-D) model based on energy- and momentum-flux conservation at the target surface, one finds that the front surface is pushed at the known HB velocity (Wilks et al. Reference Wilks, Kruer, Tabak and Langdon1992)

(2.1)\begin{equation} \frac{v_\text{HB}}{c}=\sqrt{\frac{P_L}{2m_in_ic^2}} = \sqrt{\frac{1+R}{4}\frac{Z}{A} \frac{m_e}{m_p}\frac{n_c}{n_0}}a_0\cos\theta_0, \end{equation}

where $P_L = (1+R)I\cos ^2\theta _0/c$ is the radiation pressure exerted by the laser on the target surface in the normal direction, $Z$ and $A$ are the ion charge and mass numbers, $m_i$ and $m_p$ are the ion and proton masses, $c$ the speed of light, $a_0 \simeq 0.85\sqrt {I [\text {W cm}^{-2}](\lambda _0[\mathrm {\mu }\text {m}])^2/10^{18}}$ the peak normalized vector potential, $\theta _0$ the incidence angle, $\lambda _0$ the wavelength of the laser and $R\leqslant$ 1 the reflection coefficient. We note that $R$ may be a function of $\theta _0$, $\lambda _0$, $a_0$ and target density and composition.

We distinguish between two different RPA regimes: HB (Wilks et al. Reference Wilks, Kruer, Tabak and Langdon1992; Macchi et al. Reference Macchi, Cattani, Liseykina and Cornolti2005) and LS (Esirkepov et al. Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima2004; Robinson et al. Reference Robinson, Zepf, Kar, Evans and Bellei2008; Macchi & Benedetti Reference Macchi and Benedetti2010). The acceleration regime is determined by the ratio $l_0/(v_{\rm HB} \tau _0)$, where $l_0$ is the target thickness and $\tau _0$ is the laser pulse duration. In the HB regime, $l_0 > v_\text {HB} \tau _0$ and the laser radiation pressure can only push a small fraction of the target. Ions are reflected once off the laser piston acquiring a velocity $v_i \simeq 2v_\text {HB}$, or equivalently a peak energy per nucleon $\epsilon _0 = 2m_pv_\text {HB}^2$. In the LS regime, $l_0 < v_\text {HB} \tau _0$ and the laser can push the whole target repeatedly during the laser pulse duration. In other words, the target is accelerated via multiple HB stages (Macchi & Benedetti Reference Macchi and Benedetti2010; Grech et al. Reference Grech, Skupin, Diaw, Schlegel and Tikhonchuk2011). In this case, the target experiences an acceleration (Macchi & Benedetti Reference Macchi and Benedetti2010)

(2.2)\begin{equation} a_{\rm RPA} = \frac{{\rm d}}{{\rm d}t}(v_i\gamma_i) \simeq \frac{2I}{m_in_il_0c}R\frac{1-\beta_i}{1+\beta_i} \simeq 2v_\text{HB}^2/l_0, \end{equation}

where $\beta _i=v_i/c$ and $\gamma _i$ are the normalized ion velocity and Lorentz factor. The last equality is the approximation in the non-relativistic limit. For negligible laser electron heating ($R\simeq 1$), an exact solution for the final ion velocity $\beta _{i,0}$ and the corresponding energy per nucleon $\epsilon _0$ exists (Macchi, Veghini & Pegoraro Reference Macchi, Veghini and Pegoraro2009)

(2.3a,b)\begin{equation} \beta_{i,0}=\frac{(1+\xi)^2-1}{(1+\xi)^2+1};\quad \epsilon_0=m_pc^2\frac{\xi^2}{2(\xi+1)}, \end{equation}

where $\xi =c({Zm_en_c}/{Am_pn_0})({a_0^2\tau _0}/{l_0})$.

In our simulations a laser pulse with frequency $\omega _0$ is launched along the $x_1$ direction (which is also the direction of the target normal) from the left boundary and irradiates, unless otherwise stated, an electron–proton plasma (i.e. $m_i=m_p= 1836 m_e$) with initial density $n_0\geqslant$ 40 $n_c$. A minimum density of 40 $n_c$ corresponds to that of high-repetition rate liquid hydrogen targets (Kim, Göde & Glenzer Reference Kim, Göde and Glenzer2016; Gauthier et al. Reference Gauthier2017; Curry et al. Reference Curry, Schoenwaelder, Goede, Kim, Rehwald, Treffert, Zeil, Glenzer and Gauthier2020). The plasma density follows a step-like profile with thickness $l_0$. An initial electron temperature $T_e=100$ eV is used (we have checked that our results are not sensitive to the initial temperature choice in the $10\unicode{x2013}1000$ eV range). The typical size of the simulation box in 2-D (3-D) simulations is 400 (300)$c/\omega _0$ longitudinally, and $250 c/\omega _0$ transversely in $x_2$ (and in $x_3$). The 2-D (3-D) simulations use 16 (8) particles per cell per species and a spatial resolution of 0.2 $(0.5) c/\omega _{pe}$ in each direction, where $\omega _{pe}=\sqrt {4{\rm \pi} \, {\rm e}^2n_0/m_e}$ is the electron plasma frequency. The time step is chosen according to the Courant–Friedrichs–Lewy condition. Open (absorbing) boundary conditions for both particles and fields are used in the longitudinal and transverse directions (except in the cases with a plane wave laser where the transverse boundary conditions are periodic). We have tested different resolutions and numbers of particles per cell to ensure convergence of the results and have used a third-order particle interpolation scheme for improved numerical accuracy. We have also tested different domain sizes to ensure that this domain allows capturing the electron heating and ion acceleration dynamics without the build-up of fields at the boundaries due to the absorption of current from escaping particles. Because in most practical applications the primary interest is in highly directional ion beams, for simulations with a finite laser spot the ion energy spectra are integrated within a $10^\circ$ opening angle from the laser propagation (forward) direction. We have checked that this is consistent with selecting the ions within the area of the focal spot.

To study in detail how the laser–plasma parameters affect electron heating and ion acceleration, we have performed a parameter scan in laser intensity ($a_0=5\unicode{x2013}200$), polarization (P-, S- and circular), incidence angle ($\theta _0 = 0^\circ \unicode{x2013}45^\circ$), full-width half-maximum (FWHM) duration ($\tau _0 = 30\unicode{x2013}2000 \omega _0^{-1}$), focal spot (at $1/e^2$ beam width; $w_0=4\unicode{x2013}50 c/\omega _0$ and plane wave), target composition $1\leqslant A/Z\leqslant 4$ for single-species ions, and CH, density ($n_0 = 40\unicode{x2013}500 n_c$, covering the range from liquid hydrogen to solid-density targets) and thickness ($l_0 = 0.08\unicode{x2013}40 c/\omega _0$).

3. Termination of radiation pressure acceleration due to strong electron heating

In this section, we show that for thick targets (HB regime) when there is significant electron heating at the target surface, HB gives rise to the formation of a collisionless shock that is launched into the target and dominates ion acceleration. We illustrate these results with 2-D simulations where an intense ($a_0 = 12$) laser interacts with an overdense thick target ($n_0 = 42 n_c$ and $l_0 = 12 c/\omega _0$). The laser is either a PP (with the electric field along the $x_2$ direction) or CP plane wave, with a fourth-order super-Gaussian temporal profile with $\tau _0 = 100 \omega _0^{-1}$.

In figure 1(a,b), we show the longitudinal phase spaces of electrons and protons for the PP simulation near the end of the laser–plasma interaction at $t = 150 \omega _0^{-1}$. We can clearly see hot electron bunches produced at a frequency of $2\omega _0$, which is a signature of the $\boldsymbol {J}\times \boldsymbol {B}$ heating mechanism (May et al. Reference May, Tonge, Fiuza, Fonseca, Silva, Ren and Mori2011). As a result, a significant fraction of the laser energy goes into the electron population, weakening HB. Indeed, the measured HB velocity is $v_\text {HB}\simeq 0.026 c$, consistent with a laser absorption into hot electrons of $\simeq 50\,\%$ ($R \simeq 0.5$ in (2.1)). In addition, we observe that a collisionless shock forms and dominates proton acceleration as illustrated in figure 1(a,c). The shock front detaches from the surface and propagates into the target at a nearly constant velocity $v_\mathrm {sh} \simeq 0.035 c$. The proton population at $11 c/\omega _0 \lesssim x_1\lesssim 14 c/\omega _0$ has been accelerated by the shock front to $v_i\simeq 2v_\text {sh}$. It is important to note that even after the laser interaction finishes, the collisionless shock continues to propagate through the target and reflect protons (figure 1c). It is also worth noting that due to the generation of hot electrons, a strong space-charge field develops at the rear surface of the target and leads to TNSA, which is evidenced by the proton phase space at $x_1\gtrsim 19 c/\omega _0$. At later times, the protons accelerated by the shock acquire a large energy spread when they leave the target rear surface and experience TNSA. Tailoring of the rear side density profile is required to control TNSA and enable quasimonoenergetic ion beams from CSA (Fiuza et al. Reference Fiuza, Stockem, Boella, Fonseca, Silva, Haberberger, Tochitsky, Gong, Mori and Joshi2012, Reference Fiuza, Stockem, Boella, Fonseca, Silva, Haberberger, Tochitsky, Mori and Joshi2013). In general, we have found that in configurations in which electrons become relativistic, CSA and TNSA will dominate the ion acceleration mechanisms over HB. This highlights the need to prevent or significantly mitigate electron heating in order to enable HB to be the dominant mechanism and to produce high-quality ion beams.

Figure 1. Results of 2-D PIC simulations of a P-polarized (PP) (a–c) and circularly polarized (CP) (d–f) short-pulse laser ($a_0 = 12$) interacting with an overdense target ($n_0 = 42 n_c$). Longitudinal $p_1-x_1$ phase spaces, shown at $t=150 \omega _0^{-1}$, of protons (a,d), electrons (b,e) and time evolution of the longitudinal $E_1$ electric field (c,f). The laser pulse irradiates the target from the left-hand side, has a super-Gaussian temporal profile, and ends at $t\simeq 160 \omega _0^{-1}$. For a PP laser, the electrons are heated by the laser, HB is weakened, and a collisionless shock develops, which dominates the proton acceleration. The electrostatic shock front detaches from the HB front and propagates at $v_\text {sh}\simeq 0.035 c$, which reflects the protons to a speed of $2v_\text {sh}$. The electrons in the CP case remain relatively cold and HB is the dominant ion acceleration mechanism, accelerating protons to $2v_\text {HB}$, where $v_\text {HB}\simeq 0.031 c$. After the laser ends, the HB front decays and slows down. In the phase spaces, the dotted lines denote the target front surface (with density $n \simeq n_c$) and the dashed blue line indicates the collisionless shock front. In (c) and (f), the dotted, dashed and dash–dotted lines indicate $v_\text {HB}$, $v_\text {sh}$ and the proton beam velocity ($2 v_\text {HB}$ or $2 v_\text {sh}$), respectively.

A commonly employed strategy to mitigate electron heating by both $\boldsymbol {J}\times \boldsymbol {B}$ and Brunel mechanisms in laser–plasma interactions is the use of a CP laser at near normal incidence ($\theta _0 \simeq 0^\circ$; (Macchi et al. Reference Macchi, Cattani, Liseykina and Cornolti2005)). The $\boldsymbol {J}\times \boldsymbol {B}$ mechanism relies on the standing wave created by the incoming and reflected laser field. For linear polarization, the oscillation of the magnetic field at the surface allows for electrons to escape the target and experience the electric field of the laser, being accelerated transversely and then rotated back into the target by the magnetic field. For circular polarization, the magnetic field of the standing wave at the target surface does not decrease to zero – it just rotates – and thus electrons cannot escape the target to be efficiently accelerated (May et al. Reference May, Tonge, Fiuza, Fonseca, Silva, Ren and Mori2011). The Brunel heating mechanism relies on a laser electric field component normal to the surface to directly accelerate the electrons. This is absent for normal incidence, provided that the target surface remains uniform and stable. As we will discuss in more detail in the next section, these conditions can only be maintained for very short interaction times. Figure 1(d–f) shows the results of a simulation with the same laser and plasma parameters as in figure 1(a–c) but using CP. We observe that CP is indeed capable of maintaining reduced electron heating, allowing HB to be the dominant ion acceleration mechanism. We observe that the HB velocity is $v_\text {HB} \simeq 0.031 c$, in good agreement with (2.1) for $R \simeq 1$ and that the accelerated protons have $v_1\simeq 2v_\text {HB}$. We also find that, in contrast to the case of a collisionless shock, the HB velocity abruptly slows down when the laser–plasma interaction ends and proton reflection/acceleration ceases (figure 1f at $t\simeq 165 \omega _0^{-1}$). This is an interesting difference between HB and CSA that impacts the total charge accelerated by each mechanism: while HB tends to reflect a larger fraction of the background ions, it can only do so during a shorter period when compared with CSA.

In addition to the use of a CP laser at near normal incidence ($\theta _0 \simeq 0^\circ$), it has been recently proposed in the context of collisionless shock studies (Grassi et al. Reference Grassi, Grech, Amiranoff, Macchi and Riconda2017) that S-polarization (SP) (with the electric field along the $x_3$ direction) with a large incidence angle can also achieve similar results in terms of mitigating electron heating. This is because Brunel heating is absent for SP and $\boldsymbol {J}\times \boldsymbol {B}$ heating can be significantly reduced since it scales with $\cos \theta _0$. We have explored this possibility over a large range of laser intensities ($a_0 = 5\unicode{x2013}30$) and plasma densities ($n_0 = 40\unicode{x2013}150 n_c$) and confirmed that in 2-D simulations indeed SP with $\theta _0=45^\circ$ can significantly suppress electron heating and lead to HB ion acceleration comparable to the CP case (not shown here). However, the situation changes significantly for more realistic 3-D simulations. In 3-D simulations, the laser–plasma interaction along the direction of laser polarization is effectively PP at $\theta _0 \simeq 0^\circ$ and gives rise to significant electron heating along the laser polarization. In all 3-D SP cases tested, hot electrons acquire relativistic temperature and we observe a transition from HB to CSA similar to the PP case illustrated in figure 1(a–f).

For very thin targets, corresponding to the LS regime, we have observed similar results in terms of the mitigation of electron heating, which was achieved with a CP laser with $\theta _0 \simeq 0^\circ$ and led to the acceleration of ions with a narrow energy spread. We note that for PP and SP, the onset of strong electron heating does not lead to the formation of a collisionless shock. Instead, we observe that the thin target quickly becomes transparent to the laser and the radiation pressure is no longer efficient in accelerating ions. Overall, for both HB and LS regimes, we find that the use of CP is required to significantly mitigate electron heating and optimize RPA. Non-normal incidence angles can be useful for SP, but unfortunately seem to be only effective in 2-D simulations. In the remainder of this paper, we discuss in detail the impact of the laser and plasma parameters on electron heating with CP for both HB and LS regimes in order to understand the set of conditions required for high-quality ion acceleration.

4. Development of surface corrugations

The interaction of an intense laser with an overdense target can significantly modify the shape of the target surface either due to the development of surface instabilities or from finite laser spot size effects. These modulate the surface density profile, which can ultimately trigger strong electron heating (e.g. Klimo et al. Reference Klimo, Psikal, Limpouch and Tikhonchuk2008; Dollar et al. Reference Dollar2012; Paradkar & Krishnagopal Reference Paradkar and Krishnagopal2016), even for a CP laser, and impact the quality and mechanisms of ion acceleration as discussed above. In this section, we discuss the importance of surface corrugations on HB and LS acceleration regimes with a CP laser.

4.1. Growth of surface instabilities

Previous studies have investigated the development of density ripples at the interaction surface and there has been significant discussion on which instabilities are dominant, including the Weibel instability (Sentoku et al. Reference Sentoku, Mima, Kojima and Ruhl2000), RTI (Gamaly Reference Gamaly1993; Pegoraro & Bulanov Reference Pegoraro and Bulanov2007; Palmer et al. Reference Palmer2012; Khudik et al. Reference Khudik, Yi, Siemon and Shvets2014; Eliasson Reference Eliasson2015; Sgattoni et al. Reference Sgattoni, Sinigardi, Fedeli, Pegoraro and Macchi2015), electron–ion coupling instabilities (Wan et al. Reference Wan2016; Wan et al. Reference Wan, Pai, Zhang, Li, Wu, Hua, Lu, Joshi, Mori and Malka2018) or a combination of these (Wan et al. Reference Wan, Andriyash, Lu, Mori and Malka2020). However, the correlation between these instabilities and the onset of electron heating has not been studied systematically for both HB and LS. Here, we begin by illustrating the surface dynamics and electron heating for the interaction of a CP laser with Gaussian longitudinal and transverse intensity profiles with a target at normal incidence using 2-D PIC simulations. In the HB regime, we use a laser with $a_0=27$, $\tau _0=200 \omega _0^{-1}$, $w_0=50 c/\omega _0$ with $n_0 = 40\,n_c$ and $l_0=75 c/\omega _0$; for LS, $a_0=15$, $\tau _0=105 \omega _0^{-1}$, $w_0=50 c/\omega _0$, with $n_0 = 250\,n_c$ and $l_0 =0.085 c/\omega _0$ are used. The results are shown in figure 2(a) for HB and figure 3(a) for LS, respectively.

Figure 2. Results of 2-D PIC simulations of the interaction of an intense Gaussian CP laser pulse with an overdense target in HB regime. (a,b) Longitudinal $p_1-x_1$ ion (top row) and electron (second row) phase spaces, ion density profile (third row) and local electron temperature (bottom row). The laser pulse durations are $\tau _0 = 200$ and $75 \omega _0^{-1}$ in (a) and (b), respectively. (c) Temporal evolution of $T_e$ (blue, left-hand axis) and ion beam energy spread ${\rm \Delta} \epsilon /\epsilon _0$ (black, right-hand axis). The time $t = 0$ is defined as $\tau _0/2$ before the laser peak intensity reaches the target.

Figure 3. Same as figure 2 but for LS, where the laser pulse durations are $\tau _0 = 105$ and $40 \omega _0^{-1}$ in (a) and (b), respectively. In (c) the green, rightmost axis plots $n_e/(\gamma n_c)$.

We observe that indeed, even with CP, there is the onset of strong electron heating after a relatively short interaction time. In the HB regime (figure 2a), the electron and ion phase spaces at $t=150 \omega _0^{-1}$ show that electrons have been heated to relativistic temperatures, which leads to a significant weakening of the laser radiation pressure and to the formation of a collisionless shock and development of strong TNSA field that dominate ion acceleration, similarly to what was observed for the PP case discussed in § 3. A sharp increase of the FWHM energy spread of the accelerated ions ensues, reaching ${\rm \Delta} \epsilon /\epsilon _0 > 100\,\%$ as can be seen in figure 2(c); and by $t = 225 \omega _0^{-1}$ both the electron temperature and ion energy spread have saturated at large values.

In the LS regime (figure 3a), we observe that once the electrons are significantly heated they drive the rapid expansion of the target and broadening of the ion energy spread (figure 3c) due to the associated strong space-charge field and short target thickness. By $t = 115 \omega _0^{-1}$ the target expansion leads to the onset of relativistic transparency as can be seen in figure 3(c) from the evolution of $n_e/(\gamma n_c)$ (the ratio of the electron density, $n_e$, and relativistic critical density at the target front surface, where $\gamma$ is the average Lorentz factor of the electrons). At this point, RPA is terminated and the ion energy distribution ceases to be peaked.

From the temporal evolution of the electron temperature $T_e$ (average kinetic energy of electrons) and FWHM ion energy spread ${\rm \Delta} \epsilon /\epsilon _0$ shown in figures 2(c) and 3(c) for the HB and LS regimes, respectively, we observe that they follow a similar behaviour with the ion beam energy spread increasing sharply following the rapid growth of $T_e$ in both cases. We define this time associated with the onset of strong electron heating, $\tau _\text {heating}$, as the time for which the rate of increase of the electron temperature, ${\rm d} T_e/{\rm d} t$, is maximum.

We have repeated the same simulations in both regimes but using a shorter laser pulse with $\tau _0 = 75 \omega _0^{-1}$ and $\tau _0 = 40 \omega _0^{-1}$, both $< \tau _\text {heating}$, for HB and LS, respectively, to confirm the impact of electron heating on the growth of the ion energy spread and overall target dynamics. The results are shown in figures 2(b,c) and 3(b,c). In these cases, we observe that indeed both $T_e$ and ${\rm \Delta} \epsilon /\epsilon _0$ remain low (figures 2c and 3c). The ion beam energy spread saturates at $t \simeq 2 \tau _0$ with ${\rm \Delta} \epsilon /\epsilon _0 \ll 1$ and remains stable long after the laser–plasma interaction has finished.

We have found that the onset of electron heating in both HB and LS regimes is related to the emergence of large transverse density modulations at the target front surface with a wavelength comparable to that of the laser. For the longer pulse simulations ($\tau _0 > \tau _\text {heating}$), these surface density modulations are visible in the third rows of figures 2(a) and 3(a) at the times where electron heating is also observed. Density modulations with a wavelength comparable to $\lambda _0$ allow the penetration of the laser in the lower density regions and resonant enhancement of its electric field (Eliasson Reference Eliasson2015; Sgattoni et al. Reference Sgattoni, Sinigardi, Fedeli, Pegoraro and Macchi2015), giving rise to effective electron heating, for example via the Brunel mechanism (bottom rows of figures 2a and 3a), with the temperature reached being comparable to that observed in simulations with a linearly polarized laser (not shown here). The spatial distribution of $T_e$ shows indeed that the heating is happening at the walls of these concave valleys and consistent with direct acceleration by the laser electric field. The location of the hot spots of $T_e$ oscillates (from top to bottom of the valleys) in accordance with the phase of the laser electric field. In the simulations with $\tau _0 < \tau _\text {heating}$, the amplitudes of the density modulations at the surface are much smaller during the time of laser interaction (third rows of figures 2b and 3b) leading to a much-reduced electron heating and stable ion acceleration.

In order to study the mechanism responsible for these corrugations, and isolate the effects of surface instabilities, we have performed a parameter scan of 2-D simulations with a long, plane-wave CP laser at normal incidence, with parameters varied in the following ranges: for HB, $5 \leqslant a_0 \leqslant 60, 40 n_c \leqslant n_0 \leqslant 200 n_c$; for LS, $5 \leqslant a_0 \leqslant 200, 40 n_c \leqslant n_0 \leqslant 500 n_c$ and $0.08 c/\omega _0 \leqslant l_0 \leqslant 2 c/\omega _0$. Note that for the LS regime, the initial target thickness $l_0$ is always larger than or equal to the optimal LS target thickness $l_\text {opt}$, defined as $l_\text {opt}= a_0\lambda _0n_c/(\sqrt {2}{\rm \pi} n_0)$, for which the acceleration is maximized by minimizing the total target mass while guaranteeing that the target remains relativistically opaque (Macchi et al. Reference Macchi, Veghini and Pegoraro2009). For both regimes, the target is either composed of single-species ions with $1\leqslant A/Z \leqslant 4$, or CH (plastic). We analyse the growth of ion density modulations by computing the transverse Fourier modes ($k_{x_2}$) of the longitudinal ($x_1$) displacement of the relativistic critical surface ($n \simeq \gamma _0n_c$; where $\gamma _0=\sqrt {1+a_0^2/2}$ is the electron Lorentz factor), as a function of the transverse ($x_2$) position (e.g. figure 4b,e). Previous theoretical studies of surface instabilities considered perturbations of the surface displacement and showed it will grow exponentially due to instability (e.g. Gamaly Reference Gamaly1993; Eliasson Reference Eliasson2015). Alternatively, for LS, one could obtain the Fourier modes of the amplitude of the transverse density profile by integrating over the target longitudinally. We have checked that the obtained growth rates and time scales of the different modes are consistent between both methods.

Figure 4. Development of surface corrugations in HB (a–c) and LS (d–f) regimes. (a,d) Proton density showing transverse density corrugations near the interaction surface at $t = 200 \omega _0^{-1}$ for (a) and $t = 90 \omega _0^{-1}$ for (d). (b,e) Evolution of Fourier modes at the relativistic critical surface of the corrugation amplitudes $\tilde {n}_i(k_{x_2})$. The dashed white line denotes the $k_{x_2}=k_0$ mode. (c,f) Time evolution of the $k_{x_2}=k_0$ mode and its linear fit.

Figure 4 illustrates the growth of different modes for simulations with the same laser intensity and target parameters of figures 2 and 3. Note that both targets remain opaque to the laser during the time of the analysis and thus the measurements of the growth rates and saturation levels are not affected by the onset of relativistic transparency. The fastest growing modes are observed at $k_{x_2}\gg k_0$, with $k_0=2{\rm \pi} /\lambda _0$ (e.g. $k_{x_2}\simeq 13 \omega _0/c$ in figure 4e at $t\simeq 30 \omega _0^{-1}$), and have been previously described as associated with electron–ion coupling instabilities (Wan et al. Reference Wan2016; Wan et al. Reference Wan, Pai, Zhang, Li, Wu, Hua, Lu, Joshi, Mori and Malka2018, Reference Wan, Andriyash, Lu, Mori and Malka2020). However, these modes saturate at relatively low amplitude and do not lead to significant electron heating. The dominant density modulations are associated with the mode with $k_{x_2}\simeq k_0$ and we observe the onset of strong electron heating during the linear growth and saturation of this mode. This suggests that the onset of strong electron heating is related to laser-driven RTI for which the dominant mode is $k_{x_2}\simeq k_0$ (Eliasson Reference Eliasson2015; Sgattoni et al. Reference Sgattoni, Sinigardi, Fedeli, Pegoraro and Macchi2015).

To further confirm that the RTI mode with $k_{x_2} = k_0$ is the dominant effect on the onset of strong electron heating and degradation of the ion beam quality, we have performed additional simulations with different transverse domains. We observe that for transverse domain sizes $< \lambda _0/2$, where the dominant RTI mode is prohibited, no significant electron heating is observed. In these cases, the fastest growing high-k modes are still captured and thus can still grow, but $T_e$ remains very low (figure 5). For the largest transverse domain size $\gg \lambda _0$, we see that strong electron heating starts near the saturation time of the RTI.

Figure 5. Evolution of electron temperature for different transverse simulation domain sizes (black, left-hand axis). The solid black curves correspond to the simulations in figure 4 with a transverse box size of $20\,\lambda _0$ for HB (a) and $40\,\lambda _0$ for LS (b), respectively. For these cases the growths of the $k_{x_2} = k_0$ (solid) mode are shown (blue, right-hand axis).

For the laser-driven RTI, the growth rate is $\varGamma _\text {RTI} \propto \sqrt {a_{\rm RPA}k_0}$, where $a_{\rm RPA}=2v_\text {HB}^2/l$ is the acceleration due to radiation pressure ((2.2) with $R\simeq 1$ and $\beta _i\ll 1$). For LS, the target thickness is $l_0$, whereas for HB the effective acceleration layer is characterized by the relativistic electron skin depth $l=\sqrt {a_0}c/\omega _{pe}$ (Gamaly Reference Gamaly1993). Linear fits to the measured growth rates of the $k_0$ mode from the simulations confirm the expected scaling with the following numerical factors (figure 6; see also the example fits in figure 4c,f): $\varGamma _\text {RTI}[\omega _0]\simeq 0.3a_0^{3/4}(n_0[n_c])^{-1/4}( Zm_e/(Am_p))^{1/2}$ for HB and $\varGamma _\text {RTI}[\omega _0]\simeq 0.5a_0(n_0[n_c]l_0[c/\omega _0] Am_p/(Zm_e))^{-1/2}$ for LS.

Figure 6. Scaling of the measured growth rate of the $k_{x_2}=k_0$ mode of the surface corrugations (a,c), and strong correlation between the electron heating time $\tau _\text {heating}$ and the growth time of the RTI (b,d), for both HB (a,b) and LS (c,d). Coloured symbols are measurements from 2-D PIC simulations.

We find a strong correlation between $\tau _\text {heating}$ and the growth time of the instability, with $\hat {\tau }_{0,\text {RTI}} \equiv \tau _\text {heating} \simeq 5\varGamma ^{-1}_\text {RTI}$ for HB and $\hat {\tau }_{0,\text {RTI}} \equiv \tau _\text {heating} \simeq 3\varGamma ^{-1}_\text {RTI}$ for LS, where $\hat {\tau }_{0,\text {RTI}}$ is defined as the time of the onset of strong electron heating due to the development of the RTI. This is shown in figures 6(b) and 6(d). In order to suppress or significantly mitigate electron heating, the duration of the laser pulse $\tau _0$ should then be smaller than $\hat {\tau }_{0,\text {RTI}}$, which can be written as

(4.1)\begin{equation} \hat{\tau}_{0,\text{RTI}} \text{[fs]} \simeq \begin{cases} 400 a_0^{{-}3/4}\lambda_0[\mathrm{\mu}\mathrm{m}]\left(\dfrac{n_0}{n_c}\right)^{1/4} \left(\dfrac{A}{Z}\right)^{1/2} & \text{for HB},\\ 350 a_0^{{-}1}\left(l_0 [\mathrm{\mu}\mathrm{m}]\lambda_0 [\mathrm{\mu}\mathrm{m}] \dfrac{A}{Z}\dfrac{n_0}{n_c}\right)^{1/2} & \text{for LS}. \end{cases} \end{equation}

It is important to note that although several previous works have studied the development of the RTI, the focus had been on developing strategies to mitigate the penetration of the RTI fingers on the accelerated proton species, which included the use of mixed ion species (Yu et al. Reference Yu, Pukhov, Shvets and Chen2010, Reference Yu, Pukhov, Shvets, Chen, Ratliff, Yi and Khudik2011), advanced laser configurations (Wu et al. Reference Wu, Zheng, Qiao, Zhou, Yan, Yu and He2014; Zhou et al. Reference Zhou, Yan, Mourou, Wheeler, Bin, Schreiber and Tajima2016) and curved targets (Wang, Khudik & Shvets Reference Wang, Khudik and Shvets2021). These are either challenging to implement in practice (Zhou et al. Reference Zhou, Yan, Mourou, Wheeler, Bin, Schreiber and Tajima2016; Wang et al. Reference Wang, Khudik and Shvets2021; and have not yet been proven to be effective experimentally), or still lead to significant electron heating (Yu et al. Reference Yu, Pukhov, Shvets and Chen2010; Wu et al. Reference Wu, Zheng, Qiao, Zhou, Yan, Yu and He2014). As we show here, a quantitative understanding of the detrimental effect that the instability-induced electron heating has on the ion beam quality is critical to produce ion beams with high spectral quality.

We should further note that the results presented here have considered only the regime where ions are non-relativistic, which is appropriate for most current and near-future laser systems. In the relativistic regime, the RTI can still grow as shown in previous numerical studies (Bulanov et al. Reference Bulanov, Echkina, Esirkepov, Inovenkov, Kando, Pegoraro and Korn2010; Sgattoni, Sinigardi & Macchi Reference Sgattoni, Sinigardi and Macchi2014), but how its growth rate changes and, more generally, its impact on electron heating and the spectral quality of the accelerated ions is not well established. One expects that strong electron heating will still be caused by the development of RTI in the relativistic regime – Brunel heating will still be present. The resulting space-charge fields will also lead to an increase of the ion energy spread, however, the rate at which this happens may be more moderate in the relativistic regime when compared with the non-relativistic case. Furthermore, the resulting expansion of the target will also pose limitations on the acceleration due to the onset of relativistic transparency, as in the cases discussed here. A detailed analysis of the relativistic regime is left for future work.

4.2. Finite laser spot size

The transverse variation of the laser intensity due to its spatial profile naturally leads to non-uniform HB velocities across the surface and results in a change of the surface shape over time, also triggering strong electron heating. The onset time of strong electron heating is approximately when the radiation-pressure-driven displacement $d$ of the target surface is comparable to the laser spot size $w_0$, resulting in a significant change in the local incidence angle. This effect has been previously discussed in the case of normal laser incidence ($\theta _0 = 0^\circ$ (Klimo et al. Reference Klimo, Psikal, Limpouch and Tikhonchuk2008; Wan et al. Reference Wan, Andriyash, Lu, Mori and Malka2020)). In fact, this argument can be generalized to other laser incidence angles if we consider the displacement $d$ to be along the axis of the laser resulting in a time for electron heating that is independent of $\theta _0$. We define this time as $\hat {\tau }_{0,\text {FS}}$ (‘FS’ stands for finite spot), and can estimate it as $d(\hat {\tau }_{0,\text {FS}})= w_0$, where for simplicity we assume that the laser intensity is approximately constant in time. For HB, we have $d(t)=v_\text {HB}t$. For LS, we consider first that the target experiences a constant acceleration in its rest frame – i.e. $a_{\rm RPA}\simeq 2v_\text {HB}^2/l_0$ ((2.2) with $R\simeq 1$ and $\beta _i\ll 1$). This then gives $d(t)=(c\sqrt {c^2+a_{\rm RPA}^2t^2}-c^2)/a_{\rm RPA}\simeq (cl_0/2v_\text {HB}^2)(\sqrt {c^2+4v_\text {HB}^4t^2/l_0^2}-c)$ (for all cases considered, we have found that the error introduced by taking the non-relativistic ion velocity limit is $\lesssim 5$ %). Equating $d(\hat {\tau }_{0,\text {FS}})=w_0$ yields

(4.2)\begin{equation} \hat{\tau}_{0,\text{FS}} \text{[fs]} \simeq \begin{cases} 200 a_0^{{-}1}w_0[\mathrm{\mu}\mathrm{m}]\left(\dfrac{A n_0}{Z n_c}\right)^{1/2} & \text{for HB}, \\ 200 a_0^{{-}1}\left(\dfrac{w_0[\mathrm{\mu}\mathrm{m}]l_0[\mathrm{\mu}\mathrm{m}] A n_0}{Z n_c}\right)^{1/2} & \text{for LS}. \end{cases} \end{equation}

We note that the estimate above considers near diffraction-limited laser focusing where the transverse intensity profile is smooth. If the Strehl ratio is low and speckle-like intensity distributions are present at focus, these will lead to modulations of the surface and laser incidence angle on scales comparable to the speckle size. In that case, in the threshold condition given in (4.2) we should replace $w_0$ by the size of the laser intensity speckles, which can be a significant limitation for small-scale (${\sim }\lambda _0$) speckles. We further note that with oblique incidence ($\theta _0 > 0$), the ion beam direction will be modified. This can happen due to partial absorption of the laser field, which will impart transverse (along the target surface) momentum to the ions (Macchi et al. Reference Macchi, Grassi, Amiranoff and Riconda2019) and also as the surface is modified to be nearly normal to the laser pulse causing the ion beam direction to be primarily along the laser propagation direction.

5. Laser temporal profile

In this section, we evaluate how different laser temporal profiles impact the quality of the ion beam. This can be particularly relevant for the HB regime, as different ion populations will experience different HB velocities due to $v_\text {HB}(t)\propto a_0(t)$. We have performed simulations in both HB and LS regimes, for a plane wave laser with $\tau _0<\hat {\tau }_{0,\text {RTI}}$, where the laser temporal profile is either Gaussian or a flat-top (approximated by a fourth-order super-Gaussian). The total laser energy and $a_0$ is kept the same in both cases. The range of $a_0$, $n_0$ and $l_0$ explored was similar to § 4, and we have varied the pulse duration in the range $0.2 \hat {\tau }_{0,\text {RTI}}\lesssim \tau _0 < \hat {\tau }_{0,\text {RTI}}$.

In general, we find that the accelerated proton beams have a relatively narrow energy spread with both profiles, but the energy spread in the Gaussian case is typically larger, by a factor up to $\simeq 2$. Figure 7 illustrates the typical differences between both profiles for HB and LS. One important feature that we observe in the HB regime is the development of an extended low-energy population (<3 MeV in figure 7a). This is due to the contribution from the low HB velocity phase at the edges of the laser pulse, and can be understood as follows. In a time interval $t$ to $t+{\rm d}t$, a population of ions ${\rm d}N\propto n_iv_\text {HB}(t)\,{\rm d}t$ is accelerated to a velocity $2v_\text {HB}(t)\propto a_0(t)$ (for $R\simeq 1$). Therefore, the resulting energy spectrum will be

(5.1)\begin{equation} \frac{{\rm d}N}{{\rm d}\epsilon}=\frac{{\rm d}N}{{\rm d}t} \frac{{\rm d}t}{{\rm d}\epsilon} \propto\frac{1}{\sqrt{\epsilon\ln(\tilde{\epsilon}_0/\epsilon)}}, \end{equation}

for a Gaussian pulse, where $\tilde {\epsilon }_0 = 2 m_i v_\text {HB,0}^2$ is the peak energy of the ions (with $v_\text {HB,0}$ the HB velocity associated with the peak intensity). This matches well the low-energy component of the proton spectrum in figure 7(a), which is responsible for the additional energy spread with respect to the flat-top case.

Figure 7. Proton energy spectra from 2-D PIC simulations of plane-wave CP laser pulses ($a_0=30$) with either Gaussian (blue) and flat-top (red) temporal profiles normally incident on a target ($n_0 = 136 n_c$), plotted at the time the laser ends, for (a) HB (with a semi-infinite target) and (b) LS ($l_0 = 0.5 c/\omega _0$). The Gaussian pulse has $\tau _0= 40 \omega _0^{-1}$ and the flat-top profile is approximated by a fourth-order super-Gaussian temporal profile, where the total laser energy is kept the same as the Gaussian pulse. The dotted curve in (a) describes the low-energy component of the spectrum (see (5.1)).

6. Laser prepulse

In this section, we explore the impact that the development of a preplasma induced by a laser prepulse can have in triggering early electron heating and affecting the spectral quality of ion beams accelerated via RPA. We consider a preplasma with an exponential density profile at the front surface of the target with scale length $l_g$. For the laser and target parameters considered in the short-pulse cases in figure 2(b) and 3(b), with $\tau _0 = 75 \omega _0^{-1}$ and $l_0 = 75 c/\omega _0$ for HB and $\tau _0 = 40 \omega _0^{-1}$ and $l_0 = 0.085 c/\omega _0$ for LS, we have performed additional simulations with different levels of $l_g$, which was varied in the range $0.01 \leqslant l_g/l_0 \leqslant 0.4$ (the total target mass is conserved).

Figure 8 shows that the preplasma can significantly impact electron heating and the ion energy spread for levels of $l_g \gtrsim 0.1\,l_0$. In the case of HB, the electron temperature and ion energy spread for a preplasma scale length $l_g < 0.1 l_0$ ($l_g < 1.2\ \mathrm {\mu }$m for $\lambda _0 = 1\ \mathrm {\mu }$m) are similar to the case of no preplasma, but above this level strong electron heating is triggered, leading to a very fast degradation of ion beam quality even before the pulse arrives at the main target. For LS, for a preplasma with $l_g \leq 0.1 l_0$ the results are also very similar to the no preplasma case. However, for $l_g \gtrsim 0.2 l_0$ ($l_g \gtrsim 2.7$ nm for $\lambda _0 = 1\ \mathrm {\mu }$m) both the electron temperature and ion energy spread are observed to increase to nearly twice the values of the no preplasma case. These results confirm the need to carefully control the level of preplasma to produce ion beams with high spectral quality from RPA and will help inform the laser prepulse contrast requirements for future experimental studies.

Figure 8. Results of 2-D PIC simulations of the interaction of an intense Gaussian CP laser pulse with a planar target of thickness $l_0$ and an exponential preplasma of scale length $l_g$ in the front, in both HB (a,b) and LS (c,d) regimes. (a,c) Temporal evolution of electron temperature $T_e$ and (b,d) ion beam energy spread ${\rm \Delta} \epsilon /\epsilon _0$. Here ${\rm \Delta} \epsilon /\epsilon _0$ is measured for protons within a $10^\circ$ opening angle with respect to the laser propagation direction (target normal) and the time $t = 0$ is defined as $\tau _0/2$ before the laser peak intensity reaches the main target.

7. Optimal regime of radiation pressure acceleration

Our findings make clear the importance of limiting the pulse duration to control electron heating and obtain quasimonoenergetic ion beams from RPA in both HB and LS regimes. In figure 9 we demonstrate that (4.1) is robust over a wide range of laser and target parameters even when realistic Gaussian transverse and temporal pulse profiles are considered. We observe that, indeed, the derived threshold condition marks the transition from low to high electron heating (figure 9a,c) and consequently from low to high energy spread of the accelerated ion beam (figure 9b,d). By repeating some of the simulations in the high-quality regimes using a small, but finite laser incidence angle, we have also confirmed that in general for an incidence angle $\lesssim 10^\circ$, as typically used experimentally, electron heating is still maintained at a low level and the quality of the ion beam remains similar to the case with normal incidence.

Figure 9. (a,c) Electron temperature $T_e$ and (b,d) ion beam energy spread ${\rm \Delta} \epsilon /\epsilon _0$ measured from 2-D PIC simulations of a 1 $\mathrm {\mu }$m wavelength Gaussian laser pulse with duration $\tau _0$ and spot size $w_0=7.6\ \mathrm {\mu }$m interacting with a solid target with $n_0 = 40\,n_c$, $l_0=12\ \mathrm {\mu }$m for HB (top row) and $250 n_c$, $l_0=l_\text {opt}$ for LS (bottom row). Here $T_e$ is measured at the end of the laser interaction when the maximum is observed. Here ${\rm \Delta} \epsilon /\epsilon _0$ is measured for protons within a $10^\circ$ opening angle at $t\simeq 2\tau _0$. The black curves correspond to the prediction of (4.1) and the white dots denote the parameters sampled by the simulations.

For the LS regime, combining the threshold condition for the pulse duration (see (4.1)) and the optimal target thickness condition $l_0=l_\text {opt}$ results in a new condition on the maximum laser intensity with important implications for the optimization of ion beam energy spread in LS acceleration (Chou et al. Reference Chou, Grassi, Glenzer and Fiuza2022). In particular, we obtain that the laser $a_0$ should be limited by

(7.1)\begin{equation} a_0 \lesssim \hat{a}_0 \equiv \frac{350^2}{\sqrt{2}{\rm \pi}}\frac{A}{Z}\left(\frac{\lambda_0 \text{[$\mathrm{\mu}$m]}}{\tau_0\text{[fs]}}\right)^2, \end{equation}

and consequently the maximum energy per nucleon $\epsilon _0$ of the narrow energy spread peak is

(7.2)\begin{equation} \hat{\epsilon}_0 = m_pc^2\frac{\hat{\xi}^2}{2(\hat{\xi}+1)}, \quad \text{where } \hat{\xi}\simeq 20 \frac{\lambda_0\text{[$\mathrm{\mu}$m]}}{\tau_0\text{[fs]}}. \end{equation}

In contrast to previous works, and to the common practice of pushing for higher $a_0$ to generate stable LS and higher energy ion beams (e.g. Qiao et al. Reference Qiao, Zepf, Borghesi and Geissler2009), (7.1) and (7.2) indicate the existence of a maximum $\hat {a}_0$ and $\hat {\epsilon }_0$ for high-quality LS ion beams, and show that these are determined primarily by the laser wavelength and pulse duration. In particular, it is worth highlighting that for this optimal regime of acceleration the energy per nucleon of the ion spectral peak does not depend on the target density, composition and laser energy (transverse spot size). These predictions for the LS regime have been recently validated with 3-D simulations in Chou et al. (Reference Chou, Grassi, Glenzer and Fiuza2022).

8. The 3-D simulation results

To further explore RPA in more realistic 3-D configurations and validate our model for the optimal laser duration for high-quality ion beams, we have performed 3-D simulations in both HB and LS regimes.

In the HB regime, a CP laser with $a_0=12$ ($I \simeq 2\times 10^{20} \text {W cm}^{-2}$ for $\lambda _0 = 1\ \mathrm {\mu }$m), $w_0 = 2\ \mathrm {\mu }$m is incident with $\theta _0 = 0^\circ$ on a planar electron–proton target with $n_0 = 40 n_c$ and thickness of $3.5\ \mathrm {\mu }$m, corresponding to the conditions of typical liquid hydrogen jet targets (Kim, Göde & Glenzer Reference Kim, Göde and Glenzer2016; Gauthier et al. Reference Gauthier2017; Göde et al. Reference Göde2017; Obst et al. Reference Obst2017). A fourth-order super-Gaussian temporal profile is used. For these parameters, (4.1) indicates $\tau _0 \lesssim 160$ fs for high-quality proton beams to be produced. We thus run three simulations with pulse durations of $\tau _0=80$, 160 and $265$ fs to test this criterion. In figure 10(a), we show the energy spectra of the proton beams exiting the target from the rear surface for these three cases. When the pulse duration is smaller than the predicted threshold, a quasimonoenergetic proton beam is generated, peaking at roughly $1.8$ MeV, which corresponds to $v_i=2v_\text {HB}\simeq 0.031 c$ and is consistent with (2.1) for $R\simeq 1$. When the pulse duration is comparable to the threshold, we observe that the ion spectral peak at similar energy is still visible, but less prominent. For the longer pulse duration ($\tau _0=265$ fs) we find substantial electron heating ($T_e\simeq 3$ MeV), and a strong TNSA field is generated at the rear surface, which broadens the proton energy spread. No clear spectral peak is observed at the same energy. These results confirm the validity of the derived limit on pulse duration for HB based on the development of surface corrugations and associated electron heating.

Figure 10. Proton energy spectra from 3-D simulations of an intense 1 $\mathrm {\mu }$m CP laser irradiating an overdense hydrogen target at normal incidence ($\theta _0=0^\circ$) for different pulse durations. (a) Results for HB regime with laser $a_0 = 12$ and $w_0 = 2\ \mathrm {\mu }$m and target $n_0 = 40 n_c$ and $l_0 = 3.5\ \mathrm {\mu }$m, after the proton beam has left the target rear surface ($t \simeq 310$ fs). (b) Results for LS regime with laser $a_0 = 214$ (brown, black) and $a_0 = 117$ (blue), plane wave (solid) or Gaussian transverse profile with $w_0 = 7\ \mathrm {\mu }$m (dash–dotted), and target $n_0 = 100 n_c$ and $l_0 = 350$ nm, at $t \simeq 2 \tau _0$. The dotted red curve shows the spectrum for a $\tau _0 = 15$ fs laser satisfying (7.1): $a_0 = \hat {a}_0 = 122$, with $n_0 = 250 n_c$. All the spectra are measured within a $10^\circ$ opening angle from the laser propagation direction at $t\simeq 2\tau _0$.

We note that the generation of narrow energy spread HB ion beams with similar peak energies around 1 MeV have been reported in Palmer et al. (Reference Palmer, Dover, Pogorelsky, Babzien, Dudnikova, Ispiriyan, Polyanskiy, Schreiber, Shkolnikov, Yakimenko and Najmudin2011), where the laser pulse duration appears to meet the criterion of our model. Specifically, they considered a CP CO$_2$ laser with $\lambda _0 = 10\ \mathrm {\mu }$m, $a_0\simeq 0.7$, $w_0 = 70\ \mathrm {\mu }$m and a hydrogen gas jet target with $n_0\lesssim 10 n_c$, $l_0\sim 800\ \mathrm {\mu }$m. The pulse duration used $\tau _0 \simeq 6$ ps was less than $\hat {\tau }_{0,\text {RTI}}\simeq 9$ ps as required by (4.1). The measured ion beam energies were also shown to be consistent with efficient HB.

For the LS regime, additional 3-D simulations have also been performed to illustrate the change in the ion beam quality for different pulse durations. We simulate a CP laser impinging on a planar electron–proton target with $n_0 = 100 n_c$ and $l_0 = 350$ nm at normal incidence. The laser pulse is temporally Gaussian and the transverse profile is either Gaussian with $w_0 \simeq 7\ \mathrm {\mu }$m or plane wave. These parameters are similar to those used in Qiao et al. (Reference Qiao, Zepf, Borghesi and Geissler2009), where the generation of a GeV proton beam was obtained in 2-D simulations and it was argued that extreme laser intensities were needed to obtain stable high-quality ion beams. Two cases were illustrated: unstable acceleration with $a_0 = 117$ and a stable regime with $a_0 = 214$. Under these conditions, (4.1) predicts that high-quality proton beams require $\tau _0 \lesssim 20$ fs for $a_0 = 117$ and $\tau _0\lesssim 10$ fs for $a_0 = 214$, both of which are much smaller than the duration $\tau _0\simeq 64$ fs used in the cited work.

Figure 10(b) shows the comparison of the proton spectra at the same time as figure 4 in Qiao et al. (Reference Qiao, Zepf, Borghesi and Geissler2009), which corresponds to $t\simeq 2\tau _0$ after the laser interaction. For the case with $a_0=117$, when a pulse duration $\tau _0 = 20$ fs is used, we observe the generation of a high-quality proton beam with $250$ MeV, ${\rm \Delta} \epsilon /\epsilon _0 \simeq 13$ %. For the same parameters but using $\tau _0 = 64$ fs (not shown) we observe the generation of a beam with $460$ MeV, ${\rm \Delta} \epsilon /\epsilon _0 \simeq 32$ %, similar to their green curve. This indicates that it is the onset of electron heating associated with the surface instability in the longer pulse that leads to the increase of the energy spread and to a significant reduction of the coupling efficiency.

For the highest intensity case ($a_0=214$), we observe that similarly the proton beam energy spread is improved from ${\rm \Delta} \epsilon /\epsilon _0 = 20$ % with $\tau _0=64$ fs to ${\rm \Delta} \epsilon /\epsilon _0 = 12$ % with $\tau _0=10$ fs. We further observe that for a transverse Gaussian laser profile the importance of the short pulse duration is even more dramatic. Defining the laser-to-proton energy conversion efficiency $\eta$ as the fraction of the laser energy being carried by the beam within a $10^\circ$ opening angle, for $\tau _0 = 10$ fs we obtain high-quality proton beams with ${\sim }600$ MeV, ${\rm \Delta} \epsilon /\epsilon _0 = 30$ % and $\eta = 27$ %, whereas with $\tau _0 = 64$ fs, we have ${\sim }100$ MeV, ${\rm \Delta} \epsilon /\epsilon _0 = 45$ % and $\eta = 0.5$ %. The proton beam energy obtained with the short pulse ($\tau _0 = 10$ fs) is comparable to the prediction from 1-D theory of 700 MeV (see (2.3a,b)). We note that this is slightly higher than the prediction of (7.2) because the target thickness considered in Qiao et al. (Reference Qiao, Zepf, Borghesi and Geissler2009) was $l_0\lesssim l_\text {opt}$. More importantly, these results confirm that the process that controls the stability and spectral quality of the LS accelerated ion beams is the electron heating via the development of RTI at the target surface and that choosing the appropriate pulse duration and laser intensity is critical for the acceleration of high-quality ion beams.

Figure 10(b) also includes the results of a 3-D PIC simulation for which the laser and target parameters satisfy the optimal LS regime (see (7.1)). We have chosen a set of parameters relevant for near-future short-pulse laser facilities, where a Gaussian laser pulse profile with $\tau _0 = 15$ fs, $w_0 = 5\ \mathrm {\mu }$m and $a_0 = \hat {a}_0 \simeq 122$ according to (7.1) is used. The target has $n_0 = 250 n_c$ and $l_0 = l_\text {opt}$. Under this optimal regime, we indeed observe stable acceleration of the protons via LS leading to the generation of a narrow energy spread proton beam with peak energy $\epsilon _0 \simeq 310$ MeV in very good agreement with the prediction of (7.2), ${\rm \Delta} \epsilon /\epsilon _0 \simeq$ 25 % and $\eta \simeq 2\,\%$. The total laser-to-proton energy conversion efficiency into $4{\rm \pi}$ is $\sim 20\,\%$.