2. Target normal sheath acceleration

TNSA of ions has been extensively investigated over the last decade. In fact, until now it has proved to be the most effective method for ion acceleration when highly intense laser pulses are focused onto foils with thicknesses of several micrometers. TNSA relies on the efficient conversion of laser energy into hot, relativistic electrons. These electrons propagate through the target and set up fields at the target boundaries where they exit into vacuum. The electric field is created between the expelled electrons and the surface charge that they induce on the target. Since the electric field strength is of the same relative strength as the laser electric field amplitude (TV/m) which generates the hot electrons, most of the electrons return back into the target. Hence an electron cloud (sheath) is formed. Ions at the rear surface can be accelerated by the sheath fields to multi-MeV energies in only several tens of femtoseconds. Impurities such as hydrocarbon and water are present under most experimental conditions at the solid surface. Due to their higher charge-to-mass ratio, it is the protons from these contaminants that are most readily accelerated to high energies. However, by removing the contaminants by different means, the acceleration of heavier ions can also be optimized^{[Reference Hegelich, Karsch, Pretzler, Habs, Witte, Guenther, Allen, Blazevic, Fuchs, Gauthier, Geissel, Audebert, Cowan and Roth52–Reference Henig, Kiefer, Markey, Gautier, Flippo, Letzring, Johnson, Shimada, Yin, Albright, Bowers, Fernández, Rykovanov, Wu, Zepf, Jung, Liechtenstein, Schreiber, Habs and Hegelich55]}. Due to strong spatial and temporal variations in the acceleration fields, the observed ions usually exhibit broad energy distributions extending from zero to a certain maximum cut-off energy. Reduction of this energy spread is a major challenge and some success has been achieved by micro-machining the targets in order to allow ions to be accelerated only in regions where the field is large and roughly uniform^{[Reference Hegelich, Albright, Cobble, Flippo, Letzring, Paffett, Ruhl, Schreiber, Schulze and Fernández6, Reference Schwoerer, Pfotenhauer, Jackel, Amthor, Liesfeld, Ziegler, Sauerbrey, Ledingham and Esirkepov7]}, and by the use of mass-limited targets with an appropriate ion mixture^{[Reference Ter-Avetisyan, Schnurer, Nickles, Kalashnikov, Risse, Sokollik, Sandner, Andreev and Tikhonchuk8, Reference Brantov, Tikhonchuk, Klimo, Romanov, Ter-Avetisyan, Schnürer, Sokollik and Nickles56]}.

A large number of experimental results on laser-driven ion acceleration are now available^{[Reference Schwoerer, Pfotenhauer, Jackel, Amthor, Liesfeld, Ziegler, Sauerbrey, Ledingham and Esirkepov7, Reference Snavely, Key, Hatchett, Cowan, Roth, Phillips, Stoyer, Henry, Sangster, Singh, Wilks, MacKinnon, Offenberger, Pennington, Yasuike, Langdon, Lasinski, Johnson, Perry and Campbell29, Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57–Reference Jung, Yin, Gautier, Wu, Letzring, Dromey, Shah, Palaniyappan, Shimada, Johnson, Schreiber, Habs, Fernández, Hegelich and Albright70]}, which can be used for a comparison with theoretical predictions. TNSA has been extensively studied numerically using particle-in-cell (PIC) simulations^{[Reference Pukhov71–Reference Yan, Wu, Sheng, Chen and Meyer-ter-Vehn75]}. Several analytical models have also been developed which predict the dependence of the maximum ion energy on laser and target parameters^{[Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57, Reference Fuchs, Antici, d’Humières, Lefebvre, Borghesi, Brambrink, Cecchetti, Kaluza, Malka, Manclossi, Meyroneinc, Mora, Schreiber, Toncian, Pépin and Audebert76–Reference Brantov, Tikhonchuk, Bychenkov and Bochkarev82]}. Substantial advance of the field is reported in a number of review papers available today^{[Reference Daido, Nishiuchi and Pirozhkov1, Reference Macchi, Borghesi and Passoni2, Reference Mourou, Tajima and Bulanov83]}. In particular, the ion energy should be dependent on the laser irradiance, focal spot size and laser pulse duration.

2.1. Nonrelativistic TNSA

The original Schreiber model^{[Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]} is a nonrelativistic version of TNSA and calculates the energy gain of ions in an electrostatic field determined by the transfer of laser energy to the expanding surface by the divergent beam of hot electrons. This model has two main advantages. (i) Without choosing a distribution function for the laser-accelerated electrons, the potential that they set up depends only on the absorbed laser power ${\it\eta}P_{L}$ into those electrons and the transverse size of the electron cloud $R_{s}$. (ii) Due to the consideration of the transverse dimension $R_{s}$ of the electron cloud when it exits the rear of the target, the resultant potential stays finite which is in contrast to most 1D models, where for infinite acceleration times the ion energies diverge^{[Reference Mora84]}. Although the description of the potential appears to be based on ad hoc assumptions, the model is in excellent agreement (within a factor of 1.5 or so) with experiments performed in a wide range of parameters covering foil thicknesses from one to hundreds of micrometers, laser pulse energies in the sub-joule to kilojoule level, pulse durations from 50 to 5000 fs and ion species covering a major part of the periodic table ranging from protons to tungsten ions. The electric field at the rear side of the target is set up by fast electrons produced by the laser heating at the front side of the target. It should be noted that in general the targets are much thicker than the skin depth or hole-boring depth so that the laser does not interact with the rear of the target. Let us consider that the laser produces $N_{e}$ electrons with an average energy $E_{e}$ in a bunch of length $L=c{\it\tau}_{L}$, where ${\it\tau}_{L}$ is the laser pulse duration and the electrons are assumed to propagate with the speed of light $c$. On their way through the target, the electrons spread over a circular region with radius $R_{s}$. When exiting into vacuum a positive surface charge $Qe$ is induced at the rear side of the target which yields a returning force $\propto Qe/({\it\pi}R_{s}^{2})$. Electrons run up the potential and eventually reverse their path at a distance $z_{u}$ above the surface of the foil and re-enter the foil. In an equilibrium situation $2N_{e}z_{u}/L$ electrons are permanently outside the foil. To achieve global charge neutrality we identify this number with the number of positive surface charges $Q$. The potential of the corresponding charge density is

(1)$$\begin{eqnarray}{\rm\Phi}(r,z)=\frac{Qe}{4{\it\pi}{\it\varepsilon}_{0}{\it\pi}R_{s}^{2}}\!\int _{0}^{R_{s}}\!\!\int _{0}^{2{\it\pi}}\!\!\frac{r^{\prime }\text{d}r^{\prime }\text{d}{\it\phi}^{\prime }}{\sqrt{r^{2}+z^{2}+r^{\prime 2}-2rr^{\prime }\cos {\it\phi}^{\prime }}}.\end{eqnarray}$$

In the following we will concentrate on the center of the acceleration region that is responsible for the most energetic ions, i.e., $r=0$. In that case the integration of Equation (1) reads

(2)$$\begin{eqnarray}\displaystyle & & \displaystyle -e({\rm\Phi}(0,z)-{\rm\Phi}(0,0))\nonumber\\ \displaystyle & & \displaystyle \quad \equiv -e{\rm\Phi}({\it\xi})=E_{\infty }\left(1+{\it\xi}-\sqrt{1+{\it\xi}^{2}}\right)\equiv E_{\infty }s({\it\xi})\qquad\end{eqnarray}$$

with

(3)$$\begin{eqnarray}E_{\infty }=\frac{Qe^{2}}{2{\it\pi}{\it\varepsilon}_{0}R_{s}}\end{eqnarray}$$

and ${\it\xi}=z/R_{s}$. For short distances ${\it\xi}\ll 1$ one has $-e{\rm\Phi}\cong E_{\infty }{\it\xi}$, which yields for the turning point $z_{u}=E_{e}R_{s}/E_{\infty }$. The total number of electrons is related to the laser energy $E_{L}$ by $N_{e}E_{e}={\it\eta}E_{L}$ where ${\it\eta}$ is the efficiency by which the laser energy is converted into electron energy. The electrons are provided for the duration of the laser pulse ${\it\tau}_{L}$. With $Q=2N_{e}z_{u}/(c{\it\tau}_{L})\propto P_{L}$ (the power of the laser pulse) we obtain the potential barrier

(4)$$\begin{eqnarray}E_{\infty }=2m_{e}c^{2}\sqrt{\frac{{\it\eta}P_{L}}{P_{Re}}}.\end{eqnarray}$$

We note that Equation (4) is independent of $E_{e}$ because $E_{\infty }\propto Q\propto N_{e}z_{u}=({\it\eta}E_{L}/E_{e})\cdot (E_{e}R_{s}/E_{\infty })=({\it\eta}E_{L}R_{s}/E_{\infty })$. The essential point is that laser energy is mainly converted into a large number of energetic electrons (fast enough to traverse the target) which in turn build up a dense electron sheath exhibiting extraordinary strong electric fields. It is thus not decisive what average energies $E_{e}$ are gained by the laser. This is in contrast to the model proposed by Bulanov et al. ^{[Reference Bulanov, Bychenkov, Chvykov, Kalinchenko, Litzenberg, Matsuoka, Thomas, Willingale, Yanovsky, Krushelnick and Maksimchuk85]} where ions are mainly accelerated by the longitudinal laser electric fields in a bored channel which acts as a waveguide with conducting walls. The potential of Equation (2) can be used to calculate the energy $E_{i}({\it\xi})$ an ion with charge $q_{i}$ gains between ${\it\xi}=0$ (the surface) and ${\it\xi}$:

(5)$$\begin{eqnarray}E_{i}({\it\xi})=-q_{i}e{\rm\Phi}({\it\xi}).\end{eqnarray}$$

The equation of motion yields

(6)$$\begin{eqnarray}{\it\tau}_{L}=R_{s}\int _{0}^{{\it\xi}_{m}}\frac{\text{d}{\it\xi}}{v_{i}({\it\xi})}=R_{s}\int _{0}^{s_{m}}\frac{\text{d}{\it\xi}/\text{d}s}{v_{i}(s)}\text{d}s.\end{eqnarray}$$

Now, nonrelativistically one has

(7)$$\begin{eqnarray}v_{i}(s)=v_{i,\infty }\sqrt{s}\end{eqnarray}$$

with $v_{i,\infty }=\sqrt{2E_{i,\infty }/m_{i}}=c\sqrt{2{\it\varepsilon}_{i,\infty }}$. Since

(8)$$\begin{eqnarray}\frac{\text{d}{\it\xi}}{\text{d}s}=\frac{1+(1-s)^{2}}{2(1-s)^{2}},\end{eqnarray}$$

one obtains

(9)$$\begin{eqnarray}\displaystyle {\it\tau}_{L} & = & \displaystyle {\it\tau}_{0}\int _{0}^{s_{m}}\frac{1+(1-s)^{2}}{2(1-s)^{2}\sqrt{s}}\text{d}s={\it\tau}_{0}\left[X+\int _{0}^{X}\frac{\text{d}x}{(1-x^{2})^{2}}\right]\nonumber\\ \displaystyle & \equiv & \displaystyle {\it\tau}_{0}F_{NR}(X)\end{eqnarray}$$

with ${\it\tau}_{0}=R_{s}/v_{i,\infty }$ and $X=\sqrt{s_{m}}=(E_{i,m}/E_{i,\infty })^{1/2}$. Finally, the integration yields

(10)$$\begin{eqnarray}F_{NR}(X)=X+\frac{X}{2(1-X^{2})}+\frac{1}{4}\ln \left(\frac{1+X}{1-X}\right)\end{eqnarray}$$

with $X=({\it\varepsilon}_{i,m}/{\it\varepsilon}_{i,\infty })^{1/2}$, where ${\it\varepsilon}_{i,m}=E_{i,m}/(m_{i}c^{2})$ is the normalized maximum ion energy and ${\it\varepsilon}_{i,\infty }$ defines the normalized energy an ion could gain from the potential of the sheath if it were maintained stationary, and is given by ${\it\varepsilon}_{i,\infty }({\it\tau}_{L})=q_{i}2m_{e}c^{2}\sqrt{{\it\eta}E_{L}/({\it\tau}_{L}P_{Re})}/(m_{i}c^{2})=1.1\times 10^{-3}~(q_{i}/A_{i})\sqrt{{\it\eta}E_{L}/({\it\tau}_{L}P_{Re})}$. Here, ${\it\tau}_{L}$ and $E_{L}$ are the laser pulse duration and energy respectively, $q_{i}$ is the ion charge state, $A_{i}$ is the nucleon number of the target, $P_{Re}=m_{e}c^{3}/r_{e}=8.71~\text{GW}$ is the relativistic power unit ($r_{e}=\text{classical electron radius}$, $m_{e}=\text{electron mass}$) and ${\it\eta}$ is the absorption efficiency into hot electrons. Following Refs. [Reference Yu, Jiang, Kieffer and Krol86, Reference Key, Cable, Cowan, Estabrook, Hammel, Hatchett, Henry, Hinkel, Kilkenny, Koch, Kruer, Langdon, Lasinski, Lee, MacGowan, MacKinnon, Moody, Moran, Offenberger, Pennington, Perry, Phillips, Sangster, Singh, Stoyer, Tabak, Tietbohl, Tsukamoto, Wharton and Wilks87], this efficiency is evaluated as ${\it\eta}=1.2\times 10^{-15}I_{L}^{3/4}$, where $I_{L}$ is in units of  $\text{W cm}^{-2}$, up to a maximum ${\it\eta}=0.5$. This scaling has been validated for a laser wavelength of ${\it\lambda}_{L}\sim 1~{\rm\mu}\text{m}$ and pulse durations of several hundreds of femtoseconds. We mention that a similar model has been developed by Bulanov et al. ^{[Reference Bulanov, Daido, Esirkepov, Khoroshkov, Koga, Nishihara, Pegoraro, Tajima and Yamagiwa15]} who considered a conducting prolate ellipsoid which had been charged up by $Q$ positive charges. The corresponding potential is given by Landau and Lifshitz^{[Reference Landau, Bell, Kearsley, Pitaevskii, Lifshitz and Sykes88]}. However, there exists a major difference from our model: since it is assumed by the ellipsoid model that the surface is conducting, all transverse forces at $z=0$ have to vanish, which can only be fulfilled by a specific distribution of the $Q$ charges at the surface. However, that can be in contradiction to the laser-driven charge distribution of hot electrons which might be very different from any ‘conducting’ equilibrium.

Figure 1. (a) Experimental data from Schreiber et al. ^{[Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]} and the prediction of the nonrelativistic TNSA model. (b) Optimal pulse duration for the specific example of $R_{s}=1~{\rm\mu}\text{m}$, ${\it\eta}=1$, $A_{i}/q_{i}=2$ for the nonrelativistic consideration (Equation (12), dashed) and the relativistic consideration (Equation (29), solid). The ultra-relativistic limit is given by the dash-dotted line. For larger $R_{s}$ the curves are globally shifted to respective larger optimal pulse durations.

One major result of the model presented is that for a given laser energy $E_{L}$, the shortest laser pulses and thus highest intensities are not necessarily optimal for TNSA. An important point is that the normalized maximum energy ${\it\varepsilon}_{i,\infty }$ an ion can gain depends on the pulse duration ${\it\tau}_{L}$ which has a strong consequence on the maximal ion energy $E_{i,m}$. This can be seen in Figure 1(a) where experimental proton energies $E_{i,m}$ (Ref. [Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]) are plotted for a constant laser energy $E_{L}=0.7~\text{J}$ as a function of the laser pulse duration ${\it\tau}_{L}$, showing explicitly that there exists an optimal duration ${\it\tau}_{L}^{\text{opt}}$. This behavior has been verified on other laser systems as well^{[Reference Toncian, Swantusch, Toncian, Willi, Andreev and Platonov89]} and can be understood by the following argument. The highest intensity (i.e., shortest pulse) produces the largest acceleration field, but only for a short time. On the other hand, a somewhat smaller field that is sustained over a longer duration of the laser pulse can result in higher ion energies. The acceleration is therefore optimized for some intermediate ${\it\tau}_{L}^{\text{opt}}$. The solid curve in Figure 1(a) has been obtained from Equation (9) (for details, see Ref. [Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]) and is in agreement with the experimental data, thus being able to describe an optimizing procedure. From Equation (9), for fixed laser energy and source radius, one can find the optimum using the condition $\text{d}E_{i,m}/\text{d}{\it\tau}_{L}|_{{\it\tau}_{L}^{\text{opt}}}=0$. For the nonrelativistic solution, Equation (10), this leads to

(11)$$\begin{eqnarray}\left(\frac{{\it\tau}_{L}}{{\it\tau}_{0}}\right)^{\text{opt}}=\frac{X_{\text{opt}}}{3}\left(1+\frac{1}{(1-X_{\text{opt}}^{2})^{2}}\right),\end{eqnarray}$$

where $X_{\text{opt}}$ is evaluated at ${\it\tau}_{L}={\it\tau}_{L}^{\text{opt}}$. Insertion of Equations (11) in (10) yields the solution $X_{\text{opt}}\cong 0.81$ with the corresponding value for $({\it\tau}_{L}/{\it\tau}_{0})^{\text{opt}}\cong 2.55$. On inserting the nonrelativistic characteristic time ${\it\tau}_{0}$, one obtains for the optimum pulse duration

(12)$$\begin{eqnarray}{\it\tau}_{L}^{\text{opt}}(E_{L})=195\left(\frac{R_{s}}{c}\right)^{4/3}\left(\frac{A_{i}}{q_{i}}\right)^{2/3}\left(\frac{P_{Re}}{{\it\eta}E_{L}}\right)^{1/3}.\end{eqnarray}$$

Figure 1(b) shows the optimum pulse duration as a function of the laser energy for $A_{i}/q_{i}=2$, $R_{s}=1~{\rm\mu}\text{m}$ and ${\it\eta}=1$. The dashed line is the nonrelativistic solution of Equation (12). The maximum ion energy then becomes

(13)$$\begin{eqnarray}({\it\varepsilon}_{i,m}^{\text{opt}})_{\text{TNSA}}=1.3\frac{q_{i}}{A_{i}}\sqrt{\frac{{\it\eta}E_{L}}{{\it\tau}_{L}^{\text{opt}}P_{Ri}}},\end{eqnarray}$$

where $P_{Ri}=(m_{p}/m_{e})P_{Re}=(1836)^{2}P_{Re}=29.3~\text{PW}$ is the relativistic power unit for a proton. From Equation (9) it follows that for a given laser system ${\it\tau}_{0}$ should be minimized in order to obtain the largest maximum ion energy $E_{i,m}$. This is realized for the smallest possible source size $R_{s}=r_{L}$, where $r_{L}$ is the radius of the beam spot. Therefore, it is convenient to use targets with thickness $d$ much smaller than the radius of the focal spot, which is usually of the order of some micrometers.

2.2. Relativistic TNSA

The relativistic equation of motion is

(14)$$\begin{eqnarray}\frac{\text{d}p_{i}}{\text{d}t}=q_{i}eE=-q_{i}e\frac{\text{d}{\rm\Phi}}{\text{d}z}\end{eqnarray}$$

with

(15)$$\begin{eqnarray}-q_{i}e{\rm\Phi}({\it\xi})=E_{i,\infty }\left(1+{\it\xi}-\sqrt{1+{\it\xi}^{2}}\right)\equiv E_{i,\infty }s({\it\xi}),\end{eqnarray}$$

where ${\it\xi}=z/R_{s}$ and $E_{i,\infty }=q_{i}2m_{e}c^{2}\sqrt{{\it\eta}P_{L}/P_{Re}}$, as before. The resulting two coupled first-order differential equations

(16)$$\begin{eqnarray}{\it\gamma}_{i}^{3}\frac{\text{d}{\it\beta}_{i}}{\text{d}t^{\ast }}={\it\varepsilon}_{i,\infty }\left(1-\frac{{\it\xi}}{\sqrt{1+{\it\xi}^{2}}}\right)\end{eqnarray}$$

and

(17)$$\begin{eqnarray}\frac{\text{d}{\it\xi}}{\text{d}t^{\ast }}={\it\beta}_{i}\end{eqnarray}$$

have to be solved simultaneously with $t^{\ast }=t/t_{0,R}=ct/R_{s}$ and ${\it\varepsilon}_{i,\infty }=E_{i,\infty }/(m_{i}c^{2})$. The initial condition is $t^{\ast }=0$, ${\it\xi}=0$ and ${\it\beta}_{i}=0$. The ion energy is given by ${\it\varepsilon}_{i}={\it\gamma}_{i}-1$ with ${\it\varepsilon}_{i}=E_{i}/(m_{i}c^{2})$. A first integration of Equation (16) yields

(18)$$\begin{eqnarray}\int _{0}^{{\it\beta}_{i}}{\it\gamma}_{i}^{3}{\it\beta}_{i}\text{d}{\it\beta}_{i}={\it\varepsilon}_{i,\infty }\int _{0}^{{\it\xi}}\frac{\text{d}s}{\text{d}{\it\xi}^{\prime }}\text{d}{\it\xi}^{\prime }\end{eqnarray}$$

or

(19)$$\begin{eqnarray}{\it\gamma}_{i}-1={\it\varepsilon}_{i}={\it\varepsilon}_{i,\infty }s({\it\xi}).\end{eqnarray}$$

Now, instead of Equation (7) one has

(20)$$\begin{eqnarray}v_{i}(s)=c\sqrt{1-(1+{\it\varepsilon}_{i,\infty }s)^{-2}}\end{eqnarray}$$

and thus

(21)$$\begin{eqnarray}\frac{{\it\tau}_{L}}{{\it\tau}_{0,R}}=\int _{0}^{s_{m}}\frac{1+(1-s)^{2}}{2(1-s)^{2}\sqrt{1-(1+{\it\varepsilon}_{i,\infty }s)^{-2}}}\text{d}s\end{eqnarray}$$

with ${\it\tau}_{0,R}=R_{s}/c$ and $s_{m}=E_{i,m}/E_{i,\infty }={\it\varepsilon}_{i,m}/{\it\varepsilon}_{i,\infty }=X^{2}$. It is immediately seen from Equation (21) that deviations from nonrelativistic TNSA occur for ${\it\varepsilon}_{i,\infty }s_{m}={\it\varepsilon}_{i,m}\geqslant 1$ (compare Equation (9)).The essential difference is that nonrelativistically the final ion velocity $v_{i,\infty }$ might become arbitrarily large, but special relativity limits it to the speed of light. We note that Equation (21) can be written as

(22)$$\begin{eqnarray}\displaystyle \frac{{\it\tau}_{L}}{{\it\tau}_{0}} & = & \displaystyle \sqrt{2{\it\varepsilon}_{i,\infty }}\int _{0}^{s_{m}}\frac{1+(1-s)^{2}}{2(1-s)^{2}\sqrt{1-(1+{\it\varepsilon}_{i,\infty }s)^{-2}}}\text{d}s\nonumber\\ \displaystyle & \equiv & \displaystyle F_{R}(X;{\it\varepsilon}_{i,\infty }).\end{eqnarray}$$

As a relativistic first-order correction to Equation (22) one obtains

(23)$$\begin{eqnarray}\displaystyle \frac{{\it\tau}_{L}}{{\it\tau}_{0}} & = & \displaystyle \int _{0}^{X}\frac{1+(1-x^{2})^{2}}{(1-x^{2})^{2}}\text{d}x\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\varepsilon}_{i,\infty }}{4}\int _{0}^{X}\frac{x^{2}(1+(1-x^{2})^{2})}{(1-x^{2})^{2}}\text{d}x\nonumber\\ \displaystyle & = & \displaystyle X+\frac{X}{2(1-X^{2})}+\frac{1}{4}\ln \frac{1+X}{1-X}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\varepsilon}_{i,\infty }}{4}\left(\frac{X^{3}}{3}+\frac{X}{2(1-X^{2})}-\frac{1}{4}\ln \frac{1+X}{1-X}\right).\qquad\end{eqnarray}$$

Unfortunately the simple scaling of Equation (9) is lost since now in $F_{R}$ the additional parameter ${\it\varepsilon}_{i,\infty }$ accounts for relativistic effects. Using the value of $X_{\text{opt}}\cong 0.8$ of the nonrelativistic TNSA expression as a first-order solution we obtain a condition for the optimized pulse duration in the case of relativistic TNSA:

(24)$$\begin{eqnarray}\left(\frac{{\it\tau}_{L}}{{\it\tau}_{0}}\right)^{\text{opt}}=2.5-0.18{\it\varepsilon}_{i,\infty }^{\text{opt}}\end{eqnarray}$$

or

(25)$$\begin{eqnarray}\frac{c\sqrt{a}}{R_{s}}({\it\tau}_{L}^{\text{opt}})^{3/4}=2.5-\frac{0.18a}{({\it\tau}_{L}^{\text{opt}})^{1/2}}\end{eqnarray}$$

with $a=1.1\times 10^{-3}(q_{i}/A_{i})\sqrt{{\it\eta}E_{L}/P_{Re}}$. An analytic solution of Equation (25) reads

(26)$$\begin{eqnarray}a=\frac{2.5}{{\it\alpha}_{2}}+\left(\frac{{\it\alpha}_{1}}{{\it\alpha}_{2}}\right)^{2}\left[1-\sqrt{1+\frac{5{\it\alpha}_{2}}{{\it\alpha}_{1}^{2}}}\,\right]\end{eqnarray}$$

with ${\it\alpha}_{1}=c({\it\tau}_{L}^{\text{opt}})^{3/4}/R_{s}$ and ${\it\alpha}_{2}=0.18/({\it\tau}_{L}^{\text{opt}})^{1/2}$. The nonrelativistic TNSA-limit Equation (11) is obtained for ${\it\alpha}_{2}\rightarrow 0$:

(27)$$\begin{eqnarray}a\cong \frac{25}{8{\it\alpha}_{1}^{2}}-\frac{125}{16{\it\alpha}_{1}^{4}}{\it\alpha}_{2}=\frac{3.1R_{s}^{2}}{c^{2}({\it\tau}_{L}^{\text{opt}})^{3/2}}\left(1-0.45\left(\frac{R_{s}}{c{\it\tau}_{L}^{\text{opt}}}\right)^{2}\right).\end{eqnarray}$$

Again, the nonrelativistic optimal ion energy $({\it\varepsilon}_{i,m}^{\text{opt}})_{\text{TNSA}}$ is obtained by Equation (13). The term within the bracket is the relativistic correction. A few things are worthy of note. (1) The correction due to relativistic ion motion becomes larger with decreasing optimal pulse duration, that is for larger laser energy $E_{L}$. This can be expected. However, (2) the correction decreases the value of $a$; in other words, the optimum pulse duration ${\it\tau}_{L}^{\text{opt}}$ is reduced even when the laser energy $E_{L}$ remains constant. This in turn means that the optimum ion energy becomes larger if relativistic corrections are taken into account. (3) The correction increases with increasing $R_{s}$. Nonrelativistically ${\it\tau}_{L}^{\text{opt}}$ increases with $R_{s}^{4/3}$ (see Equation (12)), but relativistically this increase is less strong.

Writing Equation (21) as

(28)$$\begin{eqnarray}\frac{{\it\tau}_{L}}{{\it\tau}_{0,R}}=\int _{0}^{s_{m}}H(s;{\it\varepsilon}_{i,\infty })\text{d}s\end{eqnarray}$$

and optimizing Equation (28) by the condition $\text{d}{\it\varepsilon}_{i,m}/\text{d}{\it\tau}_{L}|_{{\it\tau}_{L}^{\text{opt}}}=0$ yields

(29)$$\begin{eqnarray}\frac{1}{{\it\tau}_{0,R}}=\frac{\text{d}s_{m}}{\text{d}{\it\tau}_{L}}H(s_{m};{\it\varepsilon}_{i,\infty })+\int _{0}^{s_{m}}\frac{\partial }{\partial {\it\tau}_{L}}H(s,{\it\varepsilon}_{i,\infty })\text{d}s.\end{eqnarray}$$

It follows that

(30)$$\begin{eqnarray}\frac{\text{d}s_{m}}{\text{d}{\it\tau}_{L}}=\frac{\text{d}{\it\varepsilon}_{i,m}}{\text{d}{\it\tau}_{L}}\frac{1}{{\it\varepsilon}_{i,\infty }}-\frac{{\it\varepsilon}_{i,m}}{{\it\varepsilon}_{i,\infty }^{2}}\frac{\text{d}{\it\varepsilon}_{i,\infty }}{\text{d}{\it\tau}_{L}}=\frac{s_{m}}{2{\it\tau}_{L}}\end{eqnarray}$$

and

(31)$$\begin{eqnarray}\displaystyle \frac{\partial H}{\partial {\it\tau}_{L}} & = & \displaystyle \frac{1}{4{\it\tau}_{L}}\frac{1+(1-s)^{2}}{(1-s)^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{{\it\varepsilon}_{i,\infty }s}{(1+{\it\varepsilon}_{i,\infty }s)^{3}(1-(1+{\it\varepsilon}_{i,\infty }s)^{-2})^{3/2}}.\end{eqnarray}$$

One therefore has the condition

(32)$$\begin{eqnarray}\displaystyle 4\frac{{\it\tau}_{L}^{\text{opt}}}{{\it\tau}_{0,R}} & = & \displaystyle \frac{s_{m}^{\text{opt}}(1+(1-s_{m}^{\text{opt}})^{2})}{(1-s_{m}^{\text{opt}})^{2}\sqrt{1-(1+{\it\varepsilon}_{i,\infty }^{\text{opt}}s_{m}^{\text{opt}})^{-2}}}\nonumber\\ \displaystyle & & \displaystyle +\,\int _{0}^{s_{m}^{\text{opt}}}\frac{1+(1-s)^{2}}{(1-s)^{2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{{\it\varepsilon}_{i,\infty }^{\text{opt}}s}{(1+{\it\varepsilon}_{i,\infty }^{\text{opt}}s)^{3}(1-(1+{\it\varepsilon}_{i,\infty }^{\text{opt}}s)^{-2})^{3/2}}\text{d}s\nonumber\\ \displaystyle & = & \displaystyle 4\int _{0}^{s_{m}^{\text{opt}}}\frac{1+(1-s)^{2}}{2(1-s)^{2}(1-(1+{\it\varepsilon}_{i,\infty }^{\text{opt}}s)^{-2})^{1/2}}\text{d}s,\qquad\end{eqnarray}$$

where $s_{m}^{\text{opt}}={\it\varepsilon}_{i,m}^{\text{opt}}/{\it\varepsilon}_{i,\infty }^{\text{opt}}$ and ${\it\varepsilon}_{i,\infty }^{\text{opt}}=1.1\times 10^{-3}(q_{i}/A_{i})$ $\sqrt{{\it\eta}E_{L}/({\it\tau}_{L}^{\text{opt}}P_{Re})}$. A numerical solution yields $s_{m}^{\text{opt}}$ with ${\it\varepsilon}_{i,\infty }^{\text{opt}}$ as a parameter. By inserting these solutions into Equation (29) one obtains the required ${\it\tau}_{L}^{\text{opt}}/{\it\tau}_{0,R}$ (solid line in Figure 1(b)) and ${\it\varepsilon}_{i,m}^{\text{opt}}$.

In the highly relativistic regime, the ion velocity approaches the speed of light for approximately all the acceleration time, which according to Equation (6) leads to

(33)$$\begin{eqnarray}\frac{{\it\tau}_{L}}{{\it\tau}_{0,R}}={\it\xi}_{m}\end{eqnarray}$$

or

(34)$$\begin{eqnarray}{\it\varepsilon}_{i,m}=1.1\times 10^{-3}(q_{i}/A_{i})\sqrt{\frac{{\it\eta}E_{L}}{m_{e}c^{2}}\frac{R_{s}}{r_{e}}}G({\it\xi}_{m})\end{eqnarray}$$

with

(35)$$\begin{eqnarray}G({\it\xi}_{m})=\left.\left(1+{\it\xi}_{m}-\sqrt{1+{\it\xi}_{m}^{2}}\right)\right/\!\sqrt{{\it\xi}_{m}}.\end{eqnarray}$$

The function $G({\it\xi}_{m})$ has a rather broad maximum at ${\it\xi}_{m}={\it\tau}_{L}^{\text{opt}}c/R_{s}=1$. Whereas in the nonrelativistic regime the optimum pulse duration decreases with increasing laser energy $E_{L}$, see Equation (12), it becomes a constant, ${\it\tau}_{L}^{\text{opt}}=R_{s}/c$, for relativistic ion energies (dash-dotted line in Figure 1(b)). On inserting the value for $G$ one obtains from Equation (34) the optimal ion energy

(36)$$\begin{eqnarray}({\it\varepsilon}_{i,m}^{\text{opt}})_{\text{TNSA}}=0.64\times 10^{-3}(q_{i}/A_{i})\sqrt{\frac{{\it\eta}E_{L}}{m_{e}c^{2}}\frac{R_{s}}{r_{e}}}.\end{eqnarray}$$

We conclude this section with two remarks. (i) The ion energy $E_{i,m}$ is a unique function of $E_{L}$ and ${\it\tau}_{L}$, i.e., for constant laser energy a function of ${\it\tau}_{L}$ only. Thus, $E_{i,m}$ might show a maximum with respect to ${\it\tau}_{L}$ or not. However, if there exists a maximum it is unique, i.e., there is a single-valued $E_{i,m}^{\text{opt}}=E_{i,m}({\it\tau}_{L}^{\text{opt}})$ only. (ii) We propose a simple equation for the whole range,

(37)$$\begin{eqnarray}{\it\varepsilon}_{i,m}=1.3\cdot {\it\varepsilon}_{i,\infty }\left(1+\frac{{\it\tau}_{L}}{{\it\tau}^{\ast }}-\sqrt{1+\left(\frac{{\it\tau}_{L}}{{\it\tau}^{\ast }}\right)^{2}}\right),\end{eqnarray}$$

with

(38)$$\begin{eqnarray}{\it\tau}^{\ast }=\frac{R_{s}/c}{\sqrt{1-\frac{1}{(1+{\it\varepsilon}_{i,\infty })^{2}}}}.\end{eqnarray}$$

It might be that a better approximation is obtained if in the last equation ${\it\varepsilon}_{i,\infty }$ is replaced by ${\it\varepsilon}_{i,m}$. Bulanov et al. ^{[Reference Bulanov, Bychenkov, Chvykov, Kalinchenko, Litzenberg, Matsuoka, Thomas, Willingale, Yanovsky, Krushelnick and Maksimchuk85]} have performed detailed PIC studies of the generation of high energy ions in overcritical targets. For strong focusing conditions ($R_{s}\cong 0.75~{\rm\mu}\text{m}$), they obtained for $P_{L}=1~\text{PW}$ and a pulse duration of ${\it\tau}_{L}=30~\text{fs}$ (i.e., a laser energy of $E_{L}=30~\text{J}$) a proton energy of 1.3 GeV. With ${\it\varepsilon}_{i,\infty }=0.37$ and ${\it\tau}^{\ast }=3.6~\text{fs}$ we obtain from Equation (37) $E_{i,m}=430~\text{MeV}$.

3. Radiation pressure acceleration

At the intensities available with present high-intensity lasers, it seems natural to consider RPA as a means of accelerating objects to high energy. RPA offers the most promising approach for the acceleration of plasma bunches with near-solid, or at least overcritical, density to relativistic velocities. The principle of RPA is the same as was proposed to use continuous wave lasers to drive interstellar vehicles to relativistic velocities^{[Reference Simmons and McInnes31, Reference Marx50]}. According to Simmons et al. ^{[Reference Simmons and McInnes31]}, a body with rest mass comparable to the applied laser energy can be accelerated close to the speed of light. For a 5 nm thin carbon foil and a focal spot diameter of $2~{\rm\mu}\text{m}$, this would require only 5 J of energy. Unfortunately, the picture is not quite as simple due to the immense intensity of the applied laser. In Section 2, we have highlighted the importance of a high rate of absorption of laser energy into hot electrons for TNSA. For hole-boring or RPA to work efficiently, this heating must be suppressed, which may be achieved by the use of high-contrast systems with circular polarization^{[Reference Macchi, Cattani, Liseykina and Cornolti34]}. In this way the ponderomotive force that acts on the plasma electrons is only composed of a secular term which can effectively expel the electrons from the focal region and push them into the target. The space charge separation in the focal region is then maintained over the duration of the laser pulse, or even longer when the reflection front starts to move. During this time, electrons stay cold, i.e., they do not gain a large longitudinal momentum spread, so they stay bound to this initial depletion zone.

The acceleration of an object with mass $M=m_{i}n_{i}\text{d}{\it\pi}r_{L}^{2}$ by the radiation pressure, where $n_{i}$ is the ion particle density, is described by Refs. [Reference Simmons and McInnes31, Reference Macchi, Cattani, Liseykina and Cornolti34, Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima35]

(39)$$\begin{eqnarray}\frac{\text{d}({\it\gamma}{\it\beta})}{\text{d}t}=\frac{1}{t_{0}}\frac{1-{\it\beta}}{1+{\it\beta}},\end{eqnarray}$$

where $t_{0}=Mc^{2}/(2RP_{L})$ and $R$ denotes the reflectivity with which the laser is reflected. For a constant laser power $P_{L}$, a solution of Equation (39) in terms of the actual time $t$ is not of much interest because of the retardation effect. More relevant is a solution in terms of the retarded time $t_{\text{ret}}=t-\int _{0}^{t}{\it\beta}(t^{\prime })\text{d}t^{\prime }$ (Refs. [Reference Simmons and McInnes31, Reference Macchi, Cattani, Liseykina and Cornolti34]). Thus, Equation (39) becomes

(40)$$\begin{eqnarray}\frac{\text{d}{\it\beta}}{\text{d}t_{\text{ret}}}=\frac{1}{t_{0}}\frac{1-{\it\beta}}{{\it\gamma}},\end{eqnarray}$$

with the solution for the normalized ion energy

(41)$$\begin{eqnarray}{\it\varepsilon}_{i,m}^{\text{RPA}}={\it\gamma}-1=\frac{1}{2}\left(1+{\it\zeta}+\frac{1}{1+{\it\zeta}}\right)-1,\end{eqnarray}$$

where ${\it\zeta}=t_{\text{ret}}/t_{0}=2RE_{L}/(Mc^{2})$, since the light pulse is terminated at the retarded time $t_{\text{ret}}={\it\tau}_{L}$. We note that the interaction between the light and the plasma sheet may continue even when the laser has finished radiating: for relativistic velocities of the sheet the light pulse accompanies the bunch for interaction times $t=t_{\text{int}}$ much longer than the pulse duration

(42)$$\begin{eqnarray}t_{\text{int}}={\it\tau}_{L}\left(1+\frac{{\it\zeta}}{2}+\frac{{\it\zeta}^{2}}{6}\right).\end{eqnarray}$$

The essential point for the RPA mechanism is the assumption that the whole target of mass $M$ is accelerated cooperatively as a charge-neutral plasma bunch. The physical mechanism behind this process is the following. Due to their small mass the electrons are accelerated by light pressure and then due to Coulomb forces drag the ions behind them (hence the name light-sail^{[Reference Macchi, Veghini and Pegoraro43, Reference Steinke, Hilz, Schnürer, Priebe, Bränzel, Abicht, Kiefer, Kreuzer, Ostermayr, Schreiber, Andreev, Yu, Pukhov and Sandner90]}). Thus, one has to avoid the light pressure on the electrons exceeding the restoring forces due to the charge separation.

On inspecting Equation (41) it is readily seen that the ion energy $E_{i,m}^{\text{RPA}}$ increases with ${\it\zeta}$ and thus with decreasing target mass $M$, or with thinner target foils. However, a lower limit will be reached if the number of ions becomes so small that they cannot any longer retain the electrons by their Coulomb forces. One estimates the energy $E_{\infty }$ to separate an electron from a sheet of ion charge density $q_{i}n_{i}=n_{e}$ to be $E_{\infty }=n_{e}\text{d}r_{L}e^{2}/(4{\it\varepsilon}_{0})$ (Ref. [Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]). This estimate can also be derived from the ‘capacitor’ model for charge separation^{[Reference Robinson, Zepf, Kar, Evans and Bellei37, Reference Klimo, Psikal, Limpouch and Tikhonchuk38, Reference Meyer-ter-Vehn and Wu91]} assuming an effective cut-off of the induced electric field for distances larger than the lateral extension of the sheet. This assumption is equivalent to the demand to maintain the balance between charge separation and radiation pressure at least over a laser period^{[Reference Passoni and Lontano81, Reference Pegoraro and Bulanov92]}. Evidently, the charge separating energy increases with the areal density $n_{i}d$ while the radiation pressure decreases like $(n_{i}d)^{-1}$, advocating once more the use of ultra-thin targets. The ion energy can be derived from Equation (41) in the nonrelativistic case $E_{i,m}\propto {\it\zeta}^{2}\propto 1/M^{2}\propto 1/d^{2}$. For a completely ionized carbon target with a thickness of $1~{\rm\mu}\text{g cm}^{-2}$ one obtains $E_{\infty }\cong 10~\text{MeV}$. On the other hand, this estimate shows that there exists a lower limit on the electron mass $M_{e}$ that can be accelerated by radiation pressure and which remains bound to the ions. The latter demand is essential for ion acceleration since it is the electrons that pull the ions behind them. If $M_{e}$ becomes too small, the resulting fast electrons with energies $E_{e}=m_{e}c^{2}{\it\gamma}\cong m_{e}c^{2}{\it\tau}_{L}/(2t_{0})=(m_{e}/M_{e})E_{L}=E_{L}/N_{e}$ (note that at large electron energies the efficiency approaches 100%, i.e., all the laser energy is converted into the kinetic energy of $N_{e}$ electrons) can surmount the potential barrier built up by the charge separation field (note that $M_{e}=m_{e}n_{e}\text{d}r_{L}^{2}$). Thus, the minimum ion mass that prohibits charge separation and that can be accelerated as a charge-neutral plasma bunch becomes

(43)$$\begin{eqnarray}M_{\text{min}}=\frac{A_{i}}{q_{i}}\sqrt{\frac{4{\it\varepsilon}_{0}E_{L}r_{L}}{e^{2}}}m_{u}\end{eqnarray}$$

with the atomic mass unit $m_{u}$ and $A_{i}$ the mass number of the target. For a carbon target with $A_{i}/q_{i}=2$, $E_{L}=1~\text{J}$ and $r_{L}=3~{\rm\mu}\text{m}$ one obtains $M_{\text{min}}=2.2\times 10^{-16}~\text{kg}$, corresponding to an optimum thickness of 4 nm. This is in strong contrast to a TNSA model of Andreev et al. ^{[Reference Andreev, Levy, Ceccotti, Thaury, Platonov, Loch and Martin93]} who predicted an optimum thickness of about 100 nm. In Refs. [Reference Henig, Steinke, Schnürer, Sokollik, Hörlein, Kiefer, Jung, Schreiber, Hegelich, Yan, Meyer-ter-Vehn, Tajima, Nickles, Sandner and Habs51, Reference Steinke, Henig, Schnürer, Sokollik, Nickles, Jung, Kiefer, Hörlein, Schreiber, Tajima, Yan, Hegelich, Meyer-ter-Vehn, Sandner and Habs94] the maximum ion energy has been investigated as a function of the target mass $M$. As expected, the ion energy increases with $1/M$, but for masses close to $M_{\text{min}}$ the energy starts to drop rather strongly with further decreasing target masses, thus confirming the estimate of Equation (43) quantitatively. Figure 2 reproduces the experimental results of Ref. [Reference Henig, Steinke, Schnürer, Sokollik, Hörlein, Kiefer, Jung, Schreiber, Hegelich, Yan, Meyer-ter-Vehn, Tajima, Nickles, Sandner and Habs51] for the carbon energies obtained from nm-thin diamond-like carbon (DLC) foils. The solid curve has been obtained from Equation (41). Although the curve drops off faster towards thicker targets (due to the increasing domination of expansion), evidently there exists an optimal foil thickness, i.e., target mass. The estimate of Equation (43) is indicated where the solid line breaks into the dashed line. An estimate of the optimal and thus minimum foil thickness by Chen et al. ^{[Reference Chen, Pukhov, Yu and Sheng95]} is in essence the same as that of Equation (43).

Figure 2. Maximum proton and carbon ion energies for varying thicknesses of nm-thin DLC foils reported in Henig et al. ^{[Reference Henig, Steinke, Schnürer, Sokollik, Hörlein, Kiefer, Jung, Schreiber, Hegelich, Yan, Meyer-ter-Vehn, Tajima, Nickles, Sandner and Habs51]}. The solid curve represents the prediction for RPA, Equation (41), using the parameters $A_{i}/q_{i}=2$, $E_{L}=0.7~\text{J}$, $r_{L}=1.8~{\rm\mu}\text{m}$, $R=1$, ${\it\rho}_{\text{DLC}}=2.7~\text{g cm}^{-3}$. The optimum mass/thickness is indicated by the transition of the solid to a dashed curve.

In a very recent theory of laser ion acceleration from thin foils^{[Reference Yan, Tajima, Hegelich, Yin and Habs96]} the dimensionless parameter

(44)$$\begin{eqnarray}{\it\xi}=\frac{n_{e}\text{d}}{n_{c}{\it\lambda}a_{0}}\end{eqnarray}$$

has been introduced, which for small laser strength $a_{0}$ and thus ${\it\xi}\geqslant 1$ prohibits charge separation, i.e., allows collective ion acceleration induced by electrons riding ahead of the ions. Here, $n_{c}$ is the critical electron density. It is easy to show that Equation (44) can also be written as ${\it\xi}=M/M_{\text{min}}$ with $M_{\text{min}}$ of Equation (43). Thus, the condition ${\it\xi}\geqslant 1$ for an appropriate ion acceleration is equivalent to demanding $M\geqslant M_{\text{min}}$. Identifying $M_{\text{min}}$ with the optimal target mass $M_{\text{opt}}$ for RPA and inserting Equations (43) into (41) one obtains $({\it\varepsilon}_{i,m}^{\text{opt}})_{\text{RPA}}$, which can be compared with the equivalent expression for the case of optimal TNSA. In Figure 3 ${\it\varepsilon}_{i,m}^{\text{opt}}$ has been plotted for either TNSA or RPA versus the laser energy $E_{L}$. The curves hold for $R_{s}=r_{L}=1~{\rm\mu}\text{m}$ (solid lines) and $R_{s}=r_{L}=10~{\rm\mu}\text{m}$ (dashed lines), $A_{i}/q_{i}=2$ and ${\it\eta}=R=1$. For a typical high-intensity laser with a wavelength of ${\it\lambda}_{L}=1~{\rm\mu}\text{m}$, a beam spot radius of $1~{\rm\mu}\text{m}$ is close to the diffraction-limited minimum sized beam spot. The corresponding data in Figure 3 thus represent optimum ion energies also within this context. The extension of available laser energies up to 10 kJ might be optimistic but we note that in a recent paper by Tajima et al. ^{[Reference Tajima, Kando and Teshima5]} on electron acceleration up to PeV even GJ laser energies have been discussed. A laser with $E_{L}=1~\text{GJ}$ could accelerate by radiation pressure a carbon foil of optimal thickness of $300~{\rm\mu}\text{m}$ with C ions in it up to energies of 60 TeV/ion (${\it\gamma}=5\times 10^{3}$, which is close to the heavy-ion energies of CERN’s Large Hadron Collider^{[Reference Sørensen97]}). However, we also note that for a pulse duration of 1 ps^{[Reference Tajima, Kando and Teshima5]}, the interaction distance according to Equation (42) would become 5 km. The reflected radiation has a vanishingly small frequency and at the end of the acceleration process nearly all the laser pulse energy is converted into kinetic energy of the ions, $E_{L}\cong ({\it\gamma}-1)M_{\text{opt}}c^{2}$, see also Equation (45). It would not be easy to stop such an ion bunch. Its energy content is large enough to heat up and finally melt 15 tons of lead. For comparison, the 7.7 TeV protons stored in the LHC have an energy content of 0.6 GJ^[98]. An elegant solution of such a ‘beam-dump’ problem has recently been proposed by Wu et al. ^{[Reference Wu, Tajima, Habs, Chao and Meyer-ter-Vehn99]}, relying on the deceleration of energetic and dense particle bunches by collective electronic interactions in an underdense plasma, promising a compact and non-radioactive dump. However, two remarks might be made. (i) The main contribution to the energy loss of ultra-relativistic heavy ions is electron–positron pair creation and not electronic stopping (‘Bethe–Bloch’)^{[Reference Sørensen97]}. According to the description of Ref. [Reference Sørensen97] we obtain for the case discussed – 60 TeV carbon ions in lead – an electronic energy loss of $0.07~\text{TeVm}^{-1}$ and one by pair production of $0.3~\text{TeVm}^{-1}$. Energy losses due to bremsstrahlung can be neglected since the small impact parameters necessary to generate hard photon quanta lead unavoidably to a fragmentation of the projectile^{[Reference Sørensen97]}. We thus estimate for the example mentioned above a stopping length of about 570 m. In contrast, a beam-dump length of several meters only is sufficient to break up the projectile by nuclear inelastic interactions, yielding a cascade of secondary particles with individual energies much less than the primary one^{[Reference Ferrari, Stevenson and Weisse100]}. It is mainly the electromagnetic component of the cascade that finally converts the energy into heat. (ii) The essential point is not to stop the bunch within a moderate distance and thus heat the absorber but to dissipate its enormous energy concentration without any further problem. The situation for a laser-generated ion bunch is quite different from that of the LHC where the particles are more or less homogeneously distributed around the 27 km long circumference of the storage ring: C ions within a bunch length of $300~{\rm\mu}\text{m}$ compared with $5\times 10^{14}$ protons stored within 27 km. The corresponding pulse length of $90~{\rm\mu}\text{s}$ is long enough to allow kicker magnets to sweep the beam across the absorber.

Figure 3. Optimum ion energies predicted by the models for TNSA, Equations (29)–(32) (black), and RPA, Equations (41)–(43) (red). The parameters are $A_{i}/q_{i}=2$, $R={\it\eta}=1$, $r_{L}=R_{s}=1~{\rm\mu}\text{m}$ (solid) and $r_{L}=R_{s}=10~{\rm\mu}\text{m}$ (dashed). Some selected experimental results are represented by blue squares (Bin et al. ^{[Reference Bin, Ma, Allinger, Wang, Kiefer, Reinhardt, Hilz, Khrennikov, Karsch, Yan, Krausz, Tajima, Habs and Schreiber112]}, Henig et al. ^{[Reference Henig, Steinke, Schnürer, Sokollik, Hörlein, Kiefer, Jung, Schreiber, Hegelich, Yan, Meyer-ter-Vehn, Tajima, Nickles, Sandner and Habs51]}, Mackinnon et al. ^{[Reference Mackinnon, Sentoku, Patel, Price, Hatchett, Key, Andersen, Snavely and Freeman108]}, Zeil et al. ^{[Reference Zeil, Kraft, Bock, Bussmann, Cowan, Kluge, Metzkes, Richter, Sauerbrey and Schramm113]}, Ogura et al. ^{[Reference Ogura, Nishiuchi, Pirozhkov, Tanimoto, Sagisaka, Esirkepov, Kando, Shizuma, Hayakawa, Kiriyama, Shimomura, Kondo, Kanazawa, Nakai, Sasao, Sasao, Fukuda, Sakaki, Kanasaki, Yogo, Bulanov, Bolton and Kondo68]}, Jong Kim et al. ^{[Reference Kim, Pae, Kim, Kim, Sung, Lee, Yu, Choi, Lee, Nam, Nickles, Jeong and Lee114]}, Green et al. ^{[Reference Green, Robinson, Booth, Carroll, Dance, Gray, MacLellan, McKenna, Murphy, Rusby and Wilson115]}, Jung et al. ^{[Reference Jung116]}) and theoretical results obtained from PIC simulations are marked by green circles (Pukhov^{[Reference Pukhov71]}, Wang et al. ^{[Reference Wang, Lin, Zheng, Lu, Guo, He, Chen and Yan117]}, Qiao et al. ^{[Reference Qiao, Kar, Geissler, Gibbon, Zepf and Borghesi40]}, Sgattoni et al. ^{[Reference Sgattoni, Sinigardi and Macchi118]}, Yan et al. ^{[Reference Yan, Wu, Sheng, Chen and Meyer-ter-Vehn75]}, Esirkepov et al. ^{[Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima35]}). For details, see text.

It is readily seen that except for a minor factor over the whole range of laser energies both optimized theories yield the same maximum ion energies. It is therefore more a question of practicability what kind of optimization one chooses, ${\it\tau}_{L}^{\text{opt}}$ or $M_{\text{opt}}$. There might exist technical limits: it is certainly very difficult to obtain pulse durations shorter than say 1 fs. According to Equation (29) this would correspond to an upper limit of $E_{L}=3\times 10^{4}~\text{J}$, which in essence does not pose a strong limit. On the other hand, in the case of RPA a lower limit of $M_{\text{opt}}$ is reached for a monolayer of atoms. Assuming a carbon foil as the target (the most popular choice) one obtains a lower limit of $M_{\text{opt}}=1.4\times 10^{-18}~\text{kg}$, or from Equation (41) a lower limit of $E_{L}=1.3\times 10^{-4}~\text{J}$, which also does not have a practical influence. We emphasize that the optimizing procedure developed in this paper only works since the TNSA theory of Schreiber et al. ^{[Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]} does not depend on the target thickness (at least to first order) and the RPA theory of Refs. [Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima35, Reference Bulanov, Esirkepov, Migliozzi, Pegoraro, Tajima and Terranova101, Reference Pegoraro, Bulanov, Sakai and Tomassini102] does not depend on the pulse duration.

In addition, experimental results are plotted in Figure 3 demonstrating that in most cases experiments are rather far away from being optimal. Multi-J laser systems especially suffer from having too long a pulse duration. We also note that normalized ion energies ${\it\varepsilon}_{i,m}\geqslant 1$ or $E_{i,m}\geqslant 1~\text{GeV/u}$, i.e., relativistic energies, are obtained for laser energies $E_{L}\geqslant 100~\text{J}$ only and so far have only been obtained in PIC simulations. Finally, we note that the efficiency of RPA can be written as

(45)$$\begin{eqnarray}{\it\eta}_{\text{RPA}}\equiv \frac{E_{i,m}}{E_{L}}=\frac{R{\it\zeta}}{1+{\it\zeta}},\end{eqnarray}$$

i.e., in the relativistic regime with ${\it\gamma}\gg 1$ or with Equation (41) ${\it\zeta}\gg 1$ one obtains that for $R=1$, 100% of the laser energy is converted into ion energy.

Of course, if the transverse light intensity changes, different parts of the sail will be accelerated differently changing an initially plane sail to a convex one. The equation of motion of Equation (39) changes to

(46)$$\begin{eqnarray}\frac{\text{d}({\it\gamma}\vec{{\it\beta}})}{\text{d}t}=\frac{2RP_{L}}{Mc^{2}}\frac{1-{\it\beta}}{1+{\it\beta}}\vec{n},\end{eqnarray}$$

where both $P_{L}=P_{L}(y,z;t)$ and the unit vector $\vec{n}=\vec{n}(y,z;t)$ of the sail’s surface normal depend on the transverse coordinates $(y,z)$ and time $t$. Thus, a solution of Equation (46) has to account for a transverse expansion of the sail. An approximate solution of this self-consistent problem to obtain both the final velocity ${\it\beta}$ and the shape of the sail, allowing for bowing while simultaneously submitting it to the condition of constant mass, has been given in Ref. [Reference Bulanov, Echkina, Esirkepov, Inovenkov, Kando, Pegoraro and Korn103]. The essential point is that instabilities like those of the Rayleigh–Taylor type lead to strong deviations from a uniform plasma front, forming cusps at an early stage of acceleration^{[Reference Palmer, Kluge, Zeil, Bussmann, Kraft, Cowan and Schramm104]}, resulting in emission of beamlets^{[Reference Pegoraro and Bulanov105]}. To avoid such instabilities Chen et al. ^{[Reference Chen, Pukhov, Yu and Sheng95]} have proposed to use thin target foils shaped initially in the transverse direction to match the laser intensity profile. PIC simulations by Wilks et al. ^{[Reference Wilks, Kruer, Tabak and Langdon106]} performed as far back as in 1992 clearly demonstrated such instabilities for light intensities relevant to the light-sail regime. However, we remark that the ion energy differs for every beamlet. At a later stage the ‘plasma foil’ breaks into high-density clumps with diffuse lower density clouds between them^{[Reference Palmer, Kluge, Zeil, Bussmann, Kraft, Cowan and Schramm104]}, which due to its reduced mass may even accelerate ions to higher energies than the original foil^{[Reference Bulanov, Echkina, Esirkepov, Inovenkov, Kando, Pegoraro and Korn103]}. A similar result has been obtained quite recently, demonstrating by PIC simulations that ‘a relatively stable ion clump forms near the laser axis which is efficiently accelerated’^{[Reference Yan, Wu, Sheng, Chen and Meyer-ter-Vehn75]}. A detailed study of Rayleigh–Taylor instabilities which yield filamentation and their optimization with respect to maximum ion energies has been published recently^{[Reference Bulanov, Bychenkov, Chvykov, Kalinchenko, Litzenberg, Matsuoka, Thomas, Willingale, Yanovsky, Krushelnick and Maksimchuk85, Reference Mourou, Chang, Maksimchuk, Nees, Bulanov, Bychenkov, Esirkepov, Naumova, Pegoraro and Ruhl107]}.

5. Summary

Starting from experimental results, we have shown that current theories of laser–ion acceleration can and should be optimized in order to achieve maximum ion energies. Not all theories include such a possibility, but the TNSA theory of Schreiber et al. ^{[Reference Schreiber, Bell, Grüner, Schramm, Geissler, Schnürer, Ter-Avetisyan, Hegelich, Cobble, Brambrink, Fuchs, Audebert and Habs57]} can be optimized with respect to laser pulse duration and the RPA theory of Bulanov et al. ^{[Reference Esirkepov, Borghesi, Bulanov, Mourou and Tajima35, Reference Bulanov, Esirkepov, Migliozzi, Pegoraro, Tajima and Terranova101, Reference Pegoraro, Bulanov, Sakai and Tomassini102]} with respect to target thickness. It turns out that both the optimized TNSA and RPA theories yield approximately the same maximum ion energies over the range of laser energies $0.1~\text{J}<E_{L}<10~\text{kJ}$. It is thus a matter of convenience whether one adjusts the optimal pulse duration or the optimal target thickness. For both theories, the decisive laser parameter is neither the power nor the intensity but solely the laser energy. Relativistic ion energies, i.e., energies beyond 1 GeV/u, can be obtained for systems with $E_{L}>100~\text{J}$, where in addition a diffraction-limited small spot size has to be achieved.